On a theorem of Bishop and commutants of Toeplitz operators in $\mathbb {C}^n$

Abstract

We prove an approximation theorem on a class of domains in $\mathbb {C}^n$ on which the $\overline{\partial }$-problem is solvable in $L^{\infty }$. Furthermore, as a corollary, we obtain a version of the Axler–Čučković–Rao theorem in higher dimensions.

Let $\Omega$ be a domain in ${\mathbb {C}}^n$ and $\phi$ be a complex-valued function on $\Omega$. Let $H^{\infty }(\Omega )$ and $H^{\infty }(\Omega )[\phi ]$ denote the set of bounded holomorphic functions on $\Omega$ and the algebra generated by $\phi$ over $H^{\infty }(\Omega )$, respectively. In 1989, Christopher Bishop proved the following approximation theorem (see [6, Theorem 1.2]).

FormalPara Theorem

(Bishop) Let $\Omega$ be an open set in $\mathbb {C}$ and f be a bounded holomorphic function on $\Omega$ that is non-constant on every connected component of $\Omega$. Then $H^{\infty }(\Omega )[\overline{f}]$ is dense in $C(\overline{\Omega })$ in the uniform topology.

In the same paper, Christophe Bishop also proved a stronger approximation result, [6, Theorem 1.1], on a more restrictive class of domains on which $\overline{f}$ is only assumed to be a non-holomorphic harmonic function. Such a result for the unit disc goes back to Sheldon Axler and Allen Shields [4]. Recently, Guangfu Cao gave an incorrect statement [8, Theorem 5] in an attempt to give a higher dimensional version of Bishop’s Theorem. Alexander Izzo and Bo Li [14, pg 246] noticed that the statement is incorrect. Håkan Samuelsson and Erlend Wold in [24, Theorem 1.3] proved a partial extension of Bishop’s Theorem for pluriharmonic functions and $C^1$-smooth polynomially convex domains in ${\mathbb {C}}^n$.

This article is motivated by these papers and is an attempt to contribute an approximation theorem akin to Bishop’s Theorem on domains in ${\mathbb {C}}^n$. We are not able to generalize Bishop’s theorem to ${\mathbb {C}}^n$ and this is still an open problem. However, we prove approximation results under some restrictions on the functions and the domains. Furthermore, we apply our results to prove a version of the Axler–Čučković–Rao Theorem [2] in higher dimensions.

To present our first result we need to make some definitions. Let $\Omega \subset {\mathbb {C}}^n$ be a pseudoconvex domain and $CL^{\infty }_{(0,q)}(\Omega )$ denote the set of (0, q)-forms with coefficient functions that are $C^{\infty }$-smooth and bounded on $\Omega$. That is, $CL^{\infty }_{(0,q)}(\Omega )= L^{\infty }_{(0,q)}(\Omega )\cap C^{\infty }_{(0,q)}(\Omega )$. We call $\Omega$ a $L^{\infty }$-pseudoconvex domain if for $1\le q\le n$, and $f\in CL^{\infty }_{(0,q)}(\Omega )$ such that $\overline{\partial }f=0$ there exists $g\in L^{\infty }_{(0,q-1)}(\Omega )$ such that $\overline{\partial }g=f$.

The class of $L^{\infty }$-pseudoconvex domains include the products of $C^2$-smooth bounded strongly pseudoconvex domains [23], smooth bounded pseudoconvex finite type domains in ${\mathbb {C}}^2$ [22], smooth bounded finite type convex domains in ${\mathbb {C}}^n$ [12], and some infinite type smooth bounded convex domains in ${\mathbb {C}}^2$ [13].

Given a holomorphic mapping $f:\Omega \rightarrow {\mathbb {C}}^m$ (where $\Omega \subset {\mathbb {C}}^n$) and $\lambda \in {\mathbb {C}}^m$, we denote the union of all non-isolated points of $f^{-1}(\lambda )$ by $\Omega _{f,\lambda }$. Since $f^{-1}(\lambda )$ is a complex subvariety of $\Omega$ (for $\lambda$ in the range of f), it follows that $\Omega _{f,\lambda }$ is the union of all positive dimensional connected components of $f^{-1}(\lambda )$. In the case f extends smoothly up to the boundary of $\Omega ,$ we define $\Omega '_{f,\lambda }$ to be the union of all non-isolated points of $f^{-1}(\lambda )$ within $\overline{\Omega }$. Clearly $\Omega '_{f,\lambda }\subset \Omega _{f,\lambda }\cup b\Omega$ where $b\Omega$ denotes the boundary of $\Omega$. We define

$$\begin{aligned} \Omega _f=\bigcup _{\lambda \in {\mathbb {C}}^m}\Omega _{f,\lambda }. \end{aligned}$$

It is clear that $\Omega _f$ is a subset of the set where the Jacobian of f has rank strictly less than n.

Now we are ready to present our first approximation result.

FormalPara Theorem 1

Let $\Omega$ be a bounded $L^{\infty }$-pseudoconvex domain in ${\mathbb {C}}^n$ and $f_j\in H^{\infty }(\Omega )$ for $j=1,\ldots ,m$. Assume that $g\in C(\overline{\Omega })$ such that $g|_{b\Omega \cup \Omega _f}=0$ where $f=(f_1,\ldots ,f_m)$. Then g belongs to the closure of $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ in $L^{\infty }(\Omega )$.

Theorem 1 and [14, Theorem 4.2] lead to the following corollary.

