$L^p$-estimates for the wave equation with critical magnetic field in higher dimensions

Abstract

In this paper, we study the $L^p$-estimates for the solution to the wave equation with a scaling-critical magnetic potential in Euclidean $R^N$ with $N\ge 3$. Inspired by the work of \cite{L}, we show that the operators $(I+{\mathcal{L}}_{\mathbf{A}}{)}^{-\gamma }{e}^{it\sqrt{{\mathcal{L}}_{\mathbf{A}}}}$ is bounded in $L^p({\mathbb{R}}^N)$ for $1<p<+\infty$ when $\gamma >|1/p-1/2|$ and $t>0$, where ${\mathcal{L}}_{\mathbf{A}}$ is a magnetic Schrödinger operator. In particular, we derive the $L^p$-bounds for the sine wave propagator $\sin (t\sqrt{{\mathcal{L}}_{\mathbf{A}}}){\mathcal{L}}_{\mathbf{A}}^{-\frac{1}{2}}$. The key ingredient is the $L^p\to L^p$ boundedness of the analytic operator family $f_{w,t}({\mathcal{L}}_{\mathbf{A}})$.