\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 62, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/62\hfil Global solutions]
{Global solutions with $\mathcal{C}^{k}$-estimates for $\bar{\partial}$-equations
on $q$-concave intersections}

\author[S. Khidr, M.-Y. Barkatou \hfil EJDE-2013/62\hfilneg]
{Shaban Khidr, Moulay-Youssef Barkatou}  

\address{Shaban Khidr \newline
Mathematics Department, Faculty of Science,
King Abdelaziz University, North Jeddah, Jeddah, Saudi Arabia.\newline
Mathematics Department, Faculty of Science,
Beni-Suef University, Beni-Suef, Egypt}
\email{skhidr@kau.edu.sa}

\address{Moulay-Youssef Barkatou \newline
Laboratoire de Math\'ematiques et Applications\\
UMR 7348 CNRS - Universit\'e de Poitiers, T\'el\'eport 2\\
11, Boulevard Marie et Pierre Curie, BP 30179,
86962 Futuroscope Chasseneuil Cedex, France}
\email{youssef.barkatou@math.univ-poitiers.fr}

\thanks{Submitted November 20, 2012. Published February 28, 2013.}
\subjclass[2000]{32A26, 32F10, 32W05, 32W10}
\keywords{$\bar{\partial}$-equation; $q$-convexity; $q$-concave intersections;
$\mathcal{C}^{k}$-estimates}

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

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction and statement of main result}

This article is a continuation of \cite{bk} and concerns the study 
of the $\bar{\partial}$-equation on $q$-concave intersection in 
$\mathbb{C}^n$ from the viewpoint of $\mathcal{C}^{k}$-estimates 
by means of integral formulas. For this study we first solve 
the $\bar{\partial}$-equation with $\mathcal{C}^k$-estimates 
on local $q$-concave wedges in $\mathbb{C}^n$ and then we apply
the pushing out method used by Kerzman \cite{k}.

We recall the notion of $q$-convexity in the sense of 
Andreotti-Grauert \cite{bk,he,li}.

\begin{definition}\label{defn1.1} \rm
A bounded domain $G$  of class $\mathcal{C}^2$ in $\mathbb{C}^n$ is 
called strictly $q$-convex if there exist an open neighborhood $\mathbb{U}$ 
of $\partial G$ and a smooth $\mathcal{C}^2$-function
$\rho: \mathbb{U}\to\mathbb{R}$ such that
$G\cap \mathbb{U} = \{\zeta\in \mathbb{U}: \rho(\zeta)<0\}$ and the Levi form
\begin{equation*}
L_{\rho}(\zeta)t=\sum \frac {\partial^2\rho(\zeta)}{\partial
\zeta_j\partial \bar{\zeta}_{k}} t_j\bar{t}_{k},\quad
 t=(t_{1},\dots, t_{n})\in\mathbb{C}^n
\end{equation*}
has at least $q+1$ positive eigenvalues at each point $\zeta\in \mathbb{U}$.

A domain $G$ in $\mathbb{C}^n$ is said to be strictly $q$-concave
if $G$ is in the form  $G= G_{1}\setminus\overline{G}_{2}$, where  $G_{2}\Subset G_{1}$ is strictly $q$-convex and $G_{1}\Subset\mathbb{C}^n$ is strictly $(n-1)$-convex or compact. A point in $\partial G_{2}$, as a boundary point of $G$, is said to be strictly $q$-concave.
\end{definition}

Applications to the tangential Cauchy-Riemann equations require that 
Definition \ref{defn1.1} be extended to $q$-convex and $q$-concave 
domains with piecewise-smooth boundaries.

\begin{definition} \label{defn1.2}\rm
 A bounded domain $D$ in $\mathbb{C}^n$ is called a $\mathcal{C}^{d}$
$q$-convex intersection of order $N$, $d\geq 3$, if there exists a bounded
  neighborhood $U$ of $\overline{D}$ and a
  finite number of real-valued $\mathcal{C}^{d}$ functions  $\rho{_1}(z), \dots,
  \rho{_N}(z)$,  $1\leq N\leq n-1$, defined on $U$ such  that $D=\{z\in U:
  \rho_1(z)<0,\dots, \rho_N(z)<0\}$  and the following conditions are fulfilled:
\begin{itemize}
\item [(H1)] For $1\le i_1<i_2<\dots<i_{\ell} \le N$ the $1$-forms
  $d \rho_{i_1}, \dots, d \rho_{i_{\ell}}$  are $\mathbb{R}-$linearly
  independent  on the set $\cap_{j=1}^{j=\ell}\{\rho_{i_j}(z)\le 0\}$.

\item [(H2)] For $1\le i_1<i_2<\dots<i_{\ell} \le N$,   for every
$z\in \cap_{j=1}^{j=\ell}\{\rho_{i_j}(z)\le 0\}$, if  we set $I= (i_1,
  \dots,i_{\ell}),$ there  exists a linear subspace $T^I_z$  of
  $\mathbb{C}^n$ of complex dimension  at least $q+1$ such  that
  for $i\in I$  the Levi  forms $L _{\rho_i}$  restricted  on $T^I_z$
  are positive definite.
\end{itemize}
A domain $D$ in $\mathbb{C}^n$ is said to be a $\mathcal{C}^d$ $q$-concave
intersection of order $N$ if $D= D_{1}\setminus\overline{D}_{2}$, where  
$D_{2}\Subset D_{1}$ is a $\mathcal{C}^d$ $q$-convex intersection
of order $N$  and $D_{1}\Subset\mathbb{C}^n$ is a $\mathcal{C}^d$
 $(n-1)$-convex intersection. A point in $\partial D_{2}$, 
as a boundary point of $D$, is said to be strictly $q$-concave.
\end{definition}

Condition (H2)  was first introduced by Grauert \cite{gh}, it  implies  
that at every wedge the Levi forms of the corresponding $\{\rho_i\}$ 
 have their positive eigenvalues along  the same directions.

