Existence theorems for the dbar equation and Sobolev estimates on $q$-convex domains

1.   Introduction

Several complex variables involve the $\overline\partial$ problem, and Kohn solved this problem in 1963 for strongly pseudoconvex domains. It is useful to use Sobolev estimates in various areas of mathematics, such as complex geometry and partial differential equations on pseudoconvex manifolds. Introducing a sequence of subelliptic multiplier ideals, he gave a sufficient condition for subellipticity in pseudoconvex domains with real analytical boundaries. Catlin proved the most general result regarding subelliptic estimates for the $\overline\partial$-Neumann problem. In ^[1], he showed that subelliptic estimates hold for $k$-forms at $z_0$ within a smooth and bounded pseudoconvex domain. Herbig ^[2] extended Catlin's result to a weak condition for boundedness in the sense of weight functions. Hörmander ^[3] and Folland-Kohn ^[4] proved that subelliptic $\frac{1}{2}$ estimate can be estimated on non-pseudoconvex domains. For more details, we refer the readers to ^{[5,6,7,8,9,10,11,12]}.

We are motivated to give subelliptic estimates for the $\overline\partial$-Neumann problem on smooth bounded, weak $\operatorname{Z}(k)$ domains on a $\operatorname{Stein\, manifold}$ for $(\operatorname{p}, \operatorname{k})$-forms, with $k\geqslant 1$ with values in holomorphic vector bundles. Sobolev estimates of $\operatorname{N}$ on ${ M }$ for all $\overline\partial$-closed $(\operatorname{p}, \operatorname{k})$-forms. We also deduce some standard compactness consequences.

Further, if $\Xi$ is the $m$-times tensor product of holomorphic line bundle $\Xi^{\otimes \operatorname{m}}$ for integer $m > 0$, we study the $\overline\partial$ problem with support conditions in ${ M }$ for $\Xi$-valued $(\operatorname{p}, \operatorname{k})$-forms with values in $\Xi^{\otimes \operatorname{m}}$. This problem had already been discussed on domains like: Strongly $q$-convex (or concave) ^[13], pseudo-convex with $C^1$ boundary ^[14] and local Stein of the complex projective space ^[15]. We also refer the readers to ^{[13,16,17,18,19,20,21,22,23]}.

Finally, we assume that ${ M } = { M }_{1}\backslash\overline{ M }_{2}$ is an annular domain in a $\operatorname{Stein\, manifold}$, between two smooth bounded domains ${ M }_{1}$ and ${ M }_{2}$ satisfy $\overline{ M }_{2}\Subset{ M }_{1}$, ${ M }_{1}$ is weak $Z(k)$, ${ M }_{2}$ is weak $Z(n-1-k)$, $1 \leqslant k\leqslant n -2$ with $n \geqslant 3$. We prove a basic prior estimate for the weighted $\overline\partial$-Neumann problem on ${ M }$. This estimate is validated for vector bundle forms. Moreover, we also study the global boundary of $\overline\partial$ within such domains. Cho ^[24] says global boundary regularity is obtained when $M _{1}$ and $M _{2}$ are pseudoconvex manifolds. The boundary regularity and the closed range property of $\overline\partial$ were established in ^[14,25,26] for $0 < k < n-1$ and $n \geq 3$. There are also pseudoconvex and non-pseudoconvex domains in ^[15,27,28], as well as the author's results ^{[29,30,31,32,33,34,35,36,37,38,39]}. Similar results can be found in ^[40,41].

The novelty of this study is the investigation of a sufficient condition for subelliptic estimates on the weak $\operatorname{Z}(k)$ domain. Moreover, we demonstrate that $\overline\partial$-Neumann operators are compact. In addition, we examine a weighted $\overline\partial$ Neumann operator over an annular domain between two smooth-bounded domains. Despite this, all results are obtained on weak $\operatorname{Z}(k)$ domains, which contrasts to previous works that were based on strong pseudoconvex domains and non-pseudoconvex domains.

2.   Notations and definitions

Let $\operatorname{p}, \operatorname{k}\geq0$, $\operatorname{n}\geqslant 1$ be an integer and let $\operatorname{X}$ be a complex manifold of dimension $n$. Let $M \Subset\operatorname{X}$ be a subset of $\operatorname{X}$, and let $\rho$ be its defining function. Let $T^{1, 0}(b{ M })$ be the complex tangent bundle to the boundary $b{ M }$, with $T^{0, 1}(b{ M }) = \overline{T^{1, 0}(b{ M })}$. Suppose that $\Xi^*$ is the dual of a holomorphic line bundle $\Xi$ over $\operatorname{X}$. In local coordinates $(z^{1}_{j}, z^{2}_{j}, \ldots, z^{n}_{j})$ on open covering $\{\operatorname{V}_{\operatorname{j}}\}_{j\in J}$ of $\operatorname{X}$, $\Xi|_{{V}_{j}}$ is trivial. $\{f_{ab}\}$ is a transition function system of $\Xi$ in sense of $\{\operatorname{V}_{\operatorname{j}}\}_{j\in J}$. A $(\operatorname{p}, \operatorname{k})$ forms $\sigma = \{\sigma_j\}$ on $\operatorname{X}$ is given by:

where $C_{\operatorname{p}} = (c_{1}, \ldots, c_{\operatorname{p}})$ and $D_{\operatorname{k}} = (d_{1}, \ldots, d_{\operatorname{k}})$ are multiindices. A hermitian metric on $\operatorname{X}$ is

