Note on a new class of operators between some spaces of holomorphic functions

1.   Introduction

By ${\mathbb{B}}$ we denote the open unit ball in ${\mathbb C}^n$, ${\mathbb S}$ is the unit sphere in ${\mathbb C}^n$, $B(z, r)$ is the open ball centered at $z$ and with radius $r$, $d\sigma$ is the normalized rotation invariant measure on ${\mathbb S}$, $dV(z)$ is the Lebesgue measure, and $dV_{\alpha}(z): = c_{{\alpha}, n}(1-|z|^2)^{\alpha} dV(z),$ ${\alpha} > -1$, where $c_{{\alpha}, n}$ is the normalization constant such that $V_{\alpha}({\mathbb{B}}) = 1.$ The linear space of holomorphic functions on ${\mathbb{B}}$ we denote by $H({\mathbb{B}})$, whereas $S({\mathbb{B}})$ denotes the class of holomorphic self-maps of ${\mathbb{B}}$. The standard inner product between the vectors $z, w\in {\mathbb C}^n$ is denoted by $\langle z, w\rangle$, whereas $|z| = \sqrt{\langle z, z\rangle}$ is the Euclidean norm in ${\mathbb C}^n$. Many classical results on functions in $H({\mathbb{B}})$ can be found in ^[1]. If $f\in C({\mathbb{B}})$ is a positive function, then we call it a weight function, and the class of functions is denoted by $W({\mathbb{B}})$. If $p, q\in {\mathbb N}_0$, $p\le q$, then the notation $j = \overline{p, q}$ is an abbreviation for the notation $j = p, p+1, \ldots, q$. If $X$ is a Banach space, then by $B_X$ we denote the unit ball in $X$.

Each ${\varphi}\in S({\mathbb{B}})$ induces the composition operator $C_{\varphi}f(z) = f(\varphi(z))$, whereas each $u\in H({\mathbb{B}})$ induces the multiplication operator $M_{u}f(z) = u(z)f(z)$. The radial derivative of $f\in H({\mathbb{B}})$ is defined by

where $D_jf(z) = \frac{\partial f}{\partial z_j}(z), \;$$j = \overline{1, n}$ (if $n = 1$, then we regard $D_1f: = Df = f'$). There has been a huge interest in the operators and their products on subspaces of $H({\mathbb{B}})$. The first investigations have been mostly devoted to the case $n = 1$. Beside the products of the operators $C_{\varphi}$ and $M_u$, which have been studied a lot, there have been some investigations of the products of the operators $D$ and $C_{\varphi}$. For some products of these and other concrete operators, see, for example, ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]} and the related references therein. The boundedness and compactness ^[26,27] of the operators have been predominately studied so far.

The weighted Bergman space $A^p_{\alpha} = A^p_{\alpha}({\mathbb{B}})$, $p > 0$, ${\alpha} > -1$, consists of all $f\in H({\mathbb{B}})$ such that

which for $p\ge 1$ is a norm on $A^p_{\alpha}.$ With the norm the space is Banach. For some results on the space and operators on it, see, e.g., ^{[4,6,14,15,22,28,29,30,31]}.

If $\mu$ is a weight function, then the space of all $f\in H({\mathbb{B}})$ such that

is called the weighted-type space and denoted by $H_{\mu}^{\infty}(\mathbb{B}) = H_{\mu}^{\infty}$, whereas the little weighted-type space is its closed subspace consisting of all $f\in H({\mathbb{B}})$ such that $\lim_{|z|\rightarrow 1}\mu(z)|f(z)| = 0,$ and is denoted by $H_{\mu, 0}^{\infty}(\mathbb{B}) = H_{\mu, 0}^{\infty}$. There has been a huge interest in investigating the spaces, their generalizations, and linear operators on them, especially in the boundedness and compactness ^{[2,11,13,19,23,31,32,33,34]}.

