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

Abstract

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