Hardy-Stein identity for pure-jump Dirichlet forms

TL;DR

This paper establishes an $L^p$ Hardy-Stein identity for Sobolev-Bregman forms linked to pure-jump Dirichlet forms, broadening the theoretical understanding of these mathematical structures.

Contribution

It introduces an $L^p$ Hardy-Stein identity for pure-jump Dirichlet forms and provides a general abstract result for $p$-forms.

Findings

Proves the $L^p$ Hardy-Stein identity under mild assumptions.

Derives a general result for $p$-forms in an abstract setting.

Enhances theoretical framework for Sobolev-Bregman forms.

Abstract

We prove the $L^p$ variant of the Hardy-Stein identity for Sobolev-Bregman forms associated with pure-jump Dirichlet forms, under a rather mild assumptions. Along the way, we obtain a general result in terms of the $p$-form defined in a more abstract way.