Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball

1.   Introduction

Let $\mathbb{N} = \{1, 2, 3, \ldots\}$, $\mathbb{N}_{0} = \mathbb{N}\cup\{0\}$, $B(a, r) = \{z\in\mathbb{C}^{n}: |z-a| < r\}$ the open ball in the complex vector space $\mathbb{C}^{n}$ centered at $a$ with radius $r$, $\mathbb{B}^{n} = \mathbb{B} = B(0, 1)$, $\mathbb{S}$ the boundary of $\mathbb{B}$ and $n\in\mathbb{N}$. Let $z = (z_{1}, z_{2}, \ldots, z_{n})$ and $w = (w_{1}, w_{2}, \ldots, w_{n})$ be two points in $\mathbb{C}^{n}$, then $\langle z, w\rangle = z_{1}\overline{w}_{1}+z_{2}\overline{w}_{2}+\cdots+z_{n}\overline{w}_{n}$ and $|z|^{2} = \langle z, z\rangle$.

Let $H(\mathbb{B})$ be the set of all holomorphic functions on $\mathbb{B}$ and $S(\mathbb{B})$ the set of all holomorphic self-maps of $\mathbb{B}$. If $\varphi\in S(\mathbb{B})$, then by $C_{\varphi}f(z) = f(\varphi(z))$, $z\in\mathbb{B}$, is defined a operator, usually called the composition operator. If $u\in H(\mathbb{B})$, then by $M_{u}f(z) = u(z)f(z)$, $z\in\mathbb{B}$, is defined a operator, usually called the multiplication operator.

When $n = 1$, the open unit ball $\mathbb{B}$ is reduced to the open unit disk $\mathbb{D}$. Let $m\in\mathbb{N}_{0}$, the $m$th differentiation operator $D^{m}$ on some subspaces of $H(\mathbb{D})$ is defined by

where $f^{(0)} = f$. If $m = 1$, it is the classical differentiation operator, denoted by $D$. There have been a lot of studies on the products related to one of the differentiation operators. For example, products $DC_{\varphi}$ and $C_{\varphi}D$ were first studied (see, for example, ^{[5,15,16,17,18,22,33,34,35]}) containing the differentiation operator. What followed was the following six products of the operators were studied (see, for example, ^[25,40,41])

Afterwards, operators in (1.1) were studied in terms of replacing $D$ by $D^m$ (see, for example, ^[9,10,44]), that is, the following six products of the operators were studied

For some other products of the operators containing differentiation operators can be found in ^{[11,12,29,46]} and the related references therein.

There are other ways to extend differentiation operators on domains in $\mathbb{C}^{n}$. For example, the radial derivative operator

is one of natural extensions of the differentiation operators. The products of the composition, multiplication, and radial derivative operators

were studied in ^{[19,20,21,47]}. Operators $M_{u}C_{\varphi}\Re$ and $\Re M_{u}C_{\varphi}$ were also studied in ^[8]. Operators in (1.3) naturally generalize operators in (1.1) from the unit disk setting to the unit ball setting. Recently, they have been continuously studied in ^[45]. Some other products of the operators containing the radial derivative operators can be found, for example, in ^[31,49].

An advantage of the radial derivative operator is that it can be employed iteratively, that is, if $\Re^{m-1}f$ is defined for some $m\in\mathbb{N}\setminus\{1\}$, then $\Re^m f$ is naturally defined by $\Re^mf = \Re(\Re^{m-1}f)$. If $m = 0$, then we regard that $\Re^{0}f = f$. By using the $m$th iterated radial derivative operator, we obtain the related product-type operators

Operators in (1.4) are more complicated than those in (1.3). Clearly, the operator $M_{u}C_{\varphi}\Re^m$ can be regarded as the simplest one in (1.4), which was first studied and denoted by $\Re^m_{u, \varphi}$ in ^[32]. Recently, we have been reconsidered such operator in ^[38,39]. The reason why we consider the operator $\Re^m_{u, \varphi}$ again is that we need to get more information on the related function spaces in order to characterize its properties.

Operators in (1.4) generalize operators in (1.2) from the unit disk case to the unit ball case. For the unit disk case, by using the famous Faà di Bruno's formula of $(f\circ\varphi)^{(m)}$ (see ^[13]), operators $D^{m}M_{u}C_{\varphi}$, $D^{m}C_{\varphi}M_{u}$ and $M_{u}D^{m}C_{\varphi}$ have been studied (see, for example, ^[9,10]). But, we don't find any result on the operators $\Re^{m}M_{u}C_{\varphi}$, $\Re^{m}C_{\varphi}M_{u}$ and $M_{u}\Re^{m}C_{\varphi}$. By a direct calculation, it is easy to see that

Motivated by this interesting observation, we directly define the operator

on some subspaces of $H(\mathbb{B})$, where $u_{j}\in H(\mathbb{B}), j\in\{0, 1, \ldots, m\}$ and $\varphi\in S(\mathbb{B})$.

