We will discuss Hardy-Stein and Douglas identities for specific nonlinear Sobolev-Bregman integral forms which are nonlocal and have unimodal Lévy measures with Servadei-Valdinoci restriction. We will also explain that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. The results apply to the setting of $L^p$ spaces with $1<p<\infty$, and generalize earlier findings of the authors for the (quadratic) nonlocal Dirichlet forms and $L^2$ spaces. Here is the recent article: http://arxiv.org/abs/2006.01932