Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $\mathbb {C}^2$

Abstract

Let $\Omega$ be a bounded, uniformly totally pseudoconvex domain in $\mathbb {C}^2$ with smooth boundary $b\Omega$. Assume that $\Omega$ is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let $\alpha$ be a d-closed (1, 1)-form in $\Omega$. We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in $L^1(b\Omega )$ norm by applying the $L^1(b\Omega )$ estimates for $\bar{\partial }_b$-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in $\Omega$.

1 Introduction

Let $\Omega$ be a bounded domain in $\mathbb {C}^2$ with smooth boundary $b\Omega$, and let $\mathfrak {g}$ be a Nevanlinna holomorphic function on $\Omega$. In pluripotential theory, it is well-known that the zero variety $Z(\Omega , \mathfrak {g})$ associated to $\mathfrak {g}$ on $\Omega$ satisfies the Blaschke condition. Naturally, we are interested in studying the converse, that is seeking geometric conditions on $\Omega$ so that any given analytic variety is defined as the zero set of a Nevanlinna holomorphic function.

We briefly recall the illustrious history of this problem. When $\Omega$ is the unit disk on the complex plane, a well-known fact in potential theory (e.g., [8, 17]) says that if $\Omega$ satisfies the Blaschke condition, any analytic variety $M\subset \Omega$ is the zero variety of a Nevanlinna holomorphic function, or a bounded holomorphic function on $\Omega$. Actually, this is true for all simply connected domains in the complex plane by the Riemann mapping theorem.

It is more difficult when we consider the problem in $\mathbb {C}^n$, for $n\ge 2$. The existence of a Nevanlinna holomorphic function determining a given positive divisor M on the unit ball in $\mathbb {C}^n$ is well-understood, see in [29]. This is also true under certain algebraic topology conditions when $\Omega$ is a strongly pseudoconvex domain, for instance, by Gruman [10], by Henkin [14] and Skoda [33] independently. Moreover, in [21], Laville showed that if $\Omega$ is star-shaped, then there exists a Nevanlinna function $\mathfrak {g}$ determining M and $\log |\mathfrak {g}|\in L^1(\Omega )$. Another positive result was obtained by Anderson [1] when $\Omega$ is a polydisc in $\mathbb {C}^n$. The problematic situation is if $\Omega$ is a weakly pseudoconvex domain. Existence results have been obtained on some special domains: on complex ellipsoids of finite type by Bonami and Charpentier [3]; on uniformly totally pseudoconvex domains of finite type in the sense of Range in $\mathbb {C}^2$ by Shaw [31]. The large class of uniformly totally pseudoconvex/ convex domains of finite type in the sense of Range introduced in [25, 26] consists all balls, strongly pseudoconvex domains and complex ellipsoids, and convex domains with real analytic boundaries in $\mathbb {C}^2.$ In this paper, we shall give an answer to this problem on a large class of pseudoconvex domains of infinite type.

The main results are the following theorems. The first is the $L^p$ boundary regularity for solutions of the $\bar{\partial }$-equation.

Theorem 1.1

Let $\Omega$ be a smooth bounded, uniformly totally pseudoconvex domain and admit maximal type F at all boundary points for some function F (see Definition (2.2)). Assume that $\bar{\Omega }$ has a Stein neighborhood basis. Let $\varphi$ be a continuous (0, 1)-form on $\overline{\Omega }$ and satisfy $\bar{\partial }\varphi =0$ in the weak sense. Then there exists a function $u\in \ \Lambda ^f(\overline{\Omega })$ such that

$$\begin{aligned} \bar{\partial }u=\varphi , \end{aligned}$$

where

$$\begin{aligned} f(d^{-1}):=\left( \int _0^{d}\frac{\sqrt{F^*(t)}}{t}\mathrm{d}t\right) ^{-1}, \end{aligned}$$

with $F^*$ the inversion of F.

Moreover, we also have

• (i) $||u||_{L^1(\Omega )}\le C(||\varphi ||_{L^1_{(0,1)}(\Omega )}+||\varphi ||_{L^1_{(0,1)}(b\Omega )})$;

• (ii) $||u||_{L^p(b\Omega )}\le C_p||\varphi ||_{L^p_{(0,1)}(b\Omega )}$ for all $1\le p\le +\infty ;$

• (iii) $||u||_{\Lambda ^f_p(b\Omega )}\le C_p ||\varphi ||_{L^p_{(0,1)}(b\Omega )}$ for all $1\le p\le +\infty .$

Example 1.1

Let us define

$$\begin{aligned} \Omega ^{\infty }=\left\{ (z_1,z_2)\in \mathbb {C}^2:\,\exp (1+2/s)\cdot \exp \left( \frac{-1}{|z_1|^s}\right) +|z_2|^2-1<0\right\} . \end{aligned}$$

Let $\varphi$ be a continuous (0, 1)-form on $\overline{\Omega }$ and satisfy $\bar{\partial }\varphi =0$ in the weak sense. Then there exists a function $u\in \ \Lambda ^f(\overline{\Omega })$ such that

$$\begin{aligned} \bar{\partial }u=\varphi , \end{aligned}$$

where $f(t)=\frac{1024^s(1-2s)}{2s}\left( |\ln t|\right) ^{\frac{1}{2s}-1}$, for $0<s<1/2.$

Moreover, we have

• (i) $||u||_{L^1(\Omega )}\le C(||\varphi ||_{L^1_{(0,1)}(\Omega )}+||\varphi ||_{L^1_{(0,1)}(b\Omega )})$;

• (ii) $||u||_{L^p(b\Omega )}\le C_p||\varphi ||_{L^p_{(0,1)}(b\Omega )}$ for all $1\le p\le +\infty ;$

• (iii) $||u||_{\Lambda ^f_p(b\Omega )}\le C_p ||\varphi ||_{L^p_{(0,1)}(b\Omega )}$ for all $1\le p\le +\infty .$

Let $H^2(\Omega ,\mathbb {R})$ be the DeRham cohomology of the second degree on $\Omega$. The existence of solutions to the Poincaré–Lelong equation is our second result.

Theorem 1.2

Let $\Omega$ be a smooth bounded, uniformly totally pseudoconvex domain and admit maximal type F at all boundary points for some function F. Assume that $\bar{\Omega }$ has a Stein neighborhood basis, and $H^2(\Omega ,\mathbb {R})=0$. Let $\alpha$ be a positive d-closed, smooth (1, 1)-form on $\Omega$. Then the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \end{aligned}$$

admits a solution u such that

• (i) $u=\bar{u}$;

• (ii) $||u||_{L^1(b\Omega )}+||u||_{L^1(\Omega )}\le C ||\alpha ||_{L^1_{(1,1)}(\Omega )}$.

Let $H^2(\Omega ,\mathbb {Z})$ be the $\check{C}$ech cohomology group of the second degree with integer coefficients on $\Omega$. The last result is about prescribing zeros of holomorphic functions in the Nevanlinna class on $\Omega$.

Theorem 1.3

Let $\Omega$ be a smooth bounded, uniformly totally pseudoconvex domain and admit maximal type F at all boundary points for some function F. Assume that $\bar{\Omega }$ has a Stein neighborhood basis and $H^2(\Omega ,\mathbb {Z})=0$. If M is a finite area, positive divisor of $\Omega$, then we have

$$\begin{aligned} M=Z(\Omega ,\mathfrak {g}), \end{aligned}$$

for some Nevanlinna holomorphic function $\mathfrak {g}$ defined on $\Omega$.

Following the same lines in the proof of Corollary 3.3 in [31], we get a boundary property for meromorphic functions in Nevanlinna class.

Corollary 1.4

Let $\Omega$ be the same as in Theorem 1.3. Let $\mathfrak {g}$ be a meromorphic function in $\mathcal {N}(\Omega )$ such that the associated polar divisor $(M_{\mathfrak {g}})_{\infty }$ has finite area. Then there are two Nevanlinna holomorphic functions $\mathfrak {g}_1$ and $\mathfrak {g}_2$ on $\Omega$ such that $\mathfrak {g}=\mathfrak {g}_1/\mathfrak {g}_2$. Therefore, $\mathfrak {g}$ has non-tangential limit values almost everywhere on the boundary $b\Omega$.

The paper is organized as follows: In Sect. 2, we shall introduce some geometric conditions on $\Omega$ and recall the main result of [11]. Basic definitions and facts from Lelong’s theory are briefly recalled in Sect. 3. Sections 4, 5 and 6 are devoted to the proofs of the main theorems.

2 The tangential Cauchy–Riemann equation $\bar{\partial }_b u=\varphi$ on the boundary $b\Omega$

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb {C}^2$ with smooth boundary $b\Omega$. Let $\rho$ be a smooth defining function for $\Omega$ such that $\Omega =\{z\in \mathbb {C}^2:\rho (z)<0\}$ and $\nabla \rho \ne 0$ on $b\Omega =\{z\in \mathbb {C}^2: \rho (z)=0\}$, and $\nabla \rho \perp b\Omega$. The pseudoconvexity means

$$\begin{aligned} \langle \partial \bar{\partial }\rho ,L\wedge \bar{L}\rangle \ge 0\quad \text {on }\, b\Omega , \end{aligned}$$

where L is an any nonzero tangential holomorphic vector field. If the strict inequality holds on the boundary, $\Omega$ is called a strongly pseudoconvex domain.