FormalPara Corollary 1

Let $\Omega$ be a bounded $L^{\infty }$-pseudoconvex domain in ${\mathbb {C}}^n$ and $f_j\in H^{\infty }(\Omega )$ for $j=1,\ldots , m$ and $n\le m$. Then the following are equivalent.

• i. $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ is dense in $L^p(\Omega )$ for all $0< p<\infty$,

• ii. $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ is dense in $L^p(\Omega )$ for some $1\le p<\infty$,

• iii. the Jacobian of $f=(f_1,\ldots , f_m)$ has rank n for some $z\in \Omega .$

To formulate our next result we will need the following notation. The set of holomorphic functions on $\Omega$ that have smooth extensions up to the boundary is denoted by $A^{\infty }(\Omega )$. Given a compact set $K\subset \overline{\Omega },$ we will denote by $A_{\overline{\Omega }}(K)$ the norm closed subalgebra of continuous functions on K spanned by restrictions of $A^{\infty }(U\cap \Omega )$ onto K, where U runs through open neighborhoods of K.

FormalPara Theorem 2

Let $\Omega$ be a smooth bounded pseudoconvex domain in ${\mathbb {C}}^n$ and $f_j\in A^{\infty }(\Omega )$ for $j=1,\ldots ,m$. Then $g\in C(\overline{\Omega })$ belongs to the closure of $A^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ in $L^{\infty }(\Omega )$ if and only if for any $\lambda$ in the range of $f=(f_1,\ldots ,f_m)$ we have $g|_{\Omega '_{f,\lambda }}\in A_{\overline{\Omega }}(\Omega '_{f,\lambda })$.

Alexander Izzo in [15, Theorem 1.3] proved (among other things) the following interesting result.

FormalPara Theorem

(Izzo) Let A be a uniform algebra on a compact Hausdorff space X whose maximal ideal space is X and $E\subset X$ be a closed subset such that $X\setminus E$ is an m-dimensional manifold. Assume that

• i. for any $p\in X\setminus E$ there exists $f_1,\ldots , f_m\in A$ that are $C^1$-smooth on $X\setminus E$ and $df_1\wedge \cdots \wedge df_m(p)\ne 0,$

• ii. the functions in A that are $C^1$-smooth on $X\setminus E$ separate points on X.

Then $A=\lbrace g\in C(X):g|E\in A|E\rbrace .$

As pointed out to us by Alexander Izzo, a result along the lines of Theorem 1 (for a similar class of domains) can be obtained from [15] as follows. Let us take X to be the maximal ideal space (spectrum) of $H^{\infty }(\Omega )$ and $X\setminus E$ to be the set of points in $\Omega$ where the Jacobian of f has rank n with A being the closure of $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$. Then one obtains Theorem 1 if the set $\Omega _f$ is replaced by the set of points where $J_f$, the Jacobian of f, has rank strictly less than n (usually a larger set than $\Omega _f$).

Next we will present our generalization of the Axler–Čučković–Rao Theorem to ${\mathbb {C}}^n$, but first we will state the commuting problem for Toeplitz operators.

Let $A^2(\Omega )$ denote the space of square integrable holomorphic functions on $\Omega$ and $P:L^2(\Omega )\rightarrow A^2(\Omega )$ be the Bergman projection, the orthogonal projection onto $A^2(\Omega )$. For $g\in L^{\infty }(\Omega )$, the Toeplitz operator $T_g:A^2(\Omega )\rightarrow A^2(\Omega )$ is defined as $T_gf=P(gf)$ for all $f\in A^2(\Omega )$.

The commuting problem can be stated as follows: Let $\phi$ be a non-constant bounded function on $\Omega$. Determine all $\psi \in L^{\infty }(\Omega )$ such that $[T_{\phi }, T_{\psi }] = 0$.

The commuting problem was solved by Arlen Brown and Paul Halmos on the Hardy space of the unit disc in a famous paper [5]. However, on the Bergman space, the problem is still open. Many partial answers has been obtained over the years. To list a few, we refer the reader to [1, 2, 9, 20] for results over the unit disc; to [18, 19, 25] for results over the ball in ${\mathbb {C}}^n$; and to [3, 7, 11] for results on Fock spaces.

In this paper, we want to highlight the following result of Sheldon Axler, Željko Čučković, and Nagisetti Rao (see [2]).

FormalPara Theorem

(Axler–Čučković–Rao) Let $\Omega$ be a bounded domain in ${\mathbb {C}}$ and $\phi$ be a nonconstant bounded holomorphic function on $\Omega$. Assume that $\psi$ is a bounded measurable function on $\Omega$ such that $T_{\phi }$ and $T_{\psi }$ commute. Then $\psi$ is holomorphic.

As an application of our results, we get the following generalization of the Axler–Čučković–Rao Theorem.

FormalPara Corollary 2

Let $\Omega$ be a bounded $L^{\infty }$-pseudoconvex domain in ${\mathbb {C}}^n$, $g\in L^{\infty }(\Omega )$, and $f_j\in H^{\infty }(\Omega )$ for $j=1,\ldots , m$ and $n\le m$. Assume that the Jacobian of the function $f=(f_1, \ldots , f_m):\Omega \rightarrow \mathbb {C}^m$ has rank n for some $z\in \Omega$ and $T_g$ commutes with $T_{f_j}$ for $1\le j\le m$. Then g is holomorphic.

This paper is organized as follows: The next section contains relevant basic facts and results about $\overline{\partial }$-Koszul complex. Then we will present the proofs of Theorems 1 and 2. We will finish the paper with the proof of Corollaries 1 and 2.

