Uniform estimates for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Amp\`ere equation

TL;DR

This paper establishes sharp uniform estimates for supersolutions of complex degenerate elliptic equations, extending techniques from real degenerate equations and utilizing pluripotential theory and $L^p$-viscosity methods.

Contribution

It provides new uniform bounds for complex degenerate elliptic equations, combining pluripotential theory with viscosity solutions, advancing the understanding of such equations.

Findings

Sharp uniform estimates for supersolutions established

Extension of real degenerate linear equation techniques to complex setting

Integration of pluripotential theory and $L^p$-viscosity methods

Abstract

We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings rely on ideas of Kuo and Trudinger who dealt with degenerate linear equations in the real setting. We also exploit the pluripotential theory for the complex Monge-Amp\`ere operator as well as suitably tailored theory of $L^p$-viscosity subsolutions.