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On a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces
Journal of Inequalities and Applications volume 2015, Article number: 132 (2015)
Abstract
The boundedness and compactness of a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces are investigated in this paper.
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
By \(\mathcal{Z}^{\alpha}\) we denote the Zygmund-type space consisting of those functions \(f\in\mathcal{H}(\mathcal{D})\) satisfying
\(\mathcal{B}^{\alpha}\) and \(\mathcal{Z}^{\alpha}\) are Banach spaces under the norms
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 \(\|f\|_{\mathcal{B}^{\mu}}=|f(0)|+\|f\|_{\mu}\).
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
for some \(\lambda>0\) depending on f. Also, since φ is convex, it is not hard to see that Minkowski’s functional
defines a seminorm for \(\mathcal{B}^{\varphi}\), which, in this case, is known as Luxemburg’s seminorm, where
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 Ï• 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:
For \(0<\alpha<\infty\) and \(\frac{1}{2}<|a|<1\), we define the test functions (see [1])
It is easy to show that \(f_{a},h_{a}\in\mathcal{Z}^{\alpha}\) and \(f_{a}(a)=0\),
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 Cesà ro 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 \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow\mathcal {B}^{\varphi}\).
Throughout this paper,
and we use letter C to denote a positive constant whose value may change at each occurrence.
2 The boundedness of \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha }\rightarrow\mathcal{B}^{\varphi}\)
The following lemma was essentially proved in [3] and [11] (see also [1]).
Lemma 1
For \(f\in\mathcal{Z}^{\alpha}\) and \(\alpha>0\). Then:
-
(i)
For \(0<\alpha<1\), \(|f'(z)|\leq\frac{2}{1-\alpha}\|f\|_{\mathcal{Z}^{\alpha}}\) and \(|f(z)|\leq\frac{2}{1-\alpha}\|f\|_{\mathcal{Z}^{\alpha}}\).
-
(ii)
For \(\alpha=1\), \(|f'(z)|\leq\log\frac{e}{1-|z|^{2}}\|f\|_{\mathcal{Z}}\) and \(|f(z)|\leq\|f\|_{\mathcal{Z}}\).
-
(iii)
For \(\alpha>1\), \(|f'(z)|\leq\frac{2}{\alpha-1}\frac{\|f\|_{\mathcal{Z}^{\alpha }}}{(1-|z|^{2})^{\alpha-1}}\). For \(\alpha=2\), \(|f'(z)|\leq\frac{e}{1-|z|^{2}}\|f\|_{\mathcal{Z}^{2}}\).
-
(iv)
For \(1<\alpha<2\), \(|f(z)|\leq\frac{2}{(\alpha-1)(2-\alpha)}\|f\|_{\mathcal {Z}^{\alpha}}\).
-
(v)
For \(\alpha=2\), \(|f(z)|\leq2\log\frac{e}{1-|z|^{2}}\|f\|_{\mathcal{Z}^{2}}\).
-
(vi)
For \(\alpha>2\), \(|f(z)|\leq\frac{2}{(\alpha-1)(\alpha-2)}\frac{\|f\|_{\mathcal {Z}^{\alpha}}}{(1-|z|^{2})^{\alpha-2}}\).
Lemma 2
If \(DM_{u}C_{\phi}:\mathcal{Z}^{\alpha}\rightarrow \mathcal{B}^{\varphi}\) is bounded and \(0<\alpha<\infty\), then the following conditions hold:
Proof
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
Hence
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
Hence
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 \(u\in\mathcal{H}(\mathcal{D})\), Ï• be 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
Proof
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
It follows that
By \(k_{3}<\infty\), we see that
From (5) and (6), we obtain (4).
Suppose that \(k_{1}, k_{2}, k_{4}<\infty\). For each \(f\in\mathcal{Z}^{\alpha}\setminus\{0\}\), by Lemma 1(i) we have
where C is a constant such that \(C\geq k_{1}\frac{2}{1-\alpha}+k_{2}\frac{2}{1-\alpha}+k_{4}\). Here we use the fact that
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 \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be an analytic self-map of \(\mathcal{D}\). Then \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow\mathcal {B}^{\varphi}\) is bounded if and only if \(k_{1}<\infty\),
Proof
Suppose that \(DM_{u}C_{\phi}:\mathcal{Z}\rightarrow \mathcal{B}^{\varphi}\) is bounded, by Lemma 2 we know that \(k_{1}, k_{2}, k_{3}<\infty\). Let
where
for any \(a\in\mathcal{D}\) such that \(\frac{1}{2}<|a|<1\). Then we have
for \(\frac{1}{2}<|a|<1\) and \(\sup_{\frac{1}{2}<|a|<1}\|s_{a}\|_{\mathcal{Z}}<\infty\).
