A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory

Abstract

In this paper we study a mean-value property for solutions of the Laplace-Beltrami equation

$$x^{2}_{n} \Delta h - (n-1) x_n \frac{\partial h}{\partial x_n} = 0$$

(Equation 1 )

with respect to the volume and the surface integral on the Poincaré upper-half space $\mathbb{R}^{n+1}_{+} = \{(x_0,...,x_n)\ \in \mathbb{R}^{n+1} : x_n > 0\}$ with the Riemannian metric $g = \frac{dx^{2}_{0} + dx^{2}_{1} +...+ dx^{2}_{n}}{x^{2}_{n}}$. We also compute the Cauchy type kernels in terms of the hyperbolic metric.

Mathematics Subject Classification (2010). Primary 30A05; Secondary 30F45.