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 2, which turns out to be less than an integrability exponent of the fundamental solution.