% May 2, 2003. Final version. \input amstex \documentstyle{amsppt}\magnification=\magstep1 \magnification=1200 %\hcorrection{0.25in} %\vcorrection{-0.8in} \NoRunningHeads \NoBlackBoxes \baselineskip=12pt %\parindent=10pt \vsize=8.5truein \hsize=6.0truein \hoffset=0.3truein \voffset=0.2truein \loadmsam \loadbold \loadmsbm \loadeufm \loadeusm \UseAMSsymbols %\font\boldtitlefont=cmb10 scaled\magstep2 \font\sectionfont=cmb10 %scaled\magstep1 \NoRunningHeads \nologo \document \define\({\left(} \define\){\right)} \define\[{\left[} \define\]{\right]} \define\db{\bar\partial} \define\p{\parallel} \define\aint{\rlap{\kern2pt\vrule height2.5pt width9.5pt depth-2.1pt} \int_{B_{\rho}}} \nologo \topmatter \title Estimates for the $\db$-Neumann problem and nonexistence of Levi-flat hypersurfaces in $\Bbb CP^n$ \endtitle \author Jianguo Cao, Mei-Chi Shaw and Lihe Wang \endauthor % \address Jianguo Cao, %Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 %USA \endaddress %\email cao.7$\@$nd.edu\endemail %\address Mei-Chi Shaw, %Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 %USA \endaddress %\email mei-chi.shaw.1\@nd.edu \endemail %\address Lihe Wang, % Department of Mathematics, University of Iowa, % Iowa City, IA 52242 % USA % \endaddress %\email lwang\@math.uiowa.edu \endemail \author Jianguo Cao\footnote{ Partially supported by NSF grant DMS-0102552. J. Cao is grateful to the University of Michigan for the hospitality. Research at MSRI is supported in part by NSF grant DMS-9810361. \hfill{$\,$} }, Mei-Chi Shaw\footnote{ Partially supported by NSF grant DMS 01-00492. \hfill{$\,$} } and Lihe Wang\footnote{ Partially supported by NSF grant DMS 01-00679. \hfill{$\,$} } \endauthor \address Jianguo Cao, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 USA \endaddress \email cao.7$\@$nd.edu\endemail \address Mei-Chi Shaw, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 USA \endaddress \email mei-chi.shaw.1\@nd.edu \endemail \address Lihe Wang, Department of Mathematics, University of Iowa, Iowa City, IA 52242 USA \endaddress \email lwang\@math.uiowa.edu \endemail \abstract Let $\Omega$ be a {\it pseudoconvex } domain with $C^2$-smooth boundary in $\Bbb CP^n$. We prove that the $\db$-Neumann operator $N$ exists for $(p,q)$-forms on $\Omega$. Furthermore, there exists a $t_0>0$ such that the operators $N$, $\db^*N$, $\db N$ and the Bergman projection are regular in the Sobolev space $W^t (\bar{\Omega}) $ for $t 0$. \endabstract \endtopmatter \heading {\bf Introduction} \endheading One of the main results in this paper is the following: \proclaim{Theorem 1} There exists no $C^{2,\alpha}$ real Levi-flat hypersurface $M^{2n-1}$ in $\Bbb CP^n$, where $ 0<\alpha<1$ and $n\ge 2$. \endproclaim Theorem 1 is inspired by the recent papers of Siu [Siu2,3] who proved that there exists no $C^8$ Levi-flat hypersurface in $\Bbb C P^n$, $n\ge 2$. The required smoothness has been reduced to $C^4$ by Iordan [Io]. Nonexistence of real analytic Levi-flat hypersurfaces in $\Bbb CP^n$ was obtained in Lins Neto [LNe] for $n\ge 3$ and Ohsawa [Oh] for $n=2$. Our proof of Theorem 1 follows arguments along the lines of \cite{Siu2,3} who reduced the proof of Theorem 1 to the regularity of the tangential Cauchy-Riemann equations on $M$. We derive new boundary regularity results for $\db$-equation and the $\db$-Neumann problem for domains in $\Bbb CP^n$. Our Theorem 2 stated below can be applied to {\it not only} domains $\Omega$ with Levi-flat boundaries $M $ {\it but also all} other weakly pseudoconvex $C^2$ domains in $\Bbb CP^n$, when compared with the earlier work mentioned above. The boundary regularity of the solutions is interesting itself. To prove Theorem 1, Siu [Siu2,3] made the following observation: If there exists a $C^2$-smooth {\it real } Levi-flat hypersurface $M = M^{2n-1}$ in $\Bbb CP^n$ and if $\Bbb CP^n$ has the standard Fubini-Study metric, then the curvature form $i\tilde{\Theta}^N$ of its complex normal line bundle $\Cal N_{M, \Bbb CP^n}$ is {\it strictly positive definite} on $ T^{(1, 0)}(M) \oplus T^{(0, 1)}(M) $ by the Cartan-Chern-Gauss structure equation. Furthermore, one has $\tilde{\Theta}^N= d\theta$ is an exact form, where $\theta$ is the connection form of the complex normal line bundle $\Cal N_{M, \Bbb CP^n}$. If $M$ is a Levi-flat hypersurface, it is locally foliated by complex manifolds of complex dimension $n-1$. It is known that the restriction of $\tilde{\Theta}^N$ to each complex leaf of $M$, $\Theta_b = \tilde{\Theta}^N|_{[T(M)]_{\Bbb R} \cap J [T(M)]_{\Bbb R} }$ is a $(1, 1)$-form, where $J$ is the complex structure of $\Bbb CP^n$ (see Proposition 1.2 below). On the other hand, if one could show that $\Theta_b $ is $\partial_b\db_b$-exact for some continuous real-valued function $h$ (i.e., $\Theta_b= \partial_b\db_b h$ on $M$), then $i\Theta_b$ is {\it non-positive } at the maximum point of h on $M$ (cf. [Ca]). This contradicts the fact that $i\tilde{\Theta}^N$ is {\it strictly positive definite} on $ T^{(1, 0)}(M) \oplus T^{(0, 1)}(M) $, and hence Theorem 1 would follow immediately. Therefore, the proof of Theorem 1 is reduced to a problem of finding a continuous function $u $ satisfying $$ i\partial_b\db_b u = f_b = f|_{ [T(M)]_{\Bbb R} \cap J [T(M)]_{\Bbb R} } \quad \text{ on } M, \tag0.1$$ under the condition that $f = d\theta$ is an exact real-valued (1, 1)-form when restricted to $T^{(1,0)}(M) \oplus T^{(0,1)}(M)$. Equation (0.1) corresponds to the classical $\partial\db$-Lelong equation $$ i \partial\db \tilde u = \tilde f \quad \text{ in } \Bbb CP^n \tag0.2 $$ where $\tilde f = d \tilde \theta$ is an exact real-valued (1, 1)-form. Using the fact that $\tilde f$ is an exact real-valued (1,1)-form to solve equation $(0.2)$, it suffices to solve $$\db\tilde u=\tilde \theta^{(0, 1)} \quad \text{in } \Bbb CP^n,\tag 0.3$$ where $\tilde \theta^{(0, 1)} $ is the $(0,1)$ part in $\tilde\theta$, which is $\db$-closed in $\Bbb CP^n$. By the Hodge theory, % we observe that $L^2_{(0,1)}(\Bbb CP^n)\cap \text{Ker}(\triangle) = 0$. there is no nontrivial harmonic $(0,1)$-form in $\Bbb CP^n $. It follows that any $\db$-closed (0,1)-form $ \tilde \theta^{(0, 1)} $ on $\Bbb CP^n$ must be $\db$-exact, and hence equation (0.3) can be solved easily (e.g., cf. \cite{Zh}). For the proof of Theorem 1 and equation (0.1), one can similarly deduce that it suffices to solve $$\db_b u= \theta_b \quad \text{in } M,\tag 0.4$$ where $ \theta_b$ is a $(0,1)$-form in $M$ satisfying some compatibility condition. In \cite{Siu2,3}, existence and regularity of equation (0.4) on a Levi-flat boundary $M$ in $\Bbb CP^n$ is studied. In this paper, we study the more general situation when $M$ is the $C^2$ boundary of any pseudoconvex domain $\Omega$ in $\Bbb CP^n$. When $n>2$, the compatibility condition for equation (0.4) to be solvable is that $$\db_b \theta_b = 0\quad \text{on }M.\tag 0.5$$ When $n=2$, (0.5) is satisfied trivially and the compatibility condition for equation (0.4) is substituted by the following moment condition: $$\int_M \theta_b\wedge \Psi =0, \tag 0.6$$ where $\Psi$ is any $\db_b$-closed (2,0)-form on $M $ (see (9.2.12a) of [CS, p216].) To show that such a compatibility condition holds for our case with $n = 2$, we also derived several Liouville type theorems which are of independent interest. In particular, we show that on any {\it pseudoconcave} domain $\Omega $ with $C^2$ boundary in $\Bbb CP^n$, there exist no non-zero $L^2$-integrable holomorphic $(p, 0)$-forms on $\Omega $ with $p > 0$, (cf. Proposition 4.5 below). If one can extend a $(0,1)$-form $\theta_b$ satisfying (0.5) or (0.6) from the boundary $M = b\Omega$ to a $\db$-closed form $ \tilde \theta $ on the domain $\Omega$, then (0.4) can be solved by restricting the solution $\tilde u$ of the $\db$-equation (0.3) to $M = b\Omega$. To study the $\db$-closed extension from $M$ to $\Omega$, one can formulate it as a $\db$-Cauchy problem of finding solutions to the $\db$-equation (0.3) with prescribed compact support. When $\Omega$ is strictly pseudoconvex with smooth boundary, the $\db$-closed extension on $\Omega$ was pioneered in the work of Kohn and Rossi [KoR] using the boundary regularity of the $\db$-Neumann operator on strongly pseudoconvex domains. When $\Omega \subset \Bbb C^n$ is only pseudoconvex, the $\db$-closed extension for forms on $M = b\Omega$ to $\Omega$ can also be obtained if one can obtain the boundary regularity for the $\db$-Neumann problem or weighted $\db$-Neumann problem (see \cite{Sh,BSh} or \cite{CS, Chap.9}). In our case, the connected real hypersurface $M$ divides $\Bbb CP^n$ into two connected domains: $\Omega_+$ and $\Omega_-$. Both $\Omega_+$ and $\Omega_-$ are pseudoconvex and have Levi-flat boundaries. Using the boundary regularity of the $\db$-Neumann operator, a two-sided $\db$-closed extension $\tilde{\theta}_\pm$ will be constructed for any form $\theta_b$ satisfying the compatibility condition (0.5) or (0.6) on the Levi-flat hypersurface $M$, where the $\db$-closed extension $\tilde \theta = \tilde{\theta}_\pm$ is defined on the whole space $\Bbb CP^n = \overline{ \Omega_+ \cup \Omega_-}$. We let $N = N_{(p, q)} $ denote the $\db$-Neumann operator on $\Omega$, i.e., the inverse operator for the Laplace operator $\square_{(p, q)}$ acting on $(p,q)$-forms with {\it the $\db$-Neumann boundary condition} on $\Omega$. Let $W^{s}_{(p, q) }(\Omega)$ be the space of all $(p, q)$-forms whose coefficients are in the Sobolev space $W^s(\Omega) $. To prove Theorem 1, we first show that $N = N_{(p, q)}|_{\Omega}$ exists and that the operators $N$, $\db^*N$ and $\db N$ are bounded on $ W^s $ for some $s>0$, when $\Omega \subset \Bbb CP^n$ is a pseudoconvex domain with $C^2$ boundary. To formulate our main regularity result for the $\db$-Neumann operator $N$, we introduce a geometric invariant for any pseudoconvex domain $\Omega \subset \Bbb CP^n$ as follows: \proclaim{Definition 0.1} Let $\Omega$ be a pseudoconvex domain in a K\"ahler manifold with $C^2$-smooth boundary. Let $\delta(x) = \delta_{b\Omega}(x) = d(x, b \Omega)$ be a distance function from the boundary. We call $t_0 = t_0(\Omega)$ the order of plurisubharmonicity for the distance function $\delta_{b\Omega }$ if $$ t_0(\Omega) = \sup \{ 0<\epsilon \le 1 | i\partial\bar{\partial} (- \delta^{\epsilon }) \ge 0 \text{ on } \Omega\}.\tag 0.7$$ \endproclaim When $t_0(\Omega) =1$, the condition implies that there exists a plurisubharmonic defining function on $\Omega$. For a general smooth pseudoconvex domain in $\Bbb C^n$, such a plurisubharmonic defining function does not necessarily exist (cf. \cite{DF2}). However, Diederich-Fornaess \cite{DF1} showed that there exists a $ 0 0$. This corresponds to the case when $ t_0(\Omega) = 1$. For the case $ t_0(\Omega) < 1$ and $\Omega \subset \Bbb C^n$, Kohn [Ko2] and Berndtsson-Charpentier [BC] obtain the Sobolev regularity for the operators $ \bar{\partial}^*N$ and the Bergman projection $ P$. Boas-Straube [BS1] show that regularity for the Bergman projection and the $\db$-Neumann operator are equivalent for $C^\infty$-smooth pseudoconvex domains in $\Bbb C^n$ (and also for smooth pseudoconvex domains in complex manifolds with strongly plurisubharmonic functions in a neighborhood of the boundary). For pseudoconvex domains $ \Omega \subset \Bbb CP^n $, such a smooth strictly plurisubharmonic function $\phi$ might not exist in a neighborhood of the boundary $b \Omega$. Therefore, the proof of Theorem 2 has extra difficulties when compared to the case of domains in $\Bbb C^n$. We remark that our proof of the regularity for the Bergman projection and $\db^*N$ is similar to the proof in \cite{BC}, but the regularity results of the $\db$-Neumann operator $N$ and $\db N$ in Theorem 2 are new, even for domains with $C^2$-smooth boundaries in $\Bbb C^n$. The $W^s(\Omega)$-regularity result for $\db N$ is sufficient and crucial in the proof of Theorem 1. The key observation is that the $\db_b$ equation (0.4), when restricted to each complex foliation leaf of $M$, is elliptic. Thus, any solution for (0.4) has H\"older regularity on each leaf from the elliptic theory. To prove continuity of the solution for equation (0.4), one only needs to prove continuity of the solution in the transversal direction, which can be proved by using the Besov norms and a finite difference scheme. Details are given in Section 5. It should be pointed out that there exist {\it non-smooth } Levi-flat real hypersurfaces in $\Bbb CP^2$. Recall that a $C^1$ or Lipschitz real hypersurface $M^{2n-1}$ is called Levi-flat in a complex manifold $\Bbb CP^{ n}$ if $\Bbb CP^n\setminus M$ consists of two pseudoconvex domains. For example, let $$ M=\{[z_0,z_1,z_2]\in\Bbb CP^2\mid |z_0|=|z_1|\},$$ where $(z_0,z_1,z_2)$ are homogeneous coordinates in $\Bbb CP^2$. Then $M$ is Levi-flat since $\Bbb CP^2\setminus M$ consists of two pseudoconvex domains. The hypersurface $M$ is smooth except at $(0,0,1)$, where $M$ is neither $C^1$ nor Lipschitz (in the sense of a Lipschitz graph locally). It is still an open question if there exist Lipschitz or $C^2$ Levi-flat hypersurfaces in $\Bbb CP^n$. In addition, the positive curvature condition (or the irreducible condition) is needed for the possible generalization of Theorem 1. One can inspect the example $S^1 \times \Bbb CP^1$, which is a $C^\infty$-smooth Levi-flat hypersurface in $\Bbb CP^1 \times \Bbb CP^1$. Clearly, $\Bbb CP^1 \times \Bbb CP^1$ does not admit a K\"ahler metric with positive holomorphic bisectional curvature, (e.g., cf. [HSW]). Any compact K\"ahler manifold with positive holomorphic bisectional curvature must be biholomorphic to $\Bbb CP^n$ (e.g., cf. [Mok]). Although results in this paper are stated for domains in $\Bbb CP^n$, the conclusions of these results remain to be true for corresponding domains in any compact K\"ahler manifold with positive holomorphic bisectional curvature as well. % Using the complex Frobenius theorem (see H\"ormander [H\"o1], Nirenberg %[N]), one can show that a $C^k$ Levi-flat %real hypersurface $M$ is locally $C^{k-1}$-foliated by complex manifolds %of complex dimension %$n-1$ if $k \ge 2$. It follows from a result by Barrett-Fornaess [BaF], %the foliation %is actually $C^k$ for $k\ge 1$. The plan of this paper is as follows: In Section 1 we derive some preliminary results on equidistant real hypersurfaces in $\Bbb CP^n$ for the proof of Theorems 1 and 2. In Section 2 we discuss the $L^2$ existence theorems of $\db$ and the $\db$-Neumann problem on domains in $\Bbb CP^n$. In Section 3 we prove that the $\db$-Neumann operator on pseudoconvex domains with $C^2$ boundary is regular in some Sobolev spaces as stated in Theorem 2. Using the existence and regularity of the $\db$-Neumann operator, one can study the solution of the $\db$-equation with prescribed support, including the case of forms with top degrees. We construct $\db$-closed extensions from the boundary and prove the existence and regularity of equation (0.4) in Section 4. The proof of Theorem 1 is completed in Section 5. \heading {\bf 1. Equidistant hypersurfaces and preliminaries for Theorems 1-2 } \endheading In this section, we first study the geometry of equidistant real hypersurfaces which will be needed to prove Theorem 1. Afterwards we recall four preliminary results which are indispensable in the proof of Theorem 2. Suppose that $\Omega$ is a bounded domain with $C^2$-smooth boundary $b\Omega$ in $\Bbb CP^n$. Let $b\Omega_s = \{ x \in \Bbb CP^n | d(x, b\Omega) = s\}$ and $\{b\Omega_s\}_{s < \epsilon}$ be a family of equidistant real hypersurfaces from $b\Omega$ in $\Bbb CP^n$. We shall show that the connection and the curvature form of the complex line bundle of equidistant hypersurfaces $\{b\Omega_s \}_{s < \epsilon}$ are $C^{\alpha}$, as long as $b\Omega$ is a $C^{2, \alpha}$-smooth hypersurface in $\Bbb CP^n$. We will use the Cartan-Chern theory to prove this assertion as follows. Observe that, if $M = b\Omega$ is a $C^2$-smooth connected hypersurface in $\Bbb CP^n$, then $M$ divides $\Bbb CP^n$ into two connected components $\Omega_+$ and $\Omega_-$. Let $\rho$ be a $C^2$ defining function for $M$ such that $$\rho(x)=\cases &-d (z,M),\quad z\in \Omega_-,\\& d(z,M),\quad z\in \Omega_+,\endcases\tag 1.1$$ where $d(z,M)$ denotes the distance from $z$ to $M$. The defining function $\rho$ is called a signed distance function. Let $U_\epsilon(M)=\{z\in \Bbb CP^n\mid d(z,M)<\epsilon\}$ be a small tubular neighborhood of $M$. We consider the real two-dimensional normal plane bundle $$ \Cal N_{\{\nabla\rho,J\nabla \rho\}}=\{(c_1 \nabla\rho+c_2 J \nabla \rho)|_Q\mid Q\in U_\epsilon(M),\ c_1,c_2\in \Bbb R\} $$ and the complex normal line bundle $$ \Cal N_{ \widetilde{\nabla\rho} }= \Cal N_{\{\nabla\rho\otimes \Bbb C\}}=\{ \lambda(\nabla\rho-\sqrt{-1}J(\nabla\rho))|_Q\mid Q\in U _\epsilon(M),\lambda\in \Bbb C\}. $$ The Levi-Civita connection $D$ of $\Bbb CP^n$ with the standard Fubini-Study metric $\omega$ satisfies $$ D _\xi(J\eta)=J(D_\xi\eta)\tag 1.2 $$ since the metric $\omega$ is K\"ahler. We shall study the induced connections and curvature on $\Cal N_{\{\nabla\rho,J\nabla \rho\}}$ and $\Cal N_{\{\nabla\rho\otimes \Bbb C\}}$. There is also a hermitian connection $\tilde D$ on $\Bbb CT(\Bbb CP^n)=T^{1,0}(\Bbb CP^n)\oplus T^{0,1}(\Bbb CP^n)$. The two connections $D$ and $\tilde D$ are related by $$ \tilde D_{\xi+ {\sqrt{-1}} \eta}(Z+ {\sqrt{-1}} W)=(D_\xi Z-D_\eta W)+ {\sqrt{-1}}(D_\xi W+D_\eta Z) $$ for $\xi,\eta,Z,W\in [T(\Bbb CP^n)]_{\Bbb R}$. Let us consider the corresponding real orthogonal decomposition $$ \[T(\Bbb CP^n)\]_{\Bbb R}= \Cal N_{\{\nabla\rho,J\nabla \rho\}}\oplus \Cal N_{\{\nabla\rho,J\nabla \rho\}}^{\perp}.