In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$, $p\in (2,\infty)$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908488871968768 |
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| author | Feneuil, Joseph |
| author_facet | Feneuil, Joseph |
| contents | In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters \( p \in (1, \infty) \) and \( γ\in [0, 1] \) for which the quantities \( \|Δ^γf\|_p \) and \( \|\nabla f\|_p \) can be compared. In particular, we prove that in such metric spaces, the Riesz transform \( \nabla Δ^{-1/2} \) is unbounded on \( L^p \) for all \( p \in (2, \infty) \), thereby demonstrating a clear departure from the behavior observed in the Euclidean setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10837 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$, $p\in (2,\infty)$ Feneuil, Joseph Functional Analysis Probability 42B20, 43A85, 60J10, 60J60 In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters \( p \in (1, \infty) \) and \( γ\in [0, 1] \) for which the quantities \( \|Δ^γf\|_p \) and \( \|\nabla f\|_p \) can be compared. In particular, we prove that in such metric spaces, the Riesz transform \( \nabla Δ^{-1/2} \) is unbounded on \( L^p \) for all \( p \in (2, \infty) \), thereby demonstrating a clear departure from the behavior observed in the Euclidean setting. |
| title | In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$, $p\in (2,\infty)$ |
| topic | Functional Analysis Probability 42B20, 43A85, 60J10, 60J60 |
| url | https://arxiv.org/abs/2502.10837 |