1 The $\overline{\partial }$-Koszul Complex

Let $\Omega$ be a domain in ${\mathbb {C}}^n$ and V be a vector space of dimension m with a basis $\{e_1,e_2,\ldots ,e_m\}$. We define

$$\begin{aligned} \wedge ^rV =\text {span}\left\{ e_{j_1}\wedge e_{j_2}\wedge \cdots \wedge e_{j_r}:j_1<j_2<\cdots <j_r\right\} \end{aligned}$$

and $\Gamma ^{\infty }_{(r,s)}=\wedge ^rV \otimes CL^{\infty }_{(0,s)}(\Omega )$ where r and s are nonnegative integers. We note that throughout the paper we use the convention that $\Gamma ^{\infty }_{(r,s)}=\{0\}$ if $r\ge m+1$ or $s\ge n+1$. Finally, $CL^{\infty }_{(0,0)}(\Omega )=CL^{\infty }(\Omega )$.

We define the unbounded operator $\overline{\partial }: \Gamma ^{\infty }_{(r,s)}\rightarrow \Gamma ^{\infty }_{(r,s+1)}$ as $\overline{\partial }(e_J\otimes W)=e_J\otimes \overline{\partial }W$ where $e_J\in \wedge ^r V$ and $W\in CL^{\infty }_{(0,s)}(\Omega )$. The operator $\overline{\partial }$ is defined on

$$\begin{aligned} Dom_{\infty }(\overline{\partial })=\left\{ f\in \Gamma ^{\infty }_{(r,s)}:\overline{\partial }f\in \Gamma ^{\infty }_{(r,s+1)}\right\} . \end{aligned}$$

Let $f=(f_1,\ldots ,f_m):\Omega \rightarrow {\mathbb {C}}^m$ be a bounded holomorphic mapping. Then for $0\le s\le n$ and $0\le r\le m$ we define the operator

$$\begin{aligned} \mathcal {T}_f:\Gamma ^{\infty }_{(r+1,s)}\rightarrow \Gamma ^{\infty }_{(r,s)} \end{aligned}$$

with the following properties:

• (1) $\mathcal {T}_f(e_j\otimes W)=f_jW$,

• (2) $\mathcal {T}_f(A\wedge B)=\mathcal {T}_f(A)\wedge B+(-1)^{|A|_1}A\wedge \mathcal {T}_fB$ (here $|.|_1$ is the order of A in $\cup _{r=0}^m\Lambda ^rV$),

• (3) $\mathcal {T}_f \overline{\partial }=\overline{\partial }\mathcal {T}_f$ on $Dom_{\infty }(\overline{\partial })$ for $0\le s\le n$ and $0\le r\le m$,

• (4) $\mathcal {T}_f\mathcal {T}_f=0$ and $\overline{\partial }\overline{\partial }=0$.

We note that $\mathcal {T}_fW=0$ for $W\in \Gamma ^{\infty }_{(0,s)}$ and $0\le s\le n$.

Lemma 1

Let $\Omega$ be a bounded domain in ${\mathbb {C}}^n, 0\le s\le n,0\le r\le m$, and $f=(f_1,\ldots ,f_m):\Omega \rightarrow {\mathbb {C}}^m$ be a bounded holomorphic mapping. Assume that $W\in \Gamma ^{\infty }_{(r,s)}$ such that $\text {supp}(W)\subset \Omega$ and $\text {supp}(W)\cap f^{-1}(0)=\varnothing$.

• i. If $\mathcal {T}_fW=0$, then there exists $Y\in \Gamma ^{\infty }_{(r+1,s)}$ such that

  • a. $\mathcal {T}_fY=W$,

  • b. $\text {supp}(Y)\subset \Omega$ and $\text {supp}(Y)\cap f^{-1}(0)=\varnothing$.

• ii. If $\mathcal {T}_fW=0$ and $\overline{\partial }W\in \Gamma ^{\infty }_{(r,s+1)}$, then there exists $Y\in \Gamma ^{\infty }_{(r+1,s)}$ such that

  • a. $\overline{\partial }Y\in \Gamma ^{\infty }_{(r+1,s+1)}$ and $\mathcal {T}_fY=W$,

  • b. $\text {supp}(Y)\subset \Omega$ and $\text {supp}(Y)\cap f^{-1}(0)=\varnothing$.

Proof

First let us prove the lemma in case $r=m$. In this case one can show that $\mathcal {T}_fW=0$ and $\text {supp}(W)\cap f^{-1}(0)=\varnothing$ imply that $W=0$. So we can choose $Y=0\in \Gamma ^{\infty }_{(m+1,s)}$. For the rest of the proof we will assume that $0\le r\le m-1$.

Now let us prove i. Let $\chi \in C^{\infty }_0(\Omega )$ be a smooth compactly supported cut-off function such that $\chi =1$ on a neighborhood of $\text {supp}(W)$ and $\text {supp}(\chi )\cap f^{-1}(0)=\varnothing$. We define

$$\begin{aligned} g_j=\frac{\chi \overline{f_j}}{\sum _{l=1}^{m}|f_l|^2} \end{aligned}$$

and

$$\begin{aligned} X=\sum _{j=1}^m e_j\otimes g_j\in \Gamma ^{\infty }_{(1,0)}. \end{aligned}$$

Then $g_j\in C^{\infty }_0(\Omega )$ for $j=1,2,\ldots ,m$ and $\mathcal {T}_fX=1\in \Gamma ^{\infty }_{(0,0)}$ on the support of W because $\chi =1$ on a neighborhood of $\text {supp}(W)$ and $\sum _{j=1}^{m}f_j(z)g_j(z)=1$ whenever $\chi (z) =1$.

