arXiv subjects
Publications and source records attributed to Souhaibou Sambou.
In this paper, we shall use the $L^2$ existence theorems for $\partial\bar{\partial}$ provide by Guy Laville to etablish for existence theorem for the $\partial\bar{\partial}$-Neumann operator on any bounded pseudoconvex starred domaine $Ω$ in $\mathbb{C}^n$ based on the work of D. Spencer on the $\bar{\partial}$-Neumann. As a result, we obtain a new method for resolving the $\partial\bar{\partial}. $
By Hörmander's $L^2$-method, we study the operator $α\partial^k \bar{\partial}^{k} + β\bar{\partial}^k +γ\partial^k + c$ for any order $k$ with $α, β, γ\in \mathbb{R}$ such that $(α, β, γ) \neq(0,0,0)$ in the weighted Hilbert space $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$. We prove the existence of its right inverse which is also a bounded operator. Subsequently we will study two cases that arise from this operator, namely: (1) Case where $α= γ=0$: The operator $β\bar{\partial}^{k} + c$ with $\vert β\vert \geq 1$. (2) Case where $β= γ=0$: The operator $α\partial^{k} \bar{\partial}^{k} + c$ with $\vert α\vert \geq 1$.
In this paper, we consider the problem of solving the $\partial\overline{\partial}$ equation with discribed support for differential forms in a relatively compact domain $Ω$ of a complex analytic manifold $X$. And as a consequence, we have the solution of the equation $\partial\overline{\partial}$ for extendable currents without the annulation assumption of the De Rham cohomology group of the boundary.\\ \textbf{Keywords:} operator $\partial\overline{\partial}$, De Rham cohomology group, Dolbeault cohomology group, Bott-Chern cohomology group,Applie cohomology group, discribed support, extendable currents.
In this paper, we consider the problem of solving the $\partial \bar{\partial}$ with prescribed support for forms or currents in a domain $Ω$ of an complex manifold $X$.
By Hörmander's $L^2$-méthode, we study some operators in the Hilbert space of weight $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$. We prove in each case of operator the existence of its inverse which is also a bounded operator.
By the Hörmander's $L^2$-method, we study the operator $\partial^k \bar{\partial}^{k} + c$ for any order $k$ in the weighted Hilbert space $L^2(\mathbb{C}, {\rm e}^{-\vert z \vert^2})$. We prove the existence of its right inverse witch is also a bounded operator.
We solve the $\partial \bar{\partial}$-problem for the differential forms of class $C^\infty$ with boundary value in currents sense defined on a contractible completely strictly pseudoconvex domain of a complex manifold.
We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a Levi flat unbounded domain of $\mathbb{C}^n$ with the complementary is also Levi flat and unbounded.