( 1 , p ) $(1,p)$ -Sobolev spaces based on strongly local Dirichlet forms

Abstract

In the framework of quasi-regular strongly local Dirichlet form $(E,D(E\left)\right)$ on $L^2(X;m)$ admitting minimal $E$-dominant measure $\mu$, we construct a natural $p$-energy functional $(E^p,D\left(E^p\right))$ on $L^p(X;m)$ and $(1,p)$-Sobolev space $(H^{1,p}\left(X\right),\parallel ·{\parallel }_{H^{1,p}})$ for $p\in ]1,+\infty [$. In this paper, we establish the Clarkson-type inequality for $(H^{1,p}\left(X\right),\parallel ·{\parallel }_{H^{1,p}})$. As a consequence, $(H^{1,p}\left(X\right),\parallel ·{\parallel }_{H^{1,p}})$ is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of $(H^{1,p}\left(X\right),\parallel ·{\parallel }_{H^{1,p}})$, we prove that (generalized) normal contraction operates on $(E^p,D\left(E^p\right))$, which has been shown in the case of various concrete settings, but has not been proved for such a general framework. Moreover, we prove that $(1,p)$-capacity ${\mathrm{Cap}}_{1,p}\left(A\right)<\infty$ for open set $A$ admits an equilibrium potential $e_A\in D\left(E^p\right)$ with $0\le e_A\le 1$ $m$-a.e. and $e_A=1$ $m$-a.e. on $A$.