Dirichlet空间上的Differential forms和度量图上的Bakry-Émery估计

RCD（K，\infty）度量测度空间中热流的Bakry-Émery条件和Wasserstein收缩

Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces

本文证明了RCD（K，\infty）度量测度空间（X，d，m）中的线性热流关于每个L^p-Kantorovich-Rubinstein-Wasserstein距离都满足收缩性质.特别地，我们得到了两个基本解之间的最优W_ \infty -耦合关于初始点距离的精确估计.这个结果是（X，d，m）中正则Cheeger-Dirichlet型的RCD（K，\infty）Ricci下界与相应的Bakry-Émery条件之间等价的结果.关键的工具是扩展到非光滑的度量措施设置的巴克里的论点，这使得能够提高之间的交换估计的马尔可夫半群和Carré du Champ相关联的狄利克雷形式。这种扩展是基于一个新的先验估计和一个容量参数的定期和紧狄利克雷形式是独立的利益。

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-Émery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities wit…

Bakry-Emery曲率维数条件与Riemann里奇曲率界

Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds

本文的目的是弥合Bakry-Émery和Lott-Sturm-Villani方法之间的差距，以提供综合和抽象的概念较低的里奇曲率界。我们从一个强局部Dirichlet形式${\mathcal{E}}$开始，它在波兰测度空间$(X,\mathfrak{m})$中有一个Carré du champ $Γ$，并且有一个规范距离${\mathsf{d}}_{\mathcal{E}}$诱导出$X$的原始拓扑.本文首先刻画了一类特殊的黎曼能量测度空间，其中${\mathcal{E}}$与${\mathsf{d}}_{\mathcal{E}}$诱导的Cheeger能量一致，并且每个函数$f$与$Γ(f)\le1$都有连续的表示.在这样的类中，我们证明了如果${\mathcal{E}}$满足Bakry-Émery曲率维数条件$\mathrm {BE}(K,\infty)$的适当弱形式，则度量测度空间$(X,{\mathsf{d}},\mathfrak{m})$满足根据[Duke Math. J. 163（2014）1405-1490]的黎曼里奇曲率界$\mathrm {RCD}(K,\infty)$，从而表明了两个概念的等价性。然后证明了两个应用：满足Bakry-Émery $\mathrm {BE}(K,N)$条件的黎曼能量空间的张量化性质（以及相应的$\mathrm {RCD}(K,\inft…

由带号测度曲率下界驯化的 Dirichlet 空间的 Littlewood-Paley-Stein 不等式

The Littlewood-Paley-Stein inequality for Dirichlet space tamed by signed measured curvature lower bounds

本文在由分布 Ricci 曲率下界驯化的 Dirichlet 空间框架下，建立了 $L^p$-函数的 Littlewood-Paley-Stein 不等式，部分推广了 Kawabi--Miyokawa 的已有结果。

Hess-Schrader-Uhlenbrock不等式：在分布曲率下界驯化的Dirichlet空间上微分形式的热半群

Hess-Schrader-Uhlenbrock inequality for the heat semigroup on differential forms over Dirichlet spaces tamed by distributional curvature lower bounds

本文在驯化Dirichlet空间框架下，建立了1-形式的Hess-Schrader-Uhlenbrock不等式，推广了Braun在附加条件下的结果。

Tamed spaces -- Dirichlet spaces with distribution-valued Ricci bounds

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimate…