Local smoothing for the Euclidean wave equation

1 · The essentials

The central question is how much better a wave can behave when we measure its motion over time rather than at just one instant. The main manuscript settles the question described as Sogge’s local smoothing conjecture for Euclidean space in every dimension at least two. Its estimate comes arbitrarily close to the sharp required level of smoothness, while retaining an arbitrarily small extra requirement. The main result has a Lean formalization. The manuscript also derives results about reconstructing functions from frequencies and about sets containing a line segment in every direction. In particular, it states that every Euclidean Kakeya set has full Hausdorff dimension, a way of measuring size that also applies to highly irregular sets. A companion manuscript focuses on a critical estimate for waves in three-dimensional space.

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

As an imperfect image, imagine judging an uneven road from one photograph versus from a recording of a journey along it. The recording lets you assess behavior across time rather than at one instant; likewise, local smoothing concerns an estimate over time, not a claim that every individual snapshot becomes smooth.

Why it matters

The main manuscript connects a wave-equation estimate to several conjectures in real and complex analysis. Its stated consequences include Bochner–Riesz estimates, Fourier restriction estimates, and Kakeya and Nikodym maximal estimates. This makes the wave result a common starting point for questions about frequencies, evolution equations, and geometric size.

2 · Going further

The main problem is Sogge’s local smoothing conjecture for the Euclidean wave equation in spatial dimension $n \geq 2$ . The repository describes the result as an estimate with arbitrarily small loss above the sharp Sobolev threshold: the smoothness requirement can approach that threshold, but the supplied statement retains a positive loss. No explicit inequality or full range of wave-estimate exponents is provided in the abstracts.

The main manuscript organizes its conclusions around the wave estimate. From it, the manuscript derives the Bochner–Riesz conjecture, maximal estimates for Bochner–Riesz means, and sharp bounds for Stein’s square function; the repository specifies $2 \leq p < \infty$ for the latter two types of bounds. It also obtains Fourier restriction estimates for the sphere, compact paraboloid, and truncated cone throughout the interior of the conjectured range, together with local smoothing for the Schrödinger equation. The abstracts describe this chain of deductions but do not disclose the argument establishing the central wave estimate.

The manuscript further states the Kakeya and Nikodym maximal conjectures as consequences, including full Hausdorff dimension for every Euclidean Kakeya set. The companion manuscript isolates the three-dimensional case: it states the critical $L^{3}$ local smoothing estimate with every positive Sobolev loss and describes this as resolving Sogge’s Euclidean conjecture in dimension three. The main result has a Lean formalization; the supplied source does not specify whether that formalization covers every listed consequence, and it does not identify the companion manuscript as formalized.

Useful notions

The wave equation as a model of wave evolution
Measuring functions with Lᵖ norms
Smoothness measured in Sobolev spaces
Fourier analysis and frequency components
Dimension for irregular sets

What was known before / what is new

Before

The source identifies Sogge’s Euclidean local smoothing conjecture as the central problem. It also names the Bochner–Riesz, Kakeya maximal, and Nikodym maximal conjectures, and a conjectured range for Fourier restriction. It gives no account of earlier partial results or their chronology.

New

The main manuscript presents local smoothing in every spatial dimension $n \geq 2$ , with arbitrarily small loss above the sharp Sobolev threshold, and the main result has a Lean formalization. It derives the collection of frequency-analysis and geometric consequences described above. The companion manuscript presents the critical three-dimensional $L^{3}$ estimate with every positive Sobolev loss.

Limits and assumptions

The central statement concerns the Euclidean wave equation in dimensions at least two and retains an arbitrarily small positive Sobolev loss; the supplied material does not assert a zero-loss estimate. The Fourier restriction conclusions concern the interior of the conjectured range, not its boundary. The repository states the maximal Bochner–Riesz and Stein square-function bounds for $2 \leq p < \infty$ . The main result has a Lean formalization, but the source does not detail its coverage of the deductions, and no formalization is specified for the companion manuscript. The abstracts provide neither the central argument nor a list of remaining open problems.

Mini glossary

Local smoothing: An improvement in a wave-evolution estimate obtained by measuring behavior over a time interval rather than only at a fixed time.
Sobolev loss: An extra smoothness requirement above the threshold targeted by an estimate.
Fourier restriction: The study of estimates for a function’s Fourier transform when evaluated on a surface, such as the sphere.
Kakeya set: A set containing a line segment of unit length in every direction.
Hausdorff dimension: A notion of dimension for potentially irregular sets, with full dimension meaning the dimension of the surrounding space.
Lean formalization: A mathematical argument represented in Lean so that its logical steps can be checked by the system.

Keywords : Local smoothing · Fourier restriction · Kakeya sets

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)

Proves Sogge’s local smoothing conjecture in every spatial dimension $n \geq 2$ , with arbitrarily small loss above the sharp Sobolev threshold. The wave estimate yields the Bochner–Riesz conjecture and, for $2 \leq p < \infty$ , maximal bounds for these means and sharp bounds for Stein’s square function. It also gives Fourier restriction throughout the interior of the conjectured range for the sphere, compact paraboloid, and truncated cone, and local smoothing for the Schrödinger equation. The Kakeya and Nikodym maximal conjectures follow. In particular, every Euclidean Kakeya set has full Hausdorff dimension.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/079.md

Manuscripts in this family

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Local smoothing for the Euclidean wave equation

25/10/2026

Verified by Lean

We prove Sogge’s local smoothing conjecture for the Euclidean wave equation in every spatial dimension at least two. We deduce the Bochner–Riesz conjecture, maximal estimates for these means, and sharp bounds for Stein’s square function. We obtain Fourier restriction estimates for the sphere, the compact paraboloid, and the truncated cone throughout the interior of the conjectured range, as well as local smoothing for the Schrödinger equation. We also prove the Kakeya and Nikodym maximal conjectures. In particular, every Euclidean Kakeya set has full Hausdorff dimension.

PDF GitHub folder Lean formalization

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