On square functions and Fourier multipliers for nonlocal operators

TL;DR

This paper provides a straightforward proof of the Hardy-Stein identity for nonlocal operators using Itô's formula, extending the result to non-symmetric Lévy-Fourier multipliers.

Contribution

It introduces a simple proof method for Hardy-Stein identity and extends it to non-symmetric Lévy-Fourier multipliers, broadening the class of operators covered.

Findings

Proof of Hardy-Stein identity using Itô's formula for jump processes

Extension of the identity to non-symmetric Lévy-Fourier multipliers

Simplification of previous proofs in the literature

Abstract

Using It\^o's formula for processes with jumps, we give a simple direct proof of the Hardy-Stein identity proved in \cite{BBL}. We extend the proof given in that paper to non-symmetric L\'evy-Fourier multipliers.