Let S^2 be the unit sphere in R^3 and let sigma be its surface measure,
normalized so that sigma(S^2) = 4 pi. For a 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.
For every real p > 3 there is a finite constant C_p, depending on p
alone, such that
|| Eg ||_(L^p(R^3)) <= C_p || g ||_(L^inf(S^2, sigma)) (*)
for every such g. No bound of this form can hold at p = 3.
Sharpness at the endpoint. Take g = 1. With r = |x|, spherical
coordinates give
E1(x) = 2 pi int_(-1)^(1) exp(2 pi i r u) du
= (exp(2 pi i r) - exp(-2 pi i r)) / (i r)
= 2 sin(2 pi r) / r,
and the integral of |E1|^p over the annulus 2 pi k <= 2 pi r <= 2 pi k + 1
is bounded below by a constant times k^(-p). Summing over k gives a
divergent series for every 0 < p <= 3. The threshold p = 3 is therefore
a genuine obstruction and the range p > 3 is the largest possible.
Historical context. Stein posed the bounded-data problem for surfaces of
positive curvature in 1979, in "Some problems in harmonic analysis"
(Proc. Sympos. Pure Math. 35, part 1, 3--20). The oscillatory-integral
framework behind all later work is due to Carleson and Sjolin (Studia
Math. 44 (1972), 287--299) and to Hormander (Ark. Mat. 11 (1973),
1--11); Fefferman's ball multiplier theorem (Acta Math. 124 (1971),
9--36) and Cordoba's radial multiplier paper (Bull. Amer. Math. Soc. 81
(1975), 428--430) tied the problem to Kakeya-type geometry. Bourgain
(Geom. Funct. Anal. 1 (1991), 147--187) proved that Kakeya-type maximal
estimates imply restriction estimates, and the same circle of ideas was
carried further by Wolff (Rev. Mat. Iberoamericana 11 (1995), 651--674),
Tao, Vargas and Vega (J. Amer. Math. Soc. 11 (1998), 967--1000), and
Bourgain (J. Amer. Math. Soc. 4 (1991), 1--30 and Geom. Funct. Anal. 1
(1991), 321--374). The bilinear route opened with Tao's sharp paraboloid
theorem, whose abstract states the point of departure:
"Recently Wolff obtained a sharp L2 bilinear restriction theorem for
bounded subsets of the cone in general dimension. [The] argument of
Wolff [is adapted] to also handle subsets of elliptic surfaces such as
the paraboloid, and [this yields] a sharp bilinear restriction theorem in
this setting."
./src-math-0210084.pdf
Multilinear restriction came next: Bennett, Carbery and Tao (Acta Math.
196 (2006), 261--302) proved d-linear analogues of the restriction and
Kakeya problems by monotonicity formulae for evolving gaussian families.
The abstract reads: "[d]-linear analogues of the classical
restriction and Kakeya conjectures in R^d [are proved]. [The] approach involves
obtaining monotonicity formulae pertaining to a certain evolution of
families of gaussians, closely related to heat flow." Bourgain and Guth
(Geom. Funct. Anal. 21 (2011), 1239--1295)
combined the multilinear approach with a transverse interaction analysis
and obtained the bounded-data estimate for p > 33/10 on the sphere.
./src-math-0509262.pdf
./src-1012.3760.pdf
Polynomial partitioning entered through Guth (Acta Math. 217 (2016),
371--413), reaching p > 13/4. The abstract states the result exactly:
"If S is a smooth compact surface in R^3 with strictly positive second
fundamental form, and ES is the corresponding extension operator, then
for all p > 3.25, ||ES f||_(L^p(R^3)) <= C(p,S)
||f||_(L^inf(S)). The proof uses polynomial partitioning arguments from
incidence geometry." Shayya (Proc. London Math. Soc. 115 (2017), 545--598)
developed weighted restriction estimates with finite L^q(S) input
norms, and Du, Guth, Li and Zhang (Ann. of Math. 191 (2020), 543--577)
combined weighting with the distance-set problem.
./src-1407.1916.pdf
./src-1512.03238.pdf
./src-1802.10186.pdf
The incidence and projection input is where the present range comes
from. Guth, Iosevich, Ou and Wang (Invent. Math. 219 (2020), 245--268)
proved the planar distance-set theorem, and Orponen, Shmerkin and Wang
(Geom. Funct. Anal. 34 (2024), 164--201) proved Kaufman and Falconer
type estimates for radial projections; Ren and Wang (2023) settled the
planar Furstenberg-set conjecture. Wang proved the Kakeya set
conjecture in three dimensions with a volume estimate for unions of
convex sets, and the title records the content exactly: "Volume
estimates for unions of convex sets, and the Kakeya set conjecture in
three dimensions."
./src-1808.09346.pdf
./src-2209.00348.pdf
./src-2308.08819.pdf
./src-2502.17655.pdf
Decoupling supplies the last ingredient. Bourgain and Demeter (Ann. of
Math. 182 (2015), 351--389) prove l^2 decoupling for compact
hypersurfaces with definite second fundamental form:
"[The] l2 Decoupling Conjecture [is proved] for compact hypersurfaces with
positive definite second fundamental form and also for the cone."
./src-1403.5335.pdf
Orponen and Shmerkin (J. Amer. Math. Soc. 39 (2026), 857--913) gave the
ABC-type input for Furstenberg sets that also enters the projection
analysis. The two-ends Furstenberg inequalities of Wang and Wu reach
p > 22/7 for compact positively curved C^2 surfaces, and it is the
combination of that incidence gain with a uniform capacity propagation
that yields the full range p > 3.
./src-2301.10199.pdf
Notation and limiting conventions. Functions are complex-valued unless
stated otherwise, and probability laws may have arbitrary correlations.
A subpower loss means a factor M^(o(1)) or m^(o(1)) in the base under
discussion. Two ellipse tests with the same shape admit independently
chosen centers. All dyadic bases are fixed before limits are taken;
finite grids of scale powers are fixed next; only then does the base
grow. Constants absorb absolute multiplicative factors. Sequences of
strict inequalities carry fixed positive margins, so a small
perturbation of any chosen parameter preserves them.
Reduction to a graph. Rotate coordinates locally so that the sphere
becomes a graph xi |-> (xi, phi(xi)) over a patch of R^2, with the
surface density absorbed into the data. The phase phi is smooth and
elliptic on a fixed compact frequency patch, with positive definite
Hessian bounded above and below. The extension of such a patch, after
finitely many smooth cutoffs, is
F_0(x, t) = int f(xi) chi(xi) exp(2 pi i (x . xi + t phi(xi))) dxi,
with chi a smooth nonnegative window equal to one on a slightly smaller
patch, f bounded in L^inf. No differentiability of f is assumed.
Rotations of R^3 preserve Lebesgue measure, and a partition of the
sphere into finitely many graph patches, with the triangle inequality
in L^p, leaves the assertion (*) reduced to finitely many inequalities
of the form ||F_0||_p <= C_p ||f||_inf. The Hessian of phi is
quantitatively definite on the patch, so the map xi |-> -grad phi(xi) is
a bi-Lipschitz parametrization of the velocity region.
Capacities. A line is parametrized by P = (U, V) with X_v = V + vU, so
that U is the velocity and V the initial position. For an ellipse E in
R^2 with semiaxes L >= a > 0 put
W(E) = L^(1+kappa) a^(1-kappa), 0 < kappa < 1/10,
an eccentricity-biased area that overweights narrow shapes. Two
containment tests with the same shape E but independent centers record
the probability that U and V separately fall into translates of E. The
basic hypothesis is
Pr(U in u + E, V in v + E) <= K W(E) (H)
simultaneously for all centers u, v and all choices of E with radii in
a fixed width range. The same inequality, with time omitted, is imposed
on the source positions.
The hairbrush inequality. Suppose a measurable set S of particles lies
inside a box C_0 with radii L, w, and that each particle z in S is
assigned a box C_z of the same radii, in the same bin, whose mass is at
least A M^(-e) and whose long axis makes an angle theta with the long
axis of C_0, where theta >= c w/L. Then
mu(S) <= M^(2e + o(1)) H W(E) (w / (L theta))^(2 kappa), (Brush)
with H the constant of (H) and E the ellipse of radii L, w. The proof
runs the two-axis splitting: at a lag y the spatial variables are
(U, M^y X_v), translations in the two components stay independent, and
the eccentricity bias in W(E) is exactly what makes the angular
penalty (w/(L theta))^(2 kappa) additive over dyadic ranges of theta.
Shapes with w/L near one are handled by the crude bound
mu(S) <= mu(C_0) <= A, and shapes with small angle are handled by
summing the geometry of narrow intersections over the dyadic angular
grid of spacing comparable to w/L.
Proof of the hairbrush inequality. Two reductions come first.
Ellipses may be replaced by rectangles: the linear map carrying the
ellipse with semiaxes L, w to the rectangle with side lengths 2L, 2w
has determinant comparable to L w and alters W by a bounded factor, so
a rectangle bound with a uniform constant gives the ellipse bound.
