相关论文: Complex $m$-Hessian type equations in weighted ene…
In this paper, we concern with the existence of solutions of the weighted complex $m$-Hessian equation $-\chi(u)H_{m}(u)=\mu$ in the class $\mathcal{E}_{m,\chi}(f,\Omega)$ if there exists subsolution in this class, where the given boundary…
In this paper, we introduce the class $\mathcal{E}_{m,F}(\Omega)$ and solve complex $m$-Hessian equations in the class $\mathcal{E}_{m,F}(\Omega)$. Afterthat, we study subextension in the class $\mathcal{E}_{m,F}(\Omega)$ with the weighted…
In this paper, we study Hessian type equations for $m-\omega$ subharmonic functions. Using the recent results in \cite{KN23a}, \cite{KN23b}, we are able to show the existence of bounded solutions for such equations on bounded domains in…
In this paper, we first study the comparison principle for the operator $H_{\chi,m}$. This result is used to solve certain weighted complex $m-$ Hessian equations.
In this paper we study the class $\mathcal{E}_{m}(\Omega)$ of $m-$subharmonic functions introduced by Lu in \cite{L1}. We prove that the convergence in $m-$capacity implies the convergence of the associated Hessian measure for functions…
Let $\Omega$ be a $m$-hyperconvex domain of $\mathbb{C}^n$ and $\beta$ be the standard K\"{a}hler form in $\mathbb{C}^n$. We introduce finite energy classes of $m$-subharmonic functions of Cegrell type, $\mathcal{E}_m^p, p>0$ and…
Let $\Omega \Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $\mu$ a positive Borel measure with finite mass on $\Omega$. Then we solve the H\"older continuous subsolution problem for the complex…
Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix an integer $m$ such that $1\leq m\leq n$. We reformulate most relative pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the Hessian setting. As an…
Let $\Omega$ be a bounded strictly $m$-pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Hessian equations of the form $(\omega + dd^c \varphi)^m\wedge\beta^{n-m} = \mu$ in the generalized Cegrell classes…
By proving an estimate of the sublevel sets for $(\omega,m)$-subharmonic functions we obtain a Sobolev type inequality that is then used to characterize the degenerate complex Hessian equations for such functions with bounded…
In this paper, we first study the definition and the continuity of the complex Hessian operator associated to an $m$-positive closed current $T$, for some classes of unbounded $m$-subharmonic functions as well as when we consider a…
We develop potential theory for $m$-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to…
We first study subextensions of m-subharmonic functions in weighted energy classes with given boundary values. The results are used to approximate an m-subharmonic function in weighted energy classes with given boundary values by an…
Let $\Omega \subset \mathbb C^n$ be a bounded strictly $m$-pseudoconvex domain ($1\leq m\leq n$) and $\mu$ a positive Borel measure on $\Omega$. We study the Dirichlet problem for the complex Hessian equation $(dd^c u)^m \wedge \beta^{n -…
In this paper, we consider a class of Hessian type equations which include the $(n-1)$ Monge-Amp\`{e}re equation on Riemannian manifolds. The \emph{a priori} $C^2$ estimates and the existence of solutions are established.
The aim of this paper is to study Capacities and Hessians in a class of m-subharmonic functions
In this paper, we introduce finite energy classes of quaternionic $m$-plurisubharmonic functions of Cegrell type and define the quaternionic $m$-Hessian operator on some Cegrell's classes. We use the variational approach to solve the…
In this paper, we shall study the existence of weak solutions to Hessian type equations on compact Riemannian manifolds without boundary.
In this paper we establish a result on subextension of $m$-subharmonic functions in the class $\mathcal{F}_m(\Omega,f)$ without changing the hessian measures. As application, we approximate a $m$-subharmonic function with given boudary…
In this paper, we study Hessian type equations for $(\o,m)-\beta$-subharmonic functions on a ball in $\mathbb{C}^n$, where $\beta=dd^c\|z\|^2=\frac{i}{2}\sum\limits_{j=1}^n dz_j\w d\bar{z}_j$ is the flat metric on $\cn$. Using the recent…