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

TL;DR

This paper investigates the heat kernel and Riesz transform related to the flow Laplacian on homogeneous trees, providing new weighted estimates and an alternative proof of boundedness results on $L^p$ spaces.

Contribution

It offers new weighted $L^1$-estimates for the heat kernel and a novel proof of Riesz transform boundedness on homogeneous trees.

Findings

Weighted $L^1$-estimates for the heat kernel and its gradient.

Boundedness of the first order Riesz transform on $L^p( ext{measure})$ for $p o 1,2$.

Alternative proof of Riesz transform boundedness that may enable further generalizations.

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.