Local smoothing for the Euclidean wave equation

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.