Some Sharp L² Inequalities for Dirac Type Operators

Abstract

We use the spectra of Dirac type operators on the sphere Sn to produce sharp L² inequalities on the sphere. These operators include the Dirac operator on Sn, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in Rn.