Polarized Hardy--Stein identity

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