Fundamental Solutions for the Laplace–Beltrami Operator Defined by the Conformal Hyperbolic Metric and Jacobi Polynomials

Abstract

AbstractIn this paper we study fundamental solutions for the Laplace–Beltrami operator $$\begin{aligned} \Delta _\alpha f=x_n^{\frac{\alpha }{n-2}}\Big (\Delta f-\frac{\alpha }{x_n}\frac{\partial f}{\partial x_n}\Big ), \end{aligned}$$ Δ α f = x n α n - 2 ( Δ f - α x n ∂ f ∂ x n ) , defined on smooth enough functions in $$\mathbb {R}_+^{n}=\{ (x_1,\ldots ,x_n)\in \mathbb {R}^{n}: x_n>0\}$$ R + n = { ( x 1 , … , x n ) ∈ R n : x n > 0 } . We represent explicit formulas for the fundamental solutions. Moreover, we establish fundamental solutions using Jacobi polynomials when $$n=3,5,7,\ldots .$$ n = 3 , 5 , 7 , … .