$L^p$-boundedness of Riesz transforms on solvable extensions of Carnot groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910614073376768 |
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| author | Martini, Alessio Plewa, Paweł |
| author_facet | Martini, Alessio Plewa, Paweł |
| contents | Let $G=N\rtimes \mathbb{R}$, where $N$ is a Carnot group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum is a left-invariant sub-Laplacian $Δ$ on $G$. We prove that the first-order Riesz transforms $X Δ^{-1/2}$ are bounded on $L^p(G)$ for all $p\in(1,\infty)$, where $X$ is any horizontal left-invariant vector field on $G$. This extends a previous result by Vallarino and the first-named author, who obtained the bound for $p\in(1,2]$. The proof makes use of an operator-valued spectral multiplier theorem, recently proved by the authors, and hinges on estimates for products of modified Bessel functions and their derivatives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$-boundedness of Riesz transforms on solvable extensions of Carnot groups Martini, Alessio Plewa, Paweł Functional Analysis Classical Analysis and ODEs 22E30, 42B20, 42B30 Let $G=N\rtimes \mathbb{R}$, where $N$ is a Carnot group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum is a left-invariant sub-Laplacian $Δ$ on $G$. We prove that the first-order Riesz transforms $X Δ^{-1/2}$ are bounded on $L^p(G)$ for all $p\in(1,\infty)$, where $X$ is any horizontal left-invariant vector field on $G$. This extends a previous result by Vallarino and the first-named author, who obtained the bound for $p\in(1,2]$. The proof makes use of an operator-valued spectral multiplier theorem, recently proved by the authors, and hinges on estimates for products of modified Bessel functions and their derivatives. |
| title | $L^p$-boundedness of Riesz transforms on solvable extensions of Carnot groups |
| topic | Functional Analysis Classical Analysis and ODEs 22E30, 42B20, 42B30 |
| url | https://arxiv.org/abs/2409.13233 |