$L^p$ 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 $\alpha \ge s_p:=(d-1)|\frac{1}{p}-\frac{1}{2}|$. We extend this result to operators of the form $\mathcal{L}=-\sum _{j=1}^d{a}_{j+d}{\partial }_j{a}_j{\partial }_j$, 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 $d\ge 2$. 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.

35 pages

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