The study of the $\bar{\partial}$-equation on piecewise smooth intersections
 in $\mathbb{C}^n$ was initiated by Range and Siu \cite{r-s} and then followed
 by many authors (see e.g., \cite{a-h,  he,sk, he1, r-l,m, m-p, p1}).

Motivated by the same problem, Laurent-Thi\'ebaut  and  Leiterer \cite{la-le2} 
solved the $\bar{\partial}$-equation on piecewise smooth intersections of
 $q$-concave domains in $\mathbb{C}^n$ with uniform estimates for $(n, s)$-forms, $1
\le s\le q-N$; $q-N\ge1$,  where instead  of condition (H2) 
they required the following  Henkin's condition  \cite{a-h,he}:
\begin{itemize}
\item [(H3)] The Levi form  of any  nontrivial  convex  combination of 
$\{\rho_i\}_{i=1}^N$ has at  least $q+1$ positive  eigenvalues.
\end{itemize}
In addition, under slightly stronger hypotheses than those of \cite{la-le2}, 
the authors extended their results in \cite{la-le3} to the case when $s=q-N+1$.

Barkatou \cite{b} obtained  local solutions with $\mathcal{C}^k$-estimates 
for $\bar{\partial}$ on $q$-convex wedges in $\mathbb{C}^n$, his proof requires
actually  the following condition: 
\begin{quote}
there is a subdivision of the simplex $\bigtriangleup_{N}$ such that for 
every component $[a^1\dots a^N]$ in this subdivision, the Leray maps 
of $\rho_{a^1}, \dots, \rho_{a^N}$ are $q + 1$-holomorphic 
in the same directions with respect to the variable $z\in \mathbb{C}^n$,
where for $a= (\lambda_1,\dots, \lambda_N )$, $\rho_a =\sum \lambda_i \rho_{i}$
\end{quote}
which is weaker than condition (H2) and stronger than condition (H3).

Ricard \cite{r} proved weaker $\mathcal{C}^{k}$-estimates than those 
obtained in \cite{b} but for general $q$-convex ($q$-concave) wedges 
satisfying condition (3).

Recently, Barkatou and Khidr  \cite{bk} constructed a global solution 
for $\bar{\partial}$ with $\mathcal{C}^{k}$-estimates with small 
loss of smoothness for $(0, s)$-forms, $n-q\le s\le n-1$, on 
$q$-convex intersections in $\mathbb{C}^n$.

Let $V$ be a bounded open set in $\mathbb{C}^n$. We use
$\mathcal{C}^{k}_{r, s}(\overline{V})$, $k\in \mathbb{R}^+$, to 
denote the space of all  continuous $(r, s)$-forms defined on $\overline{V}$ 
and having a continuous derivatives up to $[k]$ on $\overline{V}$ 
satisfying H\"older condition of order $k-[k]$. The corresponding 
norm is denoted by $\|\cdot\|_{k, V}$.
Our main result is the following theorem.

\begin{theorem} \label{thm1.3}
Let $D$ be a $\mathcal{C}^{d}$ $q$-concave intersection 
of order $N$ in $\mathbb{C}^n$,
$d\geq 3$, and let $f\in\mathcal{C}_{n,s}^{0}(D)$, $\bar{\partial}f=0$, 
$1\leq s\leq q-N$. Then there is a form 
$g\in\mathcal{C}_{n,s-1}^{0}(D)$ such that $\bar{\partial}g=f$ on $D$. 
If $f\in\mathcal{C}^{k}_{n,s}(\overline D)$,
$1<k\leq d-2$;
 $\epsilon>0$, then $g\in \mathcal{C}^{k-\epsilon}_{n,s-1}(\overline{D})$ 
and there exists a constant $C_{k, \epsilon}>0$ such that
\begin{equation}\label{e1.1}
\| g \| _{k-\epsilon, D} \leq C_{k, \epsilon}\|f\|_{k,D}.
\end{equation}
\end{theorem}

We note that for $q=n-1$ (i.e., the pseudoconvex case) this theorem was 
proved by Michel and Perotti \cite{m-p} and for
arbitrary $q$, but smooth $\partial D$, sharp $\mathcal{C}^k$ 
estimates were obtained by Lieb and Range \cite{r-l}.

The paper is organized in the following way: 
In section 1 we introduce the definition of a $q$-concave intersection 
in $\mathbb{C}^n$ and state our main result (Theorem \ref{thm1.3}). 
In section 2 we recall the generalized Koppelman lemma which plays
 a key role in the construction of the solution operators.
 Section 3 is devoted to the construction of the local solution operators 
with $\mathcal{C}^{k}$-estimates for $\bar{\partial}$. 
The main theorem is proved in section 4. The proof is based on 
pushing out the method of Kerzman \cite{k}.


\section{Generalized Koppelman Lemma}

In this section we recall a formal identity (the generalized Koppelman lemma) 
which is essential for our purposes. The exterior
calculus we use here was developed by Harvey and Polking in \cite{ha-po}
 and Boggess \cite{bo}.

