\documentclass{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 73, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/73\hfil Solutions to $\bar{\partial}$-equations] {Solutions to $\bar{\partial}$-equations on strongly pseudo-convex domains with $L^p$-estimates} \author[O. Abdelkader \& Sh. Khidr \hfil EJDE-2004/73\hfilneg] {Osama Abdelkader \& Shaban Khidr} % in alphabetical order \address{Osama Abdelkader \hfill\break Mathematics Department, Faculty of Science, Minia University, El-Minia, Egypt} \email{usamakader882000@yahoo.com} \address{Shaban Khidr \hfill\break Mathematics Department, Faculty of Science, Cairo University, Beni- Suef, Egypt} \email{skhidr@yahoo.com} \date{} \thanks{Submitted March 01, 2004. Published May 20, 2004.} \subjclass[2000]{32F27, 32C35, 35N15} \keywords{$L^p$-estimates, $\bar{\partial}$-equation, strongly pseudo-convex, \hfill\break\indent smooth boundary, complex manifolds} \begin{abstract} We construct a solution to the $\bar{\partial}$-equation on a strongly pseudo-convex domain of a complex manifold. This is done for forms of type $(0,s)$, $s\geq 1$, with values in a holomorphic vector bundle which is Nakano positive and for complex valued forms of type $(r,s)$, $1\leq r\leq n$, when the complex manifold is a Stein manifold. Using Kerzman's techniques, we find the $L^p$-estimates, $1\leq p\leq \infty$, for the solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \section{Introduction}\label{e:Intro} The existence of solutions to the equation $\bar{\partial} g = f$, on strongly pseudo-convex domains in $\mathbb{C}^{n}$, with $L^p$-estimates when $f$ is a form of type $(0,s)$; $\bar{\partial} f = 0$, $s \geq 1$, and satisfies $L^p$-estimates, $1 \leq p \leq \infty$, has been a central theme in complex analysis for many years. \O vrelid \cite{o1} has obtained a solution with $L^p$-estimates for this equation. Abdelkader \cite{a1} has extended \O vrelid's results to forms of type $(n,s)$ on strongly pseudo-convex domains in an $n-$dimensional Stein manifold. In this paper we extend Abdelkader's results to forms of type $(r,s)$; $0 \leq r \leq n$. For this purpose, we first study the equation $\bar{\partial} g =f$, on strongly pseudo-convex domains in an $n$-dimensional complex manifold $M$ when $f$ is a form of type $(0,s)$; $\bar{\partial} f = 0$, $s \geq 1$, with values in a holomorphic vector bundle. Then, we apply this results to the vector bundle $\bigwedge^{r}T^{\star}(M)$ (the $r^{th}$-exterior product of the holomorphic cotangent vector bundle $T^{\star}(M)$) and using the fact that any $\mathbb{C}-$valued differential form of type $(r,s)$ on $M$ is a differential form of type $(0,s)$ on $M$ with values in the vector bundle $\bigwedge^{r}T^{\star}(M)$. When $r=n$, the vector bundle $K(M)=\bigwedge^{n}T^{\star}(M)$ is the canonical line bundle of $M$. Therefore it is sufficient in this case to study the equation $\bar{\partial} g = f$ for $f$ with values in a holomorphic line bundle which is the case in \cite{a1}. In fact, the main aim of this paper is to establish the following existence theorem with $L^p$-estimates: \begin{theorem}[Global theorem] \label{thm1} Let $M$ be a complex manifold of complex dimension $n$ and let $E \rightarrow M$ be a holomorphic vector bundle, of rank $N$, over $M$. Let $D \Subset M$ be a strongly pseudo-convex domain with smooth $C^{4}$-boundary. Then \noindent (1) If the holomorphic vector bundle $E$ is Nakano positive, then, there exists an integer $k_{0} = k_{0}(D) > 0$ such that for any $f \in L^{1}_{0,s}(D,E^{k})$; $\bar{\partial} f= 0$, $s \geq 1$ and $k \geq k_{0}$ there is a form $g =T^{s}_{N^{k}} f \in L^{1}_{0,s-1}(D,E^{k})$ satisfies $\bar{\partial} g = f$, where $T^{s}_{N^{k}}$ is a bounded linear operator and $E^{k} = E \otimes E \otimes \dots \otimes E$ ($k$-times). Moreover, if $f \in L^p_{0,s}(D,E^{k})$; $1 \leq p \leq \infty$, there is a constant $C^{k}_{s}$ such that $\| g \|_{L^p_{0,s-1}(D,E^{k})} \leq C_{s}^{k} \| f \|_{L^p_{0,s}(D,E^{k})}$. The constant $C^{k}_{s}$ is independent of $f$ and $p$. If $f$ is $C^{\infty}$, then $g$ is also $C^{\infty}$. \noindent (2) If $M$ is a Stein manifold, then, for any $f \in L^{1}_{r,s}(D)$; $\bar{\partial} f = 0$, $0 \leq r \leq n$, and $s \geq 1$, there is a form $g = T^{s} f \in L^{1}_{r,s-1}(D)$ such that $\bar{\partial} g = f$, where $T^{s}$ is a bounded linear operator. Moreover, if $f \in L^p_{r,s}(D)$; $1 \leq p \leq \infty$, we have $\| g \| _{L^p_{r,s-1}(D)} \leq C_{s} \| f\|_{L^p_{r,s}(D)}$. The constant $C_{s}$ is independent of $f$ and $p$. If $f$ is $C^{\infty}$, then $g$ is also $C^{\infty}$. \end{theorem} The plan of this paper is as follows: In section 1, we state the main theorem. In section 2, we set the notation and recall some useful facts. In section 3, we prove an existence theorem with $L^{2}-$ estimates. In section 4, we give local solution for the $\bar{\partial}$-equation with $L^p$-estimates for $1 \leq p \leq \infty$. In section 5, we prove the existence theorem with $L^p$-estimates. \section{Notation and Preliminaries} Let $M$ be an $n$-dimensional complex manifold and let $\pi : E \rightarrow M$ be a holomorphic vector bundle, of rank $N$, over $M$. Let $\{u_{j}\}$; $j \in I$, be an open covering of $M$ consisting of coordinates neighborhoods $u_{j}$ with holomorphic coordinates $z_{j} = (z_{j}^{1},z_{j}^{2},\dots,z_{j}^{n})$ over which $E$ is trivial, namely $\pi^{-1}(u_{j}) = u_{j} \times \mathbb{C}^{N}$. The $N$-dimensional complex vector space $E_{z} = \pi^{-1}(z)$; $z \in M$, is called the fiber of $E$ over $z$. Let $h = \{h_{j}\}$; $h_{j}=(h_{j \mu \bar{\eta}})$ be a Hermitian metric along the fibers of $E$ and let $(h_{j}^ {\mu\bar{\eta}})$ be the inverse matrix of $(h_{j \mu \bar{\eta}})$. Let $\theta=\{\theta_{j}\}$; $\theta_{j}=(\theta_{j\mu}^{\nu})$; $\theta_{j\mu}^{\nu}= \partial \log h_{j}= \sum_{\alpha=1}^{n} \sum^{N}_{\eta=1} h_{j}^{\nu \bar{\eta}}\frac{\partial h_{j\mu \bar{\eta}}}{\partial z_{j}^{\alpha}} dz_{j}^{\alpha}=\sum_{\alpha=1}^{n} \Omega_{j\mu\alpha }^{\nu}dz_{j}^{\alpha}$ and $\Theta = \{\Theta_{j}\}$; $\Theta_{j} =(\Theta_{j\mu}^{\nu})$; $\Theta_{j\mu}^{\nu}=\sqrt{-1}\bar{\partial}\partial \log h_{j}= \sqrt{-1} \sum_{\alpha,\beta=1}^{n} \Theta_{j\mu\alpha\bar{\beta}}^{\nu} dz_{j}^{\alpha} \wedge d\bar{z}_{j}^{\beta}$ be the connection and the curvature forms associated to the metric $h$ respectively, where $\Theta_{j\mu\alpha\bar{\beta}}^{\nu}=- \frac{\partial \Omega_{j\mu\alpha }^{\nu}}{\partial d \bar{z}^{\beta}_{j}}$, $1\leq\mu\leq N$; $1\leq\nu\leq N$. The associated curvature matrix is given by $$(H_{j\bar{\eta}\bar{\beta},\nu\alpha})= \big(\sum_{\mu=1}^{N}h_{j\mu\bar {\eta}}\Theta_{j\nu\alpha\bar{\beta}}^{\mu}\big).$$ Let $T(M)$ (resp. $T^{\star}(M)$) be the holomorphic tangent (resp. cotangent) bundle of $M$. \begin{definition} \label{def2.1} \rm $E$ is said to be Nakano positive, at $z \in u_{j}$, if the Hermitian form $$\sum H_{j\bar{\eta}\bar{\beta},\nu\alpha}(z) \zeta_{\alpha}^{\nu}\bar{\zeta}_{\beta}^{\eta}$$ is positive definite for any $\zeta = (\zeta^{\nu}_{\alpha}) \in E_{z} \otimes T_{z}(M)$; $\zeta \neq 0$. \end{definition} The notation $X \Subset M$ means that $X$ is an open subset of $M$ such that its closure is a compact subset of $M$. \begin{definition} \label{def2.2} \rm A domain $D \Subset M$ is said to be strongly pseudo-convex with smooth $C^{4}$-boundary if there exist an open neighborhood $U$ of the boundary $\partial D$ of $D$ and a $C^{4}$ function $\lambda : U \rightarrow \mathbb{R}$ having the following properties: \begin{itemize} \item[(i)] $D \cap U = \{z \in U ; \lambda(z) < 0\}$. \item[(ii)] $\sum_{\alpha,\beta=1}^{n} \frac{\partial^{2} \lambda(z)} {\partial z_{j}^{\alpha} \partial \bar{z}_{j}^{\beta}} \mu_{\alpha} \bar{\mu}_{\beta} \geq L(z) |\mu|^{2}$; $z \in U \cap u_{j}$, $\mu = (\mu_{1},\mu_{2},\dots,\mu_{n}) \in \mathbb{C}^{n}$ and $L(z) > 0$. \item[(iii)] The gradient $\nabla \lambda(z) = (\frac{\partial \lambda(z)}{\partial x_{j}^{1}},\frac{\partial \lambda(z)} {\partial y_{j}^{1}},\frac{\partial \lambda(Z)}{\partial x_{j}^{2}},\frac{\partial \lambda(z)}{\partial