具有非倍增流测度的树上的分析

Hardy spaces on homogeneous trees with flow measures

We consider a homogeneous tree endowed with a nondoubling flow measure $μ$ of exponential growth and a probabilistic Laplacian $\mathcal{L}$ self-adjoint with respect to $μ$. We prove that the maximal characterization in terms of the heat and the Poisson semigroup of $\mathcal{L}$ and the Riesz transform characterization of the atomic Hardy space introduced in a previous work fail.

Hardy spaces on weighted homogeneous trees

We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space V,d,μ) is nondoubling and of exponential growth, hence the classical theory of Hardy spaces does not apply in this setting. We construct an atomic Hardy space H^1 on (V,d,μ) and investigate some of its properties, focusing in particular on real interpolation properties and on boundedness of singular integrals on H^1.

BMO spaces on weighted homogeneous trees

We consider an infinite homogeneous tree $\mathcal V$ endowed with the usual metric $d$ defined on graphs and a weighted measure $μ$. The metric measure space $(\mathcal V,d,μ)$ is nondoubling and of exponential growth, hence the classical theory of Hardy and $BMO$ spaces does not apply in this setting. We introduce a space $BMO(μ)$ on $(\mathcal V,d,μ)$ and investigate some of its properties. We prove in particular that $BMO(μ)$ can be identified with the dual of a Hardy space $H^1(μ)$ introduced in a previous work and we investigate the sharp maximal function related with $BMO(μ)$.

齐次树上流拉普拉斯算子的 Riesz 变换

Riesz transform for a flow Laplacian on homogeneous trees

本文证明了齐次树上带典范流测度的流拉普拉斯算子所关联的一阶 Riesz 变换在 $p\in(1,\infty)$ 时的 $L^p$ 有界性，并研究了水平 Riesz 变换及其伴随算子的弱 $(1,1)$ 型有界性。

齐次树上的调和 Bergman 投影

Harmonic Bergman projectors on homogeneous trees

研究 q-齐次树上非倍增测度下的调和 Bergman 空间，计算 p=2 时的再生核，并证明 Bergman 投影的 Lp 有界性及弱 (1,1) 型有界性。

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

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]…