L^p-L^q ESTIMATES FOR SUBELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS

We establish the L^p-L^q-boundedness of subelliptic pseudo-differential operators on a compact Lie group G. Effectively, we deal with the L^p-L^q-bounds for operators in the sub-Riemmanian setting because the subelliptic classes are associated to a Hörmander sub-Laplacian. The Rie-mannian case associated with the Laplacian is also included as a special case. Then, applications to the L^p-L^q-boundedness of pseudo-differential operators in the Hörmander classes on G are given in the complete range 0 ≤ δ ≤ ρ ≤ 1, δ<1. This also gives the L^p-L^q-bounds in the Riemannian setting, because the later classes are associated with the Laplacian on G. In both cases, in the Riemannian and the sub-Riemannian settings, necessary and sufficient conditions for the L^p-L^q-boundedness of operators are also analysed.