y_{j}^{2}},\dots,\frac{\partial \lambda(z)}{\partial x_{j}^{n}},\frac{\partial \lambda(z)}{\partial y_{j}^{n}}) \neq 0$ for $z = (z_{j}^{1},z_{j}^{2},\dots,z_{j}^{n}) \in u_{j} \cap U$; $z_{j}^{\alpha} = x_{j}^{\alpha} + i y_{j}^{\alpha}$. \end{itemize} \end{definition} Let $\gamma = (\mu_{1},\nu_{1},\dots,\mu_{n},\nu_{n})$ be any multi-index and $|\gamma| = \sum_{i=1}^{n}(\mu_{i}+\nu_{i})$, where $\mu_{i}$ and $\nu_{i}$ are non-negative integers. Let $D^{\gamma} = \partial^{|\gamma|}/\partial x_{1}^{\mu_{1}}\partial y_{1}^{\nu_{1}}\dots\partial x_{n}^{\mu_{n}}\partial y_{n}^{\nu{n}}$. \begin{remark} \label{rmk2.3}\rm By shrinking $U$ we can assume that $U \Subset \tilde{U}$, where $\tilde{U}$ is an open, $\lambda$ is $C^{4}$ on $\tilde{U}$ and the properties (i), (ii) and (iii) of Definition \ref{def2.2} hold on $\tilde{U}$. Thus, we can choose a neighborhood $V$ of $\partial D$ such that $V \Subset U$ and for any $z \in V$ there exist positive constants $L$, $F$ and $F^{'}$ satisfy $L(z)> L$, $|\nabla \tilde{\lambda}(z)| \geq F$ and $|D^{\gamma} \tilde{\lambda}(z)| \leq F{'} < \infty$ for any multi-index $\gamma$ with $|\gamma| \leq 4$, where $\tilde{\lambda}$ is the a slight perturbation of $\lambda$. \end{remark} \begin{definition} \label{def2.4}\rm Let $X$ be an $n$-dimensional complex manifold and let $\Phi$ be an exhaustive function on $X$, that is, the sets $X_{c} = \{z \in X ; \Phi(z) < c\} \Subset X$; $c \in \mathbb{R}$ and $X = \cup\, X_{c}$. We say that $X$ is weakly $1$-complete (resp. Stein) manifold if $\Phi$ is a $C^{\infty}$ plurisubharmonic (resp. strictly plurisubharmonic), that is, if $\sum_{\alpha,\beta=1}^{n} \frac{\partial^{2} \Phi(z)} {\partial z_{j}^{\alpha} \partial\bar{z}_{j}^{\beta}}\mu^{\alpha}\bar{\mu}^{\beta}$ is positive semi-definite (resp. positive definite) on $X$ for $\mu=(\mu^{1}, \dots,\mu^{n}) \in \mathbb{C}^{n}$; $\mu\neq 0$. \end{definition} We will use the standard notation of H\"ormander \cite{o1} for differential forms. Thus a $\mathbb{C}$-valued differential form $\varphi = \{\varphi_{j}\}$ of type $(r,s)$ on $M$ can be expressed, on $u_{j}$, as $\varphi_{j}(z) = \sum_{A_{r},B_{s}} \varphi_{jA_{r}B_{s}} (z) dz_{j}^{A_{r}} \wedge d\bar{z}_{j}^{B_{s}}$, where $A_{r}$ and $B_{s}$ are strictly increasing multi-indices with lengths $r$ and $s$, respectively. An $E$-valued differential form $\varphi$ of type $(r,s)$, on $M$, is given locally by a column vector ${ }^{t}\varphi_{j} =(\varphi^{1}_{j},\varphi^{2}_{j},\dots,\varphi^{N}_{j})$ where $\varphi_{j}^{a}$, $1 \leq a \leq N$, are $\mathbb{C}$-valued differential forms of type $(r,s)$ on $u_{j}$. $\Lambda^{r,s}(M)$ denotes the space of $\mathbb{C}$-valued differential forms of type $(r,s)$ and of class $C^{\infty}$ on $M$. Let $\Lambda^{r,s}(M,E)$ (resp. $\mathcal{D}^{r,s}(M,E)$) be the space of $E$-valued differential forms (resp. with compact support) of type $(r,s)$ and of class $C^{\infty}$ on $M$. Let $h^{0} = \{h^{0}_{j}\}$, $h^{0}_{j} = (h^{0}_{j\mu\bar{\eta}})$, be the initial Hermitian metric along the fibers of $E$ and let $\Theta^{0} = \{\Theta_{j}^{0}\}$ be the associated curvature form. The induced Hermitian metric along the fibers of the line bundle $B = \bigwedge^{N}E$ is given by the system of positive $C^{\infty}$ functions $\{a^{0}_{j}\}$, where $a^{0}_{j} = \det (h_{j\mu\bar{\eta}}^{0})^{-1}$. Hence, the system $\{1/a^{0}_{j}\}$ also defines a Hermitian metric along the fibers of $B$ whose curvature matrix $(H_{j\bar{\eta}\bar{\beta},\nu\alpha})$ is given by $(1/a_{j}^{0}) (\partial^{2} \log a^{0}_{j}/\partial z_{j}^{\alpha}\partial \bar{z}_{j}^{\beta})$. If $E$ is Nakano positive, with respect to $h^{0}$, then $B$ is positive, with respect to $\{1/a_{j}^{0}\}$, that is, the Hermitian matrix $(\partial^{2} \log a_{j}^{0}/\partial z_{j}^{\alpha}\partial \bar{z}_{j}^{\beta})$ is positive definite. Hence, $$ds_{0}^{2} = \sum_{\alpha,\beta=1}^{n} g_{j\alpha\bar{\beta}}^{0}\, dz_{j}^{\alpha} d\bar{z}_{j}^{\beta}\,\,;\,\,\, g_{j\alpha\bar{\beta}}^{0} = \partial^{2} \log a_{j}^{0}/\partial