Now let \(a=\phi(\omega)\), \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega)|<1\), then
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
From this it follows that
By \(k_{3}<\infty\) we see that
From (9) and (10) we obtain \(k_{6}<\infty\).
Let
for \(\omega\in\mathcal{D}\) such that \(\frac{1}{2}<|\phi(\omega )|<1\), then, as above, we can get that \(t_{\phi(\omega)}\in\mathcal{Z}\) and
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
From this and by \(k_{6}<\infty\), we get
By \(k_{2}<\infty\) we see that
From (11) and (12) we obtain (7).
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 \(u\in\mathcal{H}(\mathcal{D})\), Ï• be 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 \(k_{1}<\infty\),
Proof
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
From this and by \(k_{8}<\infty\), we get
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 \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be 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
Proof
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
By \(k_{10}, k_{11}<\infty\) we get
By \(k_{1}<\infty\) we see that
From (18) and (19) we obtain (15).
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 \(u\in\mathcal{H}(\mathcal{D})\), Ï• be 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
Proof
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
Then we have
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 \(u\in\mathcal{H}(\mathcal{D})\), Ï• be 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 \(u\in\mathcal{H}(\mathcal{D})\), Ï• be 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,
Proof
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
Then
Hence
Therefore
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
Moreover, since \(0<\alpha<1\), by Lemma 9 we have
Hence, letting \(n\rightarrow\infty\) in (26), we get
Employing Lemma 8 the implication follows. □
Theorem 11
Let \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be 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,
Proof
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
Then
It follows that
Therefore
On the other hand, let
where
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
Then
It follows that
Letting \(n\rightarrow\infty\) in (30) and combining with (29), we can get
The implication follows from (29) and (31).
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 \(u\in\mathcal{H}(\mathcal{D})\), Ï• be 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,
Proof
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
Then
It follows that
Therefore
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 \(u\in\mathcal{H}(\mathcal{D})\) and Ï• be 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,
Proof
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
Then we have
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 \(\lim_{n\rightarrow\infty}\|DM_{u}C_{\phi}p_{n}\|_{\mathcal {B}^{\varphi}}=0\).
From (37), (38) and (39), we can get that
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 \(u\in\mathcal{H}(\mathcal{D})\), Ï• be 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,
Proof
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
Then we have
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. □
References
Esmaeili, K, Lindström, M: Weighted composition operators between Zygmund type spaces and their essential norms. Integral Equ. Oper. Theory 75, 473-490 (2013)
Li, S: Weighted composition operators from minimal Möbius invariant spaces to Zygmund spaces. Filomat 27(2), 267-275 (2013)
Li, S, Stević, S: Volterra type operators on Zygmund space. J. Inequal. Appl. 2007, Article ID 32124 (2007)
Li, S, Stević, S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 338(2), 1282-1295 (2008)
Li, S, Stević, S: Products of Volterra type operator and composition operator from \(H^{\infty}\) and Bloch spaces to the Zygmund space. J. Math. Anal. Appl. 345, 40-52 (2008)
Li, S, Stević, S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput. 206(2), 825-831 (2008)
Li, S, Stević, S: Integral-type operators from Bloch-type spaces to Zygmund-type spaces. Appl. Math. Comput. 215(2), 464-473 (2009)
Li, S, Stević, S: Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Appl. Math. Comput. 217(7), 3144-3154 (2010)
Li, S, Stević, S: On an integral-type operator from ω-Bloch spaces to μ-Zygmund spaces. Appl. Math. Comput. 215(12), 4385-4391 (2010)
Stević, S: On an integral operator from the Zygmund space to the Bloch-type space on the unit ball. Glasg. Math. J. 51, 275-287 (2009)
Stević, S: On an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball. Abstr. Appl. Anal. 2010, Article ID 198608 (2010)
Stević, S: Weighted differentiation composition operators from the mixed-norm space to the nth weighted-type space on the unit disk. Abstr. Appl. Anal. 2010, Article ID 246287 (2010)
