On extensions of local Dirichlet forms

Let E be a Dirichlet form on L2(X;μ) where (X, μ) is locally compact σ-compact measure space. Assume E is inner regular, i.e., regular in restriction to functions of compact support, and local in the sense that E(φ, ψ) = 0 for all φ, ψ∈ D(E) with φψ=0. We construct two Dirichlet forms E_m and E_M such that E_m≤ E≤ E_M. These forms are potentially the smallest and largest such Dirichlet forms. In particular E_m⊇ E_M, (EM)m=Em and (Em)M=EM. We analyze the family of local, inner regular, Dirichlet forms F which extend E and satisfy E_m≤ F≤ E_M. We prove that the latter bounds are valid if and only if F_M= E_M, or F_m= E_m, or D(E_M) is an order ideal of D(F). Alternatively the F are characterized by D(E_M) ∩ L_∞(X) being an algebraic ideal of D(F) ∩ L_∞(X). As an application we show that if E and F are strongly local then the Ariyoshi–Hino set-theoretic distance is the same for each of the forms E, E_M and F. If in addition E_m is strongly local then it also defines the same distance. Finally we characterize the uniqueness condition E_M= E_m by capacity estimates.