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 $C^{r}_{*}S^{m}_{1,delta}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $mathcal{H}^{p,s}_{FIO}(mathbb{R}^{n})$ and $mathcal{H}^{p,t}_{FIO}(mathbb{R}^{n})$ 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 $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$.