Let us define $Y=X\wedge W\in \Gamma ^{\infty }_{(r+1,s)}$. Then $\text {supp}(Y)$ is a compact subset of $\Omega$ and $\text {supp}(Y)\cap f^{-1}(0)=\varnothing$. Furthermore, $\mathcal {T}_fX=1$ on the support of W and

$$\begin{aligned} \mathcal {T}_fY=\mathcal {T}_f(X)\wedge W-X\wedge \mathcal {T}_fW=1\wedge W=W \end{aligned}$$

because $\mathcal {T}_fW=0$.

To prove ii. we observe that, in the proof of i. above, X is smooth compactly supported in $\Omega$. Therefore, if $\overline{\partial }W$ is bounded then so is $\overline{\partial }Y$ as $Y=X\wedge W$. $\square$

If $f_j\in A^{\infty }(\Omega )$ for $j=1,2,\ldots ,m$ in the lemma above, we have the following lemma.

Lemma 2

Let $\Omega$ be a bounded domain in ${\mathbb {C}}^n,V$ be an m-dimensional vector space, and $f_j\in A^{\infty }(\Omega )$ for $j=1,2,\ldots ,m$. Assume that $W\in \wedge ^rV \otimes C^{\infty }_{(0,s)}(\overline{\Omega })$ for $0\le r\le m, 0\le s\le n$, and $\text {supp}(W)\cap f^{-1}(0)=\varnothing$ where $f=(f_1,\ldots ,f_m)$. If $\mathcal {T}_fW=0$ then there exists $Y\in \wedge ^{r+1}V \otimes C^{\infty }_{(0,s)}(\overline{\Omega })$ such that $\text {supp}(Y)\cap f^{-1}(0)=\varnothing$ and $\mathcal {T}_fY=W$.

Proof

The proof of this lemma is very similar to the proof of Lemma 1. The only difference is that we choose $\chi \in C^{\infty }(\overline{\Omega })$ be a smooth function such that $\chi =1$ on a neighborhood of $\text {supp}(W)$ and $\text {supp}(\chi )\cap f^{-1}(0)=\varnothing$. $\square$

Lemma 3

Let $\Omega$ be a bounded $L^{\infty }$-pseudoconvex domain in ${\mathbb {C}}^n,f=(f_1,\ldots ,f_m):\Omega \rightarrow {\mathbb {C}}^m$ be a bounded holomorphic mapping, and $W\in \Gamma ^{\infty }_{(r,s)}$ for $0\le r\le m$ and $1\le s\le n$ such that

• i. $\text {supp}(W)\subset \Omega$ and $\text {supp}(W)\cap f^{-1}(0)=\varnothing$,

• ii. $\overline{\partial }W=0$ and $\mathcal {T}_fW=0$.

Then there exists $Y\in \Gamma ^{\infty }_{(r+1,s-1)}$ such that $Y\in Dom_{\infty }(\overline{\partial })$ and $\mathcal {T}_f\overline{\partial }Y=W$.

Proof

In case $r=m$, as in the proof of Lemma 1, one can show that if W satisfies the conditions of the lemma then $W=0$. So we can choose $Y=0$. For the rest of the proof we will assume that $0\le r\le m-1$.

First we will assume that $\Omega$ is a bounded $L^{\infty }$-pseudoconvex domain. We will use a descending induction on s to prove this lemma. So let $s=n, 0\le r\le m-1$, and $W\in \Gamma ^{\infty }_{(r,n)}$ such that $\text {supp}(W)\subset \Omega ,\text {supp}(W)\cap f^{-1}(0)=\varnothing$, and $\mathcal {T}_fW=0$ ($\overline{\partial }W=0$ as any (0, n)-form is $\overline{\partial }$-closed). Then i. in Lemma 1 implies that there exists $Y_1\in \Gamma ^{\infty }_{(r+1,n)}$ with the following properties:

• i. $\text {supp}(Y_1)\subset \Omega$ and $\text {supp}(Y_1)\cap f^{-1}(0)=\varnothing$,

• ii. $\mathcal {T}_fY_1=W$.

Furthermore, since $Y_1\in \Gamma ^{\infty }_{(r+1,n)}$ it is $\overline{\partial }$-closed. Then (since $\Omega$ is $L^{\infty }$-pseudoconvex) there exists $Y\in \Gamma ^{\infty }_{(r+1,n-1)}$ such that $\overline{\partial }Y=Y_1$. That is, $\mathcal {T}_f\overline{\partial }Y=W$.

Now we will assume that the lemma is true for $s=k+1,k+2, \ldots , n$ and $r=0,1,\ldots ,m-1$. Let $0\le r\le m-1$ and assume that $W\in \Gamma ^{\infty }_{(r,k)}$ with the following properties:

• i. $\text {supp}(W)\subset \Omega$ and $\text {supp}(W)\cap f^{-1}(0) =\varnothing$,

• ii. $\overline{\partial }W=0$ and $\mathcal {T}_fW=0$.

Then ii. in Lemma 1 implies that there exists $Y_1\in \Gamma ^{\infty }_{(r+1,k)}$ such that

• i. $\overline{\partial }Y_1\in \Gamma ^{\infty }_{(r+1,k+1)}$ and $W=\mathcal {T}_fY_1$,

• ii. $\text {supp}(Y_1)\subset \Omega$ and $\text {supp}(Y_1)\cap f^{-1}(0)=\varnothing$.

