==== TABLE OF CONTENTS ====
Local-smoothing-for-the-Euclidean-wave-equation//angular.tex
Local-smoothing-for-the-Euclidean-wave-equation//barrier.tex
Local-smoothing-for-the-Euclidean-wave-equation//calculus.tex
Local-smoothing-for-the-Euclidean-wave-equation//consequences.tex
Local-smoothing-for-the-Euclidean-wave-equation//current.tex
Local-smoothing-for-the-Euclidean-wave-equation//cylinder.tex
Local-smoothing-for-the-Euclidean-wave-equation//dirichlet.tex
Local-smoothing-for-the-Euclidean-wave-equation//energy.tex
Local-smoothing-for-the-Euclidean-wave-equation//fiber.tex
Local-smoothing-for-the-Euclidean-wave-equation//global-extension.tex
Local-smoothing-for-the-Euclidean-wave-equation//history.tex
Local-smoothing-for-the-Euclidean-wave-equation//introduction.tex
Local-smoothing-for-the-Euclidean-wave-equation//kakeya.tex
Local-smoothing-for-the-Euclidean-wave-equation//paper.tex
Local-smoothing-for-the-Euclidean-wave-equation//parabolic.tex
Local-smoothing-for-the-Euclidean-wave-equation//pencil.tex
Local-smoothing-for-the-Euclidean-wave-equation//planar.tex
Local-smoothing-for-the-Euclidean-wave-equation//propagation.tex
Local-smoothing-for-the-Euclidean-wave-equation//sharpness.tex
Local-smoothing-for-the-Euclidean-wave-equation//stein_square.tex
Local-smoothing-for-the-Euclidean-wave-equation//transfer.tex
Local-smoothing-for-the-Euclidean-wave-equation//weak.tex
Local-smoothing-for-the-Euclidean-wave-equation//bibliography.bib
===========================
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//angular.tex ====
\section{The sharp logarithm with angular regularity}\label{sec:angular}
Write \(p=\pc\), \(d=n-1\), \(\alpha=d/2\), \(I=[1,2]\), and
\(L(v)=\log(2+v)\), so that \(\alpha p=n\). Let
\(\Lambda_\omega=(1-\Delta_{S^{n-1}})^{1/2}\) act on the angular
variable on each sphere about the origin. The joint weak spacetime
norm is \(\|u\|_{p,\infty}=\sup_{\lambda>0}\lambda|\{|u|>\lambda\}|^{1/p}\).
\begin{theorem}[The angular endpoint]\label{thm:angular}
Let \(n\ge2\), \(p=\pc\), and \(B>1/2+1/p\). Then
\begin{align*}
\|U(T)f\|_{L^p(\RR^n\times I)}
&\le C_{n,B}\big(\|L(J)^{1/p}f\|_p+\|L(\Lambda_\omega)^B f\|_p\big),\\
\|U(T)f\|_{L^{p,\infty}(\RR^n\times I)}
&\le C_{n,B}\|L(\Lambda_\omega)^B f\|_p .
\end{align*}
In each inequality the angular norm on the right can be replaced by
\(\|\Lambda_\omega^\sigma f\|_p\) for any \(\sigma>0\), with a constant depending on \(\sigma\).
\end{theorem}
For radial data the angular operator acts as a scalar, so
Proposition~\ref{prop:sharpness} makes the frequency power \(1/p\) in
the strong estimate sharp. To prove the theorem we first separate
spatial frequency and angular degree on a single annulus.
Let
$\Pi_N=\chi(|D|/N)$ with $\chi\in C_c^\infty((0,\infty))$.
We use $A=2/p$ for the power in Theorem~\ref{thm:main}; the argument
applies with any positive exponent $A$ for which that bound holds.
An angular polynomial of degree at most $M$ is a function whose restriction
to almost every sphere about the origin is a sum of spherical harmonics of
degrees at most $M$.
\begin{proposition}[Bounded angular degree on a frequency annulus]\label{prop:angular-band}
For $N\ge2$ and an angular polynomial $f\in L^p(\RR^n)$ of degree at most $M$,
\begin{align}
\|U(t)\Pi_N f\|_{L^p(I\times\RR^n)}
&\lesssim [L(N)^{1/p}+L(M)^A]\|f\|_p,\label{eq:ang-polynomial}\\
\|U(t)\Pi_N f\|_{L^{p,\infty}(I\times\RR^n)}
&\lesssim L(M)^A\|f\|_p .\notag
\end{align}
The constants are independent of $M$ and $N$.
\end{proposition}
The two terms in the strong estimate come from different output radii in
physical space. When $M$ is sufficiently small relative to $N$, we split
the output at radius $(2+M)^K/N$ for a fixed $K$. The first lemma below
shows that the logarithm on the inner ball counts its radius in wavelengths,
here $(2+M)^K$, so the cost of observation is $O(L(M)^A)$. On its exterior
the input from a small ball about the origin contributes a decaying error
in the strong norm. For the remaining input, angular stationary phase at
$x=r\theta$, $r\le4$, gives profiles $r^{-\alpha}G(t\pm r,\theta)$ with
$\|G\|_{L^p(\dd v\dd\theta)}\lesssim\|f\|_p$, where $v=t\pm r$. Each
dyadic interval of $r$ contributes equally to the $p$th power of the
radial factor, since $r^{d-\alpha p}=r^{-1}$. Its weak $L^p$ norm, with
the same polar measure $r^d\dd r$, is bounded independently of the
number of intervals.
For radial data, M\"uller and Seeger reduced the wave kernel to four
one-dimensional kernels concentrated at $t\pm r\pm s=0$, with $s$ the
input radius \cite[Lemma~2.1]{MuellerSeeger1995}. Colzani, Cominardi and
Stempak obtained the corresponding weak endpoint for the radial cosine
propagator \cite[Theorem~2.1]{CCS2002}. We recover these four kernels
outside the indicated ball with constants uniform in the angular degree.
\subsection{Observation on a ball}
\begin{lemma}\label{lem:angular-ball}
For $N\ge2$, $0<r\le1$ and every $x_0\in\RR^n$,
\[
\|U(t)\Pi_N f\|_{L^p(I\times B(x_0,r))}
\lesssim \min\{1,(Nr)^{n/p}\}\,L(Nr)^A\|f\|_p .
\]
\end{lemma}
\begin{proof}
Translate to $x_0=0$ and rescale to frequency one. With $T=N$, $R=Nr$,
we must bound the norm on $[T,2T]\times B(0,R)$ by
$T^{1/p}\min(1,R^{n/p})L(R)^A\|f\|_p$.
For the kernel of $U(t)\Pi_1$, write $v=|z|$ and
$\langle v\rangle=(1+v^2)^{1/2}$. Spherical stationary phase gives,
for every $D$ and $v\ge1$,
\begin{equation}\label{eq:ang-unit-kernel}
K(t,z)=v^{-\alpha} k(t-v)+
O_D\big(v^{-\alpha-1}\langle t-v\rangle^{-D}
+v^{-\alpha}\langle t+v\rangle^{-D}\big).
\end{equation}
Here $k$ is the Fourier transform of a constant times
$\rho^\alpha\chi(\rho)$ on the positive axis. After removal of the phases
$e^{\pm i v\rho}$ the spherical amplitudes are symbols in $v\rho$;
radial integration by parts gives the stated decay. For $v<1$ the entire
kernel is $O_D(\langle t\rangle^{-D})$. In particular, at any fixed $x$,
Young's inequality in time on the input radii $T/2\le s=|y-x|\le3T$
gives
\[
\|U(t)\Pi_1 f(x)\|_{L^p([T,2T])}
\lesssim T^{1/p}\|f\|_p.
\]
Indeed the convolution acts on
$s^\alpha\int_{\mathbb S^d}|f(x+s\omega)|\dd\omega$, whose $p$th power
integrates to at most $CT\|f\|_p^p$ on these radii, since $\alpha p=d+1$.
At the remaining radii the kernel is $O_D((T+s)^{-D})$.
Integration over $x$ proves the assertion for $R\le 2$.
Suppose $2<R\le T$ and put $S=100R^{3+n/p}$.
If $T\le10S$, the scaled global estimate of Theorem~\ref{thm:main}
suffices since $L(T)\lesssim L(R)$. For $T>10S$, the input that reaches
the observation ball at time $Q$ occupies a shell of radius $Q$ and
thickness $O(R)$, up to rapidly decaying tails. We will compare its wave
with one from a shell at radius $S$, where the global bound costs only
$L(R)^A$. For this comparison take $Q\ge S$ and write the shell input as
\[
f_Q(s\omega)=s^{-\alpha}h(s-Q,\omega),\qquad
h(u,\omega)=0\quad (|u|>5R).
\]
Its contribution at $t=Q+b$, $|x|,|b|\le R$, is approximated by
\[
\mathcal T h(b,x)=\int k(b-u+x\cdot\omega)h(u,\omega)\dd u\dd\omega.
\]
This expression is independent of $Q$. In fact
$|s\omega-x|=s-x\cdot\omega+O(R^2/Q)$ and
$(s/|s\omega-x|)^\alpha=1+O(R/Q)$. Inserting these into
\eqref{eq:ang-unit-kernel}, with the polar Jacobian $s^d$, bounds the
difference kernel against $h$ by
$C_D(R^2/Q)\langle b-u+x\cdot\omega\rangle^{-D}$. Young's inequality
in $b$, followed by integration in $\omega$ and $x$, gives
\[
\|U(Q+b)\Pi_1 f_Q-\mathcal T h\|_{L^p(\{|b|,|x|\le R\})}
\lesssim (R^{2+n/p}/Q)\|h\|_{L^p(\dd u\dd\omega)} .
\]
We may therefore estimate the profile at the smaller radius $Q=S$.
Since $d-\alpha p=-1$, this input has norm
\[
\|f_S\|_p^p=\int\frac{|h(u,\omega)|^p}{S+u}\dd u\dd\omega
\asymp S^{-1}\|h\|_p^p .
\]
The factor $S^{1/p}$ in the scaled wave estimate cancels. Applying
Theorem~\ref{thm:main} on $[S/2,S]$ and $[S,2S]$ and using $L(S)\lesssim L(R)$
now gives $\|\mathcal T h\|_{L^p(\{|b|,|x|\le R\})}\lesssim L(R)^A\|h\|_p$.
Divide $[T,2T]$ into intervals of length between $R/2$ and $R$, centered at
$Q_i$. On interval $i$ take
$h_i(u,\omega)=(Q_i+u)^\alpha f((Q_i+u)\omega)\mathbf1_{|u|\le5R}$.
The input shells have bounded overlap, so
$\sum_i\|h_i\|_p^p\lesssim T\|f\|_p^p$. Applying the profile estimate
and summing gives the desired bound for these shells.
For $T/2\le s\le3T$, $|s-Q_i|>5R$, the remaining kernel is bounded by
$C_D s^{-\alpha}\mathbf1_{|t-s|>3R}\langle t-s\rangle^{-D}$:
the output displacement is at most $R$. Young's inequality in $t$ as
above bounds this contribution by $C_D R^{n/p+1-D}T^{1/p}\|f\|_p$.
The radii outside $[T/2,3T]$ again give rapid decay in $T+s$.
\end{proof}
\subsection{The polar kernel and the two endpoint norms}
At an output point $r\theta$, angular stationary phase samples each input
sphere at directions $\theta$ and $-\theta$. An angular average controls
the remainders. To bound all three terms by a single radial profile, for
$g$ on $\RR^n$ define the even function of $v\in\RR$
\[
h_g(v,\theta)=|v|^{d/p}\left(|g(|v|\theta)|+|g(-|v|\theta)|
+\int_{\mathbb S^d}|g(|v|\omega)|\dd\omega\right),\qquad
G_g(v,\theta)=\mathcal M_{\RR}h_g(v,\theta),
\]
where $\mathcal M_{\RR}$ is the Hardy--Littlewood maximal operator
in $v$. The weight $|v|^{d/p}$ incorporates the polar Jacobian, so
$\|G_g\|_{L^p(\dd v\dd\theta)}\lesssim\|g\|_p$.
\begin{lemma}\label{lem:angular-envelope}
There is an integer $K$, independent of $M,N$, with the following property.
Suppose $g$ has angular degree at most $M$, is supported on $s=|y|\ge s_0$,
and $N\min(r,s_0)\ge (2+M)^K$. Then for $t\in I$,
\begin{equation}\label{eq:ang-envelope-final}
|U(t)\Pi_N g(r\theta)|\lesssim
\sum_{\pm}G_g(t\pm r,\theta)
\begin{cases} r^{-\alpha},&r\le4,\\r^{-d/p},&r>4.\end{cases}
\end{equation}
For arbitrary $g\in L^p(\RR^n)$, the part supported in $|y|<N^{\eta-1}$, where
$\eta=\alpha/(2n)$, satisfies
\begin{equation}\label{eq:ang-small-source}
\|U(t)\Pi_N(\mathbf1_{|y|<N^{\eta-1}}g)\|_{L^p(I\times\RR^n)}
\lesssim N^{-\delta}\|g\|_p,\qquad
\delta=(\alpha/2)(1-2/p)>0 .
\end{equation}
\end{lemma}
\begin{proof}
Write $S_zF(\theta)=\int_{\mathbb S^d}e^{iz\theta\cdot\omega}
F(\omega)\dd\omega$. In polar coordinates the angular operator at
radii $r,s$ is
\[
c_n s^d\int e^{-it\rho}\chi(\rho/N)\rho^d S_{r\rho}S_{-s\rho}\dd\rho .
\]
For each sign of $z$, use rotation-invariant cutoffs to separate the
two stationary points. The expansion takes the form
\[
S_z F=|z|^{-\alpha}\sum_{\sigma=\pm1}e^{i\sigma|z|}
(c_\sigma P_\sigma F+E_{\sigma,z}F),
\]
where for either fixed sign of $z$ the two $P_\sigma$ are the identity
and the antipodal reflection, in the corresponding order.
Fix an integer $D>2+\alpha$. Then $E_{\sigma,z}F$ and its first $D$
logarithmic derivatives in $|z|$ are bounded pointwise by
$C_D|z|^{-1}\|F\|_{C^{m_D}}$. To see this, use coordinates at the pole
in which $1\mp\theta\cdot\omega=|w|^2/2$. After removing the leading
oscillation, the normalized integral has the form
$I_\lambda(a)=\lambda^{d/2}\int e^{\pm i\lambda|w|^2/2}a(w)\dd w$,
where $\lambda=|z|$. Integration by parts in $w$ gives
$\lambda\partial_\lambda I_\lambda(a)=-\frac12I_\lambda(w\cdot\nabla a)$.
Each differentiated amplitude vanishes at the pole, so its leading
stationary phase term is zero. The stationary phase remainder is
$O(\lambda^{-1})$ for each of the finitely many amplitudes, with the
stated derivative bound; off the pole one integrates by parts in angle
\cite[Chapter VIII, \S2]{Stein1993}.
On degrees at most $M$, $\|F\|_2\lesssim(2+M)^{d/2}\|F\|_1$ by the
constant diagonal of the spherical projection kernel. Sobolev embedding
and the spectrum of $-\Delta_{\mathbb S^d}$ thus bound
$\|F\|_{C^{m_D}}\lesssim(2+M)^{c_D}\|F\|_1$. Choose $K>c_D$ for the fixed $D$ above.
For $|z|\gtrsim(2+M)^K$ each normalized remainder maps this band
from $L^1$ to $L^\infty$ with uniform norm. It preserves the band by
rotation invariance. Consequently the composition of the two normalized
factors, including the needed radial derivatives, is bounded at $\theta$
by $C(|F(\theta)|+|F(-\theta)|+\|F\|_1)$.
Now integrate by parts in $\rho/N$ in each of the four phases.
The scalar kernels against the last angular majorant are bounded by
$C_D r^{-\alpha}s^\alpha N\langle N(t\pm r\pm s)\rangle^{-D}$.
The input weight left after $s^{d/p}$ is $s^{1/p}$. For $r\le4$ it is
bounded by $C\langle t\pm r\pm s\rangle^{1/p}$; for $r>4$ insert the
additional factor $r^{1/p}$ on the right. Absorb this polynomial into the
decay. For each $v=t\pm r$, the integrals with phases $v-s$ and $v+s$
are bounded by one convolution on $\RR$, since $h_g$ is even in $s$.
Its kernel is majorized by $C_D N\langle N\cdot\rangle^{-D+1/p}$, so
the convolution is bounded by $C G_g(v,\theta)$. This proves
\eqref{eq:ang-envelope-final}.
Finally \eqref{eq:ang-unit-kernel} and scaling give
$\|U(t)\Pi_N\|_{1\to\infty}\lesssim N^{\alpha+1}$ for $t\in I$, while
the $L^2$ norm is bounded. For $v=g\mathbf1_{|y|<b}$, interpolate
$\|U(t)\Pi_Nv\|_2$ and $\|U(t)\Pi_Nv\|_\infty$, then use H\"older
on the input ball. This gives
$\|U(t)\Pi_Nv\|_p\lesssim (N^{\alpha+1}b^n)^{1-2/p}\|g\|_p$.
Taking $b=N^{\eta-1}$ gives \eqref{eq:ang-small-source}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:angular-band}]
Put $H=(2+M)^K$. If $H>N^\eta$, then $L(N)\lesssim L(M)$ and the
unrestricted annular estimate gives both conclusions. Suppose $H\le N^\eta$.
On the output ball $r\le H/N$, Lemma~\ref{lem:angular-ball} gives
the strong bound $CL(H)^A\|f\|_p\lesssim L(M)^A\|f\|_p$.
On its complement we may discard the input from $s<N^{\eta-1}$ with
the strong error \eqref{eq:ang-small-source}. For the remaining input
$g$, both $Nr$ and $Ns$ are at least $H$, as required in
Lemma~\ref{lem:angular-envelope}.
Consider one profile in \eqref{eq:ang-envelope-final}. Changing variables
from $(t,r)$ to $(v,r)=(t\pm r,r)$ preserves measure. At fixed $v,\theta$,
write $G=G_g(v,\theta)$. For $N^{-1}\le r\le4$ the strong norm and the
distribution function are controlled by the two radial integrals
\begin{equation}\label{eq:ang-radial-integrals}
\int_{N^{-1}}^4(r^{-\alpha}G)^p r^d\dd r=G^p\log(4N),\qquad
\int_{\{r>0:\ r^{-\alpha}G>\lambda\}}r^d\dd r
=\frac{G^p}{n\lambda^p}\quad(\lambda>0).
\end{equation}
Integrating in $v,\theta$ gives the factor $O(L(N)^{1/p})$ for the
strong norm and a constant for the joint weak norm. For $r>4$ the
$p$th power of $r^{-d/p}$ cancels the polar Jacobian. At each $t$ the
remaining integral of $|G_g(t\pm r,\theta)|^p$ in $r,\theta$ is at
most $\|G_g\|_p^p$; this part has a strong bound uniform in $N$.
Combining the two regions proves the proposition.
\end{proof}
For $A=2/p$ and $M\le\exp(\sqrt{L(N)})$, the strong estimate therefore
costs only $L(N)^{1/p}$. For cutoffs equal to one on $[5/4,7/4]$, the
radial data of Proposition~\ref{prop:sharpness} force this power already
at angular degree zero.
\subsection{Summation over the spatial and angular frequencies}
Choose a smooth radial Littlewood--Paley decomposition $f_j$ at frequencies
$N_j=2^j$, and let $u_j=U(t)f_j$. Insert a fixed enlarged annular multiplier
$\Pi_{N_j}$ equal to one on the support of $\widehat f_j$. The bounded
frequency part satisfies a uniform fixed-time $L^p$ estimate by the
Mikhlin theorem. On the other bands, \eqref{eq:ang-small-source} allows
us to restrict the input to $s\ge N_j^{\eta-1}$, with a total strong
error $O(\|f\|_p)$ even when summed in $j$.
We group angular degrees according to the size of their logarithm.
Use smooth angular low pass operators $S_m$ at degree $M_m=2^{2^m}$,
$m\ge0$, and their differences $Q_m=S_m-S_{m-1}$ with $S_{-1}=0$.
Take a fixed smooth cutoff on degrees, equal to one up to $M_m$ and
zero beyond $2M_m$. These operators commute with all radial Fourier
multipliers by rotation invariance. Their kernels satisfy
$|S_m(\theta,\omega)|\lesssim_D M_m^d
(1+M_m\dist(\theta,\omega))^{-D}$ by smooth spherical localization
\cite[Theorem~3.5 and Corollaries~3.6, 5.3]{NPW2006}; on the circle this is
ordinary smooth Fourier summation. In particular
$|S_m g|\lesssim\mathcal M_{\mathbb S^d}g$, uniformly in $m$.
For every $B>0$,
\begin{equation}\label{eq:ang-log-projection}
\|Q_m f\|_p\lesssim_B 2^{-mB}\|L(\Om)^B f\|_p .
\end{equation}
Indeed, as a function of the harmonic degree $\nu$ on the support of
$Q_m$, $L((1+\nu(\nu+d-1))^{1/2})\asymp2^m$. The normalized symbol
$2^{mB}Q_m(\nu)L((1+\nu(\nu+d-1))^{1/2})^{-B}$ and all its logarithmic
derivatives in $\nu$ are uniformly bounded. The smooth multiplier theorem
on the sphere \cite[Theorem~4.9(1)]{BonamiClerc1973} gives \eqref{eq:ang-log-projection}
on each angular slice.
To pay the spatial logarithm only once, at fixed $r$ and $N_j$ we group
all angular bands on which stationary phase is uniform into one low pass.
Put
$H_m=(2+2M_m)^K$, and select the largest $m$ with
\begin{equation}\label{eq:ang-eligibility}
H_m\le \min(N_j^\eta,N_jr).
\end{equation}
Call this index $m(j,r)$, with value $-1$ when there is no such $m$.
For $a_j=\mathbf1_{s\ge N_j^{\eta-1}}f_j$, the exact decomposition of
the retained wave at $(t,r\theta)$ is
\[
U(t)\Pi_{N_j}a_j=U(t)\Pi_{N_j}S_{m(j,r)}a_j+
\sum_{m>m(j,r)}U(t)\Pi_{N_j}Q_ma_j.
\]
For a given output $r$, the index $m(j,r)$ is fixed on every input
sphere. If it is nonnegative, Lemma~\ref{lem:angular-envelope} applies
to the first term. Its positive majorant may use
$\mathcal M_{\mathbb S^d}f_j$ in place of $S_{m(j,r)}f_j$, eliminating
the dependence of the profile on the chosen cutoff. The vector-valued
maximal inequality, on the sphere and on the line \cite[Theorem~1.2]{GLY2009}, gives for arbitrary
nonnegative coefficients $c_j$
\[
\big\|(c_jG_{\mathcal M_{\mathbb S^d}f_j})_j
\big\|_{L^p(\dd v\dd\theta;\ell^2)}
\lesssim \|(c_jf_j)_j\|_{L^p(\RR^n;\ell^2)} .
\]
The reflection and angular integral in $h_g$ are bounded on
$L^p(\mathbb S^d;\ell^2)$ by Minkowski's inequality.
It remains to integrate these profiles along $v=t\pm r$, now taking
the square function in $j$. Write $G_j=G_{\mathcal M_{\mathbb S^d}f_j}(v,\theta)$.
On the selected region $r\ge N_j^{-1}$, so for $r\le4$ we may bound
$r^{-\alpha}$ by $C(r+N_j^{-1})^{-\alpha}$. At fixed $v,\theta$,
Minkowski's inequality in $L^{p/2}(r^d\dd r)$ gives
\begin{equation}\label{eq:ang-log-integral}
\left\|\left(\sum_j (r+N_j^{-1})^{-2\alpha}|G_j|^2\right)^{1/2}
\right\|_{L^p((0,4),r^d\dd r)}
\lesssim\left(\sum_j (1+j)^{2/p}|G_j|^2\right)^{1/2}.
\end{equation}
Indeed the $p$th power of the $j$th radial factor integrates to $O(1+j)$
by $\alpha p=n$. Integrate in $v,\theta$ and apply the maximal inequality
above with $c_j=(1+j)^{1/p}$. The smooth spatial Littlewood--Paley theorem
then bounds the strong square function by $C\|L(\J)^{1/p}f\|_p$.
For the weak norm keep the common factor $r^{-\alpha}$ and use the
distribution formula in \eqref{eq:ang-radial-integrals} with
$G=\|(G_j)\|_{\ell^2}$. The maximal inequality with $c_j=1$ gives
$C\|f\|_p$. For $r>4$ the common factor is $r^{-d/p}$; the same
integration at fixed $t$ as in the single annulus gives a strong bound
of $C\|f\|_p$ there.
It remains to estimate the unselected angular frequencies in strong
$L^p$. Fix $m$ and denote by $v_j$ the wave $U(t)\Pi_{N_j}$ applied to
$Q_m f_j\mathbf1_{s\ge N_j^{\eta-1}}$; write $b_j=\|Q_m f_j\|_p$.
When $H_m>N_j^\eta$, the entire $v_j$ is unselected. There are
$O(1+\log H_m)$ such $j$ and Theorem~\ref{thm:main} bounds each by
$CL(H_m)^A b_j$. The pointwise inequality
$\|w\|_{\ell^2}\le (\#j)^{1/2-1/p}\|w\|_{\ell^p}$ shows that
their strong square function costs
$CL(H_m)^{A+1/2-1/p}\|(b_j)\|_{\ell^p}$.
For the remaining $j$, the unselected output lies in $r<R_j=H_m/N_j$.
Let $D=\lceil\log_2(2H_m)\rceil$ and consider the disjoint shells
$R_{k+1}<r\le R_k$, with $R_k=H_m2^{-k}$. On shell $k$ only $j\le k$
contribute. For $k-D\le j\le k$ the same finite-sequence inequality
costs $(1+D)^{1/2-1/p}$. After taking $p$th powers and summing over
$k$, each $v_j$ is integrated only on its ball $r\le R_j$, where
Lemma~\ref{lem:angular-ball} gives $CL(H_m)^A b_j$. For $j<k-D$ the
shell lies within one wavelength: $N_jR_k=H_m2^{j-k}\le2^{-(k-j-D)}$.
The volume factor in the same lemma on the smaller ball $r\le R_k$ gives
$C2^{-(k-j-D)n/p}b_j$. Sum these latter bounds first in $j$ by the
triangle inequality and then in $\ell^p(k)$ by discrete Young's
inequality. Thus the unselected $m$th angular band has total strong
square-function norm at most
\[
C L(H_m)^{A+1/2-1/p}\|(b_j)\|_{\ell^p}
\lesssim 2^{m(A+1/2-1/p)}\|Q_m f\|_p.
\]
In the last step we used $p>2$ and the spatial square-function bound
for $Q_m f$. By \eqref{eq:ang-log-projection}, summing in $m$ costs
$C_B\|L(\Om)^B f\|_p$ whenever $B>A+1/2-1/p$.
These estimates control the square function of the original $u_j$:
the unselected pieces have a strong bound, and the selected pieces
have both bounds. The splitting depends on $r$, so the pieces need not
be annular separately. We recover the sum only after reassembling $u_j$:
the spatial synthesis operator
$(w_j)\mapsto\sum_j\Pi_{N_j}w_j$ is bounded from
$L^q(I\times\RR^n;\ell^2)$ to $L^q(I\times\RR^n)$ for $1<q<\infty$
by spatial Littlewood--Paley duality and Fubini
\cite[Theorem~6.1.2 and Remark~6.1.3]{Grafakos2014}. Real interpolation
gives the same assertion for joint weak $L^p$. The argument
first gives uniform bounds for finite frequency sums and then the
stated bounds by approximation. The same angular multiplier theorem
applies to $L(\Om)^{-B}$ and $L(\Om)^B\Om^{-\varepsilon}$ for every
$\varepsilon>0$: the first absorbs $\|f\|_p$ into the stated weak norm,
and the second gives the assertion with a positive angular derivative.
This proves Theorem~\ref{thm:angular}.
==== END Local-smoothing-for-the-Euclidean-wave-equation//angular.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//barrier.tex ====
\section{Estimating the energy on a fixed fiber}\label{sec:barrier}
On a fiber at fixed $(T,\tau)$, the only positive contribution to
the energy balance comes from the wave current crossing the spatial
wall $r=0$. The Gaussian is small where the current reaches the wall;
comparison with a spatial average of the wave bounds the other factor,
the boundary value of $u/r$. This gives an upper bound for the flux in
terms of the wave norm. To bound it by the energy itself, we must
recover that same norm from the interior. There we invert the Gaussian
and compare $u$ with a quadratic on each ball of radius $h$.
\begin{proposition}\label{prop:fixed-barrier}
Fix an annulus $c_1\le h|\xi|\le c_2$, with $0<c_1<c_2$. There are
constants $\gamma>0$, $K\ge1$, and $c,C>0$, depending only on $n$ and
the annulus, with the following property. Let $a=Kh$, $0<h<1/(4K)$,
and let $f$ be a trigonometric polynomial on the fixed torus, with
$\|f\|_2=1$ and frequencies in this annulus and in $|v|\le\gamma$ in
some orthonormal frame. Write $g_T=e^{-iT|D|}f$ and let
$\mathcal F(T,\tau)$ be its energy from Section~\ref{sec:energy}.
Then
\begin{align}
\mathcal F(0,\tau)&\le C a^2\|f\|_{\pc}^{\pc}
&& (|\tau|=4a),\label{eq:barrier-input}\\
\mathcal F(T,\tau)&\ge c a^2\|g_T\|_{\pc}^{\pc}
&& (|\tau|=2a),\label{eq:barrier-output}\\
\Delta_z\mathcal F(T,\tau)&\le V(s)\mathcal F(T,\tau),\qquad
V(s)=C s^{-2}e^{-c s/h}
&& (2a\le s=|\tau|\le3).\label{eq:barrier-potential}
\end{align}
The last two inequalities hold for all $T\in\mathbb R$.
\end{proposition}
\begin{proof}
Write $d=n-1$, $q=n/d=\pc/2$, and $\sigma=s/h$. Throughout the proof
$2a\le s\le3$. On the fiber
\[
r=|W|^2/w_0^2+100|Q|^2-1<0,\qquad w_0=1/10,\quad Q=\Phi/a,
\]
we use two regions. On $r=0$ the cutoff in the current vanishes unless
$|Q|\le1/20$, which forces $|W|\ge (\sqrt{3}/2)w_0$. Near the center,
where $|W|\le\gamma$ and $|Q|\le1/400$, the Gaussian multiplier can
be inverted. Figure~\ref{fig:fiber} shows the separation between these
regions. It lets us compare two exponentials with different constants
in their exponents. For now take $0<\gamma<w_0/8$. We estimate uniformly in $s$,
allowing the prefactor in the interior lower bound to depend on $\gamma$.
\input{figures/fiber}
\paragraph{A spatial average for the upper bound.}
For a dimensional constant $D\ge1$, to be fixed below, set
\[
P_*(X)=\big((J_{Ds}*|g_T|^2)(X)\big)^{1/d},\qquad
M=2n+4.
\]
Here and below convolution is in the real spatial variables $X$.
Recall that $J_\ell$ is the periodization of
$C_n\ell^{-n}(1+|X/\ell|^2)^{-M/2}$, of integral one. Differentiating this
positive kernel term by term and then taking the $d$th root gives
\begin{equation}\label{eq:barrier-average}
|\nabla_X^jP_*|\le C(Ds)^{-j}P_*\quad(j=1,2),\qquad
\int P_*^{d+1}\,dX\le\|g_T\|_{\pc}^{\pc}.
\end{equation}
The integral bound is Young's inequality with exponent $q$. The positive
kernel gives $P_*>0$ since $\|g_T\|_2=1$.
We next bound the wave current $A_{\mathrm w}$ and its regularized
density $\mu$. By Proposition~\ref{prop:current}, each scalar part of
the wave polarization is a multiplier on $g_T$ with symbol a bounded
smooth factor times
\[
(\kappa/y)^{d/2}\exp(-\Psi),\qquad
\Psi=\tfrac12\kappa y|W-v|^2,\qquad y=|Y|\asymp s.
\]
Multiply the symbol by a fixed smooth annular cutoff equal to one on the
data and supported in $|v|<w_0/3$. In the variable $\eta=h\xi$ its
derivatives through order $M$ are bounded by
$C(hs)^{-d/2}\sigma^m$ for some fixed $m$. Indeed $h\kappa$ and $v$
have uniformly bounded derivatives on this cutoff, $h\kappa$ is bounded
away from zero there, and each derivative of $\Psi$ costs at most
$C\sigma$. On $r=0$ where $g(Q)$ is nonzero, $|W-v|$ has a fixed
positive lower bound, so the same estimate gains $e^{-c_0\sigma}$ for
some $c_0>0$ independent of $\gamma$.
If $B$ bounds these derivatives of a compactly supported symbol, applying
$(1-\Delta_\eta)^{n+2}$ in its inverse Fourier transform bounds its
absolute kernel by $CB h^{-n}(1+|X/h|^2)^{-M/2}$. Periodization gives
the majorant $CBJ_h$ on the torus, also for a rotated lattice. Thus
the squared modulus of the multiplier output is bounded by
$C B^2 J_h*|g_T|^2$. There is one more convolution with $J_h$ in
$A_{\mathrm w}$. We have $J_h*J_h\le C J_h$: in the Euclidean
convolution, either $|Y|$ or $|X-Y|$ is at least $|X|/2$, and the
inequality persists after periodization. Moreover
$J_h\le(Ds/h)^nJ_{Ds}$. Using $(hs)^{-d}=s^{-2d}\sigma^d$ and
increasing $m$ when necessary, we obtain
\begin{align}
\|A_{\mathrm w}\|&\le Cs^{-2d}\sigma^m P_*^d
&&(r\le0),\nonumber\\
\|A_{\mathrm w}\|&\le Cs^{-2d}\sigma^m e^{-c_0\sigma}P_*^d
&&(r=0).\label{eq:barrier-current}
\end{align}
Here $\|A_{\mathrm w}\|$ is the matrix operator norm. Only the first
bound is needed for $\mu$. Its extra term is
$h^{50n}\det H_{\mathrm{fib}}(r)\le Ch^{50n}a^{-2}s^{-2d}$ by
Lemma~\ref{lem:fiber-geometry}. The nearest lattice term in the positive
kernel gives $\inf_X J_{Ds}(X)\ge c_D h^{M-n}$, since the torus has fixed
diameter and $h\le s\le3$. Consequently
$P_*^d\ge c_D h^{M-n}\|g_T\|_2^2=c_D h^{M-n}$, which absorbs
$h^{50n}a^{-2}$. In particular
\begin{equation}\label{eq:barrier-density}
\mu\le C s^{-2d}\sigma^m P_*^d\qquad(r\le0).
\end{equation}
\paragraph{A comparison function that vanishes on the wall.}
Put $R=H_{\mathrm{fib}}(r)$. Two features of the defining function
from Lemma~\ref{lem:fiber-geometry} will be used together:
$R\ge c s^{-2}I$ and
$G_{\mathrm{fib}}(r)G_{\mathrm{fib}}(r)^*\le C R$. It follows that
\begin{equation}\label{eq:barrier-product}
H_{\mathrm{fib}}(rP_*)\ge (1-C/D)P_*R.
\end{equation}
In fact, the product rule of Lemma~\ref{lem:hessian-calculus} gives
$P_*R$, $rH_{\mathrm{fib}}(P_*)$, and two terms pairing the gradients
of $r$ and $P_*$. For any quaternionic vector $e$ the two gradient
pairings with $e$ have moduli at most $C(e^*Re)^{1/2}$ and
$C D^{-1}P_*(e^*Re)^{1/2}$, respectively. This estimates the mixed
terms. The term involving $H_{\mathrm{fib}}(P_*)$ has relative cost
$C D^{-2}$ by \eqref{eq:barrier-average} and $|r|\le1$.
Since $r<0$, an upper bound on $u/r$ calls for a comparison function
below $u$, with determinant at least $\lambda\mu$. Fix $D$ so the
right side of \eqref{eq:barrier-product} is at least
$P_*R/2$. The background Hessian $B_{\mathrm{fib}}=\lambda
H_{\mathrm{fib}}(|\Phi|^2)$ has rank one and its positive eigenvalue
is comparable to $\lambda=h^{-30n}$. Hence for
$v_*=C_1\sigma^{m_1}rP_*$,
\[
\det\big(B_{\mathrm{fib}}+H_{\mathrm{fib}}(v_*)\big)
\ge c\lambda (C_1\sigma^{m_1}P_*s^{-2})^d\ge\lambda\mu
\]
by \eqref{eq:barrier-density} with $m_1,C_1$ large enough. The
potentials $v_*$ and $u$ have the same zero boundary data; Dirichlet
comparison (Theorem~\ref{thm:dirichlet}) yields $v_*\le u\le0$.
Division by $r<0$ and passage to the boundary give
\begin{equation}\label{eq:barrier-quotient}
0\le u/r\le C_1\sigma^{m_1}P_*\qquad(r\le0).
\end{equation}
By Lemma~\ref{lem:energy-comparison}, $\mathcal F\le\int(-u)\mu$ on the
fiber. Its imaginary volume element is comparable to
$a^2s^{2d}\,dW\,dQ$ by Lemma~\ref{lem:fiber-geometry}. Combining
\eqref{eq:barrier-average}, \eqref{eq:barrier-density} and
\eqref{eq:barrier-quotient} gives
\begin{equation}\label{eq:barrier-upper-all}
\mathcal F(T,\tau)\le Ca^2\sigma^m\|g_T\|_{\pc}^{\pc}.
