Local smoothing and Hardy spaces for Fourier integral operators

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.