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

Let Tq+1 denote the homogeneous tree of degree q + 1 with the standard graph distance d and the canonical flow measure µ. The metric measure space (Tq+1, d, µ) is of exponential growth. Let L denote the flow Laplacian, which is a probabilistic Laplacian self-adjoint on L2 (µ). In this note, we prove some weighted L1 -estimates for the heat kernel associated with L and its gradient. As a consequence, we show that the first order Riesz transform associated with the flow Laplacian on Tq+1 is bounded on Lp(µ), for p ∈ (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.