论文标题
加热内核和Riesz在同质树上的流动laplacian转换
Heat kernel and Riesz transform for the flow Laplacian on homogeneous trees
论文作者
论文摘要
令$ \ mathbb t_ {q+1} $表示均质树$ q+1 $，标准图距离$ d $和规范流量测量$μ$。度量度量空间$（\ Mathbb t_ {q+1}，d，μ）$是指数增长。令$ \ MATHCAL {L} $表示Flow Laplacian，这是$ l^2（μ）$的概率laplacian自我接合。在本说明中，我们证明了与$ \ Mathcal {l} $及其梯度相关的加热内核的加权$ l^1 $估计。结果，我们表明，一阶riesz在$ \ mathbb t_ {q+1} $上与流量laplacian相关的转换在$ l^p（μ）$上限制为$ p \ in（1,2] $ in（1,2] $和弱$（1,1）$的$ p \。
Let $\mathbb T_{q+1}$ denote the homogeneous tree of degree $q+1$ with the standard graph distance $d$ and the canonical flow measure $μ$. The metric measure space $(\mathbb T_{q+1},d,μ)$ is of exponential growth. Let $\mathcal{L}$ denote the flow Laplacian, which is a probabilistic Laplacian self-adjoint on $L^2(μ)$. 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 $L^p(μ)$, 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.
