Sharp Garding Inequality and Central Oscillating Multipliers with Global Functional Calculus on Compact Lie Groups

Abstract

We show an approximation of Hardy function on compact Lie groups by the methods of Bochner-Riesz mean .We establish the Sharp Garding inequality on compact Lie groups. The positivity condition is expressed in the non-commutative phase space in terms of the full matrix symbol, which is defined using the representation of the group. Feffermanand Steinand Lp Waingerprovedoptimal boundedness certainoscillating for multipliers on the Euclidean space. We develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provided asymbolic description of .complex powers of such operators