Schatten class Hankel and $\overline{\partial}$-Neumann operators on pseudoconvex domains in $\mathbb{C}^n$

TL;DR

This paper investigates the Schatten class properties of Hankel and $ar{ abla}$-Neumann operators on smooth pseudoconvex domains in complex space, establishing conditions under which these operators are compact or belong to specific Schatten classes.

Contribution

It proves that certain Hankel operators are in Schatten $p$-class only for constant symbols when $p o 2n$, and demonstrates the $ar{ abla}$-Neumann operator is not Hilbert-Schmidt on these domains.

Findings

Hankel operators in Schatten $p$-class imply constant symbols for $p o 2n$.

The $ar{ abla}$-Neumann operator is not Hilbert-Schmidt.

Provides conditions linking operator class membership to function constancy.

Abstract

Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt.

Equations

$${\int }_{\mathbb{D}}|{\phi }^{\prime }(z){|}^p(1-|z{|}^2{)}^{p-2}dV(z)<\infty$$

$$P=I-{\overline{\partial }}^{\ast }{N}_1\overline{\partial }.$$

$$\tilde{\phi }(z)={\int }_{\Omega }|k_z(\xi ){|}^2\phi (\xi )dV(\xi )$$

$$MO(\phi ,z)=\widetilde{|\phi {|}^2}(z)-|\tilde{\phi }(z){|}^2.$$

$$MO(\phi ,z)=\widetilde{|\phi {|}^2}(z)-|\phi (z){|}^2$$

$$P\overline{\phi }{k}_z(w)=$$

$$=$$

$$H_{\overline{\phi }}{k}_z(w)=(\overline{\phi (w)}-\overline{\phi (z)})k_z(w)$$

$${\Vert H_{\overline{\phi }}{k}_z\Vert }^2=$$

$$=$$

$${\int }_{\Omega }⟨(H_{\overline{\phi }}^{\ast }{H}_{\overline{\phi }}{)}^{p/2}{k}_z,k_z⟩K(z,z)dV(z)<\infty .$$

$${\int }_{\Omega }(MO(\phi ,z){)}^{p/2}K(z,z)dV(z)=$$

$$=$$

$$\le$$

$$<$$

$${\chi }_z(w)=\frac{1}{{\epsilon }^{2n}}\psi \left(\frac{w_1-z_1}{\epsilon }\right)\cdots \psi \left(\frac{w_n-z_n}{\epsilon }\right)\in C_0^{\infty }(\Omega )$$

$$\frac{\partial }{\partial x_j}P{\chi }_z=P\left(\frac{\partial }{\partial x_j}{\chi }_z\right)\text{ and }\frac{\partial }{\partial y_j}P{\chi }_z=P\left(\frac{\partial }{\partial y_j}{\chi }_z\right)$$

$$\Vert \frac{P{\chi }_{z+h_j}-P{\chi }_z-hP{\partial }_{x_j}{\chi }_z}{h}\Vert \le \Vert \frac{{\chi }_{z+h_j}-{\chi }_z-h{\partial }_{x_j}{\chi }_z}{h}\Vert \to 0$$

$$\left|\frac{d}{dt}\tilde{\phi }(\gamma (t))\right|\le 2\sqrt{2}\left(\frac{ds}{dt}\right)\underset{0\le t\le 1}{\sup }(MO(\phi ,\gamma (t)){)}^{1/2}.$$

$$\left|\sum _{j=1}^n{a}_j\frac{\partial \phi (z)}{\partial z_j}\right|\le 2\sqrt{2}(MO(\phi ,z){)}^{1/2}B(X,z)$$

$$H_0=\{({\xi }_1,{\xi }_2)\in {\mathbb{C}}^2:(0,\dots ,0,{\xi }_1,{\xi }_2)\in \Omega \}$$

$$B(X,z)\le C\left(\frac{|X_{\tau }|}{(d_{b\Omega }(z){)}^{1/2}}+\frac{|X_{\nu }|}{d_{b\Omega }(z)}\right)$$

$${\int }_{\Omega }(MO(\phi ,z){)}^{n}K(z,z)dV(z)<\infty .$$

$$\left|\sum _{j=1}^n{a}_j\frac{\partial \phi (z)}{\partial z_j}\right|\le 2\sqrt{2}(MO(\phi ,z){)}^{1/2}B(X,z)$$

