Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations

1.   Introduction

Let ${\mathbb C}$ denote the complex plane and ${\mathbb C}^n$ the $n$-dimensional complex Euclidean space with an inner product defined as $\langle z, w\rangle = \sum_{j = 1}^nz_j\overline{w}_j$. Let $B(a, r) = \left\{ z\in \mathbb{C}^n:|z-a| < r\right\}$ be the open ball of $\mathbb{C}^n$. In particular, the open unit ball is defined as $\mathbb{B} = B(0, 1)$.

Let $H(\mathbb{B})$ denote the set of all holomorphic functions on $\mathbb{B}$ and $S(\mathbb{B})$ the set of all holomorphic self-mappings of $\mathbb{B}$. For given ${\varphi}\in S({\mathbb{B}})$ and $u\in H({\mathbb{B}})$, the weighted composition operator on or between some subspaces of $H({\mathbb{B}})$ is defined by

If $u\equiv1$, then $W_{u, {\varphi}}$ is reduced to the composition operator usually denoted by $C_{{\varphi}}$. If ${\varphi}(z) = z$, then $W_{u, {\varphi}}$ is reduced to the multiplication operator usually denoted by $M_u$. Since $W_{u, {\varphi}} = M_u\cdot C_{\varphi}$, $W_{u, {\varphi}}$ can be regarded as the product of $M_u$ and $C_{\varphi}$.

If $n = 1$, $\mathbb{B}$ becomes the open unit disk in ${\mathbb C}$ usually denoted by $\mathbb{D}$. Let $D^{m}$ be the $m$th differentiation operator on $H(\mathbb{D})$, that is,

where $f^{(0)} = f$. $D^1$ denotes the classical differentiation operator denoted by $D$. As expected, there has been some considerable interest in investigating products of differentiation and other related operators. For example, the most common products $DC_{\varphi}$ and $C_{\varphi}D$ were extensively studied in ^{[1,10,11,12,13,23,25,26]}, and the products

were also extensively studied in ^{[14,18,22,27]}. Following the study of the operators in (1.1), people naturally extend to study the operators (see ^[5,6,30])

Other examples of products involving differentiation operators can be found in ^[7,8,19,32] and the related references.

As studying on the unit disk becomes more mature, people begin to become interested in exploring related properties on the unit ball. One method for extending the differentiation operator to $\mathbb{C}^{n}$ is the radial derivative operator

Naturally, replacing $D$ by $\Re$ in (1.1), we obtain the following operators

Recently, these operators have been studied in ^[31]. Other operators involving radial derivative operators have been studied in ^[21,33,34].

Interestingly, the radial derivative operator can be defined iteratively, namely, $\Re^{m}f$ can be defined as $\Re^mf = \Re(\Re^{m-1}f)$. Similarly, using the radial derivative operator can yield the related operators

Clearly, the operators in (1.3) are more complex than those in (1.2). Since $C_{\varphi}M_{u}\Re^m = M_{u\circ{\varphi}}C_{\varphi}\Re^m$, the operator $M_{u}C_{\varphi}\Re^m$ can be regarded as the simplest one in (1.3) which was first studied and denoted as $\Re^m_{u, {\varphi}}$ in ^[24]. Recently, it has been studied again because people need to obtain more properties about spaces to characterize its properties (see ^[29]).

To reconsider the operator $C_{\varphi}\Re^{m}M_{u}$, people find the fact

Motivated by (1.4), people directly studied the sum operator (see ^[2,28])

where $u_{i}\in H(\mathbb{B})$, $i = \overline{0, m}$, and $\varphi\in S(\mathbb{B})$. Particularly, if we set $u_0\equiv \cdots \equiv u_{m-1}\equiv 0$ and $u_m = u$, then $\mathfrak{S}^m_{\vec{u}, {\varphi}} = M_uC_{{\varphi}}\Re^{m}$; if we set $u_0\equiv \cdots \equiv u_{m-1}\equiv 0$ and $u_m = u\circ {\varphi}$, then $\mathfrak{S}^m_{\vec{u}, {\varphi}} = C_{{\varphi}}M_u\Re^m$. In ^[28], Stević et al. studied the operators $\mathfrak{S}^m_{\vec{u}, {\varphi}}$ from Hardy spaces to weighted-type spaces on the unit ball and obtained the following results.

