## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 12, Number 2 (2005), 175-192.

### Modules of solutions of the Helmholtz equation arising from eigenfunctions of the Dirac operator

Emilio Marmolejo-Olea, Salvador Pérez-Esteva, and Michael Shapiro

#### 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 $p\neq2$.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 2 (2005), 175-192.

**Dates**

First available in Project Euclid: 3 June 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1117805082

**Digital Object Identifier**

doi:10.36045/bbms/1117805082

**Mathematical Reviews number (MathSciNet)**

MR2179962

**Zentralblatt MATH identifier**

1119.35072

**Subjects**

Primary: 46E15: Banach spaces of continuous, differentiable or analytic functions 39G35

**Keywords**

Helmholtz equation reproducing kernel Dirac operator

#### Citation

Marmolejo-Olea, Emilio; Pérez-Esteva, Salvador; Shapiro, Michael. Modules of solutions of the Helmholtz equation arising from eigenfunctions of the Dirac operator. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 2, 175--192. doi:10.36045/bbms/1117805082. https://projecteuclid.org/euclid.bbms/1117805082