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$L^{p}$ and $mathcal{H}^{p}_{FIO}$ regularity for wave equations with rough coefficients, Part I

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 نشر من قبل Jan Rozendaal
 تاريخ النشر 2020
  مجال البحث
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We consider wave equations with time-independent coefficients that have $C^{1,1}$ 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 $mathcal{H}^{s,p}_{FIO}(mathbb{R}^{n})$ over the Hardy spaces $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$ 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 $1 < p < infty$. As a corollary, we obtain the optimal fixed-time $L^{p}$ regularity for such equations, generalizing work of Seeger, Sogge and Stein in the case of smooth coefficients.



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