Polarized Hardy--Stein identity

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.

32 pages, small editorial changes