Elliptic capacity propagation and Fourier restriction to the sphere

Introduction

Let $S^2 = \{\omega\in\mathbb{R}^3 : |\omega| = 1\}$, and let $\sigma$ be surface area measure, with $\sigma(S^2) = 4\pi$. For a bounded measurable function $g : S^2 \to\mathbb{C}$, define

The bounded-data restriction problem asks for the exponents $p$ for which this extension operator maps $L^\infty(S^2)$ boundedly into $L^p(\mathbb{R}^3)$. We prove the full conjectured open range.

Theorem 1.1. For every real $p > 3$, there is a finite constant $C_p$, depending only on $p$, such that

for every bounded measurable complex-valued function $g$ on $S^2$.

The exponent range is sharp. For the constant function $g = 1$, direct integration gives

The radial integral of $|E1|^p$ diverges for $0 < p \leq3$. Thus Theorem 1.1 proves exactly the conjectured open range; it makes no endpoint assertion.

The restriction problem and earlier methods

The restriction problem originates in Stein’s work, and the Tomas–Stein theory gives the classical $L^2(S^2) \to L^4(\mathbb{R}^3)$ extension estimate [37, 43, 44]. The oscillatory-integral work of Carleson–Sjölin and Hörmander [7, 17], Fefferman’s ball-multiplier obstruction [9], and Córdoba’s geometric multiplier estimates [8] established important parts of the analytic and geometric background. Bourgain’s connection between restriction and tube geometry [2] made concentration of wave packets a central issue: curvature separates packet directions, but many tubes can still pass through a small region of space.

The bilinear framework of Tao–Vargas–Vega [42] isolates interactions between separated frequency pieces and reassembles them across separation scales by rescaling and almost orthogonality. Wolff’s cone theorem [49] and Tao’s paraboloid theorem [39] developed this approach through wave packets and induction on scale. In Tao’s argument, a constraint from the Fourier analysis restricts the directions of the packets that must be counted geometrically; this supplies the additional structure needed to adapt Wolff’s counting argument to the paraboloid.

The multilinear theorem of Bennett–Carbery–Tao [1] controls interactions among surface pieces whose normals uniformly span the ambient space. In three dimensions it bounds the geometric mean of three extensions in $L^3$ on a ball, using the $L^2$ norms of the three inputs and an arbitrarily small positive power of the radius. Bourgain–Guth [5] made this useful for the linear problem by separating transverse interactions from concentration in one small frequency region and from coplanar interactions. Their argument treats the latter configurations by rescaling and square-function estimates. This progression makes the concentration of packet directions, as well as their transverse intersections, a central part of the restriction problem.

Guth’s polynomial-partitioning argument [11] divides the extension integral between cells and a neighborhood of an algebraic surface. Away from that neighborhood, each packet tube meets few of the trimmed cells, allowing induction; the remaining concentration is organized by tubes tangent to the surface. This gives the bounded-data estimate for $p > 13/4$ on a compact smooth positively curved surface $S$. Shayya’s weighted estimates [34] give estimates with finite $L^q(S)$ input norms in this output range. Kim [19] treats these finite-input estimates through rescaling and interpolation. Wang’s broom method [45] then exploits a further constraint on the truncated paraboloid: tubes that collect near one region spread toward others, which limits repeated concentration near separated surface neighborhoods. Combined with polynomial partitioning and a two-ends argument, it yields the bounded-data estimate for $p > 42/13$.

Wang and Wu [46, 47] combine incidence estimates with refined decoupling. The latter controls oscillatory interaction using the number of packets meeting each small ball, while the incidence argument retains how the occupied part of a tube is distributed along its length. Their two-ends Furstenberg inequalities use this spacing information to improve the multiplicity bound. In [47], Theorem 0.2 they obtain the diagonal estimate $L^p(S) \to L^p(\mathbb{R}^3)$ for $p > 22/7$ on compact positively curved $C^2$ surfaces, including boundary. The input norm matters in this comparison: a diagonal estimate controls all $L^p(S)$ data, whereas Theorem 1.1 is stated for bounded sphere data. The latter reaches the full open output range $p > 3$; its sphere-specific factorization consequence is discussed separately below.

Elliptic capacity and the geometric obstacle

The geometric and oscillatory estimates use the same eccentricity-biased weight. For an ellipse $E \subset\mathbb{R}^2$ with semiaxes $L \ge w > 0$, set

A joint test for two vector coordinates uses translates of the same ellipse, with the two centers chosen independently. It therefore measures concentration of their joint law without assuming independence. Equivalently, $W(E) = Lw(L/w)^\kappa$; the usual area weight is multiplied by a power of the eccentricity. Differently oriented tests intersect in smaller rectangles. In the hairbrush estimate of Lemma 2.5, the extra exponent $\kappa$ turns the resulting overlap count into a power saving. The propagation argument uses that saving to control how concentration can change as a line moves.

Let $M$ be a large dyadic scale, and consider a joint law of uniformly bounded line parameters $(U,V)$ and a time $T \in[0,1)$, with $X_t = V + tU$. Suppose its initial joint tests obey $\Pr(U \in u + E,V \in v + E) \le K W(E)$. The time hypothesis is conditional on a base label that determines $U,V$: at interval length $M^{-y}$, for $0 \le y \le1$, both the number of occupied time bins and the largest bin probability have upper bounds with exponents $sy$ and $-sy$, respectively, up to a small error. Here $s \in(0,1]$.

