Poincaré Lemma on Quaternion-like Heisenberg Groups

For smooth functions a1, a2, a3, a4 on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system X1f = a1, X2f = a2, X3f = a3, and X4f = a4. In addition, a formula for the solution function f is deduced, assuming solvability of the system.

On green functions for Dirichelet sub-Laplacians on a Quaternion Heisenberg group

In the present paper, Green functions of Dirichlet boundary value problems are constructed for sub-Laplacians on certain unbounded domains of the quaternion Heisenberg group. Also, the explicit solutions of the Dirichlet problem are presented for the sub-Laplacian with non-zero boundary datum in wedge and strip domains. We also present Hardy and Rellich inequalities for the sub-Laplacians in terms of their fundamental solutions.