Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property then ? Do we still get a integral representation of some sort. Please suggest a reference.
Thank you.
One has this type of representations for Herglotz functions under certain growth conditions. Herglotz means here that the function maps the upper half plane into the upper half plane.
This can for example be found in the spectral theory book by Teschl ( http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/index.html ). Section 3.4. seems to be the relevant one.
Of course these theorems need a growth condition. For that book for example $|F(z)| \leq \frac{M}{im(z)}$ for some constant $M > 0$. But I think one can extend this to some moderate growth at infinity. For example $|F(z)| = O(\sqrt{|z|})$ as $z\to\infty$ is ok.
Just take (x,y)->y. This is harmonic, but unbounded at infinity.
Yet any positive harmonic function h on the upper half plane can be obtained by integrating the Poisson kernel with respect to some finite measure on the boundary of the upper half plane (Riesz-Herglotz representation theorem). So one may say that there is an extension of h to the boundary in that case, and this extension is a finite positive measure.