It is well-known that there are some pseudoconvex domains not admitting any holomorphic support function, even of finite type. This phenomenon was established by Kohn and Nirenberg in [20]. Therefore, in this work, we only consider admissible domains enjoying the existence of holomorphic support functions, which were found by Range in [25].

Definition 2.1

$\Omega$ is said to be uniformly totally pseudoconvex at the point $P\in b\Omega$ if there are positive constants $\delta , c$ and a $C^1$ map $\Psi :U^{\delta }\times \Omega ^{\delta }\rightarrow \mathbb {C}$ such that for all boundary points $\zeta \in b\Omega \cap B(P,\delta )$, the following properties are satisfied:

• (1) $\Psi (\zeta ,.)$ is holomorphic on $\Omega$;

• (2) $\Psi (\zeta ,\zeta )=0$, and $d_z\Psi |_{z=\zeta }\ne 0$;

• (3) $\rho (z)>0$ for all z with $\Psi (\zeta ,z)=0$ and $0<|z-\zeta |<c$. By multiplying $\rho$ and $\Psi$ by suitable non-zero functions of $\zeta$, one may assume more

• (4) $|\partial \rho (\zeta )|=1$, and $\partial \rho (\zeta )=d_z\Psi |_{z=\zeta }$,

where $\Omega ^{\delta }=\{z\in \mathbb {C}^2:\rho (z)<\delta \}$, and $U^{\delta }=\Omega ^{\delta }{\setminus } \Omega$.

Here, $M_{\zeta }=\{z:\Psi (\zeta ,z)=0\}$ is called the supporting analytic hypersurface for $b\Omega$ at $\zeta \in b\Omega$, i.e., near $\zeta$, $\{z: \rho (z)\le 0, \Psi (\zeta ,z)=0\}=\{\zeta \}$. The following observation on $M_{\zeta }$ is needed. Let $\Omega$ be uniformly totally pseudoconvex at $P\in b \Omega$. For any $\zeta \in b\Omega \cap B(P,\delta )$, we define the map $\psi _{\zeta }: B(P,\delta )\rightarrow \mathbb {C}^2$ by $\psi _{\zeta }(z)=w=(w_1,\Psi (\zeta ,z))$ such that the Jacobian matrix of the map $\psi _{\zeta }$ at $\zeta$ is unitary. The existence of such maps is provided in [26]. Hence, after shrinking the neighborhood U of P, we could choose $c>0,d>0$ sufficiently small such that $\psi _{\zeta }$ maps $B(\zeta ,c)$ biholomorphically onto the neighborhood $\psi _{\zeta }(B(\zeta ,c))\supset B(0,d)$ of 0 in $\mathbb {C}^2$ for all $\zeta \in b\Omega \cap U$. Moreover, the analytic hypersurface $M_{\zeta }=\{z\in B(\zeta ,c):\Psi (\zeta ,z)=0\}$ is mapped by $\psi _{\zeta }$ biholomorphically into $\{w\in \mathbb {C}^2:w_2=0\}.$

Definition 2.2

Let $F:[0,\infty )\rightarrow [0,\infty )$ be a smooth, increasing function such that

• (1) $F(0)=0$;

• (2) $\int _0^{R}|\ln F(r^2)|\mathrm{d}r<\infty$ for some $R>0$;

• (3) $\dfrac{F(r)}{r}$ is increasing.

Let $\Omega \subset \mathbb {C}^2$ be uniformly totally pseudoconvex at $P\in b\Omega$. $\Omega$ is called a domain admitting maximal type F at the boundary point $P\in b\Omega$ if there are positive constants $c,c'$ such that for all $\zeta \in b\Omega \cap B(P,c')$, we have

$$\begin{aligned} \rho (z)\gtrsim F(|z_1-\zeta _1|^2),\quad \text {for all}\, z\in B(\zeta ,c)\, \mathrm{with}\, \Psi (\zeta ,z)=0. \end{aligned}$$

Here and in what follows, the notations $\lesssim$ and $\gtrsim$ denote inequalities up to a positive constant, and $\approx$ means the combination of $\lesssim$ and $\gtrsim$.

Remark 2.3

• (1) The Definition 2.2 is independent of the choice on holomorphic coordinates in a neighborhood of P and of the particular defining function $\rho$ in Definition 2.1.

• (2) The domain $\Omega$ is called a uniformly totally pseudoconvex domain and admit maximal type F if it has these above properties at every point $P\in b\Omega$, with the common function F. Actually, we could choose the common function F for all boundary points by the compactness of $b\Omega$,

For more discussions of uniformly total pseudoconvexity and its properties, the basic references are [25, 30].

Some examples will be provided to show that Definition 2.2 generalizes all uniformly totally pseudoconvex domains of finite type and a class of convex domains of infinite type in the sense of Range.

Example 2.1

• (1) Let $\Omega$ be a strongly pseudoconvex domain in $\mathbb {C}^n$ with a strictly plurisubharmonic defining function $\rho$. We define

  $$\begin{aligned} \Psi (\zeta ,z)=\sum _{j=1}^n\frac{\partial \rho }{\partial \zeta _j}(z_j-\zeta _j)+\frac{1}{2}\sum _{j,k=1}^n\frac{\partial ^2\rho }{\partial \zeta _j\partial \zeta _k}(\zeta )(z_j-\zeta _j)(z_k-\zeta _k). \end{aligned}$$

  Let us define $F(t)=t$, then $\Omega$ is in this case uniformly totally pseudoconvex of the maximal type F.

• (2) Let $\Omega \subset \mathbb {C}^2$ be pseudoconvex of strict finite type m(p) at every point $p\in b\Omega$ as defined in [19], and generalized by Range [25, 26], Shaw [30]. Let $m_0:=\sup _{p\in b\Omega }m(p)<\infty$ and $F(t)=t^{m_0/2}$. We define

  $$\begin{aligned} \Psi (\zeta ,z)=\sum _{s+t\le m_0}\frac{1}{s!t!}\frac{\partial ^{s+t}\rho }{\partial \zeta _1^s\partial \zeta _2^k}(z_1-\zeta _1)^s(z_2-\zeta _2)^k. \end{aligned}$$

  Then $\Omega$, in this case, is of the maximal type F.

• (3) Let us define

  $$\begin{aligned} \Omega ^{\infty }=\left\{ (z_1,z_2)\in \mathbb {C}^2:\,\exp (1+2/s)\cdot \exp \left( \frac{-1}{|z_1|^s}\right) +|z_2|^2-1<0\right\} . \end{aligned}$$

  Then, for $0<s<1/2$, $\Omega ^{\infty }$ is a convex domain admitting the maximal type $F(t)=\exp (\frac{-1}{32.t^s})$, see [36].

• (4) Recently, in [15], the present author et al. have considered a class of smooth, bounded domains $\Omega$ with a global defining function $\rho$ such that for any $P\in b\Omega$, there exist a coordinates $z_P =T_P(z)$ with the origin at P where $T_P$ is a linear transformation, and function $F_P$ such that

  $$\begin{aligned} \quad \qquad \Omega _P= & {} T_P(\Omega )=\{z_P=(z_{P,1},z_{P,2})\in \mathbb {C}^2:\rho (T_P^{-1}(z_P))\\= & {} F_P(|z_{P,1}|^2)+|z_{P,2}-1|^2<0\} \end{aligned}$$

  where $F_P:{\mathbb {R}}\rightarrow {\mathbb {R}}$ satisfies:

  • (i) $F_P(0)=0$;

  • (ii) $F'_P(t),F_P^{''}(t), F_P^{'''}(t)$ and $(\frac{F_P(t)}{t})'$ are non-negative on $(0,\delta )$;

  where $d_P$ is the square of the diameter of $\Omega _P$ and $\delta$ is a small number. This class of convex domains includes many examples of finite type as well as infinite type domains. Then, the support function is

  $$\begin{aligned} \Psi (\zeta ,z)=\frac{\partial \rho }{\partial \zeta _j}(\zeta )(\zeta _j-z_j). \end{aligned}$$

  By the properties of F, we have

  $$\begin{aligned} \rho (z)&\ge F(|z_1-\zeta _1|^2) \quad \text {for}\, |\zeta _1|\ge |z_1-\zeta _1|,\, \mathrm{with}\, \Psi (\zeta ,z)=0, \end{aligned}$$

  (2.1)

  where $z=(z_1,z_2)\in \Omega$, $\zeta =(\zeta _1,\zeta _2)\in \{z\in \bar{\Omega }:\rho (z)\ge -2\delta \}\cap B(0,\frac{1}{2}\epsilon )$. Therefore, $\Omega$ is uniformly totally pseudoconvex of the maximal type F at the boundary point (0, 0).