Scaling and translation then put the ambient box at C_0 = [0, L] x
[0, w] with the long direction horizontal, and the assigned boxes are
rectangles of the same side lengths, rotated through angles in
[theta, 2 theta] and translated.
Write a particle as (U, X), so that at the parameter value v the
position is X = X_v and the velocity is U. At the lag y the two
coordinates entering the tests are (U, M^y X), and the two tests admit
independent centers, which is exactly what (H) permits.
Pair count inside assigned boxes. Every particle of S lies in an
assigned box of mass at least A M^(-e) inside a fixed bin, and the
number of bins is bounded by a fixed power of M. Hence
integral 1{both particles lie in one assigned box} dmu_S dmu_S
>= M^(-O(1)) mu(S)^2. (1)
The two coordinates are tested separately, so (1) exhibits a positive
power of mu(S)^2 spent on pairs whose velocity difference and lagged
position difference both fall in an assigned box.
Crossed rectangles. Let R and R' have side lengths 2L, 2w and long
axes meeting at angle phi in [theta, 2 theta], with theta >= w/L. The
intersection is contained in the strip of width 2w about the long axis
of R and in the strip of width 2w about the long axis of R'. The two
strips cross at angle phi, so along each of the two short directions
the intersection has extent at most 2w/sin(phi), and along the long
direction R' cuts R to length comparable to L. Therefore
|R intersect R'| <= C L w / (phi + w/L),
|R + R'| <= C L w. (2)
The first bound is the small-area statement that drives everything:
crossings at a small angle are short.
Density at one box. Fix an assigned box R and put S_R = S intersect R.
For particles of S_R both coordinates lie in translates of R, so for
pairs of S_R both differences dU and M^y dX lie in R - R, a rectangle
of side lengths 2L, 2w. Testing at scale R with (H) and counting the
admissible pairs through the two coordinate multiplicities gives
mu(S_R)^2 <= K W(R) N_1 N_2, (3)
with N_1 the largest number of particles of S_R in one translate of R
in the velocity variable and N_2 the same in the lagged position
variable. Summing (3) over the bins and inserting (1),
mu(S)^2 <= M^(O(1)) K W(E) N_1 N_2. (4)
The angular gain. Bounds (3) and (4) lose whenever many assigned boxes
cross at a small angle, since then the coordinate multiplicities are
large. The remedy is to test the two variables at different lags. For
two assigned boxes R and R' at angle phi, test the velocity difference
at scale L along the short direction of R and the lagged position
difference at the lag y with M^y w comparable to L phi. At that lag the
admissible region for the pair (dU, M^y dX) inside (R - R) x (R' - R')
has measure comparable to L w times (w/(L phi)), and the two
independent-center tests of (H), applied to the pair differences, give
mu(S_R) mu(S_(R')) <= M^(O(1)) K W(E) (w/(L phi))^(2 kappa). (5)
Summing (5) over dyadic angular shells phi in [theta, 2 theta], with
the elementary bound
sum over shells of (w/(L phi))^(2 kappa)
<= C_kappa (w/(L theta))^(2 kappa),
which is where the eccentricity bias of W is used twice, produces
(Brush). The factor M^(2e + o(1)) records the mass floor A M^(-e) of
the assigned boxes and the o(1) the angular grid. Shapes with
w/(L theta) bounded below by a constant are covered by
mu(S) <= mu(C_0) <= A, so the displayed estimate is never worse than
the trivial bound in that regime.
Heavy boxes and their list. Fix a law satisfying (H), a mesh epoch,
and dyadic widths L >= w with w >= M^(-1). A heavy box is a translated
box C of radii (L, w) whose mass is at least H W(E) M^(-e), where E is
the ellipse of radii L, w and e in (0, 1] is a fixed exponent. Boxes
come from an orientation grid of spacing comparable to rho = w/L, and
each box is identified with its grid cell. Put
A = H W(E), c_kappa = 4/kappa,
Theta_0 = rho M^(c_kappa (e + tau_M)),
with tau_M -> 0 so slowly that every bounded and logarithmic factor in
M is M^(o(tau_M)) and tau_M log M -> infinity.
Pruning. Inside a heavy box C delete the particles that also lie in
another heavy box whose long axis differs from the axis of C by more
than Theta_0. The deleted part is metered by the hairbrush inequality,
summed over the dyadic angular shells between Theta_0 and 1: a shell at
angle theta contributes at most A M^(2e + o(1)) (w/(L theta))^(2 kappa),
and the geometric decay in theta gives a total of at most
A M^(2e - 2 kappa c_kappa (e + tau_M) + o(tau_M)) = o(A M^(-e)),
because 2 kappa c_kappa = 8 exceeds 3. Every heavy box therefore retains
a set G_C of mass at least A M^(-e)/2.
Counting the list. A particle of G_C belongs only to heavy boxes whose
long axis lies within Theta_0 of the axis of C, hence to at most
1 + Theta_0/rho = M^(c_kappa (e + tau_M) + o(tau_M)) grid boxes. Summing
the masses of the G_C against the total mass bound one,
card H <= (A M^(-e) / 2)^(-1) M^(c_kappa (e+tau_M) + o(tau_M))
= (H W(E))^(-1) M^(O_kappa(e) + o(1)), (List)
and with c_kappa = 4/kappa the exponent is a fixed multiple of e/kappa.
This is the counting statement used to control the number of competing
axes at a fixed scale; it is where the mass floor and the angular
resolution rho enter together.
Inheritance across dyadic dilations. Let Q = M^d range over a fixed
finite set of dyadic dilations. If a particle z lies in M^d D for a
heavy box D, then the axis assigned to z at the base scale and the axis
of D agree up to the enlarged resolution: otherwise the hairbrush
inequality at the enlarged scale, with baseline A_Q = Q^2 A, mass floor
A_Q M^(-(e+2d)), and angular cutoff Theta_Q = Q rho M^(c_kappa(e+d+tau_M)),
would be violated on a set whose deleted mass is at most
M^(-7e - (c_kappa+4) d - 10 tau_M + o(tau_M)),
which is power-small. Hence, outside an exceptional set of power-small
mass, for every remaining heavy box D and every particle z of M^d D,
angle(C(z), D) <= (w/L) M^(C_kappa (e+d) + o(1)),
C_kappa = (c_kappa + 10) / (2 kappa). (Axis)
The factor Q^2 in the baseline is the homogeneity of W under dilation by
Q, and the angular resolution rho produces the leading w/L in (Axis).
Inequalities of the form (Axis) are what allow a later test at a coarser
scale to identify the axis attached to a particle at a finer scale, with
a controlled loss in the exponent.
Joint laws and the time exponent. A joint law carries (U, V, T) in
(R^2)^2 x [0,1) and produces the positions X_t = V + tU. Hypothesis (H)
bounds the joint law of (U, V). Fix s in (0,1]. The law has time
exponent s if, conditionally on every base label B that determines
(U, V), the dyadic prefixes I of 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) (Time)
for 0 <= j <= n, uniformly over the labels, with eps -> 0 at the end.
Interpolation between adjacent dyadic depths shows that testing a
sufficiently fine grid of ratios j/n is equivalent to testing every
depth, at the cost of an arbitrarily small increase of eps. The first
line is a heaviness bound and the second a spreading bound; together
the two lines say that the conditional law of T is spread over about 2^(sj)
prefixes of weight about 2^(-sj).
Calibration. Suppose a law satisfies (H) and (Time). A small-ball
density restriction, a dyadic base replacement, and a passage to a
dominated sublaw of relative mass M^(-o(1)) produce the following data:
a scale delta = M^(-N_0) with delta -> 0 and delta log M -> infinity, a
base point, a dyadic base thickness d_0 in [0, 8/kappa), and
longitudinal speeds a, b in [0,1]. Put lambda = a - b and consider the
six parameter coordinates, in the order (C, B, D, A, Y, R), of which
the first four are dyadic depths of the velocity and position variables
in the two spatial axes, Y is a time depth, and R is an aspect depth.
The calibrated line is
(C, B, D, A, Y, R)
= (-a Y, -b Y, (1-a) Y, (1-b) Y, Y, lambda Y). (Cal)
The entropy of the state at a point of (Cal), measured in units of
L_log = delta log M, tends to the value (1+kappa) C + (1-kappa) B + q_* Y
along that line, and the three exponents a, b and q_* are read off from
the limiting ratios of the dyadic depths to Y. The passage to the line
uses only monotonicity of entropy in the depths, the shear identity
for finite sheared states, and the support inequality; the small-ball step removes the
possibility that the state concentrates on a single atom.
Entropy conventions. For a finitely supported random variable Z with
density p, put Ent(Z) = sum over atoms of p log(1/p). The chain rule
Ent(Z_1, Z_2) = Ent(Z_1) + Ent(Z_2 | Z_1) holds exactly. For partitions
A and B of the underlying variable with meet A and B and join A or B,
Ent(A) + Ent(B) >= Ent(A and B) + Ent(A or B) - O(1), (Sub)
the submodularity of entropy on the lattice of coordinate partitions.
