Riesz transform on eventually Gaussian local trees

Abstract

We study the Riesz transform $\mathcal{R}=\partial(-\Delta)^{-\frac{1}{2}}$ on uniform local trees, metric measure spaces that are locally real trees and whose canonical Dirichlet form is built from weak derivatives along the skeleton. The reference measure $m$ may be singular with respect to the length measure $\nu$, so boundedness of $\mathcal R$ is understood from $L^{p}(m)$ to $L^{p}(\nu)$. In contrast with fractal-like manifolds and cable systems, the diffusion is sub-Gaussian at small scales and Gaussian at large scales. Under uniform volume growth, two-sided heat kernel estimates and a pointwise gradient estimate for the heat kernel, we prove that a local Dini condition on the scale function implies boundedness of $\mathcal{R}$ on $L^{p}$ for every $p\in[2,\infty)$, and hence the reverse Riesz inequality for every $p\in(1,2]$. Conversely, boundedness of $\mathcal{R}$ for some $p<2$, or a reverse Riesz inequality for some $p>2$, forces the space to be one-dimensional at small scales. We show that a reverse H\"older inequality for harmonic functions yields the gradient estimate, and verify all hypotheses for spaces carrying a geometric group action whose generators have bounded displacement. As an application, for the alternating Vicsek fractafold in $\mathbb Z^{d}$ we determine the exact ranges of $p$ for which the Riesz and reverse Riesz inequalities hold.