$$B(X,z)\le \frac{C}{2\sqrt{2}}\left(\frac{|X_{\tau }|}{(d_{b\Omega }(z){)}^{1/2}}+\frac{|X_{\nu }|}{d_{b\Omega }(z)}\right)$$

$$\left|\sum _{j=1}^n{a}_j\frac{\partial \phi (z)}{\partial z_j}\right|\le C\cdot (MO(\phi ,z){)}^{1/2}\left(\frac{|X_{\tau }|}{(d_{b\Omega }(z){)}^{1/2}}+\frac{|X_{\nu }|}{d_{b\Omega }(z)}\right).$$

$$|{\partial }_b\phi (z){|}^2{d}_{b\Omega }(z)=|{\overline{\partial }}_b\overline{\phi }(z){|}^2{d}_{b\Omega }(z)\le C^2\cdot MO(\phi ,z).$$

$${\int }_{\Omega \cap U}|{\partial }_b\phi (z){|}^{2n}(d_{b\Omega }(z){)}^{n}K(z,z)dV(z)<\infty .$$

$${\int }_0^{\epsilon }\frac{dt}{t}{\int }_{b{\Omega }_t\cap \tilde{U}}|{\partial }_b\phi (z){|}^{2n}d\sigma (z)\le$$

$$\le$$

Full text

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and $A^{2}(\Omega)$ denote the Bergman space, the set of square integrable holomorphic functions on $\Omega$. We define the Hankel operator $H_{\varphi}:A^{2}(\Omega)\to L^{2}(\Omega)$ with symbol $\varphi\in L^{\infty}(\Omega)$ as follows: $H_{\varphi}f=(I-P)(\varphi f)$ for $f\in A^{2}(\Omega)$, where $I$ is the identity map and $P:L^{2}(\Omega)\to A^{2}(\Omega)$ is the Bergman projection.

In this paper we study Schatten $p$-class membership of Hankel operators. The Hankel operator $H_{\varphi}$ is said to be in the Schatten $p$-class, $S_{p}$, if the operator $(H^{*}_{\varphi}H_{\varphi})^{p/2}$ is in the trace class, $S_{1}$. We recall that a self-adjoint compact operator on a separable Hilbert space is in $S_{1}$ if its eigenvalues are absolutely summable. We note that $S_{2}$ is the class of Hilbert-Schmidt operators and we refer the reader to [Zhu07] for more information about these notions.

On the unit disc, $\mathbb{D}\subset\mathbb{C}$, Arazy-Fisher-Peetre [AFP88] (see also [Zhu07, Theorem 8.29]) showed that for $\varphi\in A^{2}(\mathbb{D})$ the Hankel operator $H_{\overline{\varphi}}$ is in the Schatten $p$-class if and only if $\varphi$ is in the Besov space $B_{p}$ consisting of holomorphic functions $\varphi$ on $\mathbb{D}$ such that

[TABLE]

where $dV$ is the Lebesgue measure.

In higher dimensions, that is $\Omega\subset\mathbb{C}^{n}$ for $n\geq 2$, the first result is due to Kehe Zhu. He [Zhu90] showed that in case $\Omega$ is the unit ball and $\varphi$ is holomorphic, $H_{\overline{\varphi}}\in S_{p}$ for $p\leq 2n$ if and only if $\varphi$ is constant. Since then Schatten $p$-class membership of Hankel operators has been studied by many authors. For example, to list a few, it has been studied on the unit ball [Zhu91, Xia02, Pau16], strongly pseudoconvex domains [Li93], finite type pseudoconvex domains in $\mathbb{C}^{2}$ [KLR97], Reinhardt domains [Le14, ÇZ13, ÇZ17], and the Fock spaces [Sch04, Sch09, SY13]. In this paper, we study it on $C^{2}$-smooth bounded pseudoconvex domains in $\mathbb{C}^{n}$ for $n\geq 2$. Throughout the paper $\mathcal{O}(\Omega)$ denotes the space of holomorphic functions on $\Omega$.

Our main result is the following theorem.

The following is a trivial corollary of Theorem 1.

