Riesz transform for a flow Laplacian on homogeneous trees

Matteo Levi, Federico Santagati, Anita Tabacco, Maria Vallarino

Published 2021-07-14Version 1

We obtain weak type $(1,1)$ and $L^p$ boundedness, for $p \in (1,\infty)$, of the first order Riesz transform and its adjoint operator on a homogeneous tree endowed with the canonical flow measure. This is a model case of measure metric space which is nondoubling, of exponential growth, does not satisfy the Cheeger isoperimetric inequality, and where the Laplacian does not have spectral gap. This complements a previous work by W. Hebisch and T. Steger.