\tag 1.3 $$ For $\xi\in \[T(\Bbb CP^n)\]_{\Bbb R},$ $\xi^N $ is the projection of $\xi$ into the normal component $\Cal N_{\{\nabla\rho,J\nabla \rho\}}$. Similarly, we have $$ T^{1,0}(\Bbb CP^n) = \Cal N_{\{\nabla\rho\otimes \Bbb C\}}\oplus \Cal N_{\{\nabla\rho\otimes\Bbb C\}}^{\perp} .\tag 1.4$$ For $\tilde\xi\in T^{1,0}(\Bbb CP^n) ,$ ${\tilde\xi}^N $ is the projection of ${\tilde\xi}$ into the complex normal component $\Cal N_{\{\nabla\rho\otimes\Bbb C\}}$. The map $$u \mapsto \tilde u = \frac 1{\sqrt{2}} (u-\sqrt{-1}J u )$$ is a linear isomorphism from $T(\Bbb CP^n)_{\Bbb R}$ to $T^{1,0}(\Bbb CP^n)$. We define $$ D^N_\xi\eta=(D_\xi\eta)^N, \qquad \eta\in \Cal N_{\{\nabla\rho,J(\nabla\rho)\}}. \tag 1.5 $$ The induced curvature tensor is defined by $$R^N(Z,W)\eta=-D_Z^ND_W^N\eta+D_W^ND_Z^N\eta+D_{[Z,W]}^N \eta,$$ for $\eta\in N_{\{\nabla\rho,J(\nabla\rho)\}}$ and $Z, W\in [T(\Bbb CP^n)]_{\Bbb R} $. Similarly, we have the induced curvature tensor $\tilde \Theta^N$ for the complex line bundle $ \Cal N_{ \widetilde{\nabla\rho} } $ on $U_\epsilon$. Since $\|\nabla \rho\|^2=1$, $$d(\|\nabla\rho\|^2)=2\langle D \nabla\rho,\nabla\rho\rangle=0.$$ Thus $D_\xi^N(\nabla\rho)$ has no component in the $\nabla\rho$ direction and $$ D_\xi^N(\nabla\rho)=\beta(\xi)J(\nabla\rho) \tag 1.6 $$ where $\beta$ on $U_\epsilon$ is defined by $$\beta(\cdot )=\text{Hess}(\rho)(\cdot ,J(\nabla\rho)).\tag 1.7 $$ The 1-form $\beta$ is the connection form for $\Cal N_{\{\nabla\rho,J(\nabla\rho)\}}.$ Recall that from (1.2), $D_\xi(J(\nabla\rho))=JD_\xi(\nabla\rho)$ and $\langle \nabla\rho,J(\nabla\rho)\rangle\equiv 0.$ Thus we have $$D_\xi^N(J(\nabla\rho))=JD_\xi^N(\nabla\rho)=J(\beta(\xi)J(\nabla\rho))=-\beta( \xi)\nabla\rho.\tag 1.8 $$ We shall study the connection form for $\Cal N_{\widetilde{\nabla\rho}}$ and its curvature form $\tilde\Theta^N$. \proclaim{Proposition 1.1} Let $M$ be a $C^m$ hypersurface in $\Bbb CP^n$ and let $\rho$ be the signed distance function for $M$, $m\ge 2$. Let $\tilde e_n=\widetilde{\nabla \rho}$ be the unit complex normal on a tubular neighborhood $U_\epsilon(M)$ for sufficiently small $\epsilon>0$. Then we have $$ \tilde D_\xi \tilde e_n=\theta(\xi)\tilde e_n \tag 1.9 $$ where $$ \theta(\xi)=\sqrt{-1} \text{Hess}(\rho)(\xi, J(\nabla\rho)) = \sqrt{-1} \beta(\xi).\tag 1.10 $$ Consequently, $\sqrt{-1}\theta$ is real-valued and the curvature form $\tilde \Theta^N$ of $\Cal N_{{ \widetilde{\nabla\rho} }}$ is an exact 2-form on $U_\epsilon(M)$: $$\tilde\Theta^N=-d\theta. \tag 1.11 $$ Furthermore, the connection $\theta$ and the curvature form $\tilde\Theta^N = - d\theta$ above are of class $C^{\alpha}$ if $M$ is a $C^{2, \alpha}$-smooth hypersurface in $\Bbb CP^n$. \endproclaim \demo{Proof} The complex line bundle $\Cal N_{{ \widetilde{\nabla\rho} }}$ is spanned by a global nowhere vanishing section $ \widetilde {\nabla\rho} =\frac 1{\sqrt 2}(\nabla\rho-\sqrt{-1}J(\nabla\rho))$, the complex normal vector on $U_\epsilon(M)$. A direct computation shows that, by (1.6) and (1.8) we have $$\aligned \tilde D_\xi^N\tilde e_n &=\frac 1{\sqrt 2}\{\tilde D_\xi^N\nabla\rho-\sqrt{-1}\tilde D_\xi^N(J(\nabla\rho))\} =\frac 1{\sqrt 2}\{\beta(\xi)J(\nabla\rho)+\sqrt{-1}\beta(\xi)(\nabla\rho) \} \\&=\frac {\sqrt{-1}} {\sqrt 2} \beta(\xi)\{(\nabla\rho)-\sqrt{-1}J(\nabla\rho) \}=\sqrt{-1}\beta(\xi)\tilde e_n =\theta(\xi)\tilde e_n, \endaligned $$ where $\theta=\sqrt{-1}\beta$ is the connection 1-form of the complex line bundle $N_{\widetilde{\nabla\rho} } $. This proves (1.9). Using the Chern formula \cite{Chern}, the curvature form $\tilde \Theta^N$ is given by $$\tilde \Theta^N=-d\theta+\theta\wedge\theta=-d\theta=-\sqrt{-1}d\beta $$ which proves (1.11). By (1.10), $\theta$ is $C^{m-2, \alpha}$ if $M$ is $C^{m, \alpha}$. It remains to prove that the curvature form $\tilde \Theta^N$ of $N_{{ \widetilde{\nabla\rho} }}$ is a 2-form with $C^{\alpha}$ coefficients on $U_\epsilon(M)$ if $M$ is $C^{2, \alpha}$-smooth. This can be proved by the generalized Gauss-Codazzi equation (the Cartan-Chern-Gauss structure equation). Let $\tilde e_1,\cdots,\tilde e_n$ be an orthonormal local basis for $T^{1,0}(U_\epsilon\cap V)$ such that $\tilde e_n=\widetilde {\nabla\rho} $, where $V$ is some neighborhood near a boundary point $p\in M$. Let $\tilde \theta=(\tilde \theta_{k\bar l})$ be the connection 1-form for $\Bbb CP^n$ defined by $$\tilde D_\xi \tilde e_j=\sum_{l=1}^n\tilde \theta_{j,\bar l}\tilde e_l. $$ We remark that though $T^{1,0}(\Bbb CP^n)$ is a holomorphic vector bundle, our unitary frame $\{\tilde e_1,\cdots,\tilde e_n\}$ is not necessarily holomorphic. Thus $\tilde \theta_{k,\bar l}$ is not necessarily of $(1,0)$ type. Let $\tilde R(X, Y) = - \tilde D_X \tilde D_Y + \tilde D_Y \tilde D_X + \tilde D_{[X, Y]}$ and $$\tilde\Theta_{k,\bar l}(Z,\overline W)=\langle \tilde R(Z,\overline W)\tilde e_k,\bar{\tilde e}_l\rangle,\quad Z,W\in T^{1,0}(V).\tag 1.12$$ The matrix-valued 2-tensor $\tilde \Theta$ is given by Chern's formula for the curvature of complex vector bundles (cf. [Chern], [KN] or [Zh]), $$\tilde \Theta=-d\tilde \theta+\tilde\theta\wedge\tilde\theta.\tag 1.13$$ In particular, we have $$\aligned \tilde \Theta_{n,\bar n}&=\langle \tilde R(\cdot,\cdot)\tilde e_n,\bar{\tilde e}_n\rangle =-d\tilde\theta_{n,\bar n}+\sum_{n=1}^n\tilde \theta_{n,\bar l}\wedge\tilde\theta_{l,\bar n} \\&=\tilde\Theta^N+\sum_{n=1}^{n-1}\tilde \theta_{n,\bar l}\wedge\tilde\theta_{l,\bar n},\endaligned\tag 1.14$$ where we have used the fact that $\tilde\theta_{n,\bar n}=\theta$ and (1.11). It remains to calculate $\tilde \theta_{n,\bar l}$ and $\tilde\theta_{l,\bar n}$. For this purpose, we use the fact that $$ \langle\tilde e_n,\bar{\tilde e}_l\rangle=\langle\tilde e_l,\bar{\tilde e}_n\rangle=0\tag 1.15 $$ where $l=1,\cdots,n-1.$ Recall that for each $\xi\in \Bbb CT(V)$, $$\tilde D_\xi\tilde e_n=\sum_{l=1}^m\tilde \theta_{n,\bar l}(\xi)\tilde e_l =\sum_{l=1}^n\langle \tilde D_\xi (\tilde e_n),\bar{\tilde e}_l\rangle \tilde e_l.$$ Thus, we have $$ \aligned \tilde \theta_{n,\bar l}(\xi)&=\langle \tilde D_\xi (\tilde e_n),\bar{\tilde e}_l\rangle =\frac 12\langle \tilde D_\xi (\nabla\rho-\sqrt{-1}J(\nabla\rho)), e_l+\sqrt{-1}Je_l \rangle \\&=\frac 12\langle \tilde D_\xi (\nabla\rho ), e_l+\sqrt{-1}Je_l \rangle \ -\frac 12\langle \sqrt{-1}J \tilde D_\xi (\nabla\rho) , e_l+\sqrt{-1}Je_l \rangle \\&= \langle \tilde D_\xi (\nabla\rho ), e_l \rangle+ {\sqrt{-1}} \langle \tilde D_\xi (\nabla\rho ), Je_l \rangle =\sqrt 2\text{Hess}(\rho)(\xi,\tilde e_l).\endaligned\tag 1.16$$ Similarly, we have $$ \aligned \tilde \theta_{l,\bar n}(\xi)&=\langle \tilde D_\xi (\tilde e_l),\bar{\tilde e}_n \rangle =-\langle \tilde e_l ,\tilde D_{ \xi} (\bar{\tilde e}_n) \rangle \\&= -\langle \tilde D_{\bar \xi} ( {\tilde e}_n), \bar{\tilde e}_l \rangle =-\sqrt 2\text{Hess}(\rho)(\bar\xi,\tilde e_l).\endaligned\tag 1.17$$ This implies that $$\tilde \theta_{l,\bar n}=-\bar{\tilde\theta}_{n,\bar l},\quad\text{ on }V.