Hankel operators, through the Kohn’s formula, are connected to the $\overline{\partial}$-Neumann operator, an important tool in several complex variables. Now we explain this connection.

Let $\overline{\partial}\overline{\partial}^{*}+\overline{\partial}^{*}\overline{\partial}$ be the complex Laplacian on $L^{2}_{(0,1)}(\Omega)$, the square integrable $(0,1)$-forms on $\Omega$. This is an unbounded, self-adjoint, closed operator. Hörmander [Hör65] showed that (see also [CS01, Theorem 4.4.1]), if $\Omega$ is a bounded pseudoconvex domain in $\mathbb{C}^{n}$, then the complex Laplacian has a bounded solution operator $N_{1}$, called the $\overline{\partial}$-Neumann operator. Furthermore, Kohn [Koh63] (see also [CS01, Theorem 4.4.5]) proved that the Bergman projection and $N_{1}$ are connected trough the following formula

[TABLE]

Therefore, one can show that if $\Omega$ is a bounded pseudoconvex domain and $\varphi\in C^{1}(\overline{\Omega})$ then $H_{\varphi}f=\overline{\partial}^{*}N_{1}(f\overline{\partial}\varphi)$ for $f\in A^{2}(\Omega)$. So it is reasonable to expect $H_{\varphi}$ to be closely connected to $N_{1}$. Indeed this is true in terms of compactness of the operators. We refer the reader to [Str10, Proposition 4.1] and [ČŞ09, ÇŞ14, ŞZ17] for some recent results in this direction, and to books [CS01, Str10, Has14] for more information about the $\overline{\partial}$-Neumann problem.

In terms of Schatten $p$-class membership of $N_{1}:L^{2}_{(0,1)}(\Omega)\to L^{2}_{(0,1)}(\Omega)$ we have the following corollary, which will be proven at the end of the paper. We note that the result in Corollary 2 below also holds for the restriction of $N_{1}$ onto $A^{2}_{(0,1)}(\Omega)$, the space of $(0,1)$-forms with square integrable holomorphic coefficients on $\Omega.$ Furthermore, while $\overline{\partial}^{*}N_{1}$ (canonical solution operator to $\overline{\partial}$) is Hilbert-Schmidt for $\Omega=\mathbb{D}\subset\mathbb{C}$, it fails to be Hilbert-Schmidt when $\Omega$ is the unit ball in $\mathbb{C}^{n}$ for $n\geq 2$. We refer the reader to [Has14, Chapter 2] and the references therein for results about Schatten $p$-class membership of $\overline{\partial}^{*}N_{1}$.

The rest of the paper is organized as follows. In the next section we will present some necessary basic results that are well known. We include them here for the convenience of the reader. In the last section we give the proofs of Theorem 1 and Corollary 2.

Preparatory Results

In this section we will include some preparatory results that will be useful in the proof of Theorem 1. We include them here for the convenience of the reader but we don’t claim any originality about these results.

Let $\Omega$ be a bounded domain and $\varphi\in L^{\infty}(\Omega)$. Then the Berezin transform of $\varphi$ is defined as

[TABLE]

where $k_{z}(\xi)=\frac{K(\xi,z)}{\sqrt{K(z,z)}}$. Furthermore, we define

[TABLE]

We denote $H^{\infty}(\Omega)=\mathcal{O}(\Omega)\cap L^{\infty}(\Omega)$. In case $\varphi\in H^{\infty}(\Omega)$ we have

[TABLE]

as $\widetilde{\varphi}=\varphi$.

Lemma 1**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and $\varphi\in H^{\infty}(\Omega)$. Then $P\overline{\varphi}k_{z}=\overline{\varphi(z)}k_{z}$ for $z\in\Omega$.

Proof.

Let $z,w\in\Omega$. Then

[TABLE]

Hence the proof of the lemma is complete. ∎

Corollary 3**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and $\varphi\in H^{\infty}(\Omega)$. Then

[TABLE]

for $z,w\in\Omega$.

Lemma 2**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and $\varphi\in H^{\infty}(\Omega)$. Then $\|H_{\overline{\varphi}}k_{z}\|^{2}=MO(\varphi,z)$.

Proof.

