Deriving harmonic functions in higher dimensional spaces
A well-known result of R. Fueter says how to construct a quaternionic monogenic function starting from a holomorphic function in an open set of the upper half complex plane, then making an appropriate change of variable and applying, finally, the Laplace operator. A later result by Sce involves certain powers of the Laplace operator, and it was extended by Sommen. The present paper deals with harmonic functions using Clifford analysis. The statements of the new results are contained in Section 2, and they are given in terms of identities in relation to multiple powers of the Laplacian on induced functions from lower dimensional harmonic functions, the identities show explicitly the harmonicity of the induced functions. The proofs of the theorems one can find in Section 3, and Section 4 presents applications, in particular, it is shown that almost all commonly used kernels, as well as generalizations of Fueter's theorem, are deducible from the results of Section 2.
scientific article; zbMATH DE number 2096446