Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II

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 $C_{\ast }^r{S}_{1,\delta }^m$ 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 is that for all $r>0$, $m=0$ and $\delta =1\mathrm{/}2$, there exists an interval of $p$ around $2$ such that $a(x,D)$ acts boundedly on ${\mathcal{H}}_{FIO}^p({\mathbb{R}}^n)$.