\tag 1.18$$ Thus $$ \tilde \theta_{n,\bar l}\wedge\tilde \theta_{l,\bar n}=-\tilde \theta_{n,\bar l}\wedge\bar{\tilde \theta}_{n,\bar l}\tag 1.19 $$ is a skew-hermitian 2-form. If $\rho$ is $C^{m, \alpha}$-smooth, then each $\tilde e_k$ is $C^{m-1, \alpha}$-smooth for $k=1,\cdots n$. Thus $\tilde\Theta_{n,\bar n}$ is $C^{m-1, \alpha}$-smooth. It follows from (1.14) and (1.16)-(1.18) that $\tilde\Theta^N$ is a $C^{m-2, \alpha}$ smooth form, since $\text{Hess}(\rho)$ is $C^{m-2, \alpha}$. The proposition is proved. \qed \enddemo The $C^{\alpha}$-regularity of $\theta$ and $\tilde\Theta^N$ will be used in the proof of Theorem 1, see Section 5 below. When $M = b\Omega$ is {\it a Levi-flat real-hypersurface}, the restriction $\Theta_b$ on $M$ of the curvature form $\tilde \Theta^N$ above has some additional properties. \proclaim{Proposition 1.2}Suppose that $M$ is a $C^2$-smooth Levi-flat hypersurface in $\Bbb CP^n$ with the signed distance function $\rho$. Let $ \widetilde{\nabla \rho}$ be the unit complex normal on a tubular neighborhood $U_\epsilon(M)$ for sufficiently small $\epsilon>0$ and let $\tilde \Theta^N$ be the curvature tensor for the complex line bundle $N_{\widetilde{\nabla \rho}}$. If $J$ is the complex structure of $\Bbb CP^n$, then $\Theta_b = \tilde{\Theta}^N|_{[T(M)]_{\Bbb R} \cap J [T(M)]_{\Bbb R} }$ is a $(1, 1)$-form. Furthermore, we have $$\sqrt{-1}\tilde \Theta^N(\tau,\bar\tau) = \sqrt{-1}\Theta_b(\tau,\bar\tau)\ge 2 \tag 1.20$$ where $\tau\in T^{1,0}(M)$ with $|\tau|=1$. \endproclaim \demo{Proof} If $M^{2n-1}$ is a Levi-flat real hypersurface, then by definition we have $$ \text{Hess}(\rho) (\bar{\tilde{Y}}, {\tilde{X}}) = (i \partial \db \rho) (\bar{\tilde{Y}}, {\tilde{X}}) = 0 \tag1.21 $$ for any pair $\{\tilde{X}, \tilde{Y}\} \in T^{(1, 0)}(M)$. It follows from (1.16)-(1.17) and (1.21) that the connection forms $\{ \tilde \theta_{n,\bar l}\}$ are $(1,0)$-forms on the Levi-flat hypersurface $M = b\Omega$. Using (1.14) and (1.18), we have $$ \tilde\Theta^N= \tilde \Theta_{n,\bar n}-\sum_{n=1}^{n-1}\tilde \theta_{n,\bar l}\wedge\tilde\theta_{l,\bar n} = \tilde \Theta_{n,\bar n}+\sum_{n=1}^{n-1}\tilde \theta_{n,\bar l}\wedge\bar{\tilde\theta}_{n,\bar l}. \tag1.22$$ Because the Fubini-Study metric $\omega$ is a K\"ahler metric, its curvature form $\tilde \Theta$ is a matrix valued (1, 1)-form with respect to any unitary frame (cf. [Chern], [Zh]). Hence, $ \tilde \Theta_{n,\bar n}$ is a (1, 1)-form. Since $\{ \tilde \theta_{n,\bar l}\}$ are $(1,0)$-forms on the Levi-flat hypersurface $M = b\Omega$, we see that $\sqrt{-1}\sum_{n=1}^{n-1}\tilde \theta_{n,\bar l}\wedge\bar{\tilde\theta}_{n,\bar l}$ is a nonnegative $(1,1)$-form. This fact together with (1.22) implies that $\Theta_b $ is a $(1,1)$-form. Furthermore, it follows that, for $\tau \in T^{(1,0)}(M)$ with $|\tau| = 1$, $$\sqrt{-1} \tilde\Theta^N(\tau,\bar\tau)\ge \sqrt{-1} \tilde \Theta_{n,\bar n}(\tau,\bar\tau)\ge 2,$$ where we used the fact that the Fubini-Study metric of $\Bbb CP^n$ has positive holomorphic bisectional curvature $\ge 2$, (e.g., cf. [KN], [Zh]). This completes the proof. \qed \enddemo Let us now recall some preliminary results, which play important roles in the proof of Theorem 2. First of all, we need to recall the so-called $\db$-Neumann boundary condition. The $L^2$-theory of the $\db$-equation $\db u = v$ and the Laplace equation $\square u = f$ are related to the formal adjoint $\vartheta$ of $\db$ on the space $L^2_{(p, q)}(\Omega)$. Let $*$ be the real Hodge star operator of the Riemannian metric and let $*$ be extended $\Bbb C$-linearly on the space of $(p, q)$-forms. Then $$ \vartheta u = - \bar{*}[ \db (\bar{*} u)] = - * \partial * u. \tag1.23 $$ For $u, v\in L^2(\Omega)$, we use $$(u,\bar v)_{L^2(\Omega)}=\int_\Omega\langle u,\bar v\rangle$$ to denote the inner product. For $q >0$, using integration by parts on the domain $\Omega$ with $C^2$-smooth boundary $b\Omega$, we have that $u \in \text{Dom}(\db^*|_\Omega)\cap C^1_{(p, q)}(\overline\Omega)$ if and only if $$ (\vartheta u, \bar v )_{L^2(\Omega)} = ( \db^* u, \bar v)_{L^2(\Omega)} = ( u, \overline{\db v} )_{L^2(\Omega)} \tag1.24 $$ holds for all $v \in L^2_{(p, q)}(\Omega)$. The condition (1.24) is equivalent to the so-called $\db$-Neumann boundary condition $$ u \llcorner_{(\db \rho)_\#} = 0 \text{ on } b\Omega, \tag1.25.1 $$ where $(\db \rho)_\# = \frac 1{\sqrt{2}} (\nabla \rho + \sqrt{-1}J \nabla \rho ) $ is the dual of $\db \rho$. In general, $u \in \text{Dom}(\db^*|_\Omega)$ implies that (1.25.1) holds as currents. Throughout this paper, we let $$\text{Dom}(\db^*) =\text{Dom}(\db^*|_\Omega) = \{ u \in L^2_{(p, q)}(\Omega) | u\llcorner_{(\db \rho)_\#} = 0 \text{ on } b\Omega \} \tag1.25.2$$ and $$ \text{Dom}(\square|_\Omega) = \{ u \in L^2_{(p, q)}(\Omega) | u\llcorner_{(\db \rho)_\#} = 0 \text{ and } (\db u)\llcorner_{(\db \rho)_\#} = 0 \text{ on } b\Omega \}. \tag1.25.3$$ There always exists the {\it weak} Hodge-Kodaira decomposition for any domain $\Omega$: $$ L^2_{(p, q)}(\Omega) =\text{ker}(\square) \oplus_{L^2} \overline{\text{Range}(\square|_\Omega)}, \tag1.26 $$ where $\overline{\text{Range}(\square|_\Omega)}$ denotes the closure of ${\text{Range}(\square|_\Omega)}$ in $L^2(\Omega)$. A necessary condition for the existence of the $\db$-Neumann operator on $L^2_{(p, q)}(\Omega)$ is the condition that the range of $\square $ must be closed, i.e., $\overline{\text{Range}(\square)} = \text{Range}(\square)$. To prove the first part of Theorem 2, by the definition of $\square = \db \db^* + \db^* \db$, it is sufficient to show that both $\text{Range}(\db)$ and $\text{Range}(\db^*)$ are closed subspaces of $L^2_{(p, q)}(\Omega)$. For this purpose (and for the convenience of the reader), we recall the following elementary but very useful fact in functional analysis (cf. [H\"o2-3] or [CS, p60]): \proclaim{Lemma 1.3} Let $\Cal F: H_1 \to H_2$ be a linear, closed, densely defined operator between two Hilbert spaces. The following conditions on $\Cal F$ are equivalent: \medskip \noindent (1) $\text{Range}(\Cal F)$ is closed; \medskip \noindent (2) There is a constant $C$ such that $$ \| u \|_1 \le C \| \Cal F u \|_2 \text{ for all } u \in \text{Dom}(\Cal F) \cap \overline{\text{Range}(\Cal F^*)}; \tag1.27.1 $$ \medskip \noindent (3) $\text{Range}(\Cal F^*)$ is closed; \medskip \noindent (4) There is a constant $C$ such that $$ \| v \|_2 \le C \| \Cal F^* v\|_2 \text{ for all } v \in \text{Dom}(\Cal F^*) \cap \overline{\text{Range}(\Cal F)}; \tag1.27.2 $$ The best constants in (1.27.1)-(1.27.2) are the same. \endproclaim Lemma 1.3 will be used in the proofs of Theorem 2.1, Theorem 2.6 and Theorems 3.4-3.5 below, which are parts of Theorem 2. By Lemma 1.3, it suffices to derive {\it a priori} estimates for the $\db$ equation and the $\square$ equation. We recall the curved version of the Morrey-Kohn-H\"ormander formula, which is a Bochner type formula with a weight function $e^{ -\phi}$. The weighted function $e^{-\phi}$ induces a perturbed adjoint operator: $$ \vartheta_\phi u = \db^*_\phi u = e^\phi \vartheta (e^{-\phi u}) = - e^\phi * \partial (e^{-\phi}*u). \tag 1.28 $$ Of course, one can formulate the weighted Laplace operator $\square_\phi = (\db \db^*_\phi + \db^*_\phi \db) $ and the corresponding weighted $\db$-Neumann operators $N_{\phi}$. Our strategy is to use the weighted $\db$-Neumann operators $ N_{\phi}$ to estimate the original $\db$-Neumann operator $N$, see Sections 2-3 below. In order to estimate the operator $\db^*_\phi N_\phi$, we let $L_1,\cdots,L_n$ be a local orthonormal frame for $T^{(1,0)}(U)$, where $U$ is a local neighborhood near some point in $\overline \Omega$. For any $u\in C_{(p,q)}(\overline\Omega)$ and $\phi\in C^2(\Omega)$, we define an operator $$ \langle( i\partial\db \phi) u, \overline u\rangle=\sum_{j,k=1}^n (i\partial\db \phi) (L_j,\overline L_k) \langle u\llcorner \overline L_j, \overline u\llcorner L_k\rangle \text{ on } \Omega. \tag1.29 $$ Notice that $\langle ( i\partial\db \phi)u, \overline u\rangle$ is independent of the choice of a local unitary basis. Similarly, for $Q\in b\Omega$, we require that $L_n = \sqrt{2} (\partial \rho)_\#$ and define $$\langle (i\partial\db \rho) u, \overline u\rangle=\sum_{j,k=1}^{n-1} (i\partial\db \rho) (L_j,\overline L_k) \langle u\llcorner \overline L_j, \overline u\llcorner L_k\rangle \text{ on } b\Omega. \tag1.30 $$ In addition, since $\Bbb CP^n$ has non-zero curvature, we set $$ \langle \Theta u, \overline u\rangle=\sum_{j,k}^n \langle \overline w^j \wedge\{ [R(L_j , \overline L_k) u]\llcorner \overline L_k\}, \overline u\rangle \text{ on } \Omega, \tag1.31 $$ where $\{w^j \}$ is the dual of the frame $\{ L_k \}$ and $R_{X, Y} = - D_X D_Y u + D_Y D_X + D_{[X, Y]}$ is the curvature operator on $(p, q)$-forms, see [Wu, Chapter 