Fourier restriction for positively curved surfaces

Claimed proofs of the full Fourier restriction range p > 3 for the sphere and for positively curved surfaces.

2 papers September 24, 2026 Explained by AI

Fourier extension builds a wave pattern in space by adding up waves whose frequencies lie on a curved surface. The restriction conjecture asks which size measures, called Lᵖ norms, keep that pattern finite. For curved surfaces in three dimensions the conjectured answer is every p above 3. One paper claims this for the sphere with bounded input. A second claims the matching Lᵖ-to-Lᵖ form for every compact positively curved surface.

First step

1975

Peter Tomas, L² restriction theorem (with Stein)

Significance

Landmark

Restriction in three dimensions is a central open problem in harmonic analysis, and these papers claim its full conjectured range.

Trust

Claimed, not checked

Field

Real and complex analysis

2 papers

The explanation

Start with the unit sphere in three-dimensional space and the constant function 1 on it. At each point x of space, add up the waves e^(2πi x·ω) over all points ω of the sphere. The total, called the extension E1(x), has an exact formula: 2 sin(2π|x|)/|x|. At x = 0 it equals 4π, about 12.57, the sphere's area. At distance 1/4 it equals 8. Far away it shrinks like 2/|x|.

Now measure its size with an Lᵖ norm: integrate |E1|ᵖ over all of space. For p = 3, each shell from radius R to 2R adds about the same amount. That amount is (128/3) × ln 2, about 29.57. Numerical integration gives 29.57 for R = 10, 20 and 40. Infinitely many shells make the total infinite. For p = 4 the shell totals halve at each doubling: about 3.77, 1.88 and 0.94. So that total is finite. The paper notes the integral diverges for every p up to 3Source 1.

The Fourier extension operator does the same for any function f on a surface. It weights each wave by the value of f. The restriction problem asks for which p this output has finite Lᵖ size, bounded by the size of f. In the bounded-data form, f is measured by its largest value. In the diagonal form, f is measured by the same Lᵖ norm on the surface. For surfaces that curve the same way in every direction, the conjectured range is every p above 3. The sphere example shows that p = 3 itself fails.

Elias Stein started the problem. The Tomas–Stein theory gives the classical bound from L² data to L⁴Source 2. Bourgain tied restriction to the geometry of thin tubes. A wave packet is a piece of the output concentrated along such a tube. Curvature separates packet directions, but many tubes can still pass through a small regionSource 3. In 2016, Guth's polynomial partitioning reached bounded data for p > 13/4, which is 3.25Source 4. Wang and Wu reached the diagonal bound for p > 22/7, about 3.143Source 5.

The sphere paper claims the full bounded-data range p > 3 for the sphereSource 6. The second paper claims the diagonal bound for every p > 3Source 7. It covers every compact smooth positively curved surface, including ones with a smooth edge. That includes the sphere, pieces of the elliptic paraboloid and smooth elliptic graphsSource 8. Positive curvature means the surface bends the same way in every direction, like part of a ball.

The diagonal proof uses two theorem inputsSource 9. One is the packet propagation theorem from the sphere paper. The other is a Kakeya maximal theorem from another paper in the same releaseSource 10. It limits how weighted tubes pile up. The second paper also gives a consequence for the free Schrödinger equation in two space dimensions. It claims local smoothing for every p > 3 above Sobolev order 2 − 6/pSource 11. A necessary condition of Rogers shows that this order cannot be loweredSource 12.

How sure is this? There is no Lean scope note for either paper, so no computer proof checker has tested them. The diagonal result depends on the sphere paper and on a Kakeya paper from the same release. No specialist review has been published.

What it doesn't show. This does not settle the restriction conjecture in four or more dimensions, or the endpoint p = 3.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark if the papers hold, since they claim a central conjecture in three dimensions. They have no Lean scope note and depend on another unreviewed paper from the same release. No specialist review is published.

Words used

Fourier extension: Building a function on space by adding up waves whose frequencies lie on a surface, each weighted by a given function.
Lᵖ norm: A measure of a function's size: the p-th root of the integral of its absolute value to the power p.
Bounded data: The form of the problem where the input function is measured by its largest value.
Diagonal estimate: A bound where input and output are measured with the same exponent p.
Wave packet: A piece of a wave pattern concentrated along a thin tube in one direction.
Positively curved surface: A surface that bends the same way in every direction at every point, like part of a ball.
Local smoothing: A gain in smoothness for a wave or Schrödinger solution when averaged over a short time interval.

The original papers (2)

Elliptic capacity propagation and Fourier restriction to the sphere (PDF, opens in a new tab)
September 24, 2026
Diagonal Fourier extension for positively curved surfaces in three dimensions (PDF, opens in a new tab)
September 24, 2026

All quoted lines (12)

The radial integral of $|E1|^p$ diverges for $0<p\le3$ .
Elliptic capacity propagation and Fourier restriction to the sphere Back to text
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.
Elliptic capacity propagation and Fourier restriction to the sphere Back to text
Bourgain's connection between restriction and tube geometry made concentration of wave packets a central issue: curvature separates packet directions, but many tubes can still pass through a small region of space.
Elliptic capacity propagation and Fourier restriction to the sphere Back to text
Guth's polynomial-partitioning argument established the bounded-data estimate for $p>13/4$ .
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text
Wang and Wu proved the diagonal bound for $p>22/7$ on compact positively curved surfaces, including surfaces with boundary, using decoupling and two-ends Furstenberg inequalities.
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text
For every real $p>3$ , there is a finite constant $C_p$ , depending only on $p$ , such that $\|Eg\|_{L^p(\R^3)} \le C_p\|g\|_{L^\infty(S^2,\sigma)}$ for every bounded measurable complex-valued function $g$ on $S^2$ .
Elliptic capacity propagation and Fourier restriction to the sphere Back to text
Let $\Sigma\subset\R^3$ be a compact smooth positively curved surface, possibly with smooth boundary.
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text
It includes the sphere, compact elliptic paraboloid patches, and general smooth elliptic graphs.
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text
The proof of Theorem … uses two theorem inputs.
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text
The Kakeya maximal theorem controls nonnegative weighted tube concentration.
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text
For every $3<p<\infty$ and $s>2-6/p$ , the free Schr\"odinger propagator, initially defined on Schwartz functions, extends boundedly from $W^{s,p}(\R^2)$ to $L^p(\R^2\times[0,1])$ : $\|e^{it\Delta}f\|_{L^p(\R^2\times[0,1])} \le C_{p,s}\|f\|_{W^{s,p}(\R^2)}.$
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text
Rogers's necessary condition shows that the Sobolev threshold cannot be lowered.
Diagonal Fourier extension for positively curved surfaces in three dimensions Back to text

Other sources

openai/math release notes (2026): model provenance manuscripts and supporting proof artifacts produced by an internal OpenAI model
P. A. Tomas, “A restriction theorem for the Fourier transform”, Bulletin of the American Mathematical Society 81 (1975), 477–478
L. Guth, “A restriction estimate using polynomial partitioning”, Journal of the American Mathematical Society 29 (2016), 371–413
H. Wang and S. Wu, “Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities” (2024)

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