Nonlinear nonlocal Douglas identity

We give Hardy–Stein and Douglas identities for nonlinear nonlocal Sobolev–Bregman integral forms with unimodal Lévy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev–Bregman spaces. As an application, we obtain the boundedness of the Dirichlet-to-Neumann operator on weighted $$L^p$$L p spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev–Bregman forms.