Sobolev algebras on nonunimodular Lie groups

Let G be a noncompact connected Lie group and ho be the right Haar measure of G. Let {X_1,dots,X_q} be a family of left invariant vector fields which satisfy H"ormander's condition, and let Delta=-sum_{i=1}^qX_i^2 be the corresponding subLaplacian. For 1= 0 we define the Sobolev space L^p_{alpha}(G)={fin L^p( ho): Delta^{alpha/2}fin L^p( ho)}, endowed with the norm |f|_{alpha,p}=|f|_{p}+|Delta^{alpha/2}f|_p. In this paper we show that for all alpha>=0 and 1