Rough pseudodifferential operators on Hardy spaces for Fourier integral operators

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 $C^{r}_{*}S^{m}_{1,delta}$ 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 implies that for $m=0$, $delta=1/2$ and $r>n-1$, $a(x,D)$ acts boundedly on $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$ for all $pin(1,infty)$.