Let $D$ be an open set in $\mathbb{C}^n\times\mathbb{C}^n$. Let
$G:D\to \mathbb{C}^n$ be a $\mathcal{C}^{1}$ map and write
$G(\zeta, z)=(g_{1}(\zeta, z), \dots, g_{n}(\zeta, z))$. We define
\begin{gather*}
\langle G(\zeta, z), \zeta-z\rangle
= \sum_{j=1}^ng_j(\zeta, z)(\zeta_j-z_j)\\
\langle G(\zeta, z), d(\zeta-z)\rangle
= \sum_{j=1}^ng_j(\zeta, z)d(\zeta_j-z_j)\\
\langle \bar{\partial}_{\zeta, z}G(\zeta, z),
d(\zeta-z)\rangle 
= \sum_{j=1}^n\bar{\partial}_{\zeta, z}g_j(\zeta, z)d(\zeta_j-z_j),
\end{gather*}
where $\bar{\partial}_{\zeta, z}=\bar{\partial}_{\zeta}+\bar{\partial}_{z}$
(in the sense of distributions).

The Cauchy-Fantappi\`{e} form $\omega^{G}$ is defined by
\begin{equation*}
\omega^{G}=\frac{\langle G(\zeta, z),\, d(\zeta-z)\rangle}{\langle G(\zeta, z),\,
  (\zeta-z)\rangle}
\end{equation*}
on the set where $\langle G(\zeta, z),\, (\zeta-z)\rangle\neq0$.

Given $m$ such maps, $G^{j}$, $1\leq j\leq m$, the generalized
 Cauchy-Fantappi\`{e} kernel is given by
\begin{align*}
&\Omega(G^{1}, \dots, G^{m})\\
&=(2\pi i)^{-n}\omega^{G^{1}}\wedge 
\dots\wedge \omega^{G^{m}}
\wedge\sum_{\alpha_{1}+\dots+\alpha_{m}=n-m}
(\bar{\partial}_{\zeta, z}\omega^{G^{1}})^{\alpha_{1}}\wedge 
\dots\wedge (\bar{\partial}_{\zeta, z}\omega^{G^{m}})^{\alpha_{m}}
\end{align*}
on the set where all the denominators are nonzero.

\begin{lemma}[generalized Koppelman lemma] \label{lem2.1} 
\begin{equation*}
\bar{\partial}_{\zeta, z}\Omega(G^{1}, \dots, G^{m})
=\sum_{j=1}^{m}(-1)^{j}\Omega(G^{1}, \dots,  \widehat{G}^{j}, \dots,G^{m})
\end{equation*}
on the set where the denominators are nonzero.
\end{lemma}

If $\beta(\zeta, z)=(\overline{\zeta_{1}-z_{1}}, \dots, 
\overline{\zeta_{n}-z_{n}})$, then $\Omega(\beta)=B(\zeta, z)$ 
is the usual Bochner-Martinelli-Koppelman kernel. Denote by $B_{r, s}(\zeta, z)$ 
the component of $B(\zeta, z)$ of type $(r, s)$ in $z$ and of type 
$(n-r, n-s-1)$ in $\zeta$. Then one has the following formula which is
known as the Bochner-Martinelli-Koppelman formula
(see e.g., \cite[Theorem 1.7]{la-le}).

\begin{theorem} \label{thm2.2}  
Let $D\Subset\mathbb{C}^n$ be a bounded domain with
$\mathcal{C}^1$-boundary, and let $f$ be a continuous $(r, s)$-form on 
$\overline{D}$ such that $\bar{\partial}f$, in the sense of distributions, 
is also continuous on $\overline{D}$, $0\leq r, s\leq n$. 
Then for any $z\in D$ we have
\begin{align*}
(-1)^{r+s}f(z)&=\int_{\zeta\in\partial D}f(\zeta)\wedge B_{r, s}(\zeta, z)
 -\int_{\zeta\in D}\bar{\partial}f(\zeta)\wedge B_{r, s}(\zeta, z)\\
&\quad + \bar{\partial}_{z}\int_{\zeta\in D}f(\zeta)\wedge B_{r, s-1}(\zeta, z).
\end{align*}
\end{theorem}

\section{Solution operators on local $q$-concave wedges}

In this section, we  construct  local solution operators $T_{s}$ on the 
complement of a $q$-convex intersection. The plan of the construction is
 similar to that of Theorem 3.1 in \cite{bk}. The main differences are due 
to the fact that in this case the function $\rho_{m+1}$ has convexity
 properties opposite to those of the functions $\rho_{1}, \dots, \rho_{m}$. 
Before we go further, we fix the following notation:
\begin{itemize}
  \item Let $J=(j_1,\dots,j_{\ell})$, $1\le \ell<\infty$, be an ordered collection of elements in $\mathbb{N}$. Then we
write $|J|=\ell$, $J(\hat \nu)=  (j_1,\dots,j_{\nu-1},j_{\nu+1},\dots,j_{\ell})$ 
for $\nu=1,\dots,\ell$, and $j\in J$ if $j\in \{j_1,\dots,j_{\ell}\}$.

\item  Let $N\ge1$ be an integer. Then we denote by $P(N)$ the set of all
 ordered collections $K=(k_1, \dots,k_{\ell})$, $\ell\ge 1$, of integers
 with $1\le k_1,\dots,k_{\ell}\le N$. We call $P'(N)$ the subset of all 
 $K=(k_1, \dots,k_{\ell})$ with $k_1<\dots<k_{\ell}$.

