676a502
DIAGONAL FOURIER EXTENSION FOR POSITIVELY CURVED SURFACES
IN THREE DIMENSIONS
A complete, self-contained exposition.
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0. WHAT THIS IS, AND WHY IT MATTERS
-----------------------------------
0.1 The Fourier transform.
For a nice function f on R^n, the Fourier transform is
f^(xi) = integral over R^n of f(x) e^{-2 pi i x . xi} dx .
Here x . xi is the dot product, and e(t) always means exp(2 pi i t).
Knowing f^(xi) for every frequency xi is the same as knowing f: the
inverse transform rebuilds f from its frequencies.
The extension operator goes the other way. Fix a surface Sigma inside
R^n and a density f living on Sigma. Build a function on all of R^n by
E_Sigma f(x) = integral over Sigma of f(xi) e^{2 pi i x . xi} dsigma(xi),
where dsigma is the surface measure of Sigma. Think of E_Sigma f as the
inverse Fourier transform of the measure "f dsigma": we have prescribed the
frequency content only on the thin set Sigma and asked what wave that
produces in physical space.
0.2 Why the problem is hard.
If the frequencies of a function were free to fill a whole open set, the
inverse transform could be arbitrarily spiky. A surface is thin (it has
measure zero), so E_Sigma f has a lot of hidden rigidity. Measuring how
large E_Sigma f can be, in terms of the size of f on Sigma, is the
restriction/extension problem.
Curvature decides everything. A flat surface (a piece of a plane) lets
wave packets pile up coherently, and the L^p bounds fail for every p not
equal to 2. A curved surface sends different pieces of Sigma in different
directions, so the packets interfere less, and L^p bounds become possible.
"Positively curved" means the surface bends the same way in both
tangential directions at every point: a sphere, an ellipsoid, a patch of a
paraboloid.
0.3 What "diagonal" means.
The natural statement pairs an input exponent with an output exponent.
The diagonal case is when both exponents are the same p:
||E_Sigma f||_{L^p(R^3)} <= C ||f||_{L^p(Sigma)} .
The theorem below proves exactly this for every 3 < p < infinity. The
threshold p = 3 is sharp for this method; at p = 2 the statement would be
false for a curved surface unless we changed the exponents.
0.4 Notation used throughout.
R, Z, C the reals, the integers, the complex numbers.
e(t) exp(2 pi i t).
y = (y', y_3) a point of R^3 split into a planar part y' in R^2
and a height y_3 in R.
D phi, D^2 phi gradient and Hessian of a function phi on R^2.
c, C positive constants. Each occurrence may stand for a
different value, but the value depends only on the
fixed data: the surface, the finitely many charts, the
windows, and the frequency regions. When a constant is
allowed to depend on an extra small parameter, we say
so explicitly.
a ~ b a and b differ by a bounded factor (used rarely).
A <= B A is at most a constant times B.
We never need the exact value of any constant; we only ever need that a
constant is finite and does not secretly depend on the scale N or on the
small exponents that will tend to zero.
0.5 The main theorem.
THEOREM 0.1 (Diagonal extension).
Let Sigma be a compact smooth surface in R^3 that is positively curved,
possibly with smooth boundary. For every 3 < p < infinity there is a
finite constant C(Sigma,p) such that
||E_Sigma f||_{L^p(R^3)} <= C(Sigma,p) ||f||_{L^p(Sigma)} (0.1)
for every complex-valued f in L^p(Sigma).
The sphere, compact elliptic paraboloid patches, and general smooth
elliptic graphs are all included. The surface and the exponent are held
fixed while the estimate is made.
0.6 Two consequences we will prove.
(a) Free Schrodinger local smoothing. On the compact paraboloid we get,
for every 3 < p < infinity and every Sobolev order s > 2 - 6/p, a bound
||e^{it Delta} f||_{L^p(R^2 x [0,1])} <= C(p,s) ||f||_{W^{s,p}(R^2)}
for the free Schrodinger propagator e^{it Delta}. The threshold s = 2 - 6/p
is sharp (Rogers). This is Corollary 6.1.
(b) Bourgain-type oscillatory integrals. For phases of the form
x . y + psi(t;y) whose time derivative of the Hessian is a scalar multiple
of the Hessian itself, a change of output variables straightens the phase
to a fixed elliptic graph, and we get the sharp N^{-3/p} decay:
|| integral e^{i N phi(x,t;y)} a(x,t;y) f(y) dy ||_{L^p(R^3_{x,t})}
<= C N^{-3/p} ||f||_{L^p(R^2)} .
This is Corollary 6.2.
0.7 How the proof is organized.
The plan is a chain of reductions, each one a smaller problem than the last.
Step 1 (Section 1). Locally a surface is a graph eta -> (eta, phi(eta))
with D^2 phi positive definite. Cut the data into finitely many compactly
supported pieces. Everything reduces to one graph estimate:
||E_phi g||_p <= C ||g||_3 , E_phi g(y) = int g(xi) e(y'.xi + y_3 phi(xi)) dxi,
for rough data g in L^3 supported in a fixed compact set.
Step 2 (Section 2). Decompose E_phi g into wave packets (localized bumps
travelling along tubes). The packet propagation theorem (external input,
stated in 2.4) controls how packets move under the "analyze at a larger
scale" operation. This gives a cubic bound on the sampled packets in
terms of an elliptic capacity.
Step 3 (Section 3). Convert the capacity factor into geometry using the
positive tube estimate, which follows from the Kakeya maximal theorem
(second external input, stated in 3.2). A decomposition by capacity lets
us apply the packet theorem to an arbitrary coefficient array.
Step 4 (Section 4). Prove a joint estimate for a sparse family of distant
balls: they share one data norm on the right, uniformly in how far apart
they sit. The key is that two far-apart translations fail to oscillate
only on a small frequency set, which we throw away.
Step 5 (Section 5). Cover each superlevel set {|F| > mu} by a controlled
number of sparse families. Then integrate in mu (layer cake) to remove the
small spatial power loss and obtain the graph estimate.
Step 6 (Section 6). Glue the finitely many charts back together, keep all
Jacobians honest, and deduce Theorem 0.1 plus the two consequences.
Two inputs are taken from outside and are used as black boxes:
PACKET THEOREM (OpenAIPacket2026), quoted as Theorem 2.4 below.
KAKEYA MAXIMAL THEOREM (OpenAIKakeya2026), quoted as inequality (3.3).
We state both precisely in the forms we use them, and we check that the
hypotheses hold for the phases we feed in. Everything after those two
inputs is proved in this document.
0.8 Background (for orientation only; not used in the proof).
Tomas-Stein gives the L^2 -> L^4 bound for positively curved surfaces.
Bilinear and multilinear methods (Tao-Vargas-Vega, Wolff, Tao,
Bennett-Carbery-Tao) brought wave packets and transversality to the front.
Guth's polynomial partitioning proved the bounded-data estimate for
p > 13/4, and Shayya, Kim, Wang, and Wang-Wu pushed the diagonal problem to
p > 22/7 and beyond. The present argument treats general smooth elliptic
charts through the packet theorem's arbitrary-array formulation together
with an explicit continuation of the phase, which is what allows the
diagonal range to reach down to p = 3. The packet input controls
oscillatory coefficients; the Kakeya input controls positive tube density.
The two roles are genuinely different.
1. LOCAL GRAPHS AND FIXED PARAMETERS
------------------------------------
1.1 Graph charts.
Near any point of Sigma we choose tangent coordinates and a normal in
which Sigma is a graph over a planar domain with a positive definite
Hessian. At a boundary point we use the standard smooth-boundary
convention: a smooth parametrization up to the boundary extends to an
open neighborhood of the parameter half-disk, projection to the tangent
plane has invertible differential there, and the inverse function theorem
gives a graph on an open planar neighborhood whose restriction
parametrizes the original piece. Thus every point, boundary included, sits
in such a graph chart.
After shrinking a chart, Lemma 2.1 below supplies a fixed invertible affine
change of ambient frequency coordinates in which the graph's phase phi is
a smooth function on all of R^2 satisfying
c I <= D^2 phi <= C I , sup_{R^2} |D^j phi| < infinity (j >= 2). (1.1)
The elementary estimates use only (1.1); the packet estimate also uses the
explicit continuation form of Lemma 2.1. The constants c, C here are fixed
once the chart is fixed.
1.2 The local extension operator.
On such a graph write xi = (eta, phi(eta)) for eta in R^2,
dsigma = w(eta) deta with w(eta) = sqrt(1 + |D phi(eta)|^2). For a datum f
on the surface define the planar density g(eta) = f(eta, phi(eta)) w(eta).
Then
E_Sigma f(x) = int g(eta) e( x'.eta + x_3 phi(eta) ) deta ,
with x = (x', x_3). So it is enough to understand, for an arbitrary
smooth phase phi satisfying (1.1),
E_phi g(y) = int_{R^2} g(xi) e( y'.xi + y_3 phi(xi) ) dxi ,
y = (y', y_3) in R^2 x R. (1.2)
All constants from now on may depend on phi and on the fixed chart data,
but not on g.
1.3 Windows and reduction to a single piece.
Fix a smooth nonnegative window chi supported in a square centered at the
origin. For a > 0 and nu in a Z^2 set chi_a^nu(xi) = chi((xi - nu)/a).
Choose chi so that the translates tile smoothly in the square sense:
sum_{nu in a Z^2} ( chi_a^nu(xi) )^2 = 1 for all xi. (1.3)
Concretely: start with a compactly supported bump positive on [-1,1]^2
and divide it by the square root of the sum of the squares of its integer
translates; that denominator is smooth, positive, and periodic. Write
D_f >= 1 for a fixed side length of a square containing the support of chi.
