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Local smoothing for the Euclidean wave equation
Proves Sogge’s local smoothing conjecture in every spatial dimension , with arbitrarily small loss above the sharp Sobolev threshold. The wave estimate yields the Bochner–Riesz conjecture and, for , 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.
Manuscripts in this family · 2
Local smoothing for the Euclidean wave equation
Oct 25, 2026·paper.pdfLean
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.
Critical local smoothing for the three-dimensional wave equation
Sep 24, 2026·paper.pdf
We prove the critical L 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.