Weak solutions to complex Hessian equations in the class $\mathcal{N}^a_m(Ω,ω,φ)$ in a ball of $\mathbb{C}^n$

Abstract: In this paper, we study weak solutions to complex Hessian equations of the form $(\omega + dd^c \varphi)^m\wedge\beta^{n-m}= F(\varphi,.)d\mu$ in the class $\mathcal{N}^a_m(\Omega,\omega,\phi)$ on a ball in $\mathbb{C}^n$, where $\omega$ is a real smooth $(1,1)$-form, $\mu$ is a positive measure which puts no mass on $m$-polar subsets, $\Omega$ is a ball in $\mathbb{C}^n$ and $F(t,z): \mathbb{R}\times\Omega\longrightarrow [0,+\infty)$ is a upper semi-continuous function on $\mathbb{R}\times\Omega$, $0\leq F(t,z)\leq G(z)$ for all $(t,z)\in\mathbb{R}\times \Omega$, $G(z)\in L^1_{loc}(\Omega, d\mu)$ and $F(t,z)$ is continuous in the first variable. Our result extends a result of Salouf in \cite{S23} in which he solved Monge-Amp`ere type equations $(\omega + dd^c \varphi)^n= F(\varphi,.)d\mu$ with $F(t,z)$ is a continuous non-decreasing function in the first variable function and $\varphi\in\mathcal{E}(\Omega)$.