Bounded $H^\infty$-calculus for vectorial-valued operators with Gaussian kernel estimates
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916855953752064 |
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| author | Addona, Davide Leone, Vincenzo Lorenzi, Luca Rhandi, Abdelaziz |
| author_facet | Addona, Davide Leone, Vincenzo Lorenzi, Luca Rhandi, Abdelaziz |
| contents | We prove that the vector-valued generator of a bounded holomorphic semigroup represented by a kernel satisfying Gaussian estimates with bounded $H^\infty$-calculus in $L^2(\mathbb R^d;\mathbb C^m)$ admits bounded $H^\infty$-calculus for every $p\in (1,\infty)$. We apply this result to the elliptic operator $-{\rm div}(Q\nabla)+V$, where the potential term V is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb R^d)$ and, for almost every $x\in \mathbb R^d$, $V(x)$ is a symmetric and nonnegative definite matrix. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16368 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded $H^\infty$-calculus for vectorial-valued operators with Gaussian kernel estimates Addona, Davide Leone, Vincenzo Lorenzi, Luca Rhandi, Abdelaziz Analysis of PDEs Primary: 47A60, Secondary: 35J47, 35K08, 47D06 We prove that the vector-valued generator of a bounded holomorphic semigroup represented by a kernel satisfying Gaussian estimates with bounded $H^\infty$-calculus in $L^2(\mathbb R^d;\mathbb C^m)$ admits bounded $H^\infty$-calculus for every $p\in (1,\infty)$. We apply this result to the elliptic operator $-{\rm div}(Q\nabla)+V$, where the potential term V is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb R^d)$ and, for almost every $x\in \mathbb R^d$, $V(x)$ is a symmetric and nonnegative definite matrix. |
| title | Bounded $H^\infty$-calculus for vectorial-valued operators with Gaussian kernel estimates |
| topic | Analysis of PDEs Primary: 47A60, Secondary: 35J47, 35K08, 47D06 |
| url | https://arxiv.org/abs/2507.16368 |