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$L^p$ and $\mathcal{H}^p_{\mathit{FIO}}$ regularity for wave equations with rough coefficients

Andrew Hassell and Jan Rozendaal

Vol. 5 (2023), No. 3, 541–599
DOI: 10.2140/paa.2023.5.541
Abstract

We consider wave equations with time-independent coefficients that have C1,1 regularity in space. We show that, for nontrivial ranges of p and s, the standard inhomogeneous initial value problem for the wave equation is well posed in Sobolev spaces FIOs,p(n) over the Hardy spaces FIOp(n) for Fourier integral operators introduced recently by the authors and Portal, following work of Smith. In spatial dimensions n = 2 and n = 3, this includes the full range 1 < p < . As a corollary, we obtain the optimal fixed-time Lp regularity for such equations, generalizing work of Seeger, Sogge and Stein in the case of smooth coefficients.

Keywords
rough wave equations, $L^{p}$ regularity, Hardy spaces for Fourier integral operators
Mathematical Subject Classification
Primary: 35R05
Secondary: 35A27, 35L05, 42B37
Milestones
Received: 4 October 2021
Revised: 8 June 2022
Accepted: 12 October 2022
Published: 24 August 2023
Authors
Andrew Hassell
Mathematical Sciences Institute
Australian National University
Canberra, ACT
Australia
Jan Rozendaal
Institute of Mathematics
Polish Academy of Sciences
Warsaw
Poland
Mathematical Sciences Institute
Australian National University
Canberra, ACT
Australia