Then

$$\begin{aligned} \mathcal {T}_f\overline{\partial }Y_1=\overline{\partial }\mathcal {T}_fY_1=\overline{\partial }W=0. \end{aligned}$$

So $\overline{\partial }Y_1$ satisfies the conditions in the lemma for $s=k+1$. That is, $\overline{\partial }Y_1\in \Gamma ^{\infty }_{(r+1,k+1)}$ such that

• i. $\text {supp}(\overline{\partial }Y_1)\subset \Omega$ and $\text {supp}(\overline{\partial }Y_1)\cap f^{-1}(0)=\varnothing$,

• ii. $\overline{\partial }\overline{\partial }Y_1=0$ and $\mathcal {T}_f\overline{\partial }Y_1=\overline{\partial }W=0$.

By the induction hypothesis, there exists $Y_2\in \Gamma ^{\infty }_{(r+2,k)}$ such that $\overline{\partial }Y_2\in \Gamma ^{\infty }_{(r+2,k+1)}$ and $\mathcal {T}_f\overline{\partial }Y_2=\overline{\partial }Y_1$. Then

$$\begin{aligned} \overline{\partial }\mathcal {T}_f Y_2=\mathcal {T}_f\overline{\partial }Y_2=\overline{\partial }Y_1. \end{aligned}$$

We define $Y_3=Y_1-\mathcal {T}_fY_2\in \Gamma ^{\infty }_{(r+1,k)}$. Then the equality above implies that

$$\begin{aligned} \mathcal {T}_fY_3=\mathcal {T}_f Y_1-\mathcal {T}_f \mathcal {T}_fY_2=W \end{aligned}$$

and $\overline{\partial }Y_3=\overline{\partial }Y_1-\overline{\partial }\mathcal {T}_fY_2=0.$ Since $\Omega$ is $L^{\infty }$-pseudoconvex domain we conclude that there exists $Y\in \Gamma ^{\infty }_{(r+1,k-1)}$ such that $\overline{\partial }Y=Y_3$. That is, $\mathcal {T}_f\overline{\partial }Y=W$. Hence the proof of Lemma 3 is complete. $\square$

Lemma 4

Let $\Omega$ be a smooth bounded pseudoconvex domain in ${\mathbb {C}}^n, V$ be an m-dimensional vector space, and $f_i\in A^{\infty }(\Omega )$ for $i=1,\ldots ,m$. Assume that $W\in \wedge ^{r}V \otimes C^{\infty }_{(0,s)}(\overline{\Omega })$ for $0\le r\le m$ and $1\le s\le n$ such that $\text {supp}(W)\cap f^{-1}(0)=\varnothing , \overline{\partial }W=0$, and $\mathcal {T}_fW=0$. Then there exists $Y\in \wedge ^{r+1}V \otimes C^{\infty }_{(0,s-1)}(\overline{\Omega })$ such that $\mathcal {T}_f\overline{\partial }Y=W$.

Proof

This proof is similar to the proof of Lemma 3 with the following changes: Instead of Lemma 1 we use Lemma 2 and, at the last step (since and $f_j\in A^{\infty }(\Omega )$), we use the following result of Joseph Kohn [16] (see also [10, Theorem 6.1.1]): Let $\Omega$ be a smooth bounded pseudoconvex domain in ${\mathbb {C}}^n, 1\le q\le n$, and $u\in C^{\infty }_{(0,q)}(\overline{\Omega })$ with $\overline{\partial }u=0$. Then there exists $f\in C^{\infty }_{(0,q-1)}(\overline{\Omega })$ such that $\overline{\partial }f=u$. $\square$

Lemma 5

Let $\Omega$ be a bounded domain in ${\mathbb {C}}^n$ and $f_j\in H^{\infty }(\Omega )$ for $j=1,\ldots , m$ such that $\sum _{j=1}^m|f_j|^2>\varepsilon$ on $\Omega$ for some $\varepsilon >0$ and $\partial f_j\in L^{\infty }_{(1,0)}(\Omega )$ for $j=1,\ldots ,m$. Assume that $W\in \Gamma ^{\infty }_{(r,s)}$ for $0\le r\le m$ and $0\le s\le n$ such that $\mathcal {T}_fW=0$ and $\overline{\partial }W\in \Gamma ^{\infty }_{(r,s+1)}$. Then there exists $Y\in \Gamma ^{\infty }_{(r+1,s)}$ such that $\overline{\partial }Y\in \Gamma ^{\infty }_{(r+1,s+1)}$ and $\mathcal {T}_fY=W$.

