Abstract
In this paper we study a mean-value property for solutions of the Laplace-Beltrami equation
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.
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Eriksson, SL., Orelma, H. (2011). A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory. In: Sabadini, I., Sommen, F. (eds) Hypercomplex Analysis and Applications. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0346-0246-4_4
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DOI: https://doi.org/10.1007/978-3-0346-0246-4_4
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Publisher Name: Springer, Basel
Print ISBN: 978-3-0346-0245-7
Online ISBN: 978-3-0346-0246-4
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