2]. Finally, we set $$ (u, \overline v)_\phi = \int_\Omega \langle u, \overline v \rangle e^{-\phi}. \tag1.32 $$ \proclaim{Proposition 1.4} (Bochner-H\"ormander-Kohn-Morrey formula) Let $\Omega$ be a compact domain $\Omega$ with $C^2$-smooth boundary $b\Omega$ and $\rho(x) = - d(x, b\Omega)$. Then, for any (p,q)-form $u\in \text{Dom}(\db)\cap\text{Dom}(\db^*)$ with $q\ge1$ on $\Omega$, we have $$ \|\db u\|_\phi^2+\|\db^*_\phi u\|_\phi^2=\|\overline\nabla u\|_\phi^2 +(\Theta u, \overline u)_\phi + ((i\partial\db \phi) u, \overline u )_\phi+\int_{b\Omega} \langle (i\partial\db \rho) u, \overline u\rangle e^{-\phi}, \tag1.33 $$ where $ \|\overline\nabla u\|_\phi^2 = \int_\Omega \sum^n_{j = 1} |D_{\bar L_j} u|^2 e^{-\phi}$ and $\{ L_1, ..., L_n\}$ is a local unitary frame of $T^{(1,0)}(\Omega)$. \endproclaim \demo{Proof} This formula is known (cf. [AV], [H\"o2] [Dem1-2], [Siu1], [Wu]) for some special cases, although it has not been stated in the literature in the form (1.33). If $u$ has compact support in the interior of $\Omega$, the formula (1.33) was proved in [AV], Chapter 8 of [Dem2] and (2.12) of [Wu]. It remains to discuss the boundary term of (1.33). For the case $\phi = 0$, the stated formula with the boundary term was proved in [Siu1]. To compute the boundary term, one notice that although the boundary terms in the proof of (1.33) involve the weight function, they have nothing to do with the Riemannian curvature $R$ nor $\Theta$. The boundary term had been computed in H\"ormander \cite{H\"o2, Chapter 3} by combining the Morrey-Kohn technique on the boundary with non-trivial weight function. If one combines the results of \cite{H\"o2} with the interior formulae discussed above, one can prove that (1.33) holds for the general case with a weight function $e^{-\phi}$ and the curvature term. \qed \enddemo In later application, we will choose $\phi = - t \log |\rho|$ in (1.33). Therefore, we need to discuss the term $ i \partial \db \phi$ with $ \phi = - t \log |\rho|$ and the curvature term $\langle \Theta u, \bar u \rangle$ in the right hand side of (1.33). \proclaim{Proposition 1.5} Let $u$ be a $(p, q)$-form of $\Omega \subset \Bbb CP^n$ with $q \ge 1$. Suppose that $\Theta$ is the curvature term defined in (1.31) with respect to the Fubini-Study metric. Then we have $$ \aligned \cases \langle \Theta u, \overline u\rangle &= 0,\quad \text{for any $(n, q)$-form u;} \\ \langle \Theta u, \overline u\rangle &\ge 0, \quad\text{ when $p \ge 1$ and $u$ is a (p, q)-form} ;\\ \langle \Theta u, \overline u\rangle &= q(2n+1) |u|^2, \quad\text{ when $u$ is a (0, q)-form}.\endcases \endaligned\tag 1.34$$ \endproclaim \demo{Proof} The assertion for $(0, q)$-forms and $(n, q)$-forms was computed in [Wu] and [Siu1]. For the curvature operator $\Theta$ acting on (1, 1)-forms $u$, [Siu1] showed that $\Theta$ is non-negative definite, but {\it not} positive definite, (see [Pe] as well). Henkin and Iordan [HI] extended this observation for all $(p, q)$-forms with $p \ge 1$. In fact, Lemma 3.3 of [HI] and its proof showed that $\Theta$ acting on $L^2_{(p, q)}(\Omega) $ is a non-negative operator. Furthermore, $ \Theta$ is {\it positive definite} on $L^2_{(p, q)}(\Omega) $ if and only if $p=0$ and $q \ge 1$. \qed\enddemo \proclaim{Proposition 1.6} If $ \Omega$ is a pseudoconvex domain with $C^2$-smooth boundary $b\Omega$ in $\Bbb CP^n$ and $\rho$ is the signed distance function, then $$ i\partial\db (-\log|\rho|)(X, \overline X) \ge \frac 12 |X|^2 \tag 1.35 $$ for any $X \in T^{(1,0)}(\Omega). $ \endproclaim Proposition 1.6 was proved in Takeuchi (e.g., cf. [Su], [Ta]). A new proof of Proposition 1.6 using comparison theorems in Riemannian geometry will appear in another paper in [CaS]. \heading {\bf 2. $L^2$ theory for $\db$ on pseudoconvex domains in $\Bbb CP^n$}\endheading Let $\Omega$ be a bounded pseudoconvex domain in $\Bbb CP^n$. We fix our Fubini-Study metric $\omega$ for $\Bbb CP^n$. In this section, we will use preliminary results of Section 1 to study the $\db$-Neumann operators $N$, $\db^* N$ and $\db N$ acting on $L^2_{(p, q)}(\Omega)$. We first prove the $L^2$ existence theorem of the $\db$-Neumann operator for the easier case when $p=0$. \proclaim{\bf Theorem 2.1} Let $\Omega$ be a pseudoconvex domain in $\Bbb CP^n$, $n\ge 2$. Then the $\db$-Neumann operator $N_{(0,q)}$ exists and is bounded on $L^2_{(0,q)}(\Omega)$, where $0\le q\le n $. For any $f\in L^2_{(0,q)}(\Omega )$, $$\aligned f&= \bar\partial \bar\partial^*N_{(0,q)}f \oplus\bar\partial^*\bar\partial N_{(0,q)}f,\quad 1\le q\le n-1.\\ f&= \bar\partial^*\bar\partial N_{(0,0)}f\oplus Pf,\quad q=0,\endaligned\tag 2.1$$ where $P:L^2(\Omega)\to L^2(\Omega)\cap\text{Ker}(\db)$ is the Bergman projection with $$P=I-\db^*N_{(0,1)}\db \tag 2.2$$ and $$N_{(0,0)}=\vartheta N_{(0,1)}^2\db.\tag 2.3$$ Moreover, $N$, $\db^* N$ and $\db N$ are bounded operators with respect to the $L^2$-norms. \endproclaim \demo{Proof} We first assume that $\Omega$ is a pseudoconvex domain in $\Bbb CP^n$ with $C^2$ boundary $M$. Let $1\le q\le n $ and let $\rho$ be a $C^2$ defining function for $\Omega$ normalized by $|d\rho|=1$ on $M$. Choosing $\phi = 0$ in Proposition 1.4 , we have for any $(0,q)$-form $u\in \text{Dom}(\db)\cap\text{Dom}(\db^*)$, $$ \|\db u\|^2+\|\db^* u\|^2=\|\overline\nabla u\|^2+\int_M\langle (i\partial\db \rho)u, \bar u\rangle +(\Theta u,\bar u) \ge \|\overline\nabla u\|^2 + (\Theta u,\bar u ),\tag 2.4.1 $$ where $\Theta$ is the Ricci form and $\overline\nabla $ is the holomorphic gradient. Since the Ricci curvature of $\Bbb CP^n$ with the Fubini-Study metric is equal to 2n + 1, (cf. [KN] [Pe] or [Zh]), by (2.4.1) we have $$ Q(u, \bar u) = \|\db u\|^2+\|\db^* u\|^2\ge q(2n + 1)\|u\|^2,\quad u\in \text{Dom}(\db) \cap \text{Dom}(\db^*). \tag2.4.2 $$ Consequently, we have $$ (\square u, \bar u) = Q(u,\bar u ) \ge q(2n + 1)\|u\|^2,\quad u\in \text{Dom}(\square_{(0, q)}|_\Omega),\tag2.4.3 $$ for $q \ge 1$. It follows from (2.4.3) that $\text{Ker}(\square_{(0, q)}) = 0$ and $\| N_{(0, q)}\|_{L^2} \le \frac{1}{q(2n+1)}$ for $q \ge 1$. Moreover, by (2.4.2)-(2.4.3) and Lemma 1.3, one can show that the three operators $\db, \db^*$ and $\square$ have closed ranges in $L^2(\Omega)$. Notice that if $\db f = 0$ and if $u = \db^* N f$, then $u$ is the solution of $\db u = f$ with the smallest $L^2$-norm. Using this fact, by (2.4.2) and Lemma 1.3, one can further show that $\db^*N$ is a bounded operator. Since $\db N = (N\db^*)^* = (\db^* N)^*$, using Lemma 1.3 again, we see that the operator $\db N$ is bounded as well. This proves Theorem 2.1 for the case of $q \ge 1$. The existence of $N_{(0, 0)}$ also follows and one can prove (2.2) and (2.3) exactly as for domains in $\Bbb C^n$. The general case for nonsmooth domains follows from exhausting a pseudoconvex domain by $C^2$ pseudoconvex domains and using (2.4) (See proofs of Theorem 4.4.1 and Theorem 4.4.3 in [CS] for details). \qed \enddemo When $00$ such that $$ i\partial\db(- \delta^t)\ge C_t \delta^t \(\omega+ \frac{i\partial\delta\wedge\db\delta }{\delta^2} \), \tag 2.5$$ where $\omega$ is the K\"ahler form of the Fubini-Study metric on $\Bbb CP^n$. \endproclaim \demo{Proof} The existence of such $t_0$ for pseudoconvex domains in $\Bbb C^n$ with $C^2$-smooth boundary in $\Bbb CP^n$ is proved in Ohsawa-Sibony [OS]. Thus, by the inequality $ i\partial\db(- \delta^{t_0}) \ge 0 $ with $0t$. Then $$ \aligned i \partial\db(- \delta^t) &= it\delta^t \(\frac {\partial\db(-\delta)}\delta+(1-t) \frac{ \partial\delta\wedge\db\delta }{\delta^2} \) \\& \ge C_t t \delta^t \(\omega+ \frac{i\partial\delta\wedge\db\delta }{\delta^2}\) \endaligned $$ where $C_t =\min( \frac 12 \epsilon_t ,(1-\epsilon_t )t_0-t)$. The lemma is proved. \qed \enddemo \medskip We will use H\"ormander's weight function method to obtain the $L^2$ existence for the $\db$-equation with weights first. Let $t $ be any real number and $\phi\in C^2(\Omega)$. Let $L^2(\delta^t)$ denote the $L^2$ space with respect to the weight function $e^{-t\phi}=\delta^t$ and $$\|f \|_{(t)}^2=\int_\Omega |f|^2e^{-t\phi}=\int_\Omega \delta^t|f|^2 . $$ We use $\db^*_{ t }$ to denote the adjoint of $\db$ with respect to the weighted space. \medskip We use the norm $\|. \|_{(t)}^2$ since it is equivalent to the Sobolev norm on a sub-space of $W^{-\frac t2}(\Omega)$, (see [CS p348] and [MS1-2]). In fact, we have $$ \|\tilde u \|_{(t)}^2 \simeq \| \tilde u \|_{W^{-\frac t2}(\Omega ) } \tag2.9 $$ when $\tilde u$ satisfies an elliptic equation such as $\db\oplus\vartheta$. In order to use Proposition 1.4 to derive the desired estimates for the operator $\db^* N$, we need to introduce the following two asymmetric weighted norms. These new norms will be used to obtain more refined estimates than those obtained in [H\"o2]. For any $(p,q)$-form $f$ on $\Omega$, we decompose $f$ into complex normal and tangential parts by setting $$\cases f^\nu&=(f\llcorner_{(\db\delta)_\#})\wedge \db\delta \\ f^\tau &=f-f^\nu.