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

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.