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.
Integral representation and asymptotic behavior of harmonic functions in half space
