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

TL;DR

This paper establishes $L^p$-estimates for solutions to the 2D wave equation with a critical magnetic potential, extending understanding of dispersive properties in magnetic Schrödinger operators.

Contribution

It provides new $L^p$-bounds for wave propagators associated with magnetic Schrödinger operators in the critical case, using kernel construction and pointwise estimates.

Findings

Boundedness of $(I+ ext{L}_A)^{- ext{γ}} e^{it ext{√L}_A}$ in $L^p$ for $1<p< ext{∞}$ when $ ext{γ}>|1/p-1/2|$

Derived $L^p$-bounds for the sine wave propagator $ ext{sin}(t ext{√L}_A) ext{L}_A^{-1/2}$

Key techniques include kernel construction and pointwise estimates of an analytic operator family.

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 $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.