\end{equation}
For the flux we need the \emph{full} gradient of $r$. The tangency in
Proposition~\ref{prop:current} gives $A_{\mathrm w}G(|\Phi|^2)=0$;
the remaining part, $G(|W|^2/w_0^2)$, has size $O(s^{-1})$ since
$W=P/Y$ and $y\asymp s$. Thus $G(r)^*A_{\mathrm w}G(r)\le
C s^{-2}\|A_{\mathrm w}\|$. The Jacobian estimate also gives
$\int\delta(r)\,dP\,dZ\le C a^2s^{2d}$ by coarea on the fixed
ellipsoid in $(W,Q)$. Applying the wall bound in
\eqref{eq:barrier-current} and \eqref{eq:barrier-quotient} to the
flux integral in Proposition~\ref{prop:energy-flux}, we conclude that
\begin{equation}\label{eq:barrier-flux-all}
\Delta_z\mathcal F(T,\tau)
\le C (a/s)^2\sigma^m e^{-c_0\sigma}\|g_T\|_{\pc}^{\pc}.
\end{equation}
\paragraph{Recovering the wave in the interior.}
Consider a center $(X,P,Z)$ with $|W|\le\gamma$ and $|Q|\le1/400$.
The ball of radius $h$ in the $3n$ real spatial coordinates about this
center lies in $|W|\le2\gamma$, $|Q|\le1/100$, provided $K\gamma$
and $K$ are sufficiently large: $W=P/Y$ changes by at most $Ch/s$
and $\Phi$ by at most $Ch$ there. The cutoff in $A_{\mathrm w}$ is
one throughout the ball.
The first scalar multiplier $\mathsf S_0$ of the time component in
\eqref{eq:current-time-multipliers} has symbol
$b_0(\kappa/y)^{d/2}e^{-\Psi}$ with
$b_0=(1+\operatorname{Re}(W)\cdot v/2)/\sqrt{2}$ bounded away from
zero. The other scalar component is orthogonal, so
$|\mathsf S_0g_T|^2\le A^0_{TT}$. Invert $\mathsf S_0$ on the frequencies of $g_T$
by multiplying its reciprocal symbol by a smooth annular cutoff equal
to one on the data and supported in $|v|\le2\gamma$. On its support
$\Psi\le C\gamma^2\sigma$. The derivatives of the reciprocal symbol in
$\eta=h\xi$ through order $M$ are bounded by
\[
B_{\rm inv}=C_\gamma(hs)^{d/2}\sigma^m e^{C\gamma^2\sigma}.
\]
Here the derivatives of the cutoff enter only the prefactor $C_\gamma$;
the constant in the exponential is independent of $\gamma$.
The kernel estimate above yields
$|g_T(X)|^2\le C B_{\rm inv}^2 (J_h*A^0_{TT})(X)$ for each choice
of the imaginary coordinates in the ball. Since $J_h(X'-Y)$ and
$J_h(X-Y)$ are comparable for $|X'-X|\le h$, the density at
\emph{every point} in the ball is at least
\begin{equation}\label{eq:barrier-ball-density}
m_X=c_\gamma(hs)^{-d}\sigma^{-m}e^{-C_2\gamma^2\sigma}|g_T(X)|^2.
\end{equation}
For the interior lower bound on $-u$ we need a comparison function
\emph{above} $u$. Let $l$ be Euclidean distance from the center. The quadratic
$v_b=\epsilon m_X^{1/d}(l^2-h^2)$ vanishes on the boundary of the ball
and has fiber Hessian $6\epsilon m_X^{1/d}I$ (three real slots per
spatial coordinate). For $m_X>0$, its determinant after adding
$B_{\mathrm{fib}}$ is at most
\[
C\big((\epsilon m_X^{1/d})^n+\lambda(\epsilon m_X^{1/d})^d\big)
\le \lambda m_X
\]
with a fixed small $\epsilon$: the density ceiling
\eqref{eq:energy-density-ceiling} and \eqref{eq:barrier-ball-density}
give $m_X^{1/d}\le\mu^{1/d}\le\lambda$. Now $u\le0=v_b$ on the ball
boundary and the determinant of $B_{\mathrm{fib}}+H_{\mathrm{fib}}(u)$
is $\lambda\mu\ge\lambda m_X$. The local comparison principle from
the proof of Theorem~\ref{thm:dirichlet} gives $u\le v_b$ in the ball.
At the center the depth is therefore at least $\epsilon h^2m_X^{1/d}$.
Multiplication by $\mu\ge m_X$ gives $(-u)\mu\ge\epsilon h^2m_X^{1+1/d}$,
also when $m_X=0$. Here $1+1/d=q$ and $m_X$ contains $|g_T(X)|^2$,
giving the required power $2q=\pc$ of the wave.
The imaginary volume of the center region is comparable to
$a^2s^{2d}\gamma^{2d}$. Integrate the center inequality there and in
$X$, use the lower bound for $\mathcal F$ in
Lemma~\ref{lem:energy-comparison}, and recall that
$h^2s^{2d}(hs)^{-dq}=\sigma^{n-2}$. Enlarging the fixed power $m$
and absorbing $\gamma^{2d}$ into the prefactor gives
\begin{equation}\label{eq:barrier-lower-all}
\mathcal F(T,\tau)\ge
c_\gamma a^2\sigma^{-m}e^{-C_3\gamma^2\sigma}\|g_T\|_{\pc}^{\pc}.
\end{equation}
Choose $\gamma$ so small that $C_3\gamma^2<c_0/2$, then fix $K$ to
meet the ball inclusions and to ensure that $h<1/(4K)$ lies in the
range of \eqref{eq:energy-density-ceiling}. Dividing the flux bound
\eqref{eq:barrier-flux-all} by the interior lower bound
\eqref{eq:barrier-lower-all} gives
\[
\frac{\Delta_z\mathcal F}{\mathcal F}
\le C_\gamma s^{-2}\sigma^{2m}e^{-(c_0-C_3\gamma^2)\sigma}.
\]
The exponential absorbs the fixed power of $\sigma$, proving
\eqref{eq:barrier-potential}. On the two circles in the $\tau$ plane
the scaled radii $4a/h=4K$ and $2a/h=2K$ are fixed. The energy
upper bound \eqref{eq:barrier-upper-all} at the first circle gives
\eqref{eq:barrier-input} for the initial data; the lower bound
\eqref{eq:barrier-lower-all} at the second gives \eqref{eq:barrier-output}
for the wave at time $T$.
\end{proof}
==== END Local-smoothing-for-the-Euclidean-wave-equation//barrier.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//calculus.tex ====
\section{The determinant and integration by parts}\label{sec:calculus}
The wave current in Proposition~\ref{prop:current} is positive and has
divergence-free real columns. We shall pair it with a Hessian. The
essential pairing is the ordinary real contraction
$A:H(v)=A^{\RR}:D^2v$: each spacetime index still has its three real
coordinates. The choice of Hessian has two further consequences. The squared modulus of a
holomorphic function contributes at most one quaternionic direction.
In $n$ spatial variables, if its nonzero eigenvalue is $\beta$, adding
$\epsilon I$ gives determinant $(\beta+\epsilon)\epsilon^{n-1}$.
This is where the denominator in $p_{\mathrm c}=2n/(n-1)$ enters.
The cofactor of the Hessian also has
divergence-free columns; this allows us to integrate the determinant
equation by parts, including at the boundary.
\subsection{The Hessian in real and complex coordinates}
Use the triples $z_a=x_a+P_a\mathbf i\in\mathcal D$ of
Section~\ref{sec:current}, with $P_a\in\CC_{\mathbf k}$. We allow a
fourth real coordinate as well: Appendix~\ref{sec:dirichlet} uses it to
rotate the normal at the boundary. Thus for each of $m$ coordinate
indices let $z_a=\sum_{\alpha=0}^{k-1} e_\alpha z_{a\alpha}$, where
$k=3$ or $4$, $(e_0,e_1,e_2,e_3)=(1,\mathbf i,\mathbf j,\mathbf k)$,
and the $z_{a\alpha}$ are real.
For a real $C^2$ function $u$ put $\partial_{a\alpha}=\partial/\partial z_{a\alpha}$
and define
\begin{equation}\label{eq:quaternionic-hessian}
G_a(u)=\sum_\alpha e_\alpha\,\partial_{a\alpha}u,\qquad
H_{ab}(u)=\sum_{\alpha,\beta} e_\alpha\bar e_\beta\,
\partial_{a\alpha}\partial_{b\beta}u .
\end{equation}
Thus $G(u)$ is the ordinary gradient written as a quaternionic column.
The symbols $\nabla$, $\Delta$ and $\partial_{\mathbf n}$ denote the
Euclidean gradient, Laplacian and outward normal derivative.
\begin{lemma}[Hessian calculus]\label{lem:hessian-calculus}
In either number of slots, $H(u)$ is Hermitian and $\tr H(u)=\Delta u$.
The linear map from the real Hessian to $H(u)$ sends
$\xi\xi^{\mathsf t}$ to $gg^*$, where
$g_a=\sum_\alpha e_\alpha\xi_{a\alpha}$. It sends nonnegative
real forms to nonnegative quaternionic forms. The product rule is
\begin{equation}\label{eq:hessian-product}
H(uv)=uH(v)+vH(u)+G(u)G(v)^*+G(v)G(u)^* .
\end{equation}
In four slots $H$ is covariant under quaternionic unitary changes of
coordinates, and in three slots under real orthogonal changes of the
coordinate index applied in all slots.
If $u$ depends only on $P\in\CC_{\mathbf k}^m$, then
\begin{equation}\label{eq:complex-hessian}
G_a(u)=2(\partial_{\bar P_a}u)\mathbf i,\qquad
H_{ab}(u)=4\partial_{\bar P_a}\partial_{P_b}u .
\end{equation}
In particular, a holomorphic map $F=(F_1,\ldots,F_l)$ with complex
Jacobian $J_{\nu a}=\partial_{P_a}F_\nu$ satisfies
\begin{equation}\label{eq:holomorphic-squares}
H\Big(\sum_\nu|F_\nu|^2\Big)=4J^*J .
\end{equation}
For every smooth real function $\rho$ of $F$,
$G(\rho\circ F)$ belongs to the quaternionic image of $J^*$.
\end{lemma}
\begin{proof}
Interchanging the real derivatives gives Hermitian symmetry. In a
diagonal entry the terms with $\alpha\ne\beta$ cancel in pairs; the
remaining terms have coefficient one. Formula~\eqref{eq:quaternionic-hessian}
applied to $\xi\xi^{\mathsf t}$ gives $gg^*$, and sums of these
rank-one forms give every nonnegative real form. The same formula and
the ordinary product rule give \eqref{eq:hessian-product}.
In four slots the real transpose of a quaternionic unitary $U$ acts
on the encoded covector as $U^*$. Thus the Hessian rule for a real
rank-one form gives $H(u\circ U)=U^*(H(u)\circ U)U$; linearity
extends this to every real Hessian. Real orthogonal matrices on the
coordinate index also preserve the three-slot subspace, giving the
stated covariance there.
For $P_a=p_a+\mathbf k q_a$ we have
$2\partial_{\bar P_a}=\partial_{p_a}+\mathbf k\partial_{q_a}$.
Since $\mathbf k\mathbf i=\mathbf j$, this gives the gradient in
\eqref{eq:complex-hessian}. The four terms in $H_{ab}$ have
coefficients $1,-\mathbf k,\mathbf k,1$ in the order
$(p_ap_b,p_aq_b,q_ap_b,q_aq_b)$, which gives its Hessian formula.
The complex chain rule now gives \eqref{eq:holomorphic-squares} and
$G(\rho\circ F)=2J^*(\partial_{\bar F_\nu}\rho)_\nu\,\mathbf i$,
with the derivatives of $\rho$ evaluated at $F$.
\end{proof}
\subsection{The Moore cofactor}
For Hermitian quaternionic matrices use the real inner product
$A:B=\Rea\tr(AB)$. The real part of the trace is cyclic; in particular
$A:(vv^*)=v^*Av$. Write $\det A$ for the Moore determinant, the product
of the $m$ real eigenvalues counted with quaternionic multiplicity.
These conventions agree with \cite[Section~1.1, especially Theorems
1.1.8--1.1.9]{Alesker2003LA}. The next identities apply also to the
full Hessian in the energy argument, whose spatial block is positive
definite but whose time entry need not have a sign.
\begin{lemma}[Matrix identities]\label{lem:quaternionic-matrices}
The Moore determinant is a homogeneous polynomial of degree $m$.
Its cofactor $C(A)$ is the Hermitian matrix defined by
\begin{equation}\label{eq:cofactor-definition}
C(A):B=\left.\frac{d}{dt}\det(A+tB)\right|_{t=0}.
\end{equation}
It is polynomial of degree $m-1$, and
\begin{equation}\label{eq:cofactor-inverse}
C(A)=(\det A) A^{-1}\quad (A>0).
\end{equation}
If $A\ge0$, then $C(A)\ge0$; if also $A$ is singular, $C(A)$ kills
the image of $A$. On positive definite matrices $\log\det$ is concave
and increasing in the matrix order.
For a unit vector $v\in\HH^m$ let $A|_{v^\perp}$ denote the
compression of the form to its quaternionic orthogonal complement.
For every Hermitian $A$ and real $t$,
\begin{equation}\label{eq:cofactor-restriction}
v^*C(A)v=\det(A|_{v^\perp}),\qquad
\det(A+t vv^*)=\det A+t\,v^*C(A)v.
\end{equation}
The determinant on the zero-dimensional space is one. Finally, for a
Hermitian $M=\bigl(\begin{smallmatrix}\ell&c^*\\ c&D\end{smallmatrix}\bigr)$
with $D>0$, put $\sigma=\ell-c^*D^{-1}c$ and let $v_0$ be the first
coordinate vector. Then
\begin{equation}\label{eq:schur-complement}
\det M=(\det D)\sigma,\qquad M-\sigma v_0v_0^*\ge0 .
\end{equation}
\end{lemma}
\begin{proof}
A Hermitian matrix has a quaternionic orthonormal eigenbasis:
maximize $v^*Av$ on the unit sphere and repeat on the orthogonal
complement. By Newton's identities its determinant is a polynomial
in the power sums $\Rea\tr(A^j)$, $1\le j\le m$. This gives
\eqref{eq:cofactor-definition} with the stated degrees.
In the real representation on $\RR^{4m}$ each eigenvalue occurs
four times. Thus for $A>0$, $\log\det A$ is one fourth of the real
log determinant. Differentiation proves \eqref{eq:cofactor-inverse},
monotonicity and concavity. On a diagonal matrix each cofactor
eigenvalue is the product of the other $m-1$ eigenvalues, first for
positive matrices by \eqref{eq:cofactor-inverse}, then for all
diagonal matrices by polynomiality. Unitary invariance proves the
claims for singular $A$.
Completing the square shows that the form of $M-\sigma v_0v_0^*$ at
$(b,w)$ is $(w+D^{-1}cb)^*D(w+D^{-1}cb)$. For $\sigma>0$ the real
Schur determinant formula gives $(\det M)^4=(\det D)^4\sigma^4$;
take the positive fourth root. Both sides of $\det M=(\det D)\sigma$
are polynomials in $\ell$, so the identity holds for every $\sigma$.
In a basis with $v$ first, adding $tvv^*$ changes only $\ell$.
For positive $A$ the change in its determinant is therefore exactly
$t\det(A|_{v^\perp})$. Differentiation in $t$ gives the first formula
in \eqref{eq:cofactor-restriction}; polynomiality extends both to
every Hermitian $A$. For $m=1$ they hold directly.
\end{proof}
\subsection{Column divergence and the boundary term}
In $k$ slots the real tensor \eqref{eq:current-real-tensor} of a
Hermitian field $A(z)$ has coefficients
\begin{equation}\label{eq:real-coefficients}
A^{\RR}_{a\alpha,b\beta}=\Rea(\bar e_\alpha A_{ab}e_\beta).
\end{equation}
This symmetric real matrix represents the Hermitian form on the
encoded real vectors: $(\nabla v)^{\mathsf t} A^{\RR}\nabla u=
\Rea(G(v)^*A G(u))$. Likewise $A:H(u)=A^{\RR}:D^2u$, where the
pairing on the right is the ordinary real matrix inner product.
Consequently, whenever the real columns have zero divergence,
\begin{equation}\label{eq:real-divergence}
A:H(u)=\Div(A^{\RR}\nabla u).
\end{equation}
One sufficient condition is the quaternionic row identity
\begin{equation}\label{eq:strong-divergence}
\sum_{a,\alpha}\bar e_\alpha\,\partial_{a\alpha}A_{ab}=0
\quad(1\le b\le m):
\end{equation}
multiply on the right by $e_\beta$ and take real parts. Also,
if $A G(w)=0$ for a real scalar function $w$, then $wA$ preserves
either form of column conservation already satisfied by $A$.
Indeed the extra row in the product rule is $G(w)^*A=0$.
Alesker showed that the real tensor representing the Moore cofactor
of a positive definite quaternionic Hessian has divergence-free columns.
This puts the linearized Monge--Ampère operator in divergence form
\cite[Lemma~4.6]{Alesker2013Flat}. The following polynomial proof
also covers the three slots used by the wave current and allows an
indefinite or singular Hessian.
\begin{lemma}[Cofactor divergence]\label{lem:cofactor-divergence}
For $k=3$ or $4$ and a smooth real $u$, the Hessian cofactor satisfies
\begin{equation}\label{eq:cofactor-divergence}
\sum_{a,\alpha}\bar e_\alpha\,\partial_{a\alpha} C(H(u))_{ab}=0
\quad(1\le b\le m).
\end{equation}
\end{lemma}
\begin{proof}
For $m=1$ the cofactor is one. Otherwise put $j_0=m-1$. Polarize
the cofactor: for Hermitian inputs $M_1,\ldots,M_{j_0}$, denote
the coefficient of $t_1\cdots t_{j_0}$ in $C(\sum_j t_jM_j)$ by
$Q(M_1,\ldots,M_{j_0})$. This is multilinear, and on the diagonal
equals $j_0!\,C(M_1)$.
Take $M_j=g_jg_j^*$. For $t_j>0$, $\sum_jt_jg_jg_j^*$ is singular:
its image is the quaternionic span of the $j_0$ columns. Its cofactor
therefore kills each $g_j$, by Lemma~\ref{lem:quaternionic-matrices}.
This polynomial identity in $t$ holds coefficient by coefficient.
Thus $Q=Q(g_1g_1^*,\ldots,g_{j_0}g_{j_0}^*)$ kills every $g_j$,
and $g_j^*Q=0$ as well since $Q$ is Hermitian.
Test the polarized identity on $u_j(z)=\exp(\xi_j\cdot z)$ with
arbitrary real covectors $\xi_j$, using the Euclidean dot product.
Set $(g_j)_a=\sum_\alpha e_\alpha(\xi_j)_{a\alpha}$. The Hessian
symbol in Lemma~\ref{lem:hessian-calculus} gives
$Q(H(u_1),\ldots,H(u_{j_0}))=\exp((\sum_j\xi_j)\cdot z)\,Q$.
After cancellation of this exponential, its row divergence is
$(\sum_jg_j)^*Q=0$. To pass to arbitrary smooth inputs, observe
that the row divergence is a multilinear differential expression with
constant coefficients: its value on these exponential functions,
divided by their product, is the polynomial whose coefficients
multiply the corresponding derivatives of the inputs. That polynomial
vanishes for all covectors, so all coefficients vanish. Taking every
input equal to $u$ proves \eqref{eq:cofactor-divergence}.
\end{proof}
Here is the boundary convention used for both the current and the
cofactor. On a smooth domain $\Omega=\{r<0\}$ assume $\nabla r\ne0$
on $\partial\Omega$. Write $\mathbf n=\nabla r/|\nabla r|$ for the
real outward unit normal and $\nu=G(r)/|\nabla r|$ for the same
vector in quaternionic coordinates. By \eqref{eq:real-divergence},
Green's identity for a field with divergence-free real columns is
\begin{equation}\label{eq:calculus-green}
\int_\Omega\!\big(v\,A:H(u)-u\,A:H(v)\big)
=\int_{\partial\Omega}\!
\Rea\!\left(v\,\nu^*A G(u)-u\,\nu^*A G(v)\right)\dd S .
\end{equation}
Volume and surface measure are Euclidean. The same formula holds in
periodic coordinates, with test functions of compact support in any
unbounded directions.
If $u$ vanishes on $\partial\Omega$, then $G(u)=\nu\,\partial_{\mathbf n}u$
there. Also $u=\rho r$ for a smooth $\rho$ near the boundary: use $r$
as one local coordinate and integrate the derivative of $u$ in that
coordinate. The cross
terms of \eqref{eq:hessian-product} then vanish on compression, giving
\begin{equation}\label{eq:boundary-restriction}
H(u)|_{\nu^\perp}=\rho\,H(r)|_{\nu^\perp}\qquad\text{on }\partial\Omega .
\end{equation}
In four slots this calculation gives the invariance of the quaternionic
Levi form under a change of defining function \cite{Alesker2003MA}.
Functions independent of the fourth slot give the same formula in three slots.
In particular the normal flux coefficient $\nu^*C(H(u))\nu$ depends
only on this compressed form, by \eqref{eq:cofactor-restriction}.
This is why the determinant boundary term can be evaluated without
knowing the second derivative of $u$ in the real normal direction.
==== END Local-smoothing-for-the-Euclidean-wave-equation//calculus.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//consequences.tex ====
\section{Bochner--Riesz means}
\label{sec:consequences}
The Bochner--Riesz mean \(B_r^\alpha f\) has Fourier multiplier
\((1-|\xi|^2/r^2)_+^\alpha\); write
\(B_*^\alpha f=\sup_{r>0}|B_r^\alpha f|\). Local smoothing controls these
means because a smooth cutoff to a shell of relative width \(\delta\)
can be written as an average of waves for times of order \(\delta^{-1}\).
Sogge observed the implication for a single radius
\cite[Equations~(4.5)--(4.6)]{Sogge1991};
Beltran, Hickman and Sogge give both the fixed-radius and maximal
reductions \cite[Propositions~39 and~40]{BHS2021}. In the range
\(p\ge2\), almost everywhere convergence at the strict orders below was
proved by Carbery, Rubio de Francia and Vega \cite{CRV1988Summability}.
\begin{theorem}[Bochner--Riesz means]\label{thm:bochner-riesz}
Let \(n\ge2\), \(1\le p\le\infty\), and put
\(\delta(p)=\max\{0,n|1/p-1/2|-1/2\}\).
If \(\alpha>\delta(p)\), then
\begin{equation}\label{eq:br-uniform}
\sup_{r>0}\|B_r^\alpha f\|_{L^p(\RR^n)}
\le C_{n,p,\alpha}\|f\|_{L^p(\RR^n)}.
\end{equation}
In this range \(B_r^\alpha f\to f\) in \(L^p\) as \(r\to\infty\) if
\(p<\infty\). If \(2\le p<\infty\) and \(\alpha>s_+(p)\), then
\begin{equation}\label{eq:br-maximal}
\|B_*^\alpha f\|_{L^p(\RR^n)}\le C_{n,p,\alpha}\|f\|_{L^p(\RR^n)},
\end{equation}
and \(B_r^\alpha f\to f\) almost everywhere as \(r\to\infty\).
\end{theorem}
Here and in the square function estimate of Section~\ref{sec:stein-square},
the strict inequality on the order of the means leaves room for a small
Sobolev loss. We rescale Theorem~\ref{thm:local-smoothing} to a unit
frequency annulus and a long time interval.
For \(2\le p<\infty\), \(\sigma>s_+(p)=\max\{0,(n-1)/2-n/p\}\),
and \(\widehat g\) supported in a fixed annulus \(c\le|\xi|\le C\),
\begin{equation}\label{eq:cs-unit-wave}
\|U(t)g\|_{L^p(\RR^n\times\{R\le|t|\le2R\})}
\lesssim_{c,C,n,p,\sigma} R^{1/p+\sigma}\|g\|_p,\qquad R\ge1.
\end{equation}
Indeed \(g(R\,\cdot)\) has frequencies of order \(R\) and
\(U(t)[g(R\,\cdot)](x)=(U(Rt)g)(Rx)\). Changing variables in space
and time contributes \(R^{(n+1)/p}\), while the input norm contributes
\(R^{-n/p}\) and the Sobolev multiplier on that annulus costs \(O(R^\sigma)\).
Conjugation gives the estimate for negative times. For \(|t|\le1\) the
unit annular wave multiplier has a uniformly integrable kernel.
\begin{lemma}[Maximal estimate for a thin shell]\label{lem:maximal-shell}
For \(0<\delta\le1/8\) and \(\psi\in C_c^\infty((1/2,2))\), set
\[
A_{\delta,r}f=\psi\big((1-|D|/r)/\delta\big) f .
\]
For \(2\le p<\infty\) and \(\sigma>s_+(p)\),
\[
\left\|\sup_{r>0}|A_{\delta,r} f|\right\|_p
\lesssim \delta^{-\sigma}\|f\|_p .
\]
The constant is uniform for \(\psi\) in bounded subsets of
\(C_c^\infty((1/2,2))\).
\end{lemma}
\begin{proof}
For \(1\le r\le2\) the shell lies in \(1/2\le|\xi|\le2\).
Choose a smooth annular symbol \(\chi\) equal to one there, and write
\(Q_k=\chi(2^{-k}D)\). For \(g=Q_0f\), Fourier inversion gives
\[
A_{\delta,r}f(x)=\frac{\delta r}{2\pi}
\int_{\RR} e^{itr}\widehat\psi(\delta rt) U(t)g(x)\dd t .
\]
Taking the supremum over \(r\in[1,2]\) bounds this by
\(C_M\delta\int (1+\delta |t|)^{-M}|U(t)g(x)|\dd t\).
On a dyadic interval \(R\le |t|\le2R\), H\"older's inequality in time
and \eqref{eq:cs-unit-wave} bound the \(L^p_x\) norm of this contribution by
\[
C_M\delta(1+\delta R)^{-M}R^{1-1/p}R^{1/p+\sigma}\|g\|_p .
\]
For \(M>1+\sigma\) its sum over dyadic \(R\ge1\) is
\(O(\delta^{-\sigma}\|g\|_p)\), by splitting at \(R=\delta^{-1}\).
The interval \(|t|\le1\) contributes \(O(\delta\|g\|_p)\).
For \(r=2^k s\), \(1\le s\le2\), set \(g_k(y)=(Q_kf)(2^{-k}y)\).
Then \(A_{\delta,r}f(2^{-k}y)=A_{\delta,s}g_k(y)\).
The factor \(2^{-kn/p}\) in the output norm cancels the factor
\(2^{kn/p}\) in \(\|g_k\|_p\). Thus the \(L^p\) norm
of the supremum in this octave is at most \(C\delta^{-\sigma}\|Q_k f\|_p\).
The supremum over all octaves is bounded by their pointwise \(\ell^p\)
norm. Since \(p\ge2\), Littlewood--Paley theory gives
\[
\left(\sum_{k\in\mathbb Z}\|Q_k f\|_p^p\right)^{1/p}
\le \left\|\left(\sum_{k\in\mathbb Z}|Q_k f|^2\right)^{1/2}\right\|_p
\lesssim \|f\|_p .
\qedhere
\]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:bochner-riesz}]
Take \(2\le p<\infty\) and choose \(s_+(p)<\sigma<\alpha\).
To decompose \((1-\rho^2)_+^\alpha\), choose
\(\eta\in C_c^\infty((1/2,2))\) with
\(\sum_{j\in\mathbb Z}\eta(2^j v)=1\) for \(v>0\), and apply the
partition to \(v=1-\rho\) on \(0\le\rho<1\). For
\(\delta=2^{-j}\le1/8\) the corresponding term is
\[
\delta^\alpha\psi_\delta((1-\rho)/\delta),
\qquad
\psi_\delta(u)=\eta(u)u^\alpha(2-\delta u)^\alpha .
\]
These profiles have uniform smooth seminorms. The remainder is a smooth
radial multiplier of compact support: it vanishes near \(\rho=1\) and
equals \((1-\rho^2)^\alpha\) near zero. The supremum of its dilations is
bounded pointwise by a constant times the Hardy--Littlewood maximal
function, since its kernel has a radial decreasing integrable majorant.
Lemma~\ref{lem:maximal-shell} bounds the other terms in \eqref{eq:br-maximal}
by the convergent sum \(C\sum_{j\ge3}2^{-j(\alpha-\sigma)}\|f\|_p\).
For \(2\le p<\infty\), \(\delta(p)=s_+(p)\), so the maximal estimate
implies \eqref{eq:br-uniform}. Each fixed \(B_r^\alpha\) is self-adjoint
on Schwartz functions, and \(\delta(p)=\delta(p')\); duality proves
\eqref{eq:br-uniform} for \(1<p<2\). At \(p=1,\infty\) the hypothesis is
\(\alpha>(n-1)/2\). The unit-radius kernel is a constant multiple of
\(|x|^{-n/2-\alpha}J_{n/2+\alpha}(|x|)\), by radial Fourier inversion.
It is bounded at zero and decays as \(O(|x|^{-(n+1)/2-\alpha})\) at
infinity. Its dilates \(r^n K(rx)\) therefore have a common \(L^1\) norm.
For Schwartz functions of compact Fourier support the multipliers converge
to one in all derivatives on the support as \(r\to\infty\); hence the means
converge in the Schwartz topology. Density and \eqref{eq:br-uniform} give
norm convergence for \(p<\infty\). To see pointwise convergence in the
maximal range, choose such functions \(f_j\) with
\(\sum_j\|f-f_j\|_p<\infty\). By \eqref{eq:br-maximal},
\(\sum_j B_*^\alpha(f_{j+1}-f_j)\) is finite almost everywhere. At every
such point \(r\mapsto B_r^\alpha f_j(x)\) converges uniformly on \(r>0\)
to a continuous representative of the means, agreeing with the \(L^p\)
extension for each fixed \(r\). Its supremum is measurable by continuity,
and Fatou's lemma extends \eqref{eq:br-maximal} to \(f\).
Since \(f_j(x)\to f(x)\) almost everywhere, uniform approximation in
\(r\) gives the asserted almost everywhere limit.
\end{proof}
==== END Local-smoothing-for-the-Euclidean-wave-equation//consequences.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//current.tex ====
\section{A conserved current for a superposition of waves}\label{sec:current}
We use the flux of a symmetric first-order system twice. It selects an
imaginary displacement that gives each mode a Gaussian weight; applied
to the continued wave, it gives a positive conserved tensor. The modes
are summed before taking the square, so their interference remains in
the time component of the tensor. We shall apply the construction on a
fixed spatial torus: an annulus then contains only finitely many modes,
and integration by parts has no boundary in the real spatial variables.
Section~\ref{sec:transfer} transfers the estimate to $\RR^n$.
\subsection{The symbol determines the Gaussian}
Choose an orthonormal frame $X=(x,b)\in\RR^d\times\RR$, where $d=n-1$,
and put
\[
t=(T+b)/\sqrt2,\qquad s_0=(b-T)/\sqrt2 .
\]
In these null coordinates the wave operator is
$2\partial_t\partial_{s_0}+\Delta_x$. At a fixed frequency $\kappa>0$
in $s_0$ this is the free Schr\"odinger operator
$2\mathrm i\kappa\partial_t+\Delta_x$ in the transverse variables.
A mode with physical phase
$X\cdot\xi-T|\xi|$ and $\xi_b+|\xi|>0$ takes the form
\begin{equation}\label{eq:current-modes}
u_{\kappa,v}
=c_{\kappa,v}\exp\!\left(\mathrm i\kappa
\left(s_0+v\cdot x-\tfrac12 |v|^2t\right)\right),
\qquad \kappa=\frac{\xi_b+|\xi|}{\sqrt2},\qquad v=\frac{\xi_x}{\kappa}.
\end{equation}
Here $\mathrm i$ denotes the ordinary Fourier imaginary unit. Throughout
the construction $u_{\rm w}=\sum_{\kappa,v}u_{\kappa,v}$ is a finite
sum with $\kappa>0$ and complex coefficients.
Write $E_{ab}$ for the elementary $n\times n$ matrix with its nonzero
entry in position $(a,b)$, indexing the states from $0$ to $d$.
For the real spacetime coordinates $\mathsf X=(t,x,s_0)$ take the symmetric matrices
\[
C^t=2E_{00},\qquad C^{x_j}=E_{0j}+E_{j0},\qquad
C^{s_0}=-\sum_{j=1}^d E_{jj},\qquad
\mathscr P=\sum_\mu C^\mu\partial_{\mathsf X^\mu}.
\]
Indeed, $\mathscr P(\partial_{s_0}\psi,\nabla_x\psi)^{\mathsf t}$
has first entry $(2\partial_t\partial_{s_0}+\Delta_x)\psi$ and all
other entries zero. At the covector $p=\kappa(-|v|^2/2,v,1)$ of
\eqref{eq:current-modes}, the symbol $C(p)=\sum_\mu p_\mu C^\mu$
satisfies
\begin{equation}\label{eq:current-wave-symbol}
-C(p)=\kappa\begin{pmatrix}|v|^2&-v^{\mathsf t}\\-v&I_d\end{pmatrix},
\qquad
-w^{\mathsf t}C(p)w=\kappa|w'-vw_0|^2 .
\end{equation}
In particular the kernel is spanned by $E(v)=(1,v)^{\mathsf t}$.
This gives a vector solution
$f_{\rm sys}=\sum_{\kappa,v}\kappa^{d/2}E(v)u_{\kappa,v}$ of
$\mathscr P f_{\rm sys}=0$.
To see the Gaussian in physical coordinates, translate $(T,X)$ by
$-\mathrm i\rho(1,\omega)$ with $\rho>0$ and $|\omega|=1$. A mode of
frequency $\xi$ acquires the factor
\[
e^{\rho(\xi\cdot\omega-|\xi|)}
=e^{-\rho|\xi|\,|\widehat\xi-\omega|^2/2},
\qquad \widehat\xi=\xi/|\xi| .
\]
For $|\xi|\asymp N$ and $\rho N\gg1$, its angular width is
$(\rho N)^{-1/2}$. The system itself supplies these light rays.
For a real state $w=(w_0,w')$ its flux $q(w)^\mu=w^{\mathsf t}C^\mu w/2$
is $(w_0^2,w_0w',-|w'|^2/2)$ in null coordinates. In physical
coordinates this gives $q(w)^T=|q(w)^X|=(w_0^2+|w'|^2/2)/\sqrt2$;
at the polarization $E(v)$ the ray is $q(E(v))=(|\xi|,\xi)/\kappa$.
Take the sum of the fluxes of two real states,
\begin{equation}\label{eq:current-real-flux}
q^\mu=q(w_1)^\mu+q(w_2)^\mu
=\tfrac12\sum_{\ell=1}^2 w_\ell^{\mathsf t}C^\mu w_\ell,
\qquad w_1,w_2\in\RR^n .
\end{equation}
At $\mathsf X-\mathrm i q$ the weight is $e^{p\cdot q}$. Pairing
corresponding entries of the states, write
\[
z=w_1+\mathbf k w_2=(R,\zeta)\in
\CC_{\mathbf k}\times\CC_{\mathbf k}^{d},\qquad
\CC_{\mathbf k}=\{a+\mathbf k b:a,b\in\RR\},\quad \mathbf k^2=-1 .
\]
Thus $q^\mu=z^*C^\mu z/2$, with $^*$ the complex conjugate transpose.
Adding the two real squares in \eqref{eq:current-wave-symbol} gives
\begin{equation}\label{eq:current-source-gaussian}
e^{p\cdot q}=\exp\!\left(-\tfrac{\kappa}{2}|\zeta-vR|^2\right).
\end{equation}
This explains the normalization in $f_{\rm sys}$: the square of
$\kappa^{d/2}e^{p\cdot q}$ integrates to $\pi^d$ in $\zeta$ for every $R$.
\subsection{Two auxiliary flows}
Recall the bilinear conservation identity for symmetric systems,
as in \citet{Friedrichs1954}. For constant real symmetric matrices
$C^\mu$, the flux of states $a,b$ is $j^\mu(a,b)=a^{\mathsf t}C^\mu b$.
Its divergence for two fields is $(\mathscr P a)^{\mathsf t}b+
a^{\mathsf t}\mathscr P b$.
The derivative of $q^\mu$ in the direction $\delta w_\ell$ is exactly
$j^\mu(w_\ell,\delta w_\ell)$. This derivative gives the physical
component of the following vector fields.
\begin{lemma}\label{lem:current-flow}
Let $\mathscr P=\sum_\mu C^\mu\partial_{\mathsf X^\mu}$ have constant
real symmetric coefficients. For a holomorphic solution
$\mathscr P f_{\rm sys}=0$ and the real displacement
\eqref{eq:current-real-flux}, put, wherever defined,
\[
G=\overline{f_{\rm sys}(\mathsf X-\mathrm i q)},\qquad
V_{\ell a}=\sum_\mu(C^\mu w_\ell)_a\,\partial_{\mathsf X^\mu},\qquad
D_{\ell a}=\partial_{w_{\ell a}}+\mathrm i V_{\ell a},\qquad
\mathcal Z_\ell=\sum_a G_aD_{\ell a}.