\item  For $I=(j_1,\dots,j_{\ell})\in P'(N)$ and 
 $j\notin \{j_1,\dots,j_{\ell}\}$, we set $I_j=(k_1, \dots,k_{\ell+1})$ if
  $\{k_1, \dots,k_{\ell+1}\}\subset\{k_1, \dots,k_{\ell}, j\}$ and  
 $k_1<\dots<k_{\ell+1}$.
\end{itemize}

\begin{theorem} \label{thm3.1}
Let $D$ be a $\mathcal{C}^{d}$ ($d\geq 3$) $q$-convex intersection
of order $n$ in  $\mathbb{C}^n$. Then for each $\xi\in \partial D$, there is a radius $R>0$
such that on the set 
$\mathcal{W}=(U\setminus \overline{D})\cap\{z\in \mathbb{C}^n: |z-\xi|<R\}$
there are linear operators
$T_{s}:\mathcal{C}^{0}_{n,s}(\mathcal{W}) \to \mathcal{C}^{0}_{n,s-1}(\mathcal{W})$
such that $\bar{\partial}T_{s}f=f$ for all 
$f\in \mathcal{C}^{0}_{n,s}(\overline{\mathcal{W}})$, 
$1\leq s\leq q-N$, with $\bar{\partial}f=0$
(in the sense of distributions) on $\mathcal{W}$. 
If $f\in \mathcal{C}^{k}_{n,s}(\overline{\mathcal{W}})$, 
 $1<k\leq d-2$; $\epsilon>0$, 
then there exists a constant $C_{k, \epsilon}>0$ (independent of $f$) satisfying 
the estimates
\begin{equation}\label{e3.1}
\| T_{s}f\| _{k-\epsilon, \mathcal{W}} \leq C_{k, \epsilon} \|f\|_{k, \mathcal{W}}.
\end{equation}
\end{theorem}

For $N=1$ (i.e., the case of local $q$-concave domains) this theorem was 
proved by Laurent-Thi\'ebaut and Leiterer \cite{la-le1}.

\begin{proof}
Let $D=\{z\in U| \rho_1(z)<0,\dots, \rho_N(z)<0\}\subset U$ be a $q$-convex 
intersection. We suppose for example that 
$E=\{\xi\in U|\rho_{1}(\xi)=\dots= \rho_{m}(\xi)=0\}$. 
If we set $\rho_{m+1}(\zeta)= |\zeta-\xi|^2 - R^2$ for $R>0$, 
it follows from \cite[Lemma 2.3]{la-le2} that 
$(E, (U\setminus \overline{D})\cap\{z\in \mathbb{C}^n: |z-\xi|<R\})$
is a local $q$-concave wedge.

Denote by $F_{\rho_i}(\zeta, .)$ the Levi polynomial of $\rho_i$ 
at $\zeta\in U$. For $\zeta\in U$, $z\in \mathbb{C}^n$,
\begin{equation*}
F_{\rho_i}(\zeta, z)=2\sum_{j=1}^n
\frac{\partial\rho_i(\zeta)}{\partial
\zeta_j}(\zeta_j-z_j)-\sum_{j, k=1}^n
\frac{\partial^2\rho_i}{\partial \zeta_j\partial
\zeta_{k}}(\zeta_j-z_j)(\zeta_{k}-z_{k}).
\end{equation*}
 By Definition \ref{defn1.2}, there exists an $(q+1)$-linear subspace
$T$ of ${\mathbb C}^n$ such that the Levi forms $L_{-\rho_i}$ at $\xi$
 are all positive definite on $T$.

Denote by ${P}$ the orthogonal projection of $\mathbb{C}^n$ onto $T$ and
set $Q:=Id-{P}$. Then it follows from Taylor's expansion theorem that there 
exist a number $R$ and two positive constants $A$ and $B$ such that 
the following estimate holds:
\begin{equation}\label{e3.2}
-\operatorname{Re}F_{\rho_i}(\zeta, z)\geq\rho_i(z)-\rho_i(\zeta)+
B|\zeta-z|^2-A|Q(\zeta-z)|^2,
\end{equation}
for every $i\in \{1,\dots,m\}$ and all $(z, \zeta)\in \mathbb{C}^n\times U$
with $|\xi-\zeta|<R$ and $|\xi-z|<R$.

