Regularity of solutions to the quaternionic Monge-Ampère equation

The regularity of solutions to the Dirichlet problem for the quaternionic Monge-Ampère equation is discussed. We prove that the solution to the Dirichlet problem is Hölder continuous under some conditions on the boundary values and the quaternionic Monge-Ampère density from Lp(Ω) for p>2. As a step towards the proof, we provide a refined version of stability for the weak solutions to this equation