\]
In the real coordinates $(\mathsf X,w_1,w_2)$, the tensor
\begin{equation}\label{eq:current-source-tensor}
B=\Rea\sum_{\ell=1}^2\mathcal Z_\ell\otimes\overline{\mathcal Z_\ell}
\end{equation}
is symmetric and positive semidefinite, and each of its columns has
zero divergence.
\end{lemma}
\begin{proof}
The fields $D_{\ell a}$ are the complex vector fields associated
with the tube coordinates $\mathsf X-\mathrm i q$ as in
\cite[Section~1]{BaouendiTreves1982}. Each annihilates every
component of $\mathsf X-\mathrm i q$, hence
$D_{\ell a}\overline G=0$. Conjugation gives
$\partial_{w_{\ell a}}G=\mathrm i V_{\ell a}G$.
Each $D_{\ell a}$ has zero divergence, so
\[
\Div\mathcal Z_\ell=2\mathrm i\sum_a V_{\ell a}G_a
=2\mathrm i\, w_\ell^{\mathsf t}\mathscr P G=0 .
\]
For the ordinary flat connection, the identity $D_{\ell a}\overline G=0$
leaves only the derivatives of the coefficients of $\overline{D_{mb}}$
when computing the acceleration:
\[
\nabla_{\mathcal Z_\ell}\overline{\mathcal Z_m}
=-\mathrm i\,\delta_{\ell m}
\sum_\mu(G^{\mathsf t}C^\mu\overline G)\partial_{\mathsf X^\mu}.
\]
The vector in the sum is real by symmetry of the matrices. The
product rule for $\Div(\mathcal Z_\ell\otimes\overline{\mathcal Z_\ell})$
now gives zero after taking real parts. Finally, if
$\mathcal Z_\ell=A_\ell+\mathrm i B_\ell$ with real fields, its
contribution to \eqref{eq:current-source-tensor} is
$A_\ell\otimes A_\ell+B_\ell\otimes B_\ell$.
\end{proof}
This cancellation also identifies the physical current in the lifted
variables: the imaginary part of the last differentiation identity at
$m=\ell$ gives
$[A_\ell,B_\ell]=\sum_\mu(G^{\mathsf t}C^\mu\overline G)
\partial_{\mathsf X^\mu}$.
For the estimate we use the four real squares in the conserved tensor
$B$. We must express them in coordinates attached to spacetime.
\subsection{Coordinates adapted to the wave}
We want one auxiliary pair for each of the $n+1$ real spacetime
coordinates. In the target, write each triple as $\mathsf X^\mu+
\omega_\mu\mathbf i$, with $\omega_\mu\in\CC_{\mathbf k}$. The
quaternions $\HH$ have units $\mathbf i,\mathbf j,\mathbf k$ satisfying
$\mathbf i^2=\mathbf j^2=\mathbf k^2=-1$ and
$\mathbf i\mathbf j=\mathbf k$, $\mathbf k\mathbf i=\mathbf j$.
Thus the target triples lie in
$\mathcal D=\operatorname{span}_{\RR}(1,\mathbf i,\mathbf j)$.
We identify the Fourier complex plane with $\CC_{\mathbf i}$.
Every quaternion is uniquely $\alpha+\mathbf k\beta$ with
$\alpha,\beta\in\CC_{\mathbf i}$; these two planes are orthogonal for
the real scalar product $\langle u,v\rangle=\Rea(u\overline v)$.
We use columns with scalars acting on the right; $^*$ on a matrix
denotes quaternionic conjugate transpose.
The representation we seek for $B$ is the following. For a quaternionic
Hermitian matrix $A$ on spacetime indices, set
\begin{equation}\label{eq:current-real-tensor}
A^{\RR}_{\mu\alpha,\nu\beta}
=\Rea(\overline{e_\alpha}A_{\mu\nu}e_\beta),\qquad
(e_0,e_1,e_2)=(1,\mathbf i,\mathbf j).
\end{equation}
If $A=\chi\chi^*$, this real tensor is the Gram matrix of the rows
$\overline{e_\alpha}\chi_\mu$. In other words the three rows over
each $\mu$ have the form $Q,-\mathbf iQ,-\mathbf jQ$.
In source coordinates, write the four squares of
\eqref{eq:current-source-tensor} as a Gram matrix by setting
$Q_I=\mathcal Z_1^I+\mathbf k\mathcal Z_2^I$ for each coordinate $I$.
Orthogonality of $\CC_{\mathbf i}$ and $\mathbf k\CC_{\mathbf i}$
gives $B^{IJ}=\Rea(Q_I\overline{Q_J})$. With
$M_{\mu a}=(C^\mu z)_a$, the rows in the real spacetime and state
directions are
\[
Q_{\mathsf X^\mu}=\sum_a M_{\mu a}\mathbf i G_a,\qquad
(Q_{w_{1a}},Q_{w_{2a}})=(G_a,\mathbf kG_a).
\]
We now choose the target auxiliary coordinates to give the desired
rows $-\mathbf i Q_{\mathsf X^\mu},-\mathbf j Q_{\mathsf X^\mu}$.
For a holomorphic coordinate $\omega_\mu(z)$ let
$L_{\mu a}=\partial_{z_a}\omega_\mu$. Multiplication by
$c=c_1+\mathbf k c_2$ on an auxiliary pair has real rows
$(c_1,-c_2),(c_2,c_1)$, so it sends $(G_a,\mathbf kG_a)$ to
$(\bar cG_a,\mathbf k\bar cG_a)$. Consequently the new pair over
$\mu$ starts with $\sum_a\overline{L_{\mu a}}G_a$. Taking $L=M$
gives exactly $-\mathbf i Q_{\mathsf X^\mu}$, since
$-\mathbf i c\,\mathbf i=\bar c$ for $c\in\CC_{\mathbf k}$.
Its partner is $\mathbf k(-\mathbf i Q_{\mathsf X^\mu})
=-\mathbf j Q_{\mathsf X^\mu}$.
\paragraph{The holomorphic flux map.}
The derivative $L_{\mu a}=(C^\mu z)_a$ integrates to the quadratic
$z^{\mathsf t}C^\mu z/2$, by symmetry. Thus the Hermitian square
$z^*C^\mu z/2$ weights the modes, while the bilinear square transforms
their Gram rows. There are $n$ complex state coordinates and $n+1$
spacetime coordinates. Introduce a passive $\eta\in\CC_{\mathbf k}$,
in whose two real directions the flows have zero component. For any
constant real symmetric system of Lemma~\ref{lem:current-flow} with
these dimensions and any fixed $c\in\RR^{n+1}$, set
\[
\Theta:(z,\eta)\longmapsto\omega,
\qquad \omega_\mu=\tfrac12 z^{\mathsf t}C^\mu z+c^\mu\eta .
\]
The map fixes the real spacetime coordinates. On a local inverse chart
its real volume Jacobian is $J=|\det_{\CC}[M,c]|^2$, where
$M_{\mu a}=(C^\mu z)_a$. Push the tensor forward as a density:
\begin{equation}\label{eq:current-pushforward}
\widetilde B\circ\Theta=J^{-1}D\Theta\,B\,D\Theta^{\mathsf t}.
\end{equation}
The two differentials transform the Gram rows as above, and $J^{-1}$
converts source volume to target volume. Hence
\begin{equation}\label{eq:current-flux-column}
\widetilde B=(\widetilde\chi\widetilde\chi^*)^{\RR},\qquad
\widetilde\chi_\mu\circ\Theta=J^{-1/2}\sum_a M_{\mu a}\mathbf i G_a .
\end{equation}
As a function of $\omega$, the passive coordinate satisfies
$(\partial_\omega\eta)M=0$: varying $z$ at fixed $\eta$ gives the
columns of $M$. Thus $(\partial_\omega\eta)\widetilde\chi=0$.
To verify conservation, test the $K$th column against the gradient of
a real smooth function $\varphi$ compactly supported in the target
chart. Changing variables in \eqref{eq:current-pushforward}, the
integral for the weak divergence becomes
$-\sum_{I,L}\int B^{IL}(\partial_L\Theta^K)\partial_I(\varphi\circ\Theta)$.
Integration by parts in source coordinates gives
\[
J(\Div\widetilde B)^K\circ\Theta
=\sum_L(\partial_L\Theta^K)(\Div B)^L
+\sum_{I,L}B^{IL}\partial_I\partial_L\Theta^K .
\]
The first sum vanishes by Lemma~\ref{lem:current-flow}. In the second
sum the two diagonal state blocks of $B$ are both
$(\Rea(G_a\overline{G_b}))_{ab}$, and the mixed state block is zero.
Each real component of the holomorphic map satisfies
$\partial_{w_{1a}w_{1b}}\Theta^K+
\partial_{w_{2a}w_{2b}}\Theta^K=0$. The remaining coordinates enter
$\Theta$ affinely, so the second sum vanishes as well.
\paragraph{The wave in these coordinates.}
For the wave matrices, add $\eta$ in the $s_0$ coordinate. In the
null frame the holomorphic flux map becomes
\begin{equation}\label{eq:current-chart}
\Theta:(R,\zeta,\eta)\longmapsto
(Y,P,S_1)=(R^2,R\zeta,\eta-\zeta\cdot\zeta/2).
\end{equation}
If $(\tau,P,Z)$ denote the target auxiliary coordinates in the physical frame $(T,x,b)$, then
$Y=(\tau+Z)/\sqrt2$ and $S_1=(Z-\tau)/\sqrt2$. Inverting
\eqref{eq:current-chart} on $Y\ne0$ gives $R=\sqrt Y$ on a local
branch and $\zeta=P/R$. The useful single-valued coordinates are the
ratio of the transverse states to $R$ and the passive variable:
\begin{equation}\label{eq:current-invariants}
W=\frac P Y=\frac{\zeta}{R},\qquad
\Phi=S_1+\frac{P\cdot P}{2Y}=\eta,\qquad y=|Y|.
\end{equation}
All dots between complex vectors here are bilinear; $|\cdot|$ denotes
the Euclidean norm. The complex Jacobian determinant of $\Theta$ is
$2R^{d+1}$, so its real volume Jacobian is
\begin{equation}\label{eq:current-volume}
J=4|R|^{2d+2}.
\end{equation}
Under the derivative $M$ in the state directions a polarization vector
$E(v)$ maps to
\begin{equation}\label{eq:current-polarization}
ME(v)=(2R,\zeta+Rv,-\zeta\cdot v)^{\mathsf t}=2R V^v,
\qquad V^v=\big(1,(v+W)/2,-W\cdot v/2\big)^{\mathsf t}.
\end{equation}
In particular $\Phi$ is constant in each such direction because it
pulls back to the passive variable. Explicitly, put
$\ell(w)=(-w\cdot w/2,w,1)$; thus $\kappa\ell(v)$ is the real phase
covector. One checks
$\ell(w)V^v=-(w-W)\cdot(w-v)/2$. The column lies in the kernels of
both $\ell(v)$ and $\partial\Phi=\ell(W)$. Differentiating $W=P/Y$
gives the second identity in
\begin{equation}\label{eq:current-directions}
(\partial\Phi)V^v=0,\qquad
(\partial W)V^v=\frac{v-W}{2Y}.
\end{equation}
These are complex differentials in $(Y,P,S_1)$. The current stays
on a level set of $\Phi$, while each mode changes $W$ in proportion to
its angular mismatch $v-W$.
For $e=Y/y$ put
$\iota_e(\alpha+\mathrm i\beta)=\alpha+e\mathbf i\beta$.
This is an isometric embedding of the Fourier complex numbers into
$\HH$: $\mathbf i c=\bar c\mathbf i$ for $c\in\CC_{\mathbf k}$,
so $(e\mathbf i)^2=-1$. Define the positive weight
\[
m_{\kappa,v}=(\kappa/y)^{d/2}
\exp\!\left(-\tfrac{\kappa y}{2}|W-v|^2\right).
\]
The factor $y^{-d/2}$ comes from the change of volume.
\begin{proposition}[The wave current]\label{prop:current}
For any finite superposition \eqref{eq:current-modes}, set
\begin{equation}\label{eq:current-wave}
\chi=\sum_{\kappa,v} V^v\,\iota_e(\overline{u_{\kappa,v}})
\,m_{\kappa,v},\qquad A^0=\chi\chi^* .
\end{equation}
On $Y\ne0$ the matrix $A^0$ is smooth and positive semidefinite,
$(\partial\Phi)A^0=0$, and each column of $(A^0)^{\RR}$ has zero
real divergence. These conclusions hold also after the orthogonal
change from null to physical coordinates. In particular,
\[
V_T^v=(1+W\cdot v/2)/\sqrt2,\qquad A^0_{TT}=|\chi_T|^2 .
\]
\end{proposition}
\begin{proof}
We identify the column in \eqref{eq:current-flux-column} on one
inverse chart. By \eqref{eq:current-source-gaussian} and
\eqref{eq:current-polarization}, the contribution of a single mode
to $\widetilde\chi$ equals
\[
V^v\,g\mathbf i\,\overline{u_{\kappa,v}}\,m_{\kappa,v},
\qquad g=R/|R| .
\]
Indeed $|\zeta-vR|^2=y|W-v|^2$ and
$2|R|/\sqrt J=y^{-d/2}$. Multiply every row on the right by the
same unit quaternion $(g\mathbf i)^{-1}$; this preserves their Gram
matrix. Conjugation by $g$ takes $\mathbf i$ to $g^2\mathbf i=e\mathbf i$.
Since $\mathbf i$ commutes with the Fourier amplitudes, the resulting
column is precisely \eqref{eq:current-wave}, with $V^v$ on the left.
The formula involves $Y$ itself, hence is smooth and independent of
the square root used in its derivation. Positivity follows from its
outer product and tangency from \eqref{eq:current-directions}. The
change back to physical coordinates acts by the same real orthogonal
matrix on each of the three coordinate slots, so it preserves the
column divergence and the row--column pairing. Finally $T=(t-s_0)/\sqrt2$
gives the stated expression for $V_T^v$.
\end{proof}
For a single $\kappa$ and real $W$, the exponential in $m_{\kappa,v}$
is a Gaussian window in transverse frequency $\kappa v$, centered at
$\kappa W$. Since $V_t^v=1$, the null component $A^0_{tt}$ is the
squared modulus of the scalar wave observed through this window:
up to normalization, its Husimi density \cite{Husimi1940}; see
\cite[Section~2.3]{KellerLasserOhsawa2016}. For the full wave,
\eqref{eq:current-wave} sums the amplitudes over $\kappa$ before
taking the square.
\subsection{The time component on a frequency cap}
At a fixed auxiliary point write $V_T^v=b_0+\mathbf k b_1$, so that
\[
b_0=\frac{1+\Rea(W)\cdot v/2}{\sqrt2},\qquad
b_1=\frac{\operatorname{Im}_{\mathbf k}(W)\cdot v}{2\sqrt2}.
\]
Here $\operatorname{Im}_{\mathbf k}(\alpha+\mathbf k\beta)=\beta$.
Let $\mathsf S_\ell$ be the ordinary scalar spatial multiplier which sends each
mode in \eqref{eq:current-modes} to $b_\ell m_{\kappa,v}u_{\kappa,v}$
at this auxiliary point. By \eqref{eq:current-wave},
\begin{equation}\label{eq:current-time-multipliers}
\chi_T=\iota_e(\overline{\mathsf S_0u_{\rm w}})
+\mathbf k\,\iota_e(\overline{\mathsf S_1u_{\rm w}}),\qquad
A^0_{TT}=|\mathsf S_0u_{\rm w}|^2+|\mathsf S_1u_{\rm w}|^2 .
\end{equation}
Indeed the two terms lie in the conjugates of the orthogonal planes
$\CC_{\mathbf i}$ and $\mathbf k\CC_{\mathbf i}$ by a square root $g$
of $e$ in $\CC_{\mathbf k}$. At $W=0$ the second multiplier vanishes:
$2A^0_{TT}$ is exactly the squared modulus of the wave with weight
$(\kappa/y)^{d/2}\exp(-\kappa y|v|^2/2)$ on each mode.
For the estimates take $u_{\rm w}=e^{-\mathrm iT|D|}f$ on the fixed
spatial torus $\RR^n/(100\mathbb Z^n)$. We choose the axis $b$ along
the center of a frequency cap, leaving the torus itself fixed. For
$0<h\le1$ suppose $\widehat f$ is supported in
$c_0/h\le|\xi|\le C_0/h$, $|v|\le\gamma$, with fixed $0<c_0<C_0$
and a small aperture $\gamma>0$ independent of $h$.
The identity $|\xi|=\kappa(1+|v|^2/2)/\sqrt2$ gives
$\kappa\asymp h^{-1}$. At $y\asymp h$ and $|W|,|v|\le1/10$,
the Gaussian and $b_0$ are bounded below, so $h^d\mathsf S_0$
has a symbol bounded above and below on the cap. For larger $y/h$
the cost of inverting it depends on the mismatch $|W-v|$.
There are $O(h^{-n})$ frequencies in the annulus, uniformly in the
frame; thus
$\sum_{\kappa,v}|c_{\kappa,v}|\lesssim h^{-n/2}\|f\|_2$ by
Cauchy--Schwarz and Parseval. The formulas in null coordinates may be
used on the covering space: the resulting current is periodic in the
original real spatial variables. We normalize nonzero cap data by
$\|f\|_2=1$ in the energy estimates that follow.
==== END Local-smoothing-for-the-Euclidean-wave-equation//current.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//cylinder.tex ====
\begin{figure}[!htbp]
\centering
\definecolor{cylinderteal}{RGB}{27,88,96}
\definecolor{cylinderorange}{RGB}{153,73,39}
\begin{tikzpicture}[
x=1cm,y=1cm,font=\small,inner sep=1.5pt,
wall/.style={draw=black!68,line width=.55pt},
guide/.style={draw=black!55,line width=.45pt,densely dashed},
panel/.style={font=\small\bfseries,anchor=west,inner sep=0pt},
ticklabel/.style={anchor=north,inner ysep=2pt}]
% The radial coordinates are schematic; the horizontal coordinates give
% the actual time windows for the two deterministic comparisons.
\node[panel] at (-.15,2.7) {(a) Outward comparison};
\fill[cylinderteal!15] (0,1.12) rectangle (4.2,1.58);
\draw[wall] (0,0) rectangle (4.2,2.05);
\draw[draw=cylinderteal,densely dashed,line width=.5pt]
(0,1.35) -- (4.2,1.35);
\node[fill=cylinderteal!15,inner xsep=3pt,text=cylinderteal]
at (2.1,1.35) {$3/4\le s\le5/4$};
\foreach \x/\t in {0/0,1.4/1,2.8/2,4.2/3}{
\draw[wall] (\x,0) -- +(0,-.09);
\node[ticklabel] at (\x,-.09) {$\t$};}
\foreach \y/\s in {0/b,1.35/1,2.05/3}{
\draw[wall] (0,\y) -- +(-.09,0);
\node[anchor=east] at (-.11,\y) {$\s$};}
\node[anchor=south] at (0,2.12) {$s$};
\node[anchor=west] at (4.32,0) {$T$};
\draw[draw=cylinderorange,line width=1.8pt] (1.4,0) -- (2.8,0);
\node[text=cylinderorange,anchor=south] at (2.1,.08) {observed wave};
\node at (2.1,-.90) {factor at $s=1$: \quad $\displaystyle\frac{\log 3}{\log(3/b)}$};
\begin{scope}[xshift=6.05cm]
\node[panel] at (-.15,2.7) {(b) Inward comparison};
\draw[wall] (0,0) rectangle (4.2,2.05);
\draw[guide] (0,.57) -- (2.1,.57) -- (2.1,0);
\foreach \x/\t in {0/{-1},2.1/0,4.2/1}{
\draw[wall] (\x,0) -- +(0,-.09);
\node[ticklabel] at (\x,-.09) {$\t$};}
\foreach \y/\s in {0/b,.57/{2b},2.05/1}{
\draw[wall] (0,\y) -- +(-.09,0);
\node[anchor=east] at (-.11,\y) {$\s$};}
\node[anchor=south] at (0,2.12) {$s$};
\node[anchor=west] at (4.32,0) {$T$};
\draw[draw=cylinderteal,line width=1.8pt] (0,2.05) -- (4.2,2.05);
\node[text=cylinderteal,anchor=north] at (2.1,1.97) {uniform lower bound};
\fill[cylinderorange] (2.1,.57) circle[radius=2.2pt];
\node[anchor=west,text=cylinderorange] at (2.23,.57) {$H(0,2b)$};
\node at (2.1,-.90) {factor at $s=2b$: \quad $\displaystyle\frac{\log 2}{\log(1/b)}$};
\end{scope}
\end{tikzpicture}
\caption{The two comparisons, with radii $s=|\tau|$ drawn schematically.
The boundary mass on $1\le T\le2$ bounds the mass in the shaded band
from below. The mean inequality on a fixed annulus supplies a lower bound
on the entire upper wall in~(b). The two displayed fractions are values
of the normalized radial harmonic functions. The comparison solutions
with potential dominate fixed multiples of these values, giving the
factor $[\log(3/b)]^{-2}$ at the dot. In the wave estimate, $b=2Kh$:
$H$ controls the wave norm on the marked segment and is controlled by
the initial data at the dot.}\label{fig:cylinder}
\end{figure}
==== END Local-smoothing-for-the-Euclidean-wave-equation//cylinder.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//dirichlet.tex ====
\section{The Dirichlet problem on a periodic fiber}
\label{sec:dirichlet}
The balance law in Section~\ref{sec:energy} uses a Dirichlet potential
smooth up to a boundary that moves with the time variables.
Section~\ref{sec:barrier} bounds this potential by comparing
determinants on the whole fiber and on small balls inside it. We prove
the required existence, comparison, and smooth dependence here;
the quantitative bounds in the wavelength $h$ come from the
comparison functions in Section~\ref{sec:barrier}. Alesker proved smooth
Dirichlet solvability on quaternionic balls \cite[Theorem~1.4]{Alesker2003MA};
Zhu proved it on bounded domains in quaternionic space in the presence of
a smooth subsolution \cite[Theorem~1.1]{Zhu2016}. Our fibers have periodic
real directions and only three real coordinates per variable. Adjoining a
fourth, periodic coordinate gives the local rotations needed to estimate
mixed derivatives at the boundary.
Let $\Lambda\subset\RR^m$ be a full lattice, $V\subset\RR^{2m}$ a
bounded domain with smooth boundary, and
$\Omega=(\RR^m/\Lambda)\times V$. The periodic variables occupy the
real slots and the coordinates of $V$ occupy the $\mathbf i,\mathbf j$
slots of the Hessian $H$ from Lemma~\ref{lem:hessian-calculus}.
\begin{theorem}[Periodic Dirichlet problem]\label{thm:dirichlet}
Let $m\ge1$. Suppose $r\in C^\infty(\overline V)$ is a defining
function, negative in $V$, with $dr\ne0$ on $\partial V$ and
$H(r)>0$ on $\overline\Omega$. Let $B_0,g\in C^\infty(\overline\Omega)$
be real and suppose
\[
H(B_0)\ge0,\qquad g>\det H(B_0)\quad\text{on }\overline\Omega.
\]
There is a unique smooth solution, up to the boundary, of
\begin{equation}\label{eq:dirichlet-problem}
\det H(B_0+u)=g,\qquad H(B_0+u)>0,\qquad
u|_{\partial\Omega}=0.
\end{equation}
There are constants $b\ge b'>0$ such that
\begin{equation}\label{eq:dirichlet-barriers}
br\le u\le b'r.
\end{equation}
For smooth finite-dimensional families of such data and domains, these
constants can be chosen locally uniformly. The solutions depend smoothly
on the parameters up to the boundary, after any smooth local
identification of the domains with a fixed domain.
\end{theorem}
\begin{proof}
Work on $\widetilde\Omega=\Omega\times(\RR/\mathbb Z)^m$, with the new
coordinates in the $\mathbf k$ slots. Extend the data constantly in these
coordinates. Until the last paragraph, $G,H$ are the full quaternionic
gradient and Hessian, which agree with the physical operators on the
extended data. Write $\mathcal H(D^2v)=H(v)$ and encode a real covector
$\xi\in\RR^{4m}$ as the quaternionic column $\widehat\xi$ in the slot
basis. Lemma~\ref{lem:hessian-calculus} gives
$\mathcal H(\xi\xi^{\mathsf t})=\widehat\xi\,\widehat\xi^*$ and
$\tr H(v)=\Delta v$.
\emph{Comparison and continuity.}
Join the known solution $u=r$ to the given density by
$g_t=(1-t)\det H(B_0+r)+tg$, $0\le t\le1$.
At a solution $\psi=B_0+u$ put $Q=H(\psi)$ and $S=\tr Q^{-1}$.
The derivative of the log determinant is
\[
Lw=Q^{-1}:H(w),\qquad
\sigma_L(\xi)=\widehat\xi^*Q^{-1}\widehat\xi .
\]
Thus $L$ is a uniformly elliptic real operator at each fixed smooth
solution with $Q>0$ on the closure. For two potentials $\psi_1,\psi_2$
with $Q_i=H(\psi_i)>0$, the corresponding difference formula is
\[
\log\det Q_1-\log\det Q_2
=\left(\int_0^1\big((1-\theta)Q_2+\theta Q_1\big)^{-1}\dd\theta\right)
:H(\psi_1-\psi_2).
\]
When the Hessians are positive on the closure, the averaged operator
is uniformly elliptic. If $\det Q_1\ge\det Q_2$ on the domain and
$\psi_1\le\psi_2$ on its boundary, the maximum principle therefore
gives $\psi_1\le\psi_2$ throughout. This also gives uniqueness.
The linear Dirichlet
operator $L:C^{k+2,\alpha}_0\to C^{k,\alpha}$ is an isomorphism
($k\ge0$, $0<\alpha<1$; the subscript denotes zero boundary trace).
Indeed the local Schauder estimates and Dirichlet theory
\cite[Theorems~6.14 and~6.19]{GilbargTrudinger} apply in a finite
cover of the flat quotient. In the continuity method from $\Delta$ to
$L$, $r$ is a common strict barrier since $H(r)\ge cI$; the maximum
principle therefore supplies the $C^0$ estimate. The starting Dirichlet
Laplacian is invertible on the compact manifold with boundary.
The implicit function theorem gives openness; the estimates below give
closedness. Their constants may depend on the fiber and its data, but
are uniform in $t$.
Both endpoints of the path have determinant strictly above $\det H(B_0)$
at every point. The gap is uniform by compactness. Hence we may choose
$0<b'<1<b$ so that
\[
\det H(B_0+b'r)<g_t<\det H(B_0+br)
\]
everywhere, using $H(r)\ge cI$ for the second inequality. Comparison
gives \eqref{eq:dirichlet-barriers}. Since $u$ vanishes on the boundary,
$u=\alpha r$ in a collar with smooth quotient $\alpha$ at each smooth
solution; division in the barriers gives $b'\le\alpha\le b$. In
particular $\nabla u=\alpha\nabla r$ on the boundary.
We also have $Lr\ge cS$ and $S\ge m g_t^{-1/m}\ge c_0>0$.
For any constant real unit vector $e$, differentiation and concavity give
\begin{equation}\label{eq:dirichlet-directional}
L\psi_e=(\log g_t)_e,\qquad L\psi_{ee}\ge(\log g_t)_{ee}.
\end{equation}
The maximum principle applied to $\pm\psi_e+Cr$ bounds the full
gradient. Applied to $\psi_{ee}+Cr$, it reduces an upper bound for the
real Hessian to the boundary.
\emph{Second derivatives at the boundary.}
Center a flat chart at a boundary point, let $\nu$ be the outward unit
normal, and identify vectors with their quaternionic columns in this
chart. For real tangent vectors $v,w$,
$D^2u(v,w)=\alpha D^2r(v,w)$ at the center. For the mixed derivatives
we adapt the tangential differentiation argument of Caffarelli, Nirenberg
and Spruck for the real Monge--Ampère equation
\cite[Section~2, (2.11)--(2.14)]{CNS1984}. Fix a real tangent unit
vector $e$. We construct a local quaternionic rigid motion
$X=(e+Ax)\cdot\nabla$ for which $Xr$ vanishes to second order along
the boundary at the center. Let
\[
v=\frac{P_{\rm tan}D^2r\,e}{|\nabla r|},\qquad
A=v\nu^*-\nu v^*-\nu(\nu^*v)\nu^* ,
\]
where $P_{\rm tan}$ is the real orthogonal tangent projection. Since
$\Rea(\nu^*v)=0$, we have $A^*=-A$ and $A\nu=v$. Now $Xr(0)=0$ and
for every tangent $w$,
\[
\partial_w(Xr)(0)=D^2r(w,e)-|\nabla r|\,w\cdot A\nu=0.
\]
Thus $|Xu|=|\alpha Xr|\le C|x|^2$ on the boundary near the center.
The flow of $X$ acts by quaternionic unitary rotations and translations.
Differentiating the covariance of $H$ under this flow gives
$L(X\psi)=X\log g_t$: the commutator from conjugating $Q$ has zero
pairing with $Q^{-1}$. This identity uses only the flat chart.
Subtracting the known $XB_0$ gives $|L(Xu)|\le CS$.
In a chart ball of fixed small radius, set $w=C_1|x|^2-C_2r$.
First choose $C_1$ so that $w\ge|Xu|$ on the true boundary by the
quadratic bound, and on the spherical boundary by the gradient bound.
Then choose $C_2$ so that $L(w\pm Xu)\le0$, using $H(|x|^2)=8I$
and $Lr\ge cS$. The maximum principle gives $|Xu|\le w$ in the
intersection. Both functions vanish at the center, so the inward
derivative bounds $|\partial_\nu Xu|$. This is $|D^2u(\nu,e)|$:
the additional term $\nabla u\cdot A\nu$ vanishes since $\nabla u$
is normal. The constants are uniform over the compact boundary.
For the remaining derivative, the product rule at the boundary reads
\[
H(u)=\alpha H(r)+G(\alpha)G(r)^*+G(r)G(\alpha)^*,\qquad
G(r)=|\nabla r|\nu .
\]
Compression to the \emph{quaternionic} orthogonal complement of $\nu$
annihilates both cross terms. On that complement $Q$ is therefore bounded
below by $b'H(r)|_{\nu^\perp}\ge cI$. The rank-one rule shows that in a
quaternionic orthonormal basis beginning with $\nu$,
\[
Q=\begin{pmatrix}\psi_{\nu\nu}+c_1&z^*\\z&D\end{pmatrix}.
\]
Here $c_1,z,D$ depend only on the real second derivatives already
bounded. For $m>1$ the Schur formula in
Lemma~\ref{lem:quaternionic-matrices} gives
$\psi_{\nu\nu}=g_t/\det D+z^*D^{-1}z-c_1$, which is bounded since
$D\ge cI$. For $m=1$ the scalar identity $Q=g_t$ suffices.
Equation~\eqref{eq:dirichlet-directional} now gives $D^2\psi\le CI$
throughout the domain. Since $\Delta\psi=\tr Q>0$, every real Hessian
eigenvalue is at least $-(4m-1)C$. This is the full $C^2$ bound.
\emph{Real elliptic regularity.}
The $C^2$ bound and $\det Q=g_t\ge c_0>0$ put all eigenvalues of $Q$
in a fixed interval $[c,C]\subset(0,\infty)$. Thus all real Hessians
$D^2\psi$ lie in the compact convex set
$\mathcal K=\{M:|M|\le C_1,\ cI\le\mathcal H(M)\le CI\}$.
On $\mathcal H(M)>0$ the smooth function $F(M)=\log\det\mathcal H(M)$
is concave. For $M\in\mathcal K$ and every real covector $\xi$,
\[
DF(M)[\xi\xi^{\mathsf t}]
=\widehat\xi^*\mathcal H(M)^{-1}\widehat\xi
\in[C^{-1}|\xi|^2,c^{-1}|\xi|^2].
\]
These are real uniform ellipticity and concavity on the range of the
solution Hessians, the hypotheses (i)$'$, (ii)$'$ of the interior and
boundary estimates \cite[Theorems~17.14 and~$17.26'$]{GilbargTrudinger}. To put
$F$ on the full real matrix space as in that theorem, multiply it by a
smooth cutoff supported in $\{\mathcal H(M)>0\}$ and equal to one on
a neighborhood of $\mathcal K$. The equation
$F(D^2\psi)-\log g_t=0$ then has uniformly smooth coefficients and
boundary data $B_0$. The cited local boundary estimates (see also
\citet{Krylov1983}) and the interior Evans--Krylov estimate
\cite{Evans1982,Krylov1982} apply in a finite
cover of the flat quotient, giving a uniform $C^{2,\alpha}$ bound for
some $\alpha>0$.
For a limit of solvable parameters, a subsequence converges in
$C^{2,\beta}$, $0<\beta<\alpha$. Positivity and the equation pass to
the limit. Schauder bootstrapping gives smoothness up to the boundary:
in flattened coordinates differentiate tangentially with the given
Dirichlet data, and recover the pure normal derivative from the equation
by uniform ellipticity. Thus the path is closed and reaches $g$.
Uniqueness on $\widetilde\Omega$ makes the solution invariant under
translations in every added periodic coordinate. It descends to
$\Omega$; any physical solution lifts, so uniqueness descends as well.
The same lift gives comparison on subdomains, in particular on the
balls in the three-slot fibers used in Section~\ref{sec:barrier}.
For moving domains pull the lifted equation back to a fixed domain.
Its coefficients and boundary data depend smoothly on the parameters,
and its linearization is conjugate to the Dirichlet isomorphism above.
The implicit function theorem in each $C^{k+2,\alpha}_0$ gives smooth
dependence up to the boundary. The strict inequalities used to choose
$b,b'$ persist locally in the parameters.
\end{proof}
==== END Local-smoothing-for-the-Euclidean-wave-equation//dirichlet.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//energy.tex ====
\section{The determinant energy and its boundary flux}
\label{sec:energy}
The time component of the wave current is a sum of squares. After
regularization it gives the density for a spatial Dirichlet potential:
a lower bound on the density throughout a ball forces a definite
depth at its center. To compare that depth at different times, we
need a scalar energy on the three real coordinates
$z=(T,\tau)\in\RR\times\CC_{\mathbf k}$ whose Laplacian is bounded
above by the wave flux through the side of the spatial fiber.
Keep the annular wave on the torus from Section~\ref{sec:current},
normalized by $\|f\|_2=1$. Put $w_0=1/10$ and assume its sector
$|v|\le\gamma$ has $\gamma<w_0/4$.
Write $s=|\tau|$ and $a=Kh$, where $K\ge1$ is fixed and $h$ is small
enough that $4a\le1$. All the constructions of this section take place
on $a<s<4$.
\subsection{The geometry of the fibers}
At fixed $z$, let $X$ run over the spatial torus and let the imaginary
spatial variables $(P,Z)\in\CC_{\mathbf k}^d\times\CC_{\mathbf k}$ vary
in the domain
\begin{equation}\label{eq:fiber-domain}
\Omega_z:\quad r=\frac{|W|^2}{w_0^2}+100|Q|^2-1<0,
\qquad Q=\Phi/a.
\end{equation}
The shape of the fiber reflects two properties of the current from
Proposition~\ref{prop:current}: it is tangent to the level sets of
$\Phi$, and its Gaussian is centered at $W=v$ for a mode of slope $v$.
Only the $W$ part of the wall normal can therefore carry wave flux.
In $(W,Q)$ the fibers are all the same ellipsoid. The definitions
$Y=(\tau+Z)/\sqrt2$, $W=P/Y$ and
$\Phi=(Z-\tau)/\sqrt2+P\cdot P/(2Y)$ give the inverse map
\begin{equation}\label{eq:fiber-inverse}
Y=\frac{\sqrt2\,\tau+aQ}{1+W\cdot W/2},\qquad P=WY,\qquad
Z=\sqrt2\,Y-\tau .
\end{equation}
The denominator is bounded away from zero on $|W|\le w_0$, and
$|aQ|\le s/10$, so $y=|Y|\asymp s$. Both the chart and its inverse
are holomorphic on neighborhoods of these closed fibers and depend
smoothly on $\tau$. On the total domain
$\Omega=\{(z,X,P,Z):a<s<4,\ (X,P,Z)\in\Omega_z\}$ we use $H,G$ for
the full Hessian and gradient from Lemma~\ref{lem:hessian-calculus}.
Their spatial parts at fixed $z$ are $H_{\rm fib},G_{\rm fib}$. Fiber
integrals use ordinary volume $\dd\zeta=\dd X\,\dd P\,\dd Z$ in the
original coordinates.
\begin{lemma}[Fiber geometry]\label{lem:fiber-geometry}
The spatial Hessian $H_{\rm fib}(r)$ is positive definite. The full
$H(r)$ is positive semidefinite with a one-dimensional quaternionic
kernel contained in both $G(r)^\perp$ and $\ker H(|\Phi|^2)$. On the
closed fibers, uniformly for $a<s<4$,
\begin{align}
H_{\rm fib}(r)&\ge c s^{-2}I,\qquad
G_{\rm fib}(r)G_{\rm fib}(r)^*\le H_{\rm fib}(r),\label{eq:fiber-curvature}\\
\dd P\,\dd Z&\asymp a^2s^{2d}\dd W\,\dd Q,\qquad
\det H_{\rm fib}(r)\asymp a^{-2}s^{-2d}.\label{eq:fiber-jacobian}
\end{align}
The spatial $H_{\rm fib}(|\Phi|^2)$ has rank one with nonzero eigenvalue
comparable to one. Also $|G(|W|^2)|\lesssim s^{-1}$.