Associate $\operatorname{G}$ with the differential form $\omega = \frac{\sqrt{-1}}{2}\sum_{\sigma, \beta = 1}^{n} \operatorname{g}_{j, \sigma\overline{\beta}} (z)\, dz_j^{\sigma} \wedge d{\overline z}_j^{\beta}$ of type $(1, 1)$. $h = \{h_a\}$ is a hermitian metric of $\Xi = \{f_{ab}\}$ in sense of $\{V_a\}_{a\in J}$, so that $h_a = |f_{ab}|^2h_b$ on $V_a\cap V_b$. ${\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is the complex vector space of ${\operatorname{C}}^{\infty}$ $\Xi$-valued $(\operatorname{p}, \operatorname{k})$-differential forms on ${ M }$. ${\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline{{ M }}, \Xi) = \{u |_{\overline{ M }}; u \in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\operatorname{X}, \Xi)\}$. The space of $\Xi$-valued $(\operatorname{p}, \operatorname{k})$-differential forms with compact support in ${ M }$ is denoted by $\operatorname{D}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$. $\divideontimes_{\Xi}:{\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(X, \Xi)\longrightarrow {\operatorname{C}}^{\infty}_{\operatorname{k}, \operatorname{p}}(\operatorname{X}, \Xi^{\ast})$ is defined by $\divideontimes_{\Xi}\sigma = \, h\, \overline\sigma,$ which commutes with the Hodge star operator $\star: {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(X, \Xi) \longrightarrow{\operatorname{C}}^{\infty}_{n-\operatorname{k}, n- \operatorname{p}}(X, \Xi)$. $\divideontimes_{\Xi^{*}}:{\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(X, \Xi^{\ast})\longrightarrow {\operatorname{C}}^{\infty}_{\operatorname{k}, \operatorname{p}}(X, \Xi)$ satisfies

with $\divideontimes_{\Xi^{*}}\sigma = \divideontimes_{\Xi}^{-1}\sigma$. $\mathcal{B}_{\operatorname{p}, \operatorname{k}}(\overline{{ M }}, \Xi) = \{\sigma \in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline{{ M }}, \Xi); \, \, \star \, \divideontimes_{\Xi}\, \sigma|_{b{ M }} = 0 \}$. The volume element related to $\operatorname{G}$ is $\operatorname{dV}$. $\overline{\partial}: {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi) \longrightarrow {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is the Cauchy-Riemann operator and $\vartheta$ its formal adjoint. ${\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(b{ M }, \Xi) = {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline{ M }, \Xi)/ \operatorname{D}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$. For $\sigma, \varrho \in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(X, \Xi)$,

is the inner product. For $\sigma, \varrho \in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(X, \Xi)$,

are the global inner product and the norm, respectively. For $\sigma \in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ and $\varrho \in \operatorname{D}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$, one obtains

$\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is the Hilbert space obtained by completing the space ${\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline{{ M }}, \Xi)$ under the norm $\| \sigma \|_{{ M }}$. The maximal closed extension of $\overline{\partial}$ is $\overline{\partial}: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi) \longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, and $\overline{\partial}^{\ast}$ its Hilbert space adjoint. $\square = \square_{\operatorname{p}, \operatorname{k}} = \overline\partial\, \overline\partial^{\ast}+ \overline\partial^{\ast}\overline\partial: \mathfrak{Dom}(\square_{\operatorname{p}, \operatorname{k}}, \Xi) \longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is the Laplace operator defined for $\Xi$-valued forms. $\mathfrak{Dom}(\square_{\operatorname{p}, \operatorname{k}}, \Xi) = \{\sigma \in \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi): \sigma\in\mathfrak{Dom}(\overline\partial, \Xi) \cap \mathfrak{Dom}(\overline\partial^{\ast}, \Xi), \overline\partial \sigma \in \mathfrak{Dom}(\overline\partial^{\ast}, \Xi)$ and $\overline\partial^{\ast}\sigma\in \mathfrak{Dom}(\overline\partial, \Xi)\}$. $\mathcal{H}_{\operatorname{p}, \operatorname{k}}(\Xi) = \{\sigma \in \mathfrak{Dom}(\square_{\operatorname{p}, \operatorname{k}}, \Xi): \overline\partial\sigma = \overline\partial^{\ast} \sigma = 0 \}$. $\operatorname{N} = \operatorname{N}_{\operatorname{p}, \operatorname{k}}:\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is the $\overline\partial$-Neumann operator and is given as

For ${\operatorname{s}}\in\Bbb{R}$, the Sobolev $\Xi$-valued of $(\operatorname{p}, \operatorname{k})$-forms is given by $\operatorname{W}^{{\operatorname{s}}}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ with $\operatorname{W}^{{\operatorname{s}}}({ M }, \Xi)$-coefficients and $\|\sigma\|_{\operatorname{W}^{{\operatorname{s}}}(\Xi)}$ their norms. The curvature form $\sum_{\sigma, \beta = 1}^{n} \left(-\frac{\partial^2\text{log}\, h_j}{\partial z_j^{\sigma}\partial \overline{z}_j^{\beta}}\right)\, dz^{\sigma} \wedge d{\overline z}^{\beta}$ of $\Xi$ provides a K$\ddot{a}$hler metric $d\sigma^2 = \sum_{\sigma, \beta = 1}^{n} \left(-\frac{\partial^2\text{log}\, h_j}{\partial z_j^{\sigma}\partial \overline{z}_j^{\beta}}\right)\, dz^{\sigma} \, d{\overline z}^{\beta}$ on $\, V$.

Definition 1. A $\epsilon$-subelliptic estimate for the $\overline{\partial}$-Neumann problem is satisfied at $z_0\in\overline{ M }$ on $k$-forms, $\epsilon > 0$, if there exists a constant $c > 0$ and a neighborhood $V \ni z_0$ such that

Definition 2. ^[9,10] A boundary $b{ M }$ is said to has the $(\operatorname{k}-\operatorname{P})$ property in $V$ if for every $\operatorname{T} > 0$, denote by $\lambda_{1}^{\phi^{{\operatorname{T}}} }\leqslant\lambda_{2}^{\phi^{{\operatorname{T}}} }\leqslant.....\leqslant\lambda_{n-1}^{\phi^{{\operatorname{T}}} }$ the eigenvalues of the Levi form $(\phi^{{\operatorname{T}}}_{ij})$, there is a function $\phi^{\operatorname{T}} \in C^\infty(\overline{ M }\cap V)$ with $|\phi^{\operatorname{T}}|\leqslant 1$ on ${ M }$ and so that

where $\epsilon > 0$ and $\operatorname{C} > 0$ does not depend on $\delta$ and $\operatorname{s}$.

