$L^p$ Maximal regularity for vector-valued Schrödinger operators
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910285182271488 |
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| author | Addona, Davide Leone, Vincenzo Lorenzi, Luca Rhandi, Abdelaziz |
| author_facet | Addona, Davide Leone, Vincenzo Lorenzi, Luca Rhandi, Abdelaziz |
| contents | In this paper we consider the vector-valued Schrödinger operator $-Δ+ V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in $L^1(\mathbb{R}^d,\mathbb{R}^m).$ Assuming further that the minimal eigenvalue of $V$ belongs to some reverse Hölder class of order $q\in(1,\infty)\cup\{\infty\}$, we obtain maximal inequality in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $p$ in between $1$ and some $q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00479 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $L^p$ Maximal regularity for vector-valued Schrödinger operators Addona, Davide Leone, Vincenzo Lorenzi, Luca Rhandi, Abdelaziz Analysis of PDEs Primary: 35K40, Secondary: 35J47, 47D06 In this paper we consider the vector-valued Schrödinger operator $-Δ+ V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in $L^1(\mathbb{R}^d,\mathbb{R}^m).$ Assuming further that the minimal eigenvalue of $V$ belongs to some reverse Hölder class of order $q\in(1,\infty)\cup\{\infty\}$, we obtain maximal inequality in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $p$ in between $1$ and some $q$. |
| title | $L^p$ Maximal regularity for vector-valued Schrödinger operators |
| topic | Analysis of PDEs Primary: 35K40, Secondary: 35J47, 47D06 |
| url | https://arxiv.org/abs/2401.00479 |