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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.

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Heat Kernel and Riesz Transform for the Flow Laplacian on Homogeneous Trees

  • 会议论文

  • 发表日期：

    2026年

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  • Alessio Martini,
  • Federico Santagati,
  • Maria Vallarino

摘要

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.