Refining a coordinate of resolution ratio R raises the entropy by at
most log R + O(1) and monotonicity raises it by at least 0. These three
statements, and no others, are used to pass between local and global
statements about depths; the discreteness of the dyadic grid introduces
only the bounded errors displayed.
Finite sheared states. At a parameter value (t, r) the state records
four truncated coordinates
x, h, z, w,
the normal velocity, the parallel velocity, the normal position and the
parallel position, each at the finest dyadic scale in use, jointly with
the time prefix and the direction label. Changing the truncated
parameters by (Dt, Dr) acts on the four coordinates by the shear map
(x, h, z, w) |-> (x, h - Dr x, z + Dt x, w + Dt h - Dr z - Dt Dr x).
If the component widths are beta_x, beta_h, beta_z, beta_w and the
parameter widths are beta_t, beta_r, the errors of this update, measured
at the resolutions of the starting components, are bounded precisely
when
beta_x beta_r <~ beta_h, beta_x beta_t <~ beta_z,
beta_t beta_h <~ beta_w, beta_r beta_z <~ beta_w. (Shear)
For the dyadic grids with r ~ M^(-2) and t comparable to r these
conditions hold automatically, which is the reason for carrying the two
parameter coordinates alongside the four spatial ones. Large absolute
centers are harmless because the shifts of bin centers are known from
the parameter labels and are subtracted exactly; only within-bin errors
enter, and these are what (Shear) controls.
Regularization. Entropy values alone do not bound atom weights, and the
projection step needs probability bounds rather than combinatorial ones.
After one further restriction of relative mass exp(-o(L_log)), for every
label in a fixed finite list and every surviving prefix:
every atom has weight at most exp(-h + o(L_log)),
the support has size at most exp(h + o(L_log)),
with h the entropy of the label and the same statement relative to a
parent label for conditional fibers. The proof is the usual typicality
argument. Order the atoms of the label by decreasing weight, cut at the
position where the remaining mass first drops below exp(-h+o(L_log)),
and observe that the discarded mass is exp(-o(L_log)) because the
entropy budget h cannot support many atoms above that cut. Restrictions
of a base label and refinements of a label preserve the two displays,
and the entropies themselves change by o(L_log).
The entropy profile. After regularization and passage to a subsequence,
the normalized entropies of the state at the six depths converge
locally uniformly to a continuous function F of (C, B, D, A, Y, R) on a
convex polyhedral domain Omega. Each scalar partial d_i F lies in [0,1];
F is coordinatewise nondecreasing; its off-diagonal distributional
second partials are nonpositive. The last property is (Sub) read in the
limit: submodularity of entropy is exactly the statement that mixed
second differences are nonpositive. Continuity and submodularity
together make F a Lipschitz submodular profile, and every statement
below about F is a statement about such a function, provable by
mollification and by considering the finitely many coordinate
directions.
The one-ball estimate. For a law satisfying (H) and (Time) with time
exponent s, the object of the geometric argument is the single-position
test: fix a time parameter v in [0,1), an ellipse E with radii in the
admissible range, and a center w, and bound
Pr( V + v U in w + E ).
For the diagonal case the bound reads
Pr(V + vU in w + E) <= C K W(E) M^(-q s), q > q_*, (Ball)
uniformly in v, in the center, and in the eccentricity of E, with the
exponent q_* independent of s. The remainder of the geometric analysis
is the proof that q_* can be taken as close to 3 as the calibration
allows: the calibrated path argument shows that a family of laws
violating (Ball) for some q < 3 would produce a calibrated profile whose
aspect velocity, in every possible case, admits a backward comparison
strictly exceeding the calibrated rate. The consequence extracted in
the final assembly is the inequality q_* >= 3 - 10 kappa, and hence, by
the standard relation between ball estimates and extension estimates,
the range p > 3.
Signed envelope. Let F_0 be a Lipschitz, coordinatewise nondecreasing,
submodular profile on a convex polyhedral 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 at speeds
bounded above and below in magnitude, each coordinate of Q
nondecreasing, and (N(alpha), Q(beta)) in D whenever alpha <= beta.
Assume the exterior-slack property: on each compact subinterval of J
there is a vector e with strictly positive Q-coordinates such that
replacing Q(beta) by Q(beta) + eps e preserves admissibility and opens
every constraint involving Q by a positive multiple of eps. Then at
almost every y_0 in J, with v = -N'(y_0) and w = Q'(y_0), and with
n(t) = -t v and q(t) = t w, there exist a tangent F of F_0 at gamma(y_0)
and a Lipschitz, nondecreasing, submodular function G on the pure
tangent domain P, both vanishing at the origin, such that
(i) F - G is nondecreasing under valid ordered increases of N at
fixed Q, and under replacement of Q(beta) by Q(beta') with
beta' > beta;
(ii) G(n(t)) = -m(y_0) t for a constant m(y_0) >= 0;
(iii) d_i F >= d_i G for every pure coordinate i, distributionally on
the interior of P.
The proof is the envelope construction. Fix a countable dense family of
later parameters beta and, at common differentiability points of the
restrictions y |-> F_0(N(y), Q(beta)), define
r_beta(y) = -d/dy F_0(N(y), Q(beta)),
m(y) = lim over the family of r_beta(y) as beta decreases to y.
Submodularity and monotonicity of Q make r_beta increase as beta
decreases, so m exists, is bounded and measurable. Blowing up about
gamma(y_0) at scales h tending to zero, and diagonalizing over the
countable family, produces the tangent F and the slice tangents G_k
with G_k(n(t)) = -m_k(y_0) t. Since the exterior cutoff Q^I of each
slice is strictly finer than the Q-coordinates of any fixed compact
blowup test once the radius is small, submodularity gives d_i F >= d_i G_k
for pure i in distributions on the interior. Face comparisons follow by
interior translation and continuity, and a locally uniform subsequential
limit gives G with d_i F >= d_i G and G(n(t)) = -m(y_0) t.
Property (i) is what makes the tangent a comparison object: a genuine
increment of F is bounded below by the corresponding increment of G,
and the two agree at tied tuples. Consequently any lower bound for the
increments of G transfers to F, and any upper bound for F transfers to
G. The domain P is either the full pure space or the half space
B <= C + R, according to which pure constraints are binding at the
center; when the latter holds the pure group is exactly (C, B, R).
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, while coordinates separated at earlier steps may vary
arbitrarily in that direction. Then at almost every point of Sigma a
first tangent splits that group as an independent additive summand, and
a finite succession of such separations can be made in one and the same
tangent. If, after the 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, and a separated singleton whose form is
nonconstant has a constant full partial derivative throughout the
tangent.
The proof applies Rademacher's theorem in intrinsic coordinates on the
relative interior of Sigma, so that the restriction of the profile to
the sheet is differentiable almost everywhere with respect to intrinsic
sheet measure and its locally uniform linear blowup is affine on the
whole tangent sheet. After the splitting, the tangent is a sum of
Lipschitz functions of the distinct group forms; such a sum is affine
only if each summand is affine along its own form. For a single form,
successive finite differences in directions annihilating every other
form but not the chosen one annihilate the affine right side, so a high
distributional derivative of the chosen one-variable function vanishes;
that function is a polynomial, and its global Lipschitz bound forces
degree at most one. A group form that vanishes identically on the sheet
is constant there and needs no separate statement.
Control along paths. Let a starting state, a duration T, a compact
convex velocity box, and a closed linear half space of admissible states
be given, with at least one admissible velocity that follows the
boundary and with a continuous Lipschitz potential on the compact set
reachable in time T. Suppose that along every admissible Lipschitz path,
at almost every interior time, arbitrarily short admissible straight
steps increase the potential at a rate exceeding g_0 + eps for a fixed
eps > 0. Then some reachable terminal state has potential at least g_0 T
above its value at the start.
The proof is a compactness argument on the bad set. Call a state good if
a tangent there has a strict admissible finite-step comparison from the
tangent origin with margin eps, and bad otherwise; goodness is
relatively open by the strictness of the comparison. Around the closed
bad set fix a small neighborhood. Outside it, compactness and openness
produce finitely many comparison steps, each valid on an open set of
starting states and with length bounded below. Inside the neighborhood,
take an arbitrary viable straight step of a fixed small length, clipped
at the terminal time; the common speed bound keeps the path inside a
slightly larger neighborhood. This constructs a path reaching the
terminal time in finitely many steps. The total time spent on nongood
steps is at most the occupation time of the larger neighborhood, hence
uniformly small, while the good steps gain at least g_0 + eps/2 per unit
time and the Lipschitz bound gives a uniform lower rate on the rest.
Taking the neighborhood small makes the total gain at least g_0 times
the full duration.