Now use the partition (1.3) at scale a = 1. Write the planar datum as
g = sum_nu (g chi_1^nu) chi_1^nu . Only finitely many terms meet the
support of g, because g is compactly supported in the chart. Each term is
E_phi( (g chi_1^nu) chi_1^nu ) = E_phi( h chi_1^{nu_*} )
after renaming nu = nu_* and folding the extra window factor into h. Thus
every graph integral is a finite sum of pieces of the single shape
F(y) = E_phi( h chi_1^{nu_*} )(y), (1.4)
where nu_* in Z^2 is fixed, and h is in L^3(R^2), supported in a fixed
compact set Q_0. The job from now on is to prove
||E_phi(h chi_1^{nu_*})||_{L^p(R^3)} <= C ||h||_{L^3(R^2)}, 3 < p < infinity,
with C independent of h.
1.4 Fixed parameters.
All constants for a fixed setup may depend on phi, Q_0, chi, D_f, and nu_*.
These objects, and every label range appearing later, are chosen before
the scale N or the small loss exponents vary. In particular (1.1) gives,
on all of R^2,
( D phi(xi) - D phi(eta) ) . (xi - eta) >= c |xi - eta|^2 . (1.5)
The velocity map U_xi = - D phi(xi) is therefore bi-Lipschitz on every
fixed bounded region and bounded there. This is exactly the separation of
directions that will make the tubes in Section 3 almost disjoint.
1.5 What remains, in one sentence.
Sections 2 through 5 prove the graph estimate (1.4) for rough L^3 data; the
small spatial power loss that appears along the way is removed at the end of
Section 5 by a level-set covering. Section 6 puts the finitely many charts
back together, restoring every Jacobian, and proves Theorem 0.1 together
with Corollaries 6.1 and 6.2.
2. WAVE PACKETS, THE CONTINUATION LEMMA, AND THE PACKET THEOREM
---------------------------------------------------------------
2.0 The idea of a wave packet.
Look at E_phi g near a frequency xi_0. Locally the phase
y'.xi + y_3 phi(xi) looks, near xi_0, like a plane wave moving in a fixed
direction plus a quadratic remainder. A "wave packet at xi_0" is a bump
that lives near a tube: it oscillates at frequency the surface normal
at xi_0, travels along the line whose velocity is determined by the bend
of phi at xi_0, and its cross-section is narrow. Decomposing g into
packets turns the oscillatory integral into a discrete sum of bumps, and
the question becomes: how much can many tubes, weighted by the sizes of
the packets, pile up? That is exactly the geometry the capacity norm and
the Kakeya input will answer.
But we cannot even set up the packets until we know the phase belongs to a
specific class. We handle that first.
2.1 The continuation class: spherical graphs.
The packet theorem is stated only for phases obtained by a particular
construction. Start with a tiny piece of the unit sphere,
xi -> (xi, -sqrt(1 - |xi|^2)), and write
q_s(xi) = -sqrt(1 - |xi|^2) (the sphere, valid for |xi| < 1)
Q_s(xi) = -1 + (1/2)|xi|^2 (the tangent paraboloid at 0)
Cut off the difference between them. For a fixed small rho > 0 fix a
smooth zeta with
0 <= zeta <= 1, zeta = 1 on B(0,rho), supp zeta inside B(0,2rho),
and set
phi(xi) = Q_s(xi) + zeta(xi) ( q_s(xi) - Q_s(xi) ). (2.1)
The product is extended by zero outside B(0,2rho). No radiality of zeta is
required. A "continued spherical phase" is any phase of the shape (2.1)
with rho small enough that D^2 phi is uniformly positive definite. Its
higher derivatives are bounded because it equals the paraboloid Q_s away
from a compact set.
Why this exact class? Because that is the hypothesis the packet theorem is
proved under. Uniform convexity alone is NOT enough; the extra structure in
(2.1) -- that the perturbation is a cutoff of the sphere-to-paraboloid
difference -- is what the source uses.
2.2 Lemma (every elliptic graph can be inserted into the class).
LEMMA 2.1. Every germ of a smooth graph with positive definite Hessian can
be carried by an invertible affine transformation of R^3 to a graph
agreeing, on an open neighborhood, with a continued spherical phase of the
form (2.1). The phase, the neighborhood, and the affine transformation can
all be fixed independently of the packet scales N and of the exponents.
PROOF. Choose a smooth cutoff zeta_0 that equals 1 on B(0,rho), is
supported in B(0,2rho), and equals 1/2 identically on a neighborhood V of
a point xi_1 with rho < |xi_1| < 2rho. (Such a cutoff is a rescaling of a
fixed cutdown with an annular plateau.) Define
q(xi) = Q_s(xi) + zeta_0(xi) ( q_s(xi) - Q_s(xi) ).
On the support of the cutoff the remainder q_s - Q_s and its first two
derivatives are O(rho^4), O(rho^3), O(rho^2). Hence D^2 q = I + O(rho^2),
and for rho small enough D^2 q is positive definite everywhere.
Translate so the target base point sits at height h_0 near zero. Put
H = h_0''(0) > 0 and B = q''(xi_1) > 0 (both positive definite 2x2
matrices). Pick an invertible real matrix A with A^T H A = B; for example
A = H^{-1/2} B^{1/2}. Near xi_1 define
psi(xi) = h_0( A (xi - xi_1) ) + q(xi_1) - h_0(0)
+ ( D q(xi_1) - A^T D h_0(0) ) . (xi - xi_1).
Then psi and q share value, gradient, and Hessian at xi_1. This change of
graph is induced by an invertible affine map of ambient frequency space
(base part xi -> xi_1 + A^{-1}x, height part = addition of affine function
of the new base coordinates), so all restriction estimates transfer with
the same exponents.
Now glue psi to q near xi_1. Take a smooth bump theta supported in
B(0,2) equal to 1 on B(0,1), 0 <= theta <= 1, and for a small fixed
epsilon > 0 put
R_epsilon(xi) = theta( (xi - xi_1)/epsilon ) ( psi(xi) - q(xi) ).
Choose epsilon so the support lies in V, where the target graph is
defined. Taylor's theorem and the agreement of the two-jets give
|D^j R_epsilon|_inf <= C epsilon^{3-j}, j = 0, 1, 2.
Thus phi = q + R_epsilon stays uniformly convex for epsilon small, and it
equals psi on B(xi_1, epsilon). Away from the perturbation it equals Q_s,
so all second-and-higher derivatives are bounded.
Finally we must check that phi can be written in the form (2.1). The
function D(xi) = q_s(xi) - Q_s(xi) is strictly positive for 0 < |xi| < 1.
Shrink V so D >= d_0 > 0 there. The ratio R_epsilon / D, extended by zero
outside V, is smooth. Set
zeta~ = zeta_0 + R_epsilon / D.
On the perturbation support zeta_0 = 1/2 and |R_epsilon/D| <= C eps^3/d_0,
and we choose epsilon so this is at most 1/4. Then 0 <= zeta~ <= 1 with the
same inner/outer support as zeta_0, and
phi = Q_s + zeta~ ( q_s - Q_s ).
Every choice used only the fixed target graph. QED.
2.3 What the packet theorem is allowed to see.
From now on the phases to which we apply the packet theorem are chosen as
in Lemma 2.1. Although the inserted graph sits in the transition annulus
rho < |xi| < 2rho, the packet theorem allows arbitrary frequency labels in
a fixed bounded region. In particular it imposes NO restriction that those
labels lie inside B(0,rho): its single-step maps act on arbitrary finite
input arrays. So all windows meeting the inserted graph, and the terminal
windows we use later, sit inside one fixed bounded frequency region, and we
may pass the whole complex input array through the theorem. Constants may
depend on the chosen phase, including its higher-derivative bounds.
2.4 Scales and the packet expansion.
Fix a dyadic integer N >= 2, and set delta = N^{-1}. Write
U_nu = - D phi(nu) for the velocity (the negative gradient; the sign is
fixed so that tubes travel along the direction (U_nu, 1) divided by its
length, see 3.3). For a dyadic factor 1 <= m <= N put
theta = m delta , r = m^{-2} , w = r theta = delta / m . (2.2)
Theta is the angular half-width of a packet, r is the time-step (a fraction
of the unit time interval), and w is the spatial width of a packet. Partition
[0,1) into intervals I of length r; write t_I for the left endpoint. The
packet indices at this scale are
(nu, z, I), nu in theta Z^2, z in D_f^{-1} w Z^2 .
The analysis operator, which turns a function into its packet coefficients,
and the synthesis operator, which rebuilds the function from coefficients,
are
(P_{m,I} f)_{nu,z} = int f(xi) chi_theta^nu(xi)
e( N^2 ( z.xi + t_I phi(xi) ) ) dxi , (2.3)
S_{m,I} d (xi) = (D_f theta)^{-2} sum_{nu,z} d_{nu,z} chi_theta^nu(xi)
e( - N^2 ( z.xi + t_I phi(xi) ) ) . (2.4)
Synthesis is defined initially on finite arrays and extended to l^2.
2.5 Lemma (exact frame identities).
LEMMA 2.2. For every f in L^2(R^2),
S_{m,I} P_{m,I} f = f , (2.5)
|| P_{m,I} f ||_{l^2}^2 = (D_f theta)^2 ||f||_2^2 , (2.6)
|| S_{m,I} ||_{l^2 -> L^2} = (D_f theta)^{-1} . (2.7)
PROOF. Because N^2 w theta = 1, the vectors N^2 z run through the
full dual lattice (D_f theta)^{-1} Z^2. For fixed nu, (2.3) is the array
of Fourier coefficients of f chi_theta^nu e(N^2 t_I phi) on a square of
side D_f theta. Parseval and Fourier inversion give
sum_z |(P_{m,I} f)_{nu,z}|^2 = (D_f theta)^2 ||f chi_theta^nu||_2^2 ,
and after synthesis we get back f (chi_theta^nu)^2. Sum over nu and use
sum_nu (chi_theta^nu)^2 = 1. The norm (2.7) follows from
S_{m,I} = (D_f theta)^{-2} P_{m,I}^*. These identities hold first for
finite partial Fourier sums in L^2, then for their limits. QED.
