Global Regularity for the ∂

Let M be a C∞ compact CR manifold of CR-codimension l≥1 and CR-dimension n-l in a complex manifold X of complex dimension n≥3. In this paper, assuming that M satisfies condition Y(s) for some s with 1≤s≤n-l-1, we prove an L2-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for (0,s)-forms on M

Holomorphic Approximation on Certain Weakly Pseudoconvex Domains in Cn

The purpose of this paper is to study the Mergelyan approximation property in L p and C k -scales on certain weakly pseudoconvex domains of finite/infinite type in C n . At the heart of our results lies the solvability of the additive Cousin problem with bounds as well as estimates of the ∂ ¯ -equation in the corresponding topologies

Ck\mathcal C^k-regularity for the ∂ˉ\bar \partial -equation with a support condition

summary:Let $D$ be a $\mathcal {C}^d$ $q$-convex intersection, $d\geq 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, $\mathcal C^k$-estimates, $k=2, 3, \dots , \infty$, for solutions to the \hbox {$\bar \partial$-equation} with small loss of smoothness are obtained for $E$-valued $(0, s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\bar \partial$-equation with a support condition in $\mathcal C^k$-spaces. More precisely, we prove that for a $\bar \partial$-closed form $f$ in $\mathcal C_{0,q}^{k}(X\setminus D, E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\varepsilon$ with $0<\varepsilon <1$ there exists a form $u$ in $\mathcal C_{0,q-1}^{k-\varepsilon }(X\setminus D, E)$ with compact support such that $\bar {\partial }u=f$ in $X\setminus \overline D$. Applications are given for a separation theorem of Andreotti-Vesentini type in $\mathcal C^k$-setting and for the solvability of the $\bar \partial$-equation for currents

Global solutions with C^k-estimates for bar-d-equations on q-concave intersections

We construct a global solution to the $ar{partial}$-equation with $mathcal{C}^k$-estimates on q-concave intersections in $mathbb{C}^n$. Our main tools are integral formulas