Define the Levi form $\mathscr{L}$ as: $\forall$ $p \in b{ M }$, with $\frac{\partial \zeta}{\partial z_{j}}(p) = 0$ $\forall$ $1 \leqslant j \leqslant n-1$.

Definition 3. ^[42] For $1 \leqslant k\leqslant n-1$, $b{ M }$ is said satisfies weak $\operatorname{Z}(k)$ if there exists a real $\Gamma \in T^{1, 1}(b{ M })$ satisfying

(1) $|\gamma|^{2} \geqslant(i \gamma \wedge \overline{\gamma})(\Gamma) \geqslant 0$ $\forall$ $\gamma \in \varrho^{1, 0}(b{ M })$.

(2) $\mu\sigma_{1}+\cdots+\mu\sigma_{k}-\mathscr{L}(\Gamma) \geqslant 0$ where $\mu\sigma_{1}, \ldots, \mu\sigma_{n-1}$ are the eigenvalues of $\mathscr{L}$.

(3) ${ M }(\Gamma) \neq k$.

Lemma 1. ^[42] For $1 \leqslant k\leqslant n-2$, let ${ M } \subset\operatorname{X}$ be a bounded domain and $B \subset \operatorname{X}$ be a bounded pseudoconvex domain satisfies $\overline{{ M }} \subset B$. Then $b{ M }$ satisfies weak $\operatorname{Z}(k)$ if and only if $b(B / \overline{{ M }})$ satisfies weak $\operatorname{Z}(n-k-1)$.

If $\mu\sigma_{1}, \ldots, \mu\sigma_{n-1}$ are the eigenvalues of $\mathscr{L}$, then one obtains

Definition 4. A form $\sigma\in \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is supported in $\overline{ M }$ if $\sigma$ vanishes on $b{ M }$.

3.   Main results

3.1.   Subelliptic estimates

Theorem 1. With a smooth boundary, let ${ M }\Subset\Bbb{\operatorname{C}}^n$ be a weak $\operatorname{Z}(k)$ domain. Suppose that $b{ M }$ has the property $(\operatorname{k}-\operatorname{P})$. Then, $\epsilon$-subelliptic estimates at $z_0$ hold for $(\operatorname{p}, \operatorname{k})$-forms. That is, there exists $c > 0$ such that

for $\sigma\in \operatorname{D}_{\operatorname{p}, \operatorname{k}}({ M })$.

Proof. Let $\operatorname{B}_\delta = \{z \in{ M }: -\delta < \rho(z)\leqslant0\}$ be a strip, where $\delta > 0$ small enough. As in Khanh and Zampieri ^[10],

for $\sigma\in \operatorname{D}_{\operatorname{p}, \operatorname{k}}(\operatorname{B}_\delta\cap{ M })$ with $k \geqslant 1$. From the compactness of $b{ M }$, using a finite covering $\{\Delta_\phi\}_{\nu = 1}$ of $b{ M }$ by neighborhoods $\Delta_\phi$ as in (3.2), we have

with $u$ is supported in $\operatorname{B}_\delta$.

If $\rho(z) \leqslant -\delta$ and $z\in{ M }\backslash \operatorname{B}_\delta$, taking $\gamma_\delta\in\operatorname{D}({ M })$ with $\gamma_\delta(z) = 1$. Using (3.3),

□

Theorem 2. Let Let $M , X$ be the same as in Theorem 1. Let $\Xi$ be a holomorphic vector bundle, of rank $r$, on $\operatorname{X}$. Suppose that $b{ M }$ has the property $(\operatorname{k}-\operatorname{P})$. Then, there exists $C > 0$ satisfies

for $\sigma\in \operatorname{D}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$.

Proof. By a local patching, one assume that $\{U_{j}\}_{j = 1}^{N}$ is a finite covering of $b M$. Extend the subelliptic estimate (3.1) to $E$-valued forms. An orthonormal basis could be $e_{1}, e_{2}, \ldots, e_{r}$ for $z\in U_{j}$; $j\in J$. Thus $\sigma(z) = \sum_{a = 1}^{r}\sigma^{a}(z)\, e_{a}(z)$, where $\sigma^{a}$ are the components of the restriction of $\sigma$ on $U_{j}$. Let $\{\zeta_{j}\}_{j = 0}^{m}$ be a partition unity. This partition of unity is $\zeta_{0}\in \mathcal{D}_{\operatorname{p}, \operatorname{k}}( M )$, $\zeta_{j}\in\mathcal{D}_{\operatorname{p}, \operatorname{k}}(U_{j})$, $j = 1, 2, ..., m$. $\sum_{j = 0}^{m}\zeta_{j}^{2} = 1\, \, \, \text{on}\, \, \, \overline M ,$ where $\{U_j\}_{j = 1, ..., m}$ is a covering of $b M$.

For a given $j_\nu\in I$, let $U$ be a neighborhood of a given boundary point $\xi_{0}\in b M$. Using $\sigma\in \mathcal{D}_{\operatorname{p}, \operatorname{k}}( M , \Xi)$, $1\leqslant k\leqslant n-2$, and $a = 1, ..., r$, we get a subelliptic estimate from (3.1), for $\sigma|_{ M \cap U}$.

Thus, subelliptic estimate for $\sigma|_{ M \cap U_j}$ is

Summing up over $j$, we get

Thus (3.4) follows. □

3.2.   Compactness estimates

Lemma 2. ^[43] Let ${ M }\Subset X$ be a weak $\operatorname{Z}(k)$ domain with $\operatorname{\operatorname{C}}^{3}$ boundary in $\operatorname{Stein\, manifold}$ $\operatorname{X}$.

