The Complex Monge-Ampère Type Equation for $\mathcal {F}$-plurisubharmonic Functions

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,α by constructing approximate 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.