$L^p$-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 Schr\"odinger operator. In particular, we derive the $L^{p}$-bounds for the sine wave propagator $\sin(t\sqrt{\mathcal{L}{\mathbf{A}}})\mathcal{L}^{-\frac12}_{\mathbf{A}}$. The key ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.