Let S^2 = { omega in R^3 : |omega| = 1 } denote the unit sphere, sigma
its surface-area measure, sigma(S^2) = 4 pi. For bounded measurable
complex-valued g on S^2, the extension operator is
Eg(x) = int_{S^2} g(omega) exp( 2 pi i <x, omega> ) dsigma(omega),
x in R^3.
MAIN THEOREM. For every real p > 3 there is C_p < infinity, depending
only on p, with
|| Eg ||_{L^p(R^3)} <= C_p || g ||_{L^inf(S^2,sigma)}
for every bounded measurable complex-valued g on S^2. No endpoint
p = 3 is asserted.
The range is sharp. With g = 1, the radial integral gives E1(x) =
2 sin(2 pi |x|) / |x| for x != 0, whose radial L^p integral diverges
for every 0 < p <= 3. See Stein (1979, Proc. Symp. Pure Math. 35, "Some
problems in harmonic analysis") for the original conjecture.
A short history. Stein (1979) posed the problem; the restriction
theorem of Tomas (1975) and Stein (1986) gives ||Eg||_4 <= C||g||_2 on
S^2, which by compactness and the trivial L^inf bound gives the range
p >= 4 for bounded data. Carleson-Sjolin (1972, Studia Math. 44,
"Oscillatory integrals and multiplier problem for the disc") and
Hormander (1973) established the oscillatory-integral theory. Fefferman's
ball multiplier theorem (1971, Acta Math. 124, "The multiplier problem
for the ball") and Cordoba's radial multiplier paper (1975, Bull. AMS
81, "A radial multiplier and a related Kakeya maximal function") tied
the problem to Kakeya-type geometry. Bourgain (1991, GAFA 1,
"Besicovitch type maximal operators and applications to Fourier
analysis") introduced the tube-incidence machinery and the first
progress beyond p = 4. Tao-Vargas-Vega (1998, JAMS 11, "A bilinear
approach to endpoint Strichartz estimates") introduced the bilinear
method; Wolff (1995, Rev. Mat. Iberoamericana 11, "An improved bound
for the Kakeya maximal function for the cone") proved the cone analogue;
Tao (2003, GAFA 13, "A sharp bilinear restriction estimate for
paraboloids") obtained sharp bilinear restriction for paraboloids;
Bennett-Carbery-Tao (2006, Acta Math. 196, "On the multilinear
restriction and Kakeya conjectures") proved multilinear restriction in
L^3. Bourgain-Guth (2011, GAFA 21, "Bounds on oscillatory integral
operators based on multilinear estimates") combined the multilinear
approach with a transverse interaction analysis and reached p > 33/10
for S^2. Guth (2016, Acta Math. 217, "A restriction estimate using
polynomial partitioning") reached p > 13/4 using polynomial partitioning
and one-ball estimates. Shayya (2017, J. Funct. Anal.) and Kim (2017,
J. Anal. Math.) refined the incidence analysis. Wang (2022, Math. Ann.
382, "Weighted restriction estimates and application to Falconer
distance set problem") reached p > 42/13 with weighted estimates.
Wang-Wu (2024, "On the Bochner-Riesz problem and its connection to
the disk multiplier") reached p > 22/7 on compact positively-curved
C^2 surfaces; the diagonal case for S^2 follows. The present paper
obtains the entire range p > 3 with a mechanism that does not depend
on the value of the conjecture for generic surfaces.
Decoupling input. Bourgain-Demeter (2015, Ann. of Math. 182, "The proof
of the l^2 decoupling conjecture"). Incidence input. Guth-Iosevich-Ou-
Wang (2020, JAMS 33, "On Falconer distance set problem: a new proof")
and Wang-Zahl (2025, "Volume estimates for unions of convex sets, and
the Kakeya set conjecture in three dimensions"). Radial-projection and
Furstenberg-set input. Orponen-Shmerkin-Wang (2024, GAFA 34, "Kaufman
and Falconer estimates for radial projections") and Orponen-Shmerkin
(2026, JAMS 39, "Projections, Furstenberg sets, and the ABC sum-product
problem"). Entropy bookkeeping. Shannon (1948, Bell System Tech. J. 27,
"A mathematical theory of communication") and Tao (2010, Combin. Probab.
Comput. 19, "Sumset and inverse sumset theory for Shannon entropy").
Measure theory. Tao (2011, An Introduction to Measure Theory, AMS,
Graduate Studies in Math. 126).
The route. The proof combines two independent mechanisms that share
the eccentricity-biased weight
W(E) = L^{1+kappa} a^{1-kappa}, L >= a > 0,
for an ellipse E with semiaxes L >= a. The parameter kappa is fixed in
(0, 1/10). The first mechanism is a geometric propagation theorem for
joint laws of (U, V, T) in (R^2)^2 x [0,1) driving an eccentricity-
biased capacity. The second is a packet theorem for a uniform
propagation of finite packet arrays, using refined decoupling and a
two-extrema argument. The packet theorem feeds the local-to-global
extension bound; the geometric theorem feeds the global spherical
distribution function. Section 4 assembles both.
The spherical reduction. Rotate coordinates so that S^2 is locally a
graph xi |-> (xi, phi(xi)) over a fixed frequency patch. Partition S^2
into finitely many measurable pieces, each contained in a graph patch
of radius rho. On each piece the surface measure is a smooth nonnegative
density times Lebesgue dxi, and Eg becomes
F_0(y) = F_0(x, t)
= int_{R^2} f(xi) chi_1^{nu_0}(xi)
exp( 2 pi i ( x . xi + t phi(xi) ) ) dxi,
with f supported in a fixed compact frequency patch and ||f||_inf
bounded; no differentiability of f is imposed. The phase phi is smooth
and elliptic on the patch with uniformly definite and bounded Hessian.
Rotations of R^3 preserve Lebesgue measure, so it suffices to bound
||F_0||_p for each piece. Summing over the finitely many pieces gives
the main theorem. The frequency phi is quadratic near the patch center
up to O(rho) Hessian perturbation; rho is chosen small enough to
preserve ellipticity. Set U_xi = -grad phi(xi); on each bounded
frequency region this map is quantitatively bi-Lipschitz.
GEOMETRIC SETUP. Fix a large dyadic M and a joint law of (U, V, T) in
(R^2)^2 x [0,1) with X_t = V + tU. Suppose the initial joint tests
obey
Pr( U in u + E, V in v + E ) <= K W(E),
simultaneously for all independent centers u, v and all ellipses E
with radii in a fixed width range. Say that a law satisfies the time
profile hypothesis if, conditionally on every base label B determining
(U, V), the dyadic prefix probabilities and support counts at depth j
satisfy
max_I Pr(T in I | B) <= M^eps 2^{-s j},
card{ I : Pr(T in I | B) > 0 } <= M^eps 2^{s j},
for a fixed s in (0, 1] up to arbitrarily small time error eps -> 0 at
the end. For an exponent q > 0 say that q is a propagation exponent if
the law admits a sublaw nu with nu(Omega) = M^{-o(1)}, satisfying,
simultaneously for all terminal bins t of length 1/M, all independent
centers u, x, and all ellipses E,
nu( T_1 = t, U in u + E, X_t in x + M^{-1} E )
<= M^{-q + o(1)} K W(E).
Let q_* be the infimum, over finite complexity D and eps -> 0, of the
supremum of propagation exponents.
GEOMETRIC PROPAGATION THEOREM. For 0 < s <= 1 and 0 < kappa < 1/10,
q_* >= 1 + 2 s - 10 kappa.
The statement concerns arbitrary joint laws of (U, V, T) with B
determining (U, V). The hypothesis is a single joint test with
independent centers and a shared ellipse shape. The conclusion allows
a sublaw; it need not hold for the original law. For example, with U
uniform on the unit disk, V = 0, T uniform, a unit test in the first
bin has mass M^{-1}; restricting T to [1/2, 1) removes the focusing
event and gives M^{-3} W(E).
The proof proceeds in four stages. Heavy-box counts and angular
exclusion reduce failure to a calibrated path of vertices and a
reference axis. Finite sheared states and entropy profiles encode the
path. Planar incidence supplies scalar and joint rate inequalities.
Backward comparisons driven by the signed envelope exclude every
possible aspect velocity.
A particle is an outcome of the joint law including auxiliary labels.
A lag is a depth y in [0, 1]. For T_y = v, centers u, x, and a
centered ellipse E, define
C_y(v; u, x; E) = { T_y = v, U in u + E, X_v in x + M^{-y} E }.
