On a compact connected group $G$, consider the infinitesimal generator $-L$
of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. Noi
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 $1\le p\le \infty$: $\Lambda_{\theta}^p$, defined via
the associated Markov semigroup, and $\mathrm L_{\theta}^p$, defined via the
intrinsic distance. In the strong sense, we prove a priori Sobolev regularity
and Lipschitz regularity in the class of $\Lambda_{\theta}^p$ space. In the
distributional sense, we further show local regularity in the class of $\mathrm
L_{\theta}^{\infty}$ space. These results require some strong assumptions on
$-L$. Our main techniques build on the differentiability of the associated
semigroup, explicit dimension-free $L^p$ ($1boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.

Questo articolo esplora i giri e le loro implicazioni.

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