Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups

Abstract

AbstractOn a compact connected group G, consider the infinitesimal generator –L of a central symmetric Gaussian convolution semigroup (μt)t>0. Using appropriate notions of distribution and smooth function spaces, we prove that L is hypoelliptic if and only if (μt)t>0 is absolutely continuous with respect to Haar measure and admits a continuous density x ⟼ μt (x), t > 0, such that limt!0t log μt(e) = 0. In particular, this condition holds if and only if any Borel measure u which is solution of Lu = 0 in an open set Ω can be represented by a continuous function in Ω. Examples are discussed.