\endcases$$ The above decomposition is well-defined for any $(p,q)$-form $f$ supported in \newline $[\Omega\setminus \text{Cut}_\Omega(b\Omega)]$, where $ \text{Cut}_\Omega(b\Omega)$ is the cut-loci of $b\Omega$ in $\Omega$, see [Cha], [CE] or [Pe]. It is well-known that the cut-loci $\text{Cut}_\Omega(b\Omega)$ of $b\Omega$ has real dimension $\le (2n-1)$ and hence has zero measure in $\Omega$, (e.g., [Cha], [CE, p90ff], [Pe]). In summary, the above decomposition exists {\it almost everywhere in $\Omega$}. We define the asymmetric weighted norm $$|f|^2_A=| f^\tau|^2+ \frac{| f^\nu|^2}{|\delta|^2}.\tag 2.10.1$$ We also define the dual norm $$|f|^2_{A'}=| f^\tau|^2+ {| f^\nu|^2}{|\delta|^2}.\tag 2.10.2$$ For any $t>0$, let $ L^{2}_A(\delta^t)$ and $ L^{ 2}_{A'}(\delta^t)$ denote the weighted $L^2$ spaces on $(p,q)$-forms defined by the norm $$|||u|||_{(t)}^2=\int_{\Omega}\delta^t |f|_A^2=\int_{\Omega}\delta^t( | f^\tau|^2+ \frac{| f^\nu|^2}{|\delta|^2}) \tag 2.10.3 $$ and $$|||u|||_{ (t)'}^2=\int_{\Omega}\delta^t |f|^2_{A'}=\int_{\Omega}\delta^t | ( f^\tau|^2+ {| f^\nu|^2}{|\delta|^2}). \tag2.10.4$$ We may assume that $\text{Diam}(\Bbb CP^n) \le 1$ up to a factor $\frac{\pi}{2}$. It is obvious that $$|||u|||_{ (t)'}^2\le \p u\p_{ (t)}^2\le |||u|||_{ (t) }^2,$$ if $\text{Diam}(\Omega) \le 1$. The following is a preliminary estimate for the operator $\db^* N$, because $u = \db^* N f$ is the solution of $\db u = f$ with the least $L^2$-norm. \proclaim{\bf Proposition 2.3} Let $\Omega\subset\subset\Bbb CP^n$ be a pseudoconvex domain with $C^2$-smooth boundary $b\Omega$. Let $t_0 = t_0(\Omega)$ be the order of plurisubharmonicity for the distance function $\delta$ defined by (0.7). For any $ 0 < t < t_0$ and any $(p,q)$-form $f\in L^{2}_{A'}(\delta^t ) $, where $0\le p\le n$ and $1\le q\le n$, such that $\db f=0$ in $\Omega$, there exists $u\in L^2_{(p,q-1)}(\delta^t)$ satisfying $\db u=f$ and $$\|u\|_{(t)}^2 \le \frac 1{C_0t}|||f|||^2_{(t)'}. \tag 2.11 $$ \endproclaim \demo{Proof} By Proposition 1.6, we have that $\phi=-\log \delta$ is strictly plurisubharmonic and $i\partial\db\phi\ge \frac 12 \omega$, where $\omega$ is the K\"ahler form of $\Bbb CP^n$ with the Fubini-Study metric. It is known that $\text{Dom}(\db^*_{t }) = \text{Dom}(\db^*)$, (e.g., [CS, Chapter 4]). Using H\"ormander's weighted estimates (cf. Proposition 1.4) and Proposition 1.5, we have the following formula: for any $(p,q)$-form $g\in \text{Dom}(\db)\cap\text{Dom}(\db^*_{t })$, $$ \|\db g\|^2_{(t)}+\|\db^*_{t } g\|^2_{(t)}\ge t((i\partial \db) \phi g,\bar g)_{(t)}. \tag 2.12.1 $$ From (2.8), the positive (1,1)-form $i\partial \db \phi$ induces a pointwise norm on $(p,q)$ forms $$ |g|^2_{i\partial\db\phi}= \langle i(\partial\db\phi) g, \bar g\rangle\ge C_0 |g |_A^2 , \tag2.12.2 $$ where $C_0=\min{(\frac 12,t_0)}$ with $t_0 = t_0(\Omega)$ as in (2.5). Thus, we have $$\|\db g\|^2_{(t)}+\|\db^*_{t } g\|^2_{(t)}\ge C_0t|||g|||_{(t)}. \tag 2.13$$ Let $(i\partial\db\phi)'$ denote the dual norm for $(p,q)$-forms induced by $i\partial\db\phi$. Using (2.13) and the same argument as in Demailly \cite{Dem1,2}, we conclude that for any $f \in L^{2}_{A'}(\delta^t )$, there exist $u\in L^2(\delta^t)$ satisfying $\db u=f$ and $$ \int_\Omega|u|^2\delta^t\le \int_\Omega |\db u|_{(i\partial\db \phi)'}^2\delta^t \le \frac 1{ C_0t} \int_\Omega |\db u|_{A'}^2\delta^t.\tag 2.14 $$ This completes the proof. \qed \enddemo Notice that the terms in (2.14) have an extra weight factor $\delta^t$. We need to remove this factor $\delta^t$ in order to obtain the desired $L^2$-estimate for all $p \ge 0$. The next two propositions have already been obtained in Berndtsson-Charpentier \cite{BC} (see also \cite{HI}). We include the proof here for the sake of completeness. \proclaim{\bf Proposition 2.4} Let $\Omega\subset\subset\Bbb CP^n$ be a pseudoconvex domain with $C^2$ boundary $b\Omega$. For any $f\in L^2_{(p,q)}(\Omega) $, where $0\le p\le n$ and $1\le q\le n$, such that $\db f=0$ in $\Omega$, there exists $u\in L^2_{(p,q-1)}(\Omega)$ satisfying $\db u=f$ with $\int_\Omega|u|^2 \le \hat{C} \int_\Omega|f|^2$. \endproclaim \demo{Proof} By Proposition 2.3, for any $t>0$ with $0 < t < t_0/4$, there exists $u\in L^2_{(p,q-1)}(\delta^t)$ satisfying $\db u=f$, such that $u$ is perpendicular to Ker$(\db)$ in $L^2(\delta^t)$ and $u$ satisfies inequality (2.11). Consider $v=u\delta^{-t}$. Then $v\in L^2(\delta^{2t})$ and $v\perp \text{Ker}(\db)$ in $L^2(\delta^{2t})$. It follows from (2.14) that the following holds: $$ \int_\Omega|u|^2 = \int_\Omega|v|^2\delta^{2 t} \le \frac 1{2C_0t}\int_\Omega|\db v|^2_{A^{'}}\delta^{2t}. \tag 2.15 $$ By the definitions of the norm $| .|_{ A'}$ and $\phi = - \log \delta$ (cf. (2.10.2)), we have $$\aligned |\db v|^2_{A^{'}}e^{-2t\phi}&\le 2|\db u|^2_{A^{'}}+2t^2|\db \phi |^2_{A^{'}}|u|^2 \\& \le C( |\db u|^2 +2t^2|u|^2).\endaligned\tag 2.16 $$ Choosing $t$ sufficiently small and substituting (2.16) into (2.15), one obtains $$\int_\Omega|u|^2 \le \hat{C}\int_\Omega|\db u|^2, \tag2.17 $$ where $ \hat{C} = \frac{C}{2 C_0 t(1 - 2 C t^2 )}$. This proves the theorem. \qed \enddemo By (2.9), the Sobolev norm $\| u \|_{W^{\frac t2}_{(p, q)}(\Omega)} $ of the solution u is related to the weighted norm $\| u \|_{(-t)}$. \proclaim{\bf Proposition 2.5} Let $\Omega\subset\subset\Bbb CP^n$ be a pseudoconvex domain with $C^2$ boundary $b\Omega$. Let $t_0 = t_0(\Omega)$ be the order of plurisubharmonicity for the distance function $\delta$ defined by (0.7). For any $0\le t< t_0$, $f\in L^2_{A'}(\delta^{-t}) $, where $0\le p\le n$ and $1\le q\le n$, such that $\db f=0$ in $\Omega$, then $u=\db^*N f$ is in $ L^2_{(p,q-1)}(\delta^{-t})$ with $\db u=f$ and $$ \|u\|_{(-t)}^2 \le \tilde C_t|||f|||^2_{(-t)'}, $$ where the weighted norms are given by (2.10.1)-(2.10.4). \endproclaim \demo{Proof} Let $v=u\delta^{-t}$. We have that $\db^*_t v=0$ if $q>1$ and $v\perp \text{Ker}(\db)$ in $L^2(\delta^{t})$ for $q=0$. This implies that $v=\db^*_tN_t\db v$. By (2.8) (with $\epsilon=0$), we have that $$|\frac {\db\delta}\delta\wedge u|^2_{(i\partial\db \phi)'}\le \frac 1{t_0}|\frac {\db\delta}\delta\wedge u|^2_{A'}\le \frac 1{t_0}|u|^2.\tag 2.18 $$ Thus for any $\epsilon>0$, by Proposition 2.3 (the proof of (2.14)), we have $$ \aligned \int_\Omega |u|^2\delta^{-t} &=\int_\Omega |v|^2\delta^t \le \frac {1}{C_0t}\int_\Omega |\db v|^2_{(i\partial\db\phi)'}\delta^t \\&\le (1+\frac 1\epsilon )\frac 1{C_0t}\int_\Omega |\db u|_{(i\partial\db\phi)'}^2\delta^{ -t}+ (1+\epsilon)\frac{t }{t_0}\int_\Omega |\frac {\db\delta}\delta\wedge u|^2_{ A '}\delta^{- t} \\&\le (1+\frac 1\epsilon )\frac 1{C_0t}\int_\Omega |\db u|_{A'}^2\delta^{ -t}+ (1+\epsilon)\frac{t }{t_0}\int_\Omega |u|^2 \delta^{- t}. \endaligned\tag 2.19$$ Choosing $\epsilon$ sufficiently small such that $(1+\epsilon)\frac{t }{t_0}=r<1$ in (2.19), we conclude that $$\int_\Omega |u|^2\delta^{-t}\le \frac{ (1+\frac 1\epsilon )\frac 1{C_0t}}{1-r} \int_\Omega |\db u|_{A'}^2\delta^{ -t}\le \tilde{ C_t}|||f|||^2_{(-t)'}.\tag 2.20$$ Proposition 2.5 is proved. \qed \enddemo We arrive at the main result of this section. \proclaim{\bf Theorem 2.6} Let $\Omega\subset\subset\Bbb CP^n$ be a pseudoconvex domain with $C^2$-smooth boundary $b\Omega$. Then $\square_{(p,q)}$ has closed range and the $\db$-Neumann operator $N_{(p,q)} : L^2_{(p,q)}(\Omega)\to L^2_{(p,q)}(\Omega)$ exists for every $ p,\ q $ such that $0\le p\le n, 0\le q\le n $. Moreover, for any $f\in L^2_{(p,q)}(\Omega )$, we have $$\aligned f&= \bar\partial \bar\partial^*N_{(p,q)}f \oplus\bar\partial^*\bar\partial N_{(0,q)}f,\quad 1\le q\le n-1.\\ f&= \bar\partial^*\bar\partial N_{(p,0)}f\oplus Pf,\quad q=0,\endaligned$$ where $P$ is the projection from $L^2_{(p,0)}(\Omega)$ onto $ L^2_{(p,0)}(\Omega)\cap\text{Ker}(\db)$ and $$N_{(p,0)}=\db^*N_{(p,1)}^2\db.