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 (ₜ) ₓ>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: _ᵖ, defined via the associated Markov semigroup, and L_ᵖ, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of _ᵖ space. In the distributional sense, we further show local regularity in the class of L_^ 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.