Let $i\in  \{1,\dots,m\}$. As $\rho_i$ is of class $\mathcal{C}^2$ on $U$,
we then can find $\mathcal{C}^{\infty}$ functions $a_i^{kj}(U)$, 
$j, k=1,\dots, n$, such that for all $\zeta\in U$,
\begin{equation}\label{e3.3}
\big|a_i^{kj}(\zeta)-\frac{\partial^2\rho(\zeta)}{\partial
\zeta_{k}\partial \zeta_j}\big|<\frac{B}{2n^2}.
\end{equation}
Denote by $Q_{kj}$ the entries of the matrix $Q$; i.e.,
\begin{equation*}
Q=(Q_{kj})_{k, j=1}^n\quad (k= \text{column index}).
\end{equation*}
We set, for $(z, \zeta)\in \mathbb{C}^n\times U$,
\begin{gather*}
g^i_j(\zeta,z)=2\frac{\partial\rho_i(\zeta)}{\partial\zeta_j}-
\sum_{k=1}^na_i^{kj}(\zeta)(\zeta_{k}-z_{k})-A\sum_{k=1}^n\overline{Q_{kj}(\zeta_{k}-z_{k})},\\
G_i(\zeta, z)=(g_i^{1}(\zeta, z), \dots, g_i^n(\zeta, z)),\\
\Phi_i(\zeta,z)=\langle G_i(\zeta, z), \zeta-z\rangle.
\end{gather*}
As $Q$ is an orthogonal projection, we then have
\begin{equation*}
\Phi_i(\zeta,z)=2\sum_{j=1}^n\frac{\partial\rho_i(\zeta)}{\partial\zeta_j}(\zeta_j-z_j)
-\sum_{k, j=1}^na_i^{kj}(\zeta)(\zeta_{k}-z_{k})(\zeta_j-z_j)-A|Q(\zeta-z)|^2.
\end{equation*}
The  estimates \eqref{e3.2} and \eqref{e3.3} imply that
\begin{equation*}
\operatorname{Re}\Phi_i(\zeta, z)\geq\rho_i(\zeta)-\rho_i(z)
+\frac{B}{2}|\zeta-z|^2
\end{equation*}
for $(z, \zeta)\in \mathbb{C}^n\times U$ with $|z_{0}-\zeta|\leq R$
and $|z_{0}-z|\leq R$.
\end{proof}

We recall that a map $f$ defined on a complex manifold $\mathcal{X}$ is 
called $k$-holomorphic if, for each point
$\xi\in \mathcal{X}$, there exist holomorphic coordinates 
$h_{1}, \dots, h_{k}$ in a neighborhood of $\xi$ such that $f$ is
 holomorphic with respect to $h_{1}, \dots, h_{k}$.

\begin{lemma}\label{lem3.2}
 For every fixed $\zeta\in U$, the maps $G_i(\zeta, z)$ and the function
$\Phi_i(\zeta, z)$  are $(q+1)$-holomorphic in the same directions in  
$z\in \mathbb{C}^n$.
\end{lemma}

\begin{proof}
Choose complex linear coordinates $h_1,\dots,h_n$ on $\mathbb {C}^n$ with
$$
\{z\in \mathbb{C}^n: Q(z)=0\}=\{z\in \mathbb {C}^n:h_{q+2}(z)=\dots=h_n(z)=0\}.
$$
The map $z\to \overline {Q(\zeta-z)}$ is then independent
of $h_1, \dots,  h_{q+1}$.
This implies  that the map $G_i(\zeta, z)$ is complex linear with
respect to $h_1, \dots,  h_{q+1}$ for all $i$, and the function
$\Phi_i(\zeta, z)$ is a quadratic complex polynomial  with
respect to $h_1, \dots , h_{q+1}$.
\end{proof}

Set
\begin{gather*}
G_{m+1}(\zeta, z)=2\Big(\frac{\partial \rho_{m+1}(\zeta)}{\partial\zeta_{1}} ,
\dots,\frac{\partial \rho_{m+1}(\zeta)}{\partial\zeta_{n}}\Big),\\
\Phi_{m+1}(\zeta,z)=\langle G_{m+1}(\zeta, z), (\zeta-z)\rangle.
\end{gather*}
As $G_{m+1}(\zeta, z)$ and $\Phi_{m+1}(\zeta,z)$ are independent of $R$, we can choose $R_{1}>0$ such that for all $R\le R_{1}$ there exists $\beta>0$ satisfying
\begin{equation*}
\operatorname{Re}\Phi_{m+1}(\zeta, z)\geq\rho_{m+1}(\zeta)
-\rho_{m+1}(z)+\beta|\zeta-z|^2
\end{equation*}
for all $(z, \zeta)\in \mathbb{C}^n\times U$ with $|z_{0}-\zeta|\leq R$
and $|z_{0}-z|\leq R$.
We define 
\[
\mathcal{W}=\{z\in U| \rho_j>0 \text{ for } j=1,
\dots, m\}\cap \{z\in \mathbb{C}^n: |z-\xi|<R\}.
\]
For $I=(j_1,\dots,j_{\ell}) \in P'(m+1)$, we define
\begin{gather*}
\widetilde{\Omega}[I]:=\Omega(G_{j_{1}}, \dots, G_{j_{\ell}}),\\
\widetilde{\Omega}[\partial I]:= \sum^{\ell}_{k=1}{(-1)^k}{\Omega(G_{j_{1}},\ldots,\widehat{G}_{j_{k}},\ldots,G_{j_{\ell}})}.
\end{gather*}
Then, we can rewrite Lemma \ref{lem2.1} in the following way.

\begin{lemma} \label{lem3.3}
 For every  $I\in P'(m+1)$, we have
$\bar{\partial}_{\zeta, z}\widetilde{\Omega}[I]=\widetilde{\Omega}[\partial I]$
outside the singularities.
\end{lemma}

For every $0\leq r\leq n$, $0\leq s\leq n-p$ ($p\ge1$) and any $I$, we define
$\widetilde{\Omega}_{r, s}[I]$ as the component of $\widetilde{\Omega}[I]$ 
which is of type $(r, s)$ in $z$. One has the following lemma.