$$ In addition, $\db N$, $\db^* N$ and $N$ are bounded linear operators on $L^2_{(p,q)}(\Omega )$. \endproclaim \demo{Proof} The $L^2$-existence theorem for the $\db$-Neumann operator $N$ on $\Omega$ follows from the $L^2$-existence of the solution $u$ for the $\db$-equation, which is proved in Proposition 2.4. The proof of Theorem 2.6 follows from Lemma 1.3 using Proposition 2.4 and (2.17) (see e.g., proofs of Theorem 4.4.1 and Theorem 4.4.3 in \cite{CS} or [H\"o3].) \qed \enddemo We remark that when $p=0$, Theorem 2.6 holds for pseudoconvex domains not necessarily with $C^2$ boundary from Theorem 2.1. \heading{\bf 3. Sobolev estimates for the $\db$-Neumann operator on pseudoconvex domains in $\Bbb CP^n$}\endheading In this section we prove that the $\db$-Neumann operator $N$ is a {\it bounded } linear operator on Sobolev spaces $W^t_{(p, q)} (\Omega)$ for small $t>0$. In order to derive some {\it a priori} estimates for the $\db $-equation, we need a variant of the Bochner-Kodaira-Morrey-Kohn-H\"ormander formula (cf. Proposition 1.4) as follows: \proclaim{Proposition 3.1} Let $\Omega$ be a domain in $\Bbb CP^n$ with $C^2$-smooth boundary $b\Omega$. Let $\delta(x) = d(x, b\Omega)$ be the distance function and $\lambda=-(\delta)^t$ for some $t > 0 $. For any $f \in C^1_{(p,q)}(\overline \Omega)\cap \text{Dom}(\db^*)$, where $0\le p\le n$ and $1\le q\le n$, we have $$\aligned \int_{\Omega} (-\lambda)|\bar\partial f|^2{} & + \int_{\Omega} (-\lambda)|\vartheta_{} f|^2{}+2\Re(\vartheta_{} f, \overline{f\llcorner_{ (\db \lambda)_\#}} ) \\&= \int_{\Omega} (-\lambda)\(| \overline\nabla f|^2+ \langle\Theta f , \bar f\rangle + \langle (i\partial\db \lambda) f, \bar f\rangle\) \endaligned \tag 3.1.1 $$ where $\Theta$ and $\overline\nabla $ are the same as in Proposition 1.4 and $\Re{( h)} $ is the real part of $h$. \endproclaim \demo{Proof} The proof follows from the same calculation as in H\"ormander \cite{H\"o2}. There is no boundary term since $t>0$. Recall that $\vartheta = \db^*$. A simple calculation shows that $$ \vartheta_\phi f = \db^*_\phi f = \db^* f - f \llcorner_{( \db \phi )_\# } \tag3.1.2 $$ holds for any $(p, q)$-form $u$ with $q \ge 1$. On the other hand, by expanding the term $\delta^t i\partial\db(-\log\delta^t )$ we have $$ \delta^t i\partial\db(-\log\delta^t) = t i\partial\db (- \delta^t) + t^2 \frac {i\partial\delta\wedge\db\delta}{\delta^2}. \tag3.1.3 $$ Substituting (3.1.2)-(3.1.3) into (1.33) of Proposition 1.4 with $\phi = - t \log \delta$, we obtain (3.1.1). \qed \enddemo Using lemma 2.2 and Proposition 3.1, we have the following estimates: \proclaim{Proposition 3.2} Let $\Omega$ be a domain in $\Bbb CP^n$ with $C^2$ boundary $b\Omega$. Let $\delta$ be the distance function from $b\Omega$. Let $00$, $$|2\Re (-\vartheta_{} f, \overline{ f\llcorner_{(\db\lambda)_\#} } )|\le \frac t\epsilon \int_{\Omega}(-\lambda)|\vartheta_{} f|^2 +t\epsilon \int_{\Omega}(-\lambda)\frac{| f\llcorner_{(\db\rho)_\#}|^2}{|\rho|^2} .\tag 3.4$$ Choosing $\epsilon$ sufficiently small, (3.2) follows from (3.1.1), (3.3) and Proposition 1.5. This proves the proposition for $C^1$-smooth forms. Since $C^1_{(p,q)}(\overline\Omega)\cap \text{Dom}(\db^*)$ is dense in $\text{Dom}(\db )\cap \text{Dom}(\db^*)$ in the graph norm (cf. Lemma 4.3.2 in \cite{CS}), we have that $C^1_{(p,q)}(\overline\Omega)\cap \text{Dom}(\db^*)$ is also dense in the graph norm with weights defined by $(\int_{\Omega}(-\lambda)|\bar\partial f|^2 + \int_{\Omega}(-\lambda)|\vartheta_{} f|^2+\int_{\Omega}(-\lambda)| f|^2)^{\frac 12}$ from the Dominated Convergence Theorem. The proposition is proved by approximation of $C^1$-smooth forms. \qed \enddemo Recall that $u = \db^* N f $ is a solution of $\db u = f$ with the least $L^2$-norm. By the estimate above, we have the following: \proclaim {\bf Corollary 3.3} Let $\Omega\subset\subset\Bbb CP^n$ and $t$ be the same as in Proposition 3.2. Then $$\p\bar\partial^* N_{(p,q)} f\p_{(t)} \le C \p f\p_{(t)}, \quad f\in\text{Ker}(\db), \ 2 \le q\le n. \tag 3.5$$ $$\p\bar\partial N_{(p,q)} f\p_{(t)} \le C \p f\p_{(t)}, \quad f\in\text{Ker}(\db^*), \ 0\le q\le n-1 . \tag 3.6$$ where $C$ depends only on $t$. \endproclaim \demo{Proof} Notice that $N = \square^{-1}$ on the range of $\square$. It follows that $N f \in \text{Dom}(\square)$. Hence, by (1.25.3), we see that $\db^*Nf $ is in $\text{Dom}(\db )\cap \text{Dom}(\db^*)$. If $q \ge 2$, for any $f\in L^2_{(p,q)}(\Omega)$, $\db^*Nf$ is a $(p,q-1)$-form. Thus for $f\in\text{Ker}(\db)$, one gets from (3.2) that $$\aligned & \int_{\Omega}(-\lambda)| f|^2 =\int_{\Omega}(-\lambda)|\bar\partial \db^*Nf|^2 + \int_{\Omega}(-\lambda)|\db^*\db^*Nf|^2{} \\ & \ge \hat C_t \int_{\Omega}(-\lambda)| \db^*Nf|^2, \endaligned $$ where we used the following facts: $f = (\db \db^* + \db^* \db) N f$, $\db^* \db^* f = 0$ and $\db N f = N \db f = 0$ with $f \in \text{ker}(\db)$. This proves inequality (3.5) by choosing $ C = \frac{1}{\hat C_t}$. Similarly, for any $0\le q\le n-1$, $\db Nf$ is in $\text{Dom}(\db )\cap \text{Dom}(\db^*)$. Substituting $\bar\partial N_{(p,q)} f$ into (3.2), for $f\in\text{Ker}(\db^*)$, we have $$\aligned & \int_{\Omega}(-\lambda)| f|^2 = \int_{\Omega}(-\lambda)|\bar\partial \db Nf|^2 + \int_{\Omega}(-\lambda)|\db^*\db Nf|^2 \\ & \ge \hat{ C}_t \int_{\Omega}(-\lambda)| \db Nf|^2. \endaligned $$ This proves the corollary by choosing $ C = \frac{1}{\hat C_t}$. \qed \enddemo Let $W^{s}_{(p, q)}(\Omega)$ be the Sobolev space with $-\frac 120$. \endproclaim \demo{Proof} From Theorem 2.6, the $\bar\partial$-Neumann operator of degree $(n-p,n-q)$ in $\Omega$, denoted by $N_{(n-p,n-q)}$, exists. Let $\star: L^2_{(p,q)}( \Omega)$ be the Hodge star operator defined as in (4.1) above. We define $$u=- {\star\bar\partial N_{(n-p,n-q)} {\star f}} = - {\bar{*}\bar\partial N_{(n-p,n-q)} {\overline{* f}}}, \tag 4.2$$ then $u\in L^2_{(p,q-1)} (\Omega) $. By the proof of Corollary 3.3, we can verify that $\star u = \db N (\star f) \in \text{Dom} (\bar\partial^*) $, because $N = \square^{-1}$ on $\text{Range}(\square)$ and $\text{Range}(N) \subset \text{Dom}(\square)$. Since $(\star u) \in \text{Dom}(\db^*)$, it follows from (1.25.2) that $u|_{T^{(0, 1) } (b \Omega ) }= 0$. Extending $u$ to $\Bbb CP^n$ by defining $u=0$ in $\Bbb CP^n \setminus \Omega$, we obtain that $u$ has support in $\overline{ \Omega}$. It follows from a theorem of Kohn-Rossi \cite{KoR} (see also Theorem 9.1.2 in Chen-Shaw \cite{CS}) that $\bar\partial u=f$ in the distribution sense in $\Bbb CP^n$. \qed \enddemo In order to solve the $\db$-equation with compact support when $q= n$, there is another compatibility condition (see (4.3) below) and we have the following result: \proclaim {Proposition 4.2} Let $\Omega$ be a pseudoconvex domain with $C^2$-smooth boundary in $\Bbb CP^n$, $n\ge 2$. For any $f\in L^2_{(p,n)}(\Bbb CP^n)$, $0\le p\le n$, such that $f$ is supported in $\overline \Omega$ and $$ \int_\Omega f\wedge g=0\qquad \text{ for every}\quad g\in L^2_{(n-p,0)} (\Omega) \cap\text{Ker}(\db),\tag 4.3$$ one can find $u\in L^2_{(p,n-1)}(\Bbb CP^n)$ such that $\bar\partial u=f$ in the distribution sense in $\Bbb CP^n$ with $u$ supported in $\overline \Omega$ and $$ \int\limits _\Omega \mid u \mid^2 dV \leq C\int\limits _\Omega \mid f \mid^2 dV, $$ for some $C>0$. \endproclaim \demo{Proof} Using Theorem 2.6, the $\db$-Neumann operator $N_{(p,0)}$ exists for any $0\le p\le n$ and we have $$N_{(p,0)}=\db^*N^2_{(p,1)}\bar\partial. \tag 4.4$$ The Bergman projection operator $P_{(p,0)}$ is given by $$\db^*\bar\partial N_{(p,0)}=I-P_{(p,0)}.\tag 4.5$$ We define $u$ by $$u=-\star {\bar\partial N_{(n-p,0)}{\star f}}.\tag 4.6$$ $$\aligned \db u&=(-1)^{p+n} {\star\bar\partial^*\bar\partial N_{(n-p,0)}\star f}\\&= f-(-1)^{p+n} {\star P_{(n-p,0)}\star f}.