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Sharp Exponents in the Fourier Restriction Conjecture

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Statement

Definitions and mathematical statement

For a Schwartz function on ${\mathbb{R}}^d$, set $\widehat f(\xi)=\int f(x)e^{-2\pi i x\cdot\xi}\,dx$, and let $d\sigma$ be surface measure on $S^{d-1}$. The restriction estimate is \[ \|\widehat f|_{S^{d-1}}\|_{L^q(d\sigma)}\le C_{d,p,q}\|f\|_{L^p({\mathbb{R}}^d)}. \] In the usual range $1\le p,q<\infty$, the conjectured non-endpoint range is $p<2d/(d+1)$ and $q\le (d-1)p'/(d+1)$, with $p'=p/(p-1)$ and the usual interpretation at $p=1$. The full target asks for the sharp range in every $d\ge2$; endpoint conventions should be taken from the source, rather than inferred from a schematic inequality.

Formulation and concept references: [S1].

Short English statement

Does Fourier restriction to a sphere satisfy every norm estimate allowed by the standard sharp necessary conditions?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 2

Math-43-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-43

Extension from curved surfaces: The extension estimate establishes the stated p>3 branch in ambient dimension three for spheres or positively curved surfaces. The question asks the sharp two-exponent restriction range in all dimensions or for general submanifolds, which is broader in dimensions, surfaces, and/or exponents.

Sphere extension and packet capacity: The extension estimate establishes the stated p>3 branch in ambient dimension three for spheres or positively curved surfaces. The question asks the sharp two-exponent restriction range in all dimensions or for general submanifolds, which is broader in dimensions, surfaces, and/or exponents.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Sharp Exponents in the Fourier Restriction Conjecture

OpenAI's claim

Extension from curved surfaces: For a compact smooth surface in $\mathbb R^3$ with positive curvature, the paper claims diagonal Fourier extension boundedness for every exponent $p>3$; it includes smooth surface boundaries and draws a two-dimensional Schrödinger smoothing consequence from the paraboloid case.

Sphere extension and packet capacity: For bounded data on the two-sphere, the announced extension bound has every output exponent larger than three; the threshold three is also asserted to be the sharp endpoint of this open range.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\Sigma\subset\mathbb R^3$ be a compact smooth surface with positive curvature, allowing smooth boundary, and use surface area $\sigma$. For $E_\Sigma f(x)=\int_\Sigma e^{2\pi ix\cdot\xi}f(\xi)\,d\sigma(\xi)$, the claim is \[\|E_\Sigma f\|_{L^p(\mathbb R^3)}\le C_{\Sigma,p}\|f\|_{L^p(\Sigma,\sigma)}\quad(3<p<\infty),\] for all complex-valued $f\in L^p(\Sigma)$. It concerns this diagonal range in three dimensions, with a surface-dependent constant.
Mathematical claim 2.
For surface area $\sigma$ on $S^2$ and $Eg(x)=\int_{S^2}e^{2\pi ix\cdot\omega}g(\omega)\,d\sigma(\omega)$, every bounded measurable complex $g$ satisfies \[\|Eg\|_{L^p(\mathbb R^3)}\le C_p\|g\|_{L^\infty(S^2,\sigma)}\quad(p>3).\] The constant depends only on $p$; the endpoint $p=3$ and other ambient dimensions are outside this theorem.

Outline of the argument

Extension from curved surfaces: Graph patches are extended to phases compatible with an elliptic-capacity propagation theorem. That theorem is combined with weighted Kakeya control to sum packet coefficients by their capacity. A joint estimate for widely separated balls removes the local spatial loss, and chart assembly transfers the estimate to the original surface.

Sphere extension and packet capacity: Joint tests of position and velocity use ellipse weights that penalize eccentricity. A geometric propagation estimate controls how this capacity evolves along lines, using overlap savings from differently directed tests. Exact packet analysis and synthesis retain the oscillatory information needed to convert the positive capacity estimate into Fourier extension bounds.

Formal verification

Extension from curved surfaces: The pinned catalogue lists no Lean formalization for this result.

Sphere extension and packet capacity: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

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Math-43-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-43

Sharp Exponents in the Fourier Restriction Conjecture

Definitions and mathematical statement

For a Schwartz function on ${\mathbb{R}}^d$, set $\widehat f(\xi)=\int f(x)e^{-2\pi i x\cdot\xi}\,dx$, and let $d\sigma$ be surface measure on $S^{d-1}$. The restriction estimate is \[ \|\widehat f|_{S^{d-1}}\|_{L^q(d\sigma)}\le C_{d,p,q}\|f\|_{L^p({\mathbb{R}}^d)}. \] In the usual range $1\le p,q<\infty$, the conjectured non-endpoint range is $p<2d/(d+1)$ and $q\le (d-1)p'/(d+1)$, with $p'=p/(p-1)$ and the usual interpretation at $p=1$. The full target asks for the sharp range in every $d\ge2$; endpoint conventions should be taken from the source, rather than inferred from a schematic inequality.

