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    \noindent Sections~\ref{sec:current}--\ref{sec:transfer} prove
    Theorem~\ref{thm:local-smoothing}; Sections~\ref{sec:kakeya}--\ref{sec:parabolic}
    give its applications.\par\smallskip
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\title{Local smoothing for the Euclidean wave equation}
\author{OpenAI}\date{October 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
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
Schr\"odinger equation. We also prove the Kakeya and Nikodym maximal
conjectures; in particular, every Euclidean Kakeya set has full
Hausdorff dimension.
\end{abstract}
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\input{sections/propagation}
\input{sections/transfer}
\input{sections/sharpness}
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\input{sections/stein_square}
\input{sections/parabolic}
\appendix
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