The product operator $\Re^m_{u, {\varphi}} = M_{u}C_{\varphi}\Re^m$ was introduced in ^[35]. For some investigations in the direction, see also ^[36]. Motivated, among others, by our investigations in ^{[14,15,16,35]}, I have introduced the operator

where $m\in {\mathbb N}$, $u_j\in H({\mathbb{B}})$, $j = \overline{0, m}$, and ${\varphi}\in S({\mathbb{B}})$, and studied it, for example, in ^[37]. For some related studies see also ^[2,3].

This note continues some of our previous investigations (for example, the ones in ^{[13,14,15,16,35,37]}), by studying the boundedness and compactness of the operators ${\mathfrak S}^m_{\vec u, {\varphi}}: A^p_{\alpha}\to H_\mu^\infty$ (or $H_{\mu, 0}^{\infty}$), where $p\ge 1$ and ${\alpha} > -1$.

By $C$ we denote some positive constants independent of essential variables and functions which may differ from line to line, whereas $a\lesssim b$ (resp. $a\gtrsim b$) means that there is $C > 0$ such that $a\leq Cb$ (resp. $a\geq Cb$). If $a\lesssim b$ and $b\lesssim a$, then we use the notation $a\asymp b$.

2.   Auxiliary results

The first result is a standard Schwartz-type lemma ^[38].

Lemma 2.1. Assume $p\ge 1$, ${\alpha} > -1$, $\mu\in W({\mathbb{B}})$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, $m\in\mathbb{N}$, $\varphi\in S(\mathbb{B})$, and that the operator ${\mathfrak S}^m_{\vec u, {\varphi}}: A^p_{\alpha}\to H_\mu^\infty$ is bounded. Then, the operator is compact if and only if for every bounded sequence $(f_k)_{k\in {\mathbb N}}\subset A^p_{\alpha}$ uniformly converging to zero on compacts of $\mathbb{B}$, we have

The following lemma was essentially proved in ^[39], so we omit the proof.

Lemma 2.2. A closed set $K$ in $H_{\mu, 0}^{\infty}$ is compact if and only if it is bounded and

The following lemma is well known (see ^[29]; for a less precise version see also ^[1]).

Lemma 2.3. Assume $p\in(0, \infty)$, ${\alpha} > -1$, and $f\in A_{\alpha}^p({\mathbb{B}})$; Then,

Lemma 2.4. Assume $p\in(0, \infty)$, ${\alpha} > -1$, and $m\in\mathbb{N}$. Then,

for every $f\in A_{\alpha}^p$ and $z\in\mathbb{B}$.

Proof. Note that it is enough to prove that for all $f\in A_{\alpha}^p$ and $z\in\mathbb{B}$,

Let $r\in(0, 1)$ be fixed. Then, the Cauchy-Schwartz and Cauchy inequalities imply

Inequality (2.1) implies that

Since $r$ is fixed, by (2.4) and (2.5) we get

that is, (2.3) holds when $m = 1$.

Assume that for a $k\in {\mathbb N}\setminus\{1\}$ and all $f\in A_{\alpha}^p$ and $z\in\mathbb{B}$ holds,

Then, since for $w\in B(z, r(1-|z|))$ we have $(1-r)^{\frac{n+{\alpha}+1}p+k-1}(1-|z|)^{\frac{n+{\alpha}+1}p+k-1}\leq (1-|w|)^{\frac{n+{\alpha}+1}p+k-1},$ from (2.7) we have

If in (2.4) we replace $f$ by $\Re^{k-1}f$, we get

Combining (2.8) and (2.9), we have

Thus, (2.3) holds for each $m\in {\mathbb N},$ implying (2.2).

The following lemma is well known.

Lemma 2.5. Let $p\ge 1$ and ${\alpha} > -1$. Then, for any $t\geq0$ and $w\in\mathbb{B}$,

belongs to $A_{\alpha}^p$ and $\sup_{w\in\mathbb{B}}\|f_{w, t}\|_{A_{\alpha}^p}\lesssim 1.$

The following lemma is from ^[34] and ^[35].

