Rough pseudodifferential operators on Hardy spaces for Fourier integral operators

Abstract

We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,\eta )$ are elements of $C_{\ast }^r{S}_{1,\delta }^m$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces ${\mathcal{H}}_{FIO}^{s,p}({\mathbb{R}}^n)$ and ${\mathcal{H}}_{FIO}^{t,p}({\mathbb{R}}^n)$ over the Hardy space for Fourier integral operators ${\mathcal{H}}_{FIO}^p({\mathbb{R}}^n)$. Our main result implies that for $m=0$, $\delta =1/2$ and $r>n-1$, $a(x,D)$ acts boundedly on ${\mathcal{H}}_{FIO}^p({\mathbb{R}}^n)$ for all $p\in (1,\infty )$.

24 pages. Minor change to the funding acknowledgment

Cited by in corpus (1)

• $L^p$ and ${\mathcal{H}}_{FIO}^p$ regularity for wave equations with rough coefficients