Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups

In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hormander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the (?, d)-Hormander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case ? = d. Interpolation properties between Besov spaces and Triebel-Lizorkin spaces are also investigated.