2.6 Elliptic capacity and the atomic norm.
We will measure how a coefficient array piles up along tubes using an
ellipse test. For 0 < kappa < 1/10, let E be a centered closed ellipse (of
any orientation) with semiaxes L >= a > 0, and write
W_kappa(E) = L^{1+kappa} a^{1-kappa} (= L a (L/a)^kappa >= L a).
For a finite array c in one bin at the scale (theta, r, w) define
K_{theta,r}(c) = sup over ellipses E, translations u, v with L >= a >= theta of
( theta^2 / W_kappa(E) )
* sum_{ U_nu in u + E ; z in v + r E } |c_{nu,z}|^2 . (2.8)
The sup runs over both the translation u of the velocity test and the
independent translation v of the position test, and there is no upper
bound on the semiaxes. The "capacity" K measures density of the array:
a thin long ellipse captures many packets if they line up; the weight
theta^2/W_kappa(E) normalizes so that an isolated cell contributes about
the square of its coefficient. The gauge is
G_{theta,r}(c)^3 = ( w^2 sum_{nu,z} |c_{nu,z}|^2 ) * K_{theta,r}(c)^{1/2} . (2.9)
Then define an atomic norm on each finite index set:
A_{theta,r}(c) = inf { sum_{l=1}^{q} G_{theta,r}(a_l) :
|c| <= sum_{l=1}^{q} |a_l| , q < infinity } , (2.10)
A_m(c)^3 = r sum_I A_{theta,r}(c_I)^3 . (2.11)
The domination in (2.10) is coordinatewise: |c_{nu,z}| <= sum_l |(a_l)_{nu,z}|
for every (nu,z). Splitting c into a few atoms each of small capacity is the
whole idea: capacity is what the packet theorem controls, and a few atoms
lose only a logarithm.
2.7 Lemma (the atomic norm controls the sample sum).
LEMMA 2.3. A_{theta,r} is a norm on every finite coordinate space, is
monotone under absolute coordinatewise domination, and is contracted by
coordinate masks. Moreover
r w^2 sum_{nu,z,I} |c_{nu,z,I}|^3 <= A_m(c)^3 . (2.12)
PROOF. The disk of radius theta, with its two translations chosen
independently, contains any prescribed index in both tests. Its weight is
theta^2, so
K_{theta,r}(c) >= ||c||_inf^2 , and so
G_{theta,r}(c)^3 >= w^2 ||c||_2^2 ||c||_inf >= w^2 sum_{nu,z}|c_{nu,z}|^3 .
For any atomic domination of c, the triangle inequality in l^3 gives
w^{2/3} ||c||_3 <= sum_l G_{theta,r}(a_l).
Take the infimum and sum its cube over bins with weight r: that is
(2.12). A singleton atom has cost w^{2/3} |c_{nu,z}|, so the infimum is
finite; the displayed lower bound makes it positive away from zero.
Homogeneity, subadditivity, absolute monotonicity, and contraction under
masks follow directly from the definition. QED.
At the initial scale m = 1 there is a single bin [0,1), and we abbreviate
K_kappa(d) = K_{delta,1}(d) , m(d) = delta^2 sum_{nu,z} |d_{nu,z}|^2 , (2.13)
so that G_{delta,1}(d)^3 = m(d) K_kappa(d)^{1/2}. The quantity m(d) is a
mass; K_kappa(d) is the peak density of the array in the elliptical sense.
2.8 The packet theorem (external input).
The cited theorem moves a coefficient array one "big step": synthesize at
factor 1, analyze in every factor-m time bin, then apply any coordinate mask.
A single-step map T_m uses the full matrix between its input indices and its
retained output indices. The admissible complexity convention fixes a finite
B and a bounded frequency-label region, requires m <= N <= m^B, allows
position labels of size at most m^B, and permits arbitrary finite arrays on
those indices, with phase and window constants fixed throughout the class.
THEOREM 2.4 (Packet propagation; OpenAIPacket2026, Thm 6.4/Eq (6.8)).
For a fixed phase of the form (2.1), every fixed admissible complexity range,
0 < kappa < 1/10, and eta > 0, there is a finite constant C such that
A_m( T_m d )^3 <= C m^{10 kappa + eta} G_{delta,1}(d)^3 (2.14)
for every single-step map in the range and every finite complex input array
d. The source also permits A_{delta,1}(d)^3 on the right. The constant may
depend on kappa, eta, the complexity range, the phase, and the windows.
2.9 Specialization to the operators we use.
We want the cubic bound for actual samples of E_phi, not for an abstract map.
Suppose the initial labels nu in delta Z^2 and z in D_f^{-1} delta Z^2 lie
in fixed bounded sets. Define
S d(xi) = (D_f delta)^{-2} sum_{nu,z} d_{nu,z} chi_delta^nu(xi)
e( - N^2 z.xi ) , (2.15)
c_{nu',z',j} = int S d(xi) chi_1^{nu'}(xi)
e( N^2 z'.xi + j phi(xi) ) dxi , (2.16)
where nu' in Z^2, z' in D_f^{-1} N^{-2} Z^2, and j = 0,...,N^2 - 1. Retain
any output coordinate mask for which nu', z' lie in fixed bounded sets.
CLAIM. There is a finite P(kappa, eta) >= 1, depending on those fixed
sets, the phase, and the windows, such that for every dyadic N >= 2 and
0 < eta <= 1,
N^{-6} sum_{nu',z',j} |c_{nu',z',j}|^3
<= P(kappa, eta) N^{10 kappa + eta} m(d) K_kappa(d)^{1/2} . (2.17)
PROOF. Set m = N. The scales (2.2) become
input : theta = delta , r = 1 , w = delta ,
output: theta = 1 , r = N^{-2}, w = N^{-2} ,
and t_I = j / N^2, with r w^2 = N^{-6} at output. Thus the synthesis
(2.15) and analysis (2.16) are exactly the companion's operators with its
window constant L_f equal to our D_f. The initial gauge is exactly the
right side of (2.17), and Lemma 2.3 gives
N^{-6} sum_{nu',z',j} |c|^3 <= A_N(c)^3 .
All bounded label sets fit one fixed admissible complexity range for
large N (since m = N); the finitely many smaller dyadic scales are
absorbed by enlarging the constant. Applying (2.14) proves (2.17). The
argument uses arbitrary input arrays, so it also survives restricting
their support. QED.
Theorem 2.4 is the oscillatory input. Its source proves (2.14); we do not
assert any rate for how P(kappa, eta) depends on kappa and eta. That is why
the final quantitative bound will display P and M as factors rather than
committing to their size.
3. FROM PACKET CAPACITY TO WEIGHTED TUBES
------------------------------------------
3.1 Tubes and their mass.
A tube is what a wave packet looks like in space-time. For an initial array
d = (d_{nu,z}) define, for (X,t) in R^2 x R,
T_{nu,z} = { (X,t) : 0 <= t <= 1, |X - z - t U_nu| <= delta } , (3.1)
H_d(X,t) = sum_{nu,z} |d_{nu,z}|^2 * indicator of T_{nu,z} ,
b_nu(d) = sum_z |d_{nu,z}|^2 .
So T_{nu,z} starts at position z at time 0, at time 1 has moved to z + U_nu,
and has circular cross-section of radius delta the whole way. H_d is the
"occupation" of space-time by tubes weighted by the squared packet sizes.
Its total mass is
int H_d = pi delta^2 * sum_{nu,z} |d_{nu,z}|^2 = pi m(d) (3.2)
(because |T_{nu,z}| = pi delta^2).
We record the maximal operator used below precisely. For rho > 0, direction
omega in S^2, and center a in R^3,
K_rho g(omega) = sup_a (1 / (pi rho^2)) int_{ T_rho(a,omega) } |g(y)| dy ,
T_rho(a,omega) = { a + r omega + w : |r| <= 1/2, w perpendicular omega, |w| <= rho } ,
a circular cylinder of length one and volume pi rho^2.
3.2 The Kakeya maximal theorem (external input).
THEOREM (OpenAIKakeya2026, Thm 1.1). For every s > 0 there is a finite
constant M(s) >= 1 such that for every g in L^3(R^3),
|| K_rho g ||_{L^3(S^2)} <= M(s) rho^{-s} ||g||_{L^3(R^3)} ,
0 < rho < 1. (3.3)
Here S^2 carries its usual surface measure. We use only 0 < s <= 1.
(3.3) is the L^3 Kakeya maximal inequality in R^3 with a tiny power loss.
3.3 Lemma (weighted tube estimate).
LEMMA 3.1. There are fixed constants C and N_0 such that for every dyadic
N >= N_0, every initial array d, and every 0 < s <= 1,
int H_d^{3/2} <= C M(s)^{3/2} delta^{-3s/2} delta^2 sum_nu b_nu(d)^{3/2} . (3.4)
The constants C, N_0 depend only on the fixed geometric setup.
PROOF. Part 1: directions are separated. By (1.5) the velocity map is
bi-Lipschitz on the fixed frequency region, where it is bounded. The map
U -> (U,1)/sqrt(1+|U|^2) is bi-Lipschitz there: its inverse is
omega -> omega'/omega_3, whose derivative is bounded where omega_3 is
bounded away from zero. Hence the directions
omega_nu = (U_nu, 1)/sqrt(1+|U_nu|^2)
are separated by a fixed positive multiple of delta. Choose spherical caps
Omega_nu centered at omega_nu of radius c_1 delta, with c_1 > 0 fixed
small enough that the caps are disjoint. Their areas satisfy
c_2 delta^2 <= |Omega_nu| <= C_2 delta^2.