Proof

The proof will be similar to the proof of Lemma 1. Let V be a vector space of dimension m and $\{e_1,e_2,\ldots ,e_m\}$ be a basis for V. We define

$$\begin{aligned} g_j=\frac{\overline{f_j}}{\sum _{l=1}^m|f_l|^2} \end{aligned}$$

and $X=\sum _{j=1}^me_j\otimes g_j\in \Gamma ^{\infty }_{(1,0)}$. Then $g_j\in L^{\infty }(\Omega )$ and

$$\begin{aligned} \overline{\partial }g_j= \frac{\overline{\partial }\overline{f_j}}{\sum _{l=1}^m|f_l|^2} -\frac{\overline{f_j}\sum _{l=1}^mf_l \overline{\partial }\overline{f_l} }{\left( \sum _{l=1}^m|f_l|^2\right) ^2} \in L^{\infty }_{(0,1)}(\Omega ). \end{aligned}$$

Furthermore, $\overline{\partial }X=\sum _{j=1}^m e_j\otimes \overline{\partial }g_j\in \Gamma ^{\infty }_{(1,1)}$. Then $Y=X\wedge W\in \Gamma ^{\infty }_{(r+1,s)}$ satisfies the following properties: $\overline{\partial }Y= \overline{\partial }X\wedge W+X\wedge \overline{\partial }W \in \Gamma ^{\infty }_{(r+1,s+1)}$ and

$$\begin{aligned} \mathcal {T}_fY=\mathcal {T}_f(X)\wedge W-X\wedge \mathcal {T}_fW=1\wedge W=W \end{aligned}$$

as $\mathcal {T}_fW=0$. $\square$

Proposition 1

Let $\Omega$ be a bounded $L^{\infty }$-pseudoconvex domain in ${\mathbb {C}}^n$ and $f_j\in H^{\infty }(\Omega )$ for $j=1,\ldots , m$ such that $\sum _{j=1}^m|f_j|^2>\varepsilon$ on $\Omega$ for some $\varepsilon >0$ and $\partial f_j\in L^{\infty }_{(1,0)}(\Omega )$ for $j=1,\ldots ,m$. Assume that $W\in \Gamma ^{\infty }_{(r,s)}$ for $0\le r\le m$ and $0\le s\le n$ such that $\overline{\partial }W=0$ and $\mathcal {T}_fW=0$. Then there exists $Y\in \Gamma ^{\infty }_{(r+1,s)}$ such that $\overline{\partial }Y=0$ and $\mathcal {T}_fY=W$.

Proof

We will use a descending induction on s as in the proof of Proposition 1. Let $s=n$. Any form of type (r, n) for $0\le r\le m$ is $\overline{\partial }$-closed. Then $\overline{\partial }Y=0$ and Lemma 5 implies that there exists $Y\in \Gamma ^{\infty }_{(r+1,n)}$ such that $\mathcal {T}_fY=W$.

Now we will assume that the lemma is true for $s=l+1,l+2, \ldots , n$ and $r=0,1,\ldots ,m$ to prove that it is also true for $s=l\le n-1$ and $0\le r\le m$.

Assume that $W\in \Gamma ^{\infty }_{(r,l)}$ such that $\overline{\partial }W=0$ and $\mathcal {T}_fW=0$. Then Lemma 5 implies that there exists $\widetilde{Y}\in \Gamma ^{\infty }_{(r+1,l)}$ such that $\overline{\partial }\widetilde{Y}\in \Gamma ^{\infty }_{(r+1,l+1)}$ and $W=\mathcal {T}_f\widetilde{Y}$. Then

$$\begin{aligned} \mathcal {T}_f\overline{\partial }\widetilde{Y} =\overline{\partial }\mathcal {T}_f\widetilde{Y}=\overline{\partial }W=0. \end{aligned}$$

So $\overline{\partial }\widetilde{Y}$ satisfies the conditions in the lemma for $s=l+1$. That is, $\overline{\partial }\widetilde{Y}\in \Gamma ^{\infty }_{(r+1,l+1)}, \overline{\partial }\overline{\partial }\widetilde{Y}=0$ and $\mathcal {T}_f\overline{\partial }\widetilde{Y}=\overline{\partial }W=0$. Then, by the induction hypothesis, there exists $Y_1\in \Gamma ^{\infty }_{(r+2,l+1)}$ such that $\overline{\partial }Y_1=0$ and $\mathcal {T}_fY_1=\overline{\partial }\widetilde{Y}$. Then since $\Omega$ is a $L^{\infty }$-pseudoconvex domain there exists $Y_2\in \Gamma ^{\infty }_{(r+2,l)}$ such that $\overline{\partial }Y_2=Y_1$. Then

$$\begin{aligned} \overline{\partial }\mathcal {T}_f Y_2=\mathcal {T}_f\overline{\partial }Y_2 =\mathcal {T}_fY_1=\overline{\partial }\widetilde{Y}. \end{aligned}$$

We define $Y=\widetilde{Y}-\mathcal {T}_fY_2\in \Gamma ^{\infty }_{(r+1,l)}$. Then the equality above implies that $\overline{\partial }Y=\overline{\partial }\widetilde{Y}-\overline{\partial }\mathcal {T}_fY_2=0$ and

$$\begin{aligned} \mathcal {T}_fY =\mathcal {T}_f \widetilde{Y}-\mathcal {T}_f\mathcal {T}_fY_2=W. \end{aligned}$$

Hence the proof of Proposition 1 is complete. $\square$

As a corollary to the previous proposition (with $W=1$ and $r=s=0$) we get the following Corona type result. We refer the reader to [17] and the references therein for more information about Corona problem on domains in ${\mathbb {C}}^n$.

Corollary 3

Let $\Omega$ be a bounded $L^{\infty }$-pseudoconvex domain in ${\mathbb {C}}^n$ and $f_j\in H^{\infty }(\Omega )$ for $j=1,\ldots , m$ such that $\sum _{j=1}^m|f_j|^2>\varepsilon$ on $\Omega$ for some $\varepsilon >0$ and $\partial f_j\in L^{\infty }_{(1,0)}(\Omega )$ for $j=1,\ldots ,m$. Then there exists $g_i\in H^{\infty }(\Omega )$ for $j=1,\ldots , m$ such that $\sum _{j=1}^mf_jg_j=1$.

