2017/10/20 by Marco M. Peloso, Peloso, Marco M., Maria Vallarino +1
Mathematics · #22E30 #43A15 #46E35 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:22E30 #msc:43A15 #msc:46E35
paper · pdf · doi:10.48550/arxiv.1710.07566
33 pages. New version with a correction in Theorem 3.1 and other minor corrections. To appear in Calculus of Variations and Partial Differential Equations
arxiv created 2018/09/12 · arxiv updated 2018/09/13
Let G be a noncompact connected Lie group and ρ be the right Haar measure of G. Let X1,...,Xq be a family of left invariant vector fields which satisfy Hörmander's condition, and let Δ=-∑i=1qXi2 be the corresponding subLaplacian. For 1≤ p<∞ and α≥ 0 we define the Sobolev space Lpα(G)=f in Lp(ρ): Δα/2f∈ Lp(ρ) , endowed with the norm ‖f‖α,p=‖f‖p+‖Δα/2f‖p, where we denote by ‖f‖p the norm of f in Lp(ρ). In this paper we show that for all α≥ 0 and p∈ (1,∞), the space L∞∩ Lpα(G) is an algebra under pointwise product. Such result was proved by T. Coulhon, E. Russ and V. Tardivel-Nachef in the case when G is unimodular. We shall prove 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 RcJ=XJ(cI+Δ)-m/2, where c>0, XJ=Xj1...Xjm and j_ℓ ∈\1,…,q\ for ℓ=1,...,m. We show that if c is sufficiently large, the Riesz transform RcJ is bounded on Lp(ρ) for every p∈ (1,∞), and prove also appropriate endpoint results involving Hardy and BMO spaces.