Improved local smoothing estimate for the wave equation in higher dimensions

TL;DR

This paper proves a sharp $k$-broad estimate for certain phase functions and applies it to improve local smoothing estimates for the half-wave operator in higher dimensions, also extending restriction estimates.

Contribution

It introduces a sharp $k$-broad estimate for convex phase functions and enhances local smoothing results for the half-wave operator in dimensions $n \\ge 3$, broadening the scope of restriction estimates.

Findings

Established sharp $k$-broad estimate for convex phase functions

Improved local smoothing estimates for half-wave operator in dimensions $n \\ge 3$

Generalized restriction estimates of Ou--Wang to more phase functions

Abstract

In this paper, we establish the sharp $k$-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions $n\ge3$. As a byproduct, we also generalize the restriction estimates of Ou--Wang to a broader class of phase functions.