Part 2: each tube is a bounded union of test cylinders. The center segment
of T_{nu,z} has Euclidean length sqrt(1+|U_nu|^2), bounded. For each
omega in Omega_nu, the distance of that segment from the line through
(z,0) in direction omega is O(delta); adding the radius-delta cross-section
preserves this. Cover the projection of the tube onto that line by a
bounded number of unit intervals; each lifts to a cylinder T_{C_4 delta}(a,omega).
Hence T_{nu,z} is contained in at most C_3 cylinders T_{C_4 delta}(a,omega),
with C_3, C_4 >= 1 fixed. Therefore for every g in L^3(R^3),
int_{T_{nu,z}} |g| <= C delta^2 K_{C_4 delta} g(omega), for all omega in Omega_nu.
Part 3: summing with weight. Average over Omega_nu, multiply by
|d_{nu,z}|^2, and sum over nu, z:
int H_d |g| <= C sum_nu b_nu(d) int_{Omega_nu} K_{C_4 delta} g(omega) domega.
Put B_d = sum_nu b_nu(d) * indicator of Omega_nu. Disjointness of the caps
gives
||B_d||_{L^{3/2}(S^2)}
<= C ( delta^2 sum_nu b_nu(d)^{3/2} )^{2/3} .
Choose N_0 so C_4 delta < 1. Holder's inequality on S^2 together with
(3.3) yields
int H_d |g| <= C M(s) delta^{-s}
( delta^2 sum_nu b_nu(d)^{3/2} )^{2/3} ||g||_3 ,
using (C_4 delta)^{-s} <= delta^{-s}, so the constant is independent of s.
Dualize between L^{3/2} and L^3 (i.e. sup over ||g||_3 <= 1) and raise to
power 3/2: that is (3.4). The sum over positions costs nothing because
K_rho already takes a supremum over cylinder locations. QED.
3.4 From one ellipse test to tube mass.
Recall W_kappa(E) = L a (L/a)^kappa >= L a. Suppose a nonzero coordinate
subarray q of d is supported on U_nu in u + E and z in v + E. Then for
0 <= t <= 1, the horizontal support of H_q(.,t) lies in
v + t u + E + t E + B(0, delta) contained in v + t u + 3 E .
Indeed B(0, delta) sits in E, and convexity plus central symmetry give
E + tE = (1+t)E. So the space-time support of H_q has measure at most
9 pi L a. By (3.2) and Holder's inequality,
int H_q^{3/2} >= ( int H_q )^{3/2} / |supp H_q|^{1/2}
>= (pi/3) * m(q) * ( m(q) / W_kappa(E) )^{1/2} . (3.5)
This bound is uniform in the eccentricity L/a, in the independent
translations u, v, and in kappa > 0. It says: a single ellipse test that
captures mass m(q) forces a definite amount of L^{3/2} tube concentration.
3.5 Lemma (decomposition by capacity).
LEMMA 3.2. Let d be a nonzero finite initial array and n = #supp d. For
every 0 < kappa < 1/10 there is a decomposition into coordinate subarrays
with pairwise disjoint supports,
d = sum_{i=1}^{J} d^{(i)} , J <= C log(2 + n) , (3.6)
such that
sum_{i=1}^{J} m(d^{(i)}) K_kappa(d^{(i)})^{1/2} <= C int H_d^{3/2} . (3.7)
Both constants are absolute, i.e. independent of the coefficient sizes
and of kappa.
PROOF. Write D = max_{nu,z} |d_{nu,z}|^2 > 0. A radius-delta disk,
independently translated to any velocity and position, is an admissible
test of weight delta^2; on the other hand every test has
W_kappa(E) >= delta^2. Hence
D <= K_kappa(d) <= n D . (3.8)
Capacity is monotone under restriction to a subset of coordinates.
Put A = D / n^2. The cost of any remaining array of capacity at most A is
controlled by one largest coefficient. Indeed, if a subarray r has
K_kappa(r) <= A, then
m(r) K_kappa(r)^{1/2} <= delta^2 n D sqrt(D/n^2) = delta^2 D^{3/2} .
A single entry of squared modulus D contributes D * indicator of T_{nu,z}
to H_d, whence int H_d^{3/2} >= pi delta^2 D^{3/2}. So the cost of any
such remainder is bounded by pi^{-1} int H_d^{3/2}, independently of which
entries remain.
Start with r_1 = d. While K_kappa(r_j) > A, choose an ellipse test whose
density reaches at least half the supremum. Let q_j be all nonzero
coordinates of r_j selected by that test, E_j its ellipse, and set
rho_j = m(q_j) / W_kappa(E_j) >= (1/2) K_kappa(r_j) ,
r_{j+1} = r_j - q_j .
(Such a test exists even if the sup is not attained.) It captures at
least one nonzero entry, so the process stops after at most n removals.
Write r_* for the terminal remainder; then K_kappa(r_*) <= A (possibly
r_* = 0).
Now group. For each integer k >= 0 put lambda_k = 2^k A. Group all q_j for
which lambda_k <= K_kappa(r_j) < 2 lambda_k. Let d_k be the union array at
this level (omit empty levels). The array d_k is a coordinate restriction
of the remainder at the FIRST removal at that level, so K_kappa(d_k) <
2 lambda_k. Every bundle at the level has rho_j >= lambda_k/2, so
additivity of m gives
m(d_k) K_kappa(d_k)^{1/2}
<= sqrt(2 lambda_k) sum_{j at level k} m(q_j)
<= 2 sum_{j at level k} m(q_j) sqrt(rho_j) .
Apply (3.5) to each selected test. Since the q_j are disjoint coordinate
subarrays, their tube functions add to something bounded by H_d. For
nonnegative numbers, sum_j x_j^{3/2} <= (sum_j x_j)^{3/2}, so
sum_k m(d_k) K_kappa(d_k)^{1/2}
<= C sum_j int H_{q_j}^{3/2}
<= C int H_d^{3/2} . (3.9)
The stopping estimate applies to r_*, so adding its cost to (3.9) gives
(3.7). At every removal the capacity lies in (A, nD], by monotonicity and
(3.8); the ratio of endpoints is n^3, so at most 1 + floor(3 log_2 n)
dyadic levels are occupied. Including the nonzero remainder,
J <= C log(2 + n). The case n = 1 gives K_kappa(d) = D = A, so no removal
is made and the remainder alone proves the claim. No lower bound on the
smallest coefficient was used. QED.
3.6 Proposition (discrete critical estimate).
PROPOSITION 3.3. Let c be the terminal array from d by the map (2.15),
(2.16) of Theorem 2.4 with any fixed permitted output mask. For every
sufficiently large dyadic N, every 0 < kappa < 1/10, 0 < eta <= 1, and
0 < s <= 1, put gamma_0 = 10 kappa + eta + 3 s / 2. Then
sum_{nu',z',j} |c_{nu',z',j}|^3
<= C N^6 P(kappa, eta) M(s)^{3/2} N^{gamma_0} log^2(2+N)
delta^2 sum_nu b_nu(d)^{3/2} . (3.10)
The constant C and the threshold for N are independent of kappa, eta, s;
they depend only on the fixed phase, windows, and label bounds. All other
parameter dependence is in P(kappa, eta) and M(s).
PROOF. If d = 0 there is nothing to prove. Otherwise, use Lemma 3.2,
and let c^{(i)} be the terminal array of d^{(i)} with the same output mask.
The sampling map is linear, so c = sum_i c^{(i)}. The triangle inequality
in l^3, Holder for the finite sum over i, and Theorem 2.4 give
N^{-6} sum |c|^3
<= J^2 sum_i N^{-6} sum |c^{(i)}|^3
<= J^2 P(kappa, eta) N^{10 kappa + eta}
sum_i m(d^{(i)}) K_kappa(d^{(i)})^{1/2}
<= C log^2(2+n) P(kappa, eta) N^{10 kappa + eta} int H_d^{3/2} .
There are O(N^2) initial frequency labels and O(N^2) initial position
labels in the fixed bounded ranges, so n <= C N^4; hence
N^{-6} sum |c|^3
<= C P(kappa, eta) N^{10 kappa + eta} log^2(2+N) int H_d^{3/2} . (3.11)
Insert Lemma 3.1 and use delta^{-3s/2} = N^{3s/2} to obtain (3.10). Every
coordinate restriction used above obeys the same label bounds as d, so the
same packet constant P(kappa, eta) applies to every piece. QED.
The factor N^{gamma_0} is a small power loss that grows with N. It is the
only obstruction to a clean L^3 -> L^p bound, and removing it (Section 5) is
where the sparse-ball estimate (Section 4) is needed.
4. THE JOINT ESTIMATE ON SPARSE BALLS
--------------------------------------
4.0 Why we need a joint estimate.
Proposition 3.3 bounds the samples on one bounded region, at the price of
the small factor N^{gamma_0}. To remove that factor we will cover a
superlevel set {|F| > mu} by many balls and add up the estimates on each.
If we added the bound per ball naively, the N^{gamma_0} would get multiplied
by the number of balls and we would lose. What we need is a SINGLE bound on
the union of many distant balls, with one norm of the data on the right and
the N^{gamma_0} appearing only once.
