Bounded solution to Hessian type equations for $(ø,m)-β$-subharmonic functions on a ball in $\mathbb C^n$
Hoang Thieu Anh, Le Mau Hai, Nguyen Quang Dieu, Nguyen Van Phu
TL;DR
This paper studies Hessian-type equations on a ball in C^n for (ω,m)-β-subharmonic functions, focusing on the Dirichlet problem with right-hand side F(u,z)μ. It develops a Schauder-free existence theory using convergence in (ω,m)-capacity and a subsolution theorem, proving existence and, under monotonicity of F in t outside polar sets, uniqueness, plus a capacity-based stability result. The results extend KN23b from Monge-Ampère-type equations to Hessian-type operators under broad, non-monotone forcing, and advance the pluripotential framework for (ω,m)-β-subharmonic functions by clarifying the role of capacity in Hessian measures and Dirichlet problems.
Abstract
In this paper, we study Hessian type equations for $(ø,m)-β$-subharmonic functions on a ball in $\mathbb{C}^n$, where $β=dd^c\|z\|^2=\frac{i}{2}\sum\limits_{j=1}^n dz_j\w d\bar{z}_j$ is the flat metric on $\cn$. Using the recent results in \cite{KN23b}, we are able to show the existence of bounded solutions for Hessian type equations.