In this paper, the boundedness and compactness of the operator $\mathfrak{S}^m_{\vec{u}, \varphi} = \sum_{j = 0}^m\Re^j_{u_{j}, \varphi}$ from the mixed-norm space to the weighted-type space on $\mathbb{B}$ are characterized. The essential norm estimate of the operator from the mixed-norm space to the weighted-type space on $\mathbb{B}$ is given, and the Hilbert-Schmidt norm of the operator on the weighted Hilbert-Bergman space is calculated. As corollaries, the corresponding results of the operator $C_{\varphi}\Re^{m}M_{u}$ from mixed-norm space to weighted-type space are also obtained. This study can be viewed as a continuation and extension of our previous work.

A positive continuous function $\phi$ on the interval $[0, 1)$ is called normal (see ^[28]), if there are $\lambda \in[0, 1)$, $a$ and $b$ ($0 < a < b$) such that

The functions $\{\phi, \psi\}$ will be called a normal pair if $\phi$ is normal and for $b$ in above definition of normal function, there exists $\beta > b$, such that

If $\phi$ is normal, then there exists $\psi$ such that $\{\phi, \psi\}$ is normal pair (see ^[28]). Note that if $\{\phi, \psi\}$ is normal pair, then $\psi$ is also normal.

For $0 < p, q < +\infty$ and a normal function $\phi$, the mixed-norm space $H(p, q, \phi)(\mathbb{B}): = H(p, q, \phi)$ consists of all $f \in H(\mathbb{B})$ such that

where

and $d\sigma$ is the normalized surface measure on $\mathbb{S}$.

If $p = q$, $\phi(r) = (1-r^{2})^{(\alpha+1)/p}$ and $\alpha > -1$, then it is equivalent with the weighted Bergman space $A_{\alpha}^{p}$ (see ^[48]), which is defined by

where $dv$ denotes the normalized volume measure on $\mathbb{B}$. Some facts can be found about the mixed-norm spaces, for example, in ^{[1,2,26,27,36]} (see also the references therein).

A positive continuous function $\mu$ on $\mathbb{B}$ is called weight. The weighted-type space $H_{\mu}^{\infty}(\mathbb{B}): = H_{\mu}^{\infty}$ consists of all $f\in H(\mathbb{B})$ such that

With the norm $\|\cdot\|_{H_{\mu}^{\infty}}$, $H^\infty_\mu$ is a Banach space. In particular, if $\mu(z) = (1-|z|^{2})^{\sigma}\; (\sigma > 0)$, then the space $H_{\mu}^{\infty}$ is called the classical weighted-type space $H_{\sigma}^{\infty}$. If $\mu\equiv 1$, then the space $H_{\mu}^{\infty}$ is reduced to the space $H^{\infty}$ of bounded holomorphic functions on $\mathbb{B}$. Many operators acting from or to the weighted-type spaces have been studied (see, for example, ^{[14,16,24,46]} and the related references therein).

Let $X$ and $Y$ be two Banach spaces. It is said that a linear operator $T: X\rightarrow Y$ is bounded if there exists a positive constant $K$ such that $\|Tf\|_Y\leq K\|f\|_X$ for all $f\in X$. The bounded operator $T: X\rightarrow Y$ is compact if it maps bounded sets into relatively compact sets. The norm of the operator $T:X\to Y$, usually denoted by $\|T\|_{X\to Y}$, is defined by

In this paper, we use the notation $j = \overline{k, l}$ instead of writing $j = k, ..., l$, where $k, l\in\mathbb{N}_{0}$ and $k\leq l$. Some positive constants are denoted by $C$, and they may differ from one occurrence to the other. The notation $a\lesssim b$ (resp. $a\gtrsim b$) means that there is a positive constant $C$ such that $a\leq Cb$ (resp. $a\geq Cb$). When $a\lesssim b$ and $b\gtrsim a$, we write $a\asymp b$.

2.   Preliminary results

Here, we give several lemmas which are used in the proofs of the main results. First, we have the following point-evaluation estimate for the functions in $H(p, q, \phi)$ (see ^[36]).

Lemma 2.1. Let $0 < p, q < +\infty$ and $\phi$ normal. Then there is a positive constant $C$ independent of $f\in H(p, q, \phi)$ and $z\in{\mathbb{B}}$ such that

Now, we cite a point-evaluation estimate for the $j$th iterated radial derivatives of the functions in $H(p, q, \phi)$ (see ^[32]).

