Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908805445451776 |
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| author | Liu, Jie Martini, Alessio |
| author_facet | Liu, Jie Martini, Alessio |
| contents | Let $Δ= \nabla^* \nabla$ be the distinguished Laplacian on a Damek-Ricci space. We prove the $L^{p}$-boundedness of the vector of first-order Riesz transforms $\nabla Δ^{-1/2}$ in the full range $p\in(1,\infty)$. The most demanding part of the proof is establishing the boundedness for $p \in (2,\infty)$; this is obtained as a consequence of an operator-valued spectral multiplier theorem for the joint functional calculus of a commuting system of self-adjoint operators, which we prove here and may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02062 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers Liu, Jie Martini, Alessio Functional Analysis Classical Analysis and ODEs 22E30, 42B20 Let $Δ= \nabla^* \nabla$ be the distinguished Laplacian on a Damek-Ricci space. We prove the $L^{p}$-boundedness of the vector of first-order Riesz transforms $\nabla Δ^{-1/2}$ in the full range $p\in(1,\infty)$. The most demanding part of the proof is establishing the boundedness for $p \in (2,\infty)$; this is obtained as a consequence of an operator-valued spectral multiplier theorem for the joint functional calculus of a commuting system of self-adjoint operators, which we prove here and may be of independent interest. |
| title | Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers |
| topic | Functional Analysis Classical Analysis and ODEs 22E30, 42B20 |
| url | https://arxiv.org/abs/2602.02062 |