$L^p$-estimates for the wave equation with partial inverse-square potentials
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911549934796800 |
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| author | Wang, Jialu Xu, Chengbin Zhang, Fang Zhang, Junyong |
| author_facet | Wang, Jialu Xu, Chengbin Zhang, Fang Zhang, Junyong |
| contents | This paper investigates $L^p$-estimates for solutions to the wave equation perturbed by a scaling-critical partial inverse-square potential. We study a model in which the singularity of the potential appears only in a subset of the variables, corresponding to the Schrödinger operator $\mathcal{H}_a = -Δ_x - Δ_y + a/|x|^2$ on $\mathbb{R}^{2+n}$. Using spectral analysis, we establish the $L^p$-boundedness of the wave propagator $(1+\sqrt{\mathcal{H}_a})^{-γ} e^{it\sqrt{\mathcal{H}_a}}$ for a range of exponents $γ$ and $p$ satisfying $|1/p -1/2| < γ/(n+1)$. The key ingredients are the spectral measure kernel of the partial inverse-square operator $\mathcal{H}_a$ and the complex interpolation argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27111 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $L^p$-estimates for the wave equation with partial inverse-square potentials Wang, Jialu Xu, Chengbin Zhang, Fang Zhang, Junyong Analysis of PDEs Functional Analysis This paper investigates $L^p$-estimates for solutions to the wave equation perturbed by a scaling-critical partial inverse-square potential. We study a model in which the singularity of the potential appears only in a subset of the variables, corresponding to the Schrödinger operator $\mathcal{H}_a = -Δ_x - Δ_y + a/|x|^2$ on $\mathbb{R}^{2+n}$. Using spectral analysis, we establish the $L^p$-boundedness of the wave propagator $(1+\sqrt{\mathcal{H}_a})^{-γ} e^{it\sqrt{\mathcal{H}_a}}$ for a range of exponents $γ$ and $p$ satisfying $|1/p -1/2| < γ/(n+1)$. The key ingredients are the spectral measure kernel of the partial inverse-square operator $\mathcal{H}_a$ and the complex interpolation argument. |
| title | $L^p$-estimates for the wave equation with partial inverse-square potentials |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2603.27111 |