Some operators associated with Rarita–Schwinger type operators

Abstract

In this article, we study some operators associated with the Rarita–Schwinger operators. They arise from the difference between the Dirac operator and the Rarita–Schwinger operators. These operators are called remaining Rarita–Schwinger operators. They are based on the Dirac operator and projection operators I − P _k , where P _k is a projection operator used to define the k-th Rarita–Schwinger type operator. The fundamental solutions of the remaining Rarita–Schwinger operators are harmonic polynomials, homogeneous of degree k. First we study the remaining Rarita–Schwinger operators and their representation theory in Euclidean space. Second, we can extend the remaining Rarita–Schwinger operators in Euclidean space to the sphere under the Cayley transformation.

Keywords:

• Clifford algebra
• Rarita–Schwinger operators
• remaining Rarita–Schwinger operators
• Cayley transformation
• Almansi–Fischer decomposition

AMS Subject Classifications::

• Primary: 30G35
• 53C27