\end{lemma}
\begin{proof}
Set $F=(W/w_0,10Q)$ and let $J$ be its complex Jacobian in
$(\tau,P,Z)$, with spatial submatrix $J_{\rm fib}$. Since $r=|F|^2-1$,
Lemma~\ref{lem:hessian-calculus} gives
\[
H(r)=4J^*J,\qquad G(r)=2J^*F\mathbf i .
\]
At fixed $\tau$ the spatial Jacobian $J_{\rm fib}$ is invertible.
In the full space we can vary $\tau$ while keeping $(W,Q)$ fixed;
the resulting direction spans $\ker J$ over $\CC_{\mathbf k}$ and
its quaternionic span is the null line of $H(r)$. This line is
orthogonal to $G(r)$, which belongs to the image of $J^*$, and lies in
$\ker H(|\Phi|^2)$ because the last row of $J$ is proportional to
$\partial\Phi$. By
\eqref{eq:fiber-inverse}, derivatives of $(P,Z)$ in $W$ are $O(s)$
and those in $Q$ are $O(a)$. Thus $\|J_{\rm fib}^{-1}\|\lesssim s$;
together with $|F|\le1$ this proves \eqref{eq:fiber-curvature}.
The complex Jacobian determinant of \eqref{eq:fiber-inverse} in $(W,Q)$
is $\sqrt2\,aY^d/(1+W\cdot W/2)$: taking the exterior product of the
differentials of $P=WY$ with $\dd Y$ leaves just $Y^d$ times the product
of the differentials of $W$ with $\dd Y$. Squaring the absolute value
gives the volume formula. The determinant of $4J_{\rm fib}^*J_{\rm fib}$
is $4^n|\det_{\CC}J_{\rm fib}|^2$, giving the second bound in
\eqref{eq:fiber-jacobian}. Finally the spatial derivatives of $\Phi$ are
bounded and $\partial_Z\Phi=(1-W\cdot W/2)/\sqrt2$ is bounded away from
zero. All first derivatives of $W=P/Y$ are $O(s^{-1})$.
\end{proof}
\subsection{The Dirichlet potential}
The time density can vanish. We first smooth the wave current in $X$
and cut it off in the passive coordinate $Q$. To make its time
density strictly positive, we then add a conserved current whose image
is the null line of $H(r)$; it has zero flux through the wall.
For $\ell>0$ let $J_\ell$ be the periodization on the spatial torus of
the probability kernel
$c_n\ell^{-n}(1+|X/\ell|^2)^{-n-2}$. Choose a smooth $0\le g\le1$ on
$\CC_{\mathbf k}$, equal to one on $|Q|\le1/100$ and zero on
$|Q|\ge1/20$. Define
\begin{equation}\label{eq:regularized-current}
A_{\rm w}=g(Q)(J_h*_X A^0),\qquad
A=A_{\rm w}+h^{50n}C(H(r)),\qquad \mu=A_{TT}.
\end{equation}
Convolution takes place only in the real spatial variable. Since
$\partial\Phi$ is independent of $X$, the averaged current still
annihilates the image of $H(|\Phi|^2)$ and hence $G(g(Q))$. The product
rule for divergence therefore preserves conservation under the cutoff.
By Lemmas~\ref{lem:quaternionic-matrices} and~\ref{lem:cofactor-divergence},
the regularizing cofactor is positive semidefinite and divergence free.
Its image is $\ker H(r)$, perpendicular to $G(r)$, and its $TT$ entry
is $\det H_{\rm fib}(r)>0$. Thus
\begin{equation}\label{eq:current-background-orthogonality}
A\ge0,\qquad \mu>0,\qquad A:H(|\Phi|^2)=0 .
\end{equation}
Put $B_0=\lambda|\Phi|^2$, $B=H(B_0)$, $B_{\rm fib}=H_{\rm fib}(B_0)$,
with $\lambda=h^{-30n}$. By \eqref{eq:current-background-orthogonality},
the full background satisfies $A:B=0$. Its spatial block $B_{\rm fib}$
has one nonzero eigenvalue $\beta\asymp\lambda$. The factor
$h^{50n}$ keeps the added density below the wave estimates
of Section~\ref{sec:barrier}; we take $\lambda$ large enough that
\begin{equation}\label{eq:energy-density-ceiling}
\mu^{1/d}\le\lambda
\end{equation}
for all sufficiently small $h$. To see this, each Gaussian amplitude
factor in $\chi$ has size at most $(\kappa/y)^{d/2}\lesssim h^{-d}$.
The columns $V^v$ are bounded and $\sum|c_{\kappa,v}|\lesssim h^{-n/2}$
by the cap estimate in Section~\ref{sec:current}, so
$\|A^0\|\lesssim h^{-n-2d}$. The regularizer adds only
$O(h^{50n}a^{-2}s^{-2d})$ to $\mu$ by \eqref{eq:fiber-jacobian}.
On each spatial fiber there is a unique smooth real $u$ satisfying
\begin{equation}\label{eq:fiber-equation}
D_u=H_{\rm fib}(B_0+u)>0,\qquad
\det D_u=\lambda\mu,\qquad u|_{\partial\Omega_z}=0 .
\end{equation}
Indeed, the spatial defining Hessian is positive definite, and
$B_{\rm fib}\ge0$ has zero determinant since $n\ge2$. Together with
$\mu>0$ these are the hypotheses of Theorem~\ref{thm:dirichlet}.
The chart \eqref{eq:fiber-inverse} and the parameter assertion of that
theorem give smooth dependence on $z$ up to the lateral boundary;
its barriers \eqref{eq:dirichlet-barriers} give $u\le0$.
To see how the equation measures depth, add $\alpha I$ to the spatial
background. Then
$\det(B_{\rm fib}+\alpha I)=(\beta+\alpha)\alpha^d$; thus for
$0<\alpha\le\lambda$ a density $\lambda\mu$ corresponds to an
isotropic comparison quadratic of curvature $\alpha\asymp\mu^{1/d}$.
Section~\ref{sec:barrier} will use such quadratics on balls.
\subsection{Cancelling the full determinant}
The quantity that measures the depth of the potential against its
density is $X_0(z)=\int_{\Omega_z}(-u)\mu\dd\zeta$. Since the real
columns of $A$ have zero divergence, Green's formula for $\Delta_z X_0$
has interior term $-\int_{\Omega_z}A:H(u)\dd\zeta$. In a related
estimate for real positive tensors, Serre uses the determinant of a
Schur complement to isolate a diagonal entry of a second tensor
\cite[Proposition~2.2]{Serre2023Mixed}. Here only the spatial block of
$H(B_0+u)$ is known to be positive. Separate off its scalar Schur
complement:
\begin{equation}\label{eq:energy-schur}
H(B_0+u)=
\begin{pmatrix}\ell&c^*\\ c&D_u\end{pmatrix}
=\sigma_0 e_Te_T^*+\Sigma,\qquad
\sigma_0=\ell-c^*D_u^{-1}c,\qquad
\Sigma=\begin{pmatrix}c^*D_u^{-1}c&c^*\\ c&D_u\end{pmatrix}\ge0 .
\end{equation}
Here $e_T$ is the time coordinate vector. Completing the square gives
$\Sigma\ge0$. In the contraction with $A$ the background term $A:B$
vanishes by \eqref{eq:current-background-orthogonality}; in the
determinant, $\det D_u=\lambda\mu$ gives
$\det H(B_0+u)=\lambda\mu\sigma_0$. Thus
\begin{equation}\label{eq:schur-energy-identity}
A:H(u)=A:\Sigma+\mu\sigma_0
=A:\Sigma+\lambda^{-1}\det H(B_0+u).
\end{equation}
The full determinant has no prescribed sign. We look for a second
fiber integral whose Laplacian cancels this term. Write
$D_v=H_{\rm fib}(B_0+v)$. Along the segment $B_0+\theta u$, the
cofactor of the full Hessian has $TT$ entry $\det D_{\theta u}$, and
its contraction with $H(u)$ differentiates the full determinant in
$\theta$. This suggests integrating the spatial determinant along the
segment. For a smooth real $v$ on $\Omega_z$ with $D_v>0$ and zero
boundary values, set
\begin{equation}\label{eq:energy-functional}
E_z(v)=\frac1\lambda\int_{\Omega_z}v
\int_0^1\det D_{\theta v}\dd\theta\,\dd\zeta,\qquad
\mathcal J_z(v)=E_z(v)-\int_{\Omega_z}v\mu\dd\zeta .
\end{equation}
The segment integral is the quaternionic counterpart of the
Aubin--Mabuchi energy: Mabuchi's functional $L$ integrates the variation
of a Kähler potential against its complex Monge--Ampère measure
\cite[(2.2.1)]{Mabuchi1986}. Here the determinant is a polynomial,
so the integrand is smooth up to the singular endpoint $D_0=B_{\rm fib}$.
Define $\mathcal F(z)=\mathcal J_z(u)=E_z(u)+X_0(z)$.
\begin{proposition}[Energy balance]\label{prop:energy-flux}
The energy $\mathcal F$ is smooth on $a<|\tau|<4$. Let $\rho$ denote
the boundary value of $u/r$, which extends smoothly from the interior.
In the three variables $z=(T,\tau)$ one has
\begin{equation}\label{eq:energy-flux}
\Delta_z\mathcal F(z)+\int_{\Omega_z} A:\Sigma\,\dd\zeta
=\int \rho\,G(r)^*A_{\rm w}G(r)\,\delta(r)\dd\zeta .
\end{equation}
The normal seen by the current is that of the moving wall in the
total space, so $G(r)$ is the \emph{full} gradient. The integration is
on one fiber: $\delta(r)\dd\zeta=\dd S_{\rm fib}/|\nabla_{\rm fib}r|$,
where the spatial gradient is nonzero by the ellipsoid chart. Since
$A:\Sigma\ge0$, the boundary integral bounds $\Delta_z\mathcal F$
from above.
\end{proposition}
\begin{proof}
Denote the boundary integral in \eqref{eq:energy-flux} by
$\mathcal B(z)$. The two identities whose sum gives the balance are
\begin{align*}
\Delta_zX_0
&=\mathcal B(z)-\int_{\Omega_z}A:\Sigma\dd\zeta
-\lambda^{-1}\int_{\Omega_z}\det H(B_0+u)\dd\zeta,\\
\Delta_z[E_z(u)]
&=\lambda^{-1}\int_{\Omega_z}\det H(B_0+u)\dd\zeta .
\end{align*}
For a fixed complex product domain, \citet[Theorem~3.1]{AhagEtAl2009}
identified the Laplacian of the slice energy, up to normalization,
with the integral of the full Monge--Ampère measure over the fiber.
Here the fiber moves, so the second identity requires a boundary calculation.
Both follow by testing with a smooth $\eta=\eta(z)$ compactly supported in the
parameter domain. The only entry of $H(\eta)$ is
$H_{TT}(\eta)=\Delta_z\eta$. For any Hermitian matrix field $M$ with
divergence-free real columns, Green's formula \eqref{eq:calculus-green}
on the \emph{total} domain gives
\begin{equation}\label{eq:green-moving-fiber}
\int_\Omega u M_{TT}\Delta_z\eta
=\int_\Omega \eta\,M:H(u)
-\int_{\partial_{\rm lat}\Omega}
\eta\,(\nu^*M\nu)\,\partial_{\mathbf n}u\dd S .
\end{equation}
Here $\partial_{\rm lat}\Omega$ is the side $r=0$, and volume on
$\Omega$ is $\dd z\,\dd\zeta$. The term containing
$u\nabla\eta$ vanishes because $u=0$ on the lateral boundary; there
are no end terms because of the support of $\eta$. The unit outward
normal $\mathbf n=\nabla r/|\nabla r|$ on the total domain has quaternionic
column $\nu=G(r)/|\nabla r|$. Since $r$ is a smooth defining function,
$u=r\rho$ near the boundary with $\rho$ smooth and $\rho\ge0$ on $r=0$.
In particular $\partial_{\mathbf n}u=\rho|\nabla r|$.
For $X_0$, take $M=A$ in \eqref{eq:green-moving-fiber} and change
sign since $X_0$ integrates $(-u)\mu$. The interior term follows
from \eqref{eq:schur-energy-identity}. On the wall the regularizer
$C(H(r))$ has image perpendicular to $\nu$, so only $A_{\rm w}$
contributes. Substituting $\partial_{\mathbf n}u=\rho|\nabla r|$ and
$\dd S=|\nabla r|\delta(r)\dd z\,\dd\zeta$ gives $\mathcal B(z)$.
For $E_z(u)$, take $M=C(H(B_0+\theta u))$, whose columns are conserved
by Lemma~\ref{lem:cofactor-divergence}. Here the entire normal
coefficient vanishes. Indeed \eqref{eq:boundary-restriction} gives,
on $r=0$,
\[
H(B_0+\theta u)|_{\nu^\perp}
=(B+\theta\rho H(r))|_{\nu^\perp}.
\]
The null line of $H(r)$ belongs to both $\nu^\perp$ and $\ker B$ by
Lemma~\ref{lem:fiber-geometry}; the restricted form is therefore
singular. The cofactor restriction formula in
Lemma~\ref{lem:quaternionic-matrices} yields $\nu^*M\nu=0$, independently
of the sign of the full Hessian. The same formula in the $e_T$ direction
gives $M_{TT}=\det D_{\theta u}$. Now integrate
\eqref{eq:green-moving-fiber} in $\theta$ and divide by $\lambda$.
On the left we obtain $\int E_z(u)\Delta_z\eta\,\dd z$; on the right,
integration of $M:H(u)=\partial_\theta\det H(B_0+\theta u)$ gives the
full determinant at $\theta=1$, since $\det B=0$. This proves the
second identity and hence the balance in distributions. The chart
\eqref{eq:fiber-inverse} makes the volume and surface integrals smooth
in $z$, so it also holds pointwise.
\end{proof}
\subsection{Size of the energy on a fiber}
Fix $z$. The segment integral in \eqref{eq:energy-functional} is a
primitive of the spatial determinant: for a smooth variation $w$ with
zero boundary values, one has
\begin{align*}
\mathrm d\mathcal J_z(v)[w]
&=\int_{\Omega_z}w(\lambda^{-1}\det D_v-\mu)\dd\zeta,\\
\mathrm d^2\mathcal J_z(v)[w,w]
&=-\lambda^{-1}\int_{\Omega_z}
G_{\rm fib}(w)^*C(D_v)G_{\rm fib}(w)\dd\zeta .
\end{align*}
For the first identity, differentiating under the integral in
\eqref{eq:energy-functional} produces a term
$\theta v C(D_{\theta v}):H_{\rm fib}(w)$. By cofactor divergence one
may interchange $v$ and $w$ in its fiber integral; both vanish on the
boundary. The integrand in $\theta$ is then
$w\,\partial_\theta(\theta\det D_{\theta v})$. Differentiating once
more and integrating by parts gives the second formula. It is strictly
negative for every nonzero $w$, since $C(D_v)>0$. Thus the Dirichlet
solution is the unique maximizer on this convex class. This is the
smooth determinant variational principle with a rank-one background.
For zero background, Wan proved the first variation and the corresponding
variational characterization in quaternionic finite energy classes
\cite[Section~4]{Wan2020}.
Since $u\le0$ and $D_{\theta u}=(1-\theta)B_{\rm fib}+\theta D_u\ge0$,
the correction $E_z(u)$ is nonpositive. At a solution the linear
term in $\mathcal F$ uses the determinant at $B_0+u$. Thus
$\lambda\mathcal F$ has the form of Aubin's $J$-functional
\cite[p.~145]{Aubin1984}, with $B_0+u$ as reference potential and $B_0$
as argument in the relative formula \cite[Section~2.1]{BBGZ2013}.
The rank-one background lets us calculate how much of $X_0$ remains
in $\mathcal F=X_0+E_z(u)$.
\begin{lemma}[Energy comparison]\label{lem:energy-comparison}
On every spatial fiber,
\begin{equation}\label{eq:energy-exact}
\mathcal F=\frac{n-1}{n+1}X_0+
\frac{1}{\lambda n(n+1)}
\int_{\Omega_z}G_{\rm fib}(u)^*C(D_u)G_{\rm fib}(u)\dd\zeta .
\end{equation}
In particular,
$\frac{n-1}{n+1}X_0\le\mathcal F\le\frac{n}{n+1}X_0$.
\end{lemma}
\begin{proof}
Because $B_{\rm fib}$ has rank one, the determinant along the segment
has just two terms:
\[
\det D_{\theta u}=\theta^n\det D_u+
(1-\theta)\theta^{n-1} C(D_u):B_{\rm fib}.
\]
Write $I=\int_{\Omega_z}(-u) C(D_u):B_{\rm fib}\dd\zeta\ge0$.
Integrating in $\theta$ gives
$\mathcal F=nX_0/(n+1)-I/[\lambda n(n+1)]$, in particular the upper
bound. Also $C(D_u):D_u=n\det D_u$, so
\[
n\lambda X_0-I
=\int_{\Omega_z}(-u)C(D_u):H_{\rm fib}(u)\dd\zeta
=\int_{\Omega_z}G_{\rm fib}(u)^*C(D_u)G_{\rm fib}(u)\dd\zeta .
\]
This last integration by parts uses $u=0$ at the boundary. Substitution
proves \eqref{eq:energy-exact} and its lower bound.
\end{proof}
==== END Local-smoothing-for-the-Euclidean-wave-equation//energy.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//fiber.tex ====
\begin{figure}[!htbp]
\centering
\definecolor{fiberteal}{RGB}{27,88,96}
\definecolor{fiberclay}{RGB}{153,73,39}
\begin{tikzpicture}[
font=\small,>=Latex,
axis/.style={->,draw=black!72,line width=.5pt},
guide/.style={draw=black!53,densely dotted,line width=.5pt},
tick/.style={draw=black!70,line width=.5pt},
inverse/.style={draw=fiberteal,line width=.8pt,dash pattern=on 3pt off 2pt},
center patch/.style={draw=fiberteal,line width=.8pt},
heading/.style={font=\small\bfseries,anchor=west,inner sep=0pt},
every node/.style={inner sep=2pt}]
% The horizontal unit is w_0, the vertical one 1/10.
% Only the ratio gamma/w_0 is schematic.
\begin{scope}[x=4.15cm,y=3.0cm]
\node[heading] at (-.11,1.48) {(a) The fiber wall};
\fill[black!3] (0,0) -- (1,0) arc[start angle=0,end angle=90,radius=1] -- cycle;
\fill[fiberteal!7] (0,0) rectangle (.22,.1);
\fill[fiberteal!28] (0,0) rectangle (.11,.025);
\draw[inverse] (0,.1) -- (.22,.1) -- (.22,0);
\draw[center patch] (0,.025) -- (.11,.025) -- (.11,0);
\draw[guide] (0,.5) -- ({sqrt(3)/2},.5);
\draw[black!78,line width=.65pt] (1,0) arc[start angle=0,end angle=90,radius=1];
\draw[fiberclay,line width=2.05pt,line cap=round]
(1,0) arc[start angle=0,end angle=30,radius=1];
\draw[axis] (-.022,0) -- (1.16,0) node[right] {$|W|$};
\draw[axis] (0,-.034) -- (0,1.15) node[above right] {$|Q|=|\Phi|/a$};
\node[below left] at (0,0) {$0$};
\draw[tick] (1,0) -- ++(0,-.025) node[below] {$w_0$};
\draw[tick] (0,.5) -- ++(-.022,0) node[left] {$1/20$};
\draw[tick] (0,1) -- ++(-.022,0) node[left] {$1/10$};
\node at (.37,.71) {$r<0$};
\node[anchor=west] at (.74,.76) {$r=0$};
\node[fiberclay,anchor=west] (walltext) at (.37,.24) {flux wall};
\draw[fiberclay,line width=.55pt] (walltext.east) -- (.80,.24) -- (.963,.257);
\draw[fiberteal,line width=.5pt] (.18,.105) -- (.115,.22);
\node[anchor=south,text=fiberteal,inner sep=1pt] at (.105,.225) {(b)};
\end{scope}
% Magnification of the lower left corner; the units are now 2 gamma
% and 1/100. The axes in either panel are moduli, not wall pieces.
\begin{scope}[xshift=6.85cm,x=3.65cm,y=2.8cm]
\node[heading] at (-.14,1.586) {(b) The recovery patch};
\fill[fiberteal!7] (0,0) rectangle (1,1);
\fill[fiberteal!28] (0,0) rectangle (.5,.25);
\draw[inverse] (0,1) -- (1,1) -- (1,0);
\draw[center patch] (0,.25) -- (.5,.25) -- (.5,0);
\draw[axis] (-.025,0) -- (1.17,0) node[right] {$|W|$};
\draw[axis] (0,-.036) -- (0,1.235) node[above right] {$|Q|$};
\node[below left] at (0,0) {$0$};
\draw[tick] (.5,0) -- ++(0,-.026) node[below] {$\gamma$};
\draw[tick] (1,0) -- ++(0,-.026) node[below] {$2\gamma$};
\draw[tick] (0,.25) -- ++(-.025,0) node[left] {$1/400$};
\draw[tick] (0,1) -- ++(-.025,0) node[left] {$1/100$};
\node[align=center] at (.50,.66) {Gaussian inversion\\[-1pt]$g(Q)=1$};
\node at (.25,.125) {centers};
\end{scope}
\end{tikzpicture}
\caption{The fixed fiber in modulus coordinates. The wave flux through
$r=0$ is supported on the heavy arc, where $|Q|\le1/20$ and hence
$|W|\ge\sqrt{3}\,w_0/2$. This separation gives Gaussian decay
$e^{-cs/h}$ even for $|v|\le w_0/3$. In the enlargement (b), once
$K\gamma$ is large, the radius-$h$ balls about the marked centers
project into the dashed patch. There $|W-v|\le4\gamma$ on the enlarged cap
$|v|\le2\gamma$, so Gaussian inversion costs at most $e^{C\gamma^2s/h}$.
The aperture $\gamma$ is fixed independently of $h$; its size relative
to $w_0$ is schematic.}
\label{fig:fiber}
\end{figure}
==== END Local-smoothing-for-the-Euclidean-wave-equation//fiber.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//global-extension.tex ====
\subsection{Global extension from the sphere and the truncated cone}
\label{sec:strict-extension}
Lemma~\ref{lem:kakeya-local-extension} gives global estimates at strictly
larger output exponents. For this step any fixed power of the logarithm
suffices; the tube argument in the next subsection will retain that power.
Write \(E\) for extension from \(S^{n-1}\).
For the truncated cone
\(\Sigma=\{(\rho\omega,\rho):1\le\rho\le2,\ \omega\in S^{n-1}\}\),
let \(E_\Sigma G(x,t)=\int_\Sigma e^{\mathrm i(x\cdot\xi+t\tau)}
G(\xi,\tau)\,\dd\Sigma(\xi,\tau)\), with induced surface measure.
\begin{corollary}[Global Fourier extension]\label{cor:spherical-restriction}\label{cor:cone-extension}
Let \(1<a\le\infty\) and \(b<\infty\). If
\[
b>\pc,\qquad b>\frac{n+1}{n-1}a',
\]
then
\begin{align}
\|Eg\|_{L^b(\RR^n)}
&\lesssim_{n,a,b} \|g\|_{L^a(S^{n-1})},\label{eq:spherical-extension-mixed}\\
\|E_\Sigma G\|_{L^b(\RR^{n+1})}
&\lesssim_{n,a,b} \|G\|_{L^a(\Sigma)}.\label{eq:cone-extension}
\end{align}
These are strong estimates, including when \(a=\infty\).
For \(b=\infty\) both estimates hold for every \(1\le a\le\infty\).
\end{corollary}
The two strict inequalities give the interior of the conjectured range
for these compact surfaces \cite[Conjecture~1.1(i)]{Nicola2009}.
The three-dimensional sphere estimate in
\cite[\S\S9.1--9.2]{OpenAI2026SphereRestriction} gives this range for
\(S^2\) and, by randomizing caps, an \(L^3\) Kakeya maximal bound with
loss \(\delta^{-\varepsilon}\).
\begin{proof}
The local spherical estimate in Lemma~\ref{lem:kakeya-local-extension},
with \(A=2/\pc\) by Theorem~\ref{thm:main},
costs \(O_\varepsilon(R^\varepsilon)\) for every \(\varepsilon>0\).
By duality it bounds restriction of functions supported in \(B(0,R)\)
from \(L^{\pc'}(\RR^n)\) to \(L^{\pc'}(S^{n-1})\) with that loss.
Since \(1<\pc'<2\), Tao's epsilon-removal theorem
\cite[Theorem~1.2]{Tao1999} gives global diagonal restriction at
every \(1<u<\pc'\). Dualizing gives \(E:L^s\to L^s\) for every
\(\pc<s<\infty\).
If \(a\ge b\), take \(s=b\) and use the finite measure of the sphere.
If \(a<b\), put \(s=1+b/a'\); then \(\pc<s<b\), since
\(\pc=1+(n+1)/(n-1)\). Interpolating \(E:L^1\to L^\infty\) with
\(E:L^s\to L^s\) at \(\theta=s/b\) gives \eqref{eq:spherical-extension-mixed}:
\(\theta/s=1/b\) and \(1-\theta+\theta/s=1/a\).
For the cone use Nicola's slicing argument
\cite[Theorem~1.2(a)(i)]{Nicola2009}. Write \(G_\rho(\omega)=G(\rho\omega,\rho)\).
Since \(\dd\Sigma=\sqrt2\,\rho^{n-1}\dd\rho\,\dd\sigma\),
\[
E_\Sigma G(x,t)=\sqrt2\int_1^2e^{\mathrm it\rho}\rho^{n-1}
E(G_\rho)(\rho x)\,\dd\rho .
\]
Hausdorff--Young in \(t\), Minkowski's inequality (\(b\ge b'\)),
and spatial dilation give
\[
\|E_\Sigma G\|_{L^b_{x,t}}
\lesssim\left(\int_1^2
[\rho^{n-1-n/b}\|E(G_\rho)\|_{L^b_x}]^{b'}\,\dd\rho\right)^{1/b'}
\lesssim\left\|\,\|G_\rho\|_{L^a(S^{n-1})}\,\right\|_{L^{b'}_\rho([1,2])}.
\]
H\"older's inequality gives \eqref{eq:cone-extension} because \(b>a'\)
implies \(a>b'\). For \(b=\infty\) the defining integrals are bounded
by the \(L^1\) norms on the two compact surfaces.
\end{proof}
==== END Local-smoothing-for-the-Euclidean-wave-equation//global-extension.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//history.tex ====
\subsection{Development of the local smoothing problem}\label{subsec:history}
The question grew out of Bourgain's theorem on maximal averages over
circles \cite{Bourgain1986Circles}. At high frequency the Fourier
transform of a circle of radius \(T\) has two leading terms, with phases
\(T|\xi|\) and \(-T|\xi|\). This explains why a spacetime estimate for
waves can help bound the supremum of circular averages over radii.
Sogge used this connection to prove a positive smoothing gain for every
\(p>2\) in two space dimensions \cite[Theorem~4.1]{Sogge1991}. Mockenhaupt, Seeger
and Sogge gave a proof of the circular maximal theorem using angular
square functions \cite{MSS1992}, and developed local smoothing for
Fourier integral operators under a curvature condition on the
associated family of cones \cite{MSS1993}.
A persistent difficulty is to put waves moving in different directions
back together. Wolff decomposed a neighborhood of the cone into thin
angular sectors and controlled their sum by induction on scales. This
gave every gain below \(1/p\) for \(p>74\) in the plane
\cite{Wolff2000}; {\L}aba and Wolff extended the argument to higher
dimensions \cite{LabaWolff2002}. Bourgain and Demeter's cone decoupling
theorem gives this gain for \(p\ge 2(n+1)/(n-1)\) in every dimension
\cite[Theorem~1.2]{BourgainDemeter2015}. A different argument using
radial Fourier multipliers attained the exact gain \(1/p\), with no
additional Sobolev loss, for \(n\ge4\) and \(p>2(n-1)/(n-3)\)
\cite[Theorem~1.2]{HeoNazarovSeeger2011}. In the plane, Guth, Wang
and Zhang reached the critical exponent \(p=4\) by controlling the
\(L^4\) norm of a wave by the spacetime square function of its angular
pieces \cite[Theorems~1.1 and~1.2]{GWZ2020}. For the canonical
sectors of a truncated cone, Cushman, Demeter and Wu give the sharp
power \(1/4\) of the logarithm in this square function inequality
\cite[Theorem~1.4]{CushmanDemeterWu2026}. To obtain a bound in
terms of \(\|f\|_4\), one must also estimate the square function from
the initial data.
In higher dimensions the distinction between a partial gain at \(\pc\)
and a full gain above \(\pc\) is substantial. At \(n=p=3\), Gao, Liu,
Miao and Xi obtained every gain below \(2/9\) by broad Fourier
restriction estimates, compared with the conjectured \(1/3\)
\cite[Corollary~1.3]{GaoLiuMiaoXi2023}. Gan, He, Li and Wu proved the
full gain up to an arbitrarily small loss for \(p\ge10/3\) in
\(\RR^3\) and on compact Riemannian three-manifolds
\cite[Theorem~1.3]{GHLW2026}; see also Wang's study of wave packet
density \cite{Wang2026v4}. Interpolating the \(p=10/3\) estimate with
the \(L^2\) identity gives every gain below \(1/4\) at \(p=3\).
Some three-dimensional metrics require
\(p\ge10/3\) for gains arbitrarily close to \(1/p\)
\cite{MinicozziSogge1997}, whereas the Euclidean exponent is \(3\).
==== END Local-smoothing-for-the-Euclidean-wave-equation//history.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//introduction.tex ====
\section{Introduction}\label{sec:introduction}
We prove Sogge's local smoothing conjecture for the Euclidean wave
equation in every spatial dimension \(n\ge2\). A wave arriving from a
thin spherical shell may have a much larger \(L^p\) norm at the instant
when its rays come to a focus than it had initially. Sogge asked how
many derivatives of the initial data are needed to control the norm
averaged over a time interval \cite[Section~4, Equation~(4.5)]{Sogge1991}.
Write \(U(T)=e^{-\mathrm iT|D|}\) for the half-wave propagator, with
\(|D|=\sqrt{-\Delta}\). Put \(I=[1,2]\), \(J=(1-\Delta)^{1/2}\), and
\[
s_+(p)=\max\left\{0,\frac{n-1}{2}-\frac np\right\}.
\]
\begin{theorem}[Euclidean local smoothing]\label{thm:local-smoothing}
For \(n\ge2\), \(2\le p<\infty\) and \(\varepsilon>0\),
\begin{equation}\label{eq:sogge}
\|U(T)f\|_{L^p(\RR^n\times I)}
\le C_{n,p,\varepsilon}\|J^{s_+(p)+\varepsilon} f\|_{L^p(\RR^n)} .
\end{equation}
At \(p=2\) one can take \(\varepsilon=0\).
\end{theorem}
At a fixed nonzero time the sharp Sobolev order is
\((n-1)(1/2-1/p)\) \cite{Peral1980,Miyachi1980,SSS1991}.
Thus for \(p\ge\pc=2n/(n-1)\), averaging in time gains every amount
less than \(1/p\) over the fixed-time estimate. For \(2<p<\pc\),
arbitrarily small positive regularity suffices. A focusing wave shows
that \(s_+(p)\) is the smallest possible threshold
(Proposition~\ref{prop:sharpness}). We state the wave estimates for complex
Schwartz data; density gives the indicated extensions, and conjugation
gives the other sign of the half wave. Unqualified \(L^p\) norms of the
data are spatial norms.
\papercontents
\subsection{Consequences for Fourier analysis and geometry}
\paragraph{Bochner--Riesz means and Stein's square function.}
The Bochner--Riesz means \(B_R^\alpha=(1-|D|^2/R^2)_+^\alpha\)
are uniformly bounded on \(L^p\) for \(1\le p\le\infty\) and every
\(\alpha>\max\{0,n|1/p-1/2|-1/2\}\), as conjectured. For
\(2\le p<\infty\) the same orders give a strong \(L^p\) bound for
the supremum over all radii (Theorem~\ref{thm:bochner-riesz}).
Stein's square function measures their variation with radius~\cite{Stein1958Square}.
Keeping this variation inside a square integral gives its conjectured
bounds for \(2\le p<\infty\) and \(\alpha>1/2+s_+(p)\)
(Theorem~\ref{thm:stein-square}).
The connection with the wave is that averaging in time against an
oscillatory weight selects a thin shell in frequency. Sogge used this
observation to pass from local smoothing to summability of Fourier
integrals \cite[Equations~(4.5)--(4.6)]{Sogge1991}; for the reductions
at fixed and variable radii see \cite[Propositions~39 and~40]{BHS2021}.
\paragraph{Fourier restriction and Schr\"odinger waves.}
Give an incoming spherical wave an amplitude that varies with
direction. Near the focus, rescaling recovers the Fourier extension
of this amplitude from the sphere. We obtain the full interior range
of global Fourier extension estimates \(L^a\to L^b\): for
\(1<a\le\infty\), \(b<\infty\), it is given by
\[
b>\frac{2n}{n-1},\qquad b>\frac{n+1}{n-1}\,a',
\qquad \frac1a+\frac1{a'}=1 .
\]
Slicing gives the same bounds for the truncated cone in \(\RR^{n+1}\)
(Corollary~\ref{cor:spherical-restriction}). Rescaling a small cap
gives the compact paraboloid estimate \eqref{eq:paraboloid-extension-mixed}
and local smoothing for Schr\"odinger waves in \(n-1\) spatial dimensions,
with the exact Sobolev order above their critical exponent
(Corollary~\ref{cor:parabolic}).
\paragraph{Kakeya and Nikodym sets.}
Random signs on
disjoint frequency caps turn the square of the extension into a
bound for overlaps of tubes. This proves the Kakeya maximal
conjecture: the operator that takes \(f\) to the largest average of
\(|f|\) on a unit \(\delta\)-tube in each direction has norm
\(O_\varepsilon(\delta^{-\varepsilon})\) from \(L^n(\RR^n)\) to
\(L^n(S^{n-1})\) for every \(\varepsilon>0\). In particular, every subset of \(\RR^n\) containing
a unit line segment in each direction has Hausdorff dimension \(n\)
(Corollary~\ref{cor:quant-kakeya}). A projective change of variables
gives the corresponding Nikodym maximal estimate for tubes through a
given point (see \eqref{eq:kakeya-nikodym}).
\input{sections/history}
\subsection{An estimate on one frequency annulus}\label{subsec:annular}
The proof is built around a single estimate at \(p=\pc\). It quantifies
the growth with frequency that the arbitrarily small Sobolev loss in
Theorem~\ref{thm:local-smoothing} allows. Our Fourier convention is
\(\widehat f(\xi)=\int_{\RR^n}e^{-\mathrm iX\cdot\xi}f(X)\dd X\),
so \(U(T)\) has phase \(X\cdot\xi-T|\xi|\) and inverse Fourier factor
\((2\pi)^{-n}\).
\begin{theorem}[Critical annular estimate]\label{thm:main}
Let \(n\ge2\), \(p=\pc\), and \(\varphi\in C_c^\infty(\RR^n\setminus\{0\})\).
For \(N\ge2\) set \(P_N=\varphi(D/N)\). Then
\begin{equation}\label{eq:main}
\|U(T)P_N f\|_{L^p(\RR^n\times I)}
\le C_{n,\varphi}[\log(2+N)]^{2/p}\|f\|_{L^p(\RR^n)} .
\end{equation}
\end{theorem}
Interpolation and summation over annuli yield
Theorem~\ref{thm:local-smoothing} (Section~\ref{sec:transfer}).
For unrestricted data on a full annulus the optimal logarithmic power in \eqref{eq:main}
lies between \(1/\pc\) and \(2/\pc\). The lower bound comes from the
focusing example: every dyadic distance from the focus between one
wavelength and unit distance contributes comparable mass to the
\(\pc\)-th power of the spacetime norm (Proposition~\ref{prop:sharpness}).
With any positive angular regularity, the power \(1/\pc\) suffices;
Appendix~\ref{sec:angular} proves a sharper result with a logarithmic
angular weight. For unrestricted data the lower bound persists even
for the weak spacetime norm (Proposition~\ref{prop:weak-obstruction}).
\subsection{A conserved current and two circles}\label{sec:idea}
Consider a finite sum of plane waves with $|\xi|\asymp N$ whose
directions lie in one small cap. We need to take a square \emph{after} summing these waves:
summing the squares would discard the interference that produces a
focus. An imaginary translation along a light ray filters the sum by
direction. Translation by $-\mathrm i\ell(1,\omega)$, with $|\omega|=1$ and
$\ell>0$, multiplies a mode of frequency $\xi$ by
$\exp[-\ell|\xi|\,|\omega-\xi/|\xi||^2/2]$. Its Gaussian width
in direction is therefore of order $(N\ell)^{-1/2}$ for $N\ell\gg1$.
The symmetric first-order wave system provides these light rays as
its quadratic fluxes. The sum of two such fluxes gives the imaginary
displacement. Two flows built from the continued wave then give a
positive semidefinite tensor. Each of its real columns has zero divergence,
and its time component still detects the original wave on the chosen cap
(Section~\ref{sec:current}).