Lemma 2.2. Let $j\in\mathbb{N}$, $0 < p, q < +\infty$ and $\phi$ normal. Then there is a positive constant $C$ independent of $f\in H(p, q, \phi)$ and $z\in{\mathbb{B}}$ such that

Remark 2.1. From (2.1) and (2.2), it follows that if $\{f_k\}$ is bounded in $H(p, q, \phi)$, then $\{f_k\}$ and $\{\Re^j f_k\}$ are uniformly bounded on any compact subset of ${\mathbb{B}}$, and if $\{f_k\}$ converges to zero in $H(p, q, \phi)$, then $\{f_k\}$ and $\{\Re^jf_k\}$ converge to zero uniformly on any compact subset of ${\mathbb{B}}$ as $k\to\infty$.

To obtain a criterion for the compactness of a bounded linear operator $T:H(p, q, \phi)\to H^\infty_{\mu}$, we need to cite the following result, which can be found in ^[3] for the unit disk version. For the unit ball case, whose proof can be directly obtained by replacing the unit disk by the unit ball, and so the proof is omitted.

Lemma 2.3. Let $X$ be a reflexive Banach space of holomorphic functions on ${\mathbb{B}}$, and $Y$ a Banach space. Then a bounded linear operator $T:X\to Y$ is compact if and only if for any bounded sequence $\{f_{k}\}$ in $X$ such that $f_{k}\rightarrow 0$ uniformly on any compact subset of ${\mathbb{B}}$ as $k\to\infty$, it follows that $\{T f_{k}\}$ converges to zero in the norm of $Y$ as $k\to\infty$.

To give the conditions such that $H(p, q, \phi)$ is reflexive, we recall some studies of the dual spaces of $H(p, q, \phi)$. Jevtić in ^[7] showed that the dual space of $H(p, q, \phi)$, where $1\leq p\leq+\infty$ and $1\leq q < +\infty$, is topologically isomorphic to $H(p', q', \psi)$, where $1/p+1/p' = 1$ and $1/q+1/q' = 1$. Shi in ^[27] considered the dual spaces of $H(p, q, \phi)$ for two cases: (i) if $0 < p\leq1$ and $1 < q\leq+\infty$, then the dual space of $H(p, q, \phi)$ is topologically isomorphic to $H(+\infty, q', \psi)$; (ii) if $0 < p\leq1$ and $0 < q\leq1$, then the dual space of $H(p, q, \phi)$ is topologically isomorphic to $H(+\infty, +\infty, \psi)$.

From above facts, we obtain the following remark.

Remark 2.2. Considering the assumption $0 < p, q < +\infty$, we see that if $1\leq p < +\infty$ and $1 < q < +\infty$, then $H(p, q, \phi)$ is reflexive.

In Lemma 2.3, Banach space $X$ is assumed to be reflexive. The author in ^[42] gave the following general result.

Lemma 2.4. Let $X$, $Y$ be Banach spaces of holomorphic functions on $\mathbb{B}$. Suppose that

(a) The point evaluation functionals on $X$ are continuous.

(b) The closed unit ball of $X$ is a compact subset of $X$ in the topology of uniform convergence on compact sets.

(c) $T:X\rightarrow Y$ is continuous when $X$ and $Y$ are given the topology of uniform convergence on compact sets.

Then the bounded operator $T:X\rightarrow Y$ is compact if and only if for every bounded sequence $\{f_{k}\}$ in $X$ such that $f_{k}\to 0$ uniformly on compact sets as $k\to\infty$, it follows that $\{Tf_{k}\}$ converges to zero in the norm of $Y$ as $k\to\infty$.

We obtain the following result, which can be proved similar to Proposition 3.11 in ^[4]. However, here we prove this result by using Lemma 2.4.

Lemma 2.5. Let $0 < p, q < +\infty$, $\phi$ normal, and $T$ be one of the operators in (1.4) and (1.6). Then the bounded operator $T:H(p, q, \phi)\to H^\infty_\mu$ is compact if and only if for any bounded sequence $\{f_{k}\}$ in $H(p, q, \phi)$ such that $f_{k}\rightarrow 0$ uniformly on any compact subset of ${\mathbb{B}}$ as $k\to\infty$, it follows that $\{T f_{k}\}$ converges to zero in the norm of $H^\infty_\mu$ as $k\to\infty$.

Proof. Without lose of generality, we choose $T = M_uC_{\varphi}\Re^m$ to prove this result. Let $X = H(p, q, \phi)$ and $Y = H_{\mu}^{\infty}$. By Remark 2.1, it is easy to see that conditions (a) and (c) in Lemma 2.4 hold (here, we would like to mention that, for an abstract bounded linear operator $T:H(p, q, \phi)\to H^\infty_\mu$, condition (c) may not be valid.)

