紧度量空间的传导齐性与p-能量的构造
Conductive homogeneity of compact metric spaces and construction of p-energy
摘要：
在$R^n$域上索伯列夫空间的一般理论中，$p$ -能量被定义为$|\nabla{f}|^p$的积分.本文试图构造紧致度量空间上的$p$ -能量作为逼近原空间的一系列图上离散$p$ -能量的标度极限。总之，我们提出了一个概念，称为导电的均匀性下，可以构建一个合理的$p$ -能量，如果$p$是大于的Ahlfors正规共形维数的空间。特别地，如果$p = 2$，那么我们构造了一个局部正则Dirichlet形式，并证明了与Dirichlet形式相关联的热核满足上下亚高斯型热核估计。作为传导性齐性空间的例子，我们提出了一类新的基于平方的自相似集和有理分歧Sierpinski交叉，其中没有扩散被构造。
英文摘要：
In the ordinary theory of Sobolev spaces on domains of $R^n$, the $p$-energy is defined as the integral of $|\nabla{f}|^p$. In this paper, we try to construct $p$-energy on compact metric spaces as a scaling limit of discrete $p$-energies on a series of graphs approximating the original space. In conclusion, we propose a notion called conductive homogeneity under which one can construct a reasonable $p$-energy if $p$ is greater than the Ahlfors regular conformal dimension of the space. In particular, if $p = 2$, then we construct a local regular Dirichlet form and show that the heat kernel associated with the Dirichlet form satisfies upper and lower sub-Gaussian type heat kernel estimates. As examples of conductively homogeneous spaces, we present a new class of square-based self-similar sets and rationally ramified Sierpinski cross, where no diffusion was constructed before.
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