Polarized Hardy--Stein identity

TL;DR

The paper extends Hardy--Stein energy disintegration to vector-valued and polarized scalar settings in the realm of symmetric jump processes. It introduces a novel Bregman co-divergence $\mathcal{J}_p$ and the polarized Sobolev--Bregman form $\mathcal{E}_p(u,v)$, and develops an $L^p$-calculus framework to derive energy identities via semigroup evolution. The main results are the Hardy--Stein identity for vector-valued functions in $L^p(\mathbb{R}^d;\mathbb{R}^n)$ and the polarized Hardy--Stein identity for pairs of scalar functions in $L^p(\mathbb{R}^d)$, including explicit integral representations and convexity-based decompositions to handle sign issues. These identities provide variational and semigroup tools for studying nonlocal operators and have potential applications to Hardy spaces, Fourier multipliers, and Dirichlet-to-Neumann maps. The work blends probabilistic semigroups, convex analysis, and $L^p$-differentiation techniques to extend classical scalar results to vector and polarized frameworks.

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.

Theorems & Definitions (47)

• Definition 2.1
• Lemma 2.1
• Lemma 2.2: Stein inequality
• Corollary 2.3
• Proposition 2.4
• Corollary 2.5
• Lemma 2.6
• Theorem 3.1
• proof
• Remark 3.2