Equivalent quasi-norms in Besov and Triebel-Lizorkin spaces on Lie groups of polynomial growth

Abstract

Let G be a Lie group of polynomial growth and X={X1,⋯,Xn} be a family of left-invariant vector fields on G satisfying the Hörmander condition. In this paper, by utilizing smooth atomic decomposition, we obtain equivalent quasi-norm characterizations for Besov and Triebel-Lizorkin spaces Fp,qs(G) on G associated with the vector fields X, for full range of indices. This extends related classical results on the Euclidean spaces.