The boundedness and compactness of a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces are investigated in this paper.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper, read and approved the final manuscript.
1 Introduction
Let \(\mathcal{D}\) denote the unit disk in the complex plane \(\mathcal {C}\), and let \(\mathcal{H}(\mathcal{D})\) be the space of all holomorphic functions on \(\mathcal{D}\) with the topology of uniform convergence on compacts of \(\mathcal{D}\).
For \(0<\alpha<\infty\), the α-Bloch space, denoted by \(\mathcal{B}^{\alpha}\), consists of all functions \(f\in\mathcal {H}(\mathcal{D})\) such that
respectively. For some results on the Zygmund-type spaces on various domains in the complex plane and \(\mathcal{C}^{n}\) and operators on them, see, for example, [1‐18]. The α-Bloch space is introduced and studied by numerous authors. For the general theory of α-Bloch or Bloch-type spaces and operators of them, see, e.g., [4, 19‐41]. Recently, many authors studied different classes of Bloch-type spaces, where the typical weight function, \(\omega(z)=1-|z|^{2}\), \(z\in\mathcal{D}\), is replaced by a bounded continuous positive function μ defined on \(\mathcal{D}\). More precisely, a function \(f\in\mathcal{H}(\mathcal{D})\) is called a μ-Bloch function, denoted by \(f\in\mathcal{B}^{\mu}\), if \(\|f\|_{\mu}=\sup_{z\in\mathcal{D}}\mu(z)|f'(z)|<\infty\). If \(\mu(z)=\omega(z)^{\alpha}\), \(\alpha>0\), \(\mathcal{B}^{\mu}\) is just the α-Bloch space \(\mathcal{B}^{\alpha}\). It is readily seen that \(\mathcal{B}^{\mu}\) is a Banach space with the norm \(\|f\|_{\mathcal{B}^{\mu}}=|f(0)|+\|f\|_{\mu}\).
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Recently, Ramos Fernández in [42] used Young’s functions to define the Bloch-Orlicz space. More precisely, let \(\varphi: [0,+\infty)\rightarrow[0,+\infty)\) be a strictly increasing convex function such that \(\varphi(0)=0\) and note that from these conditions it follows that \(\lim_{t\rightarrow+\infty}\varphi(t)=+\infty\). The Bloch-Orlicz space associated with the function φ, denoted by \(\mathcal{B}^{\varphi}\), is the class of all analytic functions f in \(\mathcal{D}\) such that
Moreover, it can be shown that \(\mathcal{B}^{\varphi}\) is a Banach space with the norm \(\|f\|_{\mathcal{B}^{\varphi}}=|f(0)|+\|f\|_{\varphi}\). We also have that the Bloch-Orlicz space is isometrically equal to a particular μ-Bloch space, where \(\mu(z)=\frac{1}{\varphi^{-1}(\frac{1}{1-|z|^{2}})}\) with \(z\in \mathcal{D}\). Thus, for any \(f\in\mathcal{B}^{\varphi}\), we have
When φ is the identity map on \([0,+\infty)\), \(\mathcal {B}^{\varphi}\) is the so-called Bloch space ℬ.
Let \(u\in\mathcal{H}(\mathcal{D})\) and ϕ be an analytic self-map of \(\mathcal{D}\). The differentiation operator D, the multiplication operator \(M_{u}\) and the composition operator \(C_{\phi}\) are defined by
There is a considerable interest in studying the above mentioned operators as well as their products (see, e.g., [1‐38, 41‐56] and the related references therein).
A product-type operator \(DM_{u}C_{\phi}\) is defined as follows:
Esmaeili and Lindström in [1] investigated weighted composition operators between Zygmund-type spaces. Ramos Fernández in [42] studied the boundedness and compactness of composition operators on Bloch-Orlicz spaces. Li and Stević in [5] investigated products of Volterra-type operator and composition operator from \(H^{\infty}\) and Bloch spaces to Zygmund spaces, and they in [8] studied products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Liu and Yu in [25] characterized the boundedness and compactness of products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball. Sharma in [27] studied the boundedness and compactness of products of composition multiplication and differentiation between Bergman and Bloch-type spaces. In [52], Stević investigated the properties of weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Stević in [13] studied weighted radial operators from the mixed-norm space to the nth weighted-type space on the unit ball. Stević et al. in [54] characterized the boundedness and compactness of products of multiplication composition and differentiation operators on weighted Bergman spaces. Zhu in [18] studied extended Cesàro operators from mixed-norm spaces to Zygmund-type spaces.
Motivated by the above papers, in this paper, we investigate the boundedness and compactness of the product-type operator \(DM_{u}C_{\phi}\) from Zygmund-type spaces to the Bloch-Orlicz space. The paper is organized as follows. In Section 2, we give some necessary and sufficient conditions for the boundedness of the operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\). In Section 3, we give some necessary and sufficient conditions for the compactness of the operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\).
