Boundedness of a Riesz transform on the homogeneous tree

Federico Santagati, Maria Vallarino

Published 2022-10-13Version 1

Let $\mathbb T_{q+1}$ denote the homogeneous tree of degree $q+1$ endowed with the standard graph distance $d$ and the canonical flow $\mu$. The metric measure space $(\mathbb T_{q+1},d,\mu)$ is of exponential growth. In this note, we prove that the first order Riesz transform associated with a flow Laplacian on $\mathbb T_{q+1}$ is bounded on $L^p(\mu)$, for $p\in (1,2]$ and of weak type $(1,1)$. This result was proved in a previous paper by Hebisch and Steger: we give a different proof which might pave the way to further generalizations.