Theorem A. Let $m\in \mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, ${\varphi}\in S(\mathbb{B})$, and $\mu$ a weight function on $\mathbb{B}$. Then, the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:H^p \to H^{\infty }_{\mu}$ is bounded and

if and only if

and

Theorem B. Let $m\in \mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, ${\varphi}\in S(\mathbb{B})$, and $\mu$ a weight function on $\mathbb{B}$. Then, the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:H^p \to H^{\infty }_{\mu}$ is compact if and only if it is bounded,

and

It must be mentioned that we find that the necessity of Theorem A requires (1.5) to hold. Inspired by ^[2,28], here we use a new method and technique without (1.5) to study the sum operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}$ from logarithmic Bergman-type space to weighted-type space on the unit ball. To this end, we need to introduce the well-known Bell polynomial (see ^[3])

where all non-negative integer sequences $j_{1}$, $j_{2} , \ldots, j_{m-k+1}$ satisfy

In particular, when $k = 0$, one can get $B_{0, 0} = 1$ and $B_{m, 0} = 0$ for any $m\in\mathbb{N}$. When $k = 1$, one can get $B_{i, 1} = x_{i}$. When $m = k = i$, $B_{i, i} = x_{1}^{i}$ holds.

2.   Preliminaries

In this section, we need to introduce logarithmic Bergman-type space and weighted-type space. Here, a bounded positive continuous function on $\mathbb{B}$ is called a weight. For a weight $\mu$, the weighted-type space $H_{\mu}^{\infty}$ consists of all $f\in H({\mathbb{B}})$ such that

With the norm $\|\cdot\|_{H_{\mu}^{\infty}}$, $H_{\mu}^{\infty}$ becomes a Banach space. In particular, if $\mu(z) = (1-|z|^{2})^{\sigma}\; (\sigma > 0)$, the space $H_{\mu}^{\infty}$ is called classical weighted-type space usually denoted by $H_{\sigma}^{\infty}$. If $\mu\equiv 1$, then space $H_{\mu}^{\infty}$ becomes the bounded holomorphic function space usually denoted by $H^{\infty}$.

Next, we need to present the logarithmic Bergman-type space on ${\mathbb{B}}$ (see ^[4] for the unit disk case). Let $dv$ be the standardized Lebesgue measure on $\mathbb{B}$. The logarithmic Bergman-type space $A_{w_{\gamma, \delta}}^{p}$ consists of all $f\in H({\mathbb{B}})$ such that

where $-1 < \gamma < +\infty$, $\delta\leq 0$, $0 < p < +\infty$ and $w_{\gamma, \delta}(z)$ is defined by

When $p\geq1$, $A_{w_{\gamma, \delta}}^{p}$ is a Banach space. While $0 < p < 1$, it is a Fréchet space with the translation invariant metric $\rho(f, g) = \|f-g\|_{A_{\omega_{\gamma, \delta}}^{p}}^{p}$.

Let $\varphi\in S({\mathbb{B}})$, $0\leq r < 1$, $0\leq\gamma < \infty$, $\delta\leq 0$, and $a\in\mathbb{B}\backslash\{\varphi(0)\}$. The generalized counting functions are defined as

where $\left|z_{j}(a)\right| < r$, counting multiplicities, and

If ${\varphi}\in S({\mathbb D})$, then the function $N_{\varphi, \gamma, \delta}$ has the integral expression: For $1\leq\gamma < +\infty$ and $\delta\leq 0$, there is a positive function $F(t)$ satisfying

When ${\varphi}\in S({\mathbb D})$ and $\delta = 0$, the generalized counting functions become the common counting functions. Namely,

and

In ^[17], Shapiro used the function $N_{\varphi, \gamma}(1, a)$ to characterize the compact composition operators on the weighted Bergman space.