At the terminal resolution let $T_1$ be the left endpoint of the time bin of length $M^{-1}$ containing $T$. A propagation exponent $q$ means that a dominated sublaw $\nu$ satisfies

simultaneously for all terminal bins and independent centers $u,x$. The retained mass is $M^{-o(1)}$, meaning that it exceeds every fixed negative power for sufficiently large $M$. Theorem 2.2 supplies every strict exponent below $1+2s-10\kappa$ when the time error is sufficiently small. That theorem specifies the input and terminal ellipse scales and all conditional quantifiers.

Inputs and methodological precedents. The planar input belongs to the projection and Furstenberg-set tradition of Marstrand and Kaufman [20, 18], Bourgain’s discretized projection theorem [3], and Oberlin’s work on exceptional projections [22]. Packing-dimension advances of Orponen and Shmerkin [26, 35], their Hausdorff-dimension improvement [27], and their work relating projections, Furstenberg sets, and the $ABC$ sum-product problem [28] provide further context for Ren–Wang’s sharp finite incidence theorem [30]. In the finite form used here, each point of a planar family is incident to a comparably sized family of thin tubes. Counts in smaller balls and angular intervals control concentration of the points and tube directions. Ren–Wang’s theorem gives a lower bound for the total number of distinct tubes. Drawing lines of constant projection through the points converts this into a bound for projected supports. We prove the weighted, conditional, and joint-coordinate consequences needed for our laws.

The proof uses finite entropy in the sense of Shannon [33] and the entropy rules organized by Tao [40]. The local-entropy approach of Hochman–Shmerkin [16] and Hochman’s inverse principles [15] are related methodological precedents; our correlated conditional laws are handled by the finite arguments below. The radial step is closely related to the reciprocal thin-tube bootstrap of Orponen–Shmerkin–Wang [29]. Hairbrush intersection counting [50, 42] and the density and factoring arguments of Wang–Zahl and Guth–Wang–Zahl [48, 13] supply further geometric context. The independently translated elliptic capacity and its propagation through arbitrary finite complex arrays and coordinate masks are the specific estimates established here. The external analytic input is Bourgain–Demeter’s $\ell^2$ decoupling theorem [4]. Section 6 proves the canonical-packet refinement needed here, following the grouping and rescaling method of Guth–Iosevich–Ou–Wang [12] and the localized extension formulation of Wang–Wu [47]. Besides these decoupling and finite-incidence inputs, the proof uses classical almost-everywhere differentiation [41, 14].

The finite packet theorem

The packet estimate underlying the sphere theorem applies to arbitrary finite complex coefficient arrays. We summarize its definitions before stating the uniform bound; Section 6 develops the frame identities and norm properties in detail.

On a graph patch of the sphere, absorbing the surface-area density into the data gives the phase $x\cdot\xi+t\phi(\xi)$, with $\xi\in\mathbb{R}^2$. Rescaling a physical region of size $N^2$ gives the factor $e^{2\pi iN^2(x\cdot\xi+t\phi(\xi))}$. A packet near frequency $\nu$, centered at $z$ at time $t_1$, is concentrated near the line

Thus its frequency and position labels specify the velocity and center tested by the elliptic capacity. The phase is continued smoothly beyond the patch as specified in the theorem below.

For a fixed dyadic $N$, a dyadic factor $q \in[1,N]$ determines the frequency width $\theta=q/N$, time-bin length $r=q^{-2}$, and spatial mesh parameter $w=(qN)^{-1}$. Choose a fixed smooth, nonnegative, compactly supported window $\chi$ whose integer translates satisfy $\sum_{n\in\mathbb{Z}^2}\chi(\xi-n)^2=1$. Let $L_f$ be a fixed side length containing its support. A bin $I$ of length $r$, with left endpoint $t_I$, has packet labels $(\nu,z)\in\theta\mathbb{Z}^2\times L_f^{-1}w\mathbb{Z}^2$. With $\chi^\nu_\theta(\xi)=\chi((\xi-\nu)/\theta)$, the exact maps for the fixed phase $\phi$ in the theorem below are

Analysis thus takes unnormalized Fourier coefficients in each window; synthesis has the corresponding inverse Fourier-series factor.

For an array $c=(c_v)$ in one bin, indexed by $v=(\nu_v,z_v)$, put

The supremum uses ellipses with both semiaxes at least $\theta$, and independent centers $u,z$; the infimum uses finite arrays and coordinatewise domination. The squared coefficients place mass $|c_v|^2$ at the velocity–position pairs $(U_{\nu_v},z_v)$. Thus $K$ measures joint concentration, $G$ combines that concentration with squared coefficient mass, and the atomic norm $A$ allows finite decompositions into arrays of controlled $G$-cost. Subscripts below specify the scale and bin.

Theorem 1.2 (Uniform propagation of finite packet arrays). Fix $0<\kappa<1/10$, a smooth elliptic continuation $\phi$ of a spherical graph patch allowed in Section 6, a bounded frequency-label region $\Omega\subset\mathbb{R}^2$, and the fixed window system used in (6.2)–(6.3). The continuation is defined on $\mathbb{R}^2$, has uniformly definite bounded Hessian, and has bounded derivatives of every order at least two.