Let $z\in\Omega$. Lemma 1 implies that $P\overline{\varphi}k_{z}=\overline{\varphi(z)}k_{z}$. Then

[TABLE]

Hence the proof of the lemma is complete. ∎

We note that even though Lemmas 1 and 2 in [Zhu91] (used in the proof below) are stated for the ball, they are actually true on any domain. The following corollary can also be deduced from [Li93, Theorem 3.1]. We present a proof here for the convenience of the reader.

Corollary 4**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n},p\geq 2$, and $\varphi\in H^{\infty}(\Omega)$. Then $H_{\overline{\varphi}}\in S_{p}$ implies that $\int_{\Omega}(MO(\varphi,z))^{p/2}K(z,z)dV(z)<\infty.$

Proof.

Let us assume that $H_{\overline{\varphi}}\in S_{p}$ for $p\geq 2$. Then $(H_{\overline{\varphi}}^{*}H_{\overline{\varphi}})^{p/2}$ is in trace class on $A^{2}(\Omega)$ (see [Zhu07, Theorem 1.26]). Then [Zhu91, Lemma 1] (see also proof of [Zhu07, Theorem 6.4]) implies that

[TABLE]

Next we use Lemma 2 and [Zhu91, Lemma 2] (see also [Zhu07, Proposition 1.31]) to conclude that

[TABLE]

Therefore, the proof of the corollary is complete. ∎

Remark 1*.*

We will use [BBCZ90, Theorem F] in the proof of Theorem 1. So we take this opportunity to comment that even though [BBCZ90, Theorem F] is stated for bounded symmetric domains, observation of the proof (see [BBCZ90, Remark on pg 321] reveals that it is actually true on all bounded domains in $\mathbb{C}^{n}$. Indeed, let $\psi:\mathbb{C}\to[0,\infty)$ be a rotation-invariant $C^{\infty}$-smooth function with $supp(\psi)\subset\mathbb{D}$ and $\int_{\mathbb{D}}\psi(\xi)dV(\xi)=1$. Then for $z\in\Omega$ and sufficiently small $\varepsilon>0$ we define

[TABLE]

where $w=(w_{1},\ldots,w_{n})$ and $z=(z_{1},\ldots,z_{n})$. Then we have $K(w,z)=P\chi_{z}(w)$ (see, for instance, [JP13, Remark 12.1.5]). To prove that $\frac{\partial}{\partial\overline{z}_{j}}K(.,z)\in A^{2}(\Omega)$, it is enough to show that

[TABLE]

where $z_{j}=x_{j}+iy_{j}$. We will show only the first equality as the second one is similar. Let $h_{j}=(0,\ldots,0,h,0,\ldots,0)$ where $h$ is a real number at the $j$th spot. Since we are dealing with holomorphic functions, it is enough to prove that $\|P\chi_{z+h_{j}}-P\chi_{z}-hP\partial_{x_{j}}\chi_{z}\|=o(h)$ where $\partial_{x_{j}}=\frac{\partial}{\partial x_{j}}$. Since $P$ is a bounded linear operator with norm equal to 1 and $\chi_{z}\in C^{\infty}_{0}(\Omega)$ we have

[TABLE]

as $h\to 0$. Therefore, $\frac{\partial}{\partial x_{j}}P\chi_{z}=P\left(\frac{\partial}{\partial x_{j}}\chi_{z}\right)$. Furthermore, using induction we conclude that $\frac{\partial^{\alpha}}{\partial\overline{z}_{j}^{\alpha}}K(.,z)\in A^{2}(\Omega)$ for any multi-index $\alpha$.

The following is a version of [BBCZ90, Theorem F] for bounded domains in $\mathbb{C}^{n}$.

Theorem 2** ([BBCZ90]).**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and $\gamma:[0,1]\to\Omega$ be a $C^{1}$-smooth curve. Assume that $s(t)$ denote the arc-length of $\gamma$ with respect to the Bergman metric of $\Omega$ and $\varphi\in L^{\infty}(\Omega)$. Then

[TABLE]

Then we have the following useful corollary.

Corollary 5**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n},\varphi\in H^{\infty}(\Omega)$, and $X=(a_{1},\ldots,a_{n})\in\mathbb{C}^{n}$. Then

[TABLE]

where $B(X,z)$ denotes the Bergman metric applied to the vector $X$ at $z$.

