Conductive Homogeneity of Compact Metric Spaces and Construction of p-Energy

In the ordinary theory of Sobolev spaces on domains of $mathbb{R}^{n}$, the $p$-energy is defined as the integral of $vertnabla fvert^{p}$. In this book, the author tries to construct a $p$-energy on compact metric spaces as a scaling limit of discrete $p$-energies on a series of graphs approximating the original space. In conclusion, the author proposes 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 he constructs a local regular Dirichlet form and shows 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, the author presents new classes of square-based, self-similar sets and rationally ramified Sierpiński crosses, where no diffusions were constructed before.

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