Local smoothing and Hardy spaces for Fourier integral operators

TL;DR

This paper introduces Hardy spaces tailored for Fourier integral operators, leading to improved local smoothing estimates for the wave equation that are nearly optimal and invariant under these operators.

Contribution

It develops Hardy spaces for Fourier integral operators and derives new, sharp local smoothing estimates that enhance existing bounds for the wave equation.

Findings

Improved local smoothing bounds for the wave equation.

Establishment of Hardy spaces invariant under Fourier integral operators.

Enhanced bounds for the local smoothing conjecture.

Abstract

We show that the Hardy spaces for Fourier integral operators form natural spaces of initial data when applying $\ell^{p}$-decoupling inequalities to local smoothing for the wave equation. This yields new local smoothing estimates which, in a quantified manner, improve the bounds in the local smoothing conjecture on $\mathbb{R}^{n}$ for $p\geq 2(n+1)/(n-1)$, and complement them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under application of Fourier integral operators, and they are essentially sharp.