$L^p-L^q$ local smoothing estimates for the wave equation via $k$-broad Fourier restriction

TL;DR

This paper establishes $L^p-L^q$ local smoothing estimates for the wave equation in higher dimensions by connecting $k$-broad Fourier restriction estimates with a Bourgain--Guth analysis, revealing non-invariance under Lorentz rescaling.

Contribution

It introduces a novel approach linking $k$-broad Fourier restriction estimates to local smoothing for the wave equation in all dimensions $n \u2265 3$, using Bourgain--Guth broad-narrow analysis.

Findings

Established sharp $L^p-L^q$ local smoothing estimates for wave solutions.

Connected $k$-broad Fourier restriction estimates with local smoothing.

Identified non-invariance of local smoothing under Lorentz rescaling.

Abstract

We explore the connection between $k$-broad Fourier restriction estimates and sharp regularity $L^p-L^q$ local smoothing estimates for the solutions of the wave equation in $\mathbb{R}^{n}\times \mathbb{R}$ for all $n \geq 3$ via a Bourgain--Guth broad-narrow analysis. An interesting feature is that local smoothing estimates for $e^{i t \sqrt{-\Delta}}$ are not invariant under Lorentz rescaling.