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 $p\in(1,\infty)$.