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Let $\mathbb T_{q+1}$ denote the homogeneous tree of degree $q+1$ with thestandard graph distance $d$ and the canonical flow measure $\mu$. The metricmeasure space $(\mathbb T_{q+1},d,\mu)$ is of exponential growth. Let$\mathcal{L}$ denote the flow Laplacian, which is a probabilistic Laplacianself-adjoint on $L^2(\mu)$. In this note, we prove some weighted$L^1$-estimates for the heat kernel associated with $\mathcal{L}$ and itsgradient. As a consequence, we show that the first order Riesz transformassociated 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 wasproved in a previous paper by Hebisch and Steger: we give a different proofthat might pave the way to further generalizations.

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