(1) $\forall$ constant $\epsilon > 0$ there exists $t_{\epsilon} > 0$ and a $\operatorname{C}_{\epsilon} > 0$ satisfy $\forall$ $t \geqslant t_{\epsilon}$ and $\sigma \in L_{\operatorname{p}, \operatorname{k}}^{2}\left({ M }, e^{-\operatorname{t}\varrho}\right) \cap \mathfrak{Dom}(\overline{\partial}) \cap \mathfrak{Dom}\left(\overline{\partial}_{{\operatorname{t}}}^{*}\right)$ we have

(2) there exists constants $C > 0$ and $\tilde{t} > 0$ satisfy $\forall$ $t \geqslant \tilde{t}$ and $\sigma \in L_{\operatorname{p}, \operatorname{k}}^{2}\left({ M }, e^{-\operatorname{t}\varrho}\right) \cap \mathfrak{Dom}(\overline{\partial}) \cap \mathfrak{Dom}\left(\overline{\partial}_{{\operatorname{t}}}^{*}\right) \cap\left(\mathcal{H}_{{\operatorname{t}}}^{k}({ M })\right)^{\perp}$ we have

(3) If $b{ M }$ is connected, $\forall$ constant $\epsilon > 0$ there exists $t_{\epsilon} > 0$ so that $\forall$ $t \geqslant t_{\epsilon}$ and $\sigma \in L_{\operatorname{p}, \operatorname{k}}^{2}\left({ M }, e^{-\operatorname{t}\varrho}\right) \cap \mathfrak{Dom}(\overline{\partial}) \cap \mathfrak{Dom}\left(\overline{\partial}_{{\operatorname{t}}}^{*}\right)$ we have

Theorem 3. Let ${ M }\Subset X$ be a weak $\operatorname{Z}(k)$ domain with $\operatorname{\operatorname{C}}^{3}$ boundary in $\operatorname{Stein\, manifold}$ $\operatorname{X}$. Then the compactness estimate for a holomorphic vector bundle $\Xi$-valued $(\operatorname{p}, \operatorname{k})$ form holds on ${ M }$. Then, $\forall$ $c > 0$, there exists a $t > 0$ and $C_{c, t} > 0$ such that

for $\sigma\in \mathscr{D}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$.

Proof. As Theorem 2, for $\sigma\in \mathscr{D}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, $1\leq k\leq n-2$, over each $\sigma^{a}$ by applying (3.5) and adding for $a = 1, ..., r$, we get compactness estimate for $\sigma\mid_{{ M }\cap U}$

For $j = 1, ..., m$, for $u|_{{ M }\cap \sigma_j}$, we have

Let's sum $j$ up

Thus (3.6) follows. □

Proposition 1. Assuming the same assumptions as Theorem 3, let us assume the following: $\ker(\square, \Xi)$ is $\operatorname{finite\, dimension}$al and $\operatorname{Ran}(\square, \Xi)$ is closed in $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ and there exists a bounded linear operator $\operatorname{N}: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ satisfies

(i) $\operatorname{Ran}(\operatorname{N}, \Xi)\subset\mathfrak{Dom}\, (\square, \Xi)$, $N \square = I -\Bbb{H}$ on $\mathfrak{Dom}\, (\square, E)$,

(ii) for $\sigma\in \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, $\sigma = \overline\partial\, \overline\partial^{\ast} N\sigma \oplus\, \overline\partial^{\ast}\overline\partial N\sigma\oplus\, \Bbb{H}\sigma,$ (iii) $N\overline\partial = \overline\partial N$, and $N\overline\partial^{\ast} = \overline\partial^{\ast} N$.

(iv) $\forall$ $\sigma\in \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$,

(v) If $\sigma\in \operatorname{L}^2_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, with $\overline\partial\sigma = 0$ (resp. $\overline\partial^{\ast}\sigma = 0$), then $\overline\partial^{\ast}\operatorname{N}\sigma$ (resp.$\overline\partial \operatorname{N}\sigma$) gives the solution of $\overline\partial u = \sigma$ (resp. $\overline\partial^{\ast} u = \sigma$) of minimal $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$(resp. $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}+1}({ M }, \Xi)$)-norm.

Proof. Applying (3.6) at $\epsilon = \frac{1}{2}$, for $\sigma\in\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\cap \mathfrak{Dom}\, (\overline\partial, \Xi)\cap\mathfrak{Dom}\, (\overline\partial^{\ast}, \Xi)$, one obtains

Then, $\operatorname{N}_{\operatorname{p}, \operatorname{k}}: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{W}^{1}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$. Following (3.7), every sequence $\{\sigma_\phi\}_{\phi = 1}^\infty$ in $\mathfrak{Dom}\, (\overline\partial, \Xi)\cap\, \mathfrak{Dom}\, (\overline\partial^{\ast}, \Xi)$ with $\|\sigma_\phi\|$ bounded, $\overline\partial \sigma_\phi\longrightarrow 0$ in $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}+1}({ M }, \Xi)$ and $\overline\partial^{\ast} \sigma_\phi\longrightarrow 0$ in $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$ as $\phi \longrightarrow\infty$, then (3.6) implies that $\left\|\sigma_\phi\right\|^{2}_{\operatorname{W}^{\frac{1}{2}}(\Xi)}\leqslant C$ for some constant $C$. Thus, the inclusion map $i_{ M }: \operatorname{W}^{\frac{1}{2}}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is compact. By Rellich Lemma, a subsequence of the sequence $\sigma_\phi$ can be extracted which converges in the $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$-norm. Thus, following Theorem 1.1.3 in ^[3], for $\sigma\in \mathfrak{Dom}\, (\overline\partial, \Xi)\cap\mathfrak{Dom}\, (\overline\partial^{\ast}, \Xi)$, $\sigma\perp\ker(\square, \Xi)$, $\ker(\square, \Xi)$ is $\operatorname{finite\, dimension}$al and one obtains

