Sogge's local smoothing conjecture for the wave equation in three spatial dimensions

Theorem 1.1: for every $\varepsilon>0$, $\|e^{it\sqrt{-\Delta}}f\|_{L^3(\mathbb R^3\times[1,2])}\le C_\varepsilon\|J^\varepsilon f\|_{L^3}$. With the $L^2$ and $L^\infty$-kernel endpoints, Corollary 10.6 gives the full strict range $2<p<\infty$, $\sigma>\max\{0,1-3/p\}$. Section 11, through Beltran-Hickman-Sogge, Tao and Wolff, derives the strict three-dimensional Bochner-Riesz range, maximal Bochner-Riesz bounds for $p\ge3$, the diagonal sphere restriction estimate $L^r(S^2)\to L^r(\mathbb R^3)$ for $r>3$, and the three-dimensional Kakeya maximal estimate. Not shown: the endpoint $\sigma=0$ at $p=3$, dimensions $n\ge4$, or the conjecture's general-$n$ form.

Theorem 1.1: for every $\varepsilon>0$, $\|e^{it\sqrt{-\Delta}}f\|_{L^3(\mathbb R^3\times[1,2])}\le C_\varepsilon\|J^\varepsilon f\|_{L^3}$. With the $L^2$ and $L^\infty$-kernel endpoints, Corollary 10.6 gives the full strict range $2<p<\infty$, $\sigma>\max\{0,1-3/p\}$. Section 11, through Beltran-Hickman-Sogge, Tao and Wolff, derives the strict three-dimensional Bochner-Riesz range, maximal Bochner-Riesz bounds for $p\ge3$, the diagonal sphere restriction estimate $L^r(S^2)\to L^r(\mathbb R^3)$ for $r>3$, and the three-dimensional Kakeya maximal estimate. Not shown: the endpoint $\sigma=0$ at $p=3$, dimensions $n\ge4$, or the conjecture's general-$n$ form.