\begin{lemma} \label{lem3.4}
 For any $I\in P'(m+1)$. For any $r\geq0$ and $s \geq n-q$ we have
\begin{itemize}
\item [(i)] $\widetilde{\Omega}_{r, s}(I)=0$,
\item [(ii)] $\bar{\partial}_{z}\widetilde{\Omega}_{r, n-q-1}(I)=0$,
 \end{itemize}
on the set where all the denominators are non-zero.
\end{lemma}

\begin{proof}
Statement (i) follows from  Lemma \ref{lem2.1} and  the fact that the 
map $z\mapsto G_{m+1}(\zeta,z)$ is holomorphic.
Lemmas \ref{lem3.3} and \ref{lem2.1} imply that
\begin{equation*}
\bar{\partial}_{z}\widetilde{\Omega}_{r, s-1}(I)=
-\bar{\partial}_{\zeta}\widetilde{\Omega}_{r, s}(I)
+\widetilde{\Omega}_{r, s}(\partial I).
\end{equation*}
Statement (ii) follows from (i).
\end{proof}

Let $\beta(\zeta, z)=(\overline{\zeta_{1}-z_{1}}, 
\dots, \overline{\zeta_{n}-z_{n}})$ be the
classical section that defines the usual Bochner-Martinelli 
kernel in $\mathbb{C}^n$ and define
\begin{equation*}
\widetilde{\Omega}_{\beta}[I]:=\Omega(\beta, G_{ j_{1} }, \dots, 
G_{j_{\ell}})
\end{equation*}
for any   $I\in P'(m+1)$. Lemma \ref{lem3.4} implies that
\begin{equation*}
\bar{\partial}_{\zeta, z}\widetilde{\Omega}_{\beta}[{I}]=
-\widetilde{\Omega}[{I}]-\widetilde{\Omega}_{\beta}[\partial {I}]
\end{equation*}
outside the singularities, where
$\widetilde{\Omega}_{\beta}[\partial {I}]:=\Omega(\beta)$ if $|I|=1$.
Define, for $|I|\geq 1$,
\begin{gather*}
K^{I}(\zeta, z)=\widetilde{\Omega}_{\beta}[{I}](\zeta, z), \\
B^{I}(\zeta, z)=-\widetilde{\Omega}_{\beta}[\partial {I}](\zeta, z).
\end{gather*}
Then we obtain the following lemma.

\begin{lemma}\label{lem3.5}
 For any $I\in P'(m+1)$,
\begin{equation*}
 \bar{\partial}_{\zeta, z}K^{I}(\zeta, z)=B^{I}(\zeta, z)-
 \widetilde{\Omega}[{I}](\zeta, z)
\end{equation*}
outside the singularities.
\end{lemma}

\begin{proof}

For every  $I=(j_1,\dots,j_{\ell}) \in  P'(m+1)$, define
\[
S_{I}=\{z\in \partial \mathcal{W}| \rho_{j_{1}}(z)=\dots=\rho_{j_{\ell}}(z)=0\}
\]
and choose the orientation of $S_{I}$ such that the orientation is skew 
symmetric in the components of $I$ and the following two equations 
hold when $\mathcal{W}$ is given the natural orientation:
\[
\partial \mathcal{W}=\sum_{j=1}^{m+1}S_{j},\quad
\partial S_{I}=\sum_{j\notin I}S_{I_{j}}.
\]
Denote by $K^{I}_{r, s}(\zeta, z)$ the component of $K^{I}(\zeta, z)$ 
which is of type $(r, s)$ in $z$.

Then Lemmas \ref{lem3.4} and \ref{lem3.5}, Theorem \ref{thm2.2}, 
and Stoke's theorem imply that the following formulas hold 
in the sense of distribution in $\mathcal{W}$ 
(see \cite[Theorem 2.7]{bc}):

For any continuous $(n,s)$-form $f$ on $\overline{\mathcal{W}}$,
 $1\leq s\leq q-m$, such that $\bar{\partial}f$ is also continuous 
on $\overline{\mathcal{W}}$. We have
\begin{align*}
&(-1)^{n+s}f(\zeta)\\
&= \sum_{I\in P'(m+1)}(-1)^{(n+s)| I|+\frac{| I|(| I|-1)}{2}}\quad
\int_{z\in S_{I}}\bar{\partial}f(z)\wedge K^{I}_{0, n-|I|-s}(\zeta, z)\\
&\quad +\sum_{I\in P'(m+1)}(-1)^{(n+s)| I|+\frac{| I|(| I|+1)}{2}+1}\quad
\bar{\partial}_{\zeta}\int_{z\in S_{I}}f(z)\wedge K^{I}_{0, n-|I|-s+1}(\zeta, z)
\\
&\quad -\int_{z\in \mathcal{W}}\bar{\partial}f(z)
\wedge B_{0, n-s-1}(\zeta, z)
+\bar{\partial}_{\zeta}\int_{z\in \mathcal{W}}f(z)
\wedge B_{0, n-s}(\zeta, z)+Lf (\zeta)
\end{align*}
where  $Lf$  is  a  linear  combination  of  the  integrals 
$  \int_{z\in S_{I_{m+1}}} f(z)\wedge \widetilde{\Omega}[{I}](\zeta, z)$,
where  $I\in P'(m)$.

