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

On a compact connected group G, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup (μₜ)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≤ ∞: Λθᵖ, defined via the associated Markov semigroup, and Lθᵖ, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of Λθᵖ space. In the distributional sense, we further show local regularity in the class of Lθ∞ space. These results require some strong assumptions on $-L$. Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free Lᵖ (1<p<∞) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.