Polarized Hardy--Stein identity

TL;DR

This paper extends the Hardy--Stein identity to vector functions and pairs of real functions in certain L^p spaces, introducing a new concept of Bregman co-divergence to facilitate the proofs.

Contribution

It introduces a novel notion of Bregman co-divergence and applies it to establish the Hardy--Stein identity in vector and scalar function settings.

Findings

Proves Hardy--Stein identity for vector functions in L^p spaces.

Establishes Hardy--Stein identity for pairs of real functions in L^p spaces.

Introduces Bregman co-divergence as a key tool in the proof.

Abstract

We prove the Hardy--Stein identity for vector functions in $L^p(\mathbb R^d;\mathbb R^n)$ with $1<p<\infty$ and for the canonical paring of two real functions in $L^p(\mathbb R^d)$ with $2\le p<\infty$. To this end we propose a notion of Bregman co-divergence and study the corresponding integral forms.