Then

Since $\square$ is self-adjoint, thus following Theorem 1.1.1 in ^[3], one obtains

According to (3.8) there's a unique bounded operator $\operatorname{N}$ on $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ that inverts $\square$ on $\ker(\square, \Xi)^{\perp}$. Extend $\operatorname{N}$ to the whole $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ space by setting $N = 0$ on $\ker(\square, \Xi)$. The rest of the proof follows Theorem 3.1.14 in ^[28]. □

Corollary 1. Assuming the same assumptions as Theorem 3, we have the following:

(i) the $\overline\partial$-Neumann operator $\operatorname{N}$ exists and $\operatorname{N}: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{W}^{1}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$.

(ii) For $\sigma\in\operatorname{W}^{\frac{1}{2}}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, there exists $u\in\operatorname{W}^{\frac{1}{2}}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$ with $\overline\partial u = \sigma$.

(iii) $\operatorname{N}: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is compact.

Proof. (ⅰ) From (3.8), for $\sigma\in \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\cap\mathfrak{Dom}\, (\overline\partial, \Xi)\cap\mathfrak{Dom}\, (\overline\partial^{\ast}, \Xi)$,

Thus, the existence of $\operatorname{N}: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{W}^{1}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ follows.

(ⅱ) From Eq (3.7), $\forall$ $\sigma\in \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\cap\ker(\overline\partial, \Xi)$ and $\sigma\perp\ker(\square, \Xi)$, there exists a $u\in\operatorname{W}^{\frac{1}{2}}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$ with $\overline\partial u = \sigma$.

(ⅲ) To prove the compactness of $\operatorname{N}$, since $N = 0$ on $\ker(\square, \Xi)$, it suffices to show compactness on $\ker(\square, \Xi)^{\perp}$. When $\sigma\in\ker(\square, \Xi)^{\perp}$ and hence $N\sigma\in\ker(\square, \Xi)^{\perp}$, the integration by parts, inequality (3.8) and the Cauchy-Schwarz inequality imply

Following (3.7)–(3.9), we get

where $K$ is a positive constant. Thus, by the Rellich Lemma, the compactness of $\operatorname{N}$ follows on $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, that is, the embedding of $\operatorname{W}^{\frac{1}{2}}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ into $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is compact. □

3.3.   Global regularity and closed range for $\overline\partial$

Lemma 3. Assuming the same assumptions as Theorem 3. Let $1\leq \operatorname{k}\leq n-2$, $n\geq3$, then there exists $C > 0$ satisfies for all $\sigma\in\mathscr{D}^{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ with $\sigma\perp\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$, we have

Proof. If for any $\nu\in\Bbb{N}$ there exists a $\sigma_{\nu}\perp\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$, with $\|\sigma_{\nu}\|_{\mathtt{t}} = 1$ so that

Combining with (3.6), we have

Then $\sigma_{\nu}\rightarrow \sigma$ in $\operatorname{L}^{2}$, where $\sigma\perp\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\operatorname{W}^{0}(\Xi)}(\Xi)$. By (3.7) we have that $\sigma\in\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$, a contradiction. Thus (3.10) must hold $\forall$ $\sigma\perp\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$. □

Using (3.10), as in ^[3,28], we have

Lemma 4. Assuming the same assumptions as Theorem 3. Let $1\leq \operatorname{k}\leq n-2$, $n\geq3$, then we have

(1) $\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$ is $\operatorname{finite\, dimension}$al.

(2) $\square^{\mathtt{t}}$ has closed range in $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$.

(3) $\overline\partial$ (resp. $\overline\partial^{\ast}_{\mathtt{t}}$) has closed range in $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ and $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}+1}({ M }, \Xi)$ (resp. $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$).

Proof. Following (3.10), every sequence $\{\sigma_{\nu}\}_{\nu = 1}^{\infty}$ in $L^2_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ with $\|\sigma_\nu\|_{\mathtt{t}}$ is bounded and $\overline\partial \sigma_\nu\longrightarrow0,$ $\overline\partial^*_{\mathtt{t}} \sigma_\nu\longrightarrow 0,$ one can extract a subsequence which converges in $L^2_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$. Since $L^2_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\hookrightarrow \operatorname{W}^{-1}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ is compact, (3.7) implies that such a subsequence is convergent in $L^2_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$. Following Theorems 1.1.3 and 1.1.2 in ^[3], implies that $\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$ is $\operatorname{finite\, dimension}$al. Thus, $\overline\partial: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}+1}({ M }, \Xi)$ and $\overline\partial^{\ast}_{\mathtt{t}}: \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)\longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$ have closed range. □

Theorem 4. Assuming the same assumptions as Theorem 3. Let $1\leq \operatorname{k}\leq n-2$, $n\geq3$, then for $\sigma\in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline { M }, \Xi)$, satisfying $\overline{\partial} \sigma = 0$ in the distribution sense in $\operatorname{X}$, there exists $u\in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}-1}(\overline{ M }, \Xi)$, satisfies $\overline{\partial} u = \sigma$ in $\operatorname{X}$.

