Some Sharp L^2 Inequalities for Dirac Type Operators

Abstract

We use the spectra of Dirac type operators on the sphere $S^n$ to produce sharp $L^2$ inequalities on the sphere. These operators include the Dirac operator on $S^n$, 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 ${mathbb R}^n$.

Keywords

• Dirac operator
• Clifford algebra
• conformal Laplacian
• Paenitz operator