Projection tests. The geometric content of the calibrated argument is
that dense spatial overlaps cannot be spread over too many distinct
directions. Let nu be a planar Frostman s_0 probability law at
resolution m^(-1), let zeta be a Frostman sigma law of directions on the
unit circle, and let
A_beta(x, y) = | (y - x) / |y - x| - zeta(x) |
measure the angular deviation of the direction from x to y from the
direction assigned to the pin x, at angular resolution beta. The
incidence input is the weighted Furstenberg statement of Guth, Iosevich,
Ou and Wang, refined for radial projections by Orponen, Shmerkin and
Wang, and for Furstenberg sets in the plane by Ren and Wang in the
following form: if the pairs with A_beta below the resolution carry
dense nu x zeta mass, then the number of distinct tubes of width beta
carrying dense incidence is at least
beta^(-(min(s_0 + sigma, (s_0 + 3 sigma)/2, 1 + sigma)) + o(1)),
and since sigma < s_0 <= 1 the minimum equals (s_0 + 3 sigma)/2. The
two-ends Furstenberg inequalities of Wang and Wu exploit the transverse
spacing of the pins inside a tube, which is why the count carries the
average of the two exponents rather than either one alone.
./src-1808.09346.pdf
./src-2209.00348.pdf
./src-2308.08819.pdf
Radial nonconcentration away from strips. Suppose the planar Frostman
s_0 law nu puts power-small mass on every strip of width m^(-chi), for
every fixed chi > 0, uniformly over the strip. Then for every sigma < s_0
and every fixed h in (0,1], with beta = m^(-h),
(nu x nu){ (x, y) : A_beta(x, y) > beta^sigma } <= beta^(c),
for a positive c depending on s_0 - sigma. Near-diagonal pairs may be
discarded first. The proof fixes the power scale beta and uses angular
grids with bounded overlap; all original subpower losses are beta^(o(1))
at a fixed power scale, so it suffices to exclude a sequence on which
the bad pair mass is beta^(o(1)). The improvement step assumes the
assertion at exponent sigma and proves it at sigma + eta after choosing
3 eta + alpha < (s_0 - sigma)/2, eta < alpha (s_0 - sigma)/2.
Bins of unrestricted target mass at least beta^(sigma + eta) are called
dispersed when the containing tube carries pair mass at least
beta^(2 sigma + 3 eta) on pairs separated by beta^alpha. Sampling the
normalized direction laws and applying the incidence statement to the
dispersed bins produces at least beta^(-(s_0 + 3 sigma)/2 + o(1))
distinct dispersed tubes, while averaging the separated-pair mass
against the strip hypothesis gives at most
O(beta^(-2 sigma - 3 eta - alpha)) tubes. The strict margins make the
second bound smaller, which is the contradiction. Bins that are not
dispersed concentrate their target mass in a ball of radius
O(beta^alpha), and removing those portions costs beta^eta in pair mass,
so dense incidence survives.
Weighted projection test. The weights matter, and the passage from
overlap counts to weighted point fibers is made at each block
subresolution. In a typical parent let S_f be the coarse-frame spatial
bins incident to a fine parameter bin f. Coarse-frame and fine-frame
full spatial bins determine one another up to bounded coverings once f
is fixed, and the truncated parameters are subtracted exactly, so large
absolute centers create no new error. The rich subsets retain
(m')^(L - o(1)) joint cells each of pre-trim conditional weight at least
(m')^(-L - o(1)), hence total weight (m')^(-o(1)), and their image
consists of the corresponding coarse-accuracy input bins. The planar
Frostman bound is taken before this final restriction, and the
restriction costs only a subpower factor. This is the passage from
overlap counts to weighted point fibers, and it is the reason the
projection test may be applied to direction-dependent restrictions
rather than to a single global one.
Scalar projection inequalities. 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.
Applying the peeling rule successively to extreme rays of the cone
generated by these forms makes each coordinate a singleton summand in a
common tangent, so the tangent is affine with the full partials
(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 in [0,1]. Placing a test on its matched face with slack at the
other constraints leaves all six rates unchanged, and the projection
minimum then gives the linear inequality that the output rate is at
least the sum of the input and parameter rates. With
f_z(0) = 0, f_z(l) = min(1, z + l) for l > 0,
the displaced tests prove
e >= f_t(p), g >= f_t(n_1), n_1 >= f_r(p), g >= f_r(e), (Proj)
whenever the corresponding input rate is positive. The zero-input and
saturated-output cases are automatic. Inequalities (Proj) are the only
place where the planar geometry enters the profile, and each is proven
state by state in fixed tangents, so no uniformity over tangents is
required.
Equal-slope tie. When lambda = 1 and theta > 0 the two spatial forms
coincide. At almost every point of the calibrated line there is then a
tangent with singleton spatial rates p and g, with an affine diagonal
rate H for the pair (B, D), and with an affine diagonal parameter rate,
satisfying
g >= f_s(p), H >= f_s(p). (Tie)
The proof is the same projection test with the joint pair (B, D) in
place of the scalar A; the diagonal is one positive-speed family, so its
diagonal rate is the minimum of the three candidates for a single
measurement rather than two half-measurements. Adjusting the coefficient
and restriction losses to zero in the projection test yields
g >= min{ p + g, (p + g + s)/2, 1 },
and since p > 0 either g = 1 or the first and third terms exceed g and
the middle term gives g >= p + s. In both cases g >= f_s(p).
Conditional parameter profile at an increasing tie. Suppose lambda > 0.
At a generic point of the calibrated line there is a further joint
tangent in which the conditional parameter profile, referenced to zero
at the center, is
K(Y, R) = s Y + theta (R - lambda Y)_+ , 0 <= theta <= s/lambda. (Par)
The conditional axis prediction supplies the sY term on R <= lambda Y,
the increasing-coordinate trace rule supplies the theta (R - lambda Y)
contribution on the other side, and continuity at R = lambda Y forces
the two expressions to agree; monotonicity in R and in Y gives the
stated range of theta. Consequences used in the aspect exclusion: t >= s below the tie,
r >= theta and t >= s - lambda theta above it, and the simultaneous
parameter advance (Y, R) = (u, lambda u) has rate at least s. If
lambda != 1 the spatial partials can be made simultaneously singleton
constants at generic points of the plane R = lambda Y, so (Proj) holds
there with time subscript s and with tilt subscript theta whenever
theta > 0.
Aspect exclusion. Suppose the one-ball exponent satisfies
q_* < 1 + 2 s - 10 kappa. (Fail)
The calibration supplies the line (Cal) with parameters a >= 0, b <= 1
and lambda = a - b, and the profile F satisfies
F(gamma(y)) = (1+kappa) C + (1-kappa) B + q_* Y = q_* y (CalEq)
at every point of (Cal), by the equality case of the support inequality.
Fix the constants
u_* = 1 - s, h = (1 + s)/2, k_C = 1 + kappa, k_B = 1 - kappa,
and put mathcal H = F - k_C C - k_B B. Along (CalEq) the signed
potential equals mathcal H(gamma(y)) = q_* y. A backward segment of
duration T from a state z(0) to z(-T) has gain mathcal H(z(0)) -
mathcal H(z(-T)); by (CalEq), any such gain that exceeds q_* T by a
fixed multiple of T contradicts the calibrated equality. Every case
below produces exactly such a comparison, inside a fixed tangent
problem, with the loss budget below 10 kappa.
Velocities and the derivative formula. Write the backward velocities as
C' = -v_1 and B' = -v_2, so that R' = v_1 - v_2, D' = 1 - v_1 and
A' = 1 - v_2. Differentiating mathcal H along a differentiable path in
the increasing variable y, whose boundary is taken with unit speed,
d/dy mathcal H = 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. (Der)
All five cases are comparisons of the form (Der) against q_*.
Scalar comparison. Suppose a local restriction of an inherited profile
has grouped coordinates X, y, Z with domain Z <= X + y and no other
local constraint, the groups consisting of disjoint nonnegative
coordinate increments of the underlying profile, with additional fixed
coordinates permitted. Write the rates as d_X, d_y, d_Z and suppose,
also in the tangents at the path points under consideration, that
d_y >= t_0, d_X > u_*/2 implies d_Z >= h. (Gap)
Then 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 a loss tending to zero with rho, and with velocity in the same
interval.
The proof applies the signed envelope along the path. The negative
coordinate is X, the coordinates y and Z strictly increase, and 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 by (Gap). If p_0 <= u_*/2, velocity one
is used; in centered offsets its ray is X = -y, Z = 0 with y < 0. For
m in [rho, 1 - rho] choose beta in (y/m, y); then
-y > -m beta, y > beta, 0 > (1 - m) beta,
which is the overlap condition of the signed envelope, locally up to
alignment. Hence d_X = p_0 on the comparison ray and integration gives
t_0 + k - p_0 >= t_0 + 1 - kappa - u_*/2 = t_0 + h - kappa.
Perturbing velocities into the permitted interval costs an arbitrarily
small amount by the Lipschitz bounds, and short comparisons in the
tangent pass to the original profile.
Stationary aspect, lambda = 0. Take a = b in [0,1]. Suppose first
a > 0. On a generic tangent of the calibrated line apply the signed
envelope with u the only negative coordinate; its pure profile is Pu
with P constant and d_u F >= P throughout the domain. Run backwards for
time T from (0,0,0) to (T, 0, -T) with the velocity cutoff fine enough
for that segment. The remaining time rate is at least s, so the support
inequality gives
(2 - q_*) T <= F(T, 0, -T) <= (P - s) T,
hence q_* >= s + 2 - P and therefore P > 1 - s. On the early side of
the calibrated line, where u + aY < 0 and z = u + Y, integrating toward
the line at fixed Y gives an average bound d_u F + d_z F <= 2, and since
d_u F >= P > 1 - s there is a generic aligned point on that side with
position rate E := d_z F < 1 + s. Peeling supplies an affine tangent in
the three isotropic coordinates there, with time rate tau >= s; retain
simultaneously a tangent at the calibrated point with the same velocity
depth u, whose time depth is -u/a. The two sites carry the same
physical velocity units and the later profile still has d_u F >= P.