Here is the estimate that the construction must deliver. Set $h=N^{-1}$,
let $a=Kh$ for a fixed large $K$, and write $p=\pc$. In addition to the
physical time $T$, the construction has two auxiliary time coordinates
$\tau\in\RR^2$. Write $H(T,s)$ for the circular mean at $|\tau|=s$
of the nonnegative energy $\mathcal F(T,\tau)$ constructed below. For $h$ small and nonzero
cap data on a fixed spatial torus normalized by $\|f\|_2=1$, we prove
\begin{align*}
\int_1^2\|U(T)f\|_p^p\dd T
&\ \lesssim\ a^{-2}\!\int_1^2 H(T,2a)\dd T\\
&\ \lesssim\ [\log(2/h)]^2 a^{-2} H(0,4a)
\ \lesssim\ [\log(2/h)]^2\|f\|_p^p .
\end{align*}
The first and last inequalities compare the energy with the wave. The
middle one uses only a scalar differential inequality.
To obtain the energy, fix $(T,\tau)$ and let the remaining real and
auxiliary spatial variables range over a bounded fiber with periodic
real directions. Write $\mu>0$ for the time component of the regularized
current. There is a complex scalar $\Phi$ constant along the current.
With $\lambda>0$ large, take on each fiber the solution with positive
definite spatial Hessian $H_{\rm fib}(\lambda|\Phi|^2+u)$ of
\[
\det H_{\mathrm{fib}}(\lambda|\Phi|^2+u)=\lambda\mu,
\qquad u|_{\partial}=0 .
\]
Here $H_{\rm fib}$ is an $n$-by-$n$ quaternionic Hermitian Hessian; its
determinant on positive matrices is the product of the eigenvalues.
The background costs nothing in the conservation law: since the
current is tangent to the levels of $\Phi$, its pairing with
$H(|\Phi|^2)$ vanishes. In the spatial determinant, however, it
supplies one direction of curvature. The Hessian of $\lambda|\Phi|^2$ has just one nonzero eigenvalue,
of size $\lambda$. Adding $\alpha I$ to this background changes its
determinant to $(\beta+\alpha)\alpha^{n-1}$, where $\beta\asymp\lambda$.
Thus density $\mu\ge m$ on a small ball forces the depth of $u$ at
its center to be at least a constant times the squared radius times
$m^{1/(n-1)}$, in the range of the construction. The integral
$\int(-u)\mu$ detects the power $m^{n/(n-1)}$. Since $\mu$ recovers
a square of the wave, the resulting exponent is $2n/(n-1)$.
A correction to $\int(-u)\mu$ makes the full determinant cancel when
one differentiates the energy in $(T,\tau)$, while leaving the corrected
energy comparable to that integral. The correction uses the integral
of the spatial determinant along the segment of potentials from
$\lambda|\Phi|^2$ to $\lambda|\Phi|^2+u$, as in the classical
Aubin--Mabuchi energy \cite{Aubin1984,Mabuchi1986}. The two integrations by parts use conservation of the
wave current and of the Hessian cofactor (Sections~\ref{sec:calculus}
and~\ref{sec:energy}). The cofactor contributes no boundary term:
its normal--normal coefficient vanishes.
Only the wave flux through the side of the fiber can give a positive
contribution to the Laplacian of the energy. A comparison function made
from a spatial average of the squared wave bounds $\mathcal F$ on the
initial circle and the normal derivative of $u$ at the side of the fiber.
On the part of the boundary that carries wave flux, the Gaussian decay
is stronger than the cost of recovering the wave in the interior.
The resulting scalar inequality is
\[
\Delta_{T,\tau}\mathcal F\le C s^{-2}e^{-cs/h}\mathcal F,
\qquad 2a\le s=|\tau|\le3
\quad\text{(Section~\ref{sec:barrier}).}
\]
It remains to compare the circles. The scalar inequality carries a
lower bound at the observed times $1\le T\le2$ from $|\tau|=2a$
to $|\tau|=1$. At that fixed radius the mean inequality carries it to
initial time; a second annulus comparison then takes it back to
$|\tau|=4a$. A radial harmonic function in the plane is affine in the
logarithm of its radius. Each passage thus loses an inverse logarithm.
Both losses can occur for a scalar supersolution even with zero
potential (Remark~\ref{rem:cylinder-power}). Taking a $p$th root and
summing a fixed number of caps gives the claimed logarithmic bound on
the torus. Section~\ref{sec:transfer} removes the spatial period to prove
Theorem~\ref{thm:main}, then sums the frequencies for the Sobolev estimate.
\paragraph{Related constructions.}
The bilinear flux of a symmetric hyperbolic system is classical
\cite{Friedrichs1954}. Analytic continuation in an imaginary spacetime
direction has also been used to study waves \cite{KaiserStreater1992}.
For positive-energy Klein--Gordon waves, Kaiser found a vector
conserved in physical spacetime at each fixed imaginary displacement.
Integration over displacements gives a current with nonnegative time density
\cite[Section~4.5, Equations~(36)--(40)]{Kaiser1990}. Here the continued wave gives a positive
symmetric tensor whose columns have zero divergence in the combined
physical and auxiliary variables.
Serre's compensated integrability estimates pair a positive
divergence-free tensor with the Hessian of a real Monge--Ampère
potential to bound a power of its determinant
\cite[Section~5.1]{Serre2018}. The usual stress tensor of a wave need
not be positive in \(n\ge2\) spatial dimensions
\cite[Section~4.1]{Serre2019}. Our lifted tensor is positive
semidefinite, but before averaging it is a sum of four real squares
and its determinant vanishes. We use its regularized time component as the prescribed
density in a quaternionic determinant equation on each spatial fiber
(Section~\ref{sec:energy}).
The main proof ends in Section~\ref{sec:transfer}; Appendix~\ref{sec:dirichlet}
supplies the Dirichlet theorem used in the construction.
Sections~\ref{sec:kakeya}--\ref{sec:parabolic} give the applications.
Appendix~\ref{sec:planar} replaces the determinant equation by a linear
Dirichlet problem on a disk in two dimensions. Appendix~\ref{sec:pencil}
extends the current construction to a simple extremal eigenvalue of a
real symmetric \(n\)-by-\(n\) pencil in \(n\) spatial variables whose
transverse Hessian is definite, and gives an example with an irreducible
cubic characteristic cone.
==== END Local-smoothing-for-the-Euclidean-wave-equation//introduction.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//kakeya.tex ====
\section{Spherical extension and Kakeya tubes}\label{sec:kakeya}
Write \(d=n-1\), \(p=\pc=2n/d\), and \(L(R)=\log(2+R)\). We keep the
logarithmic power \(A\) in the wave estimate as a parameter. To see what
happens to it in restriction, we take an incoming spherical wave whose
amplitude varies with direction. Near its focus the radial oscillation
cancels, leaving the extension of that amplitude from the unit sphere.
For \(g\) on \(S^{n-1}\) write
\(Eg(X)=\int_{S^{n-1}}e^{\mathrm iX\cdot\omega}g(\omega)\dd\sigma(\omega)\).
Let \(K_\delta f(\omega)\) be the supremum of the averages of \(|f|\)
over unit tubes of radius \(\delta\) parallel to \(\omega\).
\begin{corollary}[Spherical extension and Kakeya]\label{cor:quant-kakeya}
For \(n\ge2\), \(R\ge2\), \(z\in\RR^n\), and \(0<\delta\le1\),
\begin{align*}
\|Eg\|_{L^{\pc}(B(z,R))}
&\le C_n [\log(2+R)]^{2/\pc}\|g\|_{L^{\pc}(S^{n-1})},\\
\|K_\delta f\|_{L^n(S^{n-1})}
&\le C_n [\log(2/\delta)]^{4/\pc}\|f\|_{L^n(\RR^n)}.
\end{align*}
Every subset of \(\RR^n\) containing a unit line segment in each direction has Hausdorff dimension \(n\).
\end{corollary}
A wave estimate with logarithmic power \(A\) gives the same power for
spherical extension and power \(2A\) for the Kakeya maximal operator.
The radial example after the proof shows that the latter power must be
at least \(1-1/n\). Thus an unrestricted wave estimate with power
\(1/\pc\) would give the sharp Kakeya maximal logarithm.
In the plane, H\"ormander's estimate gives the sharp logarithmic
power $1/4$ for local circle extension: take the output exponent
$q=4+1/\log R$ in \cite[Corollary~1.3]{Hormander1973FLp} and apply
H\"older's inequality on the observation ball and on the circle.
Constant density forces this power by stationary phase. The planar directional
Kakeya maximal bound has sharp power $1/2$
\cite[Equation~(1.5)]{Bourgain1991Kakeya}; see also C\'ordoba's
theorem for rectangles \cite{Cordoba1977}. Davies proved full
Hausdorff dimension in the plane \cite{Davies1971}, and Wang and
Zahl proved it in three dimensions \cite[Theorem~1.1]{WangZahl2025};
see also \cite{GuthWangZahl2026Kakeya}. A separate three-dimensional
maximal estimate with $\delta^{-\varepsilon}$ loss and a
four-dimensional Hausdorff dimension theorem appear in
\cite[Theorem~1.1]{OpenAI2026KakeyaThree} and
\cite[Theorem~1.1]{OpenAI2026Kakeya4Hausdorff}, respectively.
Here the implication from a wave estimate to a fixed logarithmic
maximal estimate works in every dimension.
\begin{lemma}\label{lem:kakeya-local-extension}
Suppose \eqref{eq:main} holds with \(L(N)^A\) for some \(A\ge0\), for a
smooth radial cutoff \(\chi(|\xi|/N)\) with \(\chi\) supported in
\((1/2,2)\) and equal to one near \(1\). For \(R\ge2\), \(z\in\RR^n\),
and \(g\in L^p(S^{n-1})\), the spherical extension
\(Eg(X)=\int_{S^{n-1}}e^{\mathrm iX\cdot\omega}g(\omega)\dd\sigma(\omega)\)
satisfies
\begin{equation}\label{eq:kakeya-local-extension}
\|Eg\|_{L^p(B(z,R))}
\lesssim L(R)^A\|g\|_{L^p(S^{n-1})}.
\end{equation}
\end{lemma}
\begin{proof}
Choose a nonnegative nonzero \(\psi\in C_c^\infty((1,2))\).
Modulation in time selects a frequency shell of width of order one:
\[
M_N f=\int \psi(T)e^{\mathrm iNT}U(T)\chi(|D|/N)f\dd T,\qquad
\|M_N f\|_p\lesssim L(N)^A\|f\|_p
\]
by H\"older in \(T\). For \(1/2\le |Z|\le3\) its convolution kernel is
\begin{equation}\label{eq:kakeya-shell-kernel}
K_N(Z)=c_n N^{d/2}|Z|^{-d/2}e^{\mathrm iN|Z|}\psi(|Z|)
+O(N^{d/2-1}),\qquad c_n\ne0 .
\end{equation}
Indeed, the radial multiplier is \(\chi(\rho/N)\widehat\psi(\rho-N)\).
In polar coordinates the angular integral at \(u=\rho|Z|\) is
\[
\int_{S^{n-1}}e^{\mathrm iu\omega_1}\dd\sigma(\omega)
=u^{-d/2}(c_+e^{\mathrm iu}+c_-e^{-\mathrm iu})
+O(u^{-d/2-1}),\qquad c_+c_-\ne0 ,
\]
by stationary phase at the two poles. After radial integration the
remainder is \(O(N^{d/2-1})\). In the leading terms one can replace
\(\chi(\rho/N)\rho^{d/2}\) by \(N^{d/2}\) with the same error:
the prefactor, extended by zero, has derivative \(O(N^{d/2-1})\)
and \(\int |s\widehat\psi(s)|\dd s<\infty\). Fourier inversion gives
\(\psi(\pm|Z|)\); only the positive argument contributes.
We linearize the distance phase at the focus, as in H\"ormander's passage
from oscillatory integrals to extension \cite[\S3, (3.8)--(3.9)]{Hormander1973FLp};
compare Tao's deduction of local restriction from Bochner--Riesz estimates
\cite[proof of Theorem~1.1]{Tao1999}.
Take \(f_N(r\omega)=e^{-\mathrm iNr}\psi(r)g(\omega)\), so that
\(\|f_N\|_p=C_\psi\|g\|_p\). For \(|X|\le R\) and \(N\ge4R\),
\[
N|r\omega-X/N|=Nr-X\cdot\omega+O(R^2/N)
\qquad (r\in\supp\psi).
\]
Substitute this expansion in \eqref{eq:kakeya-shell-kernel} and
integrate in \(r,\omega\). The amplitude changes by \(O(R/N)\), and
the radial phases cancel:
\[
N^{-d/2}M_N f_N(X/N)
=c' Eg(-X)+O\big((1+R^2)N^{-1}\|g\|_1\big),
\qquad c'=c_n\int r^{d/2}\psi(r)^2\dd r\ne0 .
\]
Choose \(N=(4+R)^{3+n/p}\). The error has \(L^p(B(0,R))\) norm at
most \(C\|g\|_p\). On the left the change of variables costs
\(N^{n/p-d/2}=1\), and \(L(N)\lesssim_n L(R)\). This proves
\eqref{eq:kakeya-local-extension} at \(z=0\). Replacing \(g(\omega)\)
by \(e^{\mathrm iz\cdot\omega}g(\omega)\) gives every center.
\end{proof}
\input{sections/global-extension}
\subsection{From caps to tubes}
Let \(T_\delta(\omega,x)\) be the open \(\delta\)-neighborhood of
\(x+[-1/2,1/2]\omega\). Its volume is comparable to \(\delta^d\) for
\(0<\delta\le1\). Thus \(K_\delta f(\omega)\) is the supremum over
\(x\) of \(|T_\delta(\omega,x)|^{-1}\int_{T_\delta(\omega,x)}|f|\).
These averages are continuous in the center and direction for
\(f\in L^n\), by continuity of translations and rotations of the tube
indicator in \(L^{n/(n-1)}\); in particular the supremum is measurable.
A cap of radius \(\delta\) produces a wave of roughly constant modulus
on a tube of length \(\delta^{-2}\) and radius \(\delta^{-1}\).
The restriction argument of Beckner, Carbery, Semmes and Soria
\cite{BCSS1989Restriction} and Bourgain \cite{Bourgain1991Kakeya}
puts independent signs on disjoint caps. The expected squared modulus
of the sum is then the sum of the squared moduli. This accounts for
the factor of two in the logarithmic exponent.
\begin{lemma}\label{lem:kakeya-overlap}
Assume \eqref{eq:kakeya-local-extension} and set \(q=p/2=n/(n-1)\).
Let \(0<\delta\le1/2\), let the \(\omega_j\) be \(\delta\)-separated
on \(S^{n-1}\), and choose \(x_j\in\RR^n\) arbitrarily. Then for
\(T_j=T_{2\delta}(\omega_j,x_j)\) and \(a_j\ge0\),
\begin{equation}\label{eq:kakeya-overlap}
\Big\|\sum_j a_j\mathbf1_{T_j}\Big\|_{L^q(\RR^n)}
\lesssim L(1/\delta)^{2A}
\Big(\delta^d\sum_j a_j^q\Big)^{1/q}.
\end{equation}
\end{lemma}
\begin{proof}
Take the disjoint caps \(C_j=\{|\omega-\omega_j|<c\delta\}\) for a
small fixed \(c>0\), and put
\(h_j(\omega)=e^{-\mathrm i\delta^{-2}x_j\cdot\omega}\mathbf1_{C_j}(\omega)\).
For \(Y\in T_j\), write \(Y-x_j=t\omega_j+v\) with \(|t|\le1/2\),
\(|v|<2\delta\). Since
\(\omega_j\cdot(\omega-\omega_j)=-|\omega-\omega_j|^2/2\),
the phase in \(Eh_j(\delta^{-2}Y)\) differs from its value at
\(\omega_j\) by at most \(c^2/4+2c\). Thus
\(|Eh_j(\delta^{-2}Y)|\gtrsim\sigma(C_j)\asymp\delta^d\) on \(T_j\).
Group the centers by the unit lattice cubes \(Q\), with
\(J_Q=\{j:x_j\in Q\}\) (use half-open cubes). The tubes in this group
lie in the concentric cube \(5Q\), so their dilates all lie in a
ball of radius \(C_n\delta^{-2}\). For independent random signs
\(\varepsilon_j\) set \(g_\varepsilon=\sum_{j\in J_Q}
\varepsilon_j a_j^{1/2}h_j\). Disjointness of the caps gives
\(\|g_\varepsilon\|_p^p\lesssim\delta^d\sum_{j\in J_Q}a_j^q\).
At each point Jensen's inequality gives
\(\mathbb E|Eg_\varepsilon|^p\ge
(\sum_{j\in J_Q}a_j|Eh_j|^2)^q\).
Integrate on \(\delta^{-2}(5Q)\), change variables back to \(Y\),
and apply \eqref{eq:kakeya-local-extension} on the enclosing ball:
\[
\delta^{dp-2n}\int_{5Q}
\Big(\sum_{j\in J_Q}a_j\mathbf1_{T_j}\Big)^q
\lesssim L(1/\delta)^{Ap}\,\delta^d\sum_{j\in J_Q}a_j^q .
\]
The prefactor is one because \(dp=2n\). The cubes \(5Q\) have bounded
overlap, so the \(q\)-th power of the norm on the left of
\eqref{eq:kakeya-overlap} is bounded by a constant times the sum of
these integrals. Taking the \(q\)-th root gives \(Ap/q=2A\).
\end{proof}
\subsection{The maximal estimate and Hausdorff dimension}
\begin{proof}[Proof of Corollary~\ref{cor:quant-kakeya}]
Theorem~\ref{thm:main} gives \(A=2/p\) in
Lemma~\ref{lem:kakeya-local-extension}. We keep \(A\) in the next
steps to record the dependence on the wave estimate.
For \(0<\delta\le1/2\) choose a maximal \(\delta\)-separated set
\(\{\omega_j\}\). If \(|\omega-\omega_j|\le\delta\), then
\(T_\delta(\omega,x)\subset T_{2\delta}(\omega_j,x)\). Hence
\[
\|K_\delta f\|_{L^n(S^{n-1})}
\lesssim \Big(\delta^d\sum_j K_{2\delta}f(\omega_j)^n\Big)^{1/n}.
\]
To bound this discrete norm, test by \(a_j\ge0\) with
\(\delta^d\sum_j a_j^q=1\). For \emph{any} choice of the centers,
\[
\delta^d\sum_j\frac{a_j}{|T_j|}\int_{T_j}|f|
\lesssim \int |f|\sum_j a_j\mathbf1_{T_j}
\lesssim L(1/\delta)^{2A}\|f\|_n
\]
by Lemma~\ref{lem:kakeya-overlap} and H\"older. Taking suprema over
the centers and then the weights proves the maximal estimate.
For \(\delta\in[1/2,1]\) it follows from H\"older on each tube.
Suppose \(F\) contains a unit segment in every direction. Its open
\(\delta\)-neighborhood \(F_\delta\) contains the entire tube around
each such segment. Consequently
\begin{equation}\label{eq:kakeya-volume}
|F_\delta|\gtrsim L(1/\delta)^{-2An}
=L(1/\delta)^{-2(n-1)}\qquad (A=2/p).
\end{equation}
For bounded sets this already gives full lower and upper Minkowski
dimension. To obtain Hausdorff dimension we have to allow balls of
different sizes in the same cover. In fact the logarithmic rate gives
infinite Hausdorff measure for every gauge
\[
h_\gamma(r)=r^n[\log(1/r)]^{2An}
[\log\log(1/r)]^\gamma,\qquad \gamma>n-1
\quad (r\ \text{small}).
\]
Since \(h_\gamma(2r)\lesssim h_\gamma(r)\), it suffices to use ball
covers. Consider an open ball cover \(\{B_j\}\) of \(F\) with radii
\(r_j\le e^{-2^m}\). Group the radii by
\(J_k=\{j:2^k\le\log(1/r_j)<2^{k+1}\}\) as in the planar argument of
Keich \cite[Claim 3.2]{Keich1999}. Put
\(\delta_k=e^{-2^{k+1}}\), \(G_k=\bigcup_{j\in J_k}2B_j\) and
\(v_k=\sum_{j\in J_k}r_j^n\). On a segment in direction \(\omega\),
the length covered by group \(k\) is at most
\(C K_{\delta_k}\mathbf1_{G_k}(\omega)\). Indeed, at each covered
point the transverse disk of radius \(\delta_k\) lies in a doubled
ball, and disks at different points give disjoint cross sections of
the tube. Summing lengths therefore gives
\(1\lesssim\sum_{k\ge m}K_{\delta_k}\mathbf1_{G_k}(\omega)\) in
every direction. The maximal estimate and H\"older in \(k\) yield
\[
1\lesssim\sum_{k\ge m}2^{2Ak}v_k^{1/n}
\lesssim\Big(\sum_j h_\gamma(r_j)\Big)^{1/n}
\Big(\sum_{k\ge m} k^{-\gamma/(n-1)}\Big)^{(n-1)/n}.
\]
The last factor tends to zero. Thus the Hausdorff measure for this
gauge is infinite, and \(h_\gamma(r)=o(r^s)\) for every \(s<n\)
gives \(\dim_{\mathrm H}F=n\). In the plane the same calculation with
the classical $L^2$ maximal power $1/2$ cited above gives infinite
Hausdorff measure for
$r^2\log(1/r)[\log\log(1/r)]^\gamma$ whenever $\gamma>1$.
The same covering argument works if a segment in
each of a positive measure set of directions meets \(F\) in length
at least a fixed \(\lambda>0\).
\end{proof}
The radial test \(f_\delta(X)=|X|^{-1}\mathbf1_{\{2\delta<|X|<1/2\}}\)
shows the scale of a possible improvement. Its \(L^n\) norm is
comparable to \([\log(1/\delta)]^{1/n}\) for small \(\delta\), but
the average on a tube centered at the origin in any direction is at
least \(c\int_{4\delta}^{1/4}t^{-1}\dd t\). Hence the maximal
logarithm must have power at least \(1-1/n=2/p\). A wave estimate
with the sharp power \(A=1/p\) for arbitrary data would attain this
bound and give \(|F_\delta|\gtrsim L(1/\delta)^{-(n-1)}\). There are
compact Kakeya sets satisfying the opposite inequality, by
Fern\'andez-Delgado and de la Salle
\cite[Theorem 1.3]{FernandezDelgadoDeLaSalle2026}.
\subsection{Two consequences for other families of segments}
For the Nikodym maximal function \(N_\delta f(x)\), take the supremum
of the same tube averages over all unit segments \emph{through} \(x\).
The estimate just proved gives
\begin{equation}\label{eq:kakeya-nikodym}
\|N_\delta f\|_{L^n(\RR^n)}
\lesssim L(1/\delta)^{2A}\|f\|_{L^n(\RR^n)}.
\end{equation}
We give the projective argument to keep track of the logarithm;
compare Tao \cite[Theorem 4.10]{Tao1999} and Gao--Liu--Xi
\cite[\S2.2]{GaoLiuXi2025}. First consider only the
\(\rho\)-neighborhoods of the segments
\(\{x+u\omega:1/2\le u\le1\}\), and denote the supremum of their
averages by \(D_\rho f(x)\). By finitely many rotations we can
restrict to \(|\omega_n|\ge c_n\). Place \(x=(x',t_0)\) in a unit
cube \(Q\) centered at zero and fix \(t_0\). For \(\rho\) small, on each
such tube \(c_n/4\le|s-t_0|\le2\) and \(|y'|\le C_n\). On either
sign of \(s-t_0\), the map
\[
(y',s)\longmapsto (Y',Y_n)
=\big(y'/(s-t_0),\,1/(s-t_0)\big)
\]
and its inverse are uniformly Lipschitz there. It sends the line
\(y'=x'+(s-t_0)v\) to \(Y'=v+Y_n x'\), so the direction
\((x',1)/|(x',1)|\) is independent of \(v\). The image tube is
covered by a bounded number of unit tubes of width \(C_n\rho\) in
that direction. For each sign, transport \(f\) restricted to
\(5Q\cap\{c_n/4\le\pm(s-t_0)\le2\}\) by this map and extend by
zero; this is a single input for all \(x'\) in the slice. Its
\(L^n\) norm is at most \(C\|f\|_{L^n(5Q)}\) by the Jacobian
bounds. The map from \(x'\) to directions has area Jacobian bounded
below. Applying the Kakeya estimate in each slice, integrating in
\(t_0\), then translating and summing over
cubes, gives \(\|D_\rho f\|_n\lesssim L(1/\rho)^{2A}\|f\|_n\).
For \(\rho\) bounded below the same estimate follows by averaging
over a fixed ball about \(x\).
Return to an arbitrary segment through \(x\). Separate the part
within \(O(\delta)\) of \(x\), then divide each of the two rays into
distances \(s/2\le u\le s\) for dyadic \(\delta\le s\le1\).
Relative to the full tube volume, the innermost ball has weight
\(O(\delta)\), and a piece at distance \(s\) has weight \(O(s)\).
By dilation the latter's maximal average has the norm of
\(D_{\delta/s}\). Thus the operator norm is bounded by
\(\displaystyle C\delta+C\sum_s s L(s/\delta)^{2A}
\lesssim L(1/\delta)^{2A}\), proving \eqref{eq:kakeya-nikodym}.
The transverse disk proof above now applies to any Borel set meeting
a suitable unit segment through each point of a positive measure
set in length at least \(\lambda>0\). Such a set has Hausdorff
dimension \(n\). The Nikodym maximal estimate and this dimension conclusion
hold locally on a manifold of
constant sectional curvature for geodesic segments of fixed short
length in a compact subset of a straightening chart: there lengths,
tube widths and volumes are comparable before and after straightening
\cite[Proposition 2.3]{GaoLiuXi2025}.
For a different application, fix \(n\ge3\) and a real \(d\times d\)
matrix \(B\). Consider compact sets containing for every
\(y\in[-1,1]^d\) one curve
\[
\{(w(y)+(tI_d+t^2 B)y,t):|t|\le\tau_B\},
\]
where \(\tau_B>0\) is fixed sufficiently small and \(w(y)\) belongs
to the fixed sufficiently large compact ball of
\cite[Equation~(1.16)]{Nadjimzadah2026}. Nadjimzadah's lifting
theorem \cite[Theorem 1.3]{Nadjimzadah2026} now applies, since
Corollary~\ref{cor:quant-kakeya} gives full Kakeya dimension in
\(n+\rank(B^2)\) dimensions. The minimum Hausdorff dimension in
this curve family is attained and is
\[
n-\rank B+\rank(B^2)=n-\dim(\operatorname{im}B\cap\ker B);
\]
the last equality is rank--nullity for \(B\) on its image.
==== END Local-smoothing-for-the-Euclidean-wave-equation//kakeya.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//paper.tex ====
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pdfsubject={Local smoothing, Fourier restriction and summation, and Kakeya sets}}
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\begingroup
\section*{\contentsname}%
\small
\noindent Sections~\ref{sec:current}--\ref{sec:transfer} prove
Theorem~\ref{thm:local-smoothing}; Sections~\ref{sec:kakeya}--\ref{sec:parabolic}
give its applications.\par\smallskip
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\title{Local smoothing for the Euclidean wave equation}
\author{OpenAI}\date{October 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove Sogge's local smoothing conjecture for the Euclidean wave
equation in every spatial dimension at least two. We deduce the
Bochner--Riesz conjecture, maximal estimates for these means, and sharp
bounds for Stein's square function. We obtain Fourier restriction
estimates for the sphere, the compact paraboloid, and the truncated cone
throughout the interior of the conjectured range, as well as local smoothing for the
Schr\"odinger equation. We also prove the Kakeya and Nikodym maximal
conjectures; in particular, every Euclidean Kakeya set has full
Hausdorff dimension.
\end{abstract}
\bookmark[level=1,dest=paper-contents]{Contents}
\input{sections/introduction}
\input{sections/current}
\input{sections/calculus}
\input{sections/energy}
\input{sections/barrier}
\input{sections/propagation}
\input{sections/transfer}
\input{sections/sharpness}
\addtocontents{toc}{\protect\addvspace{6pt}}
\input{sections/kakeya}
\input{sections/consequences}
\input{sections/stein_square}
\input{sections/parabolic}
\appendix
\addtocontents{toc}{\protect\addvspace{6pt}}
\input{sections/dirichlet}
\input{sections/angular}
\input{sections/weak}
\input{sections/planar}
\input{sections/pencil}
\addtocontents{toc}{\protect\addvspace{6pt}}
\bibliographystyle{plainnat}
\bibliography{bibliography}
\end{document}
==== END Local-smoothing-for-the-Euclidean-wave-equation//paper.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//parabolic.tex ====
\section{The paraboloid and Schr\"odinger evolution}\label{sec:parabolic}
We return to the spherical extension estimate of
Lemma~\ref{lem:kakeya-local-extension}, with $n=d+1$. A small cap on
$S^d$ approaches the paraboloid under parabolic rescaling
\cite[\S3, (3.10)--(3.12)]{Hormander1973FLp}. The Fresnel
formula then writes a Schr\"odinger wave in $d$ spatial dimensions as
extension from that paraboloid. At $p_c=2(d+1)/d$ both changes of
variables preserve the logarithmic power. We use the relation between
paraboloid restriction and Schr\"odinger evolution developed by Rogers
\cite{Rogers2008Schrodinger} and Lee, Rogers and Seeger
\cite{LRS2013Schrodinger}.
Write $B_d$ for the unit ball in $\RR^d$ and define extension from the
compact paraboloid patch by
\[
E g(x,t)=\int_{B_d} e^{i(x\cdot v+t|v|^2)}g(v)\dd v .
\]
\begin{corollary}[Paraboloid extension and Schr\"odinger local smoothing]\label{cor:parabolic}
Let $d\ge1$ and $p_c=2(d+1)/d$. If $1<a\le\infty$, $p_c<b<\infty$
and $b>(d+2)a'/d$, then
\begin{equation}\label{eq:paraboloid-extension-mixed}
\|Eg\|_{L^b(\RR^{d+1})}\lesssim_{d,a,b}\|g\|_{L^a(B_d)} .
\end{equation}
Let $I\subset\RR$ be a compact interval. For $N\ge2$ and Schwartz $f$ with
$\supp\widehat f\subset B(0,N)\subset\RR^d$,
\begin{equation}\label{eq:parabolic-critical}
\|e^{it\Delta}f\|_{L^{p_c}(\RR^d\times I)}
\lesssim_{d,I} [\log(2+N)]^{2/p_c}\|f\|_{p_c}.
\end{equation}
For $p_c<p<\infty$ and $p\le r\le\infty$, arbitrary Schwartz data satisfy
\begin{equation}\label{eq:parabolic-exact}
\|e^{it\Delta}f\|_{L^p(\RR^d;L^r(I))}
\lesssim_{d,p,r,I}\|f\|_{B^s_{p,p}(\RR^d)}
\lesssim_{d,p,s}\|(1-\Delta)^{s/2}f\|_p,
\qquad s=d(1-2/p)-2/r .
\end{equation}
Here $B^s_{p,p}$ is the inhomogeneous Besov space. Taking $r=p$ gives
the exact spacetime Sobolev order $d-2(d+1)/p$; taking $r=\infty$
gives the corresponding maximal estimate.
\end{corollary}
\begin{proof}
We first transfer Lemma~\ref{lem:kakeya-local-extension} to $E$,
writing $A$ for the logarithmic power in that lemma. By
Theorem~\ref{thm:main} we may take $A=2/p_c$.
Parametrize a spherical cap by $(\delta v,\sqrt{1-\delta^2|v|^2})$
and evaluate its extension at $(x/\delta,-2t/\delta^2)$.
After removing a scalar phase, the new graph is
$q_\delta(v)=2(1-\sqrt{1-\delta^2|v|^2})/\delta^2$.
Choose the cap density to be $g(v)\sqrt{1-\delta^2|v|^2}$, so surface
measure times density is $\delta^d g(v)\dd v$. The image of $B(0,R)$
in physical space lies in $B(0,CR/\delta^2)$. Thus the extension
operator for $q_\delta$ has local $L^{p_c}\to L^{p_c}$ norm at most
\[
C\,\delta^{-d+(d+2)/p_c+d/p_c}
[\log(2+R/\delta^2)]^A .
\]
The power of $\delta$ is zero. Since
$q_\delta(v)=|v|^2+O(\delta^2)$, the error in replacing its extension
by $Eg$ on $B(0,R)$ is at most $C R\delta^2\|g\|_1$ pointwise.
Taking $\delta=(2R)^{-(d+2)}$ makes its $L^{p_c}$ norm
$O(\|g\|_{p_c})$ and keeps $\log(2+R/\delta^2)\lesssim_d\log(2+R)$.
Modulation of $g$ translates the observation ball, so
\begin{equation}\label{eq:local-paraboloid}
\|Eg\|_{L^{p_c}(B(z,R))}
\lesssim_{d,A} [\log(2+R)]^A\|g\|_{p_c}
\qquad(z\in\RR^{d+1},\ R\ge1).
\end{equation}
To obtain \eqref{eq:paraboloid-extension-mixed}, first note that the
loss in \eqref{eq:local-paraboloid} is $O_\varepsilon(R^\varepsilon)$
for every $\varepsilon>0$, also on a slightly larger parameter ball by
dilation. A smooth density supported on the larger patch and equal to
$\dd v$ on $B_d$ has Fourier decay $O((1+|(x,t)|)^{-d/2})$ by
stationary phase. Apply Tao's epsilon-removal theorem
\cite[Theorem~1.2]{Tao1999} in the compact hypersurface form of
\cite[Theorem~5.3]{Kim2017Restriction}, with both restriction
exponents equal to $p_c'\in(1,2)$. Duality and zero extension from
$B_d$ give $E:L^{p_c}(B_d)\to L^s(\RR^{d+1})$ for every $s>p_c$.
Since $B_d$ has finite measure, $E:L^s\to L^s$ as well. Interpolation
with $E:L^1\to L^\infty$ now gives \eqref{eq:paraboloid-extension-mixed}
as in the proof of Corollary~\ref{cor:spherical-restriction}, with $n=d+1$.
For the critical estimate write $p=p_c$, $S(t)=e^{it\Delta}$ and
$I_0=[1/2,1]$. We give the finite-scale comparison directly from the
Fresnel kernel; compare \cite[Lemma~2.2]{LRS2013Schrodinger}.
Let $\chi\in C_c^\infty(B(0,2))$ equal one on $B(0,1)$, so
$f=P_N f$ for $P_N=\chi(D/N)$. Tile $\RR^d$ by cubes $Q$ of side $N$.
For $t\in I_0$ and $|z|\ge8N$, integration by parts bounds the kernel
of $S(t)P_N$ by $C_M N^d(1+N|z|)^{-M}$: in the frequency variable
$\eta=\xi/N$ the phase gradient $Nz-2tN^2\eta$ has length at least
$N|z|/2$ on the support of $\chi$.
Thus on each $Q$ the contribution from outside the concentric cube
$30Q$ is bounded by the convolution of this majorant with $|f|$.
The same convolution works on every cube and has $L^p$ norm
$O(\|f\|_p)$ by Young's inequality for $M>d$.
Translate $Q$ to be centered at zero and set $h=f\mathbf1_{30Q}$.
This truncation may introduce higher frequencies, so the near contribution
is $S(t)P_Nh=P_NS(t)h$. The equality holds on $L^2$; at $x$ the last
expression integrates $S(t)h(x-w/N)$ against $\check\chi(w)\dd w$.
Choose $\lambda=16\sqrt d\,N$, so $g(v)=h(\lambda v)$ is supported in
the unit ball. Since $h\in L^1\cap L^2$, its Fresnel integral converges
absolutely. Substitution of $y=\lambda v$ gives
\[
|S(t)h(z)|=(4\pi t)^{-d/2}\lambda^d
\left|Eg\left(-\frac{\lambda z}{2t},\frac{\lambda^2}{4t}\right)\right|.
\]
For $z=x-w/N$, $x\in Q$ and $t\in I_0$, the arguments of $Eg$ lie in a
ball of radius $C_d(N^2+|w|)$. The change of variables from $(x,t)$
to those arguments is one-to-one, with absolute Jacobian comparable
to $\lambda^{d+2}$. Since $\|g\|_p=\lambda^{-d/p}\|h\|_p$,
\eqref{eq:local-paraboloid} gives
\[
\|S(t)h(x-w/N)\|_{L^p(Q\times I_0)}
\lesssim \lambda^{d-(2d+2)/p}
[\log(2+N^2+|w|)]^A\|h\|_p .
\]
The power of $\lambda$ is zero. Integrating against $|\check\chi(w)|\dd w$
and using its rapid decay bounds the near contribution by
$C[\log(2+N)]^A\|f\|_{L^p(30Q)}$. Sum the $p$th powers over $Q$, using
the bounded overlap of $30Q$, and add the far contribution.
This proves \eqref{eq:parabolic-critical} on $I_0$
with power $A$.
On $[a/2,a]$ parabolic scaling gives an extra $a^{1/p}$ and replaces
$N$ by $\max(2,\sqrt a\,N)$. Sum the $p$th powers over dyadic intervals
in a bounded time range and use conjugation for negative times.
For the mixed norms in \eqref{eq:parabolic-exact}, choose $p_c<a<p$ and
take $b=a$ in \eqref{eq:paraboloid-extension-mixed}. The diagonal case of
\cite[Corollary~1.2]{LRS2013Schrodinger} takes such an $L^a\to L^a$
extension bound to the first inequality in \eqref{eq:parabolic-exact}
for every $p\le r\le\infty$. The second follows from spatial
Littlewood--Paley theory and $\|\cdot\|_{\ell^p}\le\|\cdot\|_{\ell^2}$
for $p>2$.