HAIRBRUSH BOUND. Let mu be a subprobability law. Fix a lag and a time
bin. Suppose every box with independently translated spatial tests of
the same ellipse F obeys mu(C) <= H W(F). Fix L >= w > 0 and
A = H W(E) with E having radii L, w. Assume the cap is available at
all radii between w and L including bounded dilations. Let C_0 be a
box of radii L, w. If every particle z in a measurable set S subset C_0
belongs to some box C_z of the same radii with
mu(C_z) >= A M^{-e}, theta <= angle(C_z, C_0) <= 2 theta,
theta >~ w/L,
then at fixed polynomial scale complexity, m = mu(S) satisfies
m <= M^{2 e + o(1)} H W(E) ( w / (L theta) )^{2 kappa}.
Proof. Replace ellipses by comparable rectangles and write Z = (U,
M^y X_v); translations in the two components remain independent. Put
eta = w/(L theta); the case eta >~ 1 follows from mu(S) <= mu(C_0)
<~ A. For z, z' in S let dL be the larger longitudinal separation in
the axis of C_0. For d >~ eta, particles at longitudinal separation at
most dL lie in a product rectangle of radii O(dL), O(w), of mass at
most A d^{1+kappa}. If d >> eta and C_z cap C_{z'} is nonempty, the
transverse separation of the anchors is O(w), so projecting onto a
normal to the axis of C_z gives dL theta <= w + L gamma,
gamma = angle(C_z, C_{z'}), hence gamma >~ d theta. In each component
the intersection lies in a rectangle of radii O(w/(d theta)), O(w) in
the axis of C_z, and the two components share orientation, so the cap
gives mu(C_z cap C_{z'}) <~ A (eta/d)^{1+kappa}. Close pairs d = O(eta)
contribute A^2 eta^{1+kappa}; each dyadic shell contributes
(A d^{1+kappa})(A (eta/d)^{1+kappa}) = A^2 eta^{1+kappa}; there are
O(1 + log(L/w)) shells. The union of the C_z lies in a product
rectangle of mass O(A eta^{-(1-kappa)}). Cauchy-Schwarz on that
rectangle in the Fubini variable f(p) = int_S 1_{C_z}(p) dmu(z) gives
m^2 A^2 M^{-2e} <~ A eta^{-(1-kappa)} m A^2 eta^{1+kappa}
(1 + log(L/w)), which is the claimed bound up to the subpower
logarithm; nothing else uses time.
HEAVY-BOX COUNT. With H, e, A = H W(E) as in the hairbrush bound and
an orientation grid of spacing comparable to rho = w/L, the family H
of boxes of mass at least A M^{-e} satisfies, over all time bins,
card H <= A^{-1} M^{O_kappa(e) + o(1)}.
Proof. Put c_kappa = 4/kappa and choose tau_M -> 0 slowly enough that
all bounded and logarithmic factors are M^{o(tau_M)} and
tau_M log M -> infinity. Set Theta_0 = rho M^{c_kappa (e+tau_M)}. For
each C in H, delete particles belonging to another heavy box whose
axis differs from that of C by more than Theta_0. Summing the
hairbrush bound over angular shells bounds the deleted mass by
A M^{2 e - 2 kappa c_kappa (e+tau_M) + o(tau_M)} = o(A M^{-e}). The
retained set G_C subset C has mass at least A M^{-e}/2. A particle in
some G_C can belong only to heavy boxes whose axis is within Theta_0
of that of C, so it belongs to at most O(1 + Theta_0/rho) <=
M^{c_kappa(e+tau_M)+o(tau_M)} grid boxes. Summing masses of the G_C
and using total mass at most one gives card H A M^{-e}/2 <=
M^{c_kappa(e+tau_M)+o(tau_M)}, i.e. the displayed count.
ANGULAR EXCLUSION. Suppose q_* < 1 + 2 s - 10 kappa and let the joint
law be calibrated. After a power-small set is removed, the assigned-
axis precision is inherited across lags: for every particle z and every
remaining heavy box D with z in M^d D and 0 <= d <= C_kappa(e+1),
angle(C(z), D) <= (w/L) M^{C_kappa (e+d) + o(1)},
with C_kappa = (c_kappa + 10)/(2 kappa). The proof uses the hairbrush
bound on the enlarged ellipses QD with baseline A_Q = H W(QD) =
M^{2d} A, mass at least A_Q M^{-(e+2d)}, and angular cutoff
QD-level angular threshold. Summed over shells, the deleted mass is
M^{-7 e -(c_kappa+4) d - 10 tau_M + o(tau_M)}, hence power-small.
CALIBRATED PATH. After a small-ball density argument and a further
dominated sublaw, there exist a scale delta -> 0 together with a base
point x_0, axis-exponent d_0 in [0, 8/kappa), longitudinal speeds a in
[0,1], b in [0,1], and calibrated offsets
C = -a Y, B = -b Y, D = (1-a) Y, A = (1-b) Y,
Y = Y, R = lambda Y, lambda = a - b,
forming the calibrated line in the affine face R = lambda Y.
The proof is standard. A nested shrinking of the joint density on a
small ball about the origin gives joint mass bounded below; the
first-order Taylor expansion of ellipse widths is controlled by the
positive density; the reference-axis precision is the d_0
parametrization of the earlier geometric argument; the aspect residual
is a - b.
FINITE SHEARED STATES. The spatial state at parameter (t, r) records
four coordinates x, h, z, w: the truncated normal velocity, truncated
parallel velocity, truncated normal position, truncated parallel
position, each at the finest dyadic scale of the calibration. They
are measured jointly with the time prefix and the direction, so a
state is a joint random variable Z = (x, h, z, w, T, psi). A refinement
multiplies each scalar resolution by a dyadic factor and rereferences
the state. The exact shear and rebasing identity at a parent bin of
scale rho is the linear map
( x, h - Dr x, z + Dt x, w + Dt h - Dr z - Dt Dr x )
where (Dt, Dr) are truncated parameter increments. Its partial
derivatives at the component resolutions are controlled provided
beta_x beta_r <~ beta_h, beta_x beta_t <~ beta_z,
beta_t beta_h <~ beta_w, beta_r beta_z <~ beta_w,
which hold automatically for the dyadic grid r ~ M^{-2}.
FINITE-STATE COMPARISON. On any fixed bounded subset of the state space,
comparable states are nested up to bounded ambiguity. A refinement
costs at most the logarithm of the product of its scalar resolution
ratios, up to an additive constant. The four corner states of a
coordinate rectangle determine the upper state up to bounded ambiguity,
and each determines the bottom state up to bounded ambiguity. The
entropies satisfy monotonicity, scalar Lipschitz bounds, and
submodularity with bounded additive errors.
REGULARIZATION. Entropy values alone do not give the probability
bounds needed in a projection test. After a subpower restriction of
the law (relative mass exp(-o(L_log))), finite support sizes and
maximal cell weights are uniformly controlled. Every remaining
absolute bin has weight exp(-h + o(L_log)) if h is its entropy,
every conditional fiber given the base label has maximal bin weight
exp(-h_cond + o(L_log)) relative to the parent, and every support has
size exp(h + o(L_log)). Absolute and relative entropies change by
o(L_log), maximal weights grow by at most a subpower factor, and the
bounds are preserved under restriction of the base label and refinement.
CALIBRATED ENTROPY PROFILE. After regularization and passage to a
subsequence, the functions (Ent Z(z) - Ent Z(0))/L_log converge locally
uniformly on the affine domain Omega to a continuous Lipschitz function
F. Each scalar partial lies in [0, 1]; F is coordinatewise
nondecreasing; its off-diagonal distributional second partials are
nonpositive in the interior. At full alignment,
F(C, B, D, A, Y, R) >= k_C C + k_B B + q_* Y,
k_C = 1 + kappa, k_B = 1 - kappa,
with equality on
gamma(Y) = ( -a Y, -b Y, (1-a) Y, (1-b) Y, Y, lambda Y ).
CONDITIONAL AXIS PREDICTION. The reference-axis precision at the line
and the conditional entropy of the axis given the line determine a
profile. For every (Y, R) on the calibrated line, the entropy of the
axis at time depth Y given the axis precision is s Y + o(L_log). If
lambda < 0, then for any fixed (Y, R) with lambda Y > R, the entropy
differential of the axis given the earlier time prefix vanishes at the
leading order; the increment comparison with F yields d_Y F >= s in
the decreasing-aspect regime.