Let $\{f_{k}\}$ be a sequence in the closed unit ball of $X$. From (2.1), we see that $\{f_{k}\}$ is uniformly bounded on every compact subset of ${\mathbb{B}}$. Therefore, by Montel's theorem, there is a subsequence $\{f_{k_{i}}\}$ such that $f_{k_{i}}\rightarrow g$ uniformly on every compact subset of ${\mathbb{B}}$ for some $g\in H(\mathbb{B})$ as $i\to\infty$. By the definition of $M_{q}^p(f, r)$ and Fatou's theorem, we have

Hence, from (2.3) it follows that

which shows that $g\in H(p, q, \phi)$, that is, condition (b) in Lemma 2.4 holds. From Lemma 2.4, the desired result follows.

The following result was proved in ^[6] (also see ^[36]). Hence, its proof is omitted.

Lemma 2.6. Let $\tau > b$. Then for each $t\geq0$ and fixed $w\in\mathbb{B}$, the following function is in $H(p, q, \phi)$

Moreover,

We need the following result, which can be found in ^[30].

Lemma 2.7. Let $s > 0$, $w\in{\mathbb{B}}$, and

Then

where the sequences $\{a_{t}^{(k)}\}_{t = \overline{1, k}}$, $k\in\mathbb{N}$, are defined by the relations

for $k\in\mathbb{N}$ and

for $2\leq t\leq k-1, k\geq3$.

We use the idea essentially obtained in ^[30] to construct some suitable linear combinations of the functions in Lemma 2.6, which will be used in the proofs of the main results.

Lemma 2.8. Let $m\in\mathbb{N}$ and $w\in{\mathbb{B}}$, $f_{w, t}$ be the set of functions in (2.4), and the sequences $\{a_{t}^{(k)}\}_{t = \overline{1, k}}$, $k = \overline{1, m}$, are defined by the relations in (2.5) and (2.6). Then, for each $l\in\{1, 2, ..., m\}$, there are constants $c_{j} = c_{j}^{(l)}$, $j = \overline{0, m}$ such that the function

satisfies

and

Moreover,

Proof. For the sake of simplicity, we write $d_{k} = \frac{n}{q}+k+1+\tau$. By some calculations and using Lemma 2.7, we have

From (2.10), we obtain that the system consists of (2.7) and (2.8) is equivalent to the following $m+1$ linear equations

Since $d_{k} > 0$, $k = \overline{0, m}$, by Lemma 5 in ^[35], the determinant of system (2.11) is $D_{m+1}(d_{0}) = \prod_{j = 1}^{m}j!$, which is different from zero. Hence, there are unique constants $c_{j} = c_{j}^{(l)}$, $j = \overline{0, m}$ such that the system (2.11) holds. Moreover, we have that the relations in (2.7) and (2.8) hold when such obtained constants $c_{j}$, $j = \overline{0, m}$, are used in (2.11). Finally, by Lemma 2.6, it is clear that (2.9) holds. The proof is finished.

Lemma 2.9. Let $m\in\mathbb{N}$ and $w\in{\mathbb{B}}$, $f_{w, t}$ be the set of functions in (2.4), and the sequences $\{a_{t}^{(k)}\}_{t = \overline{1, k}}$, $k = \overline{1, m}$, are defined by the relations in (2.5) and (2.6). Then, there are constants $c_{j} = c_{j}^{(0)}$, $j = \overline{0, m}$, such that the function

satisfies

and

Moreover,

Proof. From the proof of Lemma 2.8, we know that the determinant of system consists of (2.12) and (2.13) is not zero. Therefore, there are unique constants $c_{j}^{(0)}$, $j = \overline{0, m}$, in the system of (2.12) and (2.13). For the same reason, we also have that the relations in (2.12) and (2.13) hold, and moreover, (2.14) holds.

3.   Boundedness and compactness of the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$

First, we need to characterize the boundedness of the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$. Although it is a folklore, we still give a proof for the completeness and benefit of the reader.

Theorem 3.1. Let $0 < p, q < +\infty$, $\phi$ normal, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu$ a weight function on $\mathbb{B}$. Then the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is bounded if and only if

Moreover, if the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is bounded, then the following asymptotic relationship holds

Proof. Assume (3.1) holds. By Lemma 2.1, for every $f \in H(p, q, \phi)$ and $z\in \mathbb{B}$, we have

By taking the supremum in inequality (3.3) over the unit ball in the space $H(p, q, \phi)$, and using (3.1), we obtain that the operator $\Re^0_{u, \varphi}: H(p, q, \phi) \rightarrow H_{\mu}^{\infty}$ is bounded. Moreover, we have

Assume that the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is bounded. Let $g_{w}(z) = f_{\varphi(w), 1}(z)$ and $w \in \mathbb{B}$. Then we have

and

From (3.5) and the boundedness of the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$, we have

From (3.6), condition (3.1) follows, and moreover,

Also (3.4) and (3.7) imply asymptotic relationship (3.2), finishing the proof.