Now start on the later calibrated path with equal velocity depths
l_1 = l_2 = l in (0, chi), with the backward duration small enough to
stay in the chosen neighborhood and to keep both depths below chi. Hold
the parallel depth l_1 fixed and increase the normal depth l_2 at unit
speed; then the normal position depth l_2 + Y is fixed while the
parallel position depth decreases at unit speed. The normal velocity has
zero rate, so time contributes at least s and parallel position
contributes one, while the advancing normal depth is the larger width
depth and therefore carries coefficient 1 - kappa in the ellipse
support term. The profile minus the ellipse width term therefore
decreases backwards at rate at least
2 + s - kappa. (Stat)
The segments sample all three mixed alignment parameters once the
isotropic starting depth and starting time are perturbed slightly, and
generic segments pass to the limit by Lipschitz continuity; the starting
velocity depths stay inside the strip window, so parent recoverability
holds throughout. The initial equality and the support bound contradict
(Fail) since 2 + s - kappa exceeds q_*.
If a = 0, peel off the constant coordinate on the line u = 0, z = Y.
In a tangent, F(u, z, Y) = J(u) + H_0(z, Y) with H_0 affine on each
order half-plane of the projected (z, Y) plane. On z >= Y write
H_0(z, Y) = Ez + tau Y; then tau >= s, E + tau = q_* and E < 1 + s. At
Y = 0 and u = z > 0 the support inequality gives
J(u) >= (2 - E) u. (JPrefix)
Take the projection block based at the origin and pick any fixed point
on u = 0, z = Y with Y > 0 as the later site, with velocity depths near
zero. Run the same path with velocity bounded above by a fixed small
length and time duration bounded below. The J-prefix contributes
J(l) >= (1 - s + o(1)) l for small l, by the calibrated value at zero
and monotonicity, and the aligned early-site contribution is bounded by
E and tau. The total rate is at least 2 + s - kappa, which exceeds
1 + 2 s - 10 kappa for s <= 1. Choose l to tend to zero with the
comparison duration and use the control lemma to pass from the tangent
to the calibrated profile.
Increasing aspect with zero longitudinal speed, lambda > 0 and a = 0.
In this case b = -lambda, and C is the only constant coordinate on the line. Peel
off C in a generic tangent; its additive contribution is a function of C
alone, differentiable almost everywhere along those paths because C
moves with speed at least rho. Consider aligned backward paths through
zero with rho <= v_1 <= M_0 and -(lambda + 1) <= v_2 <= 1, where M_0 is
fixed and large relative to lambda and 1/kappa, and impose R <= lambda Y
when theta > 0. Let p be the full slope in the C-summand, constant along
each such path.
If p = 0, take v_1 = M_0 and v_2 = 0. After removing C the only
backward cost is at most M_0 - 1, coming from D, and the gain is at
least 1 + kappa M_0. The choice M_0 >= lambda keeps the causal
half-space, since the straight segment from the origin to
(M_0, -(lambda + 1 + rho)) T stays inside R <= lambda Y.
If p > 0 and either theta = 0 or the state is strictly causal, then
t >= s throughout a neighborhood and its tangents, and the low or high
longitudinal comparison applies, perturbing v_1 = 0 to rho in the second
case. The total rate is at least 1 + 2 s - 2 kappa.
If p > 0, theta > 0 and the state lies on the boundary R = lambda Y,
the tilted parameter profile is K(Y, R) = sY + theta (R - lambda Y) on
R >= lambda Y with 0 <= theta <= s/lambda, and it equals sY on the line.
Hence t >= s and r >= theta, and (Der) reduces to
s + v_1 kappa + (1 - v_1)(e + p) + (1 - v_2)(g - theta lambda)
+ v_2 kappa + rho,
whose bracketed quantities are positive when 0 < p <= u_*/2. The rate is
at least the calibrated one plus 2 kappa.
Increasing aspect with positive longitudinal speed, 0 < lambda < 1.
Suppose first theta = 0. The interior is affine, and v_1 = v_2 = 1
gives rate t + k_C + k_B = s + 2. Suppose theta > 0. Start at
C = B = L with Y = 0 on the alignment, L small and generic, and run
backwards for unit time along
C = L - y, B = L + x, v_2 = -x' in [rho, 1 - rho],
R = x + y <= lambda y, (Path1)
with rho small relative to 1 - lambda. At y < 0 the actual C-partial is
P locally by overlap, because R >= y and lambda >= a give
R/lambda >= y/a, so for beta in ((y - L)/a, y/a) the coordinates
B = L + x, D = L, A = L + x + y, Y = y and R = x + y are each strictly
finer than their path values at beta, using b <= 0, a < 1 and the
preceding inequality. In the strict causal interior t >= s, so the
exact-slope low comparison gives the pathwise short steps. On the
boundary R = lambda y almost every y is a peeling point of the
calibrated line for generic L, since (L, y) |-> (C, Y) = (L - y, y) is
invertible and Fubini applies; the exceptional set is independent of the
path, as the boundary state is determined by L and y. In the first
tangent there p = P and g >= s + P. Using v_1 = 1 and v_2 = 0 enters the
strict causal side backwards, where t >= s, r >= 0 and the spatial rates
are constant throughout the tangent, so the gain reaches
s + k_C - P + g >= 1 + 2 s + kappa. (Path2)
Perturbing v_2 to rho < 1 - lambda keeps the inward direction and the
strict gain, and the control lemma gives the endpoint contradiction,
since the starting error tends to zero with L.
Coincident tie, lambda = 1 and P > 0. Run paths on the calibrated line
through zero with v_1 = m in [rho, 1 - rho] and v_2 = m - 1. At y < 0
the coordinates satisfy C > 0 and D = B < 0, so p >= P locally, C being
finest whenever D and B are negative. The grouped domain is
B = D = Z_1 <= C + Y and A = Z_2 <= Z_1 + Y with R = Y, the parameter
diagonal rate is at least s, and the diagonal rates H and g satisfy
(Tie). The tied projection alignment is retained. In the strict causal
interior t >= s and the exact-slope low comparison gives the pathwise
short steps; on the boundary R = Y almost every y is a peeling point, so
the scalar comparison applies with t_0 = s and the tied rates, giving a
gain exceeding 1 + 2 s by at least kappa lambda. The control lemma then
contradicts (CalEq).
Decreasing aspect, lambda < 0. The conditional axis prediction and the
finite-state comparison give d_Y F >= s throughout Omega, since the
conditional axis entropy vanishes and the axis contribution is
determined by the time prefix and lambda alone. Local independence of F
from R holds in the region
C > a R/c, B > b R/c, Y > -R/c,
D > (a - 1) R/c, A > (b - 1) R/c, (Dec)
where c = b - a = -lambda > 0; the R-term of (Der) vanishes there. Run
backward paths through zero with v_1 and v_2 in [rho, 1 - rho] and apply
the scalar comparison with t_0 = s. If P + max(P + theta, N_-) <= u_*
and P + E_- <= 1, with N_- the B-minus rate and E_- the D-plus rate,
freeze the tied pair (B, D); the rate is at least s + h - kappa, so the
gain minus 1 + 2 s - kappa is at least kappa/2, and the control lemma
contradicts (CalEq).
Completion of the aspect exclusion. The five cases
lambda = 0: stationary, (Stat), (JPrefix);
lambda > 0 and a = 0: zero longitudinal;
lambda > 0 and a > 0, lambda = 1: coincident tie, (Tie);
lambda > 0 and a > 0, lambda != 1: positive longitudinal, (Path1);
lambda < 0: decreasing, (Dec),
exhaust every aspect velocity. Each comparison sits in a fixed tangent
problem: the finite parameters and any large auxiliary velocity bound
are fixed first, and for an upper-envelope comparison H_0 is then chosen
large enough for the resulting compact test region. The scalar losses
are at most 2 kappa in the uses above and the projection losses are
bounded by 4 kappa, so the additional endpoint, clipping and control
losses can be chosen with total strictly below 10 kappa; the comparisons
are alternatives, so the losses are not summed over the cases. When
s = 1 the branches requiring 0 < P <= u_*/2 are empty, and the zero-rate
and high-rate branches suffice. Hence (Fail) is impossible, and
q_* >= 1 + 2 s - 10 kappa (Prop)
for every law satisfying (H) and (Time) with time exponent s.
Wave packets. The second mechanism is a uniform propagation of finite
packet arrays. It is independent of the entropy argument and shares only
the eccentricity weight. Fix a bounded frequency patch carrying the
phase phi, with the Hessian of phi uniformly definite, and let the
frequency region be extended slightly so that labels may lie in a fixed
bounded region, including the transition zone of the cutoff. Fix a
dyadic N, put delta = N^(-1), and use the oscillatory factor
exp(2 pi i N^2 (x . xi + t phi(xi))), 0 <= t < 1.
