arXiv subjects

Michał Gutowski

Publications and source records attributed to Michał Gutowski.

4 recordsLinked to original sources

arXiv2025

Littlewood--Paley estimates for pure-jump Dirichlet forms

We employ the recent generalization of the Hardy--Stein identity to extend the previous Littlewood--Paley estimates to general pure-jump Dirichlet forms. The results generalize those for symmetric pure-jump Lévy processes in Euclidean spaces. We also relax the assumptions for the Dirichlet form necessary for the estimates used in previous works. To overcome the difficulty that Itô's formula is not applicable, we employ the theory of Revuz correspondence and additive functionals. Meanwhile, we present a few counterexamples demonstrating that some inequalities do not hold in the generality considered in this paper. In particular, we correct errors that appear in previous works.

math.FA↗

arXiv2025

Beurling-Deny formula for Sobolev-Bregman forms

For an arbitrary regular Dirichlet form $\mathscr{E}$ and the associated symmetric Markovian semigroup $T_t$, we consider the corresponding Sobolev-Bregman form $\mathscr{E}_p(u) = -\tfrac{1}{p} \frac{d}{d t}\bigr\vert_{t = 0} \|T_t u\|_p^p$, where $p \in (1, \infty)$. We prove a variant of the Beurling-Deny formula for $\mathscr{E}_p$. As an application, we prove the corresponding Hardy-Stein identity. Our results extend previous works in this area, which either required that $\mathscr{E}$ is translation-invariant, or that $u$ is sufficiently regular.

math.AP↗

arXiv2025

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.

math.CA↗

arXiv2023

Hardy-Stein identity for pure-jump Dirichlet forms

We prove the $L^p$ variant of the Hardy-Stein identity for Sobolev-Bregman forms associated with pure-jump Dirichlet forms, under a rather mild assumptions. Along the way, we obtain a general result in terms of the $p$-form defined in a more abstract way.

math.FA↗