Rough Pseudodifferential Operators on Hardy Spaces for Fourier Integral Operators II

Abstract

We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $$a(x,\eta )$$ a ( x , η ) are elements of $$C^{r}_{*}S^{m}_{1,\delta }$$ C ∗ r S 1 , δ m classes that have limited regularity in the x variable. We show that the associated pseudodifferential operator a(x, D) maps between Sobolev spaces $${\mathcal {H}}^{s,p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO s , p ( R n ) and $${\mathcal {H}}^{t,p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO t , p ( R n ) over the Hardy space for Fourier integral operators $${\mathcal {H}}^{p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO p ( R n ) . Our main result is that for all $$r>0$$ r > 0 , $$m=0$$ m = 0 and $$\delta =1/2$$ δ = 1 / 2 , there exists an interval of p around 2 such that a(x, D) acts boundedly on $${\mathcal {H}}^{p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO p ( R n ) .