Polarized Hardy–Stein identity

Abstract

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