Beurling-Deny formula for Sobolev-Bregman forms

2023/12/17 by Michał Gutowski, Gutowski, Michał, Mateusz Kwaśnicki +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Analytic Number Theory Research #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2312.10824

openalex publication_date 2023/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an arbitrary regular Dirichlet form \mathscrE and the associated symmetric Markovian semigroup T_t, we consider the corresponding Sobolev-Bregman form \mathscrE_p(u) = -\tfrac1p (d)/(d t)\vert_{t = 0} ‖T_t u‖_p^p, where p ∈ (1, ∞). We prove a variant of the Beurling-Deny formula for \mathscrE_p. As an application, we prove the corresponding Hardy-Stein identity. Our results extend previous works in this area, which either required that \mathscrE is translation-invariant, or that u is sufficiently regular.

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