Deriving Harmonic Functions in Higher Dimensional Spaces

Abstract

For a harmonic function, by replacing its variables with norms of vectors in some multi-dimensional spaces, we may induce a new function in a higher dimensional space. We show that, after applying to it a certain power of the Laplacian, we obtain a new harmonic function in the higher dimensional space. We show that Poisson and Cauchy kernels and Newton potentials, and even heat kernels are all deducible using this method based on their forms in the lowest dimensional spaces. Fueter's theorem and its generalizations are deducible as well from our results. The latter has been used to singular integral and Fourier multiplier theory on the unit spheres and their Lipschitz perturbations of higher dimensional Euclidean spaces.