Lemma 2.6. Let $s\geq 0$, $w\in{\mathbb{B}}$ and $g_{w, s}(z) = (1-\langle z, w\rangle)^{-s}.$ Then,

where $P_k(w) = s^{k-1}w^k+p_{k-1}^{(k)}(s)w^{k-1}+\cdots+p_2^{(k)}(s)w^2+w,$ and where $p^{(k)}_j(s)$, $j = \overline{2, k-1}$, are nonnegative polynomials for $s > 0$;

where $(a_{t}^{(k)})$, $t = \overline{1, k}$, $k\in\mathbb{N},$ are defined as

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

Lemma 2.7. Assume $p\ge 1$, ${\alpha} > -1$, $m\in\mathbb{N}$, $w\in{\mathbb{B}}$, $f_{w, t}$ is defined in (2.10), and $(a_{t}^{(k)})_{t = \overline{1, k}}$, $k = \overline{1, m}$, are defined in (2.13) and (2.14). Then,

(a) for each $l\in\{1, \ldots, m\}$, there is

where $c_{k}^{(l)}$, $k = \overline{0, m}$, are numbers, such that

hold. Moreover, we have $\sup_{w\in\mathbb{B}}\|h_{w}^{(l)}\|_{A_{\alpha}^p} < +\infty;$

(b) there is

where $c_k^{(0)}$, $k = \overline{0, m}$, are numbers, such that

hold. Moreover, we have $\sup_{w\in\mathbb{B}}\|h_{w}^{(0)}\|_{A_{\alpha}^p} < +\infty.$

Proof. (a) Let $d_{k} = \frac{n+{\alpha}+1}p+k+1$, $k\in {\mathbb N}_0$. Replace the constants $c_{k}^{(l)}$ in (2.15) by $c_k$. Then, from (2.12) we get

Lemma 2.5 in ^[11] shows that the determinant of the system,

is different from zero (on the right-hand side of (2.20), the unit is in the $(l+1)$th position). Thus, there is a unique solution $c_k = c_k^{(l)}$, $k = \overline{0, m},$ to (2.20). For these $c_k$-s, function (2.15) satisfies (2.16) and (2.17). By Lemma 2.5 we have $\sup_{w\in\mathbb{B}}\|h_{w}^{(l)}\|_{A_{\alpha}^p} < +\infty.$

(b) The proof is similar, so it is omitted.

3.   Main results

Our main results are formulated and proved in this section.

Theorem 3.1. Let $p\ge 1$, ${\alpha} > -1$, $k\in {\mathbb N}$, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator $\Re^k_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu}^{\infty}$ is bounded if and only if

and if it is bounded, then we have

Proof. Assume $\Re^k_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu}^{\infty}$ is bounded. Let $g_w(z) = f_{{\varphi}(w), 1}(z)$. By Lemma 2.6 the coefficients of the polynomial $P_k$ therein are nonnegative, so we have

The boundedness, (3.3) and the fact $\sup_{w\in{\mathbb{B}}}\|g_w\|_{A_{\alpha}^p} < +\infty$, imply

Further, the fact $f_{j}(z) = z_{j}\in A_{\alpha}^p$, $j = \overline{1, n}$, implies $\Re^k_{u, {\varphi}}f_j\in H_\mu^\infty$, $j = \overline{1, n}$, from which, together with $\Re f_j = f_j,$ $j = \overline{1, n},$ we get

from which we get

Inequality (3.5) together with

implies

Combining (3.4) and (3.6), we get (3.1) and $J_k\lesssim\|\Re^k_{u, {\varphi}}\|_{A_{\alpha}^p\to H^\infty_\mu}.$

Assume (3.1) holds. Then, Lemma 2.4 implies that for any $f\in A_{\alpha}^p(\mathbb{B})$ and $z\in\mathbb{B}$,

Taking the supremum in (3.7) over $B_{A_{\alpha}^p}$, and employing (3.1), the boundedness of $\Re^k_{u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ and the relation $\|\Re^k_{u, {\varphi}}\|_{A_{\alpha}^p\rightarrow H_{\mu}^{\infty}}\lesssim J_k$ follow, implying (3.2).

