Abstract
We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$
d
s
2
=
d
x
1
2
+
⋯
+
d
x
n
2
x
n
2
α
n
-
2
in the upper half space $$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$
R
+
n
=
{
x
1
,
…
,
x
n
∈
R
n
:
x
n
>
0
}
. They are called $$\alpha $$
α
-hyperbolic harmonic. An important result is that a function f is $$\alpha $$
α
-hyperbolic harmonic íf and only if the function $$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$
g
x
=
x
n
-
2
-
n
+
α
2
f
x
is the eigenfunction of the hyperbolic Laplace operator $$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$
△
h
=
x
n
2
▵
-
n
-
2
x
n
∂
∂
x
n
corresponding to the eigenvalue $$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$
1
4
α
+
1
2
-
n
-
1
2
=
0
. This means that in case $$\alpha =n-2$$
α
=
n
-
2
, the $$n-2$$
n
-
2
-hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of $$\alpha $$
α
-hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.