$L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients

Abstract

Peral/Miyachi's celebrated theorem on fixed time $L^{p}$ estimates with loss of derivatives for the wave equation states that the operator $(I-\Delta)^{- \frac{\alpha}{2}}\exp(i \sqrt{-\Delta})$ is bounded on $L^{p}(\mathbb{R}^{d})$ if and only if $\alpha \geq s_{p}:=(d-1)|\frac{1}{p}-\frac{1}{2}|$. We extend this result to operators of the form $\mathcal{L} = -\sum \limits _{j=1} ^{d} a_{j+d}\partial_{j}a_{j}\partial_{j}$, such that, for $j=1,...,d$, the functions $a_{j}$ and $a_{j+d}$ only depend on $x_{j}$, are bounded above and below, but are merely Lipschitz continuous. This is below the $C^{1,1}$ regularity that is known to be necessary in general for Strichartz estimates in dimension $d \geq 2$. Our proof is based on an approach to the boundedness of Fourier integral operators recently developed by Hassell, Rozendaal, and the second author. We construct a scale of adapted Hardy spaces on which $\exp(i\sqrt{ \mathcal{L}} )$ is bounded by lifting $L^{p}$ functions to the tent space $T^{p,2}(\mathbb{R}^{d})$, using a wave packet transform adapted to the Lipschitz metric induced by the coefficients $a_j$. The result then follows from Sobolev embedding properties of these spaces.