The following result is known. For a more general result, see ^[31].

Theorem 3.2. Let $p\ge 1$, ${\alpha} > -1$, $\mu\in W({\mathbb{B}})$, $u\in H(\mathbb{B})$ and $\varphi\in S(\mathbb{B})$. Then, the operator $\Re^0_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu}^{\infty}$ is bounded if and only if

and if it is bounded, then $\|\Re^0_{u, {\varphi}}\|_{A_{\alpha}^p\rightarrow H_{\mu}^{\infty}}\asymp J_0.$

Theorem 3.3. Let $p\ge 1$, ${\alpha} > -1$, $m\in\mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operators $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, are bounded if and only if ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ is bounded and

Proof. Assume ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ is bounded and (3.9) holds. We need to prove

and

If ${\varphi}(w)\ne0$, then there is $h_{{\varphi}(w)}^{(m)}\in A_{\alpha}^p$ such that

and $\sup_{w\in\mathbb{B}}\|h_{{\varphi}(w)}^{(m)}\|_{A_{\alpha}^p} < +\infty$ (see Lemma 2.7 (a)). This, together with the boundedness of ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$, implies

from which it follows that

and along with

implies $I_m < +\infty$.

Assume (3.10) holds for $j = \overline{s+1, m}$, for an $s\in\{1, 2, \ldots, m-1\}$. Let $h_{{\varphi}(w)}^{(s)}(z)$ be as in Lemma 2.7 (a). Then, $\sup_{w\in{\mathbb{B}}}\|h_{{\varphi}(w)}^{(s)}\|_{A_{\alpha}^p} < +\infty$, and

from which we easily get

From (3.13) and the fact $s\ge 1$, we have

This, together with the fact

implies (3.10) for $j = s$. Thus, (3.10) holds for any $j\in\{1, \ldots, m\}$.

For any $w\in{\mathbb{B}}$, there is $h_{{\varphi}(w)}^{(0)}\in A_{\alpha}^p$ such that

and $\sup_{w\in\mathbb{B}}\|h_{{\varphi}(w)}^{(0)}\|_{A_{\alpha}^p} < +\infty$ (see Lemma 2.7 (b)).

This together with the boundedness of ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ implies

from which (3.11) follows, as claimed.

Assume $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, are bounded. Then, ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$ is also bounded. If $u$ in (3.5) is replaced by $u_j$, we get (3.9).

Theorem 3.4. Let $p\ge 1$, ${\alpha} > -1$, $k\in {\mathbb N}$, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator $\Re^k_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu}^{\infty}$ is compact if and only if it is bounded and

Proof. If $\Re^k_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu}^{\infty}$ is compact, it is also bounded. If $\|\varphi\|_{\infty} < 1$, (3.15) automatically/vacuously holds. If $\|\varphi\|_{\infty} = 1$ and $(z_j)_{j\in {\mathbb N}}\subset\mathbb{B}$ is such that $|\varphi(z_j)|\rightarrow 1$ as $j\to+\infty$, and $h_j(z) = f_{\varphi(z_j), t}(z)$, then $\sup_{j\in\mathbb{N}}\|h_j\|_{A_{\alpha}^p} < +\infty$. From $\lim_{j\to+\infty}(1-|\varphi(z_j)|^{2})^{t+1} = 0$, we have $h_j\to 0$ as $j\to+\infty$, uniformly on compacta of $\mathbb{B}$. Using Lemma 2.1, it follows that $\lim_{j\to+\infty}\|\Re^k_{u, {\varphi}}h_j\|_{H_{\mu}^{\infty}} = 0,$ from which, along with the consequence of (3.3),

which holds for sufficiently large $j$, and we easily get (3.15).