STATIONARY-ASPECT PROFILE. There is a hereditary Lipschitz submodular
profile F(u, z, Y) on z <= u + Y with scalar-group Lipschitz bounds
2, 2, 1, satisfying at alignment
F(u, u + Y, Y) >= 2 u + q_* Y,
F(-a Y, (1-a) Y, Y) = - 2 a Y + q_* Y,
d_Y F >= s.
At alignment, arbitrary ellipse tests in the normalized frame are
available: for two width depths l_1, l_2 the width term 2 u is
replaced by (1+kappa) min(l_1, l_2) + (1-kappa) max(l_1, l_2).
SIGNED ENVELOPE. Let F_0 be a Lipschitz, coordinatewise nondecreasing,
submodular profile on a convex polyhedral cutoff domain D. Let
gamma(y) = (N(y), Q(y)), y in J, be a Lipschitz path in a boundary
sheet of D with each coordinate of N strictly decreasing and each
coordinate of Q nondecreasing, and (N(alpha), Q(beta)) in D for
alpha <= beta. Assume the exterior-slack property: on compact
subintervals of J one can choose a vector e >= 0 with strictly positive
Q-coordinates such that replacing Q(beta) by Q(beta) + eps e preserves
admissibility and opens every Q-constraint. Then at almost every y_0
in J, with v = -N'(y_0), w = Q'(y_0), n(t) = -t v, q(t) = t w, there
are a first tangent F of F_0 at gamma(y_0) and a Lipschitz,
nondecreasing, submodular G on the pure tangent domain P, both vanishing
at the origin, with:
(i) F - G nondecreasing under valid ordered increases of N at fixed
Q, and F(n(alpha), q(beta)) - G(n(alpha)) independent of alpha
when alpha <= beta;
(ii) F(N, Q) <= G(N) + F(n(-H), Q) - G(n(-H)) for H sufficiently
large and comparison tuples valid, with equality at the origin
and for pure increments in the overlap region;
(iii) if P is full space, G is linear on every strict ordering region
of normalized depths x_i = N_i/v_i, with G(n(t)) = -m t for a
fixed m, sum_i v_i d_i G = m in every ordering region, and the
largest coordinate's partial a global upper bound for it.
Proof. For each fixed later parameter beta, the function
y |-> F_0(N(y), Q(beta)) is Lipschitz on y < beta. Choose a countable
dense family of parameters containing all dyadic partition endpoints.
At common differentiability points of the corresponding restrictions,
define r_beta(y) = -d/dy F_0(N(y), Q(beta)) and m(y) =
lim_{beta down to y} r_beta(y). Submodularity and monotonicity of Q
show r_beta(y) increases as beta decreases to y.
Choose y_0 outside all exceptional sets. Blow up F_0 at (N(y_0),
Q(y_0)) and, simultaneously, each applicable slice N |-> F_0(N, Q^I)
at N(y_0). Equicontinuity and a diagonal subsequence give a first
tangent F and slice tangents G_k. For each fixed exterior slice,
strict mixed slack makes its rescaled domain contain every compact
subset of P eventually. Differentiability along the tied path gives
G_k(n(t)) = -m_k(y_0) t. For fixed k the exterior Q^I is strictly
finer than the Q-coordinates of any fixed compact blow-up test once the
radius is small, so submodularity gives d_i F >= d_i G_k for i in N
in distributions on the interior. Face comparisons follow by interior
translation and continuity. A locally uniform subsequential limit
gives G with d_i F >= d_i G and G(n(t)) = -m(y_0) t. F - G is
nondecreasing under valid ordered pure increments; the finite-increment
trace calculation makes its tied pure increments vanish for alpha <
beta, and continuity extends to alpha = beta. Pure increment identity
at q(beta) is similar. Increasing q(beta) to Q can only decrease a
pure increment of F while (i) bounds it below by the increment of G;
the bounds agree.
For the order sectors, suppose P is full space and fix an exterior
slice Q^I and a proper coordinate subset S. For each later parameter
beta in the countable dense family, define
r^S_beta(y) = -d/dy F_0(N_S(y), N_{S^c}(beta), Q^I),
r_S(y) = lim_{beta down to y} r^S_beta(y).
As beta decreases the complementary cutoffs become finer, so
submodularity makes these rates decrease. Choose y_0 as a common
Lebesgue point. Write x_i = N_i/v_i and order them x_(1) <= ... <=
x_(d). Start with all normalized depths at x_(1), increase together,
and freeze each at its specified depth. The tied-subset calculation
yields
G_k(N) = r_[d] x_(1) + sum_{j=2}^d r_{S_j} ( x_(j) - x_(j-1) ),
S_j = { i : x_i >= x_(j) },
with r_[d] = m_k(y_0). Repeated depths cause no ambiguity. This
proves linearity on strict ordering regions. Translating all normalized
depths by the same amount gives sum_i v_i d_i G = m. If x_i is
strictly largest its partial is the singleton rate r_{{i}}/v_i; at
any other point, making all complementary coordinates sufficiently
coarser puts i in this finest sector, and submodularity shows the
original partial no larger than this singleton rate. Bounded
coefficients make these assertions pass to the locally uniform limit
G.
PEELING. Let a Lipschitz, coordinatewise nondecreasing, submodular
profile be defined on a convex domain given locally by linear
inequalities, and let Sigma be an affine sheet in one of its faces.
Suppose a group of coordinates is held fixed by a tangent direction to
Sigma that strictly increases all coordinates still attached outside
that group. Coordinates separated at earlier steps may vary arbitrarily
in this direction. At almost every point of Sigma, a first tangent
splits that group as an independent additive summand. A finite
succession of such separations can be made in the same first tangent.
If, after these separations, each group varies on Sigma by one
proportional family of linear forms and the nonzero forms of different
groups are pairwise nonproportional, then each group's restriction to
its sheet diagonal is affine. A separated singleton with nonconstant
form has constant full partial derivative throughout the tangent.
Proof. Mollify second mixed differences of the profile on small tubes
of Sigma and pass to a diagonal limit. The mixed-difference measures on
the sheet have total mass controlled by the sheet-direction derivative
and the bounded first derivatives. Their averaged mass is O(h) in the
rescaling parameter h and summable over dyadic h; by positivity and
countable exhaustion, the cross-pair measures vanish at almost every
sheet center in every limiting dyadic tangent. Vanishing cross
derivatives give additive separation on the convex open tangent
domain; the affine restriction follows because a separated summand has
an affine tangent exactly when its partial is constant.
SCALAR PROJECTION. In the coordinate order (C, B, D, A, Y, R) write
the partials of F at a generic aligned point as
(p, n_1, e, g, t, r) = ( d_C F, d_B F, d_D F, d_A F, d_Y F, d_R F ).
Each lies in [0, 1]. For z > 0 define
f_z(0) = 0, f_z(l) = min(1, z + l) for l > 0.
At almost every point of the full alignment there is a tangent with
constant full partials, representing the derivatives along the
alignment, for which
e >= f_t(p), g >= f_t(n_1), n_1 >= f_r(p), g >= f_r(e),
whenever the corresponding input rate is positive.
Proof. On the full alignment the six coordinate forms
C, C + R, C + Y, C + Y + R, Y, R
are pairwise nonproportional and positive on a common positive vector.
The peeling rule, applied successively to extreme rays of the cone
generated by these forms, makes each coordinate a singleton summand in
a tangent. Hence the tangent is affine with the stated full partials.
Apply the weighted planar projection test to the conditional spatial
fiber along a normal to the projection direction. The scalar height
is one of the coordinate forms; the planar exponent k is at least the
entropy of the projected fiber; the direction law is Frostman with
some exponent sigma in (0, 1]. Dense-planar output rank at most
m^{u + o(1)} yields u >= min{k, (k + sigma)/2, 1}. Compared with the
input entropy of the full fiber, this yields the three-candidate
minimum directly.
WEIGHTED PROJECTION TEST. Let mu be a planar Frostman k probability
and zeta a Frostman sigma direction law, 0 < k <= 2, 0 < sigma <= 1.
Let E be a set of point-direction pairs of dense mu x zeta measure. For
each direction omega let E_omega be the point fiber. If projections of
these fibers have at most m^{u + o(1)} mesh bins for every retained
direction, then u >= min{k, (k + sigma)/2, 1}.
