Good morning everybody.
I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ and an horizontal distribution spanned by $X=\partial_x-\frac y2\partial_z$ and $Y=\partial_y+\frac x2 \partial_z$.
I don't look for representation formulas, rather to the standard existence results like harnack maximum principles, etc..
Many thanks in advance.
Regards, Guido
I would look at:
A. Bonfiglioli, E. Lanconelli, F. Uguzzoni. Stratified Lie Groups and Potential Theory for their Sub-Laplacians. Springer, 2007. DOI 10.1007/978-3-540-71897-0
This book is always my first reference for this sort of general theory. In Section 7.1 you can find a theorem giving sufficient conditions on an open set $U$ for the Dirichlet problem $Lu=0$ on $U$, $u = \varphi$ on $\partial U$, to be solvable (here I presume you want $L = X^2 + Y^2$ to be the usual sub-Laplacian); e.g. Proposition 7.1.5. The book has a lot more material on the potential theory of $L$ as well. And in Chapter 5 you can find a development of Harnack inequalities, maximum principles, mean value theorems, etc, for functions (sub-)harmonic with respect to $L$.