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. Taking the function \(f(z)=1\in\mathcal{Z}^{\alpha}\) and using the obvious fact that \(\|f\|_{\mathcal{Z}^{\alpha}}=1\), we have that
from which it follows that (1) holds. Taking the function \(f(z)=z\in\mathcal{Z}^{\alpha}\) and using the fact that \(\|f\|_{\mathcal{Z}^{\alpha}}=1\), we obtain
From this, (1) and by the boundedness of \(\phi(z)\), condition (2) easily follows. Now taking the function \(f(z)=z^{2}\in\mathcal{Z}^{\alpha}\) and using the fact that \(\|f\|_{\mathcal{Z}^{\alpha}}=2\), we get
From this, (1), (2) and the boundedness of \(\phi(z)\), we obtain (3). □
Now, we are ready to characterize the boundedness of the product-type operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\). For this purpose we need to break the problem into five different cases: \(0<\alpha<1\), \(\alpha=1\), \(1<\alpha<2\), \(\alpha=2\) and \(\alpha>2\).
Theorem 3
Let\(u\in\mathcal{H}(\mathcal{D})\), ϕbe an analytic self-map of\(\mathcal{D}\)and\(0<\alpha<1\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is bounded if and only if\(k_{1}<\infty\), \(k_{2}<\infty\)and
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). Now we will prove (4). Let
for all \(z\in\mathcal{D}\) and \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), then \(g_{\phi(\omega)}\in\mathcal{Z}^{\alpha}\), and
By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}g_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then
Now, we can conclude that there exists a constant C such that \(\|DM_{u}C_{\phi}f\|_{\mathcal{B}^{\varphi}}\leq C\|f\|_{\mathcal {Z}^{\alpha}}\) for all \(f\in\mathcal{Z}^{\alpha}\), so the product-type operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. □
Theorem 4
Let\(u\in\mathcal{H}(\mathcal{D})\)andϕbe an analytic self-map of\(\mathcal{D}\). Then\(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\)is bounded if and only if\(k_{1}<\infty\),
By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}s_{\phi(\omega)}\|_{\mathcal{B}^{\varphi }}\leq C\), then
By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}t_{\phi(\omega)}\|_{\mathcal{B}^{\varphi }}\leq C\), then
Suppose that \(k_{1}, k_{5}, k_{6}<\infty\). Then, by Lemma 1(ii) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is bounded. □
Theorem 5
Let\(u\in\mathcal{H}(\mathcal{D})\), ϕbe an analytic self-map of\(\mathcal{D}\)and\(1<\alpha<2\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is bounded if and only if\(k_{1}<\infty\),
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). Inequality (14) can be proved as in Theorem 3. Using the test function \(f_{\phi(\omega)}(z)\) in Section 1, where \(z\in\mathcal{D}\), \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), then we have that \(f_{\phi(\omega)}\in\mathcal{Z}^{\alpha}\), and
By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}f_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then
Then, according to the former proof with \(k_{2}<\infty\), we can get (13).
Suppose that \(k_{1}, k_{7}, k_{8}<\infty\). Then, by Lemma 1(iii) and (iv) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. □
Theorem 6
Let\(u\in\mathcal{H}(\mathcal{D})\)andϕbe an analytic self-map of\(\mathcal{D}\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow \mathcal{B}^{\varphi}\)is bounded if and only if
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\) is bounded, from Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). By repeating the arguments in the proof of Theorem 3 and Theorem 5, (16) and (17) can be proved similarly. Hence we only need to show \(k_{9}<\infty\). For every \(z\in\mathcal{D}\) and \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), let \(p_{\phi(\omega)}(z)=\log\frac{e}{1-\overline{\phi(\omega)}z}\). Clearly \(p_{\phi(\omega)}\in\mathcal{Z}^{2}\), and \(p_{\phi(\omega)}(\phi(\omega))=\log\frac{e}{1-|\phi(\omega)|^{2}}\),
By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}p_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then
Suppose that \(k_{9}, k_{10}, k_{11}<\infty\). Then, by Lemma 1(iii) and (v) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\) is bounded. □
Theorem 7
Let\(u\in\mathcal{H}(\mathcal{D})\), ϕbe an analytic self-map of\(\mathcal{D}\)and\(\alpha>2\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is bounded if and only if
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). With the same argument as in Theorem 5 one can show that (21) and (22) hold.