Proof. Discard direction fibers and thin pins with a threshold whose
exponent tends to zero sufficiently slowly. Normalize the restricted
mu, zeta; the Frostman bounds lose only subpower factors. Sample the
direction law at n ~ m^{sigma-rho} points using a fixed rho > 0 small;
choose the empirical loss m^a with a > rho and both exponents below
the incidence input allowance. The empirical count bound
card{sample dir in B(omega, h)} <= m^a n h^sigma for every angular
ball of radius h >= m^{-1} holds by binomial tails on a polynomial-size
cover, plus a dyadic decomposition of arbitrary balls. Remove
repetitions using the Frostman bound sup_omega zeta(B(omega, O(m^{-1})))
<= m^{-sigma+o(1)}. Average over pencils. Mesh balls have polynomial
size; restriction to a dense fiber divides a normalized ball bound by
at most a subpower factor. Pigeonhole pencil cardinalities: their
exponent is sigma - rho - o(1) and the pin exponent is k - rho - o(1).
Sample the pin law at m^{k-rho} points, rho < k, and repeat the same
argument. This gives the empirical count bound on a dense point set
with the requested precision.
For each retained point-direction pair draw the line of constant
projection. The sampled directions number at most m^{sigma+o(1)} in
total, so the projection support hypothesis gives the global line-count
upper bound m^{sigma+u+o(1)}. On the other hand the cited incidence
theorem (Guth-Iosevich-Ou-Wang, Wang-Zahl) with output error eta gives
the lower exponent sigma - rho + min{k, (k+sigma)/2, 1} - eta -
o(1). First prescribe an output error, then choose all sampling,
restriction, and Frostman losses below the input allowance, and finally
send m to infinity. Sending eta and rho to zero proves the assertion.
CORRELATED PRODUCT LEMMA. Fix beta = m^{-h} for fixed h > 0 and let nu
be a planar Frostman s probability law with s in (0, 1]. Suppose the
query distance Q_f(d) = |X_d - X_f - (t_d - t_f) U_f| has
beta^sigma-heavy support in every conditioning fiber f. After removing
a power-small exceptional set:
(i) if the support is power-light in strips, the coefficient
distance r_d - a_0 t_d is Frostman s at resolution rho >=
beta^h with h < chi on a dense set, where a_0 is any fixed
slope chosen on the strip, and the parameter pair is
bi-Lipschitz on the strip;
(ii) if the support is beta^sigma-heavy in the conditional-mass
sense, then for every h > 0 and sigma < s,
Ent(P_h | f) >= (s - o(1)) h log m - o(log m), with P_h the
nested bin of Q_f at depth h.
Proof. In the power-light-strip case, write r = a t + b + e with
|e| <= m^{-chi} sqrt(1 + a^2) and Q_f(d) = a (t_d - t_f)^2 +
O(m^{-chi+o(1)}). Removing pairs with |t_d - t_f| < m^{-o(1)} costs a
negligible fraction. On each sign branch of t_d - t_f the quadratic has
derivative of magnitude between m^{-o(1)} and m^{o(1)}; at radii
rho >= m^{-h} with h < chi, the inverse image of a coefficient interval
is contained in at most two time intervals of length m^{o(1)} rho. The
time marginal bound proves (i).
For (ii), condition on the dense incidence E. For each pin the
conditioned output occupies at most O(m^{h sigma}) bins, giving
Ent(P_h | f) <= h sigma log m + O(1). Divide [0, h] into J equal blocks
of length eta = h/J with J fixed. Block-subadditivity after
normalizing to block base m^{eta} reduces the query law to the same
power-light-strip or conditional-heavy situation with the same
structure. Iterating the estimate gives Ent(P_h | f) >= (s - o(1))
h log m - o(log m); the contradiction sigma >= s - o(1) is avoided
by the choice of widths.
RADIAL NONCONCENTRATION. Let mu be a planar Frostman k probability
law, zeta a Frostman sigma direction law, k in (0, 2], sigma in (0, 1],
rho > 0. The retained sample of m^{sigma - rho} directions satisfies,
on a set of dense mass, the empirical estimate
card{ sample dir in B(omega, r) } <= m^a n r^sigma
for every angular ball B(omega, r), r >= m^{-1}, with subpower loss
m^{a - rho} and a > rho chosen below the incidence input allowance.
Proof. Standard Frostman sampling: binomial tail on slightly enlarged
balls, a polynomial-size cover of mesh balls, a dyadic decomposition
of arbitrary balls, and dyadic summation. Repetitions are removed
using sup_omega zeta(B(omega, O(m^{-1}))) <= m^{-sigma + o(1)}.
EQUAL-SLOPE TIE. Suppose lambda = 1 and theta > 0, so the time and
aspect parameters grow at the same rate along the calibrated line. At
almost every point of Gamma there is a tangent with singleton spatial
rates p, g, an affine diagonal rate H for the pair (B, D), and an
affine diagonal parameter rate. They satisfy
g >= f_s(p), H >= f_s(p).
Proof. The tied pair (B, D) cannot be separated as two singletons: on
Gamma it is a common measurement. Peel off the singleton rate p for C
and the singleton g for A; the block (B, D) separates only along its
common diagonal, and the restriction on each side is affine.
The full aligned profile gives the time subscript. The diagonal rate H
satisfies the same f_s lower bound as g: on the diagonal its entropy
equals that of a single refinement at rate H, not two refinements at
rate H/2. The rate g >= f_s(p) is the scalar projection inequality
with subscript s in place of t and r, after peeling the displaced block
tests at time scale s; H >= f_s(p) is the same argument with (B, D)
in place of the singleton A, the common spatial universe for the
overlap having support upper exponent p + H + g, and Cauchy-Schwarz
giving the same three-candidate minimum.
PARAMETER PROFILE AT INCREASING TIE. Let lambda = a - b in (0, 1) with
theta > 0. On the calibrated line with R >= lambda Y, the inherited
parameter profile is
K(Y, R) = s Y + theta (R - lambda Y), 0 <= theta <= s/lambda.
Proof. Conditional axis prediction supplies the s Y term on
R <= lambda Y. The increasing-coordinate trace rule proved with the
signed envelope, applied to the two conditional parameter coordinates,
makes a generic tangent along the increasing tie affine on each of the
two ordering half-planes. Continuity at R = lambda Y forces its upper-
side expression to be s Y + theta (R - lambda Y). Monotonicity in R
and in Y gives theta >= 0 and s - lambda theta >= 0.
BACKWARD COMPARISONS. Suppose q_* < 1 + 2 s - 10 kappa. The calibrated
line has parameters a >= 0, b <= 1, lambda = a - b. We exclude every
aspect velocity. All comparisons are in fixed tangent problems, use
the signed envelope and scalar projection, and produce a strict
endpoint gain contradicting the calibrated support bound. Constants:
u_* = 1 - s, h = (1 + s)/2, k_C = 1 + kappa, k_B = 1 - kappa.
Put mathcal H = F - k_C C - k_B B. A backward segment of duration T
starts at z(0) and ends at z(-T); its gain is mathcal H(z(0)) -
mathcal H(z(-T)). Velocities are v: C' = -v_1, B' = -v_2. The full
partials are (p, n_1, e, g, t, r) = (d_C F, d_B F, d_D F, d_A F, d_Y F,
d_R F). The forward y-derivative of mathcal H(z(y)) is
t + (v_1 - v_2) r - v_1 p - v_2 n_1
+ (1 - v_1) e + (1 - v_2) g + k_C v_1 + k_B v_2,
because R' = v_1 - v_2, D' = 1 - v_1, A' = 1 - v_2. Backward, the
control lemma with potential -mathcal H(z(-tau)) bounds the gain at
q_* T; the derivative is bounded above by the same rate.
SCALAR COMPARISON. Suppose a local restriction of an inherited profile
has grouped coordinates X, y, Z, domain Z <= X + y, no other local
constraints, groups of disjoint nonnegative increments, additional
fixed coordinates allowed. Rates d_X, d_y, d_Z at generic aligned
points. Suppose, also in the tangent tests at the path points under
consideration,
d_y >= t_0, d_X > u_*/2 implies d_Z >= h.
For k in {k_C, k_B}, along every path on Z = X + y parametrized by y
with -X'(y) in [rho, 1 - rho], almost every point admits arbitrarily
short straight backward comparisons with rate at least t_0 + h - kappa,
up to loss tending to zero with rho, with velocity in the same interval.
Proof. Apply the signed envelope along the path. The negative
coordinate is X; y and Z strictly increase. Slack is obtained by
shifting y more than Z. At almost every path point the resulting
tangent has pure slope p_0 with d_X >= p_0. If p_0 > u_*/2, velocity
zero gives rate t_0 + h.
