Kato smoothing and Strichartz estimates for wave equations with magnetic potentials

Abstract

Let $H$ be a selfadjoint operator and $A$ a closed operator on a Hilbert space $\mathcal{H}$. If $A$ is $H$-(super)smooth in the sense of Kato-Yajima, we prove that $AH^{-\frac14}$ is $\sqrt{H}$-(super)smooth. This allows to include wave and Klein-Gordon equations in the abstract theory at the same level of generality as Schr\"{o}dinger equations. We give a few applications and in particular, based on the resolvent estimates of Erdogan, Goldberg and Schlag \cite{ErdoganGoldbergSchlag09-a}, we prove Strichartz estimates for wave equations perturbed with large magnetic potentials on $\mathbb{R}^{n}$, $n\ge3$.