Remark 3.1. When $k\in {\mathbb N}$, from ^[32] it follows that the operator $\Re^k_{u, \varphi}: H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is bounded if and only if

Now, we consider the boundedness of the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\to H_{\mu}^{\infty}$.

Theorem 3.2. Let $m\in {\mathbb N}$, $0 < p, q < +\infty$, $\phi$ normal, $u_{j}\in H(\mathbb{B})$, $j = \overline{0, m}$, $\varphi\in S(\mathbb{B})$ and $\mu$ a weight function on $\mathbb{B}$. Then the operators $\Re^{j}_{u_{j}, \varphi}:H(p, q, \phi)\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, are bounded if and only if the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is bounded and

Proof. Assume that the operators $\Re^{j}_{u_{j}, \varphi}:H(p, q, \phi)\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, are bounded, then the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\to H_{\mu}^{\infty}$ is obviously bounded. By using the functions $f_{i}(z) = z_{i}\in H(p, q, \phi)$, $i = \overline{1, n}$, we have that $\Re^{j}_{u_{j}, \varphi} f_{i}\in H(p, q, \phi)$, $i = \overline{1, n}$. From this and since $\Re f_{i} = f_{i}$, $i = \overline{1, n}$, we have

which shows that

From (3.9), we have that (3.8) holds.

Assume that the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\to H_{\mu}^{\infty}$ is bounded and (3.8) holds. Then there exists a positive constant $C$ such that

for each $f\in H(p, q, \phi)$. By Theorem 3.1 and Remark 3.1, we need to prove

and

By Lemma 2.8, if $\varphi(w)\neq 0$, then there is a function $h_{{\varphi}(w)}^{(m)}\in H(p, q, \phi)$ such that

and

Let $L_m = \sup_{w\in\mathbb{B}}\|h_{{\varphi}(w)}^{(m)}\|_{H(p, q, \phi)}$. Then $L_m < +\infty$. From this, (3.10), (3.13) and (3.14), we have that

From (3.15), we have

From (3.16) and (3.8) with $j = m$, since $\phi$ is normal, we have

From (3.16) and (3.17), it follows that $M_m < +\infty$.

Assume that (3.11) holds for $j = \overline{s+1, m}$, for some $s\in\{1, 2, \ldots, m-1\}$. By using the function $h_{\varphi(w)}^{(s)}(z)$ in Lemma 2.8, we have $L_{s}: = \sup_{w\in\mathbb{B}}\|h_{\varphi(w)}^{(s)}\|_{H(p, q, \phi)} < +\infty$ and

from which along with (2.5) we obtain

From (3.18) and since $s\geq1$, it follows that

From (3.8) with $j = s$ and (3.19), we have

So, (3.19) and (3.20) imply that $M_{s} < +\infty$. Hence, from induction it follows that for each $j \in\{1, \ldots, m\}$, (3.11) holds.

By Lemma 2.9, we have that for each $w \in \mathbb{B}$ there exists a function $h_{\varphi(w)}^{(0)} \in H(p, q, \phi)$ such that

and

From this, (3.10) and (3.21), we have

Hence,

From (3.23), we see that $(3.12)$ holds. This completes the proof.

From Theorem 3.2 and (1.5), we obtain the following result.

Corollary 3.1. Let $m\in\mathbb{N}$, $0 < p, q < +\infty$, $\phi$ normal, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu$ a weight function on $\mathbb{B}$. Then the operators $M_{(\Re^{m-j}u)\circ{\varphi}}C_{\varphi}\Re^{j}:H(p, q, \phi)\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, are bounded if and only if the operator $C_{\varphi}\Re^{m}M_{u}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is bounded and

Remark 3.2. When $k\in {\mathbb N}$, the author in ^[32] proved that the operator $\Re^k_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is compact if and only if $\Re^k_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is bounded and

Here, we need to characterize the compactness of the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$.

Theorem 3.3. Let $0 < p, q < +\infty$, $\phi$ normal, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu$ a weight function on $\mathbb{B}$. Then the operator $\Re^0_{u, \varphi}:H(p, q, \phi)\to H_{\mu}^{\infty}$ is compact if and only if $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is bounded and

Proof. Suppose that the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is compact. Then, it is bounded. If $\|\varphi\|_{\infty} < 1$, then (3.25) holds. Let $\|\varphi\|_{\infty} = 1$ and $\{z_{i}\}_{i \in \mathbb{N}}$ be a sequence in $\mathbb{B}$ such that $\left|\varphi\left(z_{i}\right)\right| \rightarrow 1$ as $i \rightarrow \infty$, and $h_{i}(z) = f_{\varphi\left(z_{i}\right), 1}(z)$, where $f_{w, t}$ is defined in Lemma 2.6. Then $\sup _{i \in \mathbb{N}}\left\|h_{i}\right\|_{H(p, q, \phi)} < +\infty$. Since $\lim_{i\rightarrow \infty}(1-|\varphi(z_{i})|^{2})^{t+1+\tau} = 0$, we see that $h_{i} \rightarrow 0$ uniformly on compact subsets of $\mathbb{B}$ as $i \rightarrow \infty$. Hence, by Lemma 2.5 we have

On the other hand, by (3.5), we see that sufficiently large $i$

Letting $i \rightarrow \infty$ in (3.27) and using (3.26), equality (3.25) follows.

