Hardy spaces and Riesz transforms on a Lie group of exponential growth
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915441940627456 |
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| author | Sjögren, Peter Vallarino, Maria |
| author_facet | Sjögren, Peter Vallarino, Maria |
| contents | Let $G$ be the Lie group ${\Bbb{R}}^2\rtimes {\Bbb{R}}^+$ endowed with the Riemannian symmetric space structure. Take a distinguished basis $X_0,\, X_1,\,X_2$ of left-invariant vector fields of the Lie algebra of $G$, and consider the Laplacian $Δ=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_iΔ^{-1/2}$, \hskip3pt $i=0,1,2$. We first show that the atomic Hardy space $H^1$ in $G$ introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms $\mathcal R_i$. It is also proved that two of these Riesz transforms are bounded from $H^1$ to $H^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08032 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hardy spaces and Riesz transforms on a Lie group of exponential growth Sjögren, Peter Vallarino, Maria Functional Analysis Let $G$ be the Lie group ${\Bbb{R}}^2\rtimes {\Bbb{R}}^+$ endowed with the Riemannian symmetric space structure. Take a distinguished basis $X_0,\, X_1,\,X_2$ of left-invariant vector fields of the Lie algebra of $G$, and consider the Laplacian $Δ=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_iΔ^{-1/2}$, \hskip3pt $i=0,1,2$. We first show that the atomic Hardy space $H^1$ in $G$ introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms $\mathcal R_i$. It is also proved that two of these Riesz transforms are bounded from $H^1$ to $H^1$. |
| title | Hardy spaces and Riesz transforms on a Lie group of exponential growth |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2406.08032 |