What does this research mean for the field?

Solutions to the Poisson equation LU=F on a compact connected group exhibit strong a priori Sobolev and Lipschitz regularity, as well as local distributional regularity, which can be established using newly defined Lipschitz spaces and dimension-free Lp boundedness of Riesz transforms. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Investigate the regularity of solutions to the Poisson equation on compact connected groups using Riesz transforms and Lipschitz spaces.

March 29, 2026Potential Analysis0 citationsOpen Access

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

ABAlexander Bendikov LCLi Chen LSL. Saloff-Coste

Key Points

Investigate the regularity of solutions to the Poisson equation on compact connected groups using Riesz transforms and Lipschitz spaces.
Analyzed the infinitesimal generator of a Gaussian convolution semigroup.
Introduced Lipschitz spaces defined via Markov and intrinsic distances.
Established Sobolev and Lipschitz regularity in the strong sense within the class of defined spaces.
Examined local regularity in distributional sense within the specified Lipschitz spaces.
Proved a priori Sobolev and Lipschitz regularity in the class of defined Lipschitz spaces.
Demonstrated local regularity in a distributional sense for specific classes of Lipschitz spaces.
Results are contingent on strong assumptions related to the infinitesimal generator.

Abstract

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

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