Some Sharp L^2 Inequalities for Dirac Type Operators

TL;DR

This paper derives sharp L^2 inequalities for Dirac type operators on spheres and Euclidean space, utilizing spectral analysis and conformal transformations to extend results across geometries.

Contribution

It introduces new sharp L^2 inequalities for Dirac, conformal Laplacian, and Paenitz operators on spheres and Euclidean space, using spectral methods and stereographic projection.

Findings

Established sharp L^2 inequalities for Dirac operators on spheres.

Extended inequalities to Euclidean space via stereographic projection.

Connected spectral properties of Dirac operators with geometric inequalities.

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 $\mathbb{R}^{n}$.