It is  easy  to  see  that $Lf$  is  of  class  ${\mathcal C}^{d-2}$  
in  a  neighborhood  of $\xi$; moreover  if $\bar{\partial}f=0$,  
then  $\bar{\partial}  Lf = 0$.  
Let  $H$  be  the  Henkin  operator  for solving  the  
$\bar{\partial}$-equation in a  ball  $B(\xi,R')$.  
 From  the smootness  properties  of  $H$, it follows that
$H(Lf)$  is  of  class ${\mathcal C}^{{d-2}+\frac{1}{2}}$.
Note that
\begin{align*}
T_{s}(f)(\zeta)
&=\sum_{I\in P'(m+1)}(-1)^{(n+s)| I|+\frac{| I|(| I|+1)}{2}+1}
\int_{z\in S_{I}}f(z)\wedge K^{I}_{0, n-|I|-s+1}(z, \zeta)\\
&\quad +(-1)^{n+s}\int_{z\in \mathcal{W}}f(z)
 \wedge B_{0, n-s}(z, \zeta)+H(Lf)(\zeta)
\end{align*}
satisfies the equation $\bar{\partial}u=f$ on $\mathcal{W}\cap B(\xi,R')$ 
with $u=T_{s}(f)(\zeta)$.

The $\mathcal{C}^{k}$-estimates follows, as in \cite{b}, 
by using arguments similar to those in \cite{r-l}.
\end{proof}

\section{Proof of Theorem \ref{thm1.3}}

Theorem \ref{thm3.1} yields the following continuation lemma which in turn
enables us to complete the proof of Theorem \ref{thm1.3}.

\begin{lemma}[An extension lemma with bounds] \label{lem4.1}
 Let $D$ be a $\mathcal{C}^{d}$, $d\geq 3$, $q$-concave intersection 
of order $N$ in  $\mathbb{C}^n$. 
Then there exists another slightly larger $q$-concave
intersection of order $N$, $\widetilde{D}\Subset\mathbb{C}^n$
such that ${D}\Subset\widetilde{D}$
and for any $f \in \mathcal{C}^{0}_{n,s}(D)$, 
$1\leq s \leq q-N$, with $\bar{\partial}f=0$ there exist two linear operators 
$N_{1}$, $N_{2}$, a form 
$\tilde{f}=N_{1}f\in\mathcal{C}^{0}_{n,s}(\widetilde{D})$
and a form $u=N_{2}f\in\mathcal{C}^{0}_{n,s-1}(D)$ such that:
\begin{itemize}
\item[(i)] $\bar{\partial}\tilde{f} = 0$ in $\widetilde{D}$.

\item[(ii)] $\tilde{f} = f - \bar{\partial} u$ in $D$.

\item[(iii)] If $f \in \mathcal{C}^{k}_{n,s}(\overline{D})$, 
$1<k\leq  d-2$,
$\epsilon>0$, then $\tilde{f}\in \mathcal{C}^{k-\epsilon}_{n,s}(\widetilde{D})$,
 $u\in\mathcal{C}^{k-\epsilon}_{n,s-1}(\overline{D})$ and we have the estimates:
\begin{gather} \label{e4.1}
\|\tilde{f}\| _{k-\epsilon, \widetilde{D}} \leq C_{k, \epsilon} \|
f\|_{k,D},\\
\|  u \| _{k-\epsilon, D} \leq C_{k, \epsilon} \|
f\|_{k,D}.\label{e4.2}
\end{gather}
\end{itemize}
\end{lemma}

\begin{proof}
As $\partial D$ is compact, there are finitely many open neighborhoods
$(B_{{\xi_j}})_{j=1, \dots, K}$ of $\xi_j$ covering $\partial
D$. Let $(\theta_j)_{j=1, \dots, K}$ be a partition
of unity such that $\theta_j\in\mathcal{C}_{0}^{\infty}(B'_{\xi_j})$,
$B'_{\xi_j}\Subset B_{\xi_j}$, $0\leq\theta_j\leq1$, and
$\sum_{j=1}^{K}\theta_j=1$ on a neighborhood $V_{0}$ of
$\partial D$. We choose $V_{1}\Subset V_{0}\Subset U$.
We enlarge $D$ to $\widetilde{D}$ in
$K$ step as follows. For $\delta>0$, sufficiently small to be chosen 
fixed later on, and for $j=1, \dots, K$ we define
\begin{equation*}
D_j=\Big\{z\in D\cup V_{1}:\rho_{1}(z)>-\delta\sum_{k=1}^{j}
\theta_{k}(z), \dots, \rho_{N}(z)>-\delta\sum_{k=1}^{j}\theta_{k}(z)\Big\}.
\end{equation*}
We set $D_{0}=D$ and $\widetilde{D}=D_{K}$. Clearly
\begin{gather*}
D\subseteq D_j\subseteq D_{j+1}\subseteq\dots\subseteq\widetilde{D}=D_{K}.
\end{gather*}
Reducing $\delta$ if necessary, we see that all $D_j$, $j\in\{=1, \dots,K\}$
(in particular $\widetilde{D}$) are  $\mathcal{C}^{d}$ $q$-concave 
intersections.

\textbf{Claim:} For any $f_j\in \mathcal{C}^{0}_{n, s}(D_j)$ with
$\bar{\partial}f_j=0$, $j\in\{1, \dots, K-1\}$, there exist two  forms
$f_{j+1}\in \mathcal{C}^{0}_{n, s}(D_{j+1})$ and $u_j\in
\mathcal{C}^{0}_{n, s-1}(D_j)$ such that (i), (ii) and
(iii) of Lemma \ref{lem4.1} hold when $f$, $\tilde{f}$, $u$, $D$
and $\widetilde{D}$ are replaced by $f_j$, $f_{j+1}$, $u_j$,
$D_j$ and $D_{j+1}$ respectively.