Now we prove that \(k_{12}<\infty\). For every \(a, z\in\mathcal{D}\), define \(q_{a}(z)=\frac{(1-|a|^{2})^{2}}{(1-\bar{a}z)^{\alpha}}\). Then \(\sup_{z\in\mathcal{D}}(1-|z|^{2})^{\alpha}|q''_{a}(z)|\leq4\alpha \cdot2^{\alpha}\cdot(\alpha+1)\), which shows that \(q_{a}\in\mathcal{Z}^{\alpha}\). Now we let \(a=\phi (\omega)\) for every \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi (\omega)|<1\), and we have
By the boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we have \(\|DM_{u}C_{\phi}q_{\phi(\omega)}\|_{\mathcal{B}^{\varphi}}\leq C\), then
Since \(\frac{\alpha+1}{2\alpha}<1\), then by (21), (22), (23) and according to the former proof with \(k_{1}<\infty\) for \(|\phi(\omega)|\leq\frac{1}{2}\), then \(k_{12}<\infty\). Suppose that \(k_{12}, k_{13}, k_{14}<\infty\). Then, by Lemma 1(iii) and (vi) and similar to the proof of Theorem 3, we get that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is bounded. □
3 The compactness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha }\rightarrow\mathcal{B}^{\varphi}\)
In order to prove the compactness of the product-type operator \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\), we need the following lemmas. The proof of the following lemma is similar to that of Proposition 3.11 in [43]. The details are omitted.
Lemma 8
Let\(u\in\mathcal{H}(\mathcal{D})\), ϕbe an analytic self-map of\(\mathcal{D}\)and\(0<\alpha<\infty\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is compact if and only if\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is bounded and for any bounded sequence\(\{f_{n}\}_{n\in\mathcal{N}}\)in\(\mathcal{Z}^{\alpha}\)which converges to zero uniformly on compact subsets of\(\mathcal{D}\)as\(n\rightarrow\infty\), we have\(\|DM_{u}C_{\phi}f_{n}\|_{\mathcal{B}^{\varphi}}\rightarrow0\)as\(n\rightarrow\infty\).
The following lemma was essentially proved in paper [11] in Lemma 2.5.
Lemma 9
Fix\(0<\alpha<2\)and let\(\{f_{n}\}_{n\in\mathcal{N}}\)be a bounded sequence in\(\mathcal{Z}^{\alpha}\)which converges to zero uniformly on compact subsets of\(\mathcal{D}\)as\(n\rightarrow\infty \). Then\(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|f_{n}(z)|=0\). Moreover, for\(0<\alpha<1\), if\(\{f_{n}\}_{n\in\mathcal{N}}\)is a bounded sequence in\(\mathcal{Z}^{\alpha}\)which converges to zero uniformly on compact subsets of\(\mathcal{D}\)as\(n\rightarrow\infty\), then\(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|f'_{n}(z)|=0\).
Theorem 10
Let\(u\in\mathcal{H}(\mathcal{D})\), ϕbe an analytic self-map of\(\mathcal{D}\)and\(0<\alpha<1\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is compact if and only if\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is bounded,
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Taking the function
Then \(\sup_{n\in\mathcal{N}}\|g_{n}\|_{\mathcal{Z}^{\alpha }}<\infty\), and \(g_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\). Since \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}g_{n}\|_{\mathcal {B}^{\varphi}}=0\). Since \(\lim_{n\rightarrow\infty}|\phi(z_{n})|=1\), then \(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|g_{n}(z)|=0\). Moreover, we have
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded and (24) holds. Then \(k_{1}, k_{2}, k_{3}<\infty\) by Lemma 2 and for every \(\epsilon>0\), there is \(\delta\in(0,1)\) such that
whenever \(\delta<|\phi(z)|<1\). Assume that \(\{f_{n}\}_{n\in\mathcal {N}}\) is a sequence in \(\mathcal{Z}^{\alpha}\) such that \(\sup_{n\in\mathcal{N}} \|f_{n}\|_{\mathcal{Z}^{\alpha}}\leq L\), and \(f_{n}\) converges to 0 uniformly on compact subsets of \(\mathcal{D}\) as \(n\rightarrow \infty\). Let \(K=\{z\in\mathcal{D}:|\phi(z)|\leq\delta\}\). Then by \(k_{1}, k_{2}, k_{3}<\infty\) and (25) it follows that
Here we use the fact that \(\sup_{z\in\mathcal{D}}(1-|\phi(z)|^{2})^{\alpha}|f''_{n}(\phi (z))|\leq\|f_{n}\|_{\mathcal{Z}^{\alpha}}\leq L\). So we obtain
Since \(f_{n}\) converges to 0 uniformly on compact subsets of \(\mathcal {D}\) as \(n\rightarrow\infty\), Cauchy’s estimation gives that \(f'_{n}\), \(f''_{n}\) also do as \(n\rightarrow\infty\). In particular, since \(\{\omega:|\omega|\leq\delta\}\) and \(\{\phi(0)\}\) are compact, it follows that
Let\(u\in\mathcal{H}(\mathcal{D})\)andϕbe an analytic self-map of\(\mathcal{D}\). Then\(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\)is compact if and only if\(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\)is bounded,
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow \mathcal{B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in \mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Taking the function
