$L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field

Abstract

In this paper, we study the $L^p$-estimates for the solution to the $2\mathrm{D}$-wave equation with a scaling-critical magnetic potential. Inspired by the work of \cite{FZZ}, we show that the operators $(I+{\mathcal{L}}_{\mathbf{A}}{)}^{-\gamma }{e}^{it\sqrt{{\mathcal{L}}_{\mathbf{A}}}}$ is bounded in $L^p({\mathbb{R}}^2)$ 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 ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family $f_{w,t}({\mathcal{L}}_{\mathbf{A}})$.