Local smoothing in three dimensions

A claimed proof of Sogge's local smoothing conjecture for waves in 3D space, including the critical case p = 3.

1 paper September 24, 2026 Explained by AI

A wave can briefly pile up its energy in small regions, so its size at one instant can be worse than its start. Averaging over a short stretch of time is conjectured to win back most of that loss. Sogge's conjecture says exactly how much is won back in each Lᵖ size measure. This paper claims the three-dimensional case, at the critical exponent p = 3 and then all p above 2. Consequences for Bochner–Riesz means and pointwise convergence follow.

First step

1991

Christopher Sogge, who introduced local smoothing

Significance

Landmark

Local smoothing in three dimensions is a central open problem, and it implies Bochner–Riesz and other estimates in harmonic analysis.

Trust

Claimed, not checked

Field

Real and complex analysis

1 paper

The explanation

Take an initial wave f(x) = sin(Nx) with frequency N = 1000. Its derivative has size about N. A fractional derivative of order α scales it by about N^α. That is how Sobolev size measures smoothness. At one fixed time, a wave in three dimensions can lose 2(1/2 − 1/p) derivatives in Lᵖ sizeSource 1. The Lᵖ norm is the p-th root of the integral of |f|ᵖ. At p = 3 the loss is 2 × 1/6 = 1/3. For N = 1000 that costs a factor of about 1000^(1/3) = 10.

Local smoothing asks what happens when you also average over a short time interval. Sogge's conjecture says the loss drops to just above max{0, 1 − 3/p}Source 2. At p = 3 that is 0, so only an arbitrarily small loss should remain. A loss of 0.01 costs about 1000^0.01 ≈ 1.07, instead of 10. At p = 4, the loss should fall from 1/2 to 1/4. At p = 6 it should fall from 2/3 to 1/2. In each case the gain is 1/p, as the conjecture predicts for p ≥ 3Source 3.

Sogge introduced local smoothing in his work on circular maximal operatorsSource 4. Wolff, Łaba and Garrigós–Seeger built wave-packet methods for large p. A wave packet is a piece of the wave that travels along a thin tube. Bourgain and Demeter's decoupling theorem gave the conjectured gain for p ≥ 4 in three dimensionsSource 5. In two dimensions, Guth, Wang and Zhang reached the critical exponent p = 4 with a sharp square-function estimateSource 6. In three dimensions at p = 3, Gao, Liu, Miao and Xi reached every loss above 1/9Source 7. Gan, He, Li and Wu's work gives losses above 1/12 thereSource 8. The remaining small fixed loss is the gap this paper addresses.

The paper claims the estimate at p = 3 with every positive loss εSource 9. Interpolation with easier bounds then gives the full range 2 < p < ∞. Together these claim to resolve Sogge's conjecture in three space dimensionsSource 10. The lossless case ε = 0 is not addressedSource 11. The word Euclidean matters. On some curved manifolds, focusing geodesics force p ≥ 10/3 for the full gainSource 12.

The main obstacle is controlling how packets crowd together across many scales. The paper's central quantity is the entropy of a finite phase-space recordSource 13. Phase space tracks both the position and the direction of each packet. Entropy here measures how spread out those records are. The proof uses two outside inputsSource 14. They are Ren and Wang's planar Furstenberg theorem and Guth, Wang and Zhang's wave-envelope theorem.

The paper also claims consequences. Maximal Bochner–Riesz means converge almost everywhere for 3 ≤ p < ∞ when δ > 1 − 3/pSource 15. A maximal bound for the wave itself needs s > 1/3 at p = 3Source 16.

How sure is this? There is no Lean scope note for this result, so no computer proof checker has tested it. The paper is very long and relies on two recent outside theorems. No specialist review has been published.

What it doesn't show. This does not prove the lossless estimate at p = 3, or local smoothing on curved manifolds.

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 proof holds, since it claims a central conjecture in three dimensions. It is one very long paper with no Lean scope note. No specialist review is published.

Words used

Local smoothing: A gain in smoothness for a wave when its size is averaged over a short time interval.
Sobolev loss: The number of derivatives an estimate gives up, measured by powers of the frequency.
Lᵖ norm: A measure of a function's size: the p-th root of the integral of its absolute value to the power p.
Wave packet: A piece of a wave concentrated along a thin tube in one direction.
Decoupling: An inequality that bounds a sum of wave pieces using their separate sizes.
Phase space: The joint record of where a wave piece is and which direction it moves.
Bochner–Riesz means: Ways to rebuild a function from its frequencies with a softened cut-off at the edge of a ball.