Then \(\sup_{n\in\mathcal{N}}\|s_{n}\|_{\mathcal{Z}}<\infty\) by the proof of Theorem 4, and \(s_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\) by a direct calculation. Consequently, \(\lim_{n\rightarrow\infty}\sup_{z\in\mathcal{D}}|s_{n}(z)|=0\) by Lemma 9. Since \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}s_{n}\|_{\mathcal {B}^{\varphi}}=0\). Moreover, we have
such that \(\lim_{n\rightarrow\infty}c_{n}=0\). By a direct calculation, we may easily prove that \(t_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\), and \(\sup_{n\in\mathcal{N}}\| t_{n}\|_{\mathcal{Z}}<\infty\) by the proof of Theorem 4. Since \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal{B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}t_{n}\|_{\mathcal {B}^{\varphi}}=0\). Moreover, we have
Conversely, by Lemma 1(ii), Lemma 2, Lemma 8 and Lemma 9, we can prove the converse implication similar to Theorem 10, so we omit the details. □
Theorem 12
Let\(u\in\mathcal{H}(\mathcal{D})\), ϕbe an analytic self-map of\(\mathcal{D}\)and\(1<\alpha<2\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is compact if and only if\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is bounded,
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Then (33) can be proved as the method of (24) in Theorem 10, so we only need to show that (32) holds. Taking the function
Then \(\sup_{n\in\mathcal{N}}\|f_{n}\|_{\mathcal{Z}^{\alpha }}<\infty\), and \(f_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\). Since \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact, it gives \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}f_{n}\|_{\mathcal {B}^{\varphi}}=0\). Moreover, we have
By Lemma 1(iii), Lemma 2, Lemma 8 and Lemma 9, we can prove the converse implication similar to Theorem 10, so we omit the details. □
Theorem 13
Let\(u\in\mathcal{H}(\mathcal{D})\)andϕbe an analytic self-map of\(\mathcal{D}\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow \mathcal{B}^{\varphi}\)is compact if and only if\(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal {B}^{\varphi}\)is bounded,
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal{B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow\mathcal {B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Then, by repeating the arguments in the proof of Theorem 10 and Theorem 12, (35) and (36) can be proved similarly, so we only need to show that (34) holds. Taking the function
It is easy to show that \(\{p_{n}\}_{n\in\mathcal{N}}\) is a bounded sequence in \(\mathcal{Z}^{2}\), and \(p_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\). Since \(DM_{u}C_{\phi}:\mathcal{Z}^{2}\rightarrow \mathcal{B}^{\varphi}\) is compact, then \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}p_{n}\|_{\mathcal {B}^{\varphi}}=0\).
Since \(\lim_{n\rightarrow\infty} (\log\frac{e}{1-|\phi (z_{n})|^{2}} )^{-1}=0\), and by (35) and (36), we can get (34).
By Lemma 1(iii) and (v), Lemma 2 and Lemma 8, we can prove the converse implication similar to Theorem 10, so we omit the details. □
Theorem 14
Let\(u\in\mathcal{H}(\mathcal{D})\), ϕbe an analytic self-map of\(\mathcal{D}\)and\(\alpha>2\). Then\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is compact if and only if\(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\)is bounded,
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\) is compact. It is clear that \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded. By Lemma 2, we have that \(k_{1}, k_{2}, k_{3}<\infty\). Let \(\{z_{n}\}_{n\in\mathcal{N}}\) be a sequence in \(\mathcal{D}\) such that \(|\phi(z_{n})|\rightarrow1\) as \(n\rightarrow\infty\). Without loss of generality, we may suppose that \(|\phi(z_{n})|>\frac{1}{2}\) for all n. Then, by repeating the arguments in the proof of Theorem 10 and Theorem 12, (41) and (42) can be proved similarly, so we only need to show that (40) holds.
Now let \(q_{n}(z)=\frac{(1-|\phi(z_{n})|^{2})^{2}}{(1-\overline{\phi (z_{n})}z)^{\alpha}}\), then \(\sup_{n\in\mathcal{N}}\|q_{n}\|_{\mathcal{Z}^{\alpha }}<\infty\), \(q_{n}\rightarrow0\) uniformly on compact subsets of \(\mathcal{D}\), and
Since \(\frac{\alpha+1}{2\alpha}<1\), then by (41), (42) and letting \(n\rightarrow\infty\) in (43), we can get (40).
For the converse, by Lemma 1(iii) and (vi), Lemma 2 and Lemma 8, we can prove the converse implication similar to Theorem 10, so we omit the details. □
Acknowledgements
This work is supported by the NNSF of China (Nos. 11201127; 11271112) and IRTSTHN (14IRTSTHN023).
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper, read and approved the final manuscript.