Some Operators Associated to Rarita-Schwinger Type Operators

TL;DR

This paper investigates operators related to Rarita-Schwinger operators, focusing on their fundamental solutions, representation theory, and extensions from Euclidean space to the sphere via the Cayley transformation.

Contribution

It introduces and analyzes remaining operators derived from the Dirac and Rarita-Schwinger operators, including their fundamental solutions and geometric extensions.

Findings

Remaining operators have harmonic polynomial solutions

Representation theory of these operators is developed in Euclidean space

Operators are extendable to the sphere via Cayley transformation

Abstract

In this paper we study some operators associated to the Rarita-Schwinger operators. They arise from the difference between the Dirac operator and the Rarita-Schwinger operators. These operators are called remaining operators. They are based on the Dirac operator and projection operators $I-P_k.$ The fundamental solutions of these operators are harmonic polynomials, homogeneous of degree $k$. First we study the remaining operators and their representation theory in Euclidean space. Second, we can extend the remaining operators in Euclidean space to the sphere under the Cayley transformation.