Local smoothing in three dimensions

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

Manuscript

Critical local smoothing for the three-dimensional wave equation

September 24, 2026 · source · cite as \cite{OAI:Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026}

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.

Result 079 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.