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,η) are elements of C*rS1,δm classes that have limited regularity in the x variable. We show that the associated pseudodifferential operator a(x, D) maps between Sobolev spaces ℌFIOs,p(ℝn) and ℌFIOt,p(ℝn) over the Hardy space for Fourier integral operator ℌFIOp(ℝn). Our main result implies that for m = 0, δ =l/2 and r > n − 1, a(x, D) acts boundedly on ℌFIOp(ℝn) for all p ∈ (1, ∞).