The mechanism is almost-orthogonality. Two balls whose centers differ by a
vector (q,s_0) (in the space-time coordinates adapted to the packets) cause
their packet analyses to interact through the phase q.xi + s_0 phi(xi). If
the two centers are far apart, this phase oscillates rapidly in most
directions and the interaction is tiny. It fails to oscillate only on a
small set of frequencies, the "resonant set", where q + s_0 D phi(xi) is
nearly zero. Throw away those frequencies (they cost little in L^infinity),
and the balls decouple. This is Tao's epsilon-removal strategy, with
concrete estimates here.
4.1 The sparse-ball proposition.
PROPOSITION 4.1 (Sparse balls). There is a constant mu_0 in (0,1),
depending only on the fixed graph setup, with the following property.
Suppose h is supported in Q_0, ||h||_3 <= 1, and 0 < mu <= mu_0. Let B be a
finite family of balls of common radius R such that
#B <= mu^{-8} , R >= mu^{-100} , |x_B - x_B~| >= mu^{-40} R
for distinct centers B, B~ . (4.1)
Choose a dyadic integer N with 4 R <= N^2 < 16 R. For 0 < kappa < 1/10 and
0 < eta, s <= 1, put gamma_0 = 10 kappa + eta + 3 s / 2. Then
| { |F| > mu } intersect (union over B in B of B) |
<= C mu^{-3} P(kappa, eta) M(s)^{3/2} N^{gamma_0} log^2(2+N) . (4.2)
Here F = E_phi(h chi_1^{nu_*}) as in (1.4). The constant C and the
threshold mu_0 are independent of kappa, eta, s, of the centers, and of
their maximum separation.
4.2 Proof of Proposition 4.1, part 0: setup.
Write delta = N^{-1} and D = D_f. Every window chi_delta^nu meeting Q_0
lies in a fixed compact neighborhood Q_1 of Q_0 (since delta <= 1). Fix
G = sup_{Q_1} |D phi| , Z > 4 (2 + G) .
The initial spatial labels we use will satisfy |z| <= Z, and the terminal
labels |z'| <= 1. So all label bounds used in Proposition 3.3 are fixed,
independent of the small exponents. Throughout we shrink mu_0 if necessary,
only in terms of the fixed setup; in particular we may require N to exceed
any fixed lower threshold. The hypotheses (4.1) give
delta <= (1/2) mu^{50} , mu^{-8} <= N , H >= mu^{-40} R > (1/16) mu^{-40} N^2
(4.3)
whenever H is the separation of two distinct centers.
4.3 Part 1: removing resonant frequencies.
Set a_B = x_B - (0, 0, N^2/2). For a pair of distinct balls write
a_B - a_B~ = (q, s_0) in R^2 x R, with H = |(q, s_0)|. Define the
resonant set
Omega_{B,B~} = { xi in Q_0 : | q + s_0 D phi(xi) | <= H mu^{20} } . (4.4)
If Omega_{B,B~} is nonempty, then from H <= |q| + |s_0| we get
H <= H mu^{20} + (G + 1) |s_0| ,
so |s_0| >= H / (2(G+1)) for mu_0 small enough. Now take two points
xi, zeta in Omega_{B,B~}. By the velocity-monotonicity (1.5),
c |xi - zeta| <= ( | (q + s_0 D phi)(xi) | + | (q + s_0 D phi)(zeta) | ) / |s_0|
<= 2 H mu^{20} / |s_0| <= C mu^{20} .
Therefore each resonant set has area at most C mu^{40}, and the union
Omega over all pairs has area at most C mu^{24} (there are at most mu^{-16}
pairs, each of area mu^{40}: mu^{40} mu^{-16} = mu^{24}). Define
h_* = h * indicator of Q_0 \ Omega, and let F_* be the same extension
integral with h replaced by h_*. By Holder's inequality,
|| F - F_* ||_inf <= || chi_1^{nu_*} ||_inf * ||h||_3 * |Omega|^{2/3}
<= C mu^{16} <= mu / 2 . (4.5)
The specific powers 20, 40, 100 have plenty of slack for this discarded
mass and for the phase estimates below; their exact values do not matter.
4.4 Part 2: a joint bound for the initial coefficients.
Let
U = [0, D^{-1})^2 x [0, 1) , Gamma_nu = supp chi_delta^nu .
For u in U and B in B put
h_{B,u}(xi) = h_*(xi) e( (a_B + u) . (xi, phi(xi)) ) , (4.6)
d^{B,u}_{nu,z} = int h_{B,u}(xi) chi_delta^nu(xi) e( N^2 z . xi ) dxi ,
z in D^{-1} delta Z^2 ,
b_nu^{B,u} = sum_{|z| <= Z} |d^{B,u}_{nu,z}|^2 .
Only angular windows meeting Q_0 need be kept.
CLAIM:
sum_{B in B} b_nu^{B,u} <= C delta^2 int_{Gamma_nu} |h_*|^2 ,
uniformly in u. (4.7)
This is the crucial gain: instead of mu^{-8} separate bounds (one per ball),
the total over all balls is bounded by one ball's worth of mass. The reason
is that the modulations for different B are essentially orthogonal once the
resonant frequencies are removed.
Proof of (4.7). If h_* chi_delta^nu = 0 a.e. there is nothing to prove.
Otherwise choose xi_g in Gamma_nu intersect (Q_0 \ Omega). For this nu
define the operator on UNRESTRICTED L^2(R^2) by
(T_nu f)_{B,z} = int f(xi) chi_delta^nu(xi)
e( (a_B + u).(xi, phi(xi)) + N^2 z . xi ) dxi ,
|z| <= Z ,
and apply it to f = h_* * indicator of Gamma_nu. The point of enlarging the
domain is that the kernel of T_nu T_nu^* now has the SMOOTH amplitude
(chi_delta^nu)^2, with no indicator of the excised set.
Diagonal blocks. For a single ball block, Parseval on the square of side
D delta containing Gamma_nu gives
sum_{z in D^{-1} delta Z^2} |(T_nu f)_{B,z}|^2
= (D delta)^2 int |f(xi) chi_delta^nu(xi)|^2 dxi <= C delta^2 ||f||_2^2 .
(The full spatial sum is understood before restricting to |z| <= Z; its
Fourier frequencies N^2 z fill (D delta)^{-1} Z^2.) So each diagonal block
of T_nu T_nu^* has operator norm at most C delta^2.
Off-diagonal blocks. For distinct B, B~, an entry of the Gram operator is
int (chi_delta^nu(xi))^2 e( (q + N^2(z - z~)).xi + s_0 phi(xi) ) dxi .
Put xi = nu + delta v and denote the resulting phase by Psi(v). Let
v_0 = ( q + s_0 D phi(xi_g) ) / | q + s_0 D phi(xi_g) | , Lambda = delta H mu^{20} .
The direction v_0 is defined because xi_g avoids every resonant set. On
the whole support of the smooth amplitude chi(v)^2 we have the
first-derivative lower bound and the higher-derivative bounds
D_{v_0} Psi(v) >= delta ( H mu^{20} - C H delta - 2 Z N^2 ) >= Lambda / 2 , (4.8)
| partial^alpha Psi(v) | <= C_alpha H delta^2 <= C_alpha Lambda (|alpha| >= 2) ,
Lambda >= (1/16) N mu^{-20} . (4.9)
In (4.8) the two errors relative to H mu^{20} are at most C mu^{30} and
32 Z mu^{20} by (4.3); their sum is at most 1/2 after fixing mu_0. In
(4.9) we use the bounded higher derivatives of phi and delta mu^{-20} <=
(1/2) mu^{30}.
Integration by parts. Repeated integration by parts in the fixed direction
v_0 gives, for each fixed integer k >= 1,
| int (chi_delta^nu)^2 e( (q + N^2(z - z~)).xi + s_0 phi(xi) ) dxi |
<= C_k delta^2 Lambda^{-k} <= C_k delta^2 N^{-k} . (4.10)
To be explicit: apply the operator (2 pi i D_{v_0} Psi)^{-1} D_{v_0} to the
exponential k times. The lower bound (4.8) and the higher-derivative bounds
(4.9) show that each fixed directional derivative of (D_{v_0} Psi)^{-1} is
O_k(Lambda^{-1}); applying the adjoint k times to the fixed smooth bump
costs O_k(Lambda^{-k}). Notice no upper bound on the FIRST derivative is
needed, and the constants are uniform as H -> infinity because the lower
bound and the higher-derivative bounds carry the same factor H.
Operator norm. There are at most C_Z N^2 retained spatial labels in each
ball block, so every row and column of the off-diagonal block matrix has at
most C_Z mu^{-8} N^2 <= C_Z N^3 entries. Taking k = 4 in (4.10) and using
the row-and-column sum bound for the l^2 operator norm shows the
off-diagonal block has norm at most C delta^2. Together with the diagonal
block estimate, ||T_nu||_{2->2}^2 <= C delta^2. Applying to
h_* * indicator of Gamma_nu proves (4.7). QED claim.
Now sum. Windows have bounded overlap and |Gamma_nu| <= C delta^2, so
Holder's inequality gives exactly the power Proposition 3.3 needs:
delta^2 sum_{nu,B} ( b_nu^{B,u} )^{3/2}
<= delta^2 sum_nu ( sum_B b_nu^{B,u} )^{3/2}
<= C delta^5 sum_nu ( int_{Gamma_nu} |h_*|^2 )^{3/2}
<= C delta^5 sum_nu |Gamma_nu|^{1/2} int_{Gamma_nu} |h_*|^3
<= C delta^6 . (4.11)
4.5 Part 3: reconstruction and the discarded spatial tail.
Let d^{B,u} now denote the finite array in (4.6) restricted to |z| <= Z.
