Heat kernel and Riesz transform for the flow Laplacian on homogeneous trees

Abstract

Let ${\mathbb{T}}_{q+1}$ denote the homogeneous tree of degree $q+1$ with the standard graph distance $d$ and the canonical flow measure $\mu$. The metric measure space $({\mathbb{T}}_{q+1},d,\mu )$ is of exponential growth. Let $\mathcal{L}$ denote the flow Laplacian, which is a probabilistic Laplacian self-adjoint on $L^2(\mu )$. In this note, we prove some weighted $L^1$-estimates for the heat kernel associated with $\mathcal{L}$ and its gradient. As a consequence, we show that the first order Riesz transform associated with the flow Laplacian on ${\mathbb{T}}_{q+1}$ is bounded on $L^p(\mu )$, for $p\in (1,2]$ and of weak type $(1,1)$. The latter result was proved in a previous paper by Hebisch and Steger: we give a different proof that might pave the way to further generalizations.

12 pages. arXiv admin note: text overlap with arXiv:2107.06620