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 $A{H}^{-\frac{1}{4}}$ 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 Schrödinger 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 ${\mathbb{R}}^n$, $n\ge 3$.

Cited by in corpus (5)

• Attractors of Hamilton nonlinear partial differential equations
• Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials
• Uniform resolvent estimates for Schrödinger operator with an inverse-square potential
• Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials and applications
• Strichartz estimates for quadratic repulsive potentials