\end{proof}
In $d=1$ this argument gives $(\log(2+N))^{1/2}$ at $p_c=4$.
The sharp annular power on $I_0$ is $1/4$
\cite[Theorem~1.4 and \S4.1]{LRS2013Schrodinger}. It is also the sharp
power for a frequency ball: modulation moves $[-N,N]$ into $[N,3N]$
and, at each time, only translates the output and multiplies it by a phase.
==== END Local-smoothing-for-the-Euclidean-wave-equation//parabolic.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//pencil.tex ====
\section{Curved characteristic sheets of symmetric systems}\label{sec:pencil}
In the proof of Theorem~\ref{thm:main}, the wave symbol supplies both a
polarization for each plane wave and a Gaussian centered at that
polarization. A curved sheet of the largest eigenvalue of a symmetric
matrix pencil supplies the same two objects. We construct them on one
angular patch and check the two places where the phase enters the
fiber estimate: Gaussian decay on the wall and recovery of the wave
near the center.
\begin{theorem}\label{thm:pencil}
Let $C_1,\ldots,C_n$ be real symmetric $n\times n$ matrices, $n\ge2$, and
put $C(\xi)=\sum_{j=1}^n\xi_jC_j$ and $\phi(\xi)=\lambda_{\max}(C(\xi))$.
Suppose that on an open cone $\Gamma\subset\RR^n\setminus\{0\}$ this
eigenvalue is simple and $D^2\phi(\xi)$ is positive definite on $\xi^\perp$.
For every $b\in C_c^\infty(\Gamma)$, $0<h\le1$ and Schwartz function
$f$ on $\RR^n$,
\begin{equation}\label{eq:pencil-estimate}
\|e^{-\mathrm iT\phi(D)}b(hD)f\|_{L^{p_c}(\RR^n\times[1,2])}
\le C_b[\log(2/h)]^{2/p_c}\|f\|_{L^{p_c}(\RR^n)},\qquad
p_c=\frac{2n}{n-1}.
\end{equation}
The constant may depend on the pencil. The same assertion holds for a
simple smallest eigenvalue with negative definite transverse Hessian.
\end{theorem}
\begin{example}[A cubic characteristic sheet]
In three spatial dimensions take the pencil
\[
C(x,y,z)=\begin{pmatrix}z&x&y\\ x&-z&0\\ y&0&-2z\end{pmatrix}.
\]
At $(0,0,1)$ the top eigenvalue is $1$, and its transverse Hessian is
$\diag(1,2/3)$ by \eqref{eq:pencil-curvature} below. Thus
\eqref{eq:pencil-estimate} gives an $L^3$ bound on a neighborhood of this ray.
The sheet is not a piece of a quadric. Indeed its characteristic polynomial is
\[
(t-z)(t+z)(t+2z)-x^2(t+2z)-y^2(t+z).
\]
If this homogeneous cubic factored over $\CC$, it would have a linear
divisor. Since it is monic in $t$, this entails a root
$t=\alpha x+\beta y+\delta z$. At $z=0$ the polynomial is
$t(t^2-x^2-y^2)$. Since $x^2+y^2$ is not the square of a linear form,
$\alpha=\beta=0$; the coefficients of $x^2z$ and $y^2z$ would then
require $\delta=-2$ and $\delta=-1$, respectively.
Irreducibility rules out even a local quadratic equation for the sheet:
the resultant with that equation in $t$ would vanish on an open set in
$(x,y,z)$, and hence identically.
\end{example}
\subsection{Eigenvectors and the spatial fiber}
Fix a unit direction $\omega\in\Gamma$. Take the unit top eigenvector of
$C(\omega)$ as the first state coordinate $e_0$. The other state coordinates
form its orthogonal complement of dimension $d=n-1$. The isolated
eigenprojection is smooth near $\omega$ by its resolvent formula
\cite[Chapter~II]{Kato1995}. Applying it to $e_0$ and normalizing the first
component to one gives a real eigenvector
$E(\xi)=(1,v(\xi))^{\mathsf t}$, homogeneous of degree zero. Put
$S(\xi)=\phi(\xi)I-C(\xi)$ and denote its lower $d\times d$ block by
$S_\perp(\xi)$. On a sufficiently small conic neighborhood,
\[
S(\xi)E(\xi)=0,\qquad c|\xi|I\le S_\perp(\xi)\le C|\xi|I .
\]
Here and below constants are uniform on each chosen compact angular chart.
Write $Lw$ for the orthogonal projection of $C(w)e_0$ on $e_0^\perp$.
Differentiating the unit eigenvector equation along $\omega+\sigma w$
shows that the derivative of the eigenvector at $\sigma=0$ is
$S_\perp(\omega)^{-1}Lw$ in $e_0^\perp$; differentiating the eigenvalue
derivative $\phi'_w=e^{\mathsf t}C(w)e$ therefore gives
\begin{equation}\label{eq:pencil-curvature}
D^2\phi(\omega)[w,w]
=2(Lw)^{\mathsf t}S_\perp(\omega)^{-1}Lw.
\end{equation}
Consequently $L:\omega^\perp\to e_0^\perp$ is an isomorphism. So is
$Dv(\omega)=S_\perp(\omega)^{-1}L$ on $\omega^\perp$: the polarization
$v$ is a local angular coordinate. This is where the equality of the
state and spatial dimensions is used.
For $W\in\CC_{\mathbf k}^d$ near zero, let $e(W)=(1,W)^{\mathsf t}$ and set
\[
B_j(W)=\frac{e(W)^{\mathsf t} C_j e(W)}{1+W\cdot W}.
\]
The products here are bilinear. On real eigenvectors this map is the group
velocity: $B(v(\xi))=\nabla\phi(\xi)$. Its differential at zero is
$2L^{\mathsf t}$, with image $\omega^\perp$. Thus for a fixed small $w_0>0$
the projection of $B$ onto $\omega^\perp$ is a holomorphic coordinate map
on a neighborhood of $|W|\le w_0$, and the matrices $[dB(W),\omega]$
and their inverses are bounded there.
We use one further complex coordinate $\Phi$ in the normal direction
$\omega$. For the imaginary time $\tau$ and spatial coordinates
$P\in\CC_{\mathbf k}^n$ define the fiber chart by
\begin{equation}\label{eq:pencil-chart}
P=\tau B(W)+\Phi\omega,\qquad
R^2=\frac{2\tau}{1+W\cdot W},\qquad y=|R|^2\asymp s=|\tau|.
\end{equation}
Inverting the projection of $P/\tau$ onto $\omega^\perp$ recovers $W$;
$\Phi$ then follows from the component along $\omega$. At a real
polarization $W=v(\xi)$ the Rayleigh quotient $\xi\cdot B(W)$ is
stationary. Since $\xi\cdot B(W)=\phi(\xi)$ there, differentiating
\eqref{eq:pencil-chart} gives the characteristic conormal
\[
d\Phi=\frac{\xi\cdot dP-\phi(\xi)\,d\tau}{\xi\cdot\omega}
\qquad(W=v(\xi)).
\]
Let $a=Kh$, $Q=\Phi/a$, and take the spatial fiber over $(T,\tau)$ to be
the torus of Section~\ref{sec:energy} in the real variables, times
\[
r=|W|^2/w_0^2+100|Q|^2-1<0
\]
in the imaginary variables. For $a<s<4$, differentiation of
\eqref{eq:pencil-chart} gives
\begin{equation}\label{eq:pencil-scales}
|dW|\lesssim s^{-1},\qquad |d\Phi|\lesssim1,\qquad
\dd P\asymp a^2s^{2d}\dd W\,\dd Q .
\end{equation}
The derivative bounds include the time variables: solve
$dP-B\,d\tau=\tau\, dB\,dW+\omega\,d\Phi$. At fixed $\tau$ the complex
Jacobian matrix in $(W,Q)$ is $[\tau\, dB,a\omega]$, which also proves
the volume formula. In particular $\partial_\omega\Phi=1$ in the spatial
variables. If $F=(W/w_0,10Q)$, Lemma~\ref{lem:hessian-calculus} gives
$H(r)=4(dF)^*dF$. Its spatial block is $\gtrsim s^{-2}I$ and has
determinant $\asymp a^{-2}s^{-2d}$. Since $|F|\le1$, it also satisfies
$G_{\rm fib}(r)G_{\rm fib}(r)^*\le H_{\rm fib}(r)$. The full Hessian has rank $n$.
Its quaternionic null line belongs to $\ker d\Phi\cap G(r)^\perp$,
since $d\Phi$ is a multiple of the last row of $dF$ and
$G(r)=2(dF)^*F\mathbf i$. The spatial Hessian of $|\Phi|^2$ has rank one,
with nonzero eigenvalue comparable to one. These are the geometric and
boundary properties needed for the determinant energy.
\subsection{The current and its Gaussian multiplier}
Take a finite periodic Fourier sum $g_T=e^{-\mathrm iT\phi(D)}f$ in the
annulus $|\xi|\asymp h^{-1}$ in the chosen chart, with $|v(\xi)|\le\gamma$, where
$\gamma<w_0/10$ is fixed below. Write its individual terms as $u_\xi$.
The entire vector wave $\sum_\xi |\xi|^{d/2}E(\xi)u_\xi$ solves the
symmetric system with matrices $C^0=I$, $C^j=C_j$. To apply the flux
construction of Section~\ref{sec:current}, write its paired auxiliary
states as $w=w_1+\mathbf k w_2=R e(W)$. The holomorphic flux map is
\[
\Theta(w,\Phi)=
\left(\tfrac12w^{\mathsf t} C^\mu w\right)_{\mu=0}^n+(0,\omega)\Phi .
\]
Its output is exactly $(\tau,P)$ in \eqref{eq:pencil-chart}.
Writing $M_{\mu a}=(C^\mu w)_a$, its real Jacobian is
$J=|\det[M,(0,\omega)]|^2=|R|^{2n}j(W)$ with $j(W)\asymp1$:
each of the first $n$ columns has a factor $R$, and the remaining
matrix is invertible by \eqref{eq:pencil-chart}.
Let $G_{\rm sys}$ be the conjugate of the vector wave evaluated at
$(T,X)-\mathrm i q_{\RR}$, where $q_{\RR}^\mu=w^*C^\mu w/2$.
Pushing the tensor of Lemma~\ref{lem:current-flow} forward as a density
by \eqref{eq:current-pushforward}, the general row formula
\eqref{eq:current-flux-column} gives
\begin{equation}\label{eq:pencil-column}
\chi_\mu=J^{-1/2}\sum_a M_{\mu a}\mathbf i(G_{\rm sys})_a,\qquad
A^0=\chi\chi^* .
\end{equation}
The change of variables preserves column divergence: its Hessian term
pairs the two equal diagonal state blocks with
$(\partial_{w_{1a}w_{1b}}+\partial_{w_{2a}w_{2b}})\Theta^\mu=0$,
and the mixed state block vanishes. Since $\Phi$ is the passive
coordinate, $d\Phi\,M=0$ and hence $d\Phi\,\chi=0$. The choice of root
$R$ affects only the sign of $\chi$, so $A^0$ is single valued.
The attenuation of each plane wave is determined by $S(\xi)E(\xi)=0$:
\begin{equation}\label{eq:pencil-gaussian}
\exp(-w^*S(\xi)w/2)=e^{-\Psi},\qquad
\Psi=\frac{y}{2h}(W-v(\eta))^*S_\perp(\eta)(W-v(\eta)),\quad \eta=h\xi .
\end{equation}
For each row norm at fixed imaginary coordinates we may remove the common
left unit $R/|R|$ in \eqref{eq:pencil-column}. The remaining volume factor
is $|R|/\sqrt J=j(W)^{-1/2}y^{-d/2}$. With respect to the orthogonal planes
$\CC_{\mathbf i}$ and $\mathbf k\CC_{\mathbf i}$, each row is then a
pair of scalar multipliers on $g_T$, up to conjugation and multiplication
by $\mathbf i$. Their real symbols are the two components in $\CC_{\mathbf k}$ of
\begin{equation}\label{eq:pencil-symbol}
j(W)^{-1/2}e(W)^{\mathsf t}C^\mu E(\eta)\,
|\eta|^{d/2}(hy)^{-d/2} e^{-\Psi}.
\end{equation}
For the time row the coefficient is $j(W)^{-1/2}(1+W\cdot v(\eta))$,
whose real part is bounded below. Thus $A^0_{TT}\ge|\mathsf S_0 g_T|^2$,
where $\mathsf S_0$ is the multiplier with symbol the real part of
\eqref{eq:pencil-symbol} for $\mu=0$. It can be inverted on the cap.
\subsection{The fixed cap estimate}
Normalize $\|f\|_2=1$. Write $\sigma=s/h$, $p=p_c$, $q=p/2=n/d$. With
the kernels $J_\ell$ and cutoff $g(Q)$ of \eqref{eq:regularized-current} put
\[
A_{\rm w}=g(Q)(J_h*_X A^0),\qquad A=A_{\rm w}+h^{50n}C(H(r)),\qquad
\mu=A_{TT},\qquad B_0=\lambda|\Phi|^2,\quad \lambda=h^{-30n}.
\]
Since $\Phi$ is independent of $X$, the averaged current still annihilates
$G(g(Q))$, so the cutoff preserves conservation. The positive semidefinite
cofactor is conserved, takes values in $\ker H(r)\subset\ker d\Phi$, and has
$TT$ entry $\det H_{\rm fib}(r)>0$. Thus $A:H(B_0)=0$ and $\mu>0$.
The spatial background $H_{\rm fib}(B_0)$ has a single nonzero eigenvalue
$\asymp\lambda$. Theorem~\ref{thm:dirichlet} gives the solution
$u\le0$ of \eqref{eq:fiber-equation}, smooth in $(T,\tau)$. The null
line of $H(r)$ lies in $G(r)^\perp\cap\ker H(B_0)$, so both normal
cofactor terms vanish as in Proposition~\ref{prop:energy-flux}. That
balance and Lemma~\ref{lem:energy-comparison} give, for the energy
$\mathcal F=\mathcal J_z(u)$ of \eqref{eq:energy-functional},
\begin{equation}\label{eq:pencil-energy}
\mathcal F\asymp\int_{r<0}(-u)\mu\dd X\,\dd P,\qquad
\Delta_{T,\tau}\mathcal F
\le\int (u/r)G(r)^*A_{\rm w}G(r)\,\delta(r)\dd X\,\dd P .
\end{equation}
We check the two Gaussian estimates used in the proof of
Proposition~\ref{prop:fixed-barrier}, throughout $2a\le s\le3$.
First multiply each row symbol \eqref{eq:pencil-symbol} by a smooth
cutoff equal to one on the data, supported in a fixed enlarged annulus
and in $|v|<w_0/3$ inside the chart. Since $|\partial_\eta^\alpha\Psi|\lesssim_\alpha\sigma$,
the derivatives of these symbols through order $2n+4$ are bounded by
$C(hs)^{-d/2}\sigma^m$ for some $m$. On the active wall
$r=0$, $g(Q)\ne0$, we have $|W|\ge\sqrt3\,w_0/2$. The lower bound
$S_\perp(\eta)\ge cI$ gives the additional factor $e^{-c_0\sigma}$ with
$c_0>0$ independent of $\gamma$. The kernel estimate leading to
\eqref{eq:barrier-current} now gives, for $P_*=(J_{Ds}*|g_T|^2)^{1/d}$
and fixed large $D$,
\begin{equation}\label{eq:pencil-envelope}
\mu\lesssim s^{-2d}\sigma^m P_*^d \quad(r\le0),\qquad
\|A_{\rm w}\|\lesssim s^{-2d}\sigma^m e^{-c_0\sigma}P_*^d\quad(r=0).
\end{equation}
The cofactor adds $O(h^{50n}a^{-2}s^{-2d})$ to $\mu$;
the nearest lattice translate gives $P_*^d\gtrsim h^{n+4}$ since $\|g_T\|_2=1$.
For the inverse estimate take a center with $|W|\le\gamma$, $|Q|\le1/400$.
By \eqref{eq:pencil-scales}, its spatial ball of radius $h$ lies in
$|W|\le2\gamma$, $|Q|\le1/100$, where $g=1$, for $K$ large relative to
$1/\gamma$. At each imaginary point of this ball invert $\mathsf S_0$
using a smooth cutoff equal to one on the data, supported in the same
enlarged annulus and in $|v|\le2\gamma$. On its support
$\Psi\le C_1\gamma^2\sigma$, so the reciprocal with cutoff has derivatives
through order $2n+4$ bounded by
$C_\gamma(hs)^{d/2}\sigma^m e^{C_1\gamma^2\sigma}$ with $C_1$ independent
of $\gamma$. Kernel domination by $J_h$ and $A^0_{TT}\ge|\mathsf S_0g_T|^2$
give, as in \eqref{eq:barrier-ball-density},
\[
\mu\ge m_X=c_\gamma(hs)^{-d}\sigma^{-m}e^{-2C_1\gamma^2\sigma}|g_T(X)|^2
\]
on the whole ball: moving the real coordinate by at most $h$ changes $J_h$
only by a bounded factor.
The two comparison functions of Section~\ref{sec:barrier} now apply.
By \eqref{eq:barrier-product}, $H_{\rm fib}(rP_*)\ge P_*H_{\rm fib}(r)/2$.
Since $H_{\rm fib}(r)\gtrsim s^{-2}I$ and the background has rank one,
comparison with $C\sigma^{m'}rP_*$ gives $0\le u/r\lesssim\sigma^{m'}P_*$
by \eqref{eq:pencil-envelope}. On the ball the quadratic function
$\epsilon m_X^{1/d}(\ell^2-h^2)$, where $\ell$ is distance from the center,
gives $(-u)\mu\gtrsim h^2m_X^q$ at the center. This comparison uses
$m_X^{1/d}\le\lambda$: the $O(h^{-n})$ Fourier coefficients sum in modulus
to $O(h^{-n/2})$ by Parseval, and \eqref{eq:pencil-symbol} is $O(h^{-d})$.
Together with the cofactor bound this gives $\mu^{1/d}\le\lambda$ for small $h$.
Tangency removes $G(|Q|^2)$ from the flux; the remaining part of $G(r)$
is $O(s^{-1})$ by \eqref{eq:pencil-scales}. The same Jacobian estimate
gives imaginary volume $O(a^2s^{2d})$ for both the fiber and its wall with
weight $\delta(r)$, and volume $\asymp_\gamma a^2s^{2d}$ for the centers.
Using $\int P_*^{d+1}\dd X\le\|g_T\|_p^p$ and $h^2s^{2d}(hs)^{-dq}=\sigma^{n-2}$
in \eqref{eq:pencil-energy}, we get
\begin{align}
c_\gamma a^2\sigma^{-m}e^{-2q C_1\gamma^2\sigma}\|g_T\|_p^p
&\le\mathcal F\le Ca^2\sigma^m\|g_T\|_p^p,\label{eq:pencil-lower}\\
\Delta_{T,\tau}\mathcal F&\le Ca^2s^{-2}\sigma^me^{-c_0\sigma}\|g_T\|_p^p .
\label{eq:pencil-upper-flux}
\end{align}
Choose $\gamma$ so that $2q C_1\gamma^2<c_0/2$, then fix $K$. Division gives
$\Delta\mathcal F\le Cs^{-2}e^{-cs/h}\mathcal F$. At $T=0$ the upper
bound on $s=4a$ uses $g_0=f$; on $1\le T\le2$ the lower bound on $s=2a$
controls $\|g_T\|_p^p$. The values of $\sigma$ on these circles are fixed.
As in the proof of Proposition~\ref{prop:periodic-cap},
Lemma~\ref{lem:cylinder} with $b=2a$ and $r=4a$ gives
\[
\int_1^2\|g_T\|_p^p\dd T\lesssim[\log(2/h)]^2\|f\|_p^p .
\]
Homogeneity removes the normalization.
Finitely many such caps cover the directions in $\supp b$; their smooth
annular cutoffs have uniformly integrable kernels. To remove the period,
choose $M\ge1$ larger than $\sup_{\supp b}|\nabla\phi|$ and apply the
torus estimate on the same lattice to $C/M$ and $b_M(\eta)=b(\eta/M)$.
By homogeneity $|\nabla(\phi/M)|\le1$ on $\supp b_M$. For $|X|\ge3$ the
kernel phase has gradient $X-(T/M)\nabla\phi(\eta)$ of size $\gtrsim |X|$,
which gives the decay \eqref{eq:off-cone-kernel}. The cube argument of
Section~\ref{sec:transfer} gives the Euclidean estimate for $(C/M,b_M)$.
For $f_M(X)=f(MX)$ one has
\[
e^{-\mathrm iT\phi(D)/M}b_M(hD)f_M(X)
=\big(e^{-\mathrm iT\phi(D)}b(hD)f\big)(MX).
\]
Both norms gain the same factor $M^{-n/p}$, proving \eqref{eq:pencil-estimate}
for small $h$. Smooth multiplier estimates cover the remaining compact
range. Applying the result to $-C$ with cutoff $\bar b$ and conjugating
with $f(X)\mapsto\overline{f(-X)}$ gives the smallest-eigenvalue assertion.
\subsection{A lower bound on a single patch}
For every nonzero $b$ in Theorem~\ref{thm:pencil}, the logarithmic power
in \eqref{eq:pencil-estimate} must be at least $1/p_c$. Here is the
local version of the focusing test in Proposition~\ref{prop:sharpness}.
Rotate a neighborhood where $b\ne0$ into the coordinates
$\eta=\rho(\alpha,1)$, $\rho>0$, and write
$\phi(\eta)=\rho\varphi(\alpha)$. The matrix $D^2\varphi$ is positive
definite there, since the Hessian of $\phi$ has only the radial kernel.
Take $h=N^{-1}$ and
$\widehat f_N(\xi)=e^{(3/2)\mathrm i\phi(\xi)}(\psi/b)(\xi/N)$,
where $\psi$ is a nonnegative smooth bump supported in this neighborhood
and strictly positive on a smaller closed product patch. At the initial
time, stationary phase in $\alpha$ gives $\|f_N\|_\infty\lesssim N^{n-d/2}$ uniformly
in $X$, and Plancherel gives $\|f_N\|_2\lesssim N^{n/2}$. Thus
$\|f_N\|_p\lesssim N^{(n+1)/2-1/p}$ for $p\ge2$.
At time $3/2+\delta$, set
$X'=\delta\nabla\varphi(\alpha)$ and
$X_n=\delta(\varphi(\alpha)-\alpha\cdot\nabla\varphi(\alpha))+z/N$.
For $\alpha$ in the smaller patch and $|z|\le c$, this set has volume
$\asymp\delta^d/N$. Its angular stationary point is $\alpha$.
For $N\delta$ large, stationary phase now gives output of modulus at least
$cN^{n-d/2}\delta^{-d/2}$: up to a nonzero smooth factor and constant
phase the leading radial integral is
$\int e^{\mathrm i\rho z}\rho^{d/2}\psi(\rho(\alpha,1))\dd\rho$,
uniformly nonzero for small $c$. The angular remainder is smaller by
$O((N\delta)^{-1})$ \cite[Chapter~VIII]{Stein1993}. The $p$th power of
the ratio of the output and input norms thus dominates
$\int_{C/N}^{1/4}\delta^{d-dp/2}\dd\delta$. For $p=p_c$ this is
comparable to $\log N$.
==== END Local-smoothing-for-the-Euclidean-wave-equation//pencil.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//planar.tex ====
\section{A linear Dirichlet problem for the planar wave}\label{sec:planar}
When $n=2$, the coefficient of a rank-one background in the spatial
determinant is linear in the unknown Hessian. We can therefore prove the
$p_c=4$ case of Theorem~\ref{thm:main} with a linear auxiliary Dirichlet
problem on a disk. We give the reduction, including the boundary estimate
needed to apply Lemma~\ref{lem:cylinder}.
\paragraph{The disk equation.}
Fix nonzero periodic data in a single cap as in
Proposition~\ref{prop:fixed-barrier}. Retain $z=(T,\tau)$, $s=|\tau|$,
$a=Kh$, and $Q=\Phi/a$. For $a<s<4$ replace the ellipsoid by the product
\[
\Omega_z^{\rm pl}:\quad r_W=|W|^2/w_0^2-1<0,\qquad |Q|<1/10 .
\]
The real variable $X$ still runs over the fixed torus. At fixed $\tau$,
\eqref{eq:fiber-inverse} gives a holomorphic chart from this product to
the imaginary spatial variables $(P,Z)$.
Use the wave part $A=A_{\rm w}=g(Q)(J_h*_X A^0)$ of
\eqref{eq:regularized-current}, with $g=1$ on $|Q|\le1/100$ and $g=0$
on $|Q|\ge1/20$, and put $\mu=A_{TT}$. We need only the
nonnegativity of this density; no regularizing cofactor is needed.
Put $B=H(|\Phi|^2)$, let $B_{\rm fib}$ be its spatial block, and write
$L_Bu=C(B_{\rm fib}):H_{\rm fib}(u)$. With the row
$c=(\partial_P\Phi,\partial_Z\Phi)$ and the column
$e=(c_2,-c_1)^{\mathsf t}$, we have $B_{\rm fib}=4c^*c$ and
$C(B_{\rm fib})=4ee^*$. Define
$\mathcal J=\det_{\CC}\partial(W,\Phi)/\partial(P,Z)=(1+W^2/2)/(\sqrt2Y)$. Then
\begin{equation}\label{eq:planar-disk-operator}
L_Bu=4\sum_{i,j=1}^2\Rea(e_i\bar e_j)\partial_{X_iX_j}u
+4|\mathcal J|^2\Delta_Wu,\qquad |\mathcal J|\asymp s^{-1} .
\end{equation}
Indeed, the mixed derivatives between $X$ and $(P,Z)$ in the Hessian
take values in $\CC_{\mathbf k}\mathbf i$, orthogonal to the entries of
$ee^*$ in $\CC_{\mathbf k}$. The remaining imaginary block is the
complex Hessian of Lemma~\ref{lem:hessian-calculus}; under the
holomorphic chart it pairs with the vector $(\partial W\,e,c e)=(\mathcal J,0)$.
The Jacobian bound follows from \eqref{eq:fiber-inverse}.
Solve $L_Bu=\mu$ with $u=0$ on $|W|=w_0$. At each fixed $(z,\Phi)$ and
real spatial Fourier frequency $k$ this means solving
\[
4|\mathcal J|^2\Delta_W\widehat u-4|k\cdot e|^2\widehat u=\widehat\mu
\quad (|W|<w_0),\qquad \widehat u|_{|W|=w_0}=0.
\]
Here $\Delta_W$ is the ordinary planar Laplacian. After division by
$4|\mathcal J|^2$ the operator is the Dirichlet Laplacian minus a smooth
nonnegative potential. It is invertible also at $k=0$. The Dirichlet
isomorphism on H\"older spaces and the implicit function theorem give
smooth dependence on the parameters. There are only finitely many frequencies
in $\mu$. Solving before multiplication by $g(Q)$ shows that
$u=g(Q)\widetilde u$ with $\widetilde u$ smooth and real. Thus $u$
vanishes near the $Q$ wall, including the corners. Also $u\le0$:
for every $\epsilon>0$, $L_B(u+\epsilon r_W)=\mu+
16\epsilon |\mathcal J|^2/w_0^2>0$, ruling out a positive interior maximum in
$(X,W)$. Now let $\epsilon\downarrow0$.
\paragraph{The energy.}
For the full three-by-three Hessian $H=H(u)$, let
$\mathcal D_B(H)=B:C(H)$, the coefficient of $\lambda$ in
$\det(\lambda B+H)$. It is quadratic in $H$. Its derivative $M$ in
$H$ is a mixed cofactor, hence divergence free by
Lemma~\ref{lem:cofactor-divergence}. Homogeneity gives
$M:H=2\mathcal D_B(H)$; differentiation in the $TT$ entry gives
$M_{TT}=C(B_{\rm fib}):H_{\rm fib}(u)=\mu$.
On the $W$ wall, the full $H(r_W)$ has image the quaternionic line
spanned by its normal $\nu=G(r_W)/|\nabla r_W|$. Since $u=0$ there,
writing $u=(u/r_W)r_W$ and restricting the product rule to $\nu^\perp$
gives $H|_{\nu^\perp}=0$. The coefficient of $\lambda$ in
$\det((\lambda B+H)|_{\nu^\perp})$ is therefore zero, so
$\nu^*M\nu=0$ by Lemma~\ref{lem:quaternionic-matrices}. Now set
\[
X_0(z)=\int_{\Omega_z^{\rm pl}}(-u)\mu\dd\zeta,\qquad
\mathcal F_{\rm pl}(z)=\tfrac12 X_0(z),\qquad \dd\zeta=\dd X\,\dd P\,\dd Z.
\]
Green's formula \eqref{eq:green-moving-fiber} for $M$ gives
$\Delta_zX_0=-2\int\mathcal D_B(H)\dd\zeta$; the $Q$ wall contributes
nothing because $u$ vanishes in a neighborhood of it.
To apply the same formula to $A$, we claim that $A:H\ge\mathcal D_B(H)$.
The current is supported on the quaternionic plane $E=\ker B$ by
Proposition~\ref{prop:current}. Put $\beta=\tr B$, $\beta_f=\tr B_{\rm fib}$.
In $E$ take the orthonormal basis consisting of the normalized projection
of the time vector $e_T$ and the spatial vector $(0,e/|e|)$. In this
basis write $\alpha=A|_E$, $b=H|_E$, indexing rows from zero. Before
normalization the projection of $e_T$ has squared length $\beta_f/\beta$. Hence
$\alpha_{00}\beta_f/\beta=A_{TT}=\mu=\beta_f b_{11}$ by the disk
equation. The rank-one determinant identity gives
$\mathcal D_B(H)=\beta\det b$. Substituting
$\alpha_{00}=\beta b_{11}$ and completing the square gives
\[
A:H-\mathcal D_B(H)
=\beta\,|b_{01}+\alpha_{01}/\beta|^2+\det(\alpha)/\beta\ge0.
\]
Green's formula for $A$ now bounds
$\Delta_zX_0$ by $\tfrac12\Delta_zX_0$ plus the outward wave flux.
Consequently
\begin{equation}\label{eq:planar-energy}
\mathcal F_{\rm pl}\ge0,\qquad
\Delta_z\mathcal F_{\rm pl}\le
\int_{|Q|<1/10}\delta(r_W)(u/r_W)G(r_W)^*A G(r_W)\dd\zeta .
\end{equation}
\paragraph{Comparison on the product fiber.}
Write $\sigma=s/h$, $g_T=e^{-iT|D|}f$, and
$P_*=J_{Ds}*|g_T|^2$ for $D$ sufficiently large.
The multiplier proof of \eqref{eq:barrier-current} applies throughout
the product: $y\asymp s$ there and $|W|=w_0$ on the $W$ wall. It
gives $\|A\|\lesssim s^{-2}\sigma^m P_*$, with the additional factor
$e^{-c_0\sigma}$ on that wall. By \eqref{eq:planar-disk-operator} and
\eqref{eq:barrier-average},
\[
L_B(r_WP_*)=16|\mathcal J|^2P_*/w_0^2+r_W L_BP_*
\ge(c-CD^{-2})s^{-2}P_* .
\]
The two variables separate, so no mixed gradient term occurs.
For fixed large $C_1,m_1$ this gives $L_B(C_1\sigma^{m_1}r_WP_*)>\mu$.
The comparison function and $u$ both vanish on the $W$ wall.
Comparison on each fixed $Q$ slice yields $0\le u/r_W\le C_1\sigma^{m_1}P_*$ up to that wall.
The original volume is $\dd P\,\dd Z=a^2|\mathcal J|^{-2}\dd W\,\dd Q$,
while $\int\delta(r_W)\dd W=\pi w_0^2$ and the full $|G(r_W)|\lesssim s^{-1}$.
Integrating the energy and
\eqref{eq:planar-energy}, using $\int P_*^2\dd X\le\|g_T\|_4^4$, gives
\[
\mathcal F_{\rm pl}\lesssim a^2\sigma^m\|g_T\|_4^4,\qquad
\Delta_z\mathcal F_{\rm pl}\lesssim
(a/s)^2\sigma^m e^{-c_0\sigma}\|g_T\|_4^4 .
\]
For the lower bound use the centers $|W|\le\gamma$, $|Q|\le1/400$ of
Section~\ref{sec:barrier}. The inverse multiplier in the proof of
\eqref{eq:barrier-ball-density} bounds the wave density itself below
by $m_X$ there, on the entire physical spatial ball of radius $h$ about
each center; take $d=1$ in that formula. For the distance $l$ to its
center, $H_{\rm fib}(l^2)=6I$, so $L_B(l^2)=24|e|^2\le C$. For $m_X>0$, subtract $\epsilon m_X(l^2-h^2)$ from $u$, with
$\epsilon$ small. The difference has strictly positive $L_B$ on the
whole ball and is nonpositive on its boundary, so it cannot have a
positive interior maximum. This gives $(-u)\mu\ge\epsilon h^2m_X^2$
at the center, also trivially for $m_X=0$. Its integral gives
\[
\mathcal F_{\rm pl}\gtrsim_\gamma
a^2\sigma^{-m} e^{-C\gamma^2\sigma}\|g_T\|_4^4,
\]
since the imaginary center region has volume comparable to
$a^2s^2\gamma^2$ and $h^2s^2(hs)^{-2}=1$. Taking $\gamma$ small and
then $K$ large gives all three inequalities of
Proposition~\ref{prop:fixed-barrier} for $\mathcal F_{\rm pl}$.
Lemma~\ref{lem:cylinder} and Section~\ref{sec:transfer} now give the
planar $O((\log(2/h))^{1/2})$ estimate.
==== END Local-smoothing-for-the-Euclidean-wave-equation//planar.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//propagation.tex ====
\section{Propagation across a thin cylinder}\label{sec:propagation}
The energy in Proposition~\ref{prop:fixed-barrier} controls the wave on the
circle $|\tau|=2a$, $1\le T\le2$, and is controlled by the data at
$|\tau|=4a$, $T=0$. To compare these two sets, we first carry the energy
out to $|\tau|=1$, where a fixed neighborhood connects the two time
intervals. We then carry it back towards the inner wall. Each passage
costs one logarithm, as Figure~\ref{fig:cylinder} illustrates.
The radial harmonic functions used for comparison are the probabilities
that planar Brownian motion leaves an annulus through one circle or
the other \cite[Theorem~3.18]{MoertersPeres2010}. We use the maximum
principle to retain this comparison in the presence of a potential.
\begin{lemma}\label{lem:cylinder}
Let $0<b<1/4$ and $D=\log(3/b)$. Suppose a continuous potential
$V:[b,3]\to[0,\infty)$ satisfies
\begin{equation}\label{eq:cylinder-moment}
\int_b^3 s\log(s/b)V(s)\,ds\le M,\qquad
\sup_{1/2\le s\le3/2} V(s)\le M .
\end{equation}
Let $E\ge0$ be smooth on $\mathbb R_T\times
\{\tau\in\mathbb R^2:b\le|\tau|\le3\}$, with
$\Delta_{T,\tau}E\le V(|\tau|)E$ in the interior. Its circular mean
\[
H(T,s)=\frac1{2\pi}\int_0^{2\pi}
E\bigl(T,s(\cos\theta,\sin\theta)\bigr)\,d\theta
\]
satisfies, for $b\le r\le1$,
\begin{equation}\label{eq:cylinder-propagation}
H(0,r)\ \ge\ c(M)\frac{\log(r/b)}{D^2}\int_1^2 H(T,b)\,dT .
\end{equation}
In particular the factor at $r=2b$ is a constant times $D^{-2}$.
\end{lemma}
\input{figures/cylinder}
\begin{proof}
Averaging over rotations preserves the differential inequality, so
$H(T,|\tau|)$ is again a supersolution of $(-\Delta_{T,\tau}+V)H\ge0$.
We first estimate the radial solutions that will be used for both
passages. For $S=1$ or $3$ and a fixed $\kappa>0$, consider
\[
u''+s^{-1}u'=(V+\kappa^2)u,\qquad b<s<S,
\]
with endpoint values $0,1$ in either order. We claim that $u$ is bounded
below by a constant times the function affine in $\log s$ with the same
values. Indeed, $x=\log(s/b)$ changes the equation into
$u_{xx}=P(x)u$ on $[0,A]$, where $A=\log(S/b)$ and
$P(x)=s^2(V(s)+\kappa^2)$. Let $\psi$ solve this equation with
$\psi(0)=0$, $\psi'(0)=1$. Then $\psi(x)\ge x$, and integrating twice gives
\[
\frac{\psi(x)}x
=1+\int_0^x\frac{z(x-z)}x P(z)\frac{\psi(z)}z\,dz .
\]
Since $z(x-z)/x\le\min(z,A-z)$, Gronwall's inequality gives
$\psi(A)/A\le\exp(\int_0^A\min(z,A-z)P(z)\,dz)$. The contribution
of $V$ to this integral is at most $M$ by~\eqref{eq:cylinder-moment};
that of $\kappa^2$ is at most
$\kappa^2 S^2\int_0^\infty t e^{-2t}\,dt$, using $t=A-z$.
Thus $\psi(x)/\psi(A)\ge c x/A$. The integral bound is unchanged under
$x\mapsto A-x$, giving the other ordering of the boundary values.