Let f be an increasing function such that $\lim _{t\rightarrow +\infty }f(t)=+\infty$. We define the f-Hölder space on $b\Omega$ by

$$\begin{aligned} \Lambda ^f(b\Omega )=\left\{ u\in L^{\infty }(b\Omega ):||u||_{L^{\infty }}+\sup _{\begin{array}{c} x(.)\in {\mathcal {C}}\\ 0\le t\le 1 \end{array}}f(t^{-1})|u(x(t))-u(x(0))|<+\infty \right\} , \end{aligned}$$

where the class of curves ${\mathcal {C}}$ in $b\Omega$ is

$$\begin{aligned} {\mathcal {C}}=\left\{ x(t):t\in [0,1]\rightarrow x(t)\in b\Omega , \, x(t)\,\text { is}\, C^1 \,\mathrm{and}\, |x'(t)|\le 1 \right\} . \end{aligned}$$

That means $\Lambda ^f(b\Omega )$ consists all complex-valued functions u such that for each curve $x(.)\in {\mathcal {C}}$, the function $t\mapsto u(x(t))\in \Lambda ^f([0,1])$.

For $1\le p< \infty$, the f-Besov space is denoted by

$$\begin{aligned} \Lambda ^f_p(b\Omega )= & {} \left\{ u\in L^{p}(b\Omega ):||u||_{L^{p}}\right. \nonumber \\&\left. +\sup _{\begin{array}{c} 0\le t\le 1 \end{array}}f(t^{-1})\left[ \left( \int _{b\Omega }|u(x(t))-u(x(0))|^p\mathrm{d}x\right) ^{1/p}\right] <+\infty \right\} , \end{aligned}$$

where the integral is taken in $x=x(t)\in {\mathcal {C}}$ over the boundary $b\Omega$. It is obvious that $\Lambda ^f_{\infty }(b\Omega )=\Lambda ^f(b\Omega )$. Note that for each $1\le p\le \infty$, the notion of the f-Besov space $\Lambda ^f_p(b\Omega )$ includes the standard Besov space $\Lambda ^{\alpha }_p(b\Omega )$ by taking $f(t) = t^{\alpha }$ (so that $f(|h|^{-1}) = |h|^{-\alpha }$) with $0<\alpha \le 1$. The boundary regularity in standard Besov spaces for the tangential Cauchy–Riemann equation was obtained by Shaw [30, 31].

Now, let $\mathcal {A}_{(0,1)}(b\Omega )$ be the space of restrictions of (0, 1)-forms in $\mathbb {C}^2$ to $b\Omega$. Let $\mathcal {B}_{(0,1)}(b\Omega )$ be the subspace of $\mathcal {A}_{(0,1)}(b\Omega )$ which is orthogonal to the ideal generated by $\bar{\partial }\rho$ . Let $\tau$ be the projection from $\mathcal {A}_{(0,1)}(b\Omega )$ to $\mathcal {B}_{(0,1)}(b\Omega )$.

Let L be the unit holomorphic tangential vector field on $b\Omega$ and $\omega$ be its dual. The tangential Cauchy–Riemann equation $\bar{\partial }_bu=\varphi$, with $\varphi \in {\mathcal {B}}_{(0,1)}(b\Omega )$, is seeking a function u on $b\Omega$ such that $\bar{L}u=\phi$ in the sense of distributions, where $\tau (\phi \bar{\omega })=\varphi$. In this sense, the tangential Cauchy–Riemann operator could be identified by $\bar{L}$. We refer the reader to Chen–Shaw’s book [6] for a general theory of $\bar{\partial }_b$.

In [11], the present author has proved the global solvability for the tangential Cauchy–Riemann equations on the boundary $b\Omega$ in $L^p$-spaces.

Theorem 2.4

Let $\Omega$ be a smooth bounded, uniformly totally pseudoconvex domain and admit maximal type F at all boundary points for some function F. Assume that $\bar{\Omega }$ has a Stein neighborhood basis. Let $\varphi \in L^p_{(0,1)}(b\Omega )$, $1\le p\le \infty$ and $\varphi$ satisfies the compatibility condition

$$\begin{aligned} \int _{b\Omega }\varphi \wedge \alpha =0, \end{aligned}$$

for every $\bar{\partial }$-closed (2, 0)-form $\alpha$ defined continuously up to $b\Omega$.

Let $F^*$ be the inversion of F, and let

$$\begin{aligned} f(d^{-1}):=\left( \int _0^{d}\frac{\sqrt{F^*(t)}}{t}\mathrm{d}t\right) ^{-1}. \end{aligned}$$

Then, there exists a function u defined on $b\Omega$ such that $\bar{\partial }_b u=\varphi$ on $b\Omega$, and

• (1) $||u||_{\Lambda ^f(b\Omega )}\le C||\varphi ||_{L^{\infty }_{(0,1)}(b\Omega )}$, if $p=\infty$;

• (2) $||u||_{L^p(b\Omega )}\le C_p||\varphi ||_{L^p_{(0,1)}(b\Omega )}$, if $1\le p <\infty$, where $C_p>0$ independent on $\varphi$;

• (3) $||u||_{\Lambda ^f_p(b\Omega )}\le C_p ||\varphi ||_{L^p_{(0,1)}(b\Omega )}$, for every $1\le p\le \infty$.

This result is applied to prove Theorems 1.1 and 1.2.

3 Lelong’s theory

3.1 Cohomology groups

We briefly recall the definitions of the DeRham cohomology and the $\check{C}$eck cohomology groups on $\Omega$, see the Range’s fundamental book [27] for more details.

Definition 3.1

The space of d-closed 2-forms on $\Omega$ is

$$\begin{aligned} Z_2(\Omega )=\{\omega \in C^{\infty }_2(\Omega ):\mathrm{d}\omega =0\} \end{aligned}$$

and the space of d-exact forms $B_2(\Omega )=dC^{\infty }_1(\Omega )$. Then, the quotient space

$$\begin{aligned} H(\Omega ,\mathbb {R}):=\frac{Z_2(\Omega )}{B_2(\Omega )} \end{aligned}$$

is called the DeRham cohomology group of the second degree on $\Omega$. This space measures the obstruction to the solvability of the d-equation on $\Omega$.

Let $\mathcal {U}=\{U_j;j\in J\}$ be an open cover of $\Omega$. A 2-cochain f for $\mathcal {U}$ with integer coefficients is a map f which assigns to each 3-tuple $(j_0,j_1,j_2)\in J^3$ with

$$\begin{aligned} U(j_0,j_1,j_2)=U_{j_0}\cap U_{j_1}\cap U_{j_2}\ne \emptyset \end{aligned}$$

a section

$$\begin{aligned} f(j_0,j_1,j_2)\in \Gamma (U(j_0,j_1,j_2),\mathbb {Z}), \end{aligned}$$

where $\Gamma (U(j_0,j_1,j_2),\mathbb {Z})$ is the collection of all sections of $\mathbb {Z}$ over $U(j_0,j_1,j_2)$.

The set of all 2-cochains for $\mathcal {U}$ with integer coefficients is denoted by $C^2(\mathcal {U},\mathbb {Z})$. This is an abelian group. The set $C^1(\mathcal {U},\mathbb {Z}), C^3(\mathcal {U},\mathbb {Z})$ and $C^4(\mathcal {U},\mathbb {Z})$ are also defined similarly.

The coboundary map $\delta _2:C^2(\mathcal {U},\mathbb {Z})\rightarrow C^3(\mathcal {U},\mathbb {Z})$ is defined by

$$\begin{aligned} (\delta _2f)(j_0,j_1,j_2,j_3)=\sum _{k=0}^3(-1)^kf(j_0,\ldots ,\widehat{j}_k,\ldots , j_3)|_{U(j_0,j_1,j_2,j_3)}, \end{aligned}$$

where $\widehat{j}_k$ denotes the omission of the index $j_k$. We also have the similar definitions for $\delta _1,\delta _3$. We could verify straightforward that $\delta \circ \delta =0,$ where $\delta$ is one of $\delta _1,\delta _2$ or $\delta _3.$

The kernel of $\delta _2$ is called the group $Z^2(\mathcal {U},\mathbb {Z})$, and the image of $\delta _1$ in $C^2(\mathcal {U},\mathbb {Z})$ is called the group $B^2(\mathcal {U},\mathbb {Z})$.

Definition 3.2

The $\check{C}$ech cohomology group of the second degree of $\mathcal {U}$ with integer coefficients is

$$\begin{aligned} H^2(\mathcal {U},\mathbb {Z}):=\frac{Z^2(\mathcal {U},\mathbb {Z})}{B^2(\mathcal {U},\mathbb {Z})}. \end{aligned}$$

The direct limit

$$\begin{aligned} H^2(\Omega ,\mathbb {Z}):=\lim _{\overrightarrow{\mathcal {U}}}H^2(\mathcal {U},\mathbb {Z}) \end{aligned}$$

is the set of all equivalence classes in the disjoint union $\bigcup _{\mathcal {U}}H^2(\mathcal {U},\mathbb {Z})$ over all open covers $\mathcal {U}$ of $\Omega$. This abelian group is called the $\check{C}$ech cohomology group of the second degree on $\Omega$ with integer coefficients.