If $\Re^k_{u, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$ is bounded and (3.15) holds, then Theorem 3.1 implies $\mu(z)|u(z)||{\varphi}(z)|\leq J_k < +\infty,$ $z\in{\mathbb{B}},$ and (3.15) implies that for any $\varepsilon > 0$ there is $\delta\in(0, 1)$ such that when $\delta < |\varphi(z)| < 1$,

Suppose $(f_j)_{j\in {\mathbb N}}$ is a bounded sequence in $A_{\alpha}^p$ converging to zero uniformly on compacts of $\mathbb{B}$. Let $s_{\delta} = \{z\in\mathbb{B}:|\varphi(z)|\leq\delta\}$. Then, Lemma 2.4, together with the fact $\sup_{z\in{\mathbb{B}}}\mu(z)|u(z)||{\varphi}(z)| < +\infty,$ and (3.16), implies

The assumption $f_j\to 0$ on compacts along with Cauchy's estimate implies $\lim_{j\to+\infty}|\nabla\Re^{k-1}f_j| = 0$ uniformly on compacts of $\mathbb{B}$. The set $\{w:|w|\leq\delta\}$ is compact, so by letting $j\to+\infty$ in (3.17), it follows that $\limsup_{j\to+\infty}\|\Re^k_{u, {\varphi}}f_j\|_{H_{\mu}^{\infty}}\lesssim\varepsilon,$ from which it follows that $\lim_{j\to+\infty}\|\Re^k_{u, {\varphi}}f_j\|_{H_{\mu}^{\infty}} = 0.$ From this and Lemma 2.1, the compactness of $\Re^k_{u, {\varphi}}:A_{\alpha}^p\to H^\infty_\mu$ follows.

The following theorem is known. For a more general result, see ^[31].

Theorem 3.5. Let $p\ge 1$, ${\alpha} > -1$, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator $\Re^0_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu}^{\infty}$ is compact if and only if it is bounded and

Theorem 3.6. Let $p\ge 1$, ${\alpha} > -1$, $m\in\mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ is compact and (3.9) holds if and only if the operators $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$ are compact for $j = \overline{0, m}$.

Proof. If ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ is compact and (3.9) holds, then the operator is bounded, from which, together with Theorem 3.3, the boundedness of $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, follows. The previous two theorems show that it is enough to prove

and

If $\|\varphi\|_{\infty} < 1$, then (3.19) and (3.20) hold. Assume $\|\varphi\|_{\infty} = 1$. Let $(z_k)_{k\in {\mathbb N}}\subset\mathbb{B}$ be such that $\lim_{k\to+\infty}|\varphi(z_{k})| = 1$, and $h_{k}^{(s)}(z) = h_{\varphi(z_{k})}^{(s)}(z)$ for an $s\in\{1, \ldots, m\}$ (see (2.15)). Then, $\sup_{k\in\mathbb{N}}\|h_{k}^{(s)}\|_{A_{\alpha}^p} < +\infty.$ The fact $\lim_{k\to+\infty}(1-|\varphi(z_{k})|^{2})^{t+1} = 0,$ implies $\lim_{k\to+\infty}h_k^{(s)} = 0$ uniformly on any compact of $\mathbb{B}$. So, Lemma 2.1 implies

Relation (3.12) implies

for sufficiently large $k$. From (3.22) and (3.21) with $s = m$, relation (3.19) with $j = m$ follows.

If (3.19) holds for $j = \overline{s+1, m},$ for a fixed $s\in\{1, \ldots, m-1\}$, (3.13) implies

for $k$ large, from which, along with (3.21) and the hypothesis, the relation (3.19) with $j = s$ follows. Thus, (3.19) holds for any $s\in\{1, \ldots, m\}$.

