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arXiv · 2502.03151

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

Jialu Wang·Fang Zhang·Junyong Zhang·Jiqiang Zheng
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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 $1 |1/p-1/2|$ 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 $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.

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Jialu Wang, Fang Zhang, Junyong Zhang, Jiqiang Zheng. 2025-02-05. $L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field. https://arxiv.org/abs/2502.03151

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