## Some sharp $L^2$ inequalities for Dirac type operators

### Summary

Summary: 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 $R^{n}$.

### Mathematics Subject Classification

15A66, 26D10, 34L40

### Keywords/Phrases

Dirac operator, Clifford algebra, conformal Laplacian, paenitz operator