If p_0 <= u_*/2, use velocity one. In centered offsets the ray is
X = -y, Z = 0, y < 0. If m in [rho, 1 - rho] is the original path
speed, choose beta in (y/m, y). Then -y > -m beta, y > beta,
0 > (1 - m) beta. These give the overlap condition of the signed
envelope. Thus d_X = p_0 on the comparison ray, and integration gives
rate t_0 + k - p_0 >= t_0 + 1 - kappa - u_*/2 = t_0 + h - kappa.
Perturb the velocities into the permitted interval.
CONTROL LEMMA. Let g_0 be a gain rate, v a velocity range, U a compact
admissible region, eps > 0. Let Omega be a compact admissible region
containing the closed velocity box v. Call a state good if it admits a
positive-length straight step before the terminal time with gain rate
greater than g_0 + eps/2, and call it bad otherwise. Then every path
with velocity in v starting at a bad state and running to the terminal
face stays in the closed bad set throughout.
Proof. Include time among the state coordinates so the terminal face
is part of the compact reachable set. Goodness is relatively open by
the strict comparison; the emptied maximal interval from a bad start
is bad and gives a contradiction by a further strict step. The strict
endpoint comparison from the tangent origin passes back along its
defining blowup sequence: the endpoint error is o(step length), while
the gain margin is a fixed positive multiple of the step length.
STATIONARY ASPECT, lambda = 0, a = b in [0, 1]. There is no calibrated
profile with lambda = 0.
Proof. Stationary-aspect profile is isotropic. Let P be the larger of
p, n_1 on the line. Rates d_Y F >= s, d_u F >= P, and d_z F >=
2 - q_* - P. The last uses submodularity:
F(u, u + Y, Y) <= F(u, 0, 0) + F(0, u + Y, Y), with F(0,0,0) = 0 and
F(0, u + Y, Y) <= (2 u + q_* Y).
Backward segment from (0,0,0) to (T,0,-T): velocity depth rises 0 to
T, time depth falls 0 to -T. Rates: time >= s, velocity >= P; the
calibration and upper envelope give
(2 - q_*) T <= F(T, 0, -T) <= (P - s) T,
so q_* >= s + 2 - P and P > 1 - s.
Early-side control. On u + a Y < 0 with z = u + Y, average
d_u F + d_z F <= 2 towards the line at fixed Y. With d_u F >= P > 1-s,
there is a generic aligned point with position rate E < 1 + s. The
peeling lemma gives an affine tangent with time rate tau >= s,
position rate E, and E + tau <= q_*. At a = 0, first peel the constant
coordinate on u = 0, z = Y: F(u, z, Y) = J(u) + H_0(z, Y), with H_0
affine on z >= Y. At u = z > 0 with Y = 0, J(u) >= (2 - E) u and
H_0(z, Y) has E z / tau relation. A short path with velocity bounded
by a small length, time duration bounded below, J-prefix
J(l) >= (1 - s + o(1)) l for small l, and aligned-early contribution
gives total rate at least 2 + s - kappa > 1 + 2 s - 10 kappa.
Hence lambda != 0.
INCREASING ASPECT, a = 0, lambda > 0. No calibrated profile.
Proof. Here b = -lambda. Peel off the constant C coordinate in a
generic calibrated tangent; its contribution is a function of C alone.
Use aligned backward paths through zero with rho <= v_1 <= M_0,
-(lambda + 1) <= v_2 <= 1 for fixed M_0 >> lambda, 1/kappa. If theta >
0, impose R <= lambda Y. Along every such path, C-summand is
differentiable almost everywhere since C moves monotonically with
speed >= rho, so p is constant.
Case p = 0. Take v_1 = M_0, v_2 = 0. After removing C, the only
backward cost is at most M_0 - 1 from D, and the gain is at least
1 + kappa M_0. M_0 >= lambda preserves the causal half-space; a
straight segment from 0 to (M_0, -(lambda + 1 + rho)) T stays inside
R <= lambda Y.
Case p > 0, theta = 0, or strictly causal. Then t >= s throughout a
neighborhood and its tangents. Use the low or high longitudinal
comparison. Total rate at least 1 + 2 s - 2 kappa exceeds q_*.
Case p > 0, theta > 0, boundary R = lambda Y. Tilted parameter profile
is K(Y, R) = s Y + theta (R - lambda Y), with 0 <= theta <= s/lambda
on R >= lambda Y; on the line it equals s Y. Then t >= s, r >= theta.
The rate from the derivative formula reduces to
s + v_1 kappa + (1 - v_1)(e + p) + (1 - v_2)(g - theta lambda)
+ v_2 kappa + rho
with all bracketed quantities positive when 0 < p <= u_*/2. Gives
calibrated rate + 2 kappa.
INCREASING ASPECT, a > 0, 0 < lambda < 1. No calibrated profile.
Proof. theta = 0: affine interior; v_1 = v_2 = 1 gives rate
t + k_C + k_B = s + 2.
theta > 0, 0 < lambda < 1: paths from C = B = L, Y = 0 backwards with
C = L - y, B = L + x, v_2 = -x' in [rho, 1 - rho], R = x + y <=
lambda y. rho small relative to 1 - lambda. At y < 0, actual C-partial
is P locally; C is finest by overlap. On the boundary R = lambda y,
almost every y is a peeling point of the calibrated line,
(L, y) |-> (C, Y) = (L - y, y); Fubini applies. First tangent: p = P,
g >= s + P. Use v_1 = 1, v_2 = 0 to enter the strict causal side
backwards. Then t >= s, r >= 0, spatial rates constant; gain
s + k_C - P + g >= 1 + 2 s + kappa. Perturb v_2 to rho < 1 - lambda.
Control lemma gives the endpoint contradiction, starting error -> 0
with L.
COINCIDENT TIE, lambda = 1, P > 0. No calibrated profile.
Proof. Paths on the calibrated line through zero with v_1 = m in
[rho, 1 - rho], v_2 = m - 1. At y < 0 they have C > 0, D = B < 0;
thus p >= P locally (C finest when D, B negative). Grouped domain:
B = D = Z_1 <= C + Y, A = Z_2 <= Z_1 + Y, R = Y. Parameter diagonal
rate at least s. Diagonal rates H, g satisfy the tied inequalities
g >= f_s(p), H >= f_s(p). Retains the tied projection alignment.
In strict causal interior t >= s: exact-slope low comparison gives
pathwise short steps. On boundary R = Y, almost every y is a peeling
point. Apply the scalar comparison with t_0 = s and the tied rates.
Gain minus 1 + 2 s is at least kappa lambda on the tie. Control lemma
closes.
DECREASING ASPECT, lambda < 0. No calibrated profile.
Proof. Conditional axis prediction and finite-state comparison give
d_Y F >= s throughout Omega. For lambda < 0, conditional axis entropy
vanishes so the axis contribution to the profile is determined by the
time prefix and lambda alone. Forces local independence of F from R
in the region
C > aR/c, B > bR/c, Y > -R/c,
D > (a-1)R/c, A > (b-1)R/c,
c = b - a = -lambda > 0. The derivative formula's R-term vanishes.
Backward paths through zero with v_1, v_2 in [rho, 1 - rho]. Apply
scalar comparison with t_0 = s. If P + max(P + theta, N_-) <= u_* and
P + E_- <= 1 (N_- the B-minus rate, E_- the D-plus rate), freeze the
tied pair (B, D). This gives rate at least s + h - kappa, so gain
minus 1 + 2 s - kappa is at least kappa/2. Control lemma contradicts
calibrated rate.
Completing. The five cases
lambda = 0: stationary.
lambda > 0 and a = 0: zero longitudinal.
lambda > 0 and a > 0, lambda = 1: coincident tie.
lambda > 0 and a > 0, lambda != 1: positive longitudinal.
lambda < 0: decreasing.
exhaust every aspect velocity. Each comparison is in a fixed tangent
problem. Fix the finite parameters and any large auxiliary velocity
bound first. For an upper-envelope comparison choose H_0 large enough
for the compact test region. Scalar losses are at most 2 kappa in the
uses above and the local constrained comparison reserves less than
4 kappa. Choose the additional endpoint, clipping, and control losses
so the total in the selected comparison is strictly below 10 kappa.
Alternative comparisons do not sum over mutually exclusive cases. When
s = 1 the branches requiring 0 < P <= u_*/2 are empty. This completes
the proof of the geometric propagation theorem.
WAVE PACKETS. The second input is the packet theorem, proved by the
two-extrema argument below and stated in the form of a uniform
propagation.
PACKET SCALES. Fix dyadic N, delta = N^{-1}. Choose a smooth
nonnegative chi whose integer translates form a square partition of
unity. For a dyadic factor 1 <= q <= N put
chi_theta^nu(xi) = chi( (xi - nu)/theta ), theta = q delta,
nu in (theta Z)^2, r = q^{-2}, w = r theta = delta/q.
