Continuous Solutions for Degenerate Complex Hessian Equation

Abstract

Let (X,ω) be an n-dimensional compact Kähler manifold and fix an integer m such that 1 ≤ m ≤ n. Let μ be a finite Borel measure on X satisfying the conditions \({\mathscr{H}}_{m}(\delta , A,\omega )\). We study degenerate complex Hessian equations of the form (ω + dd^cφ)^m ∧ ω^n−m = F(φ,.)dμ. Under some natural conditions on F, we prove that if \(0<\delta <\frac {m}{n-m}\), then this equation has a unique continuous solution.