The Pluricomplex Green Function of the Monge–Ampère Equation for $$(n-1)$$ -Plurisubharmonic Functions and Form Type k-Hessian Equations

Abstract

In this paper, we introduce the pluricomplex Green function of the Monge–Ampère equation for \((n-1)\)-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge–Ampère and Hessian equations on a punctured domain. We prove the pluricomplex Green function is \(C^{1,\alpha }\) by constructing approximating solutions and establishing uniform a priori estimates for the gradient and the complex Hessian. The singular solutions turn out to be smooth for the k-Hessian equations for \((n-1)\)-k-admissible functions.