Abstract
We study modules of solutions of the equation $DF=F$, where $F$ is a function in the plane with values in the quaternions and $D$ is the Dirac operator. The functions $F$ will belong to the Sobolev-type space of all functions in $L^{p}(\Omega,|x|^{-3}dx)$ jointly with their angular and radial derivatives, and where $\Omega$ is the complement of the unit disk in $\mathbb{R}^{2}$. The resulting spaces are right Banach modules over the quaternions. When $p=2$ we calculate the reproducing kernel of this space and explain its reproducing properties when $p\neq2$.
Citation
Emilio Marmolejo-Olea. Salvador Pérez-Esteva. Michael Shapiro. "Modules of solutions of the Helmholtz equation arising from eigenfunctions of the Dirac operator." Bull. Belg. Math. Soc. Simon Stevin 12 (2) 175 - 192, June 2005. https://doi.org/10.36045/bbms/1117805082
Information