Riesz Transforms, Function Spaces, and Their Applications on Infinite Dimensional Compact Groups
Year: 2026
Type: article
Abstract: On a compact connected group G , consider the infinitesimal generator $$-L$$ - L of a central symmetric Gaussian convolution semigroup $$(\mu _t)_{t>0}$$ ( μ t ) t > 0 . We establish several regularity results of the solution to the Poisson equation $$LU=F$$ L U = F , both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ : $$\Lambda _{\theta }^p$$ Λ θ p , defined via the associated Markov semigroup, and $$\mathrm L_{\theta }^p$$ 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 $$\Lambda _{\theta }^p$$ Λ θ p space. In the distributional sense, we further show local regularity in the class of $$\mathrm L_{\theta }^{\infty }$$ L θ ∞ space. These results require some strong assumptions on $$-L$$ - L . Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free $$L^p$$ L p ( $$1 1 < p < ∞ ) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.
Source: Potential Analysis
Institutions: University of Wrocław, Aarhus University, Cornell University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7141549025