$L^p$-$L^q$ estimates for subelliptic pseudo-differential operators on compact Lie groups

Abstract

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 Riemannian 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 $0\le \delta \le \rho \le 1,$ $\delta <1.$ 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 anaysed.

22 Pages