# 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. Plain-English explainer with sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-079/
Last updated: 2026-10-14

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

**Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** September 24, 2026 · **Field:** Real and complex analysis · **Entry:** 079

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

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.

## At a glance

- **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 size[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}[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 ≥ 3[3].

Sogge introduced local smoothing in his work on circular maximal operators[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 dimensions[5]. In two dimensions, Guth, Wang and Zhang reached the critical exponent p = 4 with a sharp square-function estimate[6]. In three dimensions at p = 3, Gao, Liu, Miao and Xi reached every loss above 1/9[7]. Gan, He, Li and Wu's work gives losses above 1/12 there[8]. The remaining small fixed loss is the gap this paper addresses.

The paper claims the estimate at p = 3 with every positive loss ε[9]. Interpolation with easier bounds then gives the full range 2 < p < ∞. Together these claim to resolve Sogge's conjecture in three space dimensions[10]. The lossless case ε = 0 is not addressed[11]. The word Euclidean matters. On some curved manifolds, focusing geodesics force p ≥ 10/3 for the full gain[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 record[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 inputs[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/p[15]. A maximal bound for the wave itself needs s > 1/3 at p = 3[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](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf) - September 24, 2026

## Quoted lines from the paper (16)

[1] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> For $p>2$, the sharp fixed-time $L^p$ estimate has Sobolev loss $2(1/2-1/p)$ in three dimensions.

[2] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> Sogge's Euclidean local smoothing conjecture predicts, for $2<p<\infty$, the strict range $\alpha>\max\{0,1-3/p\}$; its critical exponent is $p=3$.

[3] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> 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 $1/3$.

[4] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> Sogge introduced local smoothing in his study of circular maximal operators and their variable-coefficient analogues.

[5] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> Its local-smoothing consequence gives the conjectured gain for $p\ge4$ in three spatial dimensions, leaving the critical $p=3$ estimate beyond that range.

[6] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> 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.

[7] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> In three spatial dimensions they obtain, at $p=3$, every Sobolev exponent $\alpha>1/9$, or equivalently a smoothing gain below $2/9$.

[8] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> 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.

[9] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> For every $\eps>0$ there exists $C_\eps<\infty$ such that \begin{equation} \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)). \end{equation}

[10] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> Together, these estimates resolve Sogge's Euclidean local smoothing conjecture in three spatial dimensions.

[11] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> At $p=3$, the estimate holds with every positive Sobolev loss; the lossless estimate $\eps=0$ is not addressed.

[12] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> 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.

[13] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> Our proof also studies packet concentration across scales, but its central quantity is the entropy of a finite phase-space record.

[14] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> The proof uses two external quantitative inputs: the planar Furstenberg theorem of Ren--Wang and the cone wave-envelope theorem of Guth--Wang--Zhang.

[15] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> 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 $\delta>1-3/p$.

[16] From [Critical local smoothing for the three-dimensional wave equation](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf):

> Its extra cost explains the stronger regularity hypothesis: at $p=3$, spacetime smoothing allows every positive loss, whereas this maximal conclusion requires $s>1/3$.

## Other sources

- [openai/math release notes (2026): model provenance](https://github.com/openai/math/blob/main/README.md): “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)](https://doi.org/10.1007/BF01245080)
- [T. Wolff, “Local smoothing type estimates on Lᵖ for large p”, Geometric and Functional Analysis (2000)](https://doi.org/10.1007/PL00001652)
- [L. Guth, H. Wang and R. Zhang, “A sharp square function estimate for the cone in ℝ³”, Annals of Mathematics 192 (2020)](https://doi.org/10.4007/annals.2020.192.2.6)

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