z_{j}^{\alpha} \partial\bar{z}_{j}^{\beta}$$ defines a K$\ddot{a}$hler metric on $M$. For $\varphi,\psi \in \Lambda^{r,s}(M,E)$, we define a local inner product, at $z \in u_{j}$, by $$\label{e2.1} \sum_{\nu,\mu=1}^{N} h^{0}_{j\nu\bar{\mu}} \varphi_{j}^{\nu}(z) \wedge \star \overline{\psi_{j}^{\mu}(z)} = a(\varphi(z),\psi(z)) dv_{0},$$ where the Hodge star operator $\star$ and the volume element $dv_{0}$ are defined by $ds_{0}^{2}$ and $a(\varphi,\psi)$ is a function, on $M$, independent of $j$. Let $L^p_{r,s}(M,E)$ (resp. $L^{\infty}_{r,s}(M,E)$) be the Banach space of $E$-valued differential forms $f$ on $M$, of type $(r,s)$, such that $\| f\|_{L^p_{r,s}(M,E)} = (\int _{M} |f(z)|^p dv_{0})^{1/p} < \infty$ for $1 \leq p < \infty$ (resp. $\| f \|_{L^{\infty}_{r,s}(M,E)} = \mathop{\rm ess\,sup}_{z \in M}|f(z)| < \infty$), where $|f(z)| = \sqrt{a(f(z),f(z))}$. The Hermitian metric along the fibers of $E^{k} = E \otimes E \otimes \dots \otimes E$, associated to $h^{0}$, is defined by $h^{0k} = \{h_{j}^{0k}\}$, where $h_{j}^{0k} = h_{j}^{0} h_{j}^{0}\dots h_{j}^{0}$ ($k$-factors). The transition functions of $K(M)$ are the Jacobian determinant $$k_{ij} = \frac{\partial(z_{j}^{1},z_{j}^{2},\dots,z_{j}^{n})} {\partial (z_{i}^{1},z_{i}^{2},\dots,z_{i}^{n})}$$ on $u_{i} \cap u_{j}$. We see that $|k_{ij}|^{2} = g_{i} g_{j}^{-1}$ on $u_{i} \cap u_{j}$, where $\,\,g_{i} = \det ({\partial^{2}\log a_{i}^{0}}/ {\partial z_{i}^{\alpha} \partial \bar{z}_{i}^{\beta}})\,\,$. Therefore, the system of positive $C^{\infty}$ functions $\{g^{-1}_{j}\}$ (resp. $g = \{g_{j}\}$) determines a Hermitian metric along the fibers of $K(M)$ (resp. the dual bundle $K^{-1}(M)$). \section{Existence Theorems with $L^{2}$-Estimates} Let $Y \Subset M$ be weakly $1$-complete domain of $M$ with respect to a plurisubharmonic function $\Phi$ and $\lambda (t)$ be a real $C^{\infty}$ function on $\mathbb{R}$ such that $\lambda (t) > 0$, $\lambda'(t) > 0$ and $\lambda''(t) > 0$ for $t > 0$ and $\lambda(t) = 0$ for $t \leq 0$. Let $h_{j} = e^{- \lambda (\Phi)} h_{j}^{0}$, on $u_{j} \cap Y$, and $a_{j} = \det(h_{j})^{-1}$. Thus, the Hermitian matrix $({\partial^{2} \log a_{j}}/ {\partial z_{j}^{\alpha} \partial\bar{z}_{j}^{\beta}})$ is positive definite on $u_{j} \cap Y$. Hence, $$ds^{2} = \sum_{\alpha,\beta=1}^{n} g_{j\alpha\bar{\beta}}\, dz_{j}^{\alpha} d\bar{z}_{j}^{\beta}\,\,;\,\,\, g_{j\alpha\bar{\beta}} = \partial^{2} \log a_{j}/\partial z_{j}^{\alpha} \partial\bar{z}_{j}^{\beta}$$ defines a K\"ahler metric on $Y$. The Hermitian metrics $h^{k} = \{h^{k}_{j}\}$ and $g$ induce a Hermitian metric $b^{k} = \{h_{j}^{k} g_{j}\}$; $k \geq 1$, along the fibers of $K^{-1}(M)\otimes E^{k}|_{Y}$, where $h_{j}^{k} = h_{j} h_{j} \dots h_{j}$ ($k$-factors). Let $L^{2}_{r,s}(Y,K^{-1}(M) \otimes E^{k},{\rm loc},g h^{0k},ds_{0}^{2})$ be the space of all $K^{-1}(M) \otimes E^{k}-$ valued differential forms of type $(r,s)$ which has measurable coefficients and square integrable on compact subsets of $Y$ with respect to $ds_{0}^{2}$ and $g h^{0k}$. For $\varphi,\psi \in \Lambda^{r,s}(Y,K^{-1}(M) \otimes E^{k})$ we define a local inner product $a(\varphi(z),\psi(z))_{k}dv$ by replacing $g_{j} h_{j}^{k}$ and $ds^{2}$ instead of $h_{j}^{0}$ and $ds_{0}^{2}$, respectively, in \eqref{e2.1}. For $\varphi$ or $\psi \in \mathcal{D}^{r,s}(Y,K^{-1}(M) \otimes E^{k})$, we define a global inner product by $$\label{e3.1} \langle \varphi,\psi\rangle_{k} = \int _{Y} a(\varphi,\psi)_{k}\,\, dv.$$ Let $\omega = \sqrt{-1} \sum_{\alpha,\beta=1}^{n} g_{j\alpha\bar{\beta}} dz_{j}^{\alpha} \wedge d\bar{z}_{j}^{\beta}$ be the fundamental form of $ds^{2}$ and let $L = e(\omega)$ be the wedge multiplication by $\omega$. Let $\Gamma : \Lambda^{r,s}(Y,K^{-1}(M) \otimes E^{k})\rightarrow \Lambda^{r-1,s-1}(Y,K^{-1}(M) \otimes E^{k})$ be the operator locally defined by $\Gamma= (-1)^{r+s} \star L \star$, where the $\star$ operator is defined by $ds^{2}$. Let $\vartheta_{k}$ be the formal adjoint of $\bar{\partial}: \Lambda^{r,s}(Y,K^{-1}(M) \otimes E^{k})\rightarrow \Lambda^{r,s+1}(Y,K^{-1} \otimes E^{k})$ with respect to the inner product \eqref{e3.1} and $\Box_{k} =\bar{\partial} \vartheta_{k} + \vartheta_{k} \bar{\partial}$ be the Laplace-Beltrami operator. The curvature form associated to $b^{k}$ is given by $$\Theta^{k} = \{\Theta_{j}^{k}\}; \Theta_{j}^{k} = \sqrt{-1} \bar{\partial} \partial \log b_{j}^{k} =k \Theta_{j}^{0} + \sqrt{-1}(k \partial\bar{\partial} \lambda (\Phi) - \partial\bar{\partial} \log g_{j}).