2 Proofs of results

The proofs of the theorems are mainly inspired by the proof in Christopher Bishop’s paper [6].

Proofs of Theorems1and2 The proofs of both theorems are very similar. So we will present the proof of Theorem 1 and comment on how the proof of Theorem 2 differs as we go along.

Let $\epsilon >0$ and $\lambda \in {\mathbb {C}}^m$. Since $g\in C(\overline{\Omega })$ and $g|_{b\Omega \cup \Omega _f}=0$, there exist $g^{\lambda }\in C^{\infty }(\overline{\Omega })$ such that

• i. $\sup \{|g(z)-g^{\lambda }(z)|:z\in \overline{\Omega }\}<\varepsilon$,

• ii $\text {supp}(\overline{\partial }g^{\lambda })\cap (b\Omega \cup f^{-1}(\lambda )) =\varnothing$.

In the proof Theorem 2 the second condition above is replaced by $\text {supp}(\overline{\partial }g^{\lambda })\cap f^{-1}(\lambda )=\varnothing$. This can be seen as follows: We choose an open set $U_{\varepsilon }$ in ${\mathbb {C}}^n$ containing $f^{-1}(\lambda )$ and $g_{\varepsilon }\in A^{\infty }(U_{\varepsilon }\cap \Omega )$ such that $|g-g_{\varepsilon }|<\varepsilon /2$ on $f^{-1}(\lambda )$. Then we choose $\chi _{\varepsilon }\in C^{\infty }_0(U_{\varepsilon })$ such that, $0\le \chi _{\varepsilon }\le 1,\chi _{\varepsilon }=1$ on a neighborhood of $f^{-1}(\lambda )$, and

$$\begin{aligned} \text {supp}(\chi _{\varepsilon })\cap \overline{\Omega }\subset \left\{ z\in U_{\varepsilon }\cap \overline{\Omega }:|g(z)-g_{\varepsilon }(z)|<\varepsilon \right\} . \end{aligned}$$

Then we define $g^{\lambda }=(1-\chi _{\varepsilon })g+\chi _{\varepsilon }g_{\varepsilon }$. Since $g^{\lambda }$ is holomorphic on a neighborhood of $f^{-1}(\lambda )$ we have $\overline{\partial }g^{\lambda }=0$ on the same neighborhood. Furthermore, $|g^{\lambda }(z)-g(z)|=\chi _{\varepsilon }(z)|g_{\varepsilon }(z)-g(z)|< \varepsilon$ for all $z\in \overline{\Omega }$.

Using Lemma 3 with $r=0,s=1$, and $W=\overline{\partial }g^{\lambda }$ we get $Y=\sum _{l=1}^me_l\otimes H_l\in \Gamma ^{\infty }_{(1,0)}$ such that

$$\begin{aligned} \overline{\partial }g^{\lambda }=\mathcal {T}_{f-\lambda }\overline{\partial }Y =\sum _{l=1}^m (f_l-\lambda _l)\overline{\partial }H_l^{\lambda }. \end{aligned}$$

(1)

The above equality implies that

$$\begin{aligned} G_{\lambda }=g^{\lambda }-\sum _{l=1}^m (f_l-\lambda _l)H_l^{\lambda } \end{aligned}$$

is a bounded holomorphic function.

In the proof of Theorem 2, we use Lemma 4 and get $H_l^{\lambda }\in C^{\infty }(\overline{\Omega })$ for $l=1,\ldots ,m$ in the equation (1) and $G_{\lambda }$ is smooth up to the boundary. Therefore, for $z\in \Omega$ we have

$$\begin{aligned} |G_{\lambda }(z)-g^{\lambda }(z)| \le \sum _{l=1}^m |f_l(z)-\lambda _l| \sum _{s=1}^m|H_s^{\lambda }(z)|. \end{aligned}$$

Then the above inequality implies that for $M_{\lambda }=\sum _{s=1}^m\Vert H^{\lambda }_s\Vert _{L^{\infty }(\Omega )}<\infty$ we have

$$\begin{aligned} |G_{\lambda }(z)-g^{\lambda }(z)|\le M_{\lambda }|f(z)-\lambda | \end{aligned}$$

(2)

for $z\in \Omega$.

Compactness of $\overline{f(\Omega )}$ implies that we can choose a finite collection of points $\{\lambda _j\}_{j=1}^k\subset \overline{f(\Omega )}$ such that $\{B(\lambda ^j,\epsilon M^{-1}_{\lambda ^j})\}_{j=1}^k$ forms a finite open cover for $\overline{f(\Omega )}$. Let $\{\chi _j\}_{j=1}^k$ be a smooth partition of unity on $\overline{f(\Omega )}$ such that $0\le \chi _j\le 1$ and $\text {supp}(\chi _j)\subset U_j$. Then $\{f^{-1}(B(\lambda ^j,\epsilon M^{-1}_{\lambda ^j}))\}_{j=1}^k$ is an cover for $\Omega$ and $|f(z)-\lambda ^j|< \epsilon M^{-1}_{\lambda ^j}$ for $z\in f^{-1}(B(\lambda ^j,\epsilon M^{-1}_{\lambda ^j}))$. Then for $z\in \Omega$ we have

$$\begin{aligned} \left| \sum _{j=1}^k G_{\lambda ^j}(z)\chi _j(f)(z)-g(z)\right| \le&\sum _{j=1}^k |G_{\lambda ^j}(z)-g(z)|\chi _j(f(z))\\ \le&\sum _{j=1}^k|G_{\lambda ^j}(z)-g^{\lambda ^j}|\chi _j(f(z)) +\sum _{j=1}^k|g^{\lambda ^j}(z)-g(z)|\chi _j(f(z))\\ \le&\sum _{j=1}^kM_{\lambda ^j}|f(z)-\lambda ^j|\chi _j(f(z)) +\varepsilon \sum _{j=1}^k\chi _j(f(z)) \\ \le&2\epsilon . \end{aligned}$$