For the outward passage integrate against $\sin(\pi T/3)$ on $[0,3]$.
This cutoff is positive on the observed times and its second derivative
is a multiple of itself. Writing $m(s)=\int_0^3\sin(\pi T/3)H(T,s)\,dT$,
two integrations by parts give
\[
\int_0^3\sin(\pi T/3)\,\partial_T^2 H(T,s)\,dT
=-(\pi/3)^2m(s)+(\pi/3)\bigl(H(0,s)+H(3,s)\bigr).
\]
It follows that $m''+s^{-1}m'\le(V+\pi^2/9)m$. Since $m(3)\ge0$,
the maximum principle and the radial estimate with $S=3$ give
$m(s)\ge c\,m(b)\log(3/s)/D$. In particular,
\begin{equation}\label{eq:cylinder-band}
\int_0^3\int_{3/4\le|\tau|\le5/4}H(T,|\tau|)\,d\tau\,dT
\ \ge\ c\,m(b)/D .
\end{equation}
On the fixed region $-2<T<4$, $1/2<|\tau|<3/2$, we have
$\Delta H\le MH$. The compact weak Harnack inequality there gives
$H(T,1)\ge c(M)m(b)/D$ for $|T|\le1$. To see the comparison directly,
add a real variable $v$: the function
$H(T,|\tau|)\cos(\sqrt{M+1}\,v)$ is nonnegative and superharmonic on a
fixed slab about $v=0$. The mean inequality on the four-dimensional ball of
radius $2\rho$ at $(x,0)$, integrated over $|v|<\rho$, gives on the
three-dimensional ball of radius $\rho$ about $x=(T,\tau)$ the bound $H(x)\ge c \rho^{-3}\int_{B_\rho(x)}H$ for one small
radius $\rho$ throughout a neighborhood of the compact connected band
$[-1,3]\times\{3/4\le|\tau|\le5/4\}$; here $H(x)=H(T,|\tau|)$.
Cover the band by finitely many balls of radius $\rho/4$. If two such
balls $B,B'$ overlap, then $B'\subset B_\rho(x)$ for every $x\in B$,
so $\inf_B H\ge c \rho^{-3}\int_{B'} H$. A finite chain gives the stated
lower bound from~\eqref{eq:cylinder-band}.
For the return passage take the radial solution with $S=1$,
$\kappa=\pi/2$, $u(b)=0$ and $u(1)=1$. On $|T|<1$, $b<|\tau|<1$,
the function $c(M)m(b)D^{-1}\cos(\pi T/2)u(|\tau|)$ solves
$(-\Delta_{T,\tau}+V)w=0$. It vanishes on $|T|=1$ and $|\tau|=b$,
and the preceding lower bound puts it below $H$ on $|\tau|=1$.
The maximum principle and the radial estimate for $u(r)$ therefore give
\[
H(0,r)\ge\frac{c(M)m(b)}D\,\frac{\log(r/b)}{\log(1/b)}
\qquad(b\le r\le1).
\]
This proves~\eqref{eq:cylinder-propagation}, since $\sin(\pi T/3)$ is
bounded below on $[1,2]$.
\end{proof}
\begin{remark}\label{rem:cylinder-power}
The dependence on $r$ and $b$ in Lemma~\ref{lem:cylinder} is sharp
for scalar supersolutions, even when $V=0$. Start with
$(1-\cos T)p(s)$, where $p(s)=\log(3/s)/D$ is the radial harmonic
function with boundary values $1,0$. To cancel the positive part of
its Laplacian without changing those boundary values, put
$g(s)=s^2(1+\log(3/s))/(4D)$. The radial harmonic interpolant
$L(s)=p(s)g(b)+(1-p(s))g(3)$ lies above $g$, since $\Delta_\tau g=p\ge0$.
Thus $E(T,s)=(1-\cos T)p(s)+L(s)-g(s)$ is bounded and nonnegative,
with $\Delta_{T,\tau}E=(\cos T-1)p\le0$. Its trace at $s=b$ is
$1-\cos T$, and since $g$ is increasing we have
\[
E(0,r)\le (1-p(r))g(3)=\frac94\frac{\log(r/b)}{D^2}.
\]
A fixed positive $r$ would cost one logarithm. The data control $r=2b$,
where a sharper estimate requires more information about the wave energy.
\end{remark}
\begin{proposition}\label{prop:periodic-cap}
On the spatial torus $\TT^n=\RR^n/(100\mathbb Z)^n$, let
$f$ have Fourier support in $c_0/h\le|\xi|\le C_0/h$, $0<h\le1$, and
in one cap of the fixed aperture from Proposition~\ref{prop:fixed-barrier}.
Then
\begin{equation}\label{eq:periodic-cap}
\left(\int_1^2\|\U(T)f\|_{L^{\pc}(\TT^n)}^{\pc}\,dT\right)^{1/\pc}
\ \le\ C\,[\log(2/h)]^{2/\pc}\|f\|_{L^{\pc}(\TT^n)} .
\end{equation}
The constant is uniform in $h$ and in the direction of the cap.
\end{proposition}
\begin{proof}
For $f\ne0$, first normalize $\|f\|_{L^2(\TT^n)}=1$.
Fix the constant $K$ in Proposition~\ref{prop:fixed-barrier}, put
$a=Kh$ and $b=2a$, and take $h$ small enough that $b<1/4$. Apply
Lemma~\ref{lem:cylinder} to the energy of that proposition with
$V(s)=Cs^{-2}e^{-cs/h}$. Its first bound in~\eqref{eq:cylinder-moment}
follows from
\[
\int_b^3 s\log(s/b)V(s)\,ds
\le C\int_1^\infty \frac{\log t}{t}e^{-2cKt}\,dt ,
\]
and $V$ is uniformly bounded when $1/2\le s\le3/2$. The two energy
bounds in Proposition~\ref{prop:fixed-barrier} are uniform on their
respective circles, so its circular mean satisfies
\[
H(0,2b)\le C_K a^2\|f\|_{\pc}^{\pc},\qquad
H(T,b)\ge c_K a^2\|\U(T)f\|_{\pc}^{\pc}.
\]
Take $r=2b$ in~\eqref{eq:cylinder-propagation}, substitute both bounds, cancel
$a^2$, and take $\pc$-th roots. Since $D=\log(3/(2Kh))\lesssim
\log(2/h)$ and $K$ is fixed, this gives~\eqref{eq:periodic-cap}.
Homogeneity removes the normalization after this comparison. For the
remaining $h$ the torus frequencies lie in a fixed ball, where the
estimate follows by a finite Fourier sum.
\end{proof}
==== END Local-smoothing-for-the-Euclidean-wave-equation//propagation.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//sharpness.tex ====
\section{Two obstructions to endpoint smoothing}\label{sec:sharpness}
A spherical wave that focuses at one time determines the necessary
Sobolev order. At the critical exponent the wave spends comparable
$L^{\pc}$ mass on every dyadic scale between wavelength and unit
distance from the focus. We use the radial test of Beltran, Roos, Rutar
and Seeger \cite[Section~3]{BRRS2025}, keeping the contributions from
all these scales.
\begin{proposition}[Radial focusing]\label{prop:sharpness}
Let $n\ge2$, $d=n-1$, and $I=[1,2]$. For every sufficiently large $N$
there is a nonzero radial Schwartz function $f_N$, with
$\supp\widehat f_N\subset\{5N/4<|\xi|<7N/4\}$, such that
\begin{equation}\label{eq:sharp-strong}
\frac{\|U(t)f_N\|_{L^p(\RR^n\times I)}}{\|f_N\|_{L^p(\RR^n)}}
\gtrsim_{n,p}
\begin{cases}
1,&2\le p<\pc,\\
(\log N)^{1/\pc},&p=\pc,\\
N^{d/2-n/p},&\pc<p<\infty .
\end{cases}
\end{equation}
\end{proposition}
\begin{proof}
Fix $t_0=3/2$, take a nonzero nonnegative
$\psi\in C_c^\infty((5/4,7/4))$, and set
$\widehat f_N(\xi)=e^{it_0|\xi|}\psi(|\xi|/N)$.
Plancherel gives $\|f_N\|_2\asymp N^{n/2}$. To bound $f_N$ pointwise,
write $\xi=N\sigma\omega$ in its inverse Fourier transform. If
$|x|<t_0/2$, then $t_0+x\cdot\omega\ge t_0/2$ and integration by parts
in $\sigma$ gives $O_m(N^{n-m})$ for every integer $m\ge0$.
If $|x|\ge t_0/2$, the Fourier transform of surface measure on the
unit sphere is $O((1+N\sigma|x|)^{-d/2})$. Thus
$\|f_N\|_\infty\lesssim N^{(n+1)/2}$, and
\begin{equation}\label{eq:sharp-input}
\|f_N\|_p\le\|f_N\|_2^{2/p}\|f_N\|_\infty^{1-2/p}
\lesssim N^{(n+1)/2-1/p}\qquad (2\le p<\infty).
\end{equation}
The wave at $t=t_0+u$ has phase $x\cdot\xi-u|\xi|$. Put $r=|x|$ and
$F(v)=\int \sigma^{d/2}\psi(\sigma)e^{iv\sigma}\dd\sigma$.
The two stationary points on the sphere give, uniformly for $Nr\ge1$,
\begin{multline}\label{eq:sharp-phases}
U(t_0+u)f_N(x)=c_n N^{(n+1)/2}r^{-d/2}
\Big[e^{-i\pi d/4}F\big(N(r-u)\big)\\
{}+e^{i\pi d/4}F\big(-N(r+u)\big)+O\big((Nr)^{-1}\big)\Big],
\end{multline}
where $c_n>0$. Indeed the leading terms of the angular integral have
phases $\pm N\sigma r$ and amplitudes
$(2\pi)^{d/2}(N\sigma r)^{-d/2}e^{\mp i\pi d/4}$; the remainder is
$O((N\sigma r)^{-d/2-1})$ \cite[Chapter~VIII]{Stein1993}.
Since $F(0)>0$ and $F$ is Schwartz, there are fixed $\eta>0$ small
and $A$ large such that on the outgoing shell
\[
A/N\le u\le1/4,\qquad |r-u|\le\eta/N,
\]
the first term in brackets is bounded below in modulus and dominates
the other term and the error. The shell has spatial volume
$\asymp N^{-1}u^d$. Consequently \eqref{eq:sharp-input} gives
\[
\frac{\|U(t)f_N\|_{L^p(\RR^n\times I)}^p}{\|f_N\|_p^p}
\gtrsim \int_{A/N}^{1/4}u^{d-dp/2}\dd u .
\]
The exponent of $u$ is $-1$ precisely for $p=\pc$; evaluating the
integral gives all three cases of \eqref{eq:sharp-strong}.
\end{proof}
On this annulus $J^s=(1-\Delta)^{s/2}$ has operator norm $O_s(N^s)$
on the data, for every real $s$: after dilation and division by $N^s$,
a slightly enlarged smooth annular cutoff gives a multiplier with
uniformly integrable kernel. The same argument applies to
$[\log(2+J)]^\beta$, with bound $O_\beta((\log N)^\beta)$.
Thus a Sobolev estimate requires $s\ge\max\{0,d/2-n/p\}$, and at
$\pc$ a logarithmic estimate requires $\beta\ge1/\pc$.
The functions $f_N$ are radial, so the same lower power applies to
Theorem~\ref{thm:angular} with angular derivatives on the input.
For general data, a second example concentrates a fixed amount of
$L^2$ mass on a spacetime set of small measure. Even the weak
spacetime norm detects this concentration.
\begin{proposition}[Weak spacetime obstruction]\label{prop:weak-obstruction}
For $2<p\le\pc$ and every sufficiently large $N$, there is a nonzero
Schwartz function $g_N$ with
$\supp\widehat g_N\subset\{5N/4<|\xi|<7N/4\}$ such that
\begin{equation}\label{eq:sharp-weak}
\frac{\|U(t)g_N\|_{L^{p,\infty}(\RR^n\times I)}}{\|g_N\|_{L^p(\RR^n)}}
\gtrsim_{n,p}(\log N)^{d(1/2-1/p)}.
\end{equation}
\end{proposition}
The exponent in \eqref{eq:sharp-weak} equals $1/\pc$ at $\pc$
and is positive for every $2<p<\pc$. In particular Sobolev order zero
fails throughout $2<p\le\pc$. Both lower bounds apply to
$P_N$ in Theorem~\ref{thm:main} whenever $\varphi(\eta)=1$ for
$5/4\le|\eta|\le7/4$. If the strong critical upper bound held with
power $1/\pc$ for all data, interpolation with the $L^2$ energy
bound would give exactly the exponent in \eqref{eq:sharp-weak} for
$2<p<\pc$. Appendix~\ref{sec:weak} proves
Proposition~\ref{prop:weak-obstruction} by transporting disjoint packets
into overlapping tubes.
==== END Local-smoothing-for-the-Euclidean-wave-equation//sharpness.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//stein_square.tex ====
\section{Stein's square function}\label{sec:stein-square}
Stein introduced a square function to study the convergence of
Bochner--Riesz means \cite[\S18, (18.1)]{Stein1958Square}.
For the Euclidean means \(B_r^\alpha\), with multiplier
\((1-|\xi|^2/r^2)_+^\alpha\), it measures the change with radius:
\[
\mathcal G_\alpha f(x)=
\left(\int_0^\infty |r\partial_r B_r^\alpha f(x)|^2
\frac{\dd r}{r}\right)^{1/2}.
\]
Carbery proved the sharp bounds in the plane
\cite[pp.~409--416]{Carbery1983Square}. In dimensions \(n\ge3\),
Lee, Rogers and Seeger proved them for \(p\ge 2+4/n\)
\cite[Theorem~1.1]{LRS2012Square}. Gan, Jing and Wu reached
\(p\ge13/4\) in dimension three \cite[Theorem~1.1]{GJW2024Square},
and Gan, Oh and Wu further enlarged the range in dimensions \(n\ge4\)
\cite[Theorem~1.1]{GOW2025Square}. Here the wave estimate
\eqref{eq:cs-unit-wave} gives the full range \(2\le p<\infty\).
\begin{theorem}\label{thm:stein-square}
Let \(n\ge2\) and \(2\le p<\infty\). If
\[
\alpha>\max\left\{\tfrac12,\,n\left(\tfrac12-\tfrac1p\right)\right\},
\]
then \(\|\mathcal G_\alpha f\|_p\le C_{n,p,\alpha}\|f\|_p\) for
\(f\in\mathcal S(\RR^n)\), and \(\mathcal G_\alpha\) extends to \(L^p(\RR^n)\).
\end{theorem}
The threshold can be read off from one shell. At distance \(\delta\)
from \(|\xi|=s\), differentiating the mean gives a factor
\(\delta^{\alpha-1}\). The square norm in \(s\) gives back
\(\delta^{1/2}\), while the wave estimate at times of order
\(\delta^{-1}\) costs \(\delta^{-s_+(p)-\varepsilon}\). The resulting
powers sum if \(\alpha>1/2+s_+(p)\). We first make this calculation
on \(1\le s\le2\). We then combine the dyadic intervals of radii inside
the square norm, before taking the spatial \(L^p\) norm.
\begin{lemma}[One interval of radii]\label{lem:square-shell}
Let \(2<p<\infty\), \(\varepsilon>0\), \(\sigma=s_+(p)+\varepsilon\),
and \(H=L^2([1,2],\dd s)\). Fix \(A\in C_c^\infty(\RR\times(0,\infty))\)
and \(\varphi\in C_c^\infty(\RR)\). For \(0<\delta\le1\), let
\(T_\delta f(x,s)\) be the spatial Fourier multiplier
\[
A(s,|\xi|)\,\varphi\bigl((s-|\xi|)/\delta\bigr)
\]
applied to \(f\). Then
\[
\|T_\delta f\|_{L^p_x(H)}\lesssim \delta^{1/2-\sigma}\|f\|_p,
\qquad
\|T_\delta f\|_{L^2_x(H)}\lesssim \delta^{1/2}\|f\|_2,
\]
with constants independent of \(\delta\).
\end{lemma}
\begin{proof}
Temporarily omit \(A\), allow \(s\in\RR\), and denote the resulting
operator by \(V_\delta\). For \(g\) supported in a fixed compact
annulus in frequency, Plancherel in \(s\) gives, at each \(x\),
\[
\|V_\delta g(x,\cdot)\|_2
=(2\pi)^{-1/2}\delta
\|\widehat\varphi(\delta t) U(t)g(x)\|_{L^2_t}.
\]
On \(R\le |t|\le 2R\), \(R\ge1\), H\"older in \(t\) and the rescaled
wave estimate \eqref{eq:cs-unit-wave} give
\[
\|U(t)g\|_{L^p_xL^2_t}
\lesssim R^{1/2-1/p}R^{1/p+\sigma}\|g\|_p.
\]
On \(|t|\le1\) the annular wave kernels have uniformly bounded
\(L^1\) norms. For \(b>1/2+\sigma\), summing over \(R=1,2,4,\ldots\)
now gives
\[
\|V_\delta g\|_{L^p_x L^2_s(\RR)}
\lesssim \delta\left(1+\sum_R R^{1/2+\sigma}(1+\delta R)^{-b}\right)
\|g\|_p
\lesssim \delta^{1/2-\sigma}\|g\|_p .
\]
To restore \(A\), Fourier invert it in \(s\). The multipliers
\(\widehat A_1(\eta,|D|)\), where the hat denotes transformation in
the first argument, are supported in a fixed annulus and have \(L^p\)
operator norms \(O_b((1+|\eta|)^{-b})\) for every \(b>0\), by smoothness
and compact support. Integrating
\(e^{\mathrm is\eta}V_\delta(\widehat A_1(\eta,|D|)f)\) in \(\eta\)
with measure \(\dd\eta/(2\pi)\) proves the first bound. For the second,
spatial Plancherel suffices: at each frequency the multiplier is bounded
and is supported on an \(s\)-interval of length \(O(\delta)\).
\end{proof}
To combine the intervals of radii, we use Seeger's method for passing
from bounds on individual frequency annuli to a multiplier estimate
\cite[Theorem~1]{Seeger1988Dilates}. Jones, Seeger and Wright stated
a square function version for dilates of annular multipliers
\cite[p.~6737]{JSW2008Dilates}; Guo, Roos and Yung proved it
\cite[Proposition~4.2 and \S7]{GRY2020Dilates}. Here we allow a different
input at each scale and assume decay of the Hilbert-valued kernel and
its first derivative. On each test cube only \(O(1+\log(1/\delta))\)
scales need the \(L^p\) bound. H\"older over those scales gives the
logarithmic power \(1/2-1/p\), which the strict inequality in
Theorem~\ref{thm:stein-square} absorbs.
\begin{lemma}[Combining dilates]\label{lem:square-dilates}
Let \(H\) be a Hilbert space, \(2<p<\infty\), \(0<\delta\le1\), and
\(N=n+3\). Suppose \(K:\RR^n\to H\) is continuously differentiable and
\[
\|K(x)\|_H+\|\nabla K(x)\|_{H^n}
\le C_*\delta^{-N}(1+|x|)^{-N}.
\]
Write \(S^{(k)}f=2^{nk}K(2^k\cdot)*f\), and let \(B_q\) bound the norm
of \(K*\) from \(L^q\) to \(L^q(H)\), \(q=2,p\). For every finite
sequence of Schwartz functions \(f_k\),
\[
\left\|\left(\sum_k\|S^{(k)}f_k\|_H^2\right)^{1/2}\right\|_p
\lesssim_{n,p}
\left(B_2+C_*\delta+[1+\log(1/\delta)]^{1/2-1/p}B_p\right)
\left\|\left(\sum_k|f_k|^2\right)^{1/2}\right\|_p .
\]
\end{lemma}
\begin{proof}
Put \(b_k=\|S^{(k)}f_k\|_H\),
\(h=(\sum_k b_k^2)^{1/2}\), and \(W=(\sum_k|f_k|^2)^{1/2}\).
Let \(M\) denote the Hardy--Littlewood maximal operator over cubes and
\(L=\lceil(N+2)\log_2(2/\delta)\rceil\).
On a dyadic cube \(Q\) of side \(2^{-j}\) and center \(y\), the
terms with \(k\le j-L\) are nearly constant. We compare \(h\)
with \(c_Q=(\sum_{k\le j-L}b_k(y)^2)^{1/2}\).
For \(x,z\in Q\) and \(k\le j-L\), the derivative bound gives
\[
\|S^{(k)}f_k(x)-S^{(k)}f_k(y)\|_H
\lesssim C_*\delta^{-N} 2^{k-j}MW(z).
\]
Indeed \(2^k|x-z|\lesssim1\), and convolution with
\(2^{nk}(1+2^k|\cdot|)^{-N}\) is bounded by a constant times \(M\).
Summing the squares and taking their square root costs at most
\(C C_*\delta MW(z)\).
For \(k\ge j+L\), split \(f_k\) on \(5Q\) and its complement. The
local terms satisfy
\[
\frac1{|Q|}\int_Q\sum_{k\ge j+L}
\|S^{(k)}(f_k\mathbf1_{5Q})(x)\|_H^2\dd x
\lesssim B_2^2 M(W^2)(z),\qquad z\in Q,
\]
because dilation preserves the \(L^2\) operator norm. For the terms
outside \(5Q\), summing the kernel bound on successive doubled cubes
gives, for \(x,z\in Q\),
\[
\|S^{(k)}(f_k\mathbf1_{(5Q)^c})(x)\|_H
\lesssim C_*\delta^{-N}2^{-(N-n)(k-j)}MW(z).
\]
Their square sum again has square root at most \(C C_*\delta MW(z)\).
There remain fewer than \(2L\) indices. H\"older in this finite set
bounds \(\bigl(|Q|^{-1}\int_Q\sum_{|k-j|<L} b_k^2\bigr)^{1/2}\) by
\((2L)^{1/2-1/p}F(z)\), where
\(F=(\sum_k [M(b_k^2)]^{p/2})^{1/p}\). Since \(p/2>1\), the scalar
maximal theorem and the scale invariance of \(B_p\) give
\[
\|F\|_p^p\lesssim\sum_k\|b_k\|_p^p
\le B_p^p\sum_k\|f_k\|_p^p\le B_p^p\|W\|_p^p.
\]
By the triangle inequality for the square norm and Cauchy--Schwarz on
\(Q\), the mean of \(|h-c_Q|\) is thus at most a constant times
\(
C_*\delta MW(z)+B_2 M(W^2)(z)^{1/2}+L^{1/2-1/p}F(z).
\)
Replacing \(c_Q\) by the mean \(h_Q\) costs at most a factor of two.
The dyadic sharp function inequality
\cite[Chapter~IV, \S3.6, Corollary~1]{Stein1993}
and the maximal theorem on \(L^p\) and \(L^{p/2}\) now prove the
assertion. The hypothesis of that inequality at a smaller exponent holds:
$\|h\|_2^2=\sum_k\|S^{(k)}f_k\|_{L^2(H)}^2
\le B_2^2\sum_k\|f_k\|_2^2<\infty$, and $2<p$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:stein-square}]
For \(p=2\), spatial Plancherel and \(u=|\xi|^2/r^2\) give the identity
\[
\|\mathcal G_\alpha f\|_2^2
=2\alpha^2\int_0^1 u(1-u)^{2\alpha-2}\dd u\,\|f\|_2^2
=\frac{\alpha}{2\alpha-1}\|f\|_2^2.
\]
Suppose \(p>2\). Choose \(\varepsilon>0\) so that
\(\sigma=s_+(p)+\varepsilon<\alpha-1/2\), possible since
\(1/2+s_+(p)=\max\{1/2,n(1/2-1/p)\}\).
Write \(r=2^ks\), \(1\le s\le2\), and \(\rho=2^{-k}|\xi|\).
On this interval \(\dd r/r=\dd s/s\) is comparable to \(\dd s\).
The differentiated multiplier equals
\[
m(s,\rho)=2\alpha\rho^2s^{-2\alpha}(s+\rho)^{\alpha-1}
(s-\rho)^{\alpha-1}\mathbf1_{\{\rho<s\}} ,
\]
where the last power is taken only for \(\rho<s\).
Near the boundary \(\rho=s\), split \(s-\rho>0\) into dyadic intervals
of size \(\delta=2^{-j}\), \(j\ge0\). Each piece is
\(\delta^{\alpha-1}A(s,\rho)\varphi((s-\rho)/\delta)\):
the factors preceding \((s-\rho)^{\alpha-1}\) times a smooth boundary
cutoff give a fixed amplitude \(A\) supported away from \(\rho=0\);
the power on a unit shell is absorbed in
\(\varphi\in C_c^\infty((1/2,2))\).
For fixed \(\delta\), Lemma~\ref{lem:square-shell} bounds the one-interval
norms of \(T_\delta\). Its kernel and first spatial derivatives satisfy
the hypothesis of Lemma~\ref{lem:square-dilates} with \(C_*\) independent
of \(\delta\): \(N\) integrations by parts in frequency cost at most
\(\delta^{-N}\) on a fixed compact annulus, uniformly in \(s\).
Write \(T_\delta^{(k)}\) for the spatial dilation obtained by replacing
\(|\xi|\) by \(2^{-k}|\xi|\) in the symbol of \(T_\delta\), with
\(s\in[1,2]\) unchanged. Choose smooth annular projections \(P_k\),
dilates of one fixed multiplier, which equal one on this spatial
frequency support. Lemma~\ref{lem:square-dilates} applied to \(P_kf\)
and the Littlewood--Paley inequality bound this shell, on all dyadic
radius intervals together, by
\[
C\delta^{\alpha-1}
\left(\delta^{1/2}+\delta+
[1+\log(1/\delta)]^{1/2-1/p}\delta^{1/2-\sigma}\right)\|f\|_p .
\]
The sum over \(j\) converges because \(\alpha-1/2-\sigma>0\).
The multiplier left away from the boundary is smooth, supported in
\(\rho\le2\), and vanishes to order two at zero. Split it into annuli
\(\rho\asymp 2^\ell\), \(\ell\le2\). After extracting \(2^{2\ell}\)
and rescaling to the unit annulus, the multipliers have uniformly
bounded derivatives. Their kernels satisfy Lemma~\ref{lem:square-dilates}
at \(\delta=1\) with uniform constants (also for \(B_2,B_p\), by
the integrable kernel bound). For each \(\ell\) their dilates now have
frequency scale \(2^{k+\ell}\); the same Littlewood--Paley estimate
applies, and \(\sum_{\ell\le2}2^{2\ell}<\infty\).
These estimates apply first to finite sets of \(k\) and finite shell
sums. For Schwartz \(f\) the multipliers converge in the Fourier integral
for each fixed \(r\), by the local integrability of the boundary power.
Fatou's lemma and then monotone convergence give the full estimate.
The linear map \(f\mapsto (r\partial_r B_r^\alpha f)_{r>0}\) therefore
extends by density to \(L^p_x(L^2(\dd r/r))\), giving the asserted
extension of its pointwise norm.
\end{proof}
==== END Local-smoothing-for-the-Euclidean-wave-equation//stein_square.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//transfer.tex ====
\section{Passage to Euclidean space and Sobolev estimates}\label{sec:transfer}
The aperture in Proposition~\ref{prop:periodic-cap} is fixed. A finite
partition of the sphere therefore gives the critical estimate on the
whole annulus without increasing its logarithmic power. We first remove
the spatial period and then account for the cost of summing annuli.
\subsection{Removing the period}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Write $h=N^{-1}$ and $P_h=\varphi(hD)$, with $\varphi$ supported
in a fixed compact annulus. Choose a smooth finite partition
$\varphi=\sum_\nu\varphi_\nu$ subordinate to the admissible caps.
The Euclidean kernel of $\varphi_\nu(hD)$ is
$h^{-n}\check\varphi_\nu(\cdot/h)$, and periodization does not increase
its $L^1$ norm. Apply Proposition~\ref{prop:periodic-cap} to
$\varphi_\nu(hD)f$ on the same torus for every $\nu$, and sum. This gives
\begin{equation}\label{eq:periodic-critical}
\|\U(T)P_h f\|_{L^{\pc}(I\times\TT^n)}
\lesssim [\log(2/h)]^{2/\pc}\|f\|_{L^{\pc}(\TT^n)},\qquad 0<h\le1.
\end{equation}
The Euclidean convolution kernel of $\U(T)P_h$ is
\[
K_h(T,X)=(2\pi h)^{-n}
\int e^{\mathrm i(X\cdot\eta-T|\eta|)/h}\varphi(\eta)\dd\eta .
\]
For $T\in I$ and $|X|\ge3$ the gradient of the phase before division
by $h$ has length at least $|X|-2\ge|X|/3$. Integration by parts in
the gradient direction gains $h/|X|$ each time: the amplitude is
supported away from zero and all phase derivatives of order at least
two are bounded there. Using at least $M+n$ integrations gives
\begin{equation}\label{eq:off-cone-kernel}
|K_h(T,X)|\le C_M h^M(1+|X|)^{-M},
\qquad T\in I,\quad |X|\ge3 .
\end{equation}
Tile $\RR^n$ by unit cubes $Q$, and write $10Q$ for the concentric
cube of side ten. On $Q$ the contribution of $f\mathbf1_{(10Q)^c}$
is bounded pointwise by
$C_M h^M ((1+|\cdot|)^{-M}*|f|)(X)$, since $|X-Y|\ge9/2$ for
$X\in Q$ and $Y\notin10Q$. This single function bounds the far
contributions on every cube. For $M>n$, its $L^{\pc}(I\times\RR^n)$
norm is $O(h^M)\|f\|_{\pc}$ by Young's inequality.
Translate the center of $Q$ to the origin and periodize
$f_Q=f\mathbf1_{10Q}$ on the torus of side $100$. Its torus norm is
$\|f\|_{L^{\pc}(10Q)}$. The periodic wave kernel is the periodization
of $K_h$. On $Q$, its nonzero translates see only arguments
$X-Y+100k$ with $Y\in10Q$ and $k\in\mathbb Z^n\setminus\{0\}$.
Since $|X_j-Y_j|\le5.5$, these arguments have length at least
$94.5|k|$. Summing \eqref{eq:off-cone-kernel} over $k$ bounds the
difference between the two outputs there by $C_M h^M\|f_Q\|_1$.
As $10Q$ has fixed volume, \eqref{eq:periodic-critical} now gives
\[
\|\U(T)P_h f_Q\|_{L^{\pc}(I\times Q)}
\lesssim [\log(2/h)]^{2/\pc}\|f\|_{L^{\pc}(10Q)} .
\]
Take the $\pc$-th powers and sum over $Q$. The cubes $10Q$ have
bounded overlap. Combining this sum with the far contributions gives
the asserted Euclidean estimate, since $\log(2/h)\asymp\log(2+N)$
for $N\ge2$.
\end{proof}
\subsection{Interpolation and summation}
Plancherel bounds the norm of $\U(T)P_h$ from $L^2_X$ to $L^2_{T,X}$
by $\|\varphi\|_\infty$, since $|I|=1$. At $p=\infty$ its norm is
$O(h^{-d/2})$. To see this, partition the annulus into $O(h^{-d/2})$
caps of aperture $c\sqrt h$ with bounded overlap. In coordinates
$\eta=(\sqrt h\,z,\rho)$ about the axis of a cap, its cutoff is
supported in a fixed compact set with $\rho$ bounded below and has
uniformly bounded derivatives. Such cutoffs are obtained by dividing
rescaled angular bumps by their sum; the overlapping centers have
bounded offsets in these coordinates. Removing the linear phase
$T\rho/h$ leaves
\[
\frac{T}{h}\bigl(\sqrt{\rho^2+h|z|^2}-\rho\bigr)
=\frac{T|z|^2}{\sqrt{\rho^2+h|z|^2}+\rho}.
\]
The expression on the right has uniformly bounded derivatives on the
support. Integration by parts thus bounds the $L^1$ norm of each
rescaled kernel by a constant. The affine change of frequencies and
the spatial translation from the linear phase preserve that norm.
Summing over the caps gives the claimed bound.
Set $s_+(p)=\max\{0,d/2-n/p\}$ and
$\beta(p)=\min\{d(1-2/p),2/p\}$. Interpolating
Theorem~\ref{thm:main} with the two bounds above gives
\begin{equation}\label{eq:all-p-frequency}
\|\U(T)P_h f\|_{L^p(I\times\RR^n)}
\lesssim_p h^{-s_+(p)}[\log(2/h)]^{\beta(p)}\|f\|_{L^p(\RR^n)},
\qquad 2\le p\le\infty .
\end{equation}
Indeed, above $\pc$ the power of $h^{-1}$ is
$(d/2)(1-\pc/p)=d/2-n/p$. We use Riesz--Thorin interpolation here
and the spatial Littlewood--Paley and Mihlin theorems below; see
\cite[Chapters~1 and~6]{Grafakos2014}.
\begin{proof}[Proof of Theorem~\ref{thm:local-smoothing}]
The gain of an arbitrary $\varepsilon>0$ now allows an absolute
sum over dyadic annuli. Indeed, set $g=J^{s_+(p)+\varepsilon}f$.
At frequency $2^j$, $j\ge1$, \eqref{eq:all-p-frequency} bounds the
output by $C2^{-j\varepsilon}(1+j)^{\beta(p)}\|g\|_p$: a smooth
enlargement of the annulus times $2^{j(s_+(p)+\varepsilon)}
J^{-s_+(p)-\varepsilon}$ has uniformly integrable kernel.
The coefficients are summable. On the remaining low frequencies the
cutoff times $e^{-\mathrm iT|\xi|}$ obeys uniform Mihlin bounds for
$T\in I$. At $p=2$ use Plancherel without any loss.
\end{proof}
The same summation can be made more precise if one wants an estimate
at the exact power $s_+(p)$.
\begin{corollary}[Logarithmic Sobolev estimate]\label{cor:log-sobolev}
Let $2<p<\infty$, $\delta_p=1/2-1/p$, and $B_*(p)=\beta(p)+\delta_p$.
Write $J=(1-\Delta)^{1/2}$, $\ell_1(r)=\log(2+r)$ and
$\ell_2(r)=\log(2+\ell_1(r))$. Then
\begin{equation}\label{eq:log-sobolev-summed}
\|\U(T) f\|_{L^p(I\times\RR^n)}
\lesssim_{p,B,C} \|J^{s_+(p)}\ell_1(J)^B\ell_2(J)^C f\|_{L^p(\RR^n)}
\end{equation}
whenever $B>B_*(p)$ and $C\in\RR$, or $B=B_*(p)$ and $C>\delta_p$.
At $p=\pc$, $B_*(\pc)=1/2+1/\pc$ and $\delta_{\pc}=1/(2n)$.
At $p=2$ no logarithmic factor is needed.
\end{corollary}
\begin{proof}[Proof of Corollary~\ref{cor:log-sobolev}]
Let $P_j$ be the projections of a smooth dyadic partition of the high
frequencies, $|\xi|\asymp 2^j$, $j\ge1$, and $\widetilde P_j$ an
enlarged cutoff equal to one on the support of the symbol of $P_j$.
Write $s=s_+(p)$ and
$g=J^s\ell_1(J)^B\ell_2(J)^C f$. On the $j$-th annulus put
\[
A_j=2^{js}(1+j)^B[\log(2+j)]^C\,
\widetilde P_j J^{-s}\ell_1(J)^{-B}\ell_2(J)^{-C}.
\]
If $a_j$ is the symbol of $A_j$, then the functions $a_j(2^j\eta)$
have common compact annular support and uniformly bounded derivatives.
Bounded overlap
of the original annuli therefore gives uniform Mihlin bounds for
all signed sums of the $A_j$. Randomizing these signs and using
Khintchine's inequality gives, for $p\ge2$,
\[
\Big(\sum_j\|A_jg\|_p^p\Big)^{1/p}
\le \Big\|\Big(\sum_j |A_jg|^2\Big)^{1/2}\Big\|_p
\lesssim_p \|g\|_p .
\]
The waves $u_j(T)=\U(T)P_j f$ retain their spatial frequency supports.
The reverse Littlewood--Paley inequality at each $T$, followed by the
triangle inequality in $L^{p/2}(I\times\RR^n)$, gives
\[
\Big\|\sum_j u_j\Big\|_{L^p_{T,X}}
\lesssim_p \Big(\sum_j\|u_j\|_{L^p_{T,X}}^2\Big)^{1/2}.
\]
By \eqref{eq:all-p-frequency}, the $j$-th norm on the right is at most
$C w_j\|A_jg\|_p$, where
$w_j=(1+j)^{\beta(p)-B}[\log(2+j)]^{-C}$.
With $1/r=\delta_p$, H\"older's inequality bounds this sequence in
$\ell^2$ by $\|w\|_{\ell^r}(\sum_j\|A_jg\|_p^p)^{1/p}$. The weight
belongs to $\ell^r$ when $B>B_*(p)$. At $B=B_*(p)$ it belongs to
$\ell^r$ precisely when $\sum_{j\ge2} j^{-1}(\log j)^{-Cr}<\infty$,
that is, $C>\delta_p$.
For the remaining low-frequency cutoff $\chi_0$, the symbol
$e^{-\mathrm iT|\xi|}\chi_0(\xi)$ obeys the Mihlin derivative bounds
uniformly in $T\in I$ away from $\xi=0$. The same holds after
multiplication by the inverse of the Sobolev and logarithmic weights,
which are smooth and positive near zero. This proves
\eqref{eq:log-sobolev-summed}, first for finite frequency sums and
then by density. At $p=2$ use conservation of the $L^2$ norm.
