Riesz transform, function spaces and their applications on infinite dimensional compact groups

2025/04/22 by Bendikov, Alexander, Chen, Li, Saloff-Coste, Laurent
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.15718

Abstract

On a compact connected group G, consider the infinitesimal generator -L of a central symmetric Gaussian convolution semigroup (μ_t)_t>0. We establish several regularity results of the solution to the Poisson equation LU=F, both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for 1≤ p≤ ∞: Λ_θ^p, defined via the associated Markov semigroup, and \mathrm L_θ^p, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of Λ_θ^p space. In the distributional sense, we further show local regularity in the class of \mathrm L_θ^∞ space. These results require some strong assumptions on -L. Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free L^p (1

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