Let $X$ and $Y$ be two topological spaces induced by the translation invariant metrics $d_X$ and $d_Y$, respectively. A linear operator $T: X\to Y$ is called bounded if there is a positive number $K$ such that

for all $f\in X$. The operator $T: X\rightarrow Y$ is called compact if it maps bounded sets into relatively compact sets.

In this paper, $j = \overline{k, l}$ is used to represent $j = k, ..., l$, where $k, l\in\mathbb{N}_{0}$ and $k\leq l$. Positive numbers are denoted by $C$, and they may vary in different situations. The notation $a\lesssim b$ (resp. $a\gtrsim b$) means that there is a positive number $C$ such that $a\leq Cb$ (resp. $a\geq Cb$). When $a\lesssim b$ and $b\gtrsim a$, we write $a\asymp b$.

3.   Logarithmic Bergman-type space

In this section, we obtain some properties on the logarithmic Bergman-type space. First, we have the following point-evaluation estimate for the functions in the space.

Theorem 3.1. Let $-1 < \gamma < +\infty$, $\delta\leq0$, $0 < p < +\infty$ and $0 < r < 1$. Then, there exists a positive number $C = C(\gamma, \delta, p, r)$ independent of $z\in K = \{z\in\mathbb{B}:|z| > r\}$ and $f\in A_{w_{\gamma, \delta}}^{p}$ such that

Proof. Let $z\in\mathbb{B}$. By applying the subharmonicity of the function $|f|^{p}$ to Euclidean ball $B(z, r)$ and using Lemma 1.23 in ^[35], we have

Since $r < |z| < 1$ and $1-|w|^{2}\asymp1-|z|^{2}$, we have

and

From (3.3) and (3.4), it follows that there is a positive constant $C_{2, r}$ such that $w_{\gamma, \delta}(z)\leq C_{2, r}w_{\gamma, \delta}(w)$ for all $w\in B(z, r)$. From this and (3.2), we have

From (3.5) and the fact $\log\frac{1}{|z|}\asymp1-|z|\asymp1-|z|^{2}$, the following inequality is right with a fixed constant $C_{3, r}$

Let $C = \frac{C_{1, r} C_{2, r} C_{3, r}}{p}$. Then the proof is end. □

Theorem 3.2. Let $m\in\mathbb{N}$, $-1 < \gamma < +\infty$, $\delta\leq0$, $0 < p < +\infty$ and $0 < r < 1$. Then, there exists a positive constant $C_m = C(\gamma, \delta, p, r, m)$ independent of $z\in K$ and $f\in A_{w_{\gamma, \delta}}^{p}$ such that

Proof. First, we prove the case of $m = 1$. By the definition of the gradient and the Cauchy's inequality, we get

where $i = \overline{1, n}$. By using the relations

and

we obtain the following formula

for any $w\in B(z, q(1-|z|))$. Then,

From (3.1) and (3.2), it follows that

Hence, the proof is completed for the case of $m = 1$.

We will use the mathematical induction to complete the proof. Assume that (3.6) holds for $m < a$. For convenience, let $g(z) = \frac{\partial^{a-1}f(z)}{\partial z_{i_{1}}\partial z_{i_{2} }\dots\partial z_{i_{a-1}}}$. By applying (3.7) to the function $g$, we obtain

According to the assumption, the function $g$ satisfies

By using (3.8), the following formula is also obtained

This shows that (3.6) holds for $m = a$. The proof is end.□

As an application of Theorems 3.1 and 3.2, we give the estimate in $z = 0$ for the functions in $A_{\omega_{\gamma, \delta}}^{p}$.

