On Hyperbolic Function Theory

Abstract

The hyperbolic version of the standard Clifford analysis will be considered. In this modification the power function x m becomes a solution. In more details, the Dirac operator \(Df = \sum^n_{i=0} e_i \frac{\partial f} {\partial x_i}\) with e 0 = 1, defined with respect to the Clifford algebra Cl n , is replaced by the operator \(M_kf(x) = Df (x) + \frac{k}{x_n} Q^{\prime}f(x)\), where ′ denotes the main involution in Cl n and Qf is given by the unique decomposition f(x) = Pf(x) + Qf(x)e n with Pf(x),Qf(x) ∈Cl n-1. The operator M k (k ∈R) will mainly be considered for k = 0, k = n − 1 and k = 1 − n. In case k = 0 the equation Mkf = 0 yields the well-known monogenic functions, in case k = n − 1 one obtains the so-called hypermonogenic functions introduced in [5]. Besides M k we also study the operator \(\overline{M_k}M_k = M_k\overline{M_k}\), a natural generalization of the Laplace operator Δ. Solutions of the equation \(\overline{M_{n-1}}M_{n-1}f =0\) are called hyperbolic harmonic functions. The main goal of this article is to give integral representations for hypermonogenic and hyperbolic harmonic functions in the upper half space \({\mathbb{R}}^{n+1}_+\).