Eqivalent quasi-norms in Besov and Triebel-Lizorkin spaces on Lie groups of polymial growth

Let G be a Lie group of polynomial growth and X = {X-1 , ... , X-n} be a family of left-invariant vector fields on G satisfying the Hormander condition. In this paper, by utilizing smooth atomic decomposition, we obtain equivalent quasi-norm characterizations for Besov and Triebel-Lizorkin spaces F-p,q(s)(G) on G associated with the vector fields X, for full range of indices. This extends related classical results on the Euclidean spaces. (C) 2020 Elsevier Inc. All rights reserved