$$ Since the Levi form $\sqrt{-1} \partial\bar{\partial} \lambda (\Phi)$ is positive semi-definite, $E$ is Nakano positive with respect to $h^{0}$ and $\bar{Y}$ is compact subset of $M$, there exists an integer $k_{0} = k_{0}(Y) > 0$ such that $K^{-1}(M) \otimes E^{k}|_{Y}$ is Nakano positive, with respect to $b^{k}$, for $k \geq k_{0}$. Hence as in Nakano \cite{n1} we can prove the following lemma: \begin{lemma} \label{lm3.1} Let $f \in L^{2}_{n,s}(Y,K^{-1}(M) \otimes E^{k},{\rm loc},g h^{0k},ds_{0}^{2})$; $k \geq k_{0}$, $s \geq 1$ be given, then we can choose the function $\lambda (t)$ such that $ds^{2}$ is complete, $\langle f,f\rangle_{k} < \infty$, and there is a constant $c > 0$ such that $$\label{3.2} \langle \bar{\partial}\varphi,\bar{\partial} \varphi\rangle _{k} +\langle \vartheta_{k} \varphi,\vartheta_{k} \varphi\rangle _{k} \geq c \langle \varphi,\varphi\rangle_{k},$$ for any $\varphi\in \mathcal{D}^{n,s}(Y,K^{-1}(M) \otimes E^{k})$. \end{lemma} \begin{remark} \label{rmk3.2} We note that when $E$ is a line bundle Lemma 3.1 is valid for forms in $\mathcal{D}^{r,s}(Y,K^{-1}(M) \otimes E^{k})$ with $r+s\geq n+1$. \end{remark} From Lemma 3.1 and the Hilbert space technique of H\"ormander \cite{o1}, as in the proof of \cite[Theorem 2.1]{a1}, we can prove the following theorem: \begin{theorem} \label{thm3.3} Let $Y \Subset M$ be weakly $1$-complete domain and let $E \rightarrow M$ be a holomorphic vector bundle over $M$. If $E$ is Nakano positive, over $M$, then for any $f \in L^{2}_{n,s}(Y,K^{-1}(M) \otimes E^{k},b^{k},ds^{2})$ with $\bar{\partial} f = 0$ , $s \geq 1$ and $k \geq k_{0}$ there exists a form $g =T f \in L^{2}_{n,s-1}(Y,K^{-1}(M) \otimes E^{k},b^{k},ds^{2})$ satisfies $\bar{\partial} g = f$ and two constants $C = C(Y)$ and $c_{k} = c_{k}(G,Y)$ such that \begin{gather*} \| g \|_{L^{2}_{n,s-1}(Y,K^{-1}(M) \otimes E^{k},b^{k},ds^{2})} \leq C \| f \|_{L^{2}_{n,s}(Y,K^{-1}(M) \otimes E^{k},b^{k},ds^{2})}, \\ \| g \|_{L^{2}_{n,s-1}(G,K^{-1}(M) \otimes E^{k})} \leq c_{k} \| f \|_{L^{2}_{n,s}(G,K^{-1}(M) \otimes E^{k})}, \end{gather*} where $T$ is a bounded linear operator and $G \Subset Y$. \end{theorem} \section{Local solution for the $\bar{\partial}$-equation with $L^p$-estimates} Let $D\Subset M$ be a strongly pseudo-convex domain with $\lambda$ and $U$ of Definition \ref{def2.2}. Let $x \in \partial D$ be an arbitrary fixed point and let $W_{a}$ be an open neighborhood of $x$ such that $W_{a} \Subset u_{j} \subset U$, for a certain $j \in I$, and $z_{j}(W_{a})$ is the ball $B(0,a)\Subset \mathbb{C}^{n}$, where $(u_{j},z_{j})$ is a holomorphic chart. Then, $W_{a}$ can be considered as strongly pseudo-convex domain in $\mathbb{C}^{n}$ and the volume element $dv_{0}$ can be considered as the Lebesgue measure on $B(0,a)$. \begin{theorem}[\cite{o1}] \label{thm4.1} Let $G \Subset \mathbb{C}^{n}$ be a strongly pseudo-convex domain and $u \in L^{1}_{0,s}(G)$; $s \geq 1$. Then, there exist kernels $K_{s}(\xi,z)$ such that the integral $\int_{G} u(\xi) \wedge K_{s-1}(\xi,z)d\mu(\xi)$ is absolutely convergent for almost all $z \in \bar{G}$ and the operator $T^{s}:L^p_{0,s}(G) \rightarrow L^p_{0,s-1}(G)$, defined by $T^{s} u(z) = \int_{G} u(\xi) \wedge K_{s-1}(\xi,z) d\mu(\xi)$, with norm $\leq c$\,; $1 \leq p \leq \infty$. Moreover, if $\bar{\partial} u = 0$, then, there is a form $g = T^{s}u$ satisfies $\bar{\partial}g=u$, where $d\mu(\xi)$ is the Lebesgue measure on $\mathbb{C}^{n}$. \end{theorem} Now, we extend the operator $T^{s}$ to $L^p_{0,s}(D \cap W_{a},E)$. For this purpose, we define an operator $T^{s}_{N} : f \in L^{1}_{0,s}(D \cap W_{a},E) \rightarrow