Some properties and integral transforms in higher spin Clifford analysis

TL;DR

This paper extends classical analysis properties to higher spin Clifford analysis, focusing on Rarita-Schwinger operators, integral transforms, and function space decompositions in Euclidean spaces.

Contribution

It introduces mean value properties, estimates, and Liouville's theorem for Rarita-Schwinger solutions, and develops a Hodge decomposition for higher spin Clifford analysis.

Findings

Established mean value property and Liouville's theorem for Rarita-Schwinger solutions

Analyzed boundedness of Teodorescu transform and derivatives

Derived a Hodge decomposition and generalized Bergman spaces for higher spins

Abstract

Rarita-Schwinger equation plays an important role in theoretical physics. Bure\v s et al. generalized it to arbitrary spin $k/2$ in 2002 in the context of Clifford algebras. In this article, we introduce the mean value property, Cauchy's estimates, and Liouville's theorem for null solutions to Rarita-Schwinger operator in Euclidean spaces. Further, we investigate boundednesses to the Teodorescu transform and its derivatives. This gives rise to a Hodge decomposition of an $L^2$ spaces in terms of the kernel space of the Rarita-Schwinger operator and it also generalizes Bergman spaces in higher spin cases. \end{abstract}