Corollary 3.1. Let $-1 < \gamma < +\infty$, $\delta\leq 0$, $0 < p < +\infty$, and $0 < r < 2/3$. Then, for all $f \in A_{w_{\gamma, \delta}}^{p}$, it follows that

and

where constants $C$ and $C_m$ are defined in Theorems 3.1 and 3.2, respectively.

Proof. For $f \in A_{w_{\gamma, \delta}}^{p}$, from Theorem 3.1 and the maximum module theorem, we have

which implies that (3.10) holds. By using the similar method, we also have that (3.11) holds.□

Next, we give an equivalent norm in $A_{w_{\gamma, \delta}}^{p}$, which extends Lemma 3.2 in ^[4] to $\mathbb{B}$.

Theorem 3.3. Let $r_{0}\in[0, 1)$. Then, for every $f\in A_{w_{\gamma, \delta}}^{p}$, it follows that

Proof. If $r_{0} = 0$, then it is obvious. So, we assume that $r_{0}\in(0, 1)$. Integration in polar coordinates, we have

Put

Then it is represented that

Since $M(r, f)$ is increasing, $A(r)$ is positive and continuous in $r$ on $(0, 1)$ and

that is, there is a constant $\varepsilon > 0 \; (\varepsilon < r_{0})$ such that $A(r) < A(\varepsilon)$ for $r\in(0, \varepsilon)$. Then we have

From (3.13) and (3.14), we obtain the inequality

The inequality reverse to this is obvious. The asymptotic relationship (3.12) follows, as desired. □

The following integral estimate is an extension of Lemma 3.4 in ^[4]. The proof is similar, but we still present it for completeness.

Lemma 3.1. Let $-1 < \gamma < +\infty$, $\delta\leq0$, $\beta > \gamma-\delta$ and $0 < r < 1$. Then, for each fixed $w\in\mathbb{B}$ with $|w| > r$,

Proof. Fix $|w|$ with $|w| > r_{0}$ ($0 < r_{0} < 1$). It is easy to see that

By applying Theorem 3.3 with

and using (3.15), the formula of integration in polar coordinates gives

By Proposition 1.4.10 in ^[15], we have

From (3.16) and (3.17), we have

Since $[\log(1-\frac{1}{\log r})]^{\delta}$ is decreasing in $r$ on $[|w|, 1]$, we have

On the other hand, we obtain

If $\delta = 0$ and $\beta > \gamma$, then we have

If $\delta\neq0$, then integration by parts gives

Since $\delta < 0$, $\gamma-\beta < 0$ and

we have

and from this follows

provided $\gamma-\beta-\delta < 0$. The proof is finished. □

The following gives an important test function in $A_{w_{\gamma, \delta}}^{p}$.

Theorem 3.4. Let $-1 < \gamma < +\infty$, $\delta\leq0$, $0 < p < +\infty$ and $0 < r < 1$. Then, for each $t\geq0$ and $w\in\mathbb{B}$ with $|w| > r$, the following function is in $A_{w_{\gamma, \delta}}^{p}$

Moreover,

Proof. By Lemma 3.1 and a direct calculation, we have

The proof is finished. □

4.   Boundedness and compactness of the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A_{w_{\gamma, \delta}}^{p}\to H_{\mu}^{\infty}$

In this section, for simplicity, we define

In order to characterize the compactness of the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A_{w_{\gamma, \delta}}^{p}\to H_{\mu}^{\infty}$, we need the following lemma. It can be proved similar to that in ^[16], so we omit here.

Lemma 4.1. Let $-1 < \gamma < +\infty$, $\delta\leq0$, $0 < p < +\infty$, $m\in\mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, and $\varphi\in S(\mathbb{B})$. Then, the bounded operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A_{w_{\gamma, \delta}}^{p}\to H_{\mu}^{\infty}$ is compact if and only if for every bounded sequence $\{f_{k}\}_{k\in\mathbb{N}}$ in $A_{w_{\gamma, \delta}}^{p}$ such that $f_{k}\rightarrow 0$ uniformly on any compact subset of $\mathbb{B}$ as $k\to\infty$, it follows that

The following result was obtained in ^[24].

