Weak solutions to the quaternionic Monge-Amp\`ere equation

TL;DR

This paper establishes existence results for weak solutions to the quaternionic Monge-Ampère equation with optimal integrability conditions, advancing understanding of quaternionic pluripotential theory.

Contribution

It proves the existence of solutions with boundary data in continuous functions and right hand side in L^p for p>2, the optimal bound, and analyzes integrability properties of quaternionic plurisubharmonic functions.

Findings

Existence of solutions for p>2 in the Dirichlet problem.

The local integrability exponent of quaternionic psh functions is two.

This exponent is less than that of the fundamental solution.

Abstract

We solve the Dirichlet problem for the quaternionic Monge-Amp\`ere 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.