Proofs of Theorem 1 and Corollary 2

Before we start the proof of Theorem 1 we present two results in several complex variables. We note that $B_{z_{0}}(r)$ denotes the open ball centered at $z_{0}$ with radius $r$. We will use the notion of CR functions in the following proposition. We refer the reader to [CS01, Chapter 3] for the definition and properties of CR functions.

Proposition 1**.**

Let $\Omega$ be a domain in $\mathbb{C}^{n}$ for $n\geq 2,z_{0}\in b\Omega$, and $\varphi\in\mathcal{O}(\Omega)\cap C^{2}(\overline{\Omega})$. Assume that there exists $r>0$ such that $b\Omega$ is $C^{2}$-smooth in the ball $B_{z_{0}}(r)$, the Levi form of $b\Omega$ has at least one positive eigenvalue at $z_{0}$, and $\overline{\varphi}$ is CR function on $b\Omega\cap B_{z_{0}}(r)$. Then $\varphi$ is constant.

Proof.

Using a holomorphic change of coordinates we may assume that $z_{0}$ is the origin, $y_{n}$-axis is the real normal direction and $X_{1}=(0,\ldots,0,1,0)$ is complex tangential (corresponding to a positive eigenvalue of the Levi form, and the two dimensional slice $H_{0}$) at $z_{0}$, and

[TABLE]

is strictly convex at the origin. Furthermore, since small $C^{2}$ perturbations of strictly convex surfaces are strictly convex, the slices $\{(\xi_{1},\xi_{2})\in\mathbb{C}^{2}:(z_{1},\ldots,z_{n-2},\xi_{1},\xi_{2})\in\Omega\}$ are strictly convex for sufficiently small $|z_{1}|+\ldots+|z_{n-2}|$. Then we conclude that there exists $0<c<1$ such that $\Omega\cap B_{z_{0}}(cr)$ is union of discs parallel to $z_{1}$-axis whose boundaries lie in $b\Omega\cap B_{z_{0}}(r)$.

Since $\overline{\varphi}|_{b\Omega\cap B_{z_{0}}(r)}$ is a CR function, [CS01, Theorem 3.3.2] implies that it has a holomorphic extension $\phi_{z_{0},r}$ onto $\Omega\cap B_{z_{0}}(cr)$ for some $c>0$ (here we shrink $c$ if necessary). Then the fact that $\phi_{z_{0},r}$ and $\overline{\varphi}$ are harmonic and they match on $b\Omega\cap B_{z_{0}}(r)$ imply that $\phi_{z_{0},r}=\overline{\varphi}$ on $\Omega\cap B_{z_{0}}(cr)$. Hence, $\varphi$ and $\overline{\varphi}$ are holomorphic on $\Omega\cap B_{z_{0}}(cr)$. Therefore, $\varphi$ is constant. ∎

In the following theorem (see also [Ohs02, Theorem 6.8] for a statement) $\pi(z)$ denotes the point in $b\Omega$ closest to $z$ and $d_{b\Omega}(z)$ denotes the distance from $z$ to $b\Omega$. We note that the function $\pi$ is well defined near $C^{2}$-smooth portion of the boundary.

Theorem 3** (Diederich [Die70]).**

Let $\Omega$ be a pseudoconvex domain in $\mathbb{C}^{n}$ and $z_{0}\in b\Omega$. Assume that there exists an open neighborhood $U$ of $z_{0}$ such that $b\Omega$ is $C^{2}$-smooth in $U$ and $b\Omega\cap U$ is composed of strongly pseudoconvex points. Then there exists a neighborhood $V\Subset U$ of $z_{0}$ and $C>0$ such that

[TABLE]

for $z\in V\cap\Omega$ where $X_{\tau}$ and $X_{\nu}$ denote that complex tangential and complex normal component of $X$ at $\pi(z)$, respectively.

Now we are ready to present the proof of Theorem 1. We will use the fact that every bounded $C^{2}$-smooth pseudoconvex domain has some strongly pseudoconvex boundary points (see, for instance, [Bas77]). Then we will follow the ideas in [Li93] and localize the estimate near a strongly pseudoconvex point in the boundary to get a contradiction in case $H_{\overline{\varphi}}\in S_{p}$ for $p\leq 2n$.

