Factorisations of the Helmholtz Operator, Radó’s Theorem, and Clifford Analysis

Radó’s theorem for holomorphic functions asserts that if a continuous function is holomorphic on the complement of its zero locus, then it is holomorphic everywhere. We prove in this paper an equivalent theorem for functions lying in the kernel of a first order differential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} such that the Helmholtz operator ∇2+λ can be factorized as the composition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\mathcal{D}}\mathcal{D}}$$\end{document} . We also analyse the factorisations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\mathcal{D}}\mathcal{D}}$$\end{document} of the Laplace and Helmholtz operators associated to the Clifford analysis and the representations of holomorphic function of several complex variables.