Assume that the operator $\Re^0_{u, \varphi}: H(p, q, \phi) \rightarrow H_{\mu}^{\infty}$ is bounded and (3.25) holds. By using the function $f(z)\equiv1$, we obtain that

From (3.25), we have that for each $\varepsilon > 0$ there is a $\delta \in(0, 1)$ such that

on the set $\{z\in\mathbb{B}: \delta < |\varphi(z)| < 1\}$. Suppose that $\{f_{i}\}$ is a sequence such that $\sup _{i \in \mathbb{N}}\left\|f_{i}\right\|_{H(p, q, \phi)} \leq M$, and $f_{i} \rightarrow 0$ uniformly on compacts of $\mathbb{B}$ as $i\rightarrow \infty$. Then, by Lemma 2.1 and using (3.28) and (3.29), we have

where $K = \{z\in{\mathbb{B}}:|{\varphi}(z)|\leq\delta\}$. Since $\{z:|z| \leq \delta\}$ is a compact subset of $\mathbb{B}$ and $f_{i} \rightarrow 0$ uniformly on compacts of $\mathbb{B}$ as $i\rightarrow \infty$, we have

From (3.31) and Lemma 2.5, it follows that the operator $\Re^0_{u, \varphi}: H(p, q, \phi)\to H_{\mu}^{\infty}$ is compact.

Next, we characterize the compactness of the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$.

Theorem 3.4. Let $m\in {\mathbb N}$, $0 < p, q < +\infty$, $\phi$ normal, $u_{j}\in H(\mathbb{B})$, $j = \overline{0, m}$, $\varphi\in S(\mathbb{B})$ and $\mu$ a weight function on $\mathbb{B}$. Then the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is compact and (3.8) holds if and only if the operators $\Re^{j}_{u_{j}, \varphi}:H(p, q, \phi)\to H_{\mu}^{\infty}$ are compact for $j = \overline{0, m}$.

Proof. Assume that every operator $\Re^{j}_{u_{j}, \varphi}:H(p, q, \phi)\to H_{\mu}^{\infty}$ is compact, then we have that the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is clearly compact. Furthermore, from Theorem 3.2 it follows that (3.8) holds.

Now, assume that the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\to H_{\mu}^{\infty}$ is compact and (3.8) holds. Then the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is bounded. In order to prove the operator $\Re^{j}_{u_{j}, \varphi}:H(p, q, \phi)\to H_{\mu}^{\infty}$ is compact, from Remark 3.2 and Theorem 3.3, we only need to prove

and

If $\|\varphi\|_{\infty} < 1$, then (3.32) and (3.33) are obviously true. Hence, assume that $\|\varphi\|_{\infty} = 1$. Let $\{z_{k}\}_{k \in \mathbb{N}}$ be a sequence in $\mathbb{B}$ such that $|\varphi(z_{k})|\to 1$ as $k\to\infty$, and $h_{k}^{(s)}(z) = h_{\varphi(z_{k})}^{(s)}(z)$, where $h_{w}^{(s)}$ is defined in Lemma 2.8 for a fixed $s\in\{1, 2, \ldots, m\}$. Then, we have $\sup_{k\in\mathbb{N}}\|h_{k}^{(s)}\|_{H(p, q, \phi)} < +\infty$. Since $\lim_{k\rightarrow \infty}(1-|\varphi(z_{k})|^{2})^{t+1+\tau} = 0$, we see that $h_{k}^{(s)}\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $k\rightarrow \infty$. Hence, by Lemma 2.5 we have

On the other hand, from (3.15) we have that for sufficiently large $k$

Letting $k\to\infty$ in (3.35) and using (3.34) with $s = m$, then we obtain that (3.32) holds for $j = m$.

Now, suppose that (3.32) holds for $j = \overline{s+1, m}$, for some $s\in\mathbb{N}$. Then, from (3.18), we easily have that

for sufficiently large $k$. Letting $k\rightarrow \infty$ in (3.36), using (3.34) and the induction hypothesis, we have that (3.32) holds for $j = s$, from which by induction it follows that (3.32) holds for each $s\in\{1, 2, \ldots, m\}$.

