Sharp uniform bound for the quaternionic Monge-Ampère equation on hyperhermitian manifolds

We provide the sharp \(C^0\) $C^0$ estimate for the quaternionic Monge-Ampère equation on any hyperhermitian manifold. This improves previously known results concerning this estimate in two directions. Namely, it turns out that the estimate depends only on \(L^p\) $L^p$ norm of the right hand side for any \(p>2\) $p>2$ (as suggested by the local case studied in Sroka (Anal. PDE 13(6):1755-1776, 2020)). Moreover, the estimate still holds true for any hyperhermitian initial metric - regardless of it being HKT as in the original conjecture of Alesker and Verbitsky (Isr. J. Math. 176:109-138, 2010) - as speculated by the author in Sroka (Monge-Ampére equation in hypercomplex geometry, Jagiellonian University, Kraków, 2021). For completeness, we actually provide a sharp uniform estimate for many quaternionic PDEs, in particular those given by the operator dominating the quaternionic Monge-Ampére operator, by applying the recent method of Guo and Phong (On L \(^\infty \) ${}^{\infty }$ estimates for fully nonlinear partial differential equations on hermitian manifolds).