$L^p$-estimates for the wave equation with critical magnetic field in higher dimensions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913786268483584 |
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| author | Wang, Jialu Xu, Chengbin Zhang, Fang |
| author_facet | Wang, Jialu Xu, Chengbin Zhang, Fang |
| contents | 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 $f_{w,t}(\mathcal{L}_{\mathbf{A}})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^p$-estimates for the wave equation with critical magnetic field in higher dimensions Wang, Jialu Xu, Chengbin Zhang, Fang Analysis of PDEs 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 $f_{w,t}(\mathcal{L}_{\mathbf{A}})$. |
| title | $L^p$-estimates for the wave equation with critical magnetic field in higher dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.06930 |