Some Sharp L 2 Inequalities for Dirac Type Operators ⋆

Sobolev and Hardy type inequalities play an important role in many areas of mathematics and mathematical physics. They have become standard tools in existence and regularity theories for solutions to partial differential equations, in calculus of variations, in geometric measure theory and in stability of matter. In analysis a number of inequalities like the Hardy–Littlewood– Sobolev inequality in R are obtained by first obtaining these inequalities on the compact manifold S and then using stereographic projections to R to obtain the analogous sharp inequality in that setting. See for instance [10]. This technique is also used in mathematical physics to obtain zero modes of Dirac equations in R3 (see [9]). In fact the stereographic projection corresponds to the Cayley transformation from S minus the north pole to Euclidean space. Here we shall use this Cayley transformation to obtain some sharp L2 inequalities on the sphere for a family of Dirac type operators. The main trick here is to employ a lowest eigenvalue for these operators and then use intertwining operators for the Dirac type operators to obtain analogous sharp inequalities in R. Our eventual hope is to extend the results presented here to obtain suitable L inequalities for the Dirac type operators appearing here, particularly the Dirac operator on R.