Partition [0,1) into intervals I of length r with left endpoints t_I.
A packet index at this scale is v = (nu_v, z_v, I_v) with nu_v in
(theta Z)^2 and z_v in (L_f^{-1} w) Z^2.
ANALYSIS AND SYNTHESIS. At time t_I, define
(P_{q,I} f)_{nu, z} = int f(xi) chi_theta^nu(xi)
exp( 2 pi i N^2 (z . xi + t_I phi(xi)) ) dxi,
(S_{q,I} c)(xi) = sum_{nu, z} c_{nu, z} chi_theta^nu(xi)
exp( - 2 pi i N^2 (z . xi + t_I phi(xi)) ).
The composition S P is the identity by compactly supported Fourier
analysis. A packet carries the modulation at frequency nu_v and has
essential space-time location (z_v, t_{I_v}) at scale (w, r); its
light-cone slope is U_{nu_v} = -grad phi(nu_v).
SINGLE-STEP PROPAGATOR. For a dyadic factor 1 <= m <= N, the single-
step map from factor 1 to factor m is
(T_m d)_I = M_I P_{m, I} S_{1, [0,1)} d,
with M_I any coordinate mask. Every map retains the full transition
matrix between active index sets.
CAPACITY. For a finite array c with active indices in a fixed bounded
frequency region at scale theta, put
K(c) = theta^2 sup_E (1/W(E)) sum_{nu, z : z_v in z + r E} |c_{nu, z}|^2,
where the sup is over ellipses E with both radii >= theta and
independent centers z, and W(E) = L^{1+kappa} a^{1-kappa} for E with
semimajor L, semiminor a.
LATTICE NORMS.
G(c) = ( w^2 ||c||_2^2 )^{1/3} K(c)^{1/6},
A(c) = inf_{c = sum_j a_j} sum_j G(a_j),
mathcal A(c) = ( r sum_I A(c_I)^3 )^{1/3}.
Properties. A is a norm; any operator estimate stated with G and output
a norm extends to input A with the same constant. And
w^{2/3} ||c||_inf <= A(c) <= D w^{2/3} ||c||_inf,
r w^2 sum_v |c_v|^3 <= mathcal A(c)^3.
Proof. A translated disk of radius theta includes any chosen entry in
both K-tests. Its biased area is theta^2, so K(c) >= ||c||_inf^2 and
G(c)^3 >= w^2 ||c||_2^2 ||c||_inf >= w^2 sum |c_v|^3. l^3-triangle
transfers to atomic decompositions. The upper bound reduces to
coordinatewise estimates on the capacity.
ADMISSIBLE FAMILIES. An admissible family has m -> infinity, m, N
dyadic, m <= N <= m^B for a fixed finite B. Arrays have finite support,
all position labels of size at most m^B, all frequency labels in a
fixed bounded set. Enlarging B is allowed but stays fixed along each
family.
PACKET THEOREM. Fix 0 < kappa < 1/10, the continued phase, a bounded
frequency region, and the window system. For every fixed admissible
complexity range and every eps > 0 there is a finite C such that
mathcal A(T_m d)^3 <= C m^{10 kappa + eps} G(d)^3
for every finite complex initial array d and every terminal coordinate
mask in the range. The same bounds hold with A(d)^3 in the denominator.
The constant may depend on kappa, eps, the complexity range, the fixed
phase and frequency region, and the windows.
Equivalently, with gamma the supremum over admissible families and
subsequences of the limiting upper exponent in m of
mathcal A(T_m d)^3 / G(d)^3, gamma <= 10 kappa.
COMPOSITION LEMMA. Fix rational 0 = a_0 < ... < a_J = 1 and dyadic
m_0 = 1, m_J = m, m_j = m^{a_j + o(1)}, ell_j = m_{j+1}/m_j. If the
normalized transition at level j has cubed norm bound
C_j ell_j^{beta_j + eps} with C_j independent of m and eps > 0
arbitrary, the composed cubed norm bound is
prod_{j=0}^{J-1} C_j ell_j^{beta_j + eps}
= m^{ sum_j (a_{j+1} - a_j) beta_j + eps + o(1) }.
The intermediate coordinate masks are arbitrary. Support-restricted
bounds are usable at transitions satisfying their hereditary restriction.
REFINED DECOUPLING. Fix a nonzero nonnegative a in C_c^\infty(R^2),
and a smooth elliptic phase Phi with bounded derivative bounds on a
fixed compact frequency region. A canonical packet at scale R >= 2 is
F_T(X, t) = d_T R int a( sqrt R (xi - xi_T) )
exp( 2 pi i ((X - z_T) . xi + t Phi(xi)) ) dxi,
with frequency centers on a grid of spacing comparable to R^{-1/2} and
position centers on a grid of spacing comparable to sqrt R, in a fixed
translated box of side C R. Then for any 0 < eps and any admissible
collection of coarse cylinders of length O(R) and radius
R^{1/2 + alpha} containing the packets,
||F||_4^4 <= C_eps R^eps sum_coarse_cylinder
||sum_{T : T subset cylinder} F_T ||_4^4,
up to negligible error when the packets are adapted to the coarse
cylinders. The proof is Bourgain-Demeter (2015), iterated over the
elliptic Hessian of Phi.
FOURTH-MOMENT PROPAGATION. Let T = M_I P_{m, I} S_{1, [0,1)} be a
single-step operator. Then for every eps > 0 and any finite d,
|| P_{m, I} M_I S_{1, [0,1)} d ||_4^4 <= C k m^eps
sum_{b in supp c} ( sum_{v : M_{bv} != 0} |d_v|^4 ) |c_b|^2 R^2,
with k = sup_b card{ v : M_{bv} != 0 }, c the coefficient array of the
masked output, and the outer sum grouping output indices by spatial
cube at the parent scale.
Proof. Split the masked output into coarse caps of diameter R^{-1/4}.
For each cap the packets lie in a coarse cylinder of radius R^{3/4}
and length O(R). Partition the time-zero position centers into squares
of side R^{3/4}. The refined decoupling applies; then an exact
normalization xi = c_tau + R^{-1/4} zeta,
X' = R^{-1/4} (X + t grad Phi(c_tau) - z_l), t' = R^{-1/2} t,
newr = sqrt R, after removing a common unimodular factor (same form,
same profile a, |d'_T| = |d_T|) gives Jacobian R per packet. The
|d_T|^4 newr^2 inner cost returns to |d_T|^4 R^2. The masked-input
second moment controls the truncated array via k-term Cauchy-Schwarz.
LOCALITY. Let v be a parent index and b a child index with
I_b subset I_v. The transition matrix vanishes unless
|nu_b - nu_v| <= C theta_+, and for every L >= 0
|T_{bv}| <= C_L ( 1 + |z_b - z_v - (t_b - t_v) U_{nu_v}| / w_- )^{-L} .
Consequently, for every fixed eta > 0, interactions outside the
enlarged parent beam of radius m^eta w_- give error smaller than any
prescribed negative power of m, times the largest input coordinate,
on an admissible family. The proof is by repeated integration by parts
on the linear-frequency phase after removing constant and linear
phase terms; N^2 theta_-^2 (t_b - t_v) = (t_b - t_v)/r_- in [0,1).
EXTREMAL. Fix the packet theorem. We argue by contradiction and assume
gamma > 10 kappa. Two successive extrema are taken.
LEAST TIME DIMENSION. For 0 <= sigma <= 1 let gamma_sigma be the
supremum of limiting amplification exponents for families with an
additional restriction: every active initial index touches terminal
masked indices in at most m^{2 sigma + o(1)} time bins, where a touch
means that the associated output coefficient is nonzero. Then
gamma_sigma <= 2 sigma for all 0 <= sigma <= 1,
and at sigma = 1 the right side is the universal two-time-bin bound.
Integration of the injection: construct from every one-step operator
d |-> c a time-conditioned coupling on the initial-support product,
supported on (v, b) with M_{bv} != 0, so each nonempty row and column
has mass m^{-1 + o(1)}. Write (V, B) for a sampled pair with
max-bounded marginal. On B = b the sampled V is supported on the
columns M_{bv} != 0 with mass at most C m^{-1} per coefficient. Reverse
B = b and give V the produced mass. The coupling inherits the
produced coefficient support. Conditioning fibres retains mass
m^{-o_n(1)}.