Proof of Theorem 1.

We will only prove the non-trivial direction. Since $S_{\alpha}\subseteq S_{\beta}$ for $\alpha\leq\beta$ we start the proof by assuming that $H_{\overline{\varphi}}\in S_{2n}$. Then Corollary 4 (see also [Li93, Theorem 3.1]) implies that

[TABLE]

Let $z_{0}\in b\Omega$ be a strongly pseudoconvex point and $U=B_{z_{0}}(r)$ so that all points in $B_{z_{0}}(2r)\cap b\Omega$ are strongly pseudoconvex. By Corollary 5 we have

[TABLE]

for $X=(a_{1},\ldots,a_{n})\in\mathbb{C}^{n}$. Furthermore, Theorem 3 implies that there exists $C>0$ such that

[TABLE]

for $z\in U\cap\Omega$ where $X_{\tau}$ and $X_{\nu}$ are the tangential and normal components of $X$, respectively. Combining the previous two estimates, we conclude that for any $z\in U\cap\Omega$ we have

[TABLE]

Then

[TABLE]

Combining the previous inequality with (1) we get

[TABLE]

We note that $K(z,z)$ is comparable to $(d_{b\Omega}(z))^{-n-1}$ near strongly pseudoconvex boundary points (see, for example, [Hör65, Theorem 3.5.1]). Then there exists $\widetilde{C}>0$ such that for sufficiently small $\varepsilon>0$ we get

[TABLE]

where $b\Omega_{t}=\{z\in\Omega:d_{b\Omega}(z)=t\}$ and $\widetilde{U}=B_{z_{0}}(r/2)$. Then $\int_{b\Omega\cap U}|\partial_{b}\varphi(z)|^{2n}d\sigma(z)=0$. Since $\partial_{b}\varphi$ is continuous on $b\Omega\cap U$ we conclude that $\partial_{b}\varphi=0$ on $b\Omega\cap U$. Finally, Proposition 1 implies that $\varphi$ is constant. ∎

Finally we present the proof of Corollary 2.

Proof of Corollary 2.

Let $K^{2}_{(0,q)}(\Omega)$ denote the square integrable $\overline{\partial}$-closed $(0,q)$-forms on $\Omega$ and $N_{q}$ denote the $\overline{\partial}$-Neumann operator on $L^{2}_{(0,q)}(\Omega)$. We note that $K^{2}_{(0,1)}(\Omega)$ is a closed subspace (as it is the kernel of $\overline{\partial}$) of $L^{2}_{(0,1)}(\Omega)$ and $N_{1}$ maps $K^{2}_{(0,1)}(\Omega)$ into itself (as $\overline{\partial}N_{1}=N_{2}\overline{\partial}$). Range’s Theorem (see, for instance, [Str10, p.77] and [Ran84]) implies that $N_{1}=(\overline{\partial}^{*}N_{1})^{*}\overline{\partial}^{*}N_{1}$ on $Ker(\overline{\partial})$. Furthermore, $T\in S_{p}$ if and only if $T^{*}T\in S_{p/2}$ (see [Zhu07, Theorem 1.26]). If $N_{1}$ is Hilbert-Schmidt then $\overline{\partial}^{*}N_{1}|_{A^{2}_{(0,1)}(\Omega)}\in S_{4}$ where $A^{2}_{(0,1)}(\Omega)$ is the space of $(0,1)$-forms with square integrable holomorphic coefficients. However, $H_{\overline{z}_{1}}f=\overline{\partial}^{*}N_{1}(fd\overline{z}_{1})$ for $f\in A^{2}(\Omega)$ and $H_{\overline{z}_{1}}\not\in S_{4}$. Therefore, $\overline{\partial}^{*}N_{1}\not\in S_{4}$ and $N_{1}$ is not Hilbert-Schmidt. ∎

Acknowledgment

Part of this work was done while the second author was visiting Sabancı University. He thanks this institution for its hospitality and good working conditions. He also thanks Trieu Le for fruitful discussions. We are thankful to the anonymous referee for constructive comments that improved the presentation of the paper.