$L^p$ 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 $\alpha\ge s_p:=(d-1)\left|\frac{1}{p}-\frac{1}{2}\right|$. We extend this result tooperators of the form $L=−\displaystyle\sum_{j=1}^d a_j\partial_j a_j\partial_j$, for functions $x\mapsto a_i(x_i)$ that are bounded above and below, but merely Lipschitz continuous. This is below the $C^{1,1}$ regularity that is known to be necessary in general for Strichartz estimates in dimension $d\ge2$. 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 $\exp(i\sqrt{L})$ 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 $A$. The result then follows from Sobolev embedding properties of these spaces.