Fundamental solution of Poisson equation in the Hyperbolic Plane

Question

If we consider the Poisson's equation $$ -\Delta u=f(x), \ \ \mbox{in} \ \ \mathbb{R}^n, $$ we can construct the fundamental solution $$ u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)f(y)dy, $$ where $\Gamma$ is the fundamental solution of the Laplace's equation. Now consider the problem $$ -\Delta_{\mathbb{H}^2}u=f(x), \ \ \mbox{in} \ \ \mathbb{H}^2, $$ where $\mathbb{H}^2$ is the Hyperbolic Plane and $\Delta_{\mathbb{H}^2}$ is the Laplace-Beltrami operator. Can we obtain a same type of integral representation to the solution of this problem?

Answer 1

Yes, have a look at Stoll's lectures Harmonic function theory on real hyperbolic space

where he constructs the Green function, or at Jaming's article "Harmonic functions on the real hyperbolic ball I" Colloquium Mathematicum vol 80 No 1 (1999) 63-82