Local smoothing in three dimensions

1 · The essentials

The manuscript studies how waves behave in ordinary three-dimensional space. Its question is how tightly their size can be controlled when their behavior is measured over time. The amount of initial smoothness needed depends on how strongly the measurement emphasizes large values. The manuscript claims the expected control across the full range described in the repository summary. At a particularly important measurement level, called $L^{3}$ , it allows any positive extra smoothness requirement, however small, but does not remove that requirement entirely. The manuscript describes this as resolving Sogge’s local smoothing conjecture in three dimensions. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine assessing a moving ripple through a short video rather than through a single photograph. Local smoothing concerns the control available when looking across time; the image does not mean that the wave itself necessarily becomes visibly smoother.

Why it matters

The source identifies the result as a resolution of Sogge’s Euclidean local smoothing conjecture in dimension three. Its scope is not limited to the critical $L^{3}$ estimate: the repository summary states estimates throughout the strict range $2 < p < \infty$ , with the specified smoothness requirements.

2 · Going further

The problem concerns local smoothing for the Euclidean wave equation in three spatial dimensions. Writing $s$ for the Sobolev regularity, the repository summary states the range $2 < p < \infty$ with $s > max {0, 1 - 3/ p}$ . Both inequalities are strict. At $p = 3$ , the threshold is zero, so the stated result allows every $s > 0$ , not $s = 0$ .

The main manuscript, “Critical local smoothing for the three-dimensional wave equation,” claims the critical $L^{3}$ estimate with every positive Sobolev loss and describes it as resolving Sogge’s conjecture in this setting. The supplied abstract states the result but gives no argument or proof strategy, so no mechanism for obtaining it can be explained from this source. Only one manuscript is supplied; there are no companion abstracts or relationships between manuscripts to report.

Useful notions

The wave equation
Measuring functions with $L^{p}$ norms
Sobolev regularity as a measure of smoothness
Strict inequalities and endpoint cases

What was known before / what is new

Before

The source identifies Sogge’s Euclidean local smoothing conjecture in dimension three as the problem being addressed. It supplies no account of earlier estimates, partial results, or methods.

New

The main manuscript claims the critical $L^{3}$ local smoothing estimate with every positive Sobolev loss. The repository summary states that the resulting estimates cover the full strict range $2 < p < \infty$ whenever the Sobolev regularity is above $max {0, 1 - 3/ p}$ .

Limits and assumptions

The stated setting is the Euclidean wave equation in three spatial dimensions. The summary requires $2 < p < \infty$ and $s > max {0, 1 - 3/ p}$ ; it does not include the endpoint values of $p$ or equality at the regularity threshold. In particular, the critical $L^{3}$ statement does not assert a zero-loss estimate. The supplied text gives neither a full estimate with all hypotheses nor an argument, and it does not say whether the excluded cases remain open. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Local smoothing: An estimate expressing improved smoothness control for a wave when its behavior is measured over a time interval.
$L^{p}$ norm: A measure of a function’s size whose exponent $p$ determines how strongly larger values are weighted.
Sobolev regularity: A way of quantifying a function’s smoothness using a regularity parameter such as $s$ .
Positive Sobolev loss: An extra positive smoothness requirement, which in the stated critical $L^{3}$ result may be arbitrarily small.
Endpoint: A boundary value of a stated parameter range, excluded when the defining inequality is strict.

Keywords : Local smoothing · Three-dimensional wave equation · Sobolev regularity

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range $2 < p < \infty$ , with Sobolev regularity above $max {0, 1 - 3/ p}$ . In particular, the critical L³ estimate holds with every positive Sobolev loss.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Critical local smoothing for the three-dimensional wave equation

24/09/2026

To be verified

We prove the critical L³ local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.

PDF GitHub folder