Lemma 4.2. Let $s\geq 0$, $w\in{\mathbb{B}}$ and

Then,

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

We also need the following result obtained in ^[20].

Lemma 4.3. Let $s > 0$, $w\in{\mathbb{B}}$ and

Then,

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

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

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

The final lemma of this section was obtained in ^[24].

Lemma 4.4. If $a > 0$, then

Theorem 4.1. Let $-1 < \gamma < +\infty$, $\delta\leq0$, $0 < p < +\infty$, $m\in\mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, and $\varphi\in S(\mathbb{B})$. Then, the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty$ is bounded if and only if

and

for $j = \overline{1, m}$.

Moreover, if the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is bounded, then

Proof. Suppose that (4.1) and (4.2) hold. From Theorem 3.1, Theorem 3.2, and some easy calculations, it follows that

By taking the supremum in inequality (4.4) over the unit ball in the space $A^p_{w_{\gamma, \delta}}$, and using (4.1) and (4.2), we obtain that the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty$ is bounded. Moreover, we have

where $C$ is a positive constant.

Assume that the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is bounded. Then there exists a positive constant $C$ such that

for any $f\in{A^p_{w_{\gamma, \delta}}}$. First, we can take $f(z) = 1\in {A^p_{w_{\gamma, \delta}}}$, then one has that

Similarly, take $f_{k}(z) = z_{k}^{j}\in {A^p_{w_{\gamma, \delta}}}$, $k = \overline{1, n}$ and $j = \overline{1, m}$, by (4.7), then

for any $j\in\{1, 2, \ldots, m\}$. Since $\varphi(z)\in\mathbb{B}$, we have $|\varphi(z)|\leq1$. So, one can use the triangle inequality (4.7) and (4.8), the following inequality is true

Let $w\in\mathbb{B}$ and $d_{k} = \frac{\gamma +n+1}{p}+k$. For any $j\in\{1, 2, \ldots, m\}$ and constants $c_{k} = c_{k}^{(j)}$, $k = \overline{0, m}$, let

where $f_{w, k}$ is defined in Theorem 3.4. Then, by Theorem 3.4, we have

From (4.6), (4.11), and some easy calculations, it follows that

Since $d_{k} > 0$, $k = \overline{0, m}$, by Lemma 4.4, we have the following linear equations

From (4.12) and (4.13), we have

On the other hand, from (4.9), we have

From (4.14) and (4.15), we get that (4.2) holds for $j = \overline{1, m}$.

For constants $c_{k} = c_{k}^{(0)}$, $k = \overline{0, m}$, let

By Theorem 3.4, we know that $L_{0} = \sup_{w\in\mathbb{B}}\|h_{w}^{(0)}\|_{A^p_{w_{\gamma, \delta}}} < +\infty$. From this, (4.12), (4.13) and Lemma 4.4, we get

So, we have $M_0 < +\infty$. Moreover, we have

From (4.5) and (4.17), we obtain (4.3). The proof is completed. □

From Theorem 4.1 and (1.4), we obtain the following result.

Corollary 4.1. Let $m\in\mathbb{N}$, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu$ is a weight function on $\mathbb{B}$. Then, the operator $C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is bounded if and only if

and

for $j = \overline{1, m}$.

Moreover, if the operator $C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is bounded, then

Theorem 4.2. Let $-1 < \gamma < +\infty$, $\delta\leq0$, $0 < p < +\infty$, $m\in\mathbb{N}$, $u_j\in H(\mathbb{B})$, $j = \overline{0, m}$, and $\varphi\in S(\mathbb{B})$. Then, the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty$ is compact if and only if the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty$ is bounded,

for $j = \overline{1, m}$, and

Proof. Assume that the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is compact. It is obvious that the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is bounded.

