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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.10508 |
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| _version_ | 1866911523525361664 |
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| author | Arhancet, Cédric Kriegler, Christoph |
| author_facet | Arhancet, Cédric Kriegler, Christoph |
| contents | For any non-Archimedean local field $\mathbb{K}$ and any integer $n \geq 1$, we show that the Taibleson operator admits a bounded $\mathrm{H}^\infty(Σ_θ)$ functional calculus on the Bochner space $\mathrm{L}^p(\mathbb{K}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $θ> 0$, where $Σ_θ=\{ z \in \mathbb{C}^*: |\arg z| < θ\}$ and $1 < p < \infty$. Moreover, we prove that it even admits a bounded Hörmander functional calculus of order $\frac{3}{2}$. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the $R$-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued $\mathrm{L}^p$-spaces over totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields Arhancet, Cédric Kriegler, Christoph Classical Analysis and ODEs Functional Analysis For any non-Archimedean local field $\mathbb{K}$ and any integer $n \geq 1$, we show that the Taibleson operator admits a bounded $\mathrm{H}^\infty(Σ_θ)$ functional calculus on the Bochner space $\mathrm{L}^p(\mathbb{K}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $θ> 0$, where $Σ_θ=\{ z \in \mathbb{C}^*: |\arg z| < θ\}$ and $1 < p < \infty$. Moreover, we prove that it even admits a bounded Hörmander functional calculus of order $\frac{3}{2}$. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the $R$-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued $\mathrm{L}^p$-spaces over totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator. |
| title | Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2407.10508 |