Weak solutions to the quaternionic Monge-Ampère equation

Abstract

We solve the Dirichlet problem for the quaternionic Monge-Ampère equation with a continuous boundary data and the right hand side in $L^p$ for $p>2$. This is the optimal bound on $p$. We prove also that the local integrability exponent of quaternionic plurisubharmonic functions is two which turns out to be less than an integrability exponent of the fundamental solution.

minor corrections, final version, to appear in Analysis & PDE