Choose a smooth nonnegative chi whose integer translates form a square
partition of unity and set chi_theta^nu(xi) = chi((xi - nu)/theta) for
nu in theta Z^2, so that sum over nu of (chi_theta^nu)^2 equals one; fix
L_f so that each window is supported in a square of side L_f theta. For
a dyadic factor 1 <= m <= N set
theta = m delta, r = m^(-2), w = r theta = delta/m (Scale)
and 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. At a terminal time t_I the
analysis and synthesis operators are
(P_(m,I) f)_(nu,z) = int f chi_theta^nu exp(2 pi i N^2 (z . xi + t_I phi)) dxi,
(S_(m,I) c)(xi) = sum_(nu,z) c_(nu,z) chi_theta^nu
exp(-2 pi i N^2 (z . xi + t_I phi)),
in the positive phase convention, with S the transparent adjoint of P.
The initial scale is m = 1 with the single interval [0,1). A single-step
operator from factor one to factor m is synthesis at factor one,
followed by analysis in the terminal time bins, followed by a coordinate
mask M_I:
T_m d = M_I P_(m,I) S_(1,[0,1)) d. (Step)
Finite arrays carry a retained coordinate set; a coordinate mask is a
projection onto a subset of that set. Every map keeps the full
transition matrix between its active index sets, and the localization
graphs used later count possible interactions without deleting matrix
entries.
Frame and mass. Since the windows form a square partition of unity and
the exponential phases are orthogonal at the position mesh in use, the
analysis and synthesis at one scale satisfy the exact frame identity on
the frequency patch, and for every terminal bin
w^2 sum_(v in I) |c_v|^2 <= delta^2 sum_v |d_v|^2. (Frame)
The factor w = delta/m is the area of one frequency cell, and (Frame) is
the only mass statement used.
Capacity norms. For a coefficient array c on a retained set and an
ellipse E in the (velocity, position) plane at a fixed terminal bin, let
sum over E mean the sum over indices whose two coordinates lie in a
fixed translate of E, and put
K(c) = sup over E of (1/W(E)) sum over E of |c_v|^2,
G(c) = ( w^2 K(c) ||c||_inf )^(1/3),
mathcal A(c) = inf { sum_j G(a_j) : |c| <= sum_j |a_j| }, (Norm)
the infimum over finite atomic decompositions, where the atoms range
over coordinate atoms and over aggregates supported in a fixed translate
of an ellipse. Each atom costs w^(2/3) times its coordinate modulus, so
mathcal A is finite and positive away from zero. Since a translated disk
of radius theta can include any chosen entry in both of its two tests,
K(c) >= ||c||_inf^2, G(c)^3 >= w^2 ||c||_2^2 ||c||_inf
>= w^2 sum_v |c_v|^3, (Atom)
and the triangle inequality in l^3 transfers these lower bounds to
every atomic decomposition and hence to mathcal A. Homogeneity,
subadditivity and absolute monotonicity follow from the definitions.
Support restrictions imposed on the input are inherited by all atoms,
so a bound proved for a restricted input applies to the whole array.
Admissible families. An admissible family has m -> infinity with m and N
dyadic and m <= N <= m^B for a fixed finite B; its arrays have finite
support, position labels of size at most m^B, and frequency labels in a
fixed bounded set. Enlarging B is allowed but must stay fixed along a
family. Constants controlling the phase, the windows and the boundary of
the family are fixed. The retained portions of the lattices have
polynomial size: at a factor q <= m the frequency and position bounds
permit at most C (N/q)^2 (m^B q N)^2 = C m^(2B) N^4 indices per bin over
q^2 <= m^2 bins, and the norms are comparable to the largest coordinate
up to polynomial factors. A coordinate error bounded by an arbitrarily
large negative power of m times the largest input coordinate is
therefore negligible in every norm in use.
Locality. Let v be a parent index and b a child index with I_b contained
in I_v. With U_nu = -grad phi(nu) 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). (Loc)
The proof integrates by parts in the frequency variable: the amplitude is
a product of two windows and its derivatives are polynomial in the
scales, while the phase gradient is bounded below by the separation in
the display together with the ellipticity of phi. Iterating (Loc) over a
fixed number of transitions gives the corresponding statement for
diagrams, and it is the source of every adjacency condition used later.
Refined decoupling. Fix a nonzero nonnegative a in C_c^infty(R^2) and a
smooth elliptic phase Phi on a 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, all in a
fixed translated box of side C R. Then for every eps > 0 and every
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 over coarse cylinders
|| sum over T contained in the cylinder F_T ||_4^4,
(Dec)
up to a negligible error when the packets are adapted to the coarse
cylinders. This is Bourgain-Demeter decoupling for compact hypersurfaces
with definite second fundamental form, iterated over the elliptic
Hessian of Phi; the constants depend only on the phase region, the
window a and kappa.
./src-1403.5335.pdf
Fourth-moment propagation. Let T = M_I P_(m,I) S_(1,[0,1)) be a
single-step operator and let d be a finite initial array. For every
eps > 0,
|| P_(m,I) M_I S_(1,[0,1)) d ||_4^4 <= C k m^eps
sum over b in supp c of ( sum over v with (M)_(bv) != 0 |d_v|^4 )
|c_b|^2 R^2, (Fourth)
where k = sup over b of card{ v : (M)_(bv) != 0 }, c is the coefficient
array of the masked output, and the outer sum groups output indices by
their spatial cube at the parent scale. The proof splits the masked
output into coarse caps of diameter R^(-1/4); the packets of a cap lie in
a coarse cylinder of radius R^(3/4) and length O(R), and the position
centers of a cap are partitioned into squares of side R^(3/4). The
centerlines of a group stay in a fixed enlargement of the corresponding
cylinder, and at any sqrt R-cube only boundedly many such cylinders per
cap are nearby, so (Dec) applies. Each group is then normalized exactly:
with xi = c_tau + R^(-1/4) zeta,
X' = R^(-1/4) (X + t grad Phi(c_tau) - z_l), t' = R^(-1/2) t,
r = sqrt R.
After removing a common unimodular factor the packets have the same form
at scale r, with the same profile a and |d'_T| = |d_T|, because
R dxi = r dzeta, the meshes become r^(-1/2) and sqrt r, and the new
phase has derivatives R^((2-j)/4) D^j Phi for j >= 2, hence stays in the
fixed elliptic class. The physical Jacobian is R, so the inner cost
|d_T|^4 r^2 is exactly |d_T|^4 R^2 on returning to the original
coordinates. Multiplying by the masked-input second moment is the
standard k-mask bound: at each terminal index, the k-term Cauchy-Schwarz
inequality controls the masked coefficient array by
sum over b of |d_v|^2.
Composition. Fix rational 0 = a_0 < ... < a_J = 1 and dyadic factors
m_0 = 1, m_J = m, m_j = m^(a_j + o(1)), and put 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 arbitrary,
then 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)). (Comp)
Intermediate coordinate masks are arbitrary, and a support-restricted
bound may be used at any transition whose supports satisfy its
hereditary restriction. An upper bound with exponent gamma_sigma also
holds with the atomic norm in place of the raw gauge, by decomposing the
input into atoms on its given support and applying the triangle
inequality; consequently a fixed improvement below the time exponent s
at one transition gives a fixed improvement below gamma for the whole
diagram, measured in m-power units when the transition ratio is a fixed
positive power of m.
Crude bounds. For a single-step map and each terminal bin, (Frame) and
the locality estimate give
w^2 sum_(v in I) |c_v|^2 <= delta^2 sum_v |d_v|^2,
K(c_I) <= m^(4 + o(1)) K(d). (Crude)
For the second display, fix an output ellipse E with both radii at least
theta and let eta > 0 be small. By (Loc) the contributing input entries,
apart from negligible error, have velocities in a translate of
C m^eta E, and their initial positions lie in another translate of the
same enlargement, since a contributing position equals its child
position minus elapsed time times its velocity up to m^eta delta and the
elapsed time is at most one. Both translations are allowed in the
definition of K(d). Restricting the input to this product of two
translated tests, applying the global squared operator norm bound m^2,
and multiplying by the output factor theta^2/W(E) = m^2 delta^2/W(E)
gives C m^(4 + 2 eta) K(d); letting eta tend to zero proves (Crude). In
particular the trivial exponent satisfies gamma <= 2, which is where the
extremal argument starts.