Definition 3.3

Let $\Omega$ be a bounded domain in $\mathbb {C}^2$. For each holomorphic function $\mathfrak {g}$ on $\Omega$, the zero set $Z(\Omega ,\mathfrak {g})$ of $\mathfrak {g}$ on $\Omega$ is given by

$$\begin{aligned} Z(\Omega ,\mathfrak {g})=\{(z_1,z_2)\in \Omega : \mathfrak {g}(z_1,z_2)=0\}. \end{aligned}$$

The zero set in the above definition is a one complex dimensional analytic subvariety of $\Omega$.

The following theorem is a fundamental result in the theory of several complex variables.

Theorem 3.4

(Cartan) If the cohomology group $H^2(\Omega ,\mathbb {Z})=0$, and M is a complex one-dimensional analytic subvariety of $\Omega$, then

$$\begin{aligned} M=Z(\Omega ,\mathfrak {g}) \end{aligned}$$

for some holomorphic function $\mathfrak {g}$ defined on $\Omega$.

3.2 Currents

Definition 3.5

We denote $\mathcal {D}_{(p,q)}(\Omega )$ be the space $C^{\infty }_{(p,q)}(\Omega )$ with Schwarz topology. Any continuous linear functional on the space ${\mathcal {D}}_{(p,q)}(\Omega )$ is called a current of bi-degree $(n-p,n-q)$ (or bi-dimension (p, q)) in $\Omega$.

We equip the space of currents of bi-degree $(n-p,n-q)$ with a weak-topology as follows: a sequence $T_j$ of currents of bi-degree $(n-p,n-q)$ converges to T if and only if $\lim _{j\rightarrow \infty }T_j(\phi )=T(\phi )$ for any $\phi \in {\mathcal {D}}_{(p,q)}(\Omega )$.

Let T be a current of bi-degree (p, p) in $\Omega$. If we have

$$\begin{aligned} (T,\omega )\ge 0, \end{aligned}$$

for any simple positive test form $\omega =i^p\omega _1\wedge \overline{\omega }_1\wedge \cdots \wedge \omega _p\wedge \overline{\omega }_p$, with $\omega _k$’s $\in C^{\infty }_{(1,0)}$, then T is called a positive current.

In particular, a (1, 1)-current T is positive if for every compactly support $C^{\infty }_{(0,1)}$-form $\omega$, we have

$$\begin{aligned} \int _{\Omega }T\wedge \left( \frac{\omega \wedge \bar{\omega }}{i}\right) \ge 0. \end{aligned}$$

Note that if $T=\sum _{i,j=1}^2T_{ij}\mathrm{d}z_i\wedge \mathrm{d}\bar{z}_j$ is a positive (1, 1)-current, then $T_{ij}=-T_{ji}$, i.e., $T=\bar{T}$, and all coefficients are locally finite Borel measures. A positive and d-closed (1, 1)-current is called a Lelong current. By Henkin’s result [14], if T is a Lelong (1, 1)-current, then

$$\begin{aligned} \int _{\Omega }|T(z)\wedge \partial \rho (z)\wedge \bar{\partial }\rho (z)|\mathrm{d}V(z)<\infty \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega }||\rho (z)|^{1/2}T(z)\wedge \partial \rho (z)|\mathrm{d}V(z)+\int _{\Omega }||\rho (z)|^{1/2}T(z)\wedge \bar{\partial }\rho (z)|\mathrm{d}V(z)<\infty . \end{aligned}$$

For an increasing ordered multi-index J, we denote by $J'$ the unique increasing multi-index such that $J\cup J' =\{1,2,\ldots ,n\}$ and $|J|+|J'|=n$. Let us denote by $\alpha _{JK}$ the form complementary to $\mathrm{d}z_J\wedge \mathrm{d}\bar{z}_K$, that is

$$\begin{aligned} \alpha _{JK}=\lambda \mathrm{d}z_{J'}\wedge \mathrm{d}\bar{z}_{K'}, \end{aligned}$$

where $\lambda$ is chosen so that $\mathrm{d}z_J\wedge \mathrm{d}\bar{z}_K\wedge \alpha _{JK}$ equals to the volume form $\beta _n$ in $\mathbb {C}^n$.

We could identify a current $T\in {\mathcal {D}}'_{(p,q)}(\Omega )$ with a $(n-p,n-q)$-form which has distributional coefficients, i.e.,

$$\begin{aligned} T=\sum \limits '_{|J|=n-p,|K|=n-q}T_{JK}\mathrm{d}z_J\wedge \mathrm{d}\bar{z}_K. \end{aligned}$$

The coefficients $T_{JK}$ are defined by

$$\begin{aligned} (T_{JK},\phi )=(T,\phi \alpha _{JK}). \end{aligned}$$

Moreover, all $T_{JK}$ are non-negative Radon measures if T is positive. For a current T with measure coefficients, we define

$$\begin{aligned} ||T||_{E}=\sum \limits '_{|J|=n-p,|K|=n-q}|T_{JK}|_E\quad \text {the norm of}\, T, \end{aligned}$$

where $|T_{JK}|_E$ is the total variation of $T_{JK}$ on a compact set E. We also define the wedge product of a current and a smooth form $\omega$ by setting

$$\begin{aligned} (T\wedge \omega ,\phi ):=(T,\omega \wedge \phi ) \end{aligned}$$

for any test form $\phi$. If T is positive and $\omega$ is a positive (1, 1)-form, then $T\wedge \omega$ is positive as well. In particular, for a positive (p, p)-current T, and a $(n-p,n-p)$ simple form, the current $T\wedge \omega$ is a non-negative Borel measure. We differentiate currents according to the formula

$$\begin{aligned} (\mathrm{D}T,\phi )=-(T,\mathrm{D}\phi ), \end{aligned}$$

for a first order differential operator D.

3.3 Divisors

Definition 3.6

Let $M:=\{M_j\}$ be a locally finite family of hypersurfaces of $\Omega$. The formal sum

$$\begin{aligned} \sum _{j}a_jM_j, \end{aligned}$$

with $a_j\in \mathbb {Z}$, is called a divisor of $\Omega$. For a given divisor M of $\Omega$, there are uniquely distinct irreducible hypersurfaces $\{M_j\}$ of $\Omega$ and $a_j\in \mathbb {Z}{\setminus } \{0\}$ such that we have the following irreducible decomposition

$$\begin{aligned} M=\sum _{a_j\ne 0}a_jM_j. \end{aligned}$$

If $M=\sum \nolimits _{a_j\ne 0}a_jM_j$ with $a_j>0$ for all j, we call M to be a positive divisor of $\Omega$, and write $M>0$.

For example, let h be a holomorphic function on $\Omega$. Then, the hypersurface $M_{h}:=\{z\in \Omega : h=0\}$ is a positive divisor, and

$$\begin{aligned} M_{h}=\sum _{a_j\ne 0}a_jM_j, \end{aligned}$$

where $a_j>0$ is the zero order of h on $M_j$. In this case, $M_{h}$ is also called the zero divisor of $\Omega$.

Conversely, for any positive divisor $M=\sum _{a_j\ne 0}a_jM_j$ of $\Omega$, the vanishing of the second $\check{C}$ech cohomology group $H^2(\Omega ,\mathbb {Z})$ induces the existence of a holomorphic function h on $\Omega$ such that $h=0$ of order $a_j$ on $M_j$, and $h(z)\ne 0$ for $z\notin M$. This is a consequence of Theorem 3.4.

More generally, a meromorphic function h on $\Omega$ is locally expressed by the ratio $h=h_1/h_2$ of two holomorphic functions $h_1,h_2$ with $h_2\ne 0$. By this property, the zero hypersurface $M_{h}$ is locally expressed by

$$\begin{aligned} M_{h}=(M_{h})_0+(M_{h})_{\infty }:=\sum _{a_j>0}a_jM_j+\sum _{a_j<0}a_jM_j, \end{aligned}$$

where $(M_{h})_0$ is called the zero divisor of $\Omega$ and $(M_{h})_{\infty }$ is called the polar divisor of $\Omega$ associated to h.

The following theorem asserts that every divisor $M_{h}$ locally associates to a closed (1, 1) positive current on $\Omega$.

Theorem 3.7

(Poincaré–Lelong Formula [24]) Let h be a non-zero, meromorphic function on $\Omega$ and let $\eta$ be a 2-form of $C^2$ class on $\Omega$ with compact support. Then,

$$\begin{aligned} \frac{1}{2\pi }\partial \bar{\partial }[\log |h|^2]=M_h, \end{aligned}$$

that is

$$\begin{aligned} \int _{M_h}\eta =\frac{1}{2\pi }\int _{\Omega }\log |h|^2\partial \bar{\partial }\eta =\frac{1}{2\pi }\int _{\Omega }\partial \bar{\partial }[\log |h|^2]\wedge \eta \end{aligned}$$

in this sense of currents.

