Analysis & PDE: "Wave equations with $C^{1,1}$ coefficients"

Abstract: We consider solutions to the wave equation in $\mathbb{R}^{n+1}$ with coefficients that are $C^{1,1}$ functions of space and time. We work with Hardy spaces of functions adapted to Fourier integral operators recently introduced by the speaker with Portal and Rozendaal. These are modifications of $L^p$ spaces, on which we showed classical FIOs of order zero act as bounded operators (in contrast to $L^p$ spaces, where there is a loss of derivatives for $p \neq 2$). 

I will review this result and then discuss work in progress showing that, at least in 2 spatial dimensions, the same is true for solution operators to wave equations with $C^{1,1}$ coefficients. These can be thought of as "rough Fourier integral operators". 
In 3 dimensions, we need epsilon more differentiability in the coefficients to get the same result. 

I shall also indulge in some reckless speculation about what might be true for even rougher Fourier integral operators.