Riesz Transforms, Function Spaces, and Their Applications on Infinite Dimensional Compact Groups

On a compact connected group G , consider the infinitesimal generator $$-L$$ - L of a central symmetric Gaussian convolution semigroup (μ 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≤ p≤ ∞ 1 ≤ p ≤ ∞ : Λ θᵖ Λ θ p , defined via the associated Markov semigroup, and Lθᵖ 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 θᵖ Λ θ p space. In the distributional sense, we further show local regularity in the class of Lθ∞ 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 Lᵖ L p ( 1<p<∞ 1 < p < ∞ ) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.