$L^p$-estimates for the wave equation with critical magnetic potential on conical manifolds
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909564786442240 |
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| author | Gao, Xiaofen Wang, Jialu Xu, Chengbin Zhang, Fang |
| author_facet | Gao, Xiaofen Wang, Jialu Xu, Chengbin Zhang, Fang |
| contents | In this paper, we consider a class of conical singular spaces $Σ=(0,\infty)_r\times Y$ equipped with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$ without boundary. In this context, we aim to show that the sine wave propagator $\sin\left(t\sqrt{\mathcal{L}_{\mathbf{A}}}\right)/\sqrt{\mathcal{L}_{\mathbf{A}}}$ is bounded in $L^{p}(Σ)$, where $\mathcal{L}_{\mathbf{A}}$ is a magnetic Schrödinger operator with a scaling-critical magnetic potential on metric cone $Σ$. Our main result is the generalization of the result in \cite{L}. The novel ingredient is the construction of Hadamard parametrix for $\cos\left(t\sqrt{\mathcal{L}_{\bf A}}\right)$ on $Σ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03124 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^p$-estimates for the wave equation with critical magnetic potential on conical manifolds Gao, Xiaofen Wang, Jialu Xu, Chengbin Zhang, Fang Analysis of PDEs In this paper, we consider a class of conical singular spaces $Σ=(0,\infty)_r\times Y$ equipped with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$ without boundary. In this context, we aim to show that the sine wave propagator $\sin\left(t\sqrt{\mathcal{L}_{\mathbf{A}}}\right)/\sqrt{\mathcal{L}_{\mathbf{A}}}$ is bounded in $L^{p}(Σ)$, where $\mathcal{L}_{\mathbf{A}}$ is a magnetic Schrödinger operator with a scaling-critical magnetic potential on metric cone $Σ$. Our main result is the generalization of the result in \cite{L}. The novel ingredient is the construction of Hadamard parametrix for $\cos\left(t\sqrt{\mathcal{L}_{\bf A}}\right)$ on $Σ$. |
| title | $L^p$-estimates for the wave equation with critical magnetic potential on conical manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.03124 |