Apply the synthesis-and-sampling map (2.15)-(2.16) to it, retaining only
nu' = nu_*, |z'| <= 1, and 0 <= j < N^2. Write the resulting array as
c^{B,u}_{z',j}. We must justify, uniformly in B, u, z', j,
c^{B,u}_{z',j} = F_*( a_B + u + (N^2 z', j) ) + O(N^{-3}) . (4.12)
The only issue is that we threw away the initial spatial labels with
|z| > Z. To control the loss, let d^hat^{B,u} be the initial analysis
array with the spatial grid UNTRUNCATED. Lemma 2.2 at the initial scale
gives
(D delta)^{-2} sum_{nu,z} d^hat^{B,u}_{nu,z} chi_delta^nu(xi) e(-N^2 z.xi)
= h_{B,u}(xi) in L^2(R^2) , (4.13)
and the exact mass identity
sum_{nu,z} |d^hat^{B,u}_{nu,z}|^2 = (D delta)^2 ||h_{B,u}||_2^2
= (D delta)^2 ||h_*||_2^2 <= C delta^2 . (4.14)
(These need only h in L^2(Q_0), which follows from L^3 and compact
support. The potentially huge center modulation e((a_B+u).(xi,phi)) does
not affect the bound, being unimodular.)
A matrix entry from an omitted |z| > Z to a retained output is
A_{z',j; nu,z} = (D delta)^{-2} int chi_delta^nu(xi) chi_1^{nu_*}(xi)
e( N^2 (z' - z).xi + j phi(xi) ) dxi .
Substitute xi = nu + delta v. The prefactor is D^{-2}, the amplitude has
uniformly bounded smooth derivatives, and the derivative of the phase in
the direction -z/|z| is at least
N (|z| - 1 - G) >= c N (1 + |z|) .
All phase derivatives of order >= 2 are bounded uniformly because
j delta^2 <= 1. The same integration by parts as before gives
| A_{z',j; nu,z} | <= C_k [ N (1 + |z|) ]^{-k} . (4.15)
There are O(N^2) angular labels, and on the spatial mesh D^{-1} N^{-1} Z^2,
sum_{|z| > Z} (1 + |z|)^{-2k} <= C_k N^2 for k > 2 .
So each discarded part of a matrix row has squared l^2 norm at most
C_k N^{4 - 2k}. Its pairing with d^hat^{B,u} is, by (4.14), at most
C_k N^{1 - k}. Taking k = 4 proves (4.12). All infinite sums may be
justified by the L^2 convergence (4.13); the tail pairing is absolutely
convergent by Cauchy-Schwarz. Notice we never differentiated the data nor
the sharp boundary of the excised set.
4.6 Part 4: summing samples and filling the balls.
Apply Proposition 3.3 to each retained array and sum over B. The mass
bound (4.11) cancels the factor N^6 exactly and gives
sum_{B, z', j} | c^{B,u}_{z',j} |^3
<= C P(kappa, eta) M(s)^{3/2} N^{gamma_0} log^2(2+N) .
There are O(N^4) terminal spatial labels and N^2 times per ball, so the
total number of samples is O(mu^{-8} N^6) = O(N^7). The cubed errors from
(4.12) sum to at most C N^7 N^{-9} = C N^{-2}. Using |a + b|^3 <=
4 (|a|^3 + |b|^3) and P, M >= 1, we obtain
sum_{B, z', j} | F_*( a_B + u + (N^2 z', j) ) |^3
<= C P(kappa, eta) M(s)^{3/2} N^{gamma_0} log^2(2+N) , (4.16)
uniformly in u in U.
Integrate in the common offset u. For each ball this turns the sum into an
integral over the disjoint half-open cells
a_B + (N^2 z', j) + U , |z'| <= 1, 0 <= j < N^2 . (4.17)
These cells cover B. Indeed, for y in B set r = y - a_B; then
|r'| <= R <= N^2 / 4 , N^2 / 4 <= r_3 <= 3 N^2 / 4 .
The lower corner l' of the horizontal cell containing r' in D^{-1} Z^2
satisfies |l'| <= N^2/4 + sqrt(2)/D < N^2 for the fixed large N under
consideration. Hence z' = l'/N^2 is a retained label and j = floor(r_3) is a
retained time. The change of variables has Jacobian one and U has fixed
volume D^{-2}. So (4.16) implies
sum_{B in B} int_B |F_*(y)|^3 dy
<= C P(kappa, eta) M(s)^{3/2} N^{gamma_0} log^2(2+N) .
On the set {|F| > mu}, (4.5) gives |F_*| > mu/2. Chebyshev's inequality
now proves (4.2):
| { |F| > mu } intersect union_B B |
<= (2/mu)^3 sum_B int_B |F_*|^3
<= C mu^{-3} P(kappa, eta) M(s)^{3/2} N^{gamma_0} log^2(2+N) .
Every choice made in this proof used only fixed geometric constants and
fixed orders of differentiation. Thus mu_0 is independent of kappa, eta, s,
as asserted. QED Proposition 4.1.
5. GLOBAL BOUNDS: REMOVING THE SPATIAL LOSS
--------------------------------------------
5.0 The plan.
We now put the sparse-ball estimate to work. A superlevel set {|F| > mu}
spreads over unit cubes. Basic L^4 and gradient bounds control how many
cubes are occupied. Inside that finite point set, a slow-growth argument
selects ball radii so the points fall into a controlled number of sparse
families, each meeting the hypotheses of Proposition 4.1. Summing the sparse
bounds and integrating in mu (layer cake) removes the factor N^{gamma_0}.
Throughout, all constants except the displayed dependence on small
exponents depend only on the fixed setup. The elementary estimates use only
(1.1); the estimate for 3 < p <= 4 additionally uses Proposition 4.1 and
hence the packet input.
5.1 Lemma (basic graph estimates).
LEMMA 5.1. Let g in L^2(R^2) have compact support, and let
H(y',t) = int_{R^2} g(xi) e( y'.xi + t phi(xi) ) dxi .
If D^2 phi >= c I on R^2, then
|| H ||_{L^4(R^3)} <= (pi/c)^{1/4} ||g||_2 . (5.1)
In particular, for F = E_phi(h chi_1^{nu_*}) with h supported in Q_0, there
is a fixed C_b >= 1 with
||F||_4 + ||F||_inf + ||D F||_inf <= C_b ||h||_3 . (5.2)
PROOF. Think of H as the Fourier transform of a measure on the graph of
phi. More concretely, define the finite complex measure lambda on the graph
Gamma = {(xi, phi(xi)) : xi in R^2}, with density g in the base coordinate
xi. For a test function v in C_c^inf(R^3), Cauchy-Schwarz gives
| int v d(lambda * lambda) |
<= ||g||_2^2 ( int int | v( xi + xi', phi(xi) + phi(xi') ) |^2 dxi dxi' )^{1/2} .
Change variables m = xi + xi', w = (xi - xi')/2 (Jacobian one). Write
w = r omega, r > 0, omega in S^1, and set
P_{m,omega}(r) = phi(m/2 + r omega) + phi(m/2 - r omega) .
Uniform convexity implies
P_{m,omega}'(r) = int_{-r}^{r} omega^T D^2 phi(m/2 + s omega) omega ds >= 2 c r .
So P_{m,omega} is strictly increasing for r > 0, and the substitution
s = P_{m,omega}(r) gives
int int | v( xi + xi', phi(xi) + phi(xi') ) |^2 dxi dxi'
= int_{R^2} int_{S^1} int_0^inf | v( m, P_{m,omega}(r) ) |^2 r dr domega dm
<= (pi / c) ||v||_2^2 .
(The set r = 0 has measure zero.) Hence lambda * lambda has an L^2 density
of norm at most (pi/c)^{1/2} ||g||_2^2, by the L^2 representation theorem.
Its Fourier transform is H(-.)^2 in the sense of tempered distributions.
Plancherel gives (5.1). This reasoning applies to rough data: we never
assumed in advance that the convolution has a density.
For g = h chi_1^{nu_*}, compact support and Holder bound ||g||_2 by a
fixed multiple of ||h||_3. Also
||F||_inf <= ||g||_1 , ||D F||_inf <= 2 pi int |(xi, phi(xi))| |g(xi)| dxi ,
and differentiation under the integral is valid because the integrand is
supported in a fixed compact set. Another Holder inequality gives (5.2).
QED.
5.2 Occupied unit cubes.
Normalize ||h||_3 <= 1 for the rest of the section, and write
A_mu = { y in R^3 : |F(y)| > mu } , 0 < mu <= 1 .
The half-open cubes k + [0,1)^3, k in Z^3, partition R^3. Call a cube
OCCUPIED if it meets A_mu.
LEMMA 5.2 (Finite occupancy). There is a fixed C_X >= 1 such that at most
C_X mu^{-7} unit cubes are occupied. In particular, if mu <= C_X^{-1}, we can
choose one point of A_mu in each occupied cube to obtain a finite set X with
#X <= mu^{-8}.
PROOF. Take any finite collection Q of occupied cubes, and choose
x_Q in Q intersect A_mu for each. By (5.2), on the ball B(x_Q, r_mu),
where r_mu = mu / (2 C_b) <= 1/2, one has |F| > mu/2. These balls have
overlap at most 125: a point belongs to some ball only if its center lies
within distance one, and at most 5^3 unit lattice cubes contain such
centers (separation of the selected points is NOT required). Therefore
#Q * (pi / (96 C_b^3)) mu^7
<= sum_{Q in Q} int_{B(x_Q, r_mu)} |F|^4
<= 125 ||F||_4^4 <= 125 C_b^4 .
For example C_X = max{1, 12000 C_b^7 / pi} works. This holds for every
finite subcollection of occupied cubes. An infinite collection would have
finite subcollections of arbitrarily large cardinality, contradiction; so
the full collection is finite with the same bound. Finally
C_X mu^{-7} <= mu^{-8} when mu <= C_X^{-1}. QED.