\begin{proof} 
(see \cite[p. 318]{k}): 
Fix $\delta>0$ so small that we can apply Theorem \ref{thm3.1}, we obtain 
a solution $g_j$ of $\bar{\partial}g=f_j$ defined in
$D_j\cap B_{\xi_{j+1}}$ and satisfies the estimates of the local theorem.
Let $\eta_{j+1}\in C_{0}^{\infty}(B_{\xi_{j+1}})$, $\eta_{j+1}=1$ in 
a neighborhood of the support of $\theta_{j+1}$. We set
\begin{equation*}
f_{j+1}=  \begin{cases}
    f_j-\bar{\partial}u_j    & \text{in }D_j, \\
        0 & \text{in }D_{j+1}\setminus D_j,
  \end{cases}
\qquad
u_j=  \begin{cases}
    g_j\eta_{j+1}      & \text{in }D_j\cap B_{\xi_{j+1}}, \\
    0 & \text{in } D_j\backslash B_{\xi_{j+1}}.
      \end{cases}
\end{equation*}
The estimates for $f_{j+1}$ and $u_j$ follow from those of the local theorem.
The claim is proved.
\end{proof}

Using the above claim, we can now complete the proof of Lemma \ref{lem4.1}. 
Applying the claim $K$-times, starting with
$D_{0}=D$, $f_{0}=f$ and ending with $D_{K}=\widetilde{D}$,
$f_{K}=\tilde{f}$, yield
$\tilde{f}=f-\bar{\partial}u$ in $D$, where we set
 $u=\sum_{j=0}^{K-1}u_j$. Collecting the estimates for $f_{j+1}$ and
$u_j$ in each step,
we obtain \eqref{e4.1} and
\eqref{e4.2}. Clearly $\tilde{f}$ and $u$ are linear in $f$.
\end{proof}

\begin{lemma} \label{lem4.2} 
There exists a strictly $q$-concave domain with smooth boundary
$D'\Subset\mathbb{C}^n$
satisfying
\begin{equation*}
D\Subset D'\Subset \widetilde{D}.
\end{equation*}
\end{lemma}

\begin{proof}
Let $V_{2}$ be a neighborhood of $D$ such that $V_{2}\Subset V_{1}$
and for $\tau>0$ we define 
$D_{\tau}:=\{z\in D\cup V_{2}|\, \rho_{1}(z)>\tau, \dots, \rho_{N}(z)>\tau\}$. 
Recall that $D$ is defined by the $\mathcal{C}^{d}$-functions $\rho_{1}, \dots,
\rho_{N}$. For each $\beta>0$, let $\chi_{\beta}$ be a fixed
non-negative real $C^{\infty}$ function on $\mathbb{R}$ such that,
for all $x\in \mathbb{R}$, $\chi_{\beta}(x)=\chi_{\beta}(-x)$,
$|x|\leq\chi_{\beta}(x)\leq|x|+\beta$, $|\chi'_{\beta}|\leq1$,
$\chi''_{\beta}\geq0$ and $\chi_{\beta}(x)=|x|$ if
$|x|\geq\frac{\beta}{2}$. Moreover, we assume that
$\chi'_{\beta}(x)>0$ if $x>0$ and $\chi'_{\beta}(x)<0$ if $x<0$. We
define as in \cite[Definition 4.12]{he-le}
$\max_{\beta}(t,s)=\frac{t+s}{2}+ \chi_{\beta}(\frac{t-s}{2}), t,
s\in \mathbb{R}$, and 
$ \varphi_{1}=\rho_{1}$,
$\varphi_{2}=\max_{\beta}(\rho_{1}, \rho_{2})$,
\dots, $\varphi_{N}=\max_{\beta}(\varphi_{N-1}, \rho_{N})$. Then it is
easy to compute that the Levi form of  $\varphi_{N}$ has at least
$q+1$ negative eigenvalues at each point in $U$.  For
$\tau>0$ we can choose  positive numbers 
$\beta=\frac{\tau}{2(N+1)}$,
$\gamma=\frac{\tau}{2}$ small enough and $V_{3}\Subset V_{2}$ such that
 \begin{equation*}
D\Subset D^{\ast} =\{z\in D\cup
V_{3}|\,\varphi_{N}(z)-\gamma>0\}\Subset D_{\tau}.
\end{equation*}
then  $D^{\ast}$ is a strictly $q$-concave domain. 
According to \cite[Theorem 6.6]{he-le}, there exists a strictly $q$-concave 
domain $D'$ with smooth boundary
such that $D \Subset D'\Subset D^{\ast}$.
Choose $\tau$ small enough to get $D_{\tau}\Subset\widetilde{D}$.
\end{proof}

Let $f\in \mathcal{C}^{k}_{n,s}(\overline D)$ be a $\overline\partial$-closed 
form.  Let $ \widetilde{D}$, $\tilde f$ and $u$ as in
 Lemma 4.1. Let $D'$ be given as in Lemma \ref{lem4.2} and set
 $f_{1}= \tilde f|_{D'}.$  It follows from \cite[Theorem 2]{r-l} 
that there exists $\eta\in \mathcal{C}^{k-\epsilon}_{n,s-1}(\overline{D})$ 
such that $\bar\partial \eta = f_{1}$ on $D$ and 
$\|\eta\|_{k-\epsilon, \overline{D}}\le C_{k, \epsilon}\|f_{1}\|_{k-\epsilon, D'}$. Then we have $f=\bar \partial (u+\eta)$. The form $g= u+\eta$
is a global solution that satisfies the $\mathcal{C}^k$-estimates \eqref{e1.1} 
of Theorem \ref{thm1.3}.

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\end{document}