The following definitions and their properties could be found in [24, 33].

Definition 3.8

Let $M=\sum _{a_j\ne 0}a_jM_j$ be a divisor of $\Omega$ and $d\delta$ be the surface measure on M. Then, M is said to have finite area if

$$\begin{aligned} \sum _{a_j\ne 0}a_j\int _{z\in M_j}\mathrm{d}\delta (z) \end{aligned}$$

is finite. M is said to satisfy the Blaschke condition if

$$\begin{aligned} \sum _{a_j\ne 0}a_j\int _{z\in M_j}|\rho (z)|\mathrm{d}\delta (z) \end{aligned}$$

is finite.

Definition 3.9

Let $\mathfrak {g}$ be a holomorphic function on $\Omega$. Then $\mathfrak {g}$ is called a Nevanlinna holomorphic function on $\Omega$ if

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0^+}\int _{b\Omega _{\epsilon }}\log ^+|\mathfrak {g}(z)|\mathrm{d}S_{\epsilon }(z) \end{aligned}$$

is finite, where $\log ^+|\mathfrak {g}(z)|:=\max \{\log |\mathfrak {g}(z)|,0\}$. Here, for $\epsilon >0$ small, $\Omega _{\epsilon }:=\{z\in \Omega : \rho (z)<-\epsilon \}$, and $dS_{\epsilon }$ is the Lebesgue measure of $b\Omega _{\epsilon }$. The Nevanlinna class on $\Omega$ denoted by $\mathcal {N}(\Omega )$ is the collection of all Nevanlinna holomorphic functions on $\Omega$.

Definition 3.10

A meromorphic function $\mathfrak {g}$ on $\Omega$ is said to belong to $\mathcal {N}(\Omega )$ if

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0^+}\int _{b\Omega _{\epsilon }}\log ^+|\mathfrak {g}(z)|\mathrm{d}S_{\epsilon }(z) \end{aligned}$$

is finite and the pole divisor of $\Omega$ associated to $\mathfrak {g}$ satisfying the Blaschke condition. In other words, let $\mathfrak {g}=\frac{\mathfrak {g}_1}{\mathfrak {g}_2}$ for two holomorphic functions $\mathfrak {g}_1$, $\mathfrak {g}_2$ and $\mathfrak {g}_2\ne 0$. The second condition means that we have $\int _{\Omega }(\partial \bar{\partial }|\mathfrak {g}_2|^2)(z)|\rho (z)|\mathrm{d}V(z)$ is finite by the Poincaré-Lelong Formula.

Theorem 3.11

(Henkin–Skoda Theorem) Let $\Omega$ be a smooth bounded domain in $\mathbb {C}^n$, for $n\ge 2$. Let $\mathfrak {g}$ be a Nevanlinna holomorphic function on $\Omega$, then the zero divisor $M_{\mathfrak {g}}$ of $\mathfrak {g}$ satisfies the Blaschke condition.

Moreover, if $\Omega$ is strongly pseudoconvex, and M is a positive divisor of $\Omega$ and satisfies the Blaschke condition on $\Omega$, then there exists a holomorphic function $\mathfrak {h}\in \mathcal {N}(\Omega )$ such that

$$\begin{aligned} Z(\Omega , \mathfrak {h})=M. \end{aligned}$$

4 Proof of Theorem 1.1

In this section, by applying Theorem 2.4, we prove the boundary $L^p$ estimates in Theorem 1.1. The center of the proof is based on the construction of the $\bar{\partial }$-solution by Henkin–Skoda and Range (see [11, 12, 15, 26, 27, 31, 33] for more details).

Lemma 4.1

Let $\Omega$ be a smooth bounded, uniformly totally pseudoconvex domain in $\mathbb {C}^2$. Assume that $\bar{\Omega }$ has a Stein neighborhood basis. Then there exists a $C^1$-function $\Phi (\zeta ,z)$ on $U^{\delta }\times \Omega ^{\delta }$, which is holomorphic in $z\in \Omega ^{\delta }$ and satisfies

• (1) $\Phi (\zeta ,\zeta )=0$;

• (2) $|\Phi (\zeta ,z)|\ge A>0$, for all $|\zeta -z|\ge c$;

• (3) $\Phi (\zeta ,z)=H(\zeta ,z)\Psi (\zeta ,z)$, for all $|\zeta -z|<c$;

where H is a $C^1$-function with $0<A_0\le |H|\le A_1<\infty$.

This is a consequence of the fact that $\bar{\Omega }$ has a Stein neighborhood basis, see [26]. Recently, in [35], Straube has obtained the global Sobolev regularity of the $\bar{\partial }$-Neumann problem in a class of smooth bounded pseudoconvex domains admitting good Stein neighborhood bases. The global regularity does not hold if we merely assume the existence of a standard Stein neighborhood basis. The next lemma is the key in our analysis.

Lemma 4.2

Let $\Omega \subset \mathbb {C}^2$ be a smooth bounded, uniformly totally pseudoconvex domain and admit maximal type F at $P\in b\Omega$. Assume that $\bar{\Omega }$ has a Stein neighborhood basis. Then there is a positive constant c such that the support function $\Phi (\zeta ,z)$ satisfies the following estimate

$$\begin{aligned} |\Phi (\zeta ,z)|\gtrsim |\rho (z)|+|{\text {Im}}\Phi (\zeta ,z)|+F(|z-\zeta |^2), \end{aligned}$$

(4.1)

for every $\zeta \in b\Omega \cap B(P,c)$, and $z\in \overline{\Omega }$, $|z-\zeta |<c$.

By Hefer’s Theorem in [12], we obtain the following representation

$$\begin{aligned} \Phi (\zeta ,z)=\langle P(\zeta ,z),\zeta -z\rangle , \end{aligned}$$

where $P(\zeta ,z)=(p_1(\zeta ,z),p_2(\zeta ,z))$, and each $p_j$ is $C^1$ in $\zeta$ and holomorphic in z. Here $P(\zeta ,z)$ is called a Leray map which is holomorphic in z.

To construct the Henkin solution for the $\bar{\partial }$-equation, we recall the Bochner–Martinelli kernel for (0, 1)-forms to be

$$\begin{aligned} B(\zeta ,z)=-\frac{1}{4\pi ^2 }\frac{(\overline{\zeta }_1-\bar{z}_1)d\overline{\zeta }_2-(\overline{\zeta }_2-\bar{z}_2)d\overline{\zeta }_1}{|\zeta -z|^4}, \end{aligned}$$

and

$$\begin{aligned} L(\zeta ,z)=-\frac{1}{4\pi ^2}\frac{p_1(\zeta ,z)\bar{\partial }_{\zeta ,z}p_2(\zeta ,z)-p_2(\zeta ,z)\bar{\partial }_{\zeta ,z}p_1(\zeta ,z)}{\langle P(\zeta ,z),\zeta -z \rangle ^2}, \end{aligned}$$

and

$$\begin{aligned} R(\zeta ,z,\lambda )&=-\frac{1}{4\pi ^2}\left[ \eta _1(\zeta ,z,\lambda )\wedge (\bar{\partial }_{\zeta ,z}+d_{\lambda })\eta _2(\zeta ,z,\lambda )\right. \\&\left. \quad -\eta _2(\zeta ,z,\lambda )\wedge (\bar{\partial }_{\zeta ,z}+d_{\lambda })\eta _1(\zeta ,z,\lambda )\right] , \end{aligned}$$

where

$$\begin{aligned} \eta _j(\zeta ,z,\lambda )=\lambda \frac{\bar{\zeta }_j-\bar{z}_j}{|\zeta -z|^2}+(1-\lambda )\frac{p_j(\zeta ,z)}{\langle P(\zeta ,z),\zeta -z\rangle },\quad \text {for}\, j=1,2 \,\mathrm{and}\, \lambda \in [0,1]. \end{aligned}$$

The Bochner–Martinelli–Koppelman operators acting on $\varphi \in C^1_{(0,1)}(\bar{\Omega })$ are

$$\begin{aligned} B_{\Omega }\varphi (z)&=\displaystyle \int _{\Omega }\varphi (\zeta )\wedge B(\zeta ,z)\wedge d\zeta _1\wedge d\zeta _2,\nonumber \\ R_{b\Omega }\varphi (z)&=\displaystyle \int _{b\Omega }\int _0^1\varphi (\zeta )\wedge R(\zeta ,z,\lambda )\wedge d\zeta _1\wedge d\zeta _2\nonumber \\&=\displaystyle \int _{b\Omega }\varphi (\zeta )\wedge K(\zeta ,z)\wedge d\zeta _1\wedge d\zeta _2, \end{aligned}$$

(4.2)

for $z\in \Omega$, and where

$$\begin{aligned} K(\zeta ,z)=-\frac{1}{4\pi ^2}\frac{p_1(\zeta ,z)(\bar{\zeta }_2-\bar{z}_2)-p_2(\zeta ,z)(\bar{\zeta }_1-\bar{z}_1)}{\Phi (\zeta ,z)|\zeta -z|^2}. \end{aligned}$$

Lemma 4.3

(Henkin–Skoda Theorem) Let $\varphi \in C_{(0,1)}(\overline{\Omega })$. Then, for $z\in \Omega$,

$$\begin{aligned} u(z)=B_{\Omega }\varphi (z)+R_{b\Omega }\varphi (z) \end{aligned}$$

is a solution of the equation $\bar{\partial }u=\varphi$ on $\Omega$. This solution is called the Henkin solution of the $\bar{\partial }$-equation.