Formulation and concept references: [S1].

Short English statement

Does Fourier restriction to a sphere satisfy every norm estimate allowed by the standard sharp necessary conditions?

Research attempt

Current frontier.
Literature and formulation check, 27 September 2026. The source's Conjecture 3.5 uses a sphere of dimension one less than the ambient dimension. After translating its notation, its strict radial inequality and its non-strict sloping inequality agree with the statement above. Its Theorem 3.7 gives the sharp $L^2$-restriction slice $p\le 2(d+1)/(d+3)$; this is not the full conjectured range [S1]. The two catalogue entries for arXiv:2311.03145 refer to different versions of one evolving manuscript. Version 15, dated 15 April 2026, concerns a probabilistic analogue; an estimate averaged over multipliers must not be read as a bound for every deterministic multiplier [E1]. No resolution of the full stated range is inferred from these sources. The unrestricted conjecture also has the Kakeya consequences explained in the introduction of [S1].
Attack route.
Write the extension operator as \[ Eg(x)=\int_{S^{d-1}} e^{2\pi i x\cdot\omega}g(\omega)\,d\sigma(\omega). \] Duality changes the target to $E:L^u(S^{d-1})\to L^r({\mathbb{R}}^d)$ with $u=q'$, $r=p'$. Three possible mechanisms are worth distinguishing. Convolution geometry can control an even moment exactly; randomization can remove cross terms but needs a deterministic replacement; and a multiscale argument would have to bound the surviving resonances uniformly as the cap size tends to zero. We pursue the first two, then test the proposed bridge between them. The calculations below are derived here; the recovered $L^2\to L^4$ estimate is a known special case, not a claimed improvement over Stein--Tomas or a historical novelty claim.
Attempt.
1. A cap-local convolution lemma with an explicit constant. Suppose $d\ge3$, $0<c<1$, and $g$ is supported where $\omega\cdot e\ge c$, for a unit vector $e$. For $0<|x|<2$, direct integration on the sphere gives \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-33-35-33-39-3a-63-6f-6e-76} (\sigma*\sigma)(x)=\frac{|S^{d-2}|}{|x|} \left(1-\frac{|x|^2}{4}\right)^{(d-3)/2}. \end{equation} For clarity about the normalization, set $x=\rho e_1$ and integrate $\delta(|x-\omega|-1)$ against $d\sigma(\omega)$. In the coordinate $v=\omega_1$, the spherical measure has density $|S^{d-2}|(1-v^2)^{(d-3)/2}$; the root is $v=\rho/2$ and the absolute value of the derivative of $|x-\omega|-1$ there is $\rho$. This proves (eq:30003539:conv). The density is zero outside the radius-two ball; exceptional boundary values do not affect integration.

A sum of two points from the chosen cap has $x\cdot e\ge2c$ and hence $|x|\ge2c$. Thus, on the support of the convolutions involving $g$, \[ (\sigma*\sigma)(x)\le K_{d,c}:=\frac{|S^{d-2}|}{2c}. \] Disintegrate $\sigma\otimes\sigma$ under $(\omega,\eta)\mapsto\omega+\eta$. Cauchy--Schwarz on each fiber gives, almost everywhere, \[ |(g\sigma*g\sigma)(x)|^2 \le (\sigma*\sigma)(x) \left((|g|^2\sigma)*(|g|^2\sigma)\right)(x). \] After integrating, the second convolution has total mass $\|g\|_2^4$. Consequently, \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-33-35-33-39-3a-63-61-70-2d-66-6f-75-72} \|Eg\|_4^4=\|g\sigma*g\sigma\|_2^2 \le K_{d,c}\|g\|_2^4. \end{equation} Here the equality is justified rather than assumed: the inequality first puts the convolution in $L^2$; its inverse Fourier transform is $(Eg)^2$ in distributions, so Plancherel identifies that distribution with an $L^2$ function. Since $g\sigma$ is a finite measure, $Eg$ is continuous, and this identification also gives the asserted almost-everywhere equality of functions. A finite cap cover and the $L^4$ triangle inequality yield a global $L^2\to L^4$ bound. In $d=3$ this reaches the sharp $L^2$ slice; in higher dimensions it is weaker than the quoted sharp slice.

2. An exact test of the randomization step. For arbitrary complex numbers $z_1,\ldots,z_N$ and independent random signs $\varepsilon_j$, expansion of the fourth power yields \begin{align} \mathbb E\left|\sum_j\varepsilon_jz_j\right|^4 &=2\left(\sum_j|z_j|^2\right)^2+ \left|\sum_jz_j^2\right|^2-2\sum_j|z_j|^4\notag\\ &\le3\left(\sum_j|z_j|^2\right)^2. \label{wsref-0-65-71-3a-33-30-30-30-33-35-33-39-3a-72-61-6e-64-6f-6d} \end{align} Indeed, a sign expectation vanishes unless every index occurs an even number of times. The three pairings count each all-equal quadruple three times, so two copies must be subtracted. This also proves the formula for complex, rather than only real, $z_j$.