Let $h_{k}^{(0)}(z) = h_{\varphi(z_{k})}^{(0)}(z)$ (see Lemma 2.7 (b)). Then, $\sup_{k\in\mathbb{N}}\|h_{k}^{(0)}\|_{A_{\alpha}^p} < +\infty,$ and $\lim_{k\to+\infty}h_k^{(0)}(z) = 0$ uniformly on compacts of $\mathbb{B}$. From Lemma 2.1 we have that $\lim_{k\to+\infty}\|{\mathfrak S}^m_{\vec u, {\varphi}}h_{k}^{(0)}\|_{H_{\mu}^{\infty}} = 0,$ from which, along with the consequence of (3.14),

(3.20) follows.

Assume $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, are compact. Then, ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ is also compact, and by Theorem 3.3 is obtained (3.9).

Theorem 3.7. Let $p\ge 1$, ${\alpha} > -1$, $m\in\mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu, 0}^{\infty}$ is bounded if and only if ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$ is bounded and

Proof. If ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ is bounded and (3.23) holds, then since any polynomial $p$ is represented as $p(z) = \sum_{l = 0}^tp_l(z)$, where $p_l$, $l = \overline{0, t}$ are homogeneous polynomials of degree $l$, it follows that as $|z|\to 1$,

Hence, ${\mathfrak S}^m_{\vec u, {\varphi}}p\in H_{\mu, 0}^{\infty}$. The density of the set of polynomials in $A_{\alpha}^p$, implies that for any $f\in A_{\alpha}^p$ there are polynomials $(p_k)_{k\in {\mathbb N}}$ such that $\lim_{k\to+\infty}\|f-p_{k}\|_{A_{\alpha}^p} = 0.$ From the boundedness of ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$ we have

as $k\to+\infty$. So, ${\mathfrak S}^m_{\vec u, {\varphi}}(A_{\alpha}^p)\subseteq H_{\mu, 0}^{\infty}$, implying the boundedness of ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu, 0}^{\infty}$.

If ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\to H_{\mu, 0}^{\infty}$ is bounded, then ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$ is also bounded. The fact $f_{s, l}(z) = z_s^l\in A_{\alpha}^p$, $s = \overline{1, n},$ $l\in {\mathbb N}_0$, implies ${\mathfrak S}^m_{\vec u, {\varphi}}f_{s, l}\in H_{\mu, 0}^\infty$, $s = \overline{1, n},$ $l\in {\mathbb N}_0$. Hence, for $s = \overline{1, n},$ $l\in {\mathbb N}_0$, we have

from which, along with $|{\varphi}(z)|^l\lesssim\sum_{s = 1}^n|{\varphi}_s(z)|^l$, (3.23) follows for each $l\in {\mathbb N}_0.$

Theorem 3.8. Let $p\ge 1$, ${\alpha} > -1$, $k\in {\mathbb N}$, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator $\Re^k_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu, 0}^{\infty}$ is compact if and only if

Proof. Relation (3.24) implies (3.1). Taking the supremum in (3.7) over ${\mathbb{B}}$ and $B_{A_{\alpha}^p}$, and employing (3.1), it follows that

Hence, the set ${\mathcal S} = \{\Re^k_{u, {\varphi}}f\in H^\infty_\mu : f\in B_{A_{\alpha}^p}\}$ is bounded in $H^\infty_\mu$. From (3.7) and (3.24) we easily get $\Re^k_{u, {\varphi}}f\in H^\infty_{\mu, 0}$ for any $f\in B_{A_{\alpha}^p}$, i.e., ${\mathcal S}\subset H^\infty_{\mu, 0}$. Taking the supremum in (3.7) over $B_{A_{\alpha}^p}$ and employing (3.24), it follows that

This fact and Lemma 2.2 imply the compactness of $\Re^k_{u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu, 0}^{\infty}$.