Proof. The proof follows as ^[29,30]. □

4.   Solution of the dbar problem with support conditions

Proposition 2. ^[43] Let ${ M }\Subset X$ be a weak $\operatorname{Z}(k)$ domain with smooth boundary in a complex manifold $\operatorname{X}$. Assume that $\operatorname{s} > 0$ is an integer, $\overline\nabla$ is the covariant differentiation of type $(0, 1)$ associated with the metric $\operatorname{G}$, and $\Xi^{\otimes {\operatorname{s}}}$ is the $\operatorname{s}$-times tensor product of a holomorphic line bundle $\Xi$. Suppose that there exists a strongly plurisubharmonic function on a neighborhood $U^*$ of $b{ M }$. We have

for $\sigma\in \mathcal{B}_{\operatorname{p}, \operatorname{k}}(\overline{ M }, \Xi^{\otimes {\operatorname{s}}})$, so that $\sigma$ is supported in $U^*$, and ${\operatorname{k}}\geqslant1$, where

and

is the Riemann curvature tensor,

is the Ricci curvature tensor, and the curvature tensor of $\Xi$ is given by

where $\delta_\epsilon^\sigma$ is the Kronecker's delta.

Proposition 3. ^[43] With the same assumptions as in Proposition 2, let us assume the following: There exists a constant $C > 0$ not depending on $\operatorname{s}$ and an integer ${\operatorname{s}}_0 > 0$ so that for all $\operatorname{s}\geqslant {\operatorname{s}}_0$, $\operatorname{k}\geqslant 1$, we have

where $K = { M }\backslash ({ M }\cap\, V)$ is the compact subset of ${ M }$.

Proposition 4. With the same assumptions as in Proposition 2, let us assume the following: There exists a constant ${\operatorname{s}}_* > 0$ satisfies $\forall$ $\operatorname{s}\geqslant {\operatorname{s}}_*$, the harmonic space $\mathcal{H}^{{\operatorname{s}}}_{\operatorname{p}, \operatorname{k}}(\Xi^{\otimes {\operatorname{s}}})$ has $\operatorname{finite\, dimension}$ and there exists a constant $C_{\operatorname{s}} > 0$ depending on $\operatorname{s}$ such that

for $\sigma\in\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi^{\otimes {\operatorname{s}}})\cap\mathfrak{Dom}\, (\overline\partial, \Xi^{\otimes {\operatorname{s}}})\cap\mathfrak{Dom}\, (\overline\partial^*_ {{\operatorname{s}}}, \Xi^{\otimes {\operatorname{s}}})$ with $k\geqslant 1$.

Proof. Using (4.1), the proof follows as in Saber ^[29,30]. □

Proposition 5. With the same assumptions as in Proposition 2. Assume that there exists a positive integer $m^*$ satisfies, for ${\operatorname{s}}\geqslant {\operatorname{s}}^*$, $\operatorname{k}\geqslant 1$, then there exists a bounded linear operator $\operatorname{N}^{{\operatorname{s}}}:\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi^{\otimes {\operatorname{s}}})\longrightarrow \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi^{\otimes {\operatorname{s}}})$ such that

(i) $\mathfrak{Ran}(\operatorname{N}^{{\operatorname{s}}}, \Xi^{\otimes {\operatorname{s}}})\subset\mathfrak{Dom}(\square^{{\operatorname{s}}}, \Xi^{\otimes {\operatorname{s}}})$, $\operatorname{N}^{{\operatorname{s}}} \square^{{\operatorname{s}}} = I -\Pi^{{\operatorname{s}}}$ on $\mathfrak{Dom}$ $(\square^{{\operatorname{s}}}, \Xi^{\otimes {\operatorname{s}}})$,

(ii) for $\sigma\in \operatorname{L}^2_{\operatorname{p}, \operatorname{k}}({ M }, \Xi^{\otimes {\operatorname{s}}})$, we have $\sigma = \overline\partial\, \overline{\partial}^{\ast}_{{\operatorname{s}}} \operatorname{N}^{{\operatorname{s}}}\sigma\, \oplus\, \overline{\partial}^{\ast}_{{\operatorname{s}}}\overline\partial \operatorname{N}^{{\operatorname{s}}}\sigma\, \oplus\, \Pi^{{\operatorname{s}}}\sigma,$

(iii) $\operatorname{N}^{{\operatorname{s}}}\overline\partial = \overline\partial \operatorname{N}^{{\operatorname{s}}}$ and $\operatorname{N}^{{\operatorname{s}}}\overline{\partial}^{\ast}_{{\operatorname{s}}} = \overline{\partial}^{\ast}_{{\operatorname{s}}} \operatorname{N}^{{\operatorname{s}}}$,

(iv) $\operatorname{N}^{{\operatorname{s}}}$, $\overline{\partial} \operatorname{N}^{{\operatorname{s}}}$, $\overline{\partial}^{\ast}_{{\operatorname{s}}} \operatorname{N}^{{\operatorname{s}}}$ are bounded operators on $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi^{\otimes {\operatorname{s}}})$.

Proof. The proof follows as ^[3,28]. □

Theorem 5. With the same assumptions as in Proposition 2. For $\alpha \in L^{2}_{\operatorname{p}, \operatorname{k}}(X, \Xi^{\otimes \operatorname{s}})$, $\alpha$ is supported in $\overline{ M }$, with $k\geqslant 1$, satisfying $\overline\partial\alpha = 0$ in $X$, there exists $\operatorname{w}\in L^{2}_{\operatorname{p}, \operatorname{k}-1}(X, \Xi^{\otimes \operatorname{s}})$, $\operatorname{w}$ is supported in $\overline{ M }$ such that $\overline\partial\operatorname{w} = \alpha$ in $X$.

Proof. Let $\alpha \in L^{2}_{\operatorname{p}, \operatorname{k}}(X, \Xi^{\otimes \operatorname{s}})$, $\alpha$ is supported in $\overline{ M }$, then $\alpha \in L^{2}_{\operatorname{p}, \operatorname{k}}( M , \Xi^{\otimes \operatorname{s}})$. Following Theorem 2, $\operatorname{N}^{\operatorname{s}}_{n-\operatorname{p}, n-\operatorname{k}}$ exists for $n-\operatorname{k}\geqslant1$. Define

for $\operatorname{w}\in L^{2}_{\operatorname{p}, \operatorname{k}-1}( M , \Xi^{\otimes \operatorname{s}})$. Set $\operatorname{w} = 0$ in $X\setminus \overline M$.