Taking every $z_j=1$ makes the deterministic fourth power $N^4$, whereas the random expectation is $3N^2-2N$. There is therefore no dimension-free pointwise replacement of the random average by the value at the all-positive signs. For a finite cap decomposition $g=\sum_jg_j$, one may put $z_j=Eg_j(x)$ and integrate (eq:30003539:random) over a finite ball by Tonelli. That legal interchange does not supply the missing deterministic inequality. Nonnegative cap data already have coherent phases at $x=0$. This observation tests an intermediate inference, not the conjectured whole-space norm bound.

3. Strict curvature does not eliminate resonant rectangles. A stronger attempted shortcut would assert that four unit vectors satisfying $\omega_1+\omega_2=\omega_3+\omega_4$ must coincide in pairs once all four lie in a small cap. It is false. In ${\mathbb{R}}^3$, let \[ a=\delta e_1,\quad b=\delta e_2,\quad c_0=\sqrt{1-2\delta^2}\,e_3,\qquad 0<\delta<1/\sqrt2, \] and set \[ \omega_1=c_0+a+b,\quad \omega_2=c_0-a-b,\quad \omega_3=c_0+a-b,\quad \omega_4=c_0-a+b. \] All four vectors are distinct, have norm one, and have the same positive third coordinate. Their two sums both equal $2c_0$. Letting $\delta\to0$ places them in arbitrarily small caps. The construction embeds in every higher ambient dimension. Thus the fourth-moment expansion contains genuine non-pairing resonances even on a strictly curved surface. Their removal by independent signs is a change of problem; a deterministic proof must estimate them, not declare them absent.

4. Scaling identifies the range still missing. Choose $g_\delta$ as the indicator of a spherical cap of angular radius $\delta$. After removing the common phase, phase oscillation is bounded by a small constant on a box with $d-1$ tangential side lengths comparable to $\delta^{-1}$ and one normal side length comparable to $\delta^{-2}$. Taking the implicit constants small makes the real part of the integrand positive, so \[ \|Eg_\delta\|_r\gtrsim \delta^{d-1-(d+1)/r},\qquad \|g_\delta\|_u\asymp\delta^{(d-1)/u}. \] This proves the necessary inequality \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-33-35-33-39-3a-6b-6e-61-70-70} r\ge\frac{d+1}{d-1}u' \end{equation} for finite $r$, with the usual interpretation when an exponent is infinite. Smooth cap cutoffs give the same powers. These are necessary conditions, not a proof that the conditions suffice.

For example, in $d=3$ and $u=\infty$, the target includes every $r>3$, whereas (eq:30003539:cap-four) and the elementary $L^\infty$ bound give only $r\ge4$ on the finite-measure sphere. Interpolation does not fill $3<r<4$. On ${\mathbb{R}}^3$, the bounded function $(1+|x|)^{-\alpha}$ lies in $L^4$ but not $L^r$ whenever $3/4<\alpha\le3/r$. This is not an extension-operator counterexample; it falsifies the general integrability implication that the interpolation shortcut would require.

Stress test.
The convolution argument excludes $d=2$ deliberately: the exponent in (eq:30003539:conv) would be negative and its singularity at $|x|=2$ is not removed by the cap condition. No conclusion about the planar theorem is being extracted from that argument. Antipodal singularities in $d\ge3$ are controlled locally before taking the finite cap sum. The calculation neither assumes globally bounded $\sigma*\sigma$ nor interchanges an infinite cap sum with an integral.

The random-sign identity, the resonant rectangle and the Knapp exponents are exact. None bounds the coherent resonances at arbitrarily many scales. A successful continuation needs a deterministic estimate below the obtained $L^4$ exponent, with constants independent of cap resolution and spatial radius, and with the input exponent linked to (eq:30003539:knapp). The probabilistic manuscript does not supply that missing universal quantifier [E1]. Finite sign enumeration and symbolic rectangle checks in the accompanying script test the algebra only.

Outcome.
Outcome: Exploratory approach with unresolved gap. The attempt supplies a complete cap-convolution proof of a known special case and two explicit obstructions to upgrading a randomized argument: coherent fourth moments and non-pairing resonant rectangles within arbitrarily small curved caps. These are counterexamples to proposed intermediate shortcuts, not counterexamples to Fourier restriction.

The sharp all-dimensional range remains unproved here. In particular, the argument does not produce the below-$L^4$ deterministic estimates already needed in ambient dimension three, and it makes no claim to have resolved the associated Kakeya implications or an endpoint outside the original scope.

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