In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$, $p\in (2,\infty)$

Joseph Feneuil

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arXiv · 2502.10837

In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$, $p\in (2,\infty)$

Joseph Feneuil

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Abstract

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 $ \gamma \in [0, 1] $ for which the quantities $ \|\Delta^\gamma f\|_p $ and $ \|\nabla f\|_p $ can be compared. In particular, we prove that in such metric spaces, the Riesz transform $ \nabla \Delta^{-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.

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BibTeX RIS

Joseph Feneuil. 2025-02-15. In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$, $p\in (2,\infty)$. https://arxiv.org/abs/2502.10837

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