To solve $\overline\partial\operatorname{w} = \alpha$ in $X$, first solve $\overline\partial\operatorname{w} = \alpha$ in $M$.

if $\varrho\in\text{dom}(\overline\partial, \Xi^{\ast\otimes s})$. From the fact that $\vartheta^{\operatorname{s}} = \overline\partial^{\ast}_{\operatorname{s}}$ on $\mathcal{B}_{\operatorname{p}, \operatorname{k}}(\overline M , \Xi^{\otimes \operatorname{s}})$ and the density of $\mathcal{B}_{\operatorname{p}, \operatorname{k}}(\overline{ M }, \Xi^{\otimes \operatorname{s}})$ in $\mathfrak{Dom}\, (\overline {\partial}, \Xi^{\otimes \operatorname{s}})\, \cap\, \mathfrak{Dom}\, (\overline{\partial}^{\ast}, \Xi^{\otimes \operatorname{s}})$, and from (4.3), we obtain

Thus, $\alpha$ is supported in $\overline{ M }$, implies $\overline\partial^{\ast}_{\operatorname{s}}(\divideontimes_{\Xi^{\otimes \operatorname{s}}}\, \star\alpha) = 0\, \, \, \text{ on } M .$ Proposition 5(ⅲ) implies

Thus, from (1.1), (4.3) and (4.4), one obtains

Because $\operatorname{w} = 0$ in $X\setminus M$, for $\varrho\in L^{2}_{\operatorname{p}, \operatorname{k}}(X, \Xi^{\otimes \operatorname{s}})$, one obtains

Since

Equation (4.5) gives

Thus

Thus $\overline\partial\operatorname{w} = \alpha$ in $X$. □

5.   Annular domains

5.1.   Compactness estimates

Theorem 6. Assume that ${ M } = { M }_{1}\backslash\overline{ M }_{2}$ is $\operatorname{an\, annulus\, between\, two\, smooth\, bounded\, domains}$ ${ M }_{1}$ and ${ M }_{2}$ in a $\operatorname{Stein\, manifold}$ $\operatorname{X}$ of dimension $n$ satisfy $\overline{ M }_{2}\Subset{ M }_{1}$, ${ M }_{1}$ is weak $Z(k)$, ${ M }_{2}$ is weak $Z(n-1-k)$ and $1 \leqslant k\leqslant n -2$ with $n \geqslant 3$. Let $\varrho$ be a smooth function on $\overline{ M }$ satisfy $\varrho = \mu$ in a neighborhood of $b{ M }_{1}$ and $\varrho = -\mu$ in a neighborhood of $b{ M }_{2}$. Then, there exists $c, T > 0$ satisfy for every $t \geq T$ with $C_{\mathtt{t}} > 0$, one obtains

for $\sigma\in\mathscr{D}_{\operatorname{p}, \operatorname{k}}({ M })$.

Proof. As in ^[27,29] (resp. ^[30]). Let $\sigma$ $\operatorname{be\, supported\, in\, a\, small\, neighborhood}$ $V$ of $b{ M }_{1}$. Let $(\zeta_{ij})$ be a Levi matrix of a defining function $\zeta$ of ${ M }_1$. If $\overline{U}\subset{ M }$, one obtains

Following ^[25], one obtains

If $c = t^2c'$ and using (5.2), (5.3), inequality (5.1) follows for $\sigma\in \mathscr{D}_{\operatorname{p}, \operatorname{k}}(U)$ when $\overline{U}\cap b{ M } = \varnothing$.

Since $b{ M }_1$ is weak $Z(k)$ as the boundary of ${ M }_1$ and is weak $Z(n-1-k)$ as a part of $b{ M }$. Thus, for $\sigma\in \mathscr{D}_{\operatorname{p}, \operatorname{k}}(\operatorname{U}\cap { M }_{1})$ with $1\leqslant k\leqslant n-1$ and $\forall$ $c > 0$, it follows that

Let $\Delta_{\delta_1} = \{z\in\mathbb{\operatorname{X}}:-\delta_1 < \zeta(z)\leqslant0\}$, where $\delta_1 > 0$ is a number (depend on $t$) small enough. From the compactness of $b{ M }_{1}$, by using a finite covering $\{V_{\nu}\}_{\nu = 1}^{\operatorname{s}}$ of $b{ M }_{1}$ by neighborhoods $V_{\nu}$ as in (5.4), one obtains

when $\sigma$ is supported in the strip $\Delta_{\delta_1}$.

Since $b{ M }_{2}$ is weak $Z(n-1-k)$ as the boundary of ${ M }_{2}$ and is weak $Z(k)$ as a part of $b{ M }$. Following Lemma 3,

for $\sigma\in \mathscr{D}_{\operatorname{p}, \operatorname{k}}(U\cap { M }_{2})$, $1\leqslant k\leqslant n-2$.

Let $\Delta_{\delta_2} = \{z\in X:0\leqslant\zeta(z) < \delta_2\}$, where $\delta_2 > 0$ small enough. From the compactness of $b{ M }_{2}$, by a finite covering $\{V_{\nu}\}_{\nu = 1}^{\operatorname{s}}$ of $b{ M }_{2}$ by neighborhoods $V_{\nu}$ as in (5.5),

when $\sigma$ is supported in the strip $\Delta_{\delta_2}$.