\end{proof}
The condition on $B$ and $C$ records the cost of this frequency
summation. It need not be optimal for the wave operator: for
$n\ge4$ and $p>2+4/(n-3)$, Heo, Nazarov and Seeger proved
\eqref{eq:log-sobolev-summed} with $B=C=0$
\cite[Theorem~1.2]{HeoNazarovSeeger2011}.
==== END Local-smoothing-for-the-Euclidean-wave-equation//transfer.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//weak.tex ====
\section{Disjoint packets and the weak spacetime norm}\label{sec:weak}
We prove Proposition~\ref{prop:weak-obstruction}. The geometry is the
one used in Fefferman's ball multiplier example \cite{Fefferman1971};
Wolff explains its relation to local smoothing in the discussion of
(39) in \cite{Wolff1999Kakeya}. For the quantitative compression
factor $(\log(1/\delta))^{-(n-1)}$ with disjoint translates of unit
tubes, see also Fern\'andez-Delgado and de la Salle
\cite[Theorem~1.4]{FernandezDelgadoDeLaSalle2026}. We give the short
construction needed here. The tubes are disjoint on one longitudinal
interval. On a second interval, their union is small and the sum of
their individual volumes stays bounded below. A modulated packet travels from the
first part of its tube to the second.
\paragraph{An arrangement of tubes.}
Write $x=(X,Y)\in\RR\times\RR^d$, $d=n-1$. Let
$k=\lfloor\frac14\log_2 N\rfloor$, $\delta=2^{-k}$, and put
$\alpha=1/(4\sqrt d)$, $w=\alpha\delta/16$. For
$0\le j<2^k$, write $j\delta=\sum_{\ell=1}^k\varepsilon_\ell(j)2^{-\ell}$
in binary. Use the digits to choose the intercept of a line of slope
$s_j=\alpha j\delta$:
\[
a_\ell=1-\ell/k,\qquad
L_j(X)=\alpha\sum_{\ell=1}^k\varepsilon_\ell(j)2^{-\ell}(X-a_\ell).
\]
The $\ell$th binary increment changes the slope without moving the
line at $X=a_\ell$. Near this point its displacement is small.
At $X\le-1/2$ these lines are separated. Indeed, when one passes from
$j$ to $j+1$, one digit, say $m$, changes from zero to one, and the
later digits change from one to zero. Hence
\[
L_{j+1}(X)-L_j(X)
=\alpha\delta(X-a_m)
+\alpha\sum_{\ell>m}2^{-\ell}(a_\ell-a_m)
\le-\alpha\delta/2 .
\]
The intervals of radius $w$ around the $L_j(X)$ are therefore disjoint.
For $X$ in $J=[1/4,1/2]$, let $B_X$ be the union of intervals of
radius $w/2$ around the same $L_j(X)$. To see why this union is small,
freeze the first $m=\lfloor k(1-X)\rfloor$ digits. For $\ell>m$ we
have $0\le X-a_\ell\le(\ell-m)/k$. The tail of each line therefore
ranges over an interval of length at most
$\alpha\sum_{\ell>m}2^{-\ell}(\ell-m)/k\le2\alpha 2^{-m}/k$.
There are $2^m$ choices for the frozen digits, and $m\le3k/4$, so
\[
|B_X|\le 2\alpha/k+2^m w\lesssim\alpha/k.
\]
Taking an independent choice in each transverse coordinate produces
$\delta^{-d}$ lines, indexed by $\nu=(j_1,\ldots,j_d)$. Write
$L_\nu=(L_{j_1},\ldots,L_{j_d})$ and $s_\nu=(s_{j_1},\ldots,s_{j_d})$.
The output cores
\[
S_\nu=\{(X,Y):X\in J,\ |Y_i-L_{j_i}(X)|\le w/2,\ 1\le i\le d\}
\]
all lie in $E=\{(X,Y):X\in J,\ Y\in (B_X)^d\}$. We have proved
\begin{equation}\label{eq:weak-geometry}
|E|\lesssim_n k^{-d},\qquad
\sum_\nu |S_\nu|=\delta^{-d}|J|w^d\asymp_n1.
\end{equation}
\paragraph{Transport at frequency \(N\).}
Choose $0\le\chi,\theta\le1$ smooth, with $\chi$ supported in
$(-2,-1/2)$ and equal to one on $[-3/2,-3/4]$, and $\theta$ supported
in $(-1,1)$ and equal to one on $[-1/2,1/2]$. Define
\[
b_\nu(X,Y)=\chi(X)\prod_{i=1}^d
\theta\big((Y_i-L_{j_i}(X))/w\big),\qquad
v_\nu=(1,s_\nu)(1+|s_\nu|^2)^{-1/2}.
\]
The $b_\nu$ have disjoint supports and their total support volume is
$O_n(\delta^{-d}w^d)=O_n(1)$. Set $\lambda=3N/2$,
$f_\nu(x)=e^{i\lambda v_\nu\cdot x}b_\nu(x)$, and let
$Q_N=\beta(|D|/N)$, where $\beta\in C_c^\infty((5/4,7/4))$ is one
on a neighborhood of $3/2$. We claim that, uniformly in $\nu,x$ and
$1\le t\le2$,
\begin{equation}\label{eq:weak-packet}
U(t)Q_N f_\nu(x)=
e^{i\lambda(v_\nu\cdot x-t)}\,b_\nu(x-tv_\nu)+O_n(N^{-1}w^{-2}).
\end{equation}
To verify this, write $\xi=\lambda v_\nu+\eta$. On $|\eta|\le cN$ the
cutoff is one and $|\xi|=\lambda+v_\nu\cdot\eta+O(|\eta|^2/N)$.
The absolute Fourier transform of the bump is
\[
w^d|\widehat\chi(\eta_1+s_\nu\cdot\eta')|
\prod_{i=1}^d|\widehat\theta(w\eta'_i)|,
\qquad \eta=(\eta_1,\eta').
\]
The change of variables in these arguments and $|s_\nu|\le1/4$ give
$\int|\eta|^2|\widehat b_\nu(\eta)|\dd\eta\lesssim_n w^{-2}$.
Thus the integrated phase error is $O_n(N^{-1}w^{-2})$; the absolute
integral over $|\eta|>cN$ is $O_n(N^{-2}w^{-2})$.
This proves \eqref{eq:weak-packet}. Its error tends to zero since
$w\asymp_n\delta$ and $N\ge\delta^{-4}$.
Take $I_0=[3/2,7/4]$. Translation by $tv_\nu$ preserves each
transverse difference $Y_i-L_{j_i}(X)$. Moreover, for $X\in J$ and
$t\in I_0$, we have $-3/2\le X-t(v_\nu)_1\le-3/4$, since
$4/\sqrt{17}\le(v_\nu)_1\le1$. Thus $b_\nu(x-tv_\nu)=1$ on
$S_\nu\times I_0$ and $|U(t)Q_N f_\nu|\ge1/2$ there for large $N$.
\paragraph{Choosing the signs.}
For independent random signs $\epsilon_\nu$ put
$f=\sum_\nu\epsilon_\nu f_\nu$. The disjoint initial supports give
$\|f\|_p\lesssim_{n,p}1$ for every choice of signs. Independence and
\eqref{eq:weak-geometry} give
\[
\mathbb E\int_{E\times I_0}|U(t)Q_N f(x)|^2\dd x\dd t
=\sum_\nu\int_{E\times I_0}|U(t)Q_N f_\nu(x)|^2\dd x\dd t
\ge \tfrac14|I_0|\sum_\nu|S_\nu|\gtrsim_n 1.
\]
Fix signs attaining this lower bound and set $g_N=Q_N f$.
Its Fourier support is as stated, and $\|g_N\|_p\lesssim_{n,p}1$ by
the uniform $L^1$ kernel bound for $Q_N$. For $p>2$ and a measurable
set $H$ of finite measure, integration of the decreasing rearrangement
gives
\[
\int_H |u|^2\le \frac{p}{p-2}|H|^{1-2/p}\|u\|_{L^{p,\infty}}^2 .
\]
Taking $H=E\times I_0$ and using $|E|\lesssim_n k^{-d}$ now gives
$\|U(t)g_N\|_{L^{p,\infty}}\gtrsim_{n,p}k^{d(1/2-1/p)}$.
As $k\asymp\log N$, this proves Proposition~\ref{prop:weak-obstruction}.
In fact the inputs have bounded $L^2$ and $L^\infty$ norms separately,
so interpolation also gives a bounded Lorentz $L^{p,1}$ input norm.
These packets also test the angular weight in Theorem~\ref{thm:angular}.
At $p=\pc$ a weak bound with only $[\log(2+\Om)]^B$ on the input
requires $B\ge 1/p$. Indeed, before the radial projection the input
is supported in a fixed ball and is a sum of disjoint packets. Its
first and second spatial derivatives are bounded by $C N$ and $C N^2$,
respectively, since $w^{-1}\lesssim_n N^{1/4}$. The spherical Laplacian
is a sum of squares of infinitesimal rotations $x_i\partial_j-x_j\partial_i$.
It follows that $\|(1+\Om^2/N^2)g_N\|_p\lesssim1$, since the radial
projection commutes with rotations. For $B\ge0$ the spherical multiplier
$[\log(2+\Om)]^B/(1+\Om^2/N^2)$ has norm
$O_B((\log(2+N))^B)$ by the multiplier bound used in
\eqref{eq:ang-log-projection}; for $B<0$ the logarithmic weight itself is
bounded. The preceding lower bound forces $B\ge1/p$. Theorem~\ref{thm:angular}
gives sufficiency for $B>1/2+1/p$.
==== END Local-smoothing-for-the-Euclidean-wave-equation//weak.tex ====
==== BEGIN Local-smoothing-for-the-Euclidean-wave-equation//bibliography.bib ====
% References cited in this article.
@article{AhagEtAl2009,
author = {{\AA}hag, Per and Cegrell, Urban and Ko{\l}odziej, S{\l}awomir and Ph{\d{\^a}}m, H. H. and Zeriahi, Ahmed},
title = {Partial pluricomplex energy and integrability exponents of plurisubharmonic functions},
journal = {Advances in Mathematics},
volume = {222}, number = {6}, pages = {2036--2058}, year = {2009},
doi = {10.1016/j.aim.2009.07.002}
}
@article{Alesker2003LA,
author = {Alesker, Semyon},
title = {Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables},
journal = {Bulletin des Sciences Math\'ematiques},
volume = {127}, number = {1}, pages = {1--35}, year = {2003},
doi = {10.1016/S0007-4497(02)00004-0}
}
@article{Alesker2003MA,
author = {Alesker, Semyon},
title = {Quaternionic {Monge--Amp\`ere} equations},
journal = {Journal of Geometric Analysis},
volume = {13}, number = {2}, pages = {205--238}, year = {2003},
doi = {10.1007/BF02930695}
}
@article{Alesker2013Flat,
author = {Alesker, Semyon},
title = {Solvability of the quaternionic {Monge--Amp\`ere} equation on compact manifolds with a flat {hyperK\"ahler} metric},
journal = {Advances in Mathematics},
volume = {241}, pages = {192--219}, year = {2013},
doi = {10.1016/j.aim.2013.03.021}
}
@article{Aubin1984,
author = {Aubin, Thierry},
title = {R\'eduction du cas positif de l'\'equation de {Monge--Amp\`ere} sur les vari\'et\'es {k\"ahl\'eriennes} compactes \`a la d\'emonstration d'une in\'egalit\'e},
journal = {Journal of Functional Analysis},
volume = {57}, number = {2}, pages = {143--153}, year = {1984},
doi = {10.1016/0022-1236(84)90093-4}
}
@article{BBGZ2013,
author = {Berman, Robert J. and Boucksom, S\'ebastien and Guedj, Vincent and Zeriahi, Ahmed},
title = {A variational approach to complex {Monge--Amp\`ere} equations},
journal = {Publications Math\'ematiques de l'IH\'ES},
volume = {117}, pages = {179--245}, year = {2013},
doi = {10.1007/s10240-012-0046-6}
}
@article{BCSS1989Restriction,
author = {Beckner, William and Carbery, Anthony and Semmes, Stephen and Soria, Fernando},
title = {A note on restriction of the {Fourier} transform to spheres},
journal = {Bulletin of the London Mathematical Society},
volume = {21}, number = {4}, pages = {394--398}, year = {1989},
doi = {10.1112/blms/21.4.394}
}
@incollection{BHS2021,
author = {Beltran, David and Hickman, Jonathan and Sogge, Christopher D.},
title = {Sharp local smoothing estimates for {Fourier} integral operators},
booktitle = {Geometric Aspects of Harmonic Analysis},
editor = {Ciatti, Paolo and Martini, Alessio},
series = {Springer INdAM Series}, volume = {45},
publisher = {Springer}, address = {Cham}, pages = {29--105}, year = {2021},
doi = {10.1007/978-3-030-72058-2_2}
}
@article{BRRS2025,
author = {Beltran, David and Roos, Joris and Rutar, Alex and Seeger, Andreas},
title = {A fractal local smoothing problem for the wave equation},
journal = {Bulletin of the London Mathematical Society},
volume = {57}, number = {12}, year = {2025}, pages = {3667--3690},
doi = {10.1112/blms.70214}
}
@article{BaouendiTreves1982,
author = {Baouendi, M. S. and Trèves, F.},
title = {A microlocal version of {Bochner}'s tube theorem},
journal = {Indiana University Mathematics Journal},
volume = {31}, number = {6}, pages = {885--895}, year = {1982},
doi = {10.1512/iumj.1982.31.31060}
}
@article{BonamiClerc1973,
author = {Bonami, Aline and Clerc, Jean-Louis},
title = {Sommes de {Ces\`aro} et multiplicateurs des d{\'e}veloppements
en harmoniques sph{\'e}riques},
journal = {Transactions of the American Mathematical Society}, volume = {183},
pages = {223--263}, year = {1973}, doi = {10.1090/S0002-9947-1973-0338697-5}
}
@article{Bourgain1986Circles,
author = {Bourgain, Jean},
title = {Averages in the plane over convex curves and maximal operators},
journal = {Journal d'Analyse Math\'ematique}, volume = {47},
pages = {69--85}, year = {1986}, doi = {10.1007/BF02792533}
}
@article{Bourgain1991Kakeya,
author = {Bourgain, Jean},
title = {{Besicovitch} type maximal operators and applications to {Fourier} analysis},
journal = {Geometric and Functional Analysis},
volume = {1}, number = {2}, pages = {147--187}, year = {1991},
doi = {10.1007/BF01896376}
}
@article{BourgainDemeter2015,
author = {Bourgain, Jean and Demeter, Ciprian},
title = {The proof of the {$\ell^2$} decoupling conjecture},
journal = {Annals of Mathematics},
series = {Second Series}, volume = {182}, number = {1}, pages = {351--389}, year = {2015},
doi = {10.4007/annals.2015.182.1.9}
}
@article{CCS2002,
author = {Colzani, Leonardo and Cominardi, Alice and Stempak, Krzysztof},
title = {Radial solutions to the wave equation},
journal = {Annali di Matematica Pura ed Applicata}, volume = {181},
number = {1}, pages = {25--54}, year = {2002}, doi = {10.1007/s102310100025}
}
@article{CNS1984,
author = {Caffarelli, Luis and Nirenberg, Louis and Spruck, Joel},
title = {The {Dirichlet} problem for nonlinear second-order elliptic equations. {I}. {Monge--Amp\`ere} equation},
journal = {Communications on Pure and Applied Mathematics},
volume = {37}, number = {3}, pages = {369--402}, year = {1984},
doi = {10.1002/cpa.3160370306}
}
@article{CRV1988Summability,
author = {Carbery, Anthony and Rubio de Francia, Jos{\'e} L. and Vega, Luis},
title = {Almost everywhere summability of {Fourier} integrals},
journal = {Journal of the London Mathematical Society}, series = {Second Series},
volume = {38}, number = {3}, pages = {513--524}, year = {1988},
doi = {10.1112/jlms/s2-38.3.513}
}
@article{Carbery1983Square,
author = {Carbery, Anthony},
title = {The boundedness of the maximal {Bochner--Riesz} operator on
{$L^4(\mathbb R^2)$}},
journal = {Duke Mathematical Journal},
volume = {50}, number = {2}, pages = {409--416}, year = {1983},
doi = {10.1215/S0012-7094-83-05018-4}
}
@article{Cordoba1977,
author = {C{\'o}rdoba, Antonio},
title = {The {Kakeya} maximal function and the spherical summation multipliers},
journal = {American Journal of Mathematics},
volume = {99}, number = {1}, pages = {1--22}, year = {1977},
doi = {10.2307/2374006}
}
@misc{CushmanDemeterWu2026,
author = {Cushman, Adam and Demeter, Ciprian and Wu, Shukun},
title = {Diameter-free reverse inequalities and superorthogonality},
howpublished = {Preprint, arXiv:2609.13105}, year = {2026},
url = {https://arxiv.org/abs/2609.13105}, doi = {10.48550/arXiv.2609.13105}
}
@article{Davies1971,
author = {Davies, Roy O.},
title = {Some remarks on the {Kakeya} problem},
journal = {Proceedings of the Cambridge Philosophical Society},
volume = {69}, number = {3}, pages = {417--421}, year = {1971},
doi = {10.1017/S0305004100046867}
}
@article{Evans1982,
author = {Evans, Lawrence C.},
title = {Classical solutions of fully nonlinear, convex, second-order elliptic equations},
journal = {Communications on Pure and Applied Mathematics},
volume = {35}, number = {3}, pages = {333--363}, year = {1982},
doi = {10.1002/cpa.3160350303}
}
@article{Fefferman1971,
author = {Fefferman, Charles},
title = {The multiplier problem for the ball},
journal = {Annals of Mathematics},
series = {Second Series}, volume = {94}, number = {2}, pages = {330--336}, year = {1971},
doi = {10.2307/1970864}
}
@misc{FernandezDelgadoDeLaSalle2026,
author = {Fern{\'a}ndez-Delgado, Rafael J. and de la Salle, Mikael},
title = {A construction of {Kakeya} sets in arbitrary dimension},
year = {2026}, howpublished = {Preprint, \href{https://arxiv.org/abs/2607.14824v3}{arXiv:2607.14824v3}},
doi = {10.48550/arXiv.2607.14824}
}
@article{Friedrichs1954,
author = {Friedrichs, Kurt O.},
title = {Symmetric hyperbolic linear differential equations},
journal = {Communications on Pure and Applied Mathematics},
volume = {7}, number = {2}, pages = {345--392}, year = {1954},
doi = {10.1002/cpa.3160070206}
}
@misc{GHLW2026,
author = {Gan, Shengwen and He, Danqing and Li, Xiaochun and Wu, Shukun},
title = {On local smoothing estimates for wave equations},
year = {2026},
howpublished = {Preprint, arXiv:2502.05973v2},
note = {Revised January 4, 2026},
url = {https://arxiv.org/abs/2502.05973v2},
doi = {10.48550/arXiv.2502.05973}
}
@incollection{GJW2024Square,
author = {Gan, Shengwen and Jing, Yifan and Wu, Shukun},
title = {New bounds for {Stein}'s square function in {$\mathbb R^3$}},
booktitle = {Recent Developments in Harmonic Analysis and its Applications},
series = {Contemporary Mathematics}, volume = {792}, pages = {19--65},
publisher = {American Mathematical Society}, address = {Providence, RI},
year = {2024}, doi = {10.1090/conm/792/15893}
}
@article{GLY2009,
author = {Grafakos, Loukas and Liu, Liguang and Yang, Dachun},
title = {Vector-valued singular integrals and maximal functions on spaces of homogeneous type},
journal = {Mathematica Scandinavica}, volume = {104}, number = {2},
year = {2009}, pages = {296--310}, doi = {10.7146/math.scand.a-15099}
}
@article{GOW2025Square,
author = {Gan, Shengwen and Oh, Changkeun and Wu, Shukun},
title = {New bounds for {Stein}'s square functions in higher dimensions},
journal = {Advances in Mathematics}, volume = {475}, pages = {110342},
year = {2025}, doi = {10.1016/j.aim.2025.110342}
}
@article{GRY2020Dilates,
author = {Guo, Shaoming and Roos, Joris and Yung, Po-Lam},
title = {Sharp variation-norm estimates for oscillatory integrals related
to {Carleson}'s theorem},
journal = {Analysis \& PDE}, volume = {13}, number = {5},
pages = {1457--1500}, year = {2020}, doi = {10.2140/apde.2020.13.1457}
}
@article{GWZ2020,
author = {Guth, Larry and Wang, Hong and Zhang, Ruixiang},
title = {A sharp square function estimate for the cone in {$\mathbb R^3$}},
journal = {Annals of Mathematics},
series = {Second Series}, volume = {192}, number = {2}, pages = {551--581}, year = {2020},
doi = {10.4007/annals.2020.192.2.6}
}
@article{GaoLiuMiaoXi2023,
author = {Gao, Chuanwei and Liu, Bochen and Miao, Changxing and Xi, Yakun},
title = {Improved local smoothing estimate for the wave equation in higher dimensions},
journal = {Journal of Functional Analysis}, volume = {284}, number = {9},
pages = {109879}, year = {2023}, doi = {10.1016/j.jfa.2023.109879}
}
@article{GaoLiuXi2025,
author = {Gao, Chuanwei and Liu, Diankun and Xi, Yakun},
title = {Curved {Kakeya} sets and {Nikodym} problems on manifolds},
journal = {Peking Mathematical Journal}, year = {2026},
note = {Published online February 2, 2026},
doi = {10.1007/s42543-025-00114-1}
}
@book{GilbargTrudinger,
author = {Gilbarg, David and Trudinger, Neil S.},
title = {Elliptic Partial Differential Equations of Second Order},
edition = {Second}, series = {Classics in Mathematics},
publisher = {Springer}, address = {Berlin}, year = {2001},
note = {Reprint of the revised second edition; originally published in
Grundlehren der mathematischen Wissenschaften, vol. 224},
doi = {10.1007/978-3-642-61798-0}
}
@book{Grafakos2014,
author = {Grafakos, Loukas}, title = {Classical {Fourier} Analysis},
edition = {Third}, series = {Graduate Texts in Mathematics}, volume = {249},
publisher = {Springer}, year = {2014}, doi = {10.1007/978-1-4939-1194-3}
}
@misc{GuthWangZahl2026Kakeya,
author = {Guth, Larry and Wang, Hong and Zahl, Joshua},
title = {A streamlined proof of the {Kakeya} set conjecture in {$\mathbb R^3$}},
year = {2026},
howpublished = {Preprint, \href{https://arxiv.org/abs/2601.14411v2}{arXiv:2601.14411v2}},
note = {Version 2, October 11, 2026},
doi = {10.48550/arXiv.2601.14411}
}
@article{HeoNazarovSeeger2011,
author = {Heo, Yaryong and Nazarov, F{\"e}dor and Seeger, Andreas},
title = {Radial {Fourier} multipliers in high dimensions},
journal = {Acta Mathematica}, volume = {206}, number = {1},
pages = {55--92}, year = {2011}, doi = {10.1007/s11511-011-0059-x}
}
@article{Hormander1973FLp,
author={H{\"o}rmander, Lars}, title={Oscillatory integrals and multipliers on {$FL^p$}},
journal={Arkiv f{\"o}r Matematik}, volume={11}, pages={1--11}, year={1973},
doi={10.1007/BF02388505}
}
@article{Husimi1940,
author = {Husimi, K{\^o}di},
title = {Some formal properties of the density matrix},
journal = {Proceedings of the Physico-Mathematical Society of Japan, Third Series},
volume = {22}, number = {4}, pages = {264--314}, year = {1940},
doi = {10.11429/ppmsj1919.22.4_264}
}
@article{JSW2008Dilates,
author = {Jones, Roger L. and Seeger, Andreas and Wright, James},
title = {Strong variational and jump inequalities in harmonic analysis},
journal = {Transactions of the American Mathematical Society},
volume = {360}, number = {12}, pages = {6711--6742}, year = {2008},
doi = {10.1090/S0002-9947-08-04538-8}
}
@book{Kaiser1990,
author = {Kaiser, Gerald},
title = {Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis},
series = {North-Holland Mathematics Studies}, volume = {163},
publisher = {North-Holland}, address = {Amsterdam}, year = {1990},
url = {https://arxiv.org/abs/0910.0352}
}
@incollection{KaiserStreater1992,
author = {Kaiser, Gerald and Streater, Raymond F.},
title = {Windowed {Radon} transforms, analytic signals, and the wave equation},
booktitle = {Wavelets: A Tutorial in Theory and Applications},
editor = {Chui, Charles K.}, series = {Wavelet Analysis and Its Applications},
volume = {2}, pages = {399--441}, publisher = {Academic Press},
address = {Boston}, year = {1992}, url = {https://arxiv.org/abs/math-ph/0108010}
}
@book{Kato1995,
author={Kato, Tosio}, title={Perturbation Theory for Linear Operators},
edition={Second}, series={Classics in Mathematics}, publisher={Springer},
address={Berlin}, year={1995}, doi={10.1007/978-3-642-66282-9}}
@article{Keich1999,
author = {Keich, Uri},
title = {On {$L^p$} bounds for {Kakeya} maximal functions and the
{Minkowski} dimension in {$\mathbb{R}^2$}},
journal = {Bulletin of the London Mathematical Society},
volume = {31}, number = {2}, pages = {213--221}, year = {1999},
doi = {10.1112/S0024609398005372}
}
@article{KellerLasserOhsawa2016,
author = {Keller, Johannes and Lasser, Caroline and Ohsawa, Tomoki},
title = {A new phase space density for quantum expectations},
journal = {SIAM Journal on Mathematical Analysis},
volume = {48}, number = {1}, pages = {513--537}, year = {2016},
doi = {10.1137/15M1028388}
}
@misc{Kim2017Restriction,
author = {Kim, Jongchon},
title = {Some remarks on {Fourier} restriction estimates},
year = {2017},
howpublished = {Preprint, \href{https://arxiv.org/abs/1702.01231}{arXiv:1702.01231}},
doi = {10.48550/arXiv.1702.01231}
}
@article{Krylov1982,
author = {Krylov, Nikolai V.},
title = {Boundedly nonhomogeneous elliptic and parabolic equations},
journal = {Mathematics of the USSR-Izvestiya},
volume = {20}, number = {3}, pages = {459--492}, year = {1983},
note = {Russian original: Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487--523},
doi = {10.1070/IM1983v020n03ABEH001360}
}
@article{Krylov1983,
author = {Krylov, Nikolai V.},
title = {Boundedly nonhomogeneous elliptic and parabolic equations in a domain},
journal = {Mathematics of the USSR-Izvestiya},
volume = {22}, number = {1}, pages = {67--97}, year = {1984},
note = {Russian original: Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75--108},
doi = {10.1070/IM1984v022n01ABEH001434}
}
@article{LRS2012Square,
author = {Lee, Sanghyuk and Rogers, Keith M. and Seeger, Andreas},
title = {Improved bounds for {Stein}'s square functions},
journal = {Proceedings of the London Mathematical Society},
series = {Third Series}, volume = {104}, number = {6},
pages = {1198--1234}, year = {2012},
doi = {10.1112/plms/pdr067}
}
@article{LRS2013Schrodinger,
author = {Lee, Sanghyuk and Rogers, Keith M. and Seeger, Andreas},
title = {On space-time estimates for the {Schr\"odinger} operator},
journal = {Journal de Math\'ematiques Pures et Appliqu\'ees},
volume = {99}, number = {1}, pages = {62--85}, year = {2013},
doi = {10.1016/j.matpur.2012.06.003}
}
@article{LabaWolff2002,
author = {{\L}aba, Izabella and Wolff, Thomas},
title = {A local smoothing estimate in higher dimensions},
journal = {Journal d'Analyse Math\'ematique},
volume = {88}, pages = {149--171}, year = {2002},
doi = {10.1007/BF02786576}
}
@article{MSS1992,
author = {Mockenhaupt, Gerd and Seeger, Andreas and Sogge, Christopher D.},
title = {Wave front sets, local smoothing and {Bourgain}'s circular maximal theorem},
journal = {Annals of Mathematics},
series = {Second Series}, volume = {136}, number = {1}, pages = {207--218}, year = {1992},
doi = {10.2307/2946549}
}
@article{MSS1993,
author = {Mockenhaupt, Gerd and Seeger, Andreas and Sogge, Christopher D.},
title = {Local smoothing of {Fourier} integral operators and {Carleson--Sj\"olin} estimates},
journal = {Journal of the American Mathematical Society},
volume = {6}, number = {1}, pages = {65--130}, year = {1993},
doi = {10.1090/S0894-0347-1993-1168960-6}
}
@article{Mabuchi1986,
author = {Mabuchi, Toshiki},
title = {{$K$}-energy maps integrating {Futaki} invariants},
journal = {Tohoku Mathematical Journal},
series = {Second Series}, volume = {38}, number = {4},
pages = {575--593}, year = {1986},
doi = {10.2748/tmj/1178228410}
}
@article{MinicozziSogge1997,
author = {Minicozzi, II, William P. and Sogge, Christopher D.},
title = {Negative results for {Nikodym} maximal functions and related oscillatory integrals in curved space},
journal = {Mathematical Research Letters},
volume = {4}, number = {2--3}, pages = {221--237}, year = {1997},
doi = {10.4310/MRL.1997.v4.n2.a5}
}
@article{Miyachi1980,
author = {Miyachi, Akihiko},
title = {On some estimates for the wave equation in {$L^p$} and {$H^p$}},
journal = {Journal of the Faculty of Science, University of Tokyo, Section IA, Mathematics},
volume = {27}, number = {2}, pages = {331--354}, year = {1980}
}
@book{MoertersPeres2010,
author = {M{\"o}rters, Peter and Peres, Yuval},
title = {{Brownian} Motion},
series = {Cambridge Series in Statistical and Probabilistic Mathematics},
volume = {30}, publisher = {Cambridge University Press},
address = {Cambridge}, year = {2010},
doi = {10.1017/CBO9780511750489}
}
@article{MuellerSeeger1995,
author = {M{\"u}ller, Detlef and Seeger, Andreas},
title = {Inequalities for spherically symmetric solutions of the wave equation},
journal = {Mathematische Zeitschrift}, volume = {218},
year = {1995}, pages = {417--426}, doi = {10.1007/BF02571912}
}
@article{NPW2006,
author = {Narcowich, F. J. and Petrushev, P. and Ward, J. D.},
title = {Localized tight frames on spheres},
journal = {SIAM Journal on Mathematical Analysis}, volume = {38},
number = {2}, pages = {574--594}, year = {2006}, doi = {10.1137/040614359}
}
@misc{Nadjimzadah2026,
author = {Nadjimzadah, Arian},
title = {Lifting curved {Kakeya} sets to linear {Kakeya} sets},
year = {2026},
howpublished = {Preprint, \href{https://arxiv.org/abs/2609.06299v2}{arXiv:2609.06299v2}},
note = {Revised October 20, 2026},
doi = {10.48550/arXiv.2609.06299}
}
@article{Nicola2009,
author = {Nicola, Fabio},
title = {Slicing surfaces and the {Fourier} restriction conjecture},
journal = {Proceedings of the Edinburgh Mathematical Society},
series = {Second Series}, volume = {52}, number = {2},
pages = {515--527}, year = {2009},
doi = {10.1017/S0013091507000995}
}
@misc{OpenAI2026Kakeya4Hausdorff,
author = {{OpenAI}},
title = {{Every Four-Dimensional Kakeya Set Has Full Hausdorff Dimension}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf}{OAI:Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026}},
year = {2026}
}
@misc{OpenAI2026KakeyaThree,
author = {{OpenAI}},
title = {{The Kakeya maximal conjecture in three dimensions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf}{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026}},
year = {2026}
}
@misc{OpenAI2026SphereRestriction,
author = {{OpenAI}},
title = {{Elliptic capacity propagation and Fourier restriction to the sphere}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf}{OAI:Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026}},
year = {2026}
}
@article{Peral1980,
author = {Peral, Juan C.},
title = {{$L^p$} estimates for the wave equation},
journal = {Journal of Functional Analysis},
volume = {36}, number = {1}, pages = {114--145}, year = {1980},
doi = {10.1016/0022-1236(80)90110-X}
}
@article{Rogers2008Schrodinger,
author = {Rogers, Keith M.},
title = {A local smoothing estimate for the {Schr\"odinger} equation},
journal = {Advances in Mathematics},
volume = {219}, number = {6}, pages = {2105--2122}, year = {2008},
doi = {10.1016/j.aim.2008.08.008}
}
@article{SSS1991,
author = {Seeger, Andreas and Sogge, Christopher D. and Stein, Elias M.},
title = {Regularity properties of {Fourier} integral operators},
journal = {Annals of Mathematics},
series = {Second Series}, volume = {134}, number = {2}, pages = {231--251}, year = {1991},
doi = {10.2307/2944346}
}
@article{Seeger1988Dilates,
author = {Seeger, Andreas},
title = {Some inequalities for singular convolution operators in {$L^p$}-spaces},
journal = {Transactions of the American Mathematical Society},
volume = {308}, number = {1}, pages = {259--272}, year = {1988},
doi = {10.1090/S0002-9947-1988-0955772-3}
}
@article{Serre2018,
author = {Serre, Denis},
title = {Divergence-free positive symmetric tensors and fluid dynamics},
journal = {Annales de l'Institut Henri Poincar\'e C, Analyse non lin\'eaire},
volume = {35}, number = {5}, pages = {1209--1234}, year = {2018},
doi = {10.1016/j.anihpc.2017.11.002}
}
@article{Serre2019,
author = {Serre, Denis},
title = {Compensated integrability. {Applications} to the {Vlasov--Poisson} equation and other models in mathematical physics},
journal = {Journal de Math\'ematiques Pures et Appliqu\'ees},
volume = {127}, pages = {67--88}, year = {2019},
doi = {10.1016/j.matpur.2018.06.025}
}
@article{Serre2023Mixed,
author = {Serre, Denis},
title = {Mixed determinants, compensated integrability, and new a priori estimates in gas dynamics},
journal = {Quarterly of Applied Mathematics},
volume = {81}, number = {2}, pages = {281--295}, year = {2023},
doi = {10.1090/qam/1640}
}
@article{Sogge1991,
author = {Sogge, Christopher D.},
title = {Propagation of singularities and maximal functions in the plane},
journal = {Inventiones Mathematicae},
volume = {104}, number = {2}, pages = {349--376}, year = {1991},
doi = {10.1007/BF01245080}
}
@article{Stein1958Square,
author = {Stein, Elias M.},
title = {Localization and summability of multiple {Fourier} series},
journal = {Acta Mathematica},
volume = {100}, pages = {93--147}, year = {1958},
doi = {10.1007/BF02559603}
}
@book{Stein1993,
author = {Stein, Elias M.},
title = {Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals},
series = {Princeton Mathematical Series}, volume = {43},
publisher = {Princeton University Press}, address = {Princeton, NJ}, year = {1993},
note = {With the assistance of Timothy S. Murphy},
doi = {10.1515/9781400883929}
}
@article{Tao1999,
author = {Tao, Terence},
title = {The {Bochner--Riesz} conjecture implies the restriction conjecture},
journal = {Duke Mathematical Journal},
volume = {96}, number = {2}, pages = {363--375}, year = {1999},
doi = {10.1215/S0012-7094-99-09610-2}
}
@article{Wan2020,
author = {Wan, Dongrui},
title = {A variational approach to the quaternionic {Monge--Amp\`ere} equation},
journal = {Annali di Matematica Pura ed Applicata},
volume = {199}, number = {6}, pages = {2125--2150}, year = {2020},
doi = {10.1007/s10231-020-00960-z}
}
@misc{Wang2026v4,
author = {Wang, Xiangyu},
title = {Wave packet density in local smoothing: refinements and limitations},
howpublished = {Preprint, arXiv:2609.34675v4}, year = {2026},
note = {Revised October 16, 2026}, url = {https://arxiv.org/abs/2609.34675v4},
doi = {10.48550/arXiv.2609.34675}
}
@misc{WangZahl2025,
author = {Wang, Hong and Zahl, Joshua},
title = {Volume estimates for unions of convex sets, and the {Kakeya} set conjecture in three dimensions},
year = {2025},
howpublished = {Preprint, \href{https://arxiv.org/abs/2502.17655}{arXiv:2502.17655}},
doi = {10.48550/arXiv.2502.17655}
}
@incollection{Wolff1999Kakeya,
author = {Wolff, Thomas},
title = {Recent work connected with the {Kakeya} problem},
booktitle = {Prospects in Mathematics (Princeton, NJ, 1996)},
editor = {Rossi, Hugo},
publisher = {American Mathematical Society},
address = {Providence, RI}, year = {1999}, pages = {129--162}
}
@article{Wolff2000,
author = {Wolff, Thomas},
title = {Local smoothing type estimates on {$L^p$} for large {$p$}},
journal = {Geometric and Functional Analysis},
volume = {10}, number = {5}, pages = {1237--1288}, year = {2000},
doi = {10.1007/PL00001652}
}
@article{Zhu2016,
author = {Zhu, Jingyong},
title = {{Dirichlet} problem of quaternionic {Monge--Amp\`ere} equations},
journal = {Israel Journal of Mathematics},
volume = {214}, number = {2}, pages = {597--619}, year = {2016},
doi = {10.1007/s11856-016-1329-2}
}
==== END Local-smoothing-for-the-Euclidean-wave-equation//bibliography.bib ====