5.3 Lemma (sparse covering, no diameter bound).
The finite-growth and coloring argument is the covering step of the
sparse-ball method. We keep explicit scale choices to track dependence on the
output exponent.
Fix once and for all a number mu_0 > 0 no larger than the threshold in
Proposition 4.1, than C_X^{-1}, and than 1/2. Then for 0 < mu <= mu_0,
mu^{-60} >= 8 , mu^{-100} > 2 . (5.3)
This choice is independent of all small exponents below.
LEMMA 5.3. Let 0 < tau <= 1, 0 < mu <= mu_0, and let X be the set of
Lemma 5.2. Set
beta = tau / 4 , J = ceil(32 / tau) + 1 , R_j = mu^{-100(j+1)}
for 0 <= j <= J . (5.4)
Then the union of the occupied cubes is covered by at most
3 J ( 1 + 8 log_2(1/mu) ) mu^{-beta} (5.5)
families of balls. Each family has at most mu^{-8} balls, all of radius
R = 4 R_j for some 0 <= j < J, and its centers are separated by at least
mu^{-40} R. For the dyadic N chosen in Proposition 4.1,
4 R <= N^2 < 16 R , N < 8 mu^{-50 J} . (5.6)
PROOF. If X is empty there is nothing to cover. Otherwise, using open
balls throughout, set
n_j(x) = # ( X intersect B(x, R_j) ) , x in X .
For each x there is an index 0 <= j < J with
n_{j+1}(x) <= mu^{-beta} n_j(x) . (5.7)
Indeed, failure at every index would imply
n_J(x) > mu^{-beta J} n_0(x) >= mu^{-beta J} > mu^{-8}, since beta J > 8,
contradicting #X <= mu^{-8}.
Assign each x one such index j, and the unique integer k >= 0 with
2^k <= n_j(x) < 2^{k+1}. There are at most J (1 + 8 log_2(1/mu)) nonempty
groups with a common pair (j, k).
In one group, take a maximal pairwise disjoint collection of the open balls
B(x, R_j); its centers form the set Y (finite). Every unselected ball
meets a selected one, so every center in the group lies within distance
2 R_j of a point of Y. A point of the unit cube containing that center is
within a further sqrt(3). Since R_j >= mu^{-100} >= 1, the balls
{ B(y, 4 R_j) : y in Y } cover all cubes assigned to the group.
Coloring. Make a finite graph on Y, joining distinct x, y when
|x - y| < R_{j+1}/2. For a fixed vertex x, the original balls B(y, R_j)
of all its neighbors are disjoint and lie in B(x, R_{j+1}) (by (5.3),
R_j < R_{j+1}/2). Each neighbor ball contains at least 2^k points of X,
whereas (5.7) gives n_{j+1}(x) < mu^{-beta} 2^{k+1}. Hence the degree of x
is less than 2 mu^{-beta}. A greedy coloring therefore uses at most
3 mu^{-beta} colors. Centers of the same color are separated by at least
R_{j+1}/2, and
R_{j+1} / 2 = (mu^{-100} / 2) R_j >= mu^{-40} (4 R_j)
by (5.3). So each color gives a family satisfying the separation
condition, with at most #X <= mu^{-8} balls and R = 4 R_j >= mu^{-100}.
Multiplying group and color counts proves (5.5). A dyadic N with
4 R <= N^2 < 16 R exists and satisfies N < 4 sqrt(R) = 8 mu^{-50(j+1)}
<= 8 mu^{-50 J}. The argument used neither the absolute locations of the
points nor the diameter of X: different clusters may be arbitrarily far
apart. Open balls resolve boundary cases (unselected balls meet selected
ones by maximality, and equality at the distance threshold keeps the weak
separation inequality). QED.
5.4 Theorem (global graph estimate).
THEOREM 5.4. Fix a phase and window setup for which Theorem 2.4 and
Proposition 4.1 apply. For 3 < p <= 4, put
tau = p - 3 , theta = 10^{-6} tau^2 . (5.8)
There is a constant C, independent of p in (3,4], such that
|| E_phi(h chi_1^{nu_*}) ||_p
<= C [ P(theta, theta) M(theta)^{3/2} tau^{-7} ]^{1/p} ||h||_3 (5.9)
for all h in L^3 supported in Q_0. For 4 < p < infinity, (5.2) alone gives
|| E_phi(h chi_1^{nu_*}) ||_p <= C_b ||h||_3.
PROOF. By homogeneity take ||h||_3 <= 1. In Proposition 4.1 choose
kappa = eta = s = theta. These lie in the permitted ranges, and
gamma_0 = 10 kappa + eta + (3/2) s = (25/2) theta .
For J in (5.4),
J <= 34 / tau , 50 J gamma_0 <= (17/800) tau < tau / 4 . (5.10)
Consequently every family from Lemma 5.3 has scale
N^{gamma_0} <= 8^{gamma_0} mu^{-tau/4} <= 2 mu^{-tau/4} ,
log(2 + N) <= 53 J ( 1 + log(1/mu) ) . (5.11)
Apply Proposition 4.1 to each family and sum over the cover. For
0 < mu <= mu_0,
|A_mu| <= C P(theta, theta) M(theta)^{3/2}
J^3 ( 1 + log(1/mu) )^3 mu^{-3 - tau/2} . (5.12)
Here the colors cost mu^{-beta} = mu^{-tau/4}, the scale factor costs at
most mu^{-tau/4}, the group count contributes one factor of
J (1 + log(1/mu)), and the log-squared in (4.2) contributes the other two.
All constants and mu_0 are independent of tau.
Layer cake. The identity, valid by Tonelli even before the norm is known
finite, is
||F||_p^p = p int_0^inf mu^{p-1} |A_mu| dmu .
For the portion below mu_0, (5.12) and p = 3 + tau reduce the needed
integral to
int_0^1 mu^{-1 + tau/2} ( 1 + log(1/mu) )^3 dmu
= 2/tau + 12/tau^2 + 48/tau^3 + 96/tau^4 <= 158 tau^{-4} . (5.13)
(The substitution u = log(1/mu) turns it into int_0^inf e^{-tau u/2}
(1+u)^3 du, then integrate each monomial.) Since p <= 4 and
J^3 <= 34^3 tau^{-3}, the small-level contribution is at most
C P(theta,theta) M(theta)^{3/2} tau^{-7}.
For the remaining levels, again by Tonelli,
p int_{mu_0}^inf mu^{p-1} |A_mu| dmu
<= int_{ {|F| > mu_0} } |F|^p
<= mu_0^{p-4} ||F||_4^4 <= mu_0^{-1} C_b^4 .
This is uniform for 3 < p <= 4 and is absorbed into the previous bound
because P, M >= 1 and tau <= 1. Take the pth root and restore homogeneity:
that proves (5.9). If p > 4, use directly
||F||_p^p <= ||F||_inf^{p-4} ||F||_4^4 <= C_b^p ||h||_3^p . QED.
The factor tau^{-7} records the elementary loss-removal argument explicitly.
The hypotheses give finite P(theta,theta) and M(theta), but no rate as
theta -> 0. So (5.9) asserts no polynomial upper bound in (p-3)^{-1} for the
full extension constant.
5.5 Proposition (the best constant must diverge at p = 3).
PROPOSITION 5.5. For every nonempty compact smooth surface Sigma in the
stated class there is c_Sigma > 0 such that the best constant in (0.1)
satisfies
C_{Sigma,p}^{best} >= c_Sigma (p-3)^{-1/p} , 3 < p <= 4 . (5.14)
This conclusion does not use the Kakeya or packet estimates.
PROOF. Choose a nonempty interior chart; a nonempty smooth surface with
boundary also has one. Write it in affine graph form
Psi(xi) = v_0 + L ( xi, phi(xi) ) , xi in Dom ,
with L invertible, and with surface measure dsigma = J_Sigma(xi) dxi on the
chart. Take a fixed nonzero g in C_c^inf(Dom) and put f(Psi(xi)) =
g(xi) / J_Sigma(xi) on the chart, f = 0 elsewhere. This is a smooth surface
datum with
B_f := sup_{3 <= p <= 4} ||f||_{L^p(Sigma)} < infinity .
Let
G(y', t) = int g(xi) e( y'.xi + t phi(xi) ) dxi , A = ||g||_2^2 > 0 .
Horizontal Plancherel and its derivative identity give, for t >= 0,
int_{R^2} |G(y', t)|^2 dy' = A ,
int_{R^2} |y'|^2 |G(y', t)|^2 dy'
= (1/(2 pi)^2) || D( g e(t phi) ) ||_2^2 <= B_g (1 + t)^2 ,
where B_g = 2 (2 pi)^{-2} ||D g||_2^2 + 2 ||g D phi||_2^2 > 0 works.
Put K = max{1, (2 B_g / A)^{1/2}}. The integral outside the disk
|y'| < K(1 + t) is at most B_g / K^2 <= A/2, so at least A/2 of the
squared norm lies inside that disk. Holder on the disk gives
int_{R^2} |G(y', t)|^p dy'
>= (A/2)^{p/2} (pi K^2)^{1 - p/2} (1 + t)^{2 - p}
>= c_g (1 + t)^{2 - p} . (5.15)
uniformly for 3 <= p <= 4, with the explicit positive
c_g = min{ (A/2)^{3/2} (pi K^2)^{-1/2} , (A/2)^2 (pi K^2)^{-1} } .
Integrating (5.15) over t >= 0 gives ||G||_p^p >= c_g / (p - 3).
The affine coordinates introduce exactly the factors
E_Sigma f(x) = e(x . v_0) G(L^T x) , ||E_Sigma f||_p^p = |det L|^{-1} ||G||_p^p .
