Gaussian bounds for derivatives of central Gaussian semigroups on compact groups (Q2781361)

In a recent series of papers the authors investigated continuous, central, symmetric Gaussian convolution semigroups \((\mu_t)_{t\geq 0}\) with \(\mu_0= \delta_e\) on second countable, comapct connected groups \(G\). The main objects in the paper under review are such semigroups, that admit continuous densities \(f_t\) for \(t> 0\) w.r.t. the Haar measure with \(\lim_{t\to 0} t\cdot\ln f_t(e)= 0\). This latter condition is closely related to hypoellipticity of the generator. It is shown that the densities \(f_t\) admit space derivatives of all orders in certain directions. Moreover, Gaussian bounds for the time and space derivatives are derived under certain further restrictions. The proofs rely on the fact that \(G\) is a projective limit of compact Lie groups.