Let $h_{k}^{(0)}(z) = h_{\varphi(z_{k})}^{(0)}(z)$, where $h_{w}^{(0)}$ is defined in Lemma 2.9. Then we have that $\sup_{k\in\mathbb{N}}\|h_{k}^{(0)}\|_{H(p, q, \phi)} < +\infty$ and $h_{k}^{(0)}\to 0$ uniformly on any compact subset of $\mathbb{B}$ as $k\rightarrow \infty$. Hence, by Lemma 2.5 we have

From (3.22), for sufficiently large $k$ it follows that

Letting $k\rightarrow \infty$ in (3.38) and using (3.37), (3.33) follows. The proof is finished.

We also have the following result.

Corollary 3.2. Let $m\in\mathbb{N}$, $0 < p, q < +\infty$, $\phi$ normal, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu$ a weight function on $\mathbb{B}$. Then the operator $C_{\varphi}\Re^{m}M_{u}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is compact and (3.24) holds if and only if the operators $M_{(\Re^{m-j}u)\circ{\varphi}}C_{\varphi}\Re^{j}:H(p, q, \phi)\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, are compact.

4.   Essential norm estimate of the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\to H_{\mu}^{\infty}$

In this section, we estimate the essential norm of $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\to H_{\mu}^{\infty}$. Let us recall the definition of the essential norm of the bounded linear operators. Assume that $X$ and $Y$ are Banach spaces and $T: X \rightarrow Y$ is a bounded linear operator, then the essential norm of the operator $T: X \rightarrow Y$ is defined by

where $\mathcal{K}$ denotes the set of all compact linear operators from $X$ to $Y$. It is easy to see that $\|T\|_{e, X \rightarrow Y} = 0$ if and only if the bounded operator $T: X \rightarrow Y$ is compact.

Theorem 4.1. Let $m\in {\mathbb N}$, $1\leq p < +\infty$, $1 < q < +\infty$, $u_{j}\in H(\mathbb{B})$, $j = \overline{0, m}$, $\varphi\in S(\mathbb{B})$, $\phi$ normal, $\mu$ a weight function on $\mathbb{B}$ and (3.8) hold. If the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}:H(p, q, \phi)\to H_{\mu}^{\infty}$ is bounded, then

Proof. Assume that $\left\{z_{i}\right\}_{i\in\mathbb{N}}$ is a sequence in $\mathbb{B}$ such that $\left|\varphi\left(z_{i}\right)\right| \rightarrow 1$ as $i \rightarrow \infty$. Take the function $h_{i}^{(s)} = h_{\varphi(z_{i})}^{(s)}$, where $h_{w}^{(s)}$ is defined in Lemma 2.8 for a fixed $s\in\{1, 2, \ldots, m\}$. Then we have $\sup_{i \in \mathbb{N}}\|h_{i}^{(s)}\|_{H(p, q, \phi)} < +\infty$ and $h_{i}^{(s)} \rightarrow 0$ uniformly on compacts of $\mathbb{B}$ as $i \rightarrow \infty$. Hence, by Lemma 2.3 and Remark 2.2 we have $\|K h_{i}^{(s)}\|_{H(p, q, \phi)} \rightarrow 0$ as $i \rightarrow \infty$. Thus, from (3.15) it follows that

From induction and combining these inequalities, it follows that

Take the function $h_{i}^{(0)} = h_{\varphi(z_{i})}^{(0)}$, where $h_{w}^{(0)}$ is defined in Lemma 2.9. Then we have that $\sup_{i \in \mathbb{N}}\big\|h_{i}^{(0)}\big\|_{H(p, q, \phi)} < +\infty$, and $h_{i}^{(0)}\rightarrow 0$ uniformly on compacts of $\mathbb{B}$ as $i \rightarrow \infty$. Then from (3.22), we have

By taking the infimum in (4.1) and (4.2) over the set of all compact operators $K: H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$, we have

Now, assume that $\{r_{i}\}_{i\in\mathbb{N}}$ is a positive sequence which increasingly converges to 1. For each $r_{i}$, we define the operator by

Since the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is bounded, by Theorem 3.2 one can obtain that the operator $\mathfrak{S}^m_{\vec{u}, r_{i}{\varphi}}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is bounded. Since $\left|r_{i} \varphi(z)\right| \leq r_{i} < 1$, by Lemma 2.5 the operator $\mathfrak{S}^m_{\vec{u}, r_{i}{\varphi}}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is also compact. Hence, from Lemmas 2.1 and 2.2, we have

For each $f\in H(\mathbb{B})$ and $|\varphi(z)|\leq \delta$, we have

where $\theta_{k}(t) = (1-t)r_{i}\varphi_{k}(z)+t\varphi_{k}(z)$. Let $g = \nabla\Re^{j-1}f$, $j = \overline{1, m}$, from (4.4) and $|\varphi(z)|\leq \delta$, we have that

By letting $i\rightarrow \infty$ in (4.3), from (3.8), (3.28), (4.4) and (4.5), we obtain

as $i\rightarrow \infty$. Since

we finish the proof.

