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

Fuente: arXiv
Saved in:
Bibliographic Details
Main Authors: Wang, Jialu, Zhang, Fang, Zhang, Junyong, Zheng, Jiqiang
Format: Preprint
Published: 2025
Subjects:
Online Access:
Tags: Add Tag
No Tags, Be the first to tag this record!
_version_ 1866910815203885056
author Wang, Jialu
Zhang, Fang
Zhang, Junyong
Zheng, Jiqiang
author_facet Wang, Jialu
Zhang, Fang
Zhang, Junyong
Zheng, Jiqiang
contents 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 $(I+\mathcal{L}_{\mathbf{A}})^{-γ}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}$ is bounded in $L^{p}(\mathbb{R}^{2})$ 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 ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.
format Preprint
id arxiv_https___arxiv_org_abs_2502_03151
institution arXiv
publishDate 2025
record_format arxiv
spellingShingle $L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field
Wang, Jialu
Zhang, Fang
Zhang, Junyong
Zheng, Jiqiang
Analysis of PDEs
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 $(I+\mathcal{L}_{\mathbf{A}})^{-γ}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}$ is bounded in $L^{p}(\mathbb{R}^{2})$ 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 ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.
title $L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field
topic Analysis of PDEs
url https://arxiv.org/abs/2502.03151