Nonlinear nonlocal Douglas identity

We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal L'evy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev-Bregman forms.