So, setting d_g = |det L|^{-1} c_g > 0,
C_{Sigma,p}^{best}
>= B_f^{-1} d_g^{1/p} (p - 3)^{-1/p}
>= B_f^{-1} min{ d_g^{1/3}, d_g^{1/4} } (p - 3)^{-1/p} .
That proves (5.14), with every chart and datum fixed independently of p.
QED.
6. PASSAGE TO THE SURFACE AND THE TWO CONSEQUENCES
---------------------------------------------------
6.1 From the graph to Sigma.
We now deduce Theorem 0.1 from Theorem 5.4. Cover Sigma by finitely many
sufficiently small charts. By Lemma 2.1, after fixed affine normalizations
each chart lies in a graph whose phase is admissible for Theorem 2.4. The
construction and its constants are fixed per chart and independent of p. A
measurable partition subordinate to the cover avoids assuming the data
extends smoothly across chart boundaries.
Write one chart in ambient frequency coordinates as
xi = v_0 + L( zeta, phi(zeta) ) , zeta in A in R^2 ,
with L invertible and A bounded, possibly cut off by the boundary of Sigma.
Its surface measure is J_Sigma(zeta) d(zeta), J_Sigma smooth and strictly
positive on a neighborhood of the chart. For data supported on this piece,
set
g(zeta) = f( v_0 + L(zeta, phi(zeta)) ) J_Sigma(zeta) for zeta in A ,
g(zeta) = 0 for zeta not in A .
Then
E_Sigma f(x) = e(x . v_0) int g(zeta)
e( (L^T x)' . zeta + (L^T x)_3 phi(zeta) ) d(zeta) .
The change of physical variables y = L^T x contributes |det L|^{-1/p} to the
L^p norm. Moreover, on the chart,
||g||_3^3 <= (sup_A J_Sigma)^2 int_{Sigma_chart} |f|^3 dsigma ,
||f||_{L^3(Sigma_chart)}
<= sigma(Sigma_chart)^{1/3 - 1/p} ||f||_{L^p(Sigma_chart)} for p > 3 .
After the finite window partition of (1.3), Theorem 5.4 applies to every
resulting term. All determinant, area, window, and chart factors are bounded
uniformly for 3 < p <= 4. Summing the finitely many terms by the triangle
inequality gives (0.1) in this range, with the quantitative bound (5.9).
Here P_Sigma is the maximum of the finitely many packet constants, enlarged
to be at least one.
For p > 4, the local L^4 and L^infinity bounds give the result directly, as
in Theorem 5.4, and the same finite chart argument applies. No smoothness of
f, nor of its extension by zero on the parameter plane, was used. Boundary
charts are therefore included.
Proposition 5.5 supplies (5.14) for every nonempty surface by choosing a
nonzero smooth datum in an interior chart. This completes the proof of
Theorem 0.1 and the stated dependence on p.
6.2 Corollary (free Schrodinger local smoothing).
COROLLARY 6.1. For every 3 < p < infinity and s > 2 - 6/p, the free
Schrodinger propagator, initially on Schwartz functions, extends boundedly
from W^{s,p}(R^2) to L^p(R^2 x [0,1]):
|| e^{it Delta} f ||_{L^p(R^2 x [0,1])} <= C(p,s) ||f||_{W^{s,p}(R^2)} . (6.1)
PROOF. The extension estimate for a compact paraboloid controls all
frequencies of the propagator through a transfer theorem; then a strict
Sobolev inequality supplies summability across frequency scales.
On the compact paraboloid Sigma = {(y, |y|^2) : |y| <= 1}, write
dsigma = w(y) dy, w(y) = sqrt(1 + 4 |y|^2). Apply Theorem 0.1 to the datum
g(y)/w(y) on Sigma. A fixed change of output variables gives
|| int_{|y| <= 1} g(y) e^{ i( tau |y|^2 - xi . y ) } dy ||
_{L^p(R^3_{xi,tau})} <= C_p ||g||_{L^p({|y| <= 1})} .
This is the compact-frequency extension estimate of Lee-Rogers-Seeger with
both exponents equal to p. Their Theorem 1.1, with spatial dimension two and
Besov summation exponent two, yields
|| e^{it Delta} f ||_{L^p(R^2 x [0,1])} <= C_p ||f||_{B^p_{alpha,2}(R^2)} ,
alpha = 2 (1 - 2/p) - 2/p = 2 - 6/p .
For the inhomogeneous Littlewood-Paley partition (P_k)_{k >= 0}, the
uniform annular multiplier bound ||P_k f||_p <= C_s 2^{-ks}
||(I - Delta)^{s/2} f||_p gives
||f||_{B^p_{alpha,2}}
<= C_s ( sum_{k >= 0} 2^{-2k(s - alpha)} )^{1/2}
||f||_{W^{s,p}} .
The series converges for s > alpha. That proves (6.1) on Schwartz
functions; density gives the stated bounded extension. QED.
6.3 Corollary (translation-invariant Bourgain phases).
A second application concerns phases whose curvature changes only by a
scalar factor in time. Such a phase is straightened by changing the output
variables alone, reducing the estimate to one fixed elliptic graph. This is
the translation-invariant setting of Gao-Liu-Xi. We give the short argument
for the two-dimensional input here, then allow a general smooth amplitude by
Fourier expansion.
COROLLARY 6.2. Let
phi(x, t; y) = x . y + psi(t; y) , x, y in R^2 , t in R , psi(0;y) = 0 ,
be a fixed real smooth phase near the origin. Suppose throughout this
neighborhood that
M(t,y) = D_y^2 partial_t psi(t;y) is definite ,
partial_t M(t,y) = c(t,y) M(t,y)
for a smooth real scalar c. On a sufficiently small fixed product chart, let
a be any fixed smooth amplitude compactly supported in its interior. For
every 3 < p < infinity,
|| int e^{i N phi(x,t;y)} a(x,t;y) f(y) dy ||_{L^p(R^3_{x,t})}
<= C N^{-3/p} ||f||_{L^p(R^2)} , N >= 1 , (6.2)
where the operator is zero outside the output support of a. The constant may
depend on the fixed chart, phase, amplitude, and p, but not on N or f.
PROOF. Shrink the product neighborhood so its input domain is a ball.
Since M is a Hessian in y, commuting derivatives in partial_t M = c M gives
(partial_{y_k} c) M_{ij} = (partial_{y_i} c) M_{kj} .
Multiplying by (M^{-1})_{j l} and summing over j yields
(partial_{y_k} c) delta_{i l} = (partial_{y_i} c) delta_{k l} .
For each k choose l = i != k; hence D_y c = 0. The input ball is connected,
so c = c(t). Set
rho(t) = exp( int_0^t c(v) dv ) , h(y) = partial_t psi(0; y) ,
S(t) = int_0^t rho(v) dv .
The equation for M now gives D_y^2( partial_t psi - rho h ) = 0. So
partial_t psi = rho(t) h(y) + b(t) . y + d(t) for smooth b, d. Integrating
and using psi(0;y) = 0,
psi(t;y) = S(t) h(y) + B(t) . y + D(t) ,
B(t) = int_0^t b , D(t) = int_0^t d .
Since S' = rho > 0, S has a smooth local inverse t = t(s). The output
change kappa(u, s) = ( u - B(t(s)), t(s) ) therefore satisfies
phi( kappa(u,s); y ) = u . y + s h(y) + r(s) , r(s) = D(t(s)) .
Here D_y^2 h = M(0, y) is definite. No input coordinate change is needed.
Put z = (u, s) and A(z, y) = a( kappa(z); y ). Extend this fixed amplitude
smoothly by zero, and periodically in z across a fixed cube Q of side L
whose interior contains its output support. On Q,
A(z, y) = sum_{k in Z^3} e( k . z / L ) A_k(y) ,
sum_k ||A_k||_inf < infinity .
Indeed integration by parts in z gives ||A_k||_inf <= C_m (1 + |k|)^{-m}
for any m > 3. All coefficients have a common compact frequency support,
and the constants are independent of N.
Choose a compact graph patch Sigma parametrized by omega(y) = (y, h(y)),
with smooth boundary and parameter domain containing this support, and
write dsigma = w(y) dy, w(y) = sqrt(1 + |D h(y)|^2). For
g_k(omega(y)) = A_k(y) f(y) / w(y),
||g_k||_{L^p(Sigma)}^p = int |A_k f|^p w^{1-p} dy
<= ||A_k||_inf^p ||f||_p^p .
The factor e^{i N r(s)} is unimodular, and each remaining Fourier term is
e( k . z / L ) E_Sigma g_k( N z / (2 pi) ). Minkowski's inequality and
Theorem 0.1 applied to each term bound their sum on Q by
(2 pi / N)^{3/p} C_Sigma,p sum_k ||A_k||_inf ||f||_p .
Changing back by kappa contributes only the bounded factor
sup |det D kappa|^{1/p} on the fixed output support. Absolute coefficient
summability and the local L^1 integrability of f justify the series and its
interchange with the integral. That proves (6.2), with no power loss in N.
QED.
6.4 Closing remark.
The only external inputs are the packet propagation theorem (Theorem 2.4)
and the L^3 Kakeya maximal theorem (3.3). From those two we proved the
diagonal extension conjecture for compact positively curved surfaces in R^3
(Theorem 0.1), together with the Schrodinger local smoothing and
translation-invariant oscillatory-integral consequences. The dependence on
p is displayed honestly: a global constant C(Sigma) times the explicit
elementary loss (p-3)^{-7/p}, times the packet and Kakeya constants evaluated
at the tiny exponent theta = 10^{-6}(p-3)^2. No rate for those two constants
is claimed, and Proposition 5.5 shows the best constant really does diverge
like (p-3)^{-1/p} as p -> 3.