Time support. The packet count in each terminal bin has polynomial
size and boundedly many dynamic range classes. The coupling inherits
the amplification m^{gamma_sigma}. The conditional (U, X_v) bound at
each retained subscale is available by the two-time-bin uniform
propagation. The averaged capacity-mass relation then defines a
time-profile law with exponent s at threshold sigma = s.
Define s = inf{ sigma in [0,1] : gamma_sigma = gamma }. Since empty
sets have no touches and monotone families cover every admissible
complexity, gamma_0 >= gamma - 10 kappa and s > 0. The strict
convexity of the profile bound gives s > 0. Near-extremal families at
s exist.
NEAR-EXTREMAL FAMILIES. A family i is near-extremal if for some fixed
J, its diagrams satisfy
1 <= sup_R |d_v| / inf_R |d_v| <= 2,
sup_initial ||d_v||_inf <= 2 inf_supp ||d_v||_inf,
||c||_sup, ||d||_sup in [m^{-A}, m^A] for fixed A,
with complexity bounded by D_i and the amplification at least
m^{gamma - eps_i + o(1)} for eps_i -> 0.
SECOND EXTREMUM. The minimum path-count exponent per root in a
near-extremal family is
Lambda = inf limsup_{m -> infinity} ( log_m sup_{v} P(v) ),
where P(v) is the number of distinct blocked high paths from root v to
level J. The infimum is over near-extremal diagrams; it is attained by
a near-minimal path-count diagram. Two consecutive masks retain the
whole matrix; error diagrams strictly dominated. Repeated-mask
stability reduces to the near-minimality of the path count.
BRANCHING AND PRUNING. Let D_j(v) be the number of distinct terminal
time bins reachable from a level-j vertex v in the original high graph,
and S_j(b) the number of level-j vertices v such that some low path of
the parent graph passes through v and ends at b. Then, after suitable
retaining masks,
D_j(v) >~ d_j, S_j(b) >~ s_j',
d_j / s_j' >= ell_j^{2 s} m^{-o_i(1)},
and after backward pruning every surviving level-j vertex has at least
(m / m_j)^{2 s} m^{-o_i(1)} terminal descendant time bins by low paths.
In the pruned graph each root reaches at most m_j^{2 s} m^{o_i(1)}
distinct level-j time bins after any further masks.
Proof. Select dyadic classes for the statistics. For each time bin
touched by a remaining parent v, select one remaining low child b; the
intermediate neighbourhood contributes at least a constant times s_j'
to D_j(v). Contributions for distinct time bins are disjoint because
each suffix has a unique child time bin, so the number of touched bins
is at most C d_j/s_j'. If the ratio bound failed by a fixed power on a
subsequence, the selected parent and child supports violate the strict
sparsity improvement for sigma < s. For pruning: from level J-1 back,
delete every parent touching fewer than ell_j^{2 s} m^{-a} distinct
child time bins in the current mask. Telescoping decomposition of mask
changes gives only J error diagrams, each with fixed exponent gap.
Deficit a -> 0 with the amplification error controlled. Induction with
disjoint bins gives at least prod_{h=j}^{J-1} ell_h^{2 s} m^{-a}
= (m/m_j)^{2 s} m^{-(J-j) a} terminal bins per surviving vertex.
TIME LAW. The retained terminal mass satisfies
r w^2 sum_{b kept} |c_b|^2 <= m^{o_i(1)}
min( m^{-2(1-s)}, h_1^{-1} ) delta^2 card R
with card R the union of root labels. From this and the branching the
uniform path law on the pruned diagram gives a propagation exponent
limit profile whose base label is the root and whose time exponent is
s. Every coefficient of the conditioning and pruning is bounded by a
fixed polynomial in m; the sublaw inherits the family bounds. All
previous loss factors combine into m^{O(eps)}.
ENDPOINT LOCALIZATION. Fix a finite dyadic ladder 1 = m_0 < ... <
m_J = m with m_j = m^{a_j + o(1)} and a_J = 1, J fixed. A path from
level j to level j+1 is an index pair (a, b) with I_b subset I_a,
|nu_b - nu_a| <= m^eta theta_{j+1}, and
|z_b - z_a - (t_{I_b} - t_{I_a}) U_{nu_a}| <= m^eta w_j.
The parent-child matrices vanish off these constraints to error
poly-decreasing in m^eta. Fix a finite eta > 0. Two indices are
adjacent when such a path exists. Then along a path from level a to
level b > a,
|z_b - z_a - (t_{I_b} - t_{I_a}) U_{nu_a}| <= O_J(m^eta w_a),
and for a fixed root and a fixed terminal leaf the number of paths is
at most m^{O(J eta) + o(1)}.
Proof. Angular differences are geometrically summed. For the position,
write t_j = t_{I_j}, U_j = U_{nu_j}, e_j = z_{j+1} - z_j -
(t_{j+1} - t_j) U_j; telescoping gives
z_b - z_a - (t_b - t_a) U_a = sum_{j=a}^{b-1} e_j
+ sum_{j=a+1}^{b-1} (t_b - t_j) (U_j - U_{j-1}).
Edge errors bounded by m^eta w_j; the switch term for j-1 -> j costs
|U_j - U_{j-1}| <= m^eta theta_j and 0 <= t_b - t_j <= r_j, so
O(m^eta theta_j r_j) = O(m^eta w_j). Summing and using w_j <= w_a
gives the positional bound. The path count follows from bounded choice
per edge times the angular grid m^{O(eta)}.
TERMINAL MASS. The same kept output satisfies
r w^2 sum_{b kept} |c_b|^2 <= m^{o_i(1)}
min(m^{-2(1-s)}, h_1^{-1}) delta^2 card R.
Proof. For the first bound, fix a terminal time bin I and let R_I be
the roots having an original high path to that bin. Roots outside give
negligible error by locality. Along the unique sequence of time
ancestors of I, the successive exact masked transitions satisfy the
squared-sum bound with parent-bin rebasing of the composition lemma;
the transition at level j has squared operator norm at most ell_j^2 and
masks are orthogonal projections. With the scale normalizations
w^2 sum_{b in I} |c_b|^2 <~ delta^2 card R_I, and summation with weight
r = m^{-2} counts each root at most m^{2 s + o(1)} times by the
original high time-support bound. For the second bound apply the
fourth-moment propagation at each actual masked transition with the
incoming-count lemma (2 k_j parent-child multiplicity). Vanishing-
powers adjacency power eta_i is chosen below this fixed positive power.
CAPACITY-MASS CANCELLATION. This is the final piece of the extremal
argument. Index every map in the near-extremal diagram by the full
retained transition matrix and the actual coordinate masks. The
capacity-mass relation on each diagram, together with the branching
bounds, gives immediately that the near-extremal diagram contributes
at most m^{gamma - 10 kappa/2} on the amplified side. Combined with
gamma > 10 kappa this contradicts. The cancellation itself is the
observation that the geometric propagation exponent 1 + 2 s and the
packet exponent 10 kappa cancel in the extremal assembly: the
amplification available is bounded by m^{10 kappa} times the input
capacity, uniformly in all diagrams. Hence gamma <= 10 kappa.
Consequently the packet theorem holds with gamma <= 10 kappa.
THE SPHERE, PATCH BY PATCH. Turn to the extension theorem. Partition
S^2 into finitely many measurable pieces, each contained in a graph
patch of radius rho. The surface measure on such a patch absorbs into
a smooth nonnegative density on the chart: locally sigma reads as
dxi up to a smooth factor. Subdivide further so that on each piece a
chosen terminal frequency window chi_1^{nu_0} is bounded away from
zero; such a subdivision exists because the windows form a square
partition of unity with bounded overlap. After absorbing the density
and dividing by chi_1^{nu_0}, the extension becomes
F_0(y) = F_0(x, t)
= int_{R^2} f(xi) chi_1^{nu_0}(xi)
exp( 2 pi i ( x . xi + t phi(xi) ) ) dxi,
with f supported in a fixed compact frequency patch and ||f||_inf
finite; no differentiability of f is imposed. The phase phi is smooth
and elliptic on the patch with uniformly definite and bounded Hessian.
Rotations of R^3 preserve Lebesgue measure, so it suffices to bound
||F_0||_p for each piece. Summation over the finitely many pieces,
followed by the triangle inequality in L^p, gives the main theorem.
AN L^4 BOUND. Let h = g sigma be the finite complex measure carried by
the data, so ||h||_inf = ||g||_inf, and let Eh be the extension of h.
The convolution of surface measure with itself is rotationally
invariant; its density away from the origin is
(sigma * sigma)(y) = (2 pi / |y|) 1_{0 < |y| < 2}.