If $\Re^k_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu, 0}^{\infty}$ is compact, then $\Re^k_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu}^{\infty}$ is also compact. From Theorem 3.4 we have that (3.15) and (3.16) hold. The fact $f_{j}(z) = z_{j}\in A_{\alpha}^p$, $j = \overline{1, n}$, implies $\Re^k_{u, {\varphi}}f_j\in H_{\mu, 0}^\infty$, $j = \overline{1, n}$, from which we have $\lim_{|z|\to 1}\mu(z)|u(z)||{\varphi}_j(z)| = 0,$ $j = \overline{1, n}.$ Hence,

From (3.26) together with (3.16) we obtain (3.24) in a standard way.

The following result is known. For a more general result, see ^[31].

Theorem 3.9. Let $p\ge 1$, ${\alpha} > -1$, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator $\Re^0_{u, {\varphi}}: A_{\alpha}^p\to H_{\mu, 0}^{\infty}$ is compact if and only if

Theorem 3.10. Let $p\ge 1$, ${\alpha} > -1$, $m\in\mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, $\varphi\in S(\mathbb{B})$ and $\mu\in W({\mathbb{B}})$. Then, the operator ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu, 0}^{\infty}$ is compact and

if and only if $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu, 0}^{\infty}$ are compact for $j = \overline{0, m}$.

Proof. Suppose ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu, 0}^{\infty}$ is compact and (3.28) holds. For the compactness of $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu, 0}^{\infty}$, $j = \overline{0, m},$ it is enough to prove (see Theorems 3.8 and 3.9),

and

Note that ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu}^{\infty}$ is compact, whereas (3.9) follows from (3.28). The compactness of $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu}^{\infty}$, $j = \overline{0, m}$, follows from Theorem 3.6. Hence, we have (3.19) and (3.20). Therefore, for every ${\varepsilon} > 0$ there is $\delta\in(0, 1)$ such that for ${\delta} < |{\varphi}(z)| < 1$,

From (3.28) and (3.31), (3.29) easily follows. From the fact $f_0(z)\equiv1\in A_{\alpha}^p$ it follows that ${\mathfrak S}^m_{\vec u, {\varphi}}1 = u_0\in H_{\mu, 0}^\infty$, from which, together with (3.31), we similarly get (3.30).

If $\Re^j_{u_j, {\varphi}}:A_{\alpha}^p\to H_{\mu, 0}^{\infty}$, $j = \overline{0, m}$, are compact, then ${\mathfrak S}^m_{\vec u, {\varphi}}:A_{\alpha}^p\rightarrow H_{\mu, 0}^{\infty}$ is also compact. Beside this (3.26) holds when $u$ is replaced by $u_j$ for each $j\in\{1, 2, \ldots, m\},$ that is, (3.28) also holds.

Remark 3.1. The quantities $J_0$ and $J_k,$ $k\in {\mathbb N}$, in Theorems 3.1 and 3.2, are essentially obtained by using the point evaluations in (2.1) and (2.2), respectively. Since the numerator of the right-hand side in (2.1) does not contain the term $|z|$, the quantity $J_0$ does not contain the term $|{\varphi}(z)|$, unlike the quantities $J_k$, $k\in {\mathbb N}.$ This is connected with the definition of the radial derivative operator.

4.   Conclusions

Motivated, among others, by our investigations in ^{[14,15,16,35]}, in 2016 I came up with an idea of studying finite sums of the weighted differentiation composition operators and introduced several operators of this form acting on spaces of holomorphic functions on the unit disk or on the unit ball. One of them was the operator in (1.1). In ^[37] we have studied the operator from Hardy spaces to weighted-type spaces on the unit ball. Here we complement the main results therein by characterizing the boundedness and compactness of the operator from the weighted Bergman space to the weighted-type spaces on the unit ball. The methods, ideas and tricks presented here, with some modifications, can be used in some other settings, which should lead to some further investigations in the direction.

Acknowledgments

The paper was made during the investigation supported by the Ministry of Education, Science and Technological Development of Serbia, contract no. 451-03-68/2022-14/200029.

Conflict of interest

The author declare no conflict of interest.