arXiv subjects
Publications and source records attributed to Le Mau Hai.
In this paper, we study Hessian type equations for $m-ω$ 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 $\mathbb{C}^n$.
In this note, we establish several results concerning the continuity (or weak convergence) of the complex Monge-Ampère operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Ampère equation on a compact Hermitian manifold under the assumption of the existence of a smooth subsolution.
In this paper, we establish the weak*-convergence of a sequence of the complex Hessian measures $H_m(u_j)$ to the complex Hessian measure $H_m(u)$ in the class $\mathcal{E}(X,ω,m)$ under hypotheses that $u_j$ is convergent to $u$ in $(ω,m)$-capacity.
In this paper, we investigate the continuity of solutions to the Dirichlet problem for complex Hessian-type equations associated with $(ω, m)-β$-subharmonic functions on a ball in $\mathbb{C}^n$, where $ β=d d^c\|z\|^2=\frac{i}{2} \sum_{j=1}^n d z_j \wedge d \bar{z}_j $ is denoted the flat metric on $\mathbb{C}^n$.
Let $Ω$ be a bounded strictly $m$-pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Hessian equations of the form $(ω+ dd^c φ)^m\wedgeβ^{n-m} = μ$ in the generalized Cegrell classes $\mathcal{K}_m(Ω,ω,ϕ)$, where $ϕ\in \mathcal{E}_m(Ω)$ is a $m$-maximal function, $ω$ is a smooth real $(1,1)$-form defined in a neighborhood of $\barΩ$ and $μ$ is a positive Radon measure which is dominated by a Hessian measure of $m$-subharnomic functions in Cegrell class.
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.
In this paper, we introduce the notion of strong locally irreducible complex spaces $\widetilde{X}$. Based on this notion we prove the equality $\barν_φ(x)=$ mult$(\widetilde{X},x). ν_φ(x)$ for all $x\in \widetilde{X}$, where $\barν_φ(x)$ is the projective mass of a plurisubharmonic function $φ$ at $x$ and mult$(\widetilde{X},x)$ is the multiplicity of $\widetilde{X}$ at $x$ and $ν_φ(x)$ is Lelong number of $φ$ at $x$. Moreover, we show that the closure of the upper-level sets $\{z\in \widetilde{X}:ν_φ(z)\geq c\}$ of a plurisubharmonic function $φ$ on a strong locally irreducible complex space $\widetilde{X}$ is a subvariety of $\widetilde{X}$ for all $c\geq 0$.
In this paper we study the relation between the weighted energy class $\mathcal{E}_χ$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_χ$ and, as an application, we prove a slight version of existence of solutions of Monge-Ampère type equation in the class $\mathcal{E}_χ(H,Ø)$.