\endaligned \tag 4.7$$ From (4.3), we get for any $g\in L^2_{(n-p,0)} (\Omega) \cap\text{Ker}(\db)$, $$ (\star f, \bar{g})=(-1)^{p+n}\overline{ \int_\Omega g\wedge f}=0. $$ Thus $P_{(n-p,0)}( {\star f)}=0$ and $\bar\partial u=f$ in $\Omega$. Using $\star u\in \text{Dom}(\db^*)$ and extending $u$ to be zero outside $\Omega$, then $\bar\partial u=f$ in $\Bbb CP^n$ in the distribution sense. \qed \enddemo Let us now summarize the necessary and sufficient condition on $f \in W^{\frac 12}_{(p,q)}(b\Omega)$ to have a $\db $-closed extension $F$ on $\Omega$. \proclaim{Proposition 4.3} Let $\Omega\subset\subset\Bbb CP^n$ be a pseudoconvex domain with $C^2$ boundary $M = b\Omega$. Let $f\in W^{\frac 12}_{(p,q)}(b\Omega)$, where $0\le p\le n$, $ 1\le q\le n-1$. We assume that $$ \db_b f =0, \qquad \text{if }1\le q0$, the Fermi-map $h|_{M \times [-\epsilon, \epsilon]}$ is an embedding map. Let us now choose an extension $w \in L^2_{(p, q)}(\Bbb CP^n)$ of $f$ as follows: for $0 \ge t \ge -\epsilon$, let $$ w(h(Q_0, t)) = f(h(Q_0, -t)). \tag4.12 $$ Thus, $w$ is well-defined on $U_\epsilon(M)$. We further extend $w $ to $ [\Omega_- \setminus U_{\frac{\epsilon}{2}}(M)] $ so that $w|_{\overline{\Omega}_+ } = f$ and $w \in W^1_{(p,q)}(\Bbb CP^n)$. Recall that $\Omega_-$ is pseudoconvex by the assumption. Applying Proposition 4.1 to the equation $$ \db u = \db w \quad \text{ on } \Omega_-, $$ we get a solution $u$ with support in $\overline{\Omega}_-$ by choosing $$ u = - {\star\bar\partial N_{(n-p,n-q-1)} {(\star \db w)}} \quad \quad \text{ on } \Omega_-. \tag4.13 $$ Inspired by (4.0), we let $$ F_- = w - u = w + {\star\bar\partial N_{(n-p,n-q-1)} {(\star \db w)}} \quad\quad \text{ on } \Omega_- , \tag4.14 $$ where $u $ is given by (4.13). A direct computation shows that $$ \db u = \db w - (-1)^{p + q} \star P_{(n-p, n-q)} \star (\db w), \tag4.15 $$ where $P_{(n-p, n-q)}: L^2_{(p, q)}(\Omega_-) \to \text{Ker}(\db )$ is the Bergman projection. When $w$ is a $(p, q)$-form with $q < n-1$, $\db w \in\text{Ker}(\db)$ and hence $P(\star(\db w)) = 0$. By (4.15) and $P(\star(\db w)) = 0$, we have that $\db u = \db w$ and $\db F_- = 0$. By the proof of Proposition 4.1, we know that $u$ has compact support in $\overline{\Omega}_-$. Define $$F=\cases &f,\quad x \in \bar{\Omega}_+,\\&F_- ,\quad x\in \Omega_-.\endcases \tag 4.16$$ Then $F\in L^2_{(p,q)}(\Bbb CP^n)$, $F=f$ on $\bar{\Omega}_+$ and $\db F=0$ in $\Bbb CP^n$ in the distribution sense. This proves the Proposition. \qed \enddemo In order to show that (4.9) holds, it is sufficient to derive a new Liouville type Theorem, i.e., to show that $ L^2_{(n-p,0)}(\Omega) \cap \text{Ker}( \db) = 0 $ for a pseudoconcave domain and $n-p >0$. \proclaim{Proposition 4.5} Let $\Omega_+\subset\subset\Bbb CP^n$ be a pseudoconcave domain with $C^2$-smooth boundary $b\Omega_+ $. Then $L^2_{(p,0)}( \Omega_+)\cap \text{Ker}(\db)=\{0\}$ for every $0< p\le n$; and $L^2_{(0,0)}( \Omega_+)\cap \text{Ker}(\db)=\Bbb C$. \endproclaim \demo{Proof} We will extend $f$ to be a $\db$-closed form in $\Bbb CP^n$ in a similar way as in the proof of Proposition 4.4. If $f\in L^2_{(p,0)}( \Omega_+)\cap \text{Ker}(\db)$, then there exists a $\db$-closed extension $F \in L^2 $ on the whole $\Bbb CP^n$. To see this, we extend $f$ to be $\tilde f$ in $\Bbb CP^n$ such that $f_1\equiv\db \tilde f\in L^2_{(p,1)}(\Bbb CP^n)$ with support $f_1\subset\overline{\Omega}_-$, where $\Omega_-=\Bbb CP^n\setminus \overline\Omega_+$. Such an $L^2$-extension $\tilde f$ can be carried out, because $b\Omega$ is a $C^2$-smooth real hypersurface. Here is the detail for the construction of $\tilde f$ and $f_1$. Let $h: b\Omega \times [-\epsilon, \epsilon] \to \Bbb CP^n $ be the Fermi-map given by (4.11). For sufficiently small $\epsilon$, $h$ is an embedding map which gives rise to the so-called Fermi coordinate along $b\Omega$. We now define $ \tilde f(h(Q, t)) = f (h(Q, -t))$ for $h(Q, t) \in \Omega_-$ and $|t| < \epsilon/2 $. Outside the annuli neighborhood $h( b\Omega \times [-\epsilon, \epsilon] ) $, we require $\tilde f$ to be $C^2$-smooth. Therefore, the $(p, 1)$-form $f_1\equiv\db \tilde f \in L^2_{(p,1)}(\Bbb CP^n)$ with support $f_1\subset\overline{\Omega}_-$. Using Propositions 4.1-4.4, we see that there exists a solution $v_0\in L^2_{( p,0)}(\Omega_-)$ such that $\db v_0=f_1$ in $\Bbb CP^n$ if we extend $v_0$ to be zero on $\Omega_+$. Thus, the extension form $F =\tilde f -v_0$ is $\db$-closed in the whole space $\Bbb CP^n$. We observe that \newline $L^2_{( p, 0)}(\Bbb CP^n)\cap \text{Ker}(\db) =\{0\}$ for all $0< p \le n$; and $L^2_{( 0, 0)}(\Bbb CP^n) \cap \text{Ker}(\db) = \Bbb C$. Hence, $f = 0$ when $p> 0$, and $f$ must be a constant when it is a $L^2$-holomorphic function on $\Omega_+$. This proves Proposition 4.5. \qed \enddemo Some related results similar to Propositions 4.4-4.5 were obtained by Henkin-Iordan [HI] earlier via a different argument. Let us now apply our results above to a $C^2$-smooth Levi-flat hypersurface $M$ if it exists. \proclaim{Corollary 4.6} Let $M\subset\Bbb CP^n$ be a $C^2$-smooth Levi-flat hypersurface. For any $f\in W^{\frac 12}_{(p,n-1)}(M)$, where $0\le p< n$, there exists a $\db$-closed form $F\in L^2_{(p,n-1)}(\Bbb CP^n)$ such that $F$ is a $\db$-closed extension of $f$. \endproclaim \demo{Proof} Let $M$ be a compact connected $C^2$-smooth real hypersurface in $\Bbb CP^n$. It is well-known that $f \in W^{\frac 12}_{(p,q)}(M)$ if and only if there exists an extension $\tilde f$ of $f$ with $\tilde f \in W^{1}_{(p,q)}(\Bbb CP^n)$. Since $\Bbb CP^n$ is simply-connected, $\Bbb CP^n\setminus M = \Omega_+ \cup \Omega_-$ has exactly two connected components: $\Omega_+$ and $\Omega_-$. Since $\Omega_{\pm}$ is pseudoconvex, we let $$ F_{\pm} = \tilde f + \star[\bar\partial N_{(n-p,0)}|_{\Omega_\pm}] {(\star \db \tilde f)} \quad \quad \text{ on } \Omega_{\pm} \tag4.17 $$ and $ v_{\pm} = - \star\bar\partial N_{(n-p,0)} {(\star \db \tilde f)}$. By Theorem 2.6, we have $F_\pm \in L^2_{(p,n-1)}(\Bbb CP^n)$. Since $M$ is Levi-flat, we see that both $\Omega_+$ and $\Omega_-$ are pseudoconcave as well. By Proposition 4.5, the form $f_1 = \db \tilde f $ satisfies (4.3) and (4.9) on each of $\Omega_\pm$. By the proofs of Propositions 4.1-4.5, we conclude that $F$ is the $\db $-closed extension of $f$ on the whole $\Bbb CP^n$. \qed \enddemo We are now ready to study the inhomogeneous equation (0.4). \proclaim{Corollary 4.7} Let $M\subset\Bbb CP^n$ be a $C^2$-smooth Levi-flat hypersurface. Let $f\in W^{\frac 12}_{(p,q)}(M)$, where $0\le p\le n $, $1\le q\le n-1$ and $p\neq q$. We assume that $f$ is $\db_b$-closed in $M$ if $q 0$, see Lemmas 5.1-5.2 below. \proclaim{Lemma 5.1} Let $M\subset\Bbb CP^n$ be a $C^2$-smooth Levi-flat hypersurface and let $f\in W^{\frac 12}_{(p,q)}(M)$, where $0\le p\le n $, $1\le q\le n-1$ and $p\neq q$. Suppose that $\epsilon_0= \frac 12 \min\{t_0(\Omega_+), t_0(\Omega_-)\}$ and $t_0(\Omega)$ is the order of plurisubharmonicity of $\Omega$ given by Definition 0.1. We further assume that $f\in W^{\frac 12+\epsilon}_{(p,q)}(M)$ is $\db_b$-closed in $M$ if $q2$, $$\db_b\beta^{0,1}=0\quad \text{in } M. \tag 5.5$$ When $n =2$, $\beta^{0,1}$ satisfies the compatibility condition (4.9) by Proposition 4.5. We claim that there exists a solution $u$ of (5.4) such that $u\in W^{\frac 12+\epsilon}(M)$ for some $\epsilon>0$. If $\alpha\ge \frac 12$, we can use Lemma 5.1 directly. If $\alpha<\frac 12$, we note that the Cartan-Chern-Gauss structure equation holds in a tubular neighborhood $U = U_\epsilon(M)$ of $M$. Thus, $\beta^{0,1}\in C^\alpha(U)\subset W^\alpha(U)$. Using the last assertion of Proposition 1.1, $\db \beta^{0,1}$ is in $C^\alpha(U)$ since it is the $(0,2)$-component of the curvature form $\tilde\Theta^N$, which is in $C^\alpha$. Thus $ \beta^{0,1}$ has an extension to $\Bbb CP^n $ with $\db\beta^{0,1}$ in $W^{\alpha}_{(0,2)}(\Omega_+)$ and $W^{\alpha}_{(0,2)}(\Omega_-)$ respectively. Repeating the same arguments as in the proof of Proposition 4.4 for $q0$ independent of $t$ such that $$\|u\|_{L^2 (\Sigma_t)}\le C_2 \|u\|_{W^{\frac 12+\epsilon} (M)}. \tag 5.10$$ Combining (5.9) and (5.10), we get $$ |u|_{L^\infty(V)}\le \underset{|t| < \mu}\to\sup |u|_{C^{2,\alpha}(\Sigma_t)}\le C_3.\tag 5.11 $$ Thus we have already proved that $u$ is bounded. It remains to show that $u$ is H\"older continuous in $t$ as well. To do this we differentiate the equation (5.8) in $t$ with order $0 < \epsilon < 1$. Let $D_{t,h}^\epsilon $ denote the finite difference $$D_{t,h}^\epsilon u= \frac{ u(z',t+h)- u(z',t) }{ |h|^\epsilon} $$ and let $|\ |_{ s}$ be the Besov norm given by $$|u|_s=\underset{0<|h|<\eta_0}\to\sup \frac{ \|u(x+h)-u(x)\|}{|h|^s}.$$ The Besov norm is weaker than the Sobolev norm $\|u\|_s$. It is easier to use the Besov norm than the Sobolev norm (see e.g. H\"ormander \cite{H\"o4}) and we have $$\|u\|_{s'}\le |u|_s\le \|u\|_s\tag 5.12$$ for any $s'