Proof of Theorem 1.1

Part 1: The existence in $\Lambda ^f(\Omega )$.

For any f such that $0<f(d^{-1})<d^{-1}$, by Lemma 1.15 in [27], we always have

$$\begin{aligned} ||B_{\Omega }\varphi ||_{L^{\infty }(\Omega )}\lesssim ||\varphi ||_{L^{\infty }(\Omega )}\quad \text {and}\quad ||B_{\Omega }\varphi ||_{\Lambda ^f(\Omega )}\lesssim ||\varphi ||_{L^{\infty }(\Omega )}. \end{aligned}$$

(4.3)

Hence, we only concentrate on the boundary term $R_{b\Omega }\varphi$. It is necessary to recall the General Hardy-Littlewood Lemma proved by Khanh [18]. $\square$

Lemma 4.4

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^m$ and let $\delta _{b\Omega }(x)$ denote the distance function from x to the boundary $b\Omega$ of $\Omega$. Let $G:\mathbb {R}^+\rightarrow \mathbb {R}^+$ be an increasing function such that $\frac{G(t)}{t}$ is decreasing and the integral $\int _0^d\frac{G(t)}{t}\mathrm{d}t$ is finite for some sufficiently small $d>0$. If $u\in C^1(\Omega )$ such that

$$\begin{aligned} |\nabla u(x)|\lesssim \frac{G(\delta _{b\Omega })(x)}{\delta _{b\Omega }(x)}\quad \text {for every}\, x\in \Omega , \end{aligned}$$

(4.4)

then $u\in \Lambda ^f(\Omega )$ in which $f(d^{-1}):=\left( \int _0^d\frac{G(t)}{t}\mathrm{d}t\right) ^{-1}$.

By (4.2) and the calculus quotient rule, we have

$$\begin{aligned} |\nabla _zR_{b\Omega }\varphi (z)|&\le ||\varphi ||_{L^{\infty }}.\int _{b\Omega }|\nabla _zK(\zeta ,z)|\mathrm{d}\sigma (\zeta )\nonumber \\&\lesssim ||\varphi ||_{L^{\infty }}.\int _{b\Omega }\left( \frac{1}{|\Phi (\zeta ,z)|.|\zeta -z|^2}+\frac{1}{|\Phi (\zeta ,z)|^2.|\zeta -z|}\right) \mathrm{d}\sigma (\zeta ).\quad \quad \end{aligned}$$

(4.5)

Now, for each fixed $z\in \Omega$, by the condition (2) in Lemma 4.1, it is enough to consider the integral (4.5) over $b\Omega \cap B(z,c)$. For convenience, we put

$$\begin{aligned} I_1(z):=\int _{b\Omega \cap B(z,c)}\frac{1}{|\Phi (\zeta ,z)|.|\zeta -z|^2}\mathrm{d}\sigma (\zeta ) \end{aligned}$$

and

$$\begin{aligned} I_2(z):=\int _{b\Omega \cap B(z,c)}\frac{1}{|\Phi (\zeta ,z)|^2.|\zeta -z|}\mathrm{d}\sigma (\zeta ). \end{aligned}$$

To estimate these integrals, we recall a real coordinate system $t=(t',t_3)=(t_1,t_2,t_2)$ introduced by Henkin, where

$$\begin{aligned} {\left\{ \begin{array}{ll} t_1={\text {Re}}\,(\zeta _1-z_1),\\ t_2={\text {Im}}\,(\zeta _1-z_1),\\ t_3={\text {Im}}\,\Phi (\zeta ,z). \end{array}\right. } \end{aligned}$$

Since $|\zeta -z|\ge |t'|+|\rho (z)|$, we have

$$\begin{aligned} I_1(z)\lesssim \int _{|t|\le R, t_3\ge 0}\frac{1}{(|\rho (z)|+t_3+F(|t'|^2)).(|t'|+|\rho (z)|)^2}\mathrm{d}t_1\mathrm{d}t_2\mathrm{d}t_3 \end{aligned}$$

and

$$\begin{aligned} I_2(z)\lesssim \int _{|t|\le R, t_3\ge 0}\frac{1}{(|\rho (z)|+t_3+F(|t'|^2))^2.|t'|}\mathrm{d}t_1\mathrm{d}t_2\mathrm{d}t_3'. \end{aligned}$$

Since $|\rho (z)|\approx \delta _{b\Omega }(z)$, after some simple calculations, we obtain

$$\begin{aligned} I_1(z)\lesssim |\ln ( |\rho (z)|)|^2\lesssim \frac{G(\delta _{b\Omega })(z)}{\delta _{b\Omega },(z)} \end{aligned}$$

(4.6)

for any G satisfying Lemma 4.4.

Moreover, we also have

$$\begin{aligned} I_2(z)&\lesssim \int _0^R\frac{1}{|\rho (z)|+F(r^2)}\mathrm{d}r\nonumber \\&=\int _0^{\sqrt{F^*(|\rho (z)|)}}\frac{1}{|\rho (z)|+F(r^2)}\mathrm{d}r\nonumber \\&\quad +\int _{\sqrt{F^*(|\rho (z)|)}}^R\frac{1}{|\rho (z)|+F(r^2)}\mathrm{d}r, \end{aligned}$$

(4.7)

where $F^*$ is the inversion of F.

The hypothesis that $\frac{F(r)}{r}$ is increasing implies

$$\begin{aligned} \frac{F(r^2)}{|\rho (z)|}\ge \frac{r^2}{F^*(|\rho (z)|)}\quad \text {for all}\, r\ge \sqrt{F^*(|\rho (z)|)}, \end{aligned}$$

and so

$$\begin{aligned} \int _{\sqrt{F^*(|\rho (z)|)}}^R\frac{1}{|\rho (z)|+F(r^2)}\mathrm{d}r\le \frac{\pi }{4}\frac{\sqrt{F^*(|\rho (z)|)}}{|\rho (z)|}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \int _0^{\sqrt{F^*(|\rho (z)|)}}\frac{1}{|\rho (z)|+F(r^2)}\mathrm{d}r\le \frac{\sqrt{F^*(|\rho (z)|)}}{|\rho (z)|}, \end{aligned}$$

and then we obtain

$$\begin{aligned} I_2(z)\lesssim \frac{\sqrt{F^*(|\rho (z)|)}}{|\rho (z)|}. \end{aligned}$$

The last step in this proof is to check the function $G(t):=\sqrt{F^*(t)}$ satisfies all conditions in Lemma 4.4. Then, by (4.3), we have

$$\begin{aligned} I_1(z)+I_2(z)\lesssim \frac{\sqrt{F^*(|\rho (z)|)}}{|\rho (z)|}, \end{aligned}$$

and by (4.6), $u\in \Lambda ^f(\Omega )$ in which $f(d^{-1}):=\left( \int _0^{d}\frac{\sqrt{F^*(t)}}{t}\mathrm{d}t\right) ^{-1},$ for small $d>0$.

Now, since $\sqrt{F^*(t)}$ is increasing and $\frac{\sqrt{F^*(t)}}{t}$ is decreasing, for some small $R>0$, $|\ln (F(t^2))|$ is decreasing for all $0\le t\le R$. Thus, by the hypothesis (2) of F, we have

$$\begin{aligned} |\ln F(\eta ^2)|\eta \le \int _0^{\eta }|\ln F(t^2)|\mathrm{d}t\le \int _0^R|\ln F(t^2)|\mathrm{d}t<\infty , \end{aligned}$$

for all $0\le \eta \le R$. As a consequence, $\sqrt{F^*(t)}|\ln t|$ is finite for all $0\le t\le \sqrt{F^*(R)}$ and $\lim _{t\rightarrow 0}t|\ln F(t^2)|$ is zero. These facts, and the second hypothesis of F imply

$$\begin{aligned} \int _0^d\frac{\sqrt{F^*(t)}}{t}\mathrm{d}t=\int _0^{\sqrt{F^*(d)}}y(\ln F(y^2))'\mathrm{d}y=\sqrt{F^*(d)}\ln d-\int _0^{\sqrt{F^*(d)}}(\ln F(y^2))\mathrm{d}y<\infty , \end{aligned}$$

for $d>0$ small enough.

This completes the proof of the first part.

Part 2: The estimates (i), (ii), (iii).

By Lemma 4.3, to prove the estimates in Theorem 1.1, we estimate $B_{\Omega }\varphi$ and $R_{b\Omega }\varphi$.