T^{s}_{N} f \in L^{1}_{0,s-1}(D \cap W_{a},E)$; $s \geq 1$, by $$\label{e4.1} T^{s}_{N} f(z) = \sum_{\lambda=1}^{N} T^{s}f^{\lambda}(z)\,\, b_{\lambda}(z),$$ where $f(z) = \sum_{\lambda=1}^{N} f^{\lambda}(z)b_{\lambda}(z)$, that is, $f^{\lambda}(z)$ are the components of $f|_{u_{j}}$ with respect to an orthonormal basis $b_{\lambda}(z)$ on $E_{z}$; $z \in u_{j}$. We consider the following situation: In the notation of Definition \ref{def2.2}, from Remark \ref{rmk2.3}, let $y \in \partial V^{-}$, where $V^{-} = \{z \in V; \tilde{\lambda}(z) < 0\}$ and let $W_{a}$ be a neighborhood of $y$ such that $W_{a} \Subset u_{j} \subset V$, for a certain $j \in I$, and $z_{j}(W_{a})$ is the ball $B(0,a) \subset \mathbb{C}^{n}$, $a \leq \tilde{a}$, where $\tilde{a}$ depends continuously on $L$, $F$, $F^{'}$ and the distance $d(y,{\it C} V)$ from $y$ to the complement of $V$. In the above notation, as the local theorem in \cite{k1}, we can prove the following theorem: \begin{theorem}[Local theorem] \label{thm4.3} Let $T^{s}_{N^{k}}$ be the linear operator defined by \eqref{e4.1} and let $f \in L^{1}_{0,s}(V^{-},E^{k})$; $\bar{\partial} f = 0$, where $N^{k}$ is the rank of $E^{k}$. Then, there is a form $g = T^{s}_{N} f \in L^{1}_{0,s-1}(V^{-} \cap W_{a},E^{k})$ such that $\bar{\partial} g = f$. If $f$ is $C^{\infty}$, then so is $g$. If $f \in L^p_{0,s}(V^{-},E^{k})$, then $g \in L^p_{0,s-1}(V^{-} \cap W_{a},E^{k})$ and satisfies $$\| g \|_{L^p_{0,s-1}(V^{-} \cap W_{a},E^{k})} \leq C \| f \|_{L^p_{0,s}(V^{-},E^{k})};\,\, 1 \leq p \leq \infty,$$ where $C = C(s,k,N)$ is a constant which depends continuously on $L$, $F$, $F^{'}$ and $a$. \end{theorem} \section{Global solution for the $\bar{\partial}$-equation with $L^p$-estimates} The local result yields Lemma 5.1 (An extension lemma) which in turn enables one to solve $\bar{\partial}\eta=\hat{f}$ (with bounds) in a strongly pseudoconvex domain $\hat{D}$ which is larger than $D$, $\bar{D}\subseteq \hat{D}$. Here we make use of the $L^{2}-$estimates for solutions of the $\bar{\partial}-$equation as presented in Theorem \ref{thm3.3}. \begin{lemma}[An extension lemma] \label{lm5.1} Let $D \Subset M$ be a strongly pseudo-convex domain with smooth $C^{4}$-boundary. Then, there exists another slightly larger strongly pseudo-convex domain $\hat{D} \Subset M$ with the following properties: $\bar{D} \Subset \hat{D}$, for any $f \in L^{1}_{0,s}(D,E^{k})$ with $s \geq 1$ and $\bar{\partial} f = 0$, there exist two bounded linear operators $L_{1}$, $L_{2}$, a form $\hat{f} = L_{1} f \in L^{1}_{0,s}(\hat{D},E^{k})$ and a form $u = L_{2} f \in L^{1}_{0,s-1}(D,E^{k})$ such that: \begin{itemize} \item[(i)] $\bar{\partial}\hat{f} = 0$ in $\hat{D}$. \item[(ii)] $\hat{f} = f - \bar{\partial} u$ in $D$. \item[(iii)] If $f \in L^p_{0,s}(D,E^{k})$, then $\hat{f} \in L^p_{0,s}(\hat{D},E^{k})$ and $u \in L^p_{0,s-1}(D,E^{k})$ with the estimates \begin{gather} \label{e5.1} \| \hat{f} \|_{L^p_{0,s}(\hat{D},E^{k})} \leq C_{1} \| f \|_{L^p_{0,s}(D,E^{k})}, \\ \| u \|_{L^p_{0,s-1}(D,E^{k})} \leq C_{2} \| f \|_{L^p_{0,s}(D,E^{k})}\,\quad 1 \leq p \leq \infty, \label{e5.2} \end{gather} where the constants $C_{1}$ and $C_{2}$ are independent of $f$ and $p$. \end{itemize} If $f$ is $C^{\infty}$ in $D$, then $\hat{f}$ is $C^{\infty}$ in $\hat{D}$ and $u$ is $C^{\infty}$ in $D$. \end{lemma} Since $\partial D$ is compact, we can Cover $\partial D$ by finitely many neighborhoods $W_{i, a_{i}}$ of $x_{i} \in \partial D$, $i = 1,2,\dots,m$, such that for each $x_{i}$ we have $W_{i, a_{i}}\Subset u_{j}\Subset V\Subset U$ for a certain $i\in I$. Put $a=\min_{1\leq i\leq m}a_{i}$. Then as Lemma 2.3.3 and the Claim on page 321 in Kerzman \cite{k1} (see also \cite[Proposition 3.2]{a1}), we can prove the following proposition: \begin{proposition} \label{prop5.2} Let $\hat{D}$ be as in the extension lemma and let $W_{i,a}$ be an open set of $\hat{D}$ such that $W_{i,a} \Subset u_{j} \subset \hat{D}$, for a certain $j \in I$ and $z_{j}(W_{i,a})$ is the ball $B(0,a) \Subset \mathbb{C}^{n}$. Then, for any $f \in L^{1}_{0,s}(W_{i,a},E^{k})$; $\bar{\partial} f = 0$ there is $\alpha = T f \in L^{1}_{0,s-1}(W_{i,a/2},E^{k})$ such that $\bar{\partial} \alpha = f$, where $T$ is a bounded linear operator. If $f \in L^p_{0,s}(W_{i,a},E^{k})$; $1 \leq p \leq 2$, then, we have $\alpha \in L^{p+1/4n}_{0,s-1}(W_{i,a/2},E^{k})$ and $$\| \alpha \|_{L^{p+1/4n}_{0,s-1}(W_{i,a/2},E^{k})} \leq c \| f \|_{L^p_{0,s}(W_{i,a},E^{k})},$$ and for any $p$, $1 \leq p \leq \infty$, we have $$\| \alpha \|_{L^p_{0,s-1}(W_{i,a/2},E^{k})} \leq c \| f \|_{L^p_{0,s}(W_{i,a},E^{k})},$$ where $c = c(n,a,k,N)$ is a constant independent of $f$ and $p$. \end{proposition} The proof of Proposition \ref{prop5.2} is purely local. Using Proposition \ref{prop5.2}, as \cite[Proposition 3.2]{a1}, we prove the following proposition: \begin{proposition} \label{prop5.3} Let $\hat{D}$ be as in the extension lemma. Then, there is a strongly pseudo-convex domain $D_{1} \Subset \hat{D}$ such that for every $\hat{f} \in L^{1}_{0,s}(\hat{D},E^{k})$; $\bar{\partial}\hat{f} = 0$, there are two bounded linear operators $L_{1}$ and $L_{2}$ and two forms $f_{1} = L_{1} \hat{f} \in L^{1}_{0,s}(D_{1},E^{k})$ and $\eta_{1} = L_{2} \hat {f} \in L^{1}_{0,s-1}(D_{1},E^{k})$ such that: \begin{itemize} \item[(i)] $\bar{\partial} f_{1} = 0$ on $D_{1}$, \item[(ii)] $\hat{f} = f_{1} + \bar{\partial} \eta_{1}$ on $D_{1}$, \item[(iii)] $\| f_{1}\|_{L^{p+1/4n}_{0,s}(D_{1},E^{k})} \leq c \| \hat{f} \|_{L^p_{0,s}(\hat{D},E^{k})}$ for $\hat{f} \in L^p_{0,s}(\hat{D},E^{k})$; $1 \leq p \leq 2$, \item[(iv)] For every open set $W \Subset D_{1}$ and for every $p$, $1 \leq p \leq \infty$, we have \begin{gather*} \| f_{1} \|_{L^p_{0,s}(W,E^{k})} \leq c \| \hat{f} \|_{L^p_{0,s}(\hat{D},E^{k})}, \\ \| \eta_{1} \|_{L^p_{0,s-1}(W,E^{k})} \leq c \| \hat{f}\|_{L^p_{0,s}(\hat{D},E^{k})}, \end{gather*} where $c = c(\hat{D},W,n,k,N)$ is a constant independent of $\hat{f}$ and $p$. \end{itemize} \end{proposition} Since every strongly pseudo-convex domain is weakly $1$-complete and noting that $\Lambda^{n,s}(D,K^{-1}(M) \otimes E^{k}) \equiv \Lambda^{0,s}(D,E^{k})$; $k \geq 1$. Then, using Theorem \ref{thm3.3}, Proposition \ref{prop5.3}, and the interior regularity properties of the $\bar{\partial}$-operator, as \cite[Theorem 3.1]{a1}, we prove the following theorem: \begin{theorem} \label{thm5.4} Let $\hat{D}$ be the strongly pseudo-convex domain of the extension lemma and $W \Subset \hat{D}$. Then, for any form $\hat{f} \in L^{1}_{0,s}(\hat{D},E^{k})$ with $\bar{\partial} \hat{f} = 0$, there exists a form $\eta \in L^{1}_{0,s-1}(W,E^{k})$, $\eta = T \hat{f}$ such that $\bar{\partial} \eta = \hat{f}$, where $T$ is a bounded linear operator. If $\hat{f} \in L^p_{0,s}(\hat{D},E^{k})$ with $1 \leq p \leq \infty$ and $k \geq k_{0}$, then $\eta \in L^p_{0,s-1}(W,E^{k})$ and $$\| \eta \|_{L^p_{0,s-1}(W,E^{k})} \leq C \| \hat{f} \|_{L^p_{0,s}(\hat{D},E^{k})}$$ where $C = C(\hat{D},W,k)$ is a constant independent of $\hat{f}$ and $p$. If $\hat{f}$ is $C^{\infty}$, then $\eta$ is $C^{\infty}$. \end{theorem} \begin{proof} Proposition \ref{prop5.3} yields $D_{1}$. A new application of Proposition \ref{prop5.3} to $D_{1}$ yields $D_{2}$. We iterate $4n$ times and obtain $$\hat{D}\supseteq D_{1}\supseteq D_{2}\supseteq \dots \supseteq D_{4n}\Supset \overline{W}$$ Hence, for any $f\in L^{1}_{0,s}(\hat{D},E^{k}); \Bar{\partial}f=0$, there exist $f_{j}\in L^{1}_{0,s}(D_{j},E^{k})$ and $\upsilon_{j}\in L^{1}_{0,s-1}(D_{j},E^{k})$; $j=1,2,\dots, 4n$. Clearly, we have: $$\hat{f}=f_{1}+\Bar{\partial}\upsilon_{1}= f_{2}+\Bar{\partial}\upsilon_{1} +\Bar{\partial}\upsilon_{2}=f_{3}+\Bar{\partial}\upsilon_{1} +\Bar{\partial}\upsilon_{2}+\Bar{\partial}\upsilon_{3}=\dots=f_{4n} +\Bar{\partial}(\sum_{j=1}^{4n}\upsilon_{j})$$ in $D_{4n}$, $f_{4n}\in L^{2}_{0,s}(D_{4n},E^{k})$ and $\| f_{4n} \|_{L^{2}_{0,s-1}(D_{4n},E^{k})} \leq K \| \hat{f} \|_{L^{1}_{0,s}(\hat{D},E^{k})}$. Now we apply Theorem \ref{thm3.3} with $\hat{D}=D_{4n}$ and $\overline{W}\subset Y\Subset D_{4n}$. Let $\upsilon$ be the solution of $\Bar{\partial}\upsilon=f_{4n}$ obtained from Theorem \ref{thm3.3}, with $$\| \upsilon \|_{L^{2}_{0,s-1}(Y,E^{k})} \leq K \| f_{4n} \|_{L^{2}_{0,s}(D_{4n},E^{k})}\leq K \| \hat{f} \|_{L^{1}_{0,s}(\hat{D},E^{k})}.