LP-BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS

Given a compact Lie group G, in this paper we establish L-p-bounds for pseudo-differential operators in L-p(G). The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space G x (G) over cap, where (G) over cap is the unitary dual of G. We obtain two different types of L-p bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using I-rho,delta(m) (G) classes which are a suitable extension of the well-known (rho,delta) ones on the Euclidean space. The results herein extend classical L-p bounds established by C. Fefferman on R-n. While Fefferman's results have immediate consequences on general manifolds for rho > max{delta, 1 -delta}, our results do not require the condition rho >1-delta. Moreover, one of our results also does not require p > delta. Examples are given for the case of SU(2) congruent to S-3 and vector fields/sub-Laplacian operators when operators in the classes I-0,0(m) and I-1/2,0(m) naturally appear, and where conditions p > delta and p > 1 - delta fail, respectively.