Riesz transform and spectral multipliers for the flow Laplacian on nonhomogeneous trees
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| Format: | Preprint |
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2023
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| _version_ | 1866915773281206272 |
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| author | Martini, Alessio Santagati, Federico Tabacco, Anita Vallarino, Maria |
| author_facet | Martini, Alessio Santagati, Federico Tabacco, Anita Vallarino, Maria |
| contents | Let $T$ be a locally finite tree equipped with a flow measure $m$. Let $\mathcal L$ be the flow Laplacian on $(T,m)$. We prove that the first order Riesz transform $\nabla \mathcal L^{-1/2}$ is bounded on $L^p(m)$ for $p\in (1,\infty)$. Moreover, we prove a sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type for $\mathcal L$. In the case where $m$ is locally doubling, we also prove corresponding weak type and Hardy space endpoint bounds. This generalises results by Hebisch and Steger for the canonical flow Laplacian on homogeneous trees to the setting of nonhomogeneous trees with arbitrary flow measures. The proofs rely on approximation and perturbation arguments, which allow one to transfer to any flow tree a number of $L^p$ bounds that hold on homogeneous trees of arbitrarily large degree and are uniform in the degree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_09113 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Riesz transform and spectral multipliers for the flow Laplacian on nonhomogeneous trees Martini, Alessio Santagati, Federico Tabacco, Anita Vallarino, Maria Functional Analysis Classical Analysis and ODEs 05C05, 05C21, 42B20, 43A99 Let $T$ be a locally finite tree equipped with a flow measure $m$. Let $\mathcal L$ be the flow Laplacian on $(T,m)$. We prove that the first order Riesz transform $\nabla \mathcal L^{-1/2}$ is bounded on $L^p(m)$ for $p\in (1,\infty)$. Moreover, we prove a sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type for $\mathcal L$. In the case where $m$ is locally doubling, we also prove corresponding weak type and Hardy space endpoint bounds. This generalises results by Hebisch and Steger for the canonical flow Laplacian on homogeneous trees to the setting of nonhomogeneous trees with arbitrary flow measures. The proofs rely on approximation and perturbation arguments, which allow one to transfer to any flow tree a number of $L^p$ bounds that hold on homogeneous trees of arbitrarily large degree and are uniform in the degree. |
| title | Riesz transform and spectral multipliers for the flow Laplacian on nonhomogeneous trees |
| topic | Functional Analysis Classical Analysis and ODEs 05C05, 05C21, 42B20, 43A99 |
| url | https://arxiv.org/abs/2310.09113 |