$$ Set $\eta=\upsilon+\sum_{j=1}^{4n}\upsilon_{j}$, then we obtain $\Bar{\partial}\eta=\Bar{\partial}\upsilon +\Bar{\partial}(\sum_{j=1}^{4n}\upsilon_{j}) =f_{4n}+\Bar{\partial}(\sum_{j=1}^{4n}\upsilon_{j})=\hat{f}$ in $Y$ (hence in $W$). Using $(iv)$ of Proposition \ref{prop5.3}, collecting estimates and the estimates $\| . \|_{L^{1}_{0,s}(\hat{D},E^{k})}\leq K\| . \|_{L^p_{0,s}(\hat{D},E^{k})}$ (since $\hat{D}$ is bounded), we obtain: $$\label{e5.3} \|\eta\|_{L^{1}{0,s-1}(Y,E^{k})}\leq K\| \hat{f}\|_{L^p_{0,s}(\hat{D}, E^{k})},\,\,\,\ 1\leq p\leq \infty.$$ Finally, an application of the interior regularity properties for solutions of the elliptic $\Bar{\partial}-$operator yields $$\|\eta\|_{L^p_{0,s-1}(W,E^{k}}) \leq K(\| \eta \|_{L^{1}_{0,s-1}(Y,E^{k}})+ \| \hat{f}\|_{L^p_{0,s}(Y,E^{k})}),\quad 1\leq p\leq \infty,$$ which together with \eqref{e5.3} give the estimates in Theorem \ref{thm5.4}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] Let $\hat{D}\supseteq \bar{D}$ be the strongly pseudo-convex domain furnished by Lemma 5.1 (An extension lemma). If $f \in L^{1}_{0,s}(D, E^{k})$ with $s \geq 1$ and $\bar{\partial} f = 0$, then Lemma 5.1 yields a form $\hat{f} = L_{1} f \in L^{1}_{0,s}(\hat{D},E^{k})$ and a form $u = L_{2} f \in L^{1}_{0,s-1}(D,E^{k})$ such that: $\bar{\partial}\hat{f} = 0$; $\hat{f} = f - \bar{\partial} u$ in $D$, and $(i)$, $(ii)$, $(iii)$, \eqref{e5.1}, \eqref{e5.2} in that lemma are valid. We solve $\bar{\partial} \eta=\hat{f}$ using Theorem \ref{thm5.4} (with $W=D$). Hence, $\eta \in L^{1}_{0,s-1}(D, E^{k})$ and $$\bar{\partial}\eta=\hat{f}=f - \bar{\partial}u \quad\text{in } D.$$ the desired solution is $g=\eta+u$. The estimates in the first part of Theorem \ref{thm1} follows from those in Lemma \ref{lm5.1} and Theorem \ref{thm5.4}. $\eta$ and $u$ are linear in $f$ and they are $C^{\infty}$ if $f$ is $C^{\infty}$. The first part of Theorem \ref{thm1} is proved. \end{proof} Now, we prove the second part of Theorem \ref{thm1}. In fact, Theorem \ref{thm4.3}, Lemma \ref{lm5.1}, Proposition \ref{prop5.2}, and Proposition \ref{prop5.3} are valid if we replace the vector bundle $E^{k}$ by the vector bundle $\bigwedge^{r}T^{\star}(M)$. If $M$ is a Stein manifold, then every strongly pseudo-convex domain of $M$ is also a Stein manifold. Hence, as \cite[Theorem 5.2.4]{h1}, we can prove the following auxiliary theorem: \begin{theorem} \label{thm5.5} Let $M$ be a Stein manifold of complex dimension $n$ and let $D \Subset M$ be strongly pseudo-convex domain. Then, for every $f \in L^{2}_{r,s}(D,E^{k},{\rm loc})$ with $\bar{\partial} f = 0$, $0 \leq r \leq n$ and $s \geq 1$ there exists a form $g = T f \in L^{2}_{r,s-1}(D,E^{k}, {\rm loc})$; $\bar{\partial} g = f$, and a constant $c = c(D)$ such that $$\| g \|_{L^{2}_{r.s-1}(D,E^{k}, {\rm loc})} \leq c \| f\|_{L^{2}_{r,s}(D,E^{k}, {\rm loc})},$$ where $T$ is a bounded linear operator. Moreover, for any $G \Subset D$ there exists a constant $c_{1} = c_{1}(G,D)$ such that $$\| g\|_{L^{2}_{r,s-1}(G,E^{k}, {\rm loc})} \leq c_{1} \| f\|_{L^{2}_{r,s}(D,E^{k})}.$$ \end{theorem} Then, we can apply the result of Theorem \ref{thm5.5} instead of that of Theorem \ref{thm3.3}, we conclude that Theorem \ref{thm5.4} is valid if we replace $E^{k}$ by $\bigwedge^{r}T^{\star}(M)$; $0 \leq r \leq n$. Using this result and the identity $$\Lambda^{r,s}(M)\equiv \Lambda^{0,s}(M, \wedge^{r}T^{\star}(M)),\quad 1\leq r\leq n$$ we obtain the second part of our results. \begin{thebibliography}{00} \bibitem {a1} O. Abdelkader, \emph{$L^p$-estimates for solution of $\bar{\partial}$-equation in strongly pseudo-convex domains}, Tensor. N. S, {\bf 58}, (1997), 128-136. \bibitem {h1} L. H\"ormander, \emph{An Introduction to Complex Analysis in Several Variables}, Van Nostrand, Princeton. N. J., (1990). \bibitem{k1} N. 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