Local smoothing for rough wave equations

We prove local smoothing estimates for wave equations with $C^r$ coefficients, relying on bilinear restriction estimates associated with rough wave propagation. In two and three dimensions, for $C^{1,1}$ coefficients we recover the sharp $L^p$-$L^q$ local smoothing bounds previously established for smooth coefficients. These are implied by stronger $L^p$-$L^q$ decoupling inequalities with endpoint derivative loss. As a corollary, we obtain non-trivial $L^p$-$L^p$ local smoothing estimates as well.