From Theorem 4.1 and (1.5), we obtain the following result.

Corollary 4.1. Let $m\in {\mathbb N}$, $1\leq p < +\infty$, $1 < q < +\infty$, $\phi$ normal, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$, $\mu$ a weight function on $\mathbb{B}$ and (3.24) hold. If the operator $C_{\varphi}\Re^{m}M_{u}:H(p, q, \phi)\rightarrow H_{\mu}^{\infty}$ is bounded, then

5.   Hilbert-Schmidt norm of the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}: A_{\alpha}^{2}\rightarrow A_{\alpha}^{2}$

In this section, we calculate Hilbert-Schmidt norm of the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}: A_{\alpha}^{2} \rightarrow A_{\alpha}^{2}$. For some related results, one can see ^[37,43]. If $\mathcal{H}$ is a separable Hilbert space, then the Hilbert-Schmidt norm $\|T\|_{\mathrm{HS}, \mathcal{H}\to\mathcal{H}}$ of an operator $T: \mathcal{H} \rightarrow \mathcal{H}$ is defined by

where $\left\{e_{n}\right\}_{n \in \mathbb{N}}$ is an orthonormal basis on $\mathcal{H}$. The right-hand side in (5.1) does not depend on the choice of basis. Hence, we have that $\|T\|_{\mathrm{HS}, \mathcal{H}\to\mathcal{H}}\geq\|T\|_{\mathcal{H}\to\mathcal{H}}$ the operator norm.

Theorem 5.1. Let $m\in\mathbb{N}$ and $\alpha > -1$. Then Hilbert-Schmidt norm of the operator $\mathfrak{S}_{\vec{u}, {\varphi}}^{m}$ on $A_{\alpha}^{2}$ is

Proof. From Proposition 1.4.9 in ^[23] (or Lemma 1.11 in ^[48]), for each multi-index $\beta = \left(\beta_{1}, \ldots, \beta_{n}\right)$, we have

where $\beta! = \beta_{1}! \cdots \beta_{n}!$, and

From this, we have that the vectors

form an orthonormal basis in $A_{\alpha}^{2}$. By using the definition of the Hilbert-Schmidt norm and the monotone convergence theorem, we have

For each $f \in H(\mathbb{B})$, by using the Taylor expansion

and taking the $j$th radial derivatives, we have

where $\beta = \left(\beta_{1}, \beta_{2}, \ldots, \beta_{n}\right)$ is a multi-index, $|\beta| = \beta_{1}+\cdots+\beta_{n}$ and $z^{\beta} = z_{1}^{\beta_{1}}\cdots z_{n}^{\beta_{n}}$.

On the other hand, from Proposition 1.14 in ^[48] (also see ^[32]), we have

Hence, from (5.3) and (5.4), we have

From (5.2) and (5.5), the desired result follows.

From Theorem 5.1 and (1.5), we obtain the following result.

Corollary 5.1. Let $m \in \mathbb{N}$ and $\alpha > -1$. Then Hilbert-Schmidt norm of the operator $C_{\varphi}\Re^{m}M_{u}$ on $A_{\alpha}^{2}$ is

6.   Conclusions

In this paper, we define the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}} = \sum_{j = 0}^mM_{u_j}C_{\varphi}\Re^j$ on some subspaces of $H({\mathbb{B}})$, where $u_{j}\in H(\mathbb{B}), j\in\{0, 1, \ldots, m\}$ and $\varphi\in S(\mathbb{B})$. We completely characterized the boundedness and compactness of the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:H(p, q, \phi)\to H^\infty_\mu$ in terms of the behaviours of the symbols $u_j$ and ${\varphi}$. In order to study the essential norm estimate of the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:H(p, q, \phi)\to H^\infty_\mu$, we considered the conditions for the reflexivity of $H(p, q, \phi)$. By using a criterion of the compactness for a bounded linear operator $T:X\to Y$, where $X$ is a reflexive Banach space of holomorphic functions on ${\mathbb{B}}$ and $Y$ is any Banach space, we obtained the essential norm estimate of the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:H(p, q, \phi)\to H^\infty_\mu$. Moreover, we also calculated the Hilbert-Schmidt norm of the operator on the weighted Bergman space $A^2_{\alpha}$. As an application, the corresponding results of the operator $C_{\varphi}\Re^{m}M_{u}:H(p, q, \phi)\to H^\infty_\mu$ are obtained. This paper can be viewed as a continuation and extension of our previous work. We hope that this study can attract people's more attention for such operators and mixed-norm spaces.

Acknowledgments

This work was supported by the Innovation Fund of Postgraduate, Sichuan University of Science and Engineering (Y2021098) and the Key Project of Zigong City (2020YGJC24).

Conflict of interest

The authors declare that they have no competing interests.