Polarized Hardy--Stein identity

We prove the Hardy--Stein identity for vector functions in $L^p(\mathbb R^d;\mathbb R^n)$ with $1<p<\infty$ 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.