$L^p$-estimates for the wave equation with critical magnetic field in higher dimensions
Jialu Wang, Chengbin Xu, Fang Zhang
TL;DR
This work establishes $L^p$ bounds for wave evolutions under a scaling-critical magnetic potential in $\mathbb{R}^N$ by developing an analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$ and leveraging a Hadamard parametrix on $\mathbb{S}^{N-1}$. It proves that for $1<p<\infty$ and $\gamma>|1/p-1/2|$, the operator $(I+\mathcal{L}_{\mathbf{A}})^{-\gamma/2} e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}$ is bounded on $L^p$, and it derives corresponding bounds for the sine propagator $\sin(t\sqrt{\mathcal{L}_{\mathbf{A}}})\,\mathcal{L}_{\mathbf{A}}^{-1/2}$. The approach combines spectral multiplier theory, imaginary powers bounds, Hadamard parametrix arguments, and Stein interpolation to handle the high-dimensional, singular magnetic setting under a transversality condition. These results extend dispersive-type $L^p$ estimates to wave equations with critical magnetic fields and provide tools for spectral analysis and multiplier theorems in singular magnetic contexts.
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 $N\geq3$. Inspired by the work of \cite{L}, we show that the operators $(I+\mathcal{L}_{\mathbf{A}})^{-γ}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}$ is bounded in $L^{p}(\mathbb{R}^{N})$ for $1<p<+\infty$ when $γ>|1/p-1/2|$ 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 $\sin(t\sqrt{\mathcal{L}_{\mathbf{A}}})\mathcal{L}^{-\frac12}_{\mathbf{A}}$. The key ingredient is the $L^p\rightarrow L^p$ boundedness of the analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.