Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II

TL;DR

This paper improves bounds for rough pseudodifferential operators acting on Hardy spaces associated with Fourier integral operators, demonstrating boundedness under limited regularity conditions in the symbol's x-variable.

Contribution

It establishes boundedness of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators, extending previous results to symbols with limited regularity.

Findings

Boundedness of pseudodifferential operators on Hardy spaces for FIOs for all r>0

Operators map between specific Sobolev spaces over Hardy spaces

Existence of p-interval around 2 for boundedness

Abstract

We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,\eta)$ are elements of $C^{r}_{*}S^{m}_{1,\delta}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ and $\mathcal{H}^{t,p}_{FIO}(\mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. Our main result is that for all $r>0$, $m=0$ and $\delta=1/2$, there exists an interval of $p$ around $2$ such that $a(x,D)$ acts boundedly on $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$.