Sobolev algebras on nonunimodular Lie groups

Let G be a noncompact connected Lie group and. be the right Haar measure of G. Let X = {X1,..., Xq} be a family of left invariant vector fields which satisfy Hormander's condition, and let Delta = -Sigma(q)(i=1) X-i(2) be the corresponding subLaplacian. For 1 = p < 8 and a = 0 we define the Sobolev space L-alpha(p) (G) = {f is an element of L-p (.) : a/2 f. L-p (.)}, endowed with the norm vertical bar vertical bar f vertical bar vertical bar a, p = vertical bar vertical bar f vertical bar vertical bar(p) + vertical bar vertical bar Delta(a/2) f vertical bar vertical bar(p), where we denote by f p the norm of f in L-p(.). In this paper we show that for all a = 0 and p. (1,8), the space L 8 n L-p a (G) is an algebra under pointwise product, that is, there exists a positive constant Ca, p such that for all f, g. L 8n L-p a (G), f g. L 8n L-p a (G) and vertical bar vertical bar fg vertical bar vertical bar a,p <= Ca,p (vertical bar vertical bar f vertical bar vertical bar a,p vertical bar vertical bar g vertical bar vertical bar(infinity) + vertical bar vertical bar f vertical bar vertical bar(infinity) vertical bar vertical bar g vertical bar vertical bar a, p). Such estimates were proved by Coulhon, Russ and Tardivel-Nachef in the case when G is unimodular. We shalL(p)rove it on Lie groups, thus extending their result to the nonunimodular case. In order to prove our main result, we need to study the boundedness of local Riesz transforms Rc J = XJ (cI + ) -m/2, where c > 0, XJ = X j1... X jm and j . {1,..., q} for = 1,..., m. We show that if c is sufficiently large, the Riesz transform Rc J is bounded on L-p(rho) for every p is an element of (1, infinity), and prove also appropriate endpoint results involving Hardy and BMO spaces.