Finally, the Stone-Weierstrass Theorem implies that $\chi _j(f)$ can be approximated uniformly on $\overline{\Omega }$ by elements of $\mathbb {C}[f_1,\ldots , f_m, \overline{f_1},\ldots ,\overline{f_m}]$. Hence the proofs of Theorems 1 and 2 are complete. $\square$

Hartogs Extension Theorem together Theorem 2 lead to the following corollary.

Corollary 4

Let $\Omega$ be a bounded $L^{\infty }$-pseudoconvex domain in ${\mathbb {C}}^n$. Assume that $f=(f_1,\ldots ,f_m):\Omega \rightarrow {\mathbb {C}}^m$ be a bounded holomorphic mapping and $g\in C(\overline{\Omega })$ such that $\overline{\partial }g$ is supported away from $b\Omega$ and the set of points at which the Jacobian of f has rank strictly less than n. Then g belongs to the closure of $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ in $L^{\infty }(\Omega )$.

Proof

Since $\overline{\partial }g$ vanishes near the boundary of $\Omega$, Hartogs Extension Theorem implies that there exists $g_1\in H^{\infty }(\Omega )$ such that $g=g_1$ near the boundary of $\Omega$. Then $g_2=g-g_1\in C(\overline{\Omega })$ and $g_2$ is compactly supported in $\Omega$. Furthermore, $g_2$ is holomorphic on a neighborhood of the set where the Jacobian of f has rank strictly less than n. Therefore, Theorem 2 implies that $g_2$ can be approximated in the sup-norm by functions in $H^{\infty }(\Omega )[\overline{f_1}, \ldots , \overline{f_m}]$. This completes the proof of the corollary. $\square$

Next we provide the proof of Corollary 1.

Proof of Corollary 1

Obviously i. implies ii. So to prove that ii. implies iii., let us assume that $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ is dense in $L^p(\Omega )$ for some $1\le p<\infty$. Let $B\subset \Omega$ be a ball such that $\overline{B}\subset \Omega$. Then, the algebra $H^{\infty }(B)[\overline{f_1},\ldots , \overline{f_m}]$ is dense in $L^p(B)$ for some $1\le p<\infty$. Moreover, the algebra generated by $\{z_1,\ldots ,z_n\}$ is dense in $H^{\infty }(B)$ and $f_1,\ldots , f_m$ are holomorphic on a neighborhood of $\overline{B}.$ Next we adopt [14, Theorem 4.2] to our set-up. Namely, [14, Theorem 4.2] implies that if the algebra generated by $\{z_1,\ldots , z_n,\overline{f}_1,\ldots , \overline{f}_m\}\subset C^{\infty }(B)$ is dense in $L^p(B)$ for some $1\le p<\infty$ then the real Jacobian of $\{z_1,\ldots , z_n,\overline{f}_1,\ldots , \overline{f}_m\}$ is of full rank on a dense open set in B. Hence the rank of $J_f$ is n on a dense open subset in B and (by identity principle) in $\Omega$. Hence, we have iii.

Finally, to prove iii. implies i. we assume that the rank of $J_f$ is n for some $z\in \Omega$. Then, the set of points at which $J_f$ has rank strictly less than n is a closed set of measure 0 (see [21, Theorem 3.7]). One can show that $X_f$, the set of smooth functions with compact support in $\Omega$ and vanish where $J_f$ has rank strictly less than n, is dense in $L^p(\Omega )$ for all $0<p<\infty$. On the other hand, Theorem 1 implies that any function in $X_f$ is in the closure of $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ in $L^{\infty }(\Omega )$. Therefore, $H^{\infty }(\Omega )[\overline{f_1},\ldots , \overline{f_m}]$ is dense in $L^p(\Omega ).$ Hence, we have i. $\square$

We finally end the paper with the proof of Corollary 2.

Proof of Corollary 2

We will use the fact that $T_g$ can be defined by the following formula

$$\begin{aligned} \langle T_g\phi , \psi \rangle _{A^2(\Omega )}=\langle g\phi , \psi \rangle _{L^2(\Omega )} \end{aligned}$$

for all $\phi , \psi \in A^2(\Omega ).$ Since $T_g$ commutes with $T_{P(f)}$, for any holomorphic polynomial P, we have

$$\begin{aligned} \langle gP(f),\psi \rangle = \langle T_gT_{P(f)}(1), \psi \rangle =\langle P(f)T_g(1),\psi \rangle \end{aligned}$$

for all $\psi \in A^2(\Omega )$. Then $\langle T_{g}(1)-g, \overline{P(f)}\psi \rangle =0$ for all $\psi \in A^2(\Omega )$. Since, by Corollary 1, the subspace generated by $\{\overline{P(f)}\psi : \psi \in A^2(\Omega )\}$ is dense in $L^2(\Omega ),$ we conclude that $T_g(1)=g$. That is, g is holomorphic. $\square$