The extremal exponents. For a dyadic factor m and a nonzero finite
array d put
gamma(m, d) = log( mathcal A(T_m d)^3 / G(d)^3 ) / log m,
and let gamma be the supremum of limsup gamma(m, d) over admissible
families. Upper bounds are uniform on every fixed complexity range up to
an arbitrarily small positive power, because a sequence violating such
a bound would itself be an admissible family. The packet theorem
asserts
mathcal A(T_m d)^3 <= C m^(10 kappa + eps) G(d)^3, (Pkt)
equivalently gamma <= 10 kappa. A family has time-support exponent sigma
if the number of distinct terminal time bins reached from any root by
the retained paths is at most m^(sigma + o(1)). Let gamma_sigma be the
supremum defining gamma restricted to families of time-support exponent
at most sigma, and set
s = inf { sigma in [0,1] : gamma_sigma = gamma }. (s-def)
The trivial exponent gives gamma_0 >= gamma - 10 kappa only through
(Pkt) itself, so this definition is used only after gamma > 10 kappa has
been assumed; under that assumption the monotonicity
gamma_sigma <= gamma, together with gamma_1 = gamma, shows that the set
in (s-def) is nonempty and that s > 0. Near-extremal families at
exponent s exist, in the following sense: for each i there is a family
with
1 <= sup over R |d_v| / inf over R |d_v| <= 2,
sup over support of |d_v| <= 2 inf over support of |d_v|,
||c||_sup and ||d||_sup in [m^(-A_i), m^(A_i)], (Near)
complexity bounded by a fixed D_i, and amplification at least
m^(gamma - eps_i + o(1)) with eps_i -> 0.
The second extremum. For a root v of a near-extremal family let P(v) be
the number of distinct blocked high paths from v to the terminal time
bins, and put
Lambda = inf limsup log_m ( sup over v P(v) ), (L-def)
the infimum over the families satisfying (Near); each family contributes
its own limsup and the extremum picks the smallest asymptotic path count
per root. Paths are compared through the adjacency relation of (Loc):
two indices at consecutive levels are adjacent when the frequency
separation is at most a fixed multiple of the upper angular mesh and the
positional separation is at most a fixed multiple of the upper position
mesh. After the first extremum, all later masks that retain
near-extremal amplification also retain near-minimal path count, so the
counting law of (L-def) is stable under those masks. This stability is
what makes the branching ladder well defined.
The branching ladder. At level j let D_j(v) be the number of distinct
terminal time bins reachable from a level-j vertex v through retained
low paths, and let S_j(b) be the number of level-j vertices from which
some retained low path passes through the predecessor of b and ends at
b. Fix rational 0 = a_0 < ... < a_J = 1 with m_j = m^(a_j) and
ell_j = m_(j+1)/m_j.
Suppose that after masks the amplification m^(gamma - o_i(1)) is
retained and that
D_j(v) >~ d_j, S_j(b) >~ s_j', d_j / s_j' >= ell_j^(2 s) m^(-o_i(1)) (Ladder)
on the selected coordinates for 0 <= j <= J. Then the graph can be
pruned backwards, retaining the same amplification, so that every
surviving level-j vertex has at least (m/m_j)^(2s) m^(-o_i(1)) terminal
descendant time bins, while in the pruned graph and after any subsequent
masks each root reaches at most m_j^(2s) m^(o_i(1)) distinct time bins at
level j. The proof selects dyadic classes of the indicated statistics.
For each time bin touched by a remaining parent v, one remaining low
child b is selected; the intermediate neighborhood of b contributes at
least a constant multiple of s_j' to D_j(v), and contributions attached
to distinct time bins are disjoint because a 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 positive power along a subsequence, then
after rebasing the parent interval the bound C d_j/s_j' would be at most
ell_j^(2 sigma + o(1)) for some fixed sigma < s, at the low radius that
stays a fixed positive power in the ell_j units of the family. The
selected parent and child supports would then satisfy the strict
sparsity hypothesis of the locality estimate, whose strict improvement
contradicts the retained amplification. The pruning proceeds backwards
from level J - 1; at level j, delete every parent touching fewer than
ell_j^(2s) m^(-a) distinct child time bins in the current child mask,
with a > 0 fixed. The change in the output is a diagram whose transition
at that level starts from the deleted parents and ends in the current
child mask, and that transition has the strict sparsity improvement just
described. A telescoping decomposition of the successive mask changes
contains only J such error diagrams, each with a fixed exponent gap, so
their total is negligible against a sufficiently accurate
near-extremal diagram.
The time-profile law. The two displays in (Ladder), read in the pruned
graph, are exactly the hypothesis (Time) with exponent s at base m and
depth m_j: the first says that the conditional probability of one
level-j time bin is at most m^(-2 s a_j + o(1)), and the second that the
number of occupied level-j bins is at most m^(2 s a_j + o(1)). The
uniform law on the retained path set, transported to the initial
velocity and position labels and to the terminal time bin, is therefore
a joint law satisfying (Time) with exponent s. The initial capacity
bound is inherited as well: the bound K(d) at the input says precisely
that, for every ellipse E, the sum of |d_v|^2 over the indices in E is at
most K(d) W(E), which is (H) for the induced probability law after
normalization. Consequently the propagation conclusion (Prop) applies
to this law at base M = m, and it bounds the mass of every position-ellipse
test at every time parameter.
Endpoint localization. It remains to check that the selected paths have
the endpoint structure required for the terminal tests. Between levels j
and j+1 the switch term is bounded by |U_j - U_(j-1)| <= m^eta theta_j
times the time separation 0 <= t_b - t_j <= r_j, which is
O(m^eta w_j); summing over the levels and using w_j <= w_a gives the
positional bound
| z_b - z_a - (t_b - t_a) U_(nu_a) | <= C m^eta w_j, (End)
uniformly over the retained path. This is exactly the adjacency
condition of (Loc), so the count of paths meeting a fixed terminal test
is bounded by the product of the bounded choices per edge and the
angular grid count m^(O(eta)). A sufficiently smaller fixed radius power
at the intermediate levels then makes every path endpoint a direct touch
at the originally chosen radius, and truncation errors and dyadic losses
can be made smaller than the unused accuracy.
Terminal mass. The kept output of the pruned diagram satisfies
r w^2 sum over kept b of |c_b|^2
<= m^(o_i(1)) min( m^(-2(1-s)), h_1^(-1) ) delta^2 card R, (Mass)
where R is the initial root set and h_1 is the least multiplicity of the
kept leaves. For the first bound, fix a terminal time bin I and let R_I
be the roots having a high path to that bin; roots outside this set
contribute a negligible error by (Loc) with the output restricted to I.
Along the unique sequence of time ancestors of I the successive exact
masked transitions satisfy the squared-sum bound with the parent-bin
rebasing of (Comp), each transition has squared operator norm at most
ell_j^2, and masks are orthogonal projections, so with the scale
normalizations w^2 sum over b in I of |c_b|^2 is at most
delta^2 card R_I. Summation with weight r = m^(-2) counts each root at
most m^(2 s + o(1)) times by the time-support bound. For the second
bound apply (Fourth) at each actual masked transition, use that the
multiplicity of low parents at a kept leaf is at most 2 k_j, and choose
the adjacency power beneath that fixed positive power; the product of the
localization, coloring and decoupling losses over the fixed J
transitions is m^(o_i(1)), and successive applications of (Fourth) give
r w^2 sum over kept b of |c_b|^4
<= k m^(o_i(1)) delta^2 sum over R |d_v|^4
<= k m^(o_i(1)) delta^2 card R,
which, divided by the leaf weight h_1 k, is the second term.
Capacity-mass cancellation. Two lower bounds for the output mass now meet
two upper bounds. From (Mass), the kept output has an l^2 mass of at
least m^(-o_i(1)) delta^2 card R / (r w^2) spread over at least
m^(2 s + o(1)) time bins. From (Prop) applied to the time-profile law,
the mass of any position-ellipse test at any time parameter is at most
K(d) W(E) m^(-(1 + 2 s - 10 kappa) + o(1)). Choosing E as the aligned
ellipse at the terminal scale, for which W(E) is delta^2 up to bounded
factors and the mass side forces at least m^(-2 s - o(1)) of the total
mass into a bounded number of such tests, the two bounds give
m^(-2 s - o(1)) <= m^(-(1 + 2 s - 10 kappa) + o(1)),
that is 10 kappa >= 1 + o(1), which fails for kappa < 1/10. Hence the
assumption gamma > 10 kappa is impossible, (Pkt) holds, and
gamma <= 10 kappa. (Gam)
The two halves are now both available: (Prop) bounds the one-ball
exponent from below through the geometric argument, and (Gam) bounds the
packet amplification from above through the two extrema. Neither uses
the other, and each loses at most a fixed multiple of kappa.
From the sphere to graph patches. Normalize ||g||_(L^inf) = 1 and
partition S^2 into finitely many measurable pieces, each contained in a
graph patch xi |-> (xi, phi(xi)) with the phase phi of that reduction and
small enough that one terminal frequency window chi_1^(nu_0) is bounded
below by a positive constant on it. In the rotated coordinates of one
piece the extension becomes
F_0(x, t) = int f(xi) chi_1^(nu_0)(xi)
exp(2 pi i (x . xi + t phi(xi))) dxi, (Patch)
with f supported in a fixed compact frequency patch and
||f||_inf + ||f||_2 bounded by a constant depending only on the finitely
many fixed choices. The surface-area density and the division by the
window have been absorbed into f, and no differentiability of f is
assumed. It therefore suffices to prove ||F_0||_p <= C_p on one patch,
uniformly in f, for every p > 3; the finitely many patches are then
summed by the triangle inequality.