For the interior term $B_{\Omega }\varphi$.

Applying the following basic estimate

$$\begin{aligned} |B(\zeta ,z)|\lesssim \frac{1}{|\zeta -z|^3}, \end{aligned}$$

the operator $B_{\Omega }\varphi$ is bounded from $L^1(\Omega )\rightarrow L^{\frac{4}{3}-\epsilon }(\Omega )$ for all small $\epsilon >0$. Hence, for $\epsilon =1/3$, in particular, we have

$$\begin{aligned} ||B_{\Omega }\varphi ||_{L^1(\Omega )}\lesssim ||\varphi ||_{L^1_{(0,1)}(\Omega )}. \end{aligned}$$

For the boundary term $R_{b\Omega }\varphi$.

We know that for each fixed $\zeta$, the set of singularities of the kernel $K(\zeta ,z)$ is the surface $\{z=\zeta \}$. Hence, for any ball $B(\zeta ,\epsilon )$ centered at $\zeta$, with radius $\epsilon$, the following estimate

$$\begin{aligned} \int _{\Omega {\setminus } B(\zeta ,\epsilon )}|K(\zeta ,z)|\mathrm{d}V(z)\lesssim \int _{\Omega {\setminus } B(\zeta ,\epsilon )}\frac{\mathrm{d}V(z)}{|\Phi (\zeta ,z)|\cdot |\zeta -z|}\lesssim 1 \end{aligned}$$

(4.8)

holds uniformly in $\zeta \in b\Omega$.

Therefore, the problematic point is to estimate the integral on the ball $B(\zeta ,\epsilon )$ containing the singularities of $K(\zeta ,z)$. Again, applying the Henkin setting up above, we recall a special real coordinate chart $(t',t_3,y)=(t_1,t_2,t_3,y)$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} y&{}=|\rho (z)|\\ t_1&{}= {\text {Re}}(z_1-\zeta _1)\\ t_2&{}={\text {Im}}(z_1-\zeta _1)\\ t_3&{}=|{\text {Im}}(\Phi (\zeta ,z))|. \end{array}\right. } \end{aligned}$$

Thus, in this special coordinate chart, it follows from Lemma 4.2 that

$$\begin{aligned} |\Phi (\zeta ,z)|\gtrsim y+t_3+F(|t'|^2). \end{aligned}$$

(4.9)

Then, for a sufficient large $R>0$, we obtain

$$\begin{aligned} \int _{\Omega \cap B(\zeta ,\epsilon )}|K(\zeta ,z)|\mathrm{d}V(z)&\le \displaystyle \int _{\Omega \cap B(\zeta ,\epsilon )}\frac{\mathrm{d}V(z)}{|\Phi (\zeta ,z)|\cdot |\zeta _1-z_1|}\nonumber \\&\lesssim \displaystyle \int _{|(t,y)|\le R}\frac{1}{(y+t_3+F(|t'|^2))|t'|}\mathrm{d}t_1\mathrm{d}t_2\mathrm{d}t_3dy\nonumber \\&\lesssim \displaystyle \int _{|t|\le R}\frac{1}{(t_3+F(|t'|^2))|t'|}\mathrm{d}t_1\mathrm{d}t_2\mathrm{d}t_3\nonumber \\&\lesssim \displaystyle \int _{|t'|\le R}\frac{\ln F(|t'|^2)}{|t'|}\mathrm{d}t_1\mathrm{d}t_2. \end{aligned}$$

(4.10)

Using the polar coordinates $(t_1,t_2)=r(\cos \theta , \sin \theta )$, we have

$$\begin{aligned} \int _{\Omega \cap B(\zeta ,\epsilon )}|K(\zeta ,z)|\mathrm{d}V(z) \lesssim \int _0^{R}\ln F(r^2)\mathrm{d}r\le C <\infty \end{aligned}$$

(4.11)

uniformly in $\zeta \in b\Omega .$

Now, (4.8) and (4.11) imply

$$\begin{aligned} ||R_{b\Omega }\varphi ||_{L^1(\Omega )}&\le \displaystyle \int _{\Omega }\int _{b\Omega }|K(\zeta ,z)||\varphi (\zeta )|dS(\zeta )\mathrm{d}V(z)\nonumber \\&\le \int _{b\Omega }\left( \displaystyle \int _{\Omega }|K(\zeta ,z)|\mathrm{d}V(z)|\varphi (\zeta )|\right) dS(\zeta )\nonumber \\&\lesssim \displaystyle \int _{b\Omega }|\varphi (\zeta )|\mathrm{d}S(\zeta )\nonumber \\&\lesssim ||\varphi ||_{L^1(b\Omega )}. \end{aligned}$$

(4.12)

Finally, we have the first inequality

$$\begin{aligned} ||u||_{L^1(\Omega )}\lesssim ||\varphi ||_{L^1(\Omega )}+||\varphi ||_{L^1(b\Omega )}. \end{aligned}$$

(4.13)

To estimate the boundary norms of u in (ii) and (iii), we convert the interior term $B_{\Omega }(\varphi )$ into a suitable boundary manner. This manner was introduced by Shaw in [31]. Let us define the following kernel

$$\begin{aligned} R^*(\zeta ,z,\lambda )&=R(z,\zeta ,\lambda ). \end{aligned}$$

(4.14)

This kernel is well-defined on $(\zeta ,z)\in \Omega \times U^{\delta }$. Then, we have

Lemma 4.5

([31], page 414) For $z\in b\Omega$, we have

$$\begin{aligned} u(z)=R_{b\Omega }\varphi (z)-R^*_{b\Omega }\varphi (z), \end{aligned}$$

where

$$\begin{aligned} R^*_{b\Omega }\varphi (z)=\int _{b\Omega }\int _0^1\varphi (\zeta )\wedge R^*(\zeta ,z,\lambda )\wedge \mathrm{d}\zeta _1\wedge \mathrm{d}\zeta _2. \end{aligned}$$

Now, for $z\in b\Omega$, let $\varphi (z)=\varphi _t(z)+\varphi _n(z)$, where $\varphi _t$ defined on $b\Omega$ is the tangential part of $\varphi$, which is orthogonal to $\bar{\partial }\rho$, and $\varphi _n(z)=g(z)\bar{\partial }\rho (z)$ is the corresponding normal part, for a function g defined on $b\Omega$. And since $d\rho \perp b\Omega$, we have

$$\begin{aligned} R_{b\Omega }\varphi _n(z)&=\displaystyle \int _{b\Omega }g(\zeta )\bar{\partial }\rho (\zeta )\wedge K(\zeta ,z)\wedge \mathrm{d}\zeta _1\wedge \mathrm{d}\zeta _2\nonumber \\&=\displaystyle \int _{b\Omega }g(\zeta ) d\rho (\zeta )\wedge K(\zeta ,z)\wedge \mathrm{d}\zeta _1\wedge \mathrm{d}\zeta _2\nonumber \\&=0. \end{aligned}$$

(4.15)

That is $R_{b\Omega }\varphi (z)=R_{b\Omega }\varphi _t(z)$ for all $z\in b\Omega$. Similarly, we obtain $R^*_{b\Omega }\varphi (z)=R^*_{b\Omega }\varphi _t(z)$ for all $z\in b\Omega .$

Therefore, we have

$$\begin{aligned} u(z)=R_{b\Omega }\varphi _t(z)-R^*_{b\Omega }\varphi _t(z),\quad \text {for}\, z\in b\Omega , \end{aligned}$$

(4.16)

where the right-hand side only depends on the tangential part of $\varphi$ on the boundary $b\Omega$.

The right-hand side in (4.16) agrees with the term after the operator $\bar{\partial }_b$ in the formula (3.8) of Lemma 3.6 in [11]. That means u given by (4.16) solves the tangential Cauchy–Riemann

$$\begin{aligned} \bar{\partial }_bu=\varphi _t \end{aligned}$$

on the boundary $b\Omega$.

Therefore, using the estimates (1), (2) and (3) in Theorem 2.4, we obtain (i) and (ii) in Theorem 1.1.

Hence, the first main theorem is completely proved.

5 Proof of Theorem 1.2

Solving the Poincaré–Lelong equation $i\partial \bar{\partial }u=\alpha$ is based on solutions to the d-equations on star-shaped domains and Theorem 1.1. Hence, we first assume that $\Omega$ is a star-shaped domain and contains the origin.