The original paper

Critical local smoothing for the three-dimensional wave equation (PDF, opens in a new tab)
September 24, 2026

All quoted lines (16)

For $p>2$ , the sharp fixed-time $L^p$ estimate has Sobolev loss $2(1/2-1/p)$ in three dimensions.
Critical local smoothing for the three-dimensional wave equation Back to text
Sogge's Euclidean local smoothing conjecture predicts, for $2<p<\infty$ , the strict range $\alpha>\max\{0,1-3/p\}$ ; its critical exponent is $p=3$ .
Critical local smoothing for the three-dimensional wave equation Back to text
In three dimensions it asks to recover almost $1/p$ derivatives when $p\ge3$ ; at $p=3$ , time averaging should remove all but an arbitrarily small part of the fixed-time loss of $1/3$ .
Critical local smoothing for the three-dimensional wave equation Back to text
Sogge introduced local smoothing in his study of circular maximal operators and their variable-coefficient analogues.
Critical local smoothing for the three-dimensional wave equation Back to text
Its local-smoothing consequence gives the conjectured gain for $p\ge4$ in three spatial dimensions, leaving the critical $p=3$ estimate beyond that range.
Critical local smoothing for the three-dimensional wave equation Back to text
There Bourgain–Demeter's theorem gives the full smoothing gain for $p\ge6$ ; Guth–Wang–Zhang reached the critical exponent $p=4$ through a sharp cone square-function estimate.
Critical local smoothing for the three-dimensional wave equation Back to text
In three spatial dimensions they obtain, at $p=3$ , every Sobolev exponent $\alpha>1/9$ , or equivalently a smoothing gain below $2/9$ .
Critical local smoothing for the three-dimensional wave equation Back to text
Interpolation of the $p=10/3$ estimate with the $L^2$ energy bound gives $\alpha>1/12$ at $p=3$ , improving the preceding $1/9$ threshold.
Critical local smoothing for the three-dimensional wave equation Back to text
For every $\eps>0$ there exists $C_\eps<\infty$ such that $\norm{Uf}_{L^3(\R^3\times[1,2])} \le C_\eps\norm{J^\eps f}_{L^3(\R^3)} \qquad (f\in\mathcal S(\R^3)).$
Critical local smoothing for the three-dimensional wave equation Back to text
Together, these estimates resolve Sogge's Euclidean local smoothing conjecture in three spatial dimensions.
Critical local smoothing for the three-dimensional wave equation Back to text
At $p=3$ , the estimate holds with every positive Sobolev loss; the lossless estimate $\eps=0$ is not addressed.
Critical local smoothing for the three-dimensional wave equation Back to text
Minicozzi–Sogge constructed metrics with concentrating families of geodesics; in spatial dimension three these examples require $p\ge10/3$ for the full smoothing gain on general manifolds.
Critical local smoothing for the three-dimensional wave equation Back to text
Our proof also studies packet concentration across scales, but its central quantity is the entropy of a finite phase-space record.
Critical local smoothing for the three-dimensional wave equation Back to text
The proof uses two external quantitative inputs: the planar Furstenberg theorem of Ren–Wang and the cone wave-envelope theorem of Guth–Wang–Zhang.
Critical local smoothing for the three-dimensional wave equation Back to text
Section … applies them to (…): Theorem … bounds $B_*^\delta$ strongly on $L^p$ and gives $B_R^\delta f\to f$ almost everywhere for $3\le p<\infty$ and $\delta>1-3/p$ .
Critical local smoothing for the three-dimensional wave equation Back to text
Its extra cost explains the stronger regularity hypothesis: at $p=3$ , spacetime smoothing allows every positive loss, whereas this maximal conclusion requires $s>1/3$ .
Critical local smoothing for the three-dimensional wave equation Back to text

Other sources

openai/math release notes (2026): model provenance manuscripts and supporting proof artifacts produced by an internal OpenAI model
C. D. Sogge, “Propagation of singularities and maximal functions in the plane”, Inventiones Mathematicae (1991)
T. Wolff, “Local smoothing type estimates on Lᵖ for large p”, Geometric and Functional Analysis (2000)
L. Guth, H. Wang and R. Zhang, “A sharp square function estimate for the cone in ℝ³”, Annals of Mathematics 192 (2020)

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