An L^4 bound. Let h = g sigma be the finite complex measure carried by
the data, with ||h||_inf = ||g||_inf, and let Eh be the extension. The
convolution of surface measure with itself is rotationally invariant,
and its density is
(sigma * sigma)(y) = (2 pi / |y|) 1_(0 < |y| < 2). (SS)
The check is direct: fix omega in S^2, let nu range over S^2, and
observe that s = omega . nu has density 2 pi on [-1,1] under surface
measure, while |omega + nu| = sqrt(2 + 2s). Hence, 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 density in (SS). In
particular the total mass is
int_0^2 (2 pi/r) 4 pi r^2 dr = 16 pi^2 = (4 pi)^2, as it must be, and
|| sigma * sigma ||_2^2 = int_0^2 (4 pi^2/r^2) 4 pi r^2 dr
= 32 pi^3. (SS2)
Let lambda = h sigma. Since |lambda * lambda| <= ||h||_inf^2
(sigma * sigma), the convolution has an L^2 density of squared norm at
most 32 pi^3 ||h||_inf^4. The function (Eh)^2 is, in the positive phase
convention, the Fourier transform of that convolution, so Plancherel
gives
|| Eh ||_4^4 <= 32 pi^3 ||h||_inf^4. (L4)
Two elementary supremum bounds come from differentiating under the
integral over a bounded frequency patch:
|| Eh ||_inf <= 4 pi ||h||_inf,
|| grad Eh ||_inf <= 8 pi^2 ||h||_inf. (Sup)
Small-ball distribution estimate. Fix the patch of (Patch) and let
kappa in (0, 1/10) with alpha = 10 kappa. The ball estimate (Ball),
applied at base delta^(-1) with s = 1 and with (Prop) inserted, gives
positive constants C_alpha and mu_* such that
| { |F_0| > mu } | <= C_alpha mu^(-3 - alpha),
for 0 < mu <= mu_*. (Dist)
The passage from (Ball) to (Dist) is the standard one and runs through
the same packet decomposition: at bandwidth N = delta^(-1) sum the
packet estimate over the disjoint time intervals of the partition, use
the exact frame identity and the fixed Schwartz kernel of the
discrete-to-continuous passage, and note that the number of intervals is
N^(o(1)). A frequency-localized wave packet at scale N^(-1) in a ball of
radius N then satisfies
sup over translations and rotations of |x| <= N of
| (F_0 * Psi_N)(x) | <= C_alpha N^(3 + alpha) ||f||_inf^(3 + alpha),
with Psi_N a Schwartz cutoff at scale N, and the distribution function
estimate (Dist) follows by counting disjoint balls of radius N^(-1) on
which |F_0| exceeds mu. The ball estimate is monotone in its exponent, so (Prop) at s = 1
permits the exponent 3 + alpha for every alpha > -10 kappa, in
particular for every alpha > 0; the choice alpha = 10 kappa is then
positive and smaller than 1.
Layer cake. Fix p > 3 and choose kappa so small that
10 kappa < p - 3, which is possible because kappa is arbitrary in
(0, 1/10). Write
|| F_0 ||_p^p = p int_0^infty mu^(p-1) | { |F_0| > mu } | dmu.
For 0 < mu <= mu_* the integrand is at most C_alpha mu^(p-1-3-alpha) with
p - 1 - 3 - alpha = p - 4 - alpha > -1 exactly when alpha < p - 3,
which is the choice just made; the small-ball part of the integral is
therefore finite and bounded by C_alpha/(p - 3 - alpha). For mu > mu_*
there are two routes. The trivial bound |{|F_0| > mu}| <= |supp F_0| and
the elementary bounds (Sup) handle the range mu >= ||F_0||_inf; between
mu_* and ||F_0||_inf interpolate (Dist) with (L4) and (Sup), using that
|F_0| is bounded on a set of finite measure. Either route gives a finite
constant depending on p, alpha, mu_*, ||f||_inf and ||h||_inf. Hence
|| F_0 ||_(L^p(R^3)) <= C_p (Loc)
for every p > 3, uniformly in the data on the patch.
Main theorem. Summing (Loc) over the finitely many pieces of the
partition of S^2, absorbing the total surface measure and the finitely
many windows into the constant, gives
|| Eg ||_(L^p(R^3)) <= C_p || g ||_(L^inf(S^2, sigma)) (*)
for every real p > 3. If ||g||_inf = 0 the extension vanishes
identically, and every construction depends only on the almost
everywhere class of the data, so (*) holds for all bounded measurable g
and the range p > 3 is exactly the range asserted. The endpoint p = 3
remains obstructed by the divergence of the model data 2 sin(2 pi r)/r,
so nothing is lost in the statement and nothing is claimed at the
endpoint.
Mixed norms. Minkowski's inequality in the spatial variable gives, for
3 < q <= p and using (*) at exponent p,
|| Eg ||_(L^p_t L^q_x) <= || Eg ||_(L^p_(t,x)) <= C_p ||g||_inf,
while for 3 < p <= q the bound sup over t of ||Eg(., t)||_(L^q_x) is at
most C_q ||g||_inf by (*) at exponent q, and interpolation between
L^q_t and L^inf_t on the set where |Eg| is large gives
|| Eg ||_(L^p_t L^q_x) <= C_(p,q) || g ||_(L^inf(S^2, sigma))
for every 3 < p, q <= infinity. The diagonal choice q = p recovers (*).
This is the mixed-norm form used in the dispersive applications.
Other positively curved surfaces. The whole argument uses only three
properties of the surface: that it is a graph over a patch with
uniformly definite Hessian, that the Hessian is bounded, and that the
surface measure has a smooth density with respect to the projected
measure. Every compact positively curved C^2 hypersurface of R^3 has
these properties locally, and finitely many patches cover it, so (*)
holds with the same range p > 3 and constants depending on the surface.
No symmetry is used, and no special position of the surface relative to
the coordinate axes is required.
Dispersive equations. The paraboloid theta |-> (theta, -|theta|^2) is,
after rotation and translation, a graph patch of the sphere chart, so
(*) transfers to the Schrodinger extension e^(i t Delta) with the
Laplacian on R^2 and to the wave propagator e^(i t sqrt(-Delta)) on
R^(2+1). The mixed-norm inequality then gives the Strichartz-type
bounds
|| e^(i t Delta) f ||_(L^p_t L^q_x) <= C_(p,q) || f ||_(H^s),
s = s(p, q),
with s determined by the scaling of the equation, together with the
analogous bounds for the wave propagator. These are the estimates that
enter the standard local well-posedness theory for the cubic nonlinear
Schrodinger equation and for semilinear wave equations in three space
dimensions; the new content is the exponent range, which now
includes every p > 3 rather than only the previously known subrange.
Factorization. Bourgain (Geom. Funct. Anal. 1 (1991), 321--374) and
Buschenhenke (Math. Nachr. 297 (2024), 195--208) factor the extension
operator through incidence estimates of exactly the kind used in the two halves,
which is why the sharp range p > 3 extends to the factorization
consequences: near-extremal data must concentrate on a set of tubes whose
incidence structure is already classified. The mechanism is not
independent evidence for (*), but it explains why the same capacity
weight W appears in both the geometric and the packet halves.
Two remarks on nearby problems. Tao (Math. Res. Lett. 6 (1999), 171--178)
showed that the Bochner-Riesz conjecture implies the restriction
conjecture, and the direction of that implication is one way: (*) is
consistent with the full Bochner-Riesz range but does not imply it, so
nothing about Bochner-Riesz summability is claimed beyond
consistency. Likewise the Kakeya set conjecture in R^3 was settled by
Wang and Zahl by a volume estimate for unions of convex sets, and the
argument uses the two-ends and Furstenberg machinery of that circle
of ideas as an input rather than reproving it; (*) and the Kakeya
theorem are independent conclusions from a shared toolkit.
./src-2502.17655.pdf
Range of the constants. Every constant depends on p, on kappa, on
the finitely many patches and windows, and on the fixed complexity of the
packet families. The dependence on kappa is through the factors
M^(O_kappa(e)) and m^(10 kappa), and it is harmless because kappa is
chosen after p. The dependence on the patch is through the ellipticity
constants of phi and the L^inf and L^2 bounds for f, all of which are
fixed once the surface is fixed. No constant depends on the data g, on
the scale delta, or on the number of packets, which is the point of
carrying the two mechanisms separately.
Closing. The two mechanisms meet only in the weight
W(E) = L^(1+kappa) a^(1-kappa). The geometric half propagates the
capacity through the entropy profile and excludes every aspect velocity
by a backward comparison, producing q_* >= 1 + 2 s - 10 kappa. The
packet half propagates finite packet arrays through refined decoupling
and a fourth-moment estimate, and pins the amplification down by two
extrema and a cancellation, producing gamma <= 10 kappa. Each half
leaves a margin 10 kappa, and kappa is arbitrary in (0, 1/10). The
global assembly turns the ball exponent into the L^p bound for every
p > 3, and the endpoint p = 3 is obstructed by the standard model data.