Let $\Delta_{\delta} = \Delta_{\delta_1}\cup \Delta_{\delta_2}$ with $\delta =$ min$\{\delta_1, \delta_2\}$. Thus, by (5.5) and (5.7), one obtains

The integral on $M \backslash \Delta_{\delta}$ can be estimated by choosing $\gamma_{\delta}\in \mathscr{D}({ M })$ with $\gamma_{\delta}(z) = 1$, $\zeta(z)\leqslant-\delta$ and $z\in { M }\backslash \Delta_{\delta}$ as

Because $\operatorname{Q}^{{\operatorname{t}}}$ is elliptic, by Gårding's inequality ^[28],

Thus, from (5.8)–(5.10), one obtains

Thus, from (5.10), (5.11), we get

Thus (5.1) follows by choosing $c+\mathtt{k}\mathtt{t} < \frac{C}{2}$ and $C'_{\mathtt{t}}+\frac{t}{k} < \frac{C_{\mathtt{t}}}{2}$. □

Theorem 7. Let $\operatorname{X}$, $\Xi$, $M$ be as in Theorem 6. Then, the compactness estimate of the weighted $\overline\partial$-Neumann problem holds on ${ M }$ for a holomorphic vector bundle $\Xi$-valued $(\operatorname{p}, \operatorname{k})$ form. Then, for all $c > 0$, there exists a $\mathtt{t} > 0$ and $C_{c, \mathtt{t}} > 0$ such that

for $\sigma\in \mathscr{D}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$.

Proof. The proof follows as Theorem 2. □

5.2.   Global regularity up to the boundary

Lemma 5. With the same assumptions as in Theorem 7, let us assume the following: For $1\leq \operatorname{k}\leq n-2$, $n\geq3$, there exists $C > 0$ satisfies$\forall$ $\sigma\in\mathscr{D}^{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ with $\sigma\perp\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$, we have

Proof. The proof follows as Lemma 3. □

By using (5.12), as Proposition 3.5 in ^[3], we prove the following theorem:

Lemma 6. With the same assumptions as in Theorem 7, let us assume the following: for $1\leq \operatorname{k}\leq n-2$, $n\geq3$, we have

(1) $\overline\partial$ (resp. $\overline\partial^{\ast}_{\mathtt{t}}$) has closed range in $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ and $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}+1}({ M }, \Xi)$ (resp. $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$),

(2) $\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$ is $\operatorname{finite\, dimension}$al,

(3) $\square^{\mathtt{t}}$ has closed range in $\operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$,

(4) $\mathfrak{Ran}(N^{\mathtt{t}})\subset {Dom}\, \square^{\mathtt{t}}$, $N^{\mathtt{t}} \square^{\mathtt{t}} = I-\mathcal{H}^{\operatorname{p}, \operatorname{k}}_{\mathtt{t}}(\Xi)$ on ${Dom}(\square^{\mathtt{t}}, \Xi)$,

(5) for $\sigma\in \operatorname{L}^{2}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, we have $\sigma = \overline\partial\, \overline\partial^{\star}_{\mathtt{t}} N \sigma \oplus\, \overline\partial^{\star}\overline\partial N^{\mathtt{t}} \sigma\, \oplus\, \Bbb{H}_{\mathtt{t}} \sigma.$

Proof. The proof follows as in ^[28]. □

By Lemma 6 (ⅱ) and the density of ${\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline { M }, \Xi)$ in $\operatorname{W}^{k}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, the following result is immediate.

Lemma 7. ^[44] With the same assumptions as in Theorem 7, let us assume the following: If $f\in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline{ M }, \Xi)$ with $1\leq \operatorname{k}\leq n-2$, $n\geq3$ and $N^{\mathtt{t}}f\in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline{ M }, \Xi)$, then for all $s\geq0$, there exists constants $C_{s}$ and $T_{s}$ such that

One can prove the following theorem by using the elliptic regularization method used in ^[44]:

Lemma 8. Assuming the same assumption as in Theorem 7, for every integer $s\geq0$ and real $t > T > 0$, $N^{\mathtt{t}}$ is bounded from $\operatorname{W}^{s}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$ into itself for $1\leq \operatorname{k}\leq n-2$, $n\geq3$.

By Lemma and the density of ${\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline{ M }, \Xi)$ in $\operatorname{W}^{s}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, the following is immediate.

Corollary 2. Let ${ M }$, $\Xi$ and $\operatorname{X}$ be the same as in Theorem 7. Then, if $f\in \operatorname{W}^{s}_{\operatorname{p}, \operatorname{k}}({ M }, \Xi)$, $s = 0, 1, 2, 3, ....$ satisfies $\overline\partial f = 0$, where $1\leq \operatorname{k}\leq n-2$, $n\geq3$, there exits $\sigma\in \operatorname{W}^{s}_{\operatorname{p}, \operatorname{k}-1}({ M }, \Xi)$ so that $\overline\partial \sigma = f$ on ${ M }$ and

Theorem 8. With the same assumptions as in Theorem 7, let us assume the following: for $\sigma\in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}}(\overline { M }, \Xi)$, $1\leq \operatorname{k}\leq n-2$, $n\geq3$, satisfying $\overline\partial \sigma = 0$, there exists $\operatorname{w}\in {\operatorname{C}}^{\infty}_{\operatorname{p}, \operatorname{k}-1}(\overline{ M }, \Xi)$, satisfies $\overline\partial \operatorname{w} = \sigma$ in $\operatorname{X}$.

Proof. The proof follows as ^[30,44]. □

6.   Conclusions

In this paper, we are concerned with the Sobolev estimates of the $\overline{\partial}$-Neumann operator N and the resulting results (Compactness and Global regularity, etc.). Existence theorems and Sobolev estimates for the $\overline{\partial}$ and the $\overline{\partial}$-Neumann operator on the weak $\operatorname{Z}(k)$ domain with ${\operatorname{C}}^{3}$ boundary in an $n$-dimensional Stein manifold $\operatorname{X}$ are fundamental results in complex analysis. In this way, we can gain a deeper understanding of holomorphic functions, and we can implement tools to solve the $\overline{\partial}$-equation more efficiently.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgment

The authors are thankful to the Deanship of Scientific Research at Najran University for funding this work under the Research Priorities and Najran Research funding program, grant code (NU/NRP/SERC/12/11).

Conflict of interest

The authors declare that they have no conflicts of interest.