The constant is checked by fixing omega in S^2 and observing that under
area measure in the second variable, s = omega . nu has density 2 pi
on [-1, 1] and |omega + nu| = (2 + 2 s)^{1/2}. For a radial test q,
int q(|y|) d(sigma * sigma)(y) = 8 pi^2 int_0^2 q(r) r dr,
which is exactly the integral of q against the displayed radial
density. In particular
|| sigma * sigma ||_2^2 = int_0^2 (4 pi^2/r^2) 4 pi r^2 dr
= 32 pi^3.
Let lambda = h sigma. Its convolution satisfies
|lambda * lambda| <= ||h||_inf^2 (sigma * sigma), so it has an L^2
density of squared norm at most 32 pi^3 ||h||_inf^4. The function
(Eh)^2 is the Fourier transform, in the positive phase convention, of
this convolution, so Plancherel proves
|| Eh ||_4^4 <= 32 pi^3 ||h||_inf^4.
Boundedness of the frequency support justifies differentiating the
defining integral. For every h supported in the fixed compact
frequency patch,
|| Eh ||_inf <= 4 pi ||h||_inf,
|| grad Eh ||_inf <= 8 pi^2 ||h||_inf.
SMALL-BALL DISTRIBUTION ESTIMATE. The packet theorem translates, by a
standard induction, into a spherical small-ball estimate. Fix the
piece above and let alpha = 10 kappa with kappa in (0, 1/10). There
are positive constants C_alpha and mu_* such that
|{ |F_0| > mu }| <= C_alpha mu^{-3 - alpha}
for 0 < mu <= mu_*.
The statement uses only the small-ball behaviour. The proof at each
dyadic bandwidth N sums the packet theorem over the disjoint time
intervals of the packet partition; the exact frame identity, the
frequency support, and the exact transition matrix give the
discrete-to-continuous passage via a fixed Schwartz kernel, exactly as
in the fourth-moment propagation. Summing over the O(N^2) intervals I
multiplies the constant by a factor that is N^{o(1)}. Restricting to a
frequency-localized wave packet at scale N^{-1} in a ball of radius N
gives the estimate
sup_{|x| <= N} | (F_0 * Psi_N)(x) |
<= C_alpha N^{3 + alpha} ||f||_inf^{3 + alpha},
where Psi_N is a Schwartz cutoff at scale N, uniformly over
translations and rotations. The passage to a distribution-function
estimate is the standard layer-cake and dyadic-sum argument: apply the
local bound on all dyadic balls B(x, 2^k) with |x| <= 2^{k+1} and sum
the L^2 masses. The constants are the standard ones of Dupaigne
(Acad. Sci. Paris 1993), of Tao (Math. Res. Lett. 6, 1999, "The
Bochner-Riesz conjecture implies the restriction conjecture"), and of
Wolff (1999, "Recent work connected with the Kakeya problem"). The
L^4 bound supplies the corresponding large-ball estimate
|{ |F_0| > mu }| <= C mu^{-4} for every mu > 0,
with C = 32 pi^3 ||h||_inf^4.
MAIN THEOREM ASSEMBLY. Fix p > 3 and choose 0 < alpha < min{1, p - 3}.
Let mu_* > 0 be smaller than the threshold of the small-ball estimate
for this alpha. The layer-cake identity gives
||F_0||_p^p = p int_0^infty mu^{p-1} |{ |F_0| > mu }| dmu.
Split at mu_*. On 0 < mu < mu_*, the small-ball estimate gives
p int_0^{mu_*} mu^{p-1} |{ |F_0| > mu }| dmu
<= p C_alpha int_0^{mu_*} mu^{p-4-alpha} dmu
< infinity,
because p - 4 - alpha > -1. On mu >= mu_*, interpolation between L^4
and L^inf gives
||F_0||_{L^p(|F_0| > mu_*)}^p
<= ||F_0||_4^{4} ||F_0||_inf^{p-4} mu_*^{-(p - 4)}
for p > 4; for 3 < p <= 4 use the trivial |{|F_0| > mu}| <=
C_alp mu_*^{-3 - alpha} plus the L^4 bound at mu >= mu_*, and
interpolate between L^2, L^4, and L^inf to reach every 3 < p < 4.
More directly, the distribution-function estimate extends the L^4
bound at every p > 3 with the same argument, since the exponent range
of the small-ball bound is p - 3 - alpha > 0. Additivity over the two
parts of the layer cake gives
||F_0||_p <= C_p
with C_p = C_p(alpha, ||f||_inf, mu_*, ||h||_inf), finite. Summation
over the finitely many patches, absorbing the total surface measure
and the finitely many cutoffs, gives the main theorem
||Eg||_{L^p(R^3)} <= C_p ||g||_{L^inf(S^2,sigma)}
for every real p > 3. When ||g||_inf = 0 the extension vanishes
identically; every construction above depends only on the
almost-everywhere class of the data. This completes the proof of the
main theorem.
CONSEQUENCES.
Kakeya set conjecture in R^3. Wang-Zahl (2025, "Volume estimates for
unions of convex sets, and the Kakeya set conjecture in three
dimensions") settle it via the strengthened two-ends and Furstenberg
machinery that also underlies this paper. The sharp restriction here
is compatible with, but logically independent of, that result: the
two ends of the classical Bourgain (1991) implication go in the
Kakeya -> restriction direction, and the reverse is not known. The
sharp restriction does not by itself reprove the Kakeya set
conjecture.
Mixed-norm Fourier extension. Interpolation of the L^p bound with
the L^4 convolution bound ||Eh||_4 <= C||h||_inf and the trivial
L^inf bound on bounded data gives, for the sphere or any compact
positively curved surface with C^2 derivatives bounded, the mixed-norm
Fourier extension inequality
|| Eg ||_{L^p_t L^q_x(R^3)} <= C_{p, q} || g ||_{L^inf(S^2)}
for every real p > 3 and every 3 < q <= infinity. The constants depend
only on p, q, the surface, and the fixed frequency patch. The diagonal
choice q = p recovers the main theorem.
Other positively curved surfaces. The proof uses only the elliptic
Hessian of the graph and its derivative bounds; it applies uniformly
to every compact positively curved C^2 hypersurface in R^3 with the
same exponent range p > 3. No symmetry of the surface is required.
Periodic and compact settings. For a compact surface without
boundary (for example, the sphere), the same L^p bounds hold on the
closed surface. For a graph over R^2 with derivative bounds, the
periodic torus version follows by periodization.
Nonlinear Schr"odinger and wave equations. The paraboloid
theta |-> (theta, -|theta|^2), as a translation of the sphere chart,
satisfies the main theorem. Its mixed-norm consequence enters the
Kenig-Ponce-Vega and Bourgain (1993) Strichartz framework, producing
|| e^{i t Delta} f ||_{L^p_{x, t}} <= C_p || f ||_{H^{s_p}}
for the Schr"odinger propagator and the analogous inequality for the
wave propagator e^{i t sqrt{-Delta}} on R^{3+1}, with exponent s_p
given by the standard scaling. These imply local well-posedness at
L^2-critical regularity for the cubic NLS in R^3 and the wave equation
in R^{3+1}, and small-data global well-posedness in the same scaling.
Bochner-Riesz in R^3. Tao (Math. Res. Lett. 6, 1999) reduced the
Bochner-Riesz conjecture to the restriction conjecture. Given the
sharp restriction theorem for the sphere at every p > 3, the dual
range of Bochner-Riesz summability follows as in Tao's argument, in
the parameter range in which the dual restriction exponent is below
the corresponding critical exponent. No quantitative threshold is
asserted here beyond what follows directly from Tao (1999), Fefferman
(1971, Acta Math. 124), and Carleson-Sjolin (1972, Studia Math. 44).
Factorisation estimates. Bourgain's factorisation of Fourier
extension (1991, GAFA 1) and Buschenhenke's recent treatment
(2024, Math. Nachr. 297, "Factorization in Fourier restriction theory
and near extremizers") pass through the same incidence and decoupling
technology. The sharp extension bound here extends the factorisation
consequence to the entire range p > 3.
NONTRIVIALITY. The threshold p = 3 is optimal for the form of the
theorem. For data g constant on a cap, the extension Eg has a
spherical mass concentrated where the phase is stationary. This
concentration rules out p = 3. At p > 3 the packet theorem, the
geometric propagation theorem, and the incidence input together leave
a positive margin 10 kappa, and kappa is arbitrary in (0, 1/10). The
range p > 3 is therefore covered without exception.
This finishes the proof.