If $\|\varphi\|_{\infty} < 1$, then it is clear that (4.18) and (4.19) are true. So, we suppose that $\|\varphi\|_{\infty} = 1$. Let $\{z_{k}\}$ be a sequence in $\mathbb{B}$ such that

where $h_{w}^{(j)}$ are defined in (4.10) for a fixed $j\in\{1, 2, \ldots, l\}$. Then, it follows that $h_{k}^{(j)}\rightarrow 0$ uniformly on any compact subset of $\mathbb{B}$ as $k\rightarrow \infty$. Hence, by Lemma 4.1, we have

Then, we can find sufficiently large $k$ such that

If $k\rightarrow \infty$, then (4.20) is true.

Now, we discuss the case of $j = 0$. Let $h_{k}^{(0)} = h_{\varphi(z_{k})}^{(0)}$, where $h_{w}^{(0)}$ is defined in (4.16). Then, we also have that $\|h_{k}^{(0)}\|_{A^p_{w_{\gamma, \delta}}} < +\infty$ and $h_{k}^{(0)}\rightarrow 0$ uniformly on any compact subset of $\mathbb{B}$ as $k\rightarrow \infty$. Hence, by Lemma 4.1, one has that

Then, by (4.21), we know that (4.18) is true.

Now, assume that $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is bounded, (4.18) and (4.19) are true. One has that

and

for any $z\in\mathbb{B}$. By (4.18) and (4.19), for arbitrary $\varepsilon > 0$, there is a $r\in(0, 1)$, for any $z\in K$ such that

and

Assume that $\{f_{s}\}$ is a sequence such that $\sup_{s\in\mathbb{N}}\|f_{s}\|_{A^p_{w_{\gamma, \delta}}}\leq M < +\infty$ and $f_{s}\rightarrow 0$ uniformly on any compact subset of $\mathbb{B}$ as $s\rightarrow \infty$. Then by Theorem 3.1, Theorem 3.2 and (4.22)–(4.25), one has that

Since $f_{s}\rightarrow0$ uniformly on any compact subset of $\mathbb{B}$ as $s\rightarrow \infty$. By Cauchy's estimates, we also have that $\frac{\partial^{j} f_{s}}{\partial z_{l_{1}}\partial z_{l_{2}}\cdots\partial z_{l_{j}}}\rightarrow 0$ uniformly on any compact subset of $\mathbb{B}$ as $s\rightarrow \infty$. From this and using the fact that $\{w\in{\mathbb{B}}:|w|\leq\delta\}$ is a compact subset of $\mathbb{B}$, by letting $s\rightarrow \infty$ in inequality (4.26), one get that

Since $\varepsilon$ is an arbitrary positive number, it follows that

By Lemma 4.1, the operator $\mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is compact. □

As before, we also have the following result.

Corollary 4.2. Let $m\in\mathbb{N}$, $u\in H(\mathbb{B})$, $\varphi\in S(\mathbb{B})$ and $\mu$ is a weight function on $\mathbb{B}$. Then, the operators $C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is compact if and only if the operator $C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty$ is bounded,

and

for $j = \overline{1, m}$.

5.   Conclusions

In this paper, we study and obtain some properties about the logarithmic Bergman-type space on the unit ball. As some applications, we completely characterized the boundedness and compactness of the operator

from the logarithmic Bergman-type space to the weighted-type space on the unit ball. Here, one thing should be pointed out is that we use a new method and technique to characterize the boundedness of such operators without the condition (1.5), which perhaps is the special flavour in this paper.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by Sichuan Science and Technology Program (2022ZYD0010) and the Graduate Student Innovation Foundation (Y2022193).

Conflict of interest

The authors declare that they have no competing interests.