Vol. 13, No. 6, 2020

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Weak solutions to the quaternionic Monge–Ampère equation

Marcin Sroka

Vol. 13 (2020), No. 6, 1755–1776
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.

Keywords
Monge–Ampère equation, pluripotential theory, quaternionic plurisubharmonic functions
Mathematical Subject Classification 2010
Primary: 32U05, 32U15, 35D30, 35J60