Dynkin's formula and large coupling convergence

Let E be an m-symmetric regular Dirichlet form in L2(E,m), Γ a closed subset of E with finite capacity and μΓ the equilibrium measure of Γ. If H-1 is the selfadjoint operator associated to E and J:D(E)⟶L2(E,μΓ) is defined by Jf=f˜μΓ-a.e., f∈D(E), then(1)∥D∞J-DβJ∥⩽11+β,β>0.Here f˜ denotes any quasi-continuous representative of f. Estimate (1) is sharp.