Let $\mathcal {K}$ be the Poincaré–Cartan homotopy operator defined in [7, page 36]. Let $\alpha =\sum _{ij}\alpha _{ij}dz_i\wedge d\bar{z}_j$ be a positive, smooth (1, 1)-form on $\Omega$ such that $d\alpha =0$, then

$$\begin{aligned} \mathcal {K}\alpha (z)=\sum _{j}\left( \sum _i\int _0^1t\alpha _{ij}(tz)\mathrm{d}t z_i\right) d\bar{z}_j-\sum _{i}\left( \sum _j\int _0^1t\alpha _{ij}(tz)\mathrm{d}t \bar{z}_j\right) dz_i. \end{aligned}$$

(5.1)

By Proposition 2.13.2 in [7], we have

$$\begin{aligned} d\mathcal {K}\alpha (z)=\alpha (z). \end{aligned}$$

Because of the positivity of $\alpha$, we obtain

$$\begin{aligned} \mathcal {K}\alpha (z)=\sum _{j}\left( \sum _i\int _0^1t\alpha _{ij}(tz)\mathrm{d}t z_i\right) \mathrm{d}\bar{z}_j-\overline{\sum _{j}\left( \sum _i\int _0^1t\alpha _{ij}(tz)\mathrm{d}t z_i\right) \mathrm{d}\bar{z}_j}. \end{aligned}$$

(5.2)

In short, $\mathcal {K}\alpha (z)=\mathcal {F}(z)+\overline{\mathcal {F}(z)}$, where

$$\begin{aligned} \mathcal {F}(z)=\sum _{j}\left( \sum _i\int _0^1t\alpha _{ij}(tz)\mathrm{d}t z_i\right) \mathrm{d}\bar{z}_j. \end{aligned}$$

Moreover, as a consequence of the d-closed property of $\alpha$,

$$\begin{aligned} \bar{\partial }\mathcal {F}=\partial \mathcal {F}=0. \end{aligned}$$

(5.3)

By a changing coordinates $b\Omega \times [0,1]\rightarrow \Omega$, we also obtain

$$\begin{aligned} ||\mathcal {F}||_{L^1(b\Omega )}\lesssim ||\alpha ||_{L^1(\Omega )}\quad \text {and}\quad ||\mathcal {F}||_{L^1(\Omega )}\le ||\alpha ||_{L^1(\Omega )}. \end{aligned}$$

(5.4)

Applying the estimates (5.3), (5.4) and the existence in Theorem 1.1, there is a function $v\in L^1(\bar{\Omega })$ solving the equation $\bar{\partial }v=\mathcal {F}$ on $\bar{\Omega }$, and satisfying

$$\begin{aligned} ||v||_{L^1(\Omega )}+||v||_{L^1(b\Omega )}&\lesssim ||\mathcal {F}||_{L^1(\Omega )}+||\mathcal {F}||_{L^1(b\Omega )}\nonumber \\&\lesssim ||\alpha ||_{L^1(\Omega )}. \end{aligned}$$

(5.5)

Now, we define $u=\frac{v-\bar{v}}{i}$, then $u=\bar{u}$, and

$$\begin{aligned} ||u||_{L^1(b\Omega )}+||u||_{L^1(\Omega )}\lesssim ||\alpha ||_{L^1(\Omega )}, \end{aligned}$$

and

$$\begin{aligned} \alpha&=d(\mathcal {K}\alpha )=\partial \mathcal {F}+\bar{\partial }\bar{\mathcal {F}}\nonumber \\&=\partial (\bar{\partial }v)+\bar{\partial }(\partial \bar{v})\nonumber \\&=i\partial \bar{\partial }\left( \frac{v-\bar{v}}{i}\right) \nonumber \\&=i\partial \bar{\partial }u. \end{aligned}$$

(5.6)

Thus, the theorem is proved in the case that $\Omega$ is a star-shaped domain.

Generally, when $\Omega$ is a domain in $\mathbb {C}^2$ such that the DeRham cohomology of the second degree $H^2(\Omega ,\mathbb {R})=0$, we could apply the well-known global construction of Weil [37] for $H^2(\Omega ,\mathbb {R})$ to obtain the Poincaré-Cartan Lemma in a local sense. Then, Theorem 1.2 is proved.

6 Proof of Theorem 1.3

Applying a smooth approximation and the Poincaré–Lelong Formula, Theorem 1.3 follows from Theorems 1.1 and 1.2.

Indeed, by Theorem 3.7, let $\alpha _M$ be a d-closed (1, 1) positive current associated with M. That means, for some holomorphic function h which has zero set M on $\Omega$, we have

$$\begin{aligned} \alpha _M=\frac{1}{\pi }\partial \bar{\partial }[\log |h|] \end{aligned}$$

in the sense of currents.

Let

$$\begin{aligned} V_{\epsilon }(z)=\log |h|*\chi _{\epsilon }(z) \end{aligned}$$

be the smooth regularity of $\log |h(z)|$, where for each $\epsilon >0$, and $\chi _{\epsilon } \in C^{\infty }_c(\mathbb {R})$ is a non-negative function such that $\chi _{\epsilon }$ is supported on $[-\epsilon /2,\epsilon /2]$, and $\int _{\mathbb {R}}\chi _{\epsilon }(x)\mathrm{d}x=1$. Then, $V_{\epsilon }$ is smooth on $\Omega _{\epsilon }=\{\rho (z)<-\epsilon \}\Subset \Omega$ and $V_{\epsilon }(z)\rightarrow \log |h(z)|$ as $\epsilon \rightarrow 0^+.$

For convenience, we also denote $V_{\epsilon }$ by the smooth extension of $V_{\epsilon }$ to a neighborhood of $\Omega$, so $V_{\epsilon }(z)\rightarrow \log |h(z)|$ almost everywhere as $\epsilon \rightarrow 0^+.$ Then the smooth regularity of $\alpha _M$ is $\alpha _{\epsilon }=\frac{1}{\pi }\partial \bar{\partial }V_{\epsilon }\in C^{\infty }_{(1,1)}(\bar{\Omega })$, and $\alpha _{\epsilon }$ is also d-closed and positive. Moreover, $\alpha _{\epsilon }\rightarrow \alpha _M$ in the sense of currents. Thus, applying Theorem 1.2 to each $\pi \alpha _{\epsilon }$, we could seek a function $u_{\epsilon }$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll}\displaystyle u_{\epsilon }=\bar{u_{\epsilon }},\\ \frac{1}{\pi }\partial \bar{\partial }u_{\epsilon }=\alpha _{\epsilon },\\ ||u_{\epsilon }||_{L^1(b\Omega )}+||u_{\epsilon }||_{L^1(\Omega )}\lesssim ||\alpha _{\epsilon }||_{L^1(\Omega )}. \end{array}\right. } \end{aligned}$$

As a consequence, for some constant $C>0$, we have

$$\begin{aligned} \int _{\Omega }|u_{\epsilon }(z)|\mathrm{d}V(z)<C,\quad \text {uniformly in}\, \epsilon >0. \end{aligned}$$

(6.1)

The plurisubharmonicity implies that $\log |h(z)|$ is locally integrable. Hence, for any compact subset $K\subset \Omega$, we have

$$\begin{aligned} \int _K|V_{\epsilon }(z)|\mathrm{d}V(z)<C_K,\quad C_K>0\quad \text {depends only on}\, K. \end{aligned}$$

(6.2)

Next, we define

$$\begin{aligned} g_{\epsilon }=u_{\epsilon }-V_{\epsilon }. \end{aligned}$$

It is easy to see that $g_{\epsilon }$ is a pluriharmonic function on $\Omega$. Since $\Omega$ is a domain, $g_{\epsilon }=\text {Re}[G_{\epsilon }]$, where $G_{\epsilon }$ is holomorphic on $\Omega$.

Using (6.1), (6.2) and Montel’s Theorem applied to $g_{\epsilon }$, there exists a subsequence $\{g_{\epsilon _n}\}$ of $\{g_{\epsilon }\}$ that converges to a pluriharmonic function g uniformly on every compact set of $\Omega$, where $\lim _{n\rightarrow \infty }\epsilon _n=0$. Moreover, we also have

$$\begin{aligned} g=\lim _{n\rightarrow \infty }g_{\epsilon _n}=\lim _{n\rightarrow \infty }\text {Re}[G_{\epsilon _n}]=\text {Re}[G], \end{aligned}$$

for some holomorphic function G on $\Omega$. Now, let $u=\log [|h|]+g=\log [|h|]+\text {Re}[G]=\log [|he^G|]$, then we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \lim _{n\rightarrow \infty }u_{\epsilon _n}=u,&{}\quad \text {in}\, L^1(\overline{\Omega }),\\ \frac{1}{\pi }\partial \bar{\partial }u=\alpha _M&{}\quad \text {in the sense of currents},\\ u\in L^1(\overline{\Omega }),&{}\quad \text {by Theorem 1.2}. \end{array}\right. } \end{aligned}$$

On the other hand, let $\mathfrak {g}(z)=he^G(z)$ since $\frac{1}{\pi }\partial \bar{\partial }\log [|h|]=\frac{1}{\pi }\partial \bar{\partial }\log [|\mathfrak {g}|]=\alpha _M$, the zero set of $\mathfrak {g}$ is the same as the zero set of h. Finally, $\mathfrak {g}\in \mathcal {N}(\Omega )$ since $u=\log [|\mathfrak {g}|]\in L^1(\overline{\Omega })$. Thus, we complete the proof.

Remark. In the next paper, we will apply the present technique to construct a bounded holomorphic function which defines the given positive divisor in $\mathbb {C}^2$.