Stević, S: Weighted radial operator from the mixed-norm space to the nth weighted-type space on the unit ball. Appl. Math. Comput. 218(18), 9241-9247 (2012)
Stević, S, Sharma, AK: Composition operators from weighted Bergman-Privalov spaces to Zygmund type spaces on the unit disk. Ann. Pol. Math. 105(1), 77-86 (2012)
Ye, S, Hu, Q: Weighted composition operators on the Zygmund spaces. Abstr. Appl. Anal. 2012, Article ID 462482 (2012)
Zhu, X: Volterra type operators from logarithmic Bloch spaces to Zygmund type space. Int. J. Mod. Math. 3(3), 327-336 (2008)
Zhu, X: Integral-type operators from iterated logarithmic Bloch spaces to Zygmund-type spaces. Appl. Math. Comput. 215(3), 1170-1175 (2009)
Zhu, X: Extended Cesà ro operators from mixed norm spaces to Zygmund type spaces. Tamsui Oxf. J. Math. Sci. 26(4), 411-422 (2010)
Krantz, SG, Stević, S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal. 71, 1772-1795 (2009)
Li, S: Differences of generalized composition operators on the Bloch space. J. Math. Anal. Appl. 394(2), 706-711 (2012)
Li, S, Stević, S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 9(2), 195-205 (2007)
Li, S, Stević, S: Weighted composition operators from Bergman-type spaces into Bloch spaces. Proc. Indian Acad. Sci. Math. Sci. 117(3), 371-385 (2007)
Li, S, Stević, S: Composition followed by differentiation from mixed-norm spaces to α-Bloch spaces. Sb. Math. 199(12), 1847-1857 (2008)
Li, S, Stević, S: Products of composition and integral type operators from \(H^{\infty}\) to the Bloch space. Complex Var. Elliptic Equ. 53(5), 463-474 (2008)
Liu, Y, Yu, Y: Products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball. J. Math. Anal. Appl. 423(1), 76-93 (2015)
Madigan, K, Matheson, A: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347(7), 2679-2687 (1995)
Sharma, AK: Products of composition multiplication and differentiation between Bergman and Bloch type spaces. Turk. J. Math. 35, 275-291 (2011)
Stević, S: Norms of some operators from Bergman spaces to weighted and Bloch-type space. Util. Math. 76, 59-64 (2008)
Stević, S: On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball. Discrete Dyn. Nat. Soc. 2008, Article ID 154263 (2008)
Stević, S: Integral-type operators from a mixed norm space to a Bloch-type space on the unit ball. Sib. Math. J. 50(6), 1098-1105 (2009)
Stević, S: Norm of weighted composition operators from α-Bloch spaces to weighted-type spaces. Appl. Math. Comput. 215, 818-820 (2009)
Stević, S: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 354, 426-434 (2009)
Stević, S: Norm of an integral-type operator from Dirichlet to Bloch space on the unit disk. Util. Math. 83, 301-303 (2010)
Stević, S: On an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball. Appl. Math. Comput. 215(11), 3817-3823 (2010)
Stević, S: On operator \(P_{\varphi}^{g}\) from the logarithmic Bloch-type space to the mixed-norm space on unit ball. Appl. Math. Comput. 215, 4248-4255 (2010)
Stević, S: Characterizations of composition followed by differentiation between Bloch-type spaces. Appl. Math. Comput. 218(8), 4312-4316 (2011)
Stević, S: On some integral-type operators between a general space and Bloch-type spaces. Appl. Math. Comput. 218(6), 2600-2618 (2011)
Stević, S: Boundedness and compactness of an integral-type operator from Bloch-type spaces with normal weights to \(F(p,q,s)\) space. Appl. Math. Comput. 218(9), 5414-5421 (2012)
Zhao, R: A characterization of Bloch-type spaces on the unit ball of \(C^{n}\). J. Math. Anal. Appl. 330(1), 291-297 (2007)
Zhu, K: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23(3), 1143-1177 (1993)
Zhu, X: Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces. J. Korean Math. Soc. 46(6), 1219-1232 (2009)
Ramos Fernández, JC: Composition operators on Bloch-Orlicz type spaces. Appl. Math. Comput. 217(7), 3392-3402 (2010)
Cowen, CC, MacCluer, BD: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Roton (1995)
Li, S, Stević, S: Weighted composition operators from \(H^{\infty}\) to the Bloch space on the polydisc. Abstr. Appl. Anal. 2007, Article ID 48478 (2007)
Li, S, Stević, S: Composition followed by differentiation between \(H^{\infty}\) and α-Bloch spaces. Houst. J. Math. 35(1), 327-340 (2009)
Sehba, B, Stević, S: On some product-type operators from Hardy-Orlicz and Bergman-Orlicz spaces to weighted-type spaces. Appl. Math. Comput. 233, 565-581 (2014)
Stević, S: Essential norms of weighted composition operators from the α-Bloch space to a weighted-type space on the unit ball. Abstr. Appl. Anal. 2008, Article ID 279691 (2008)
Stević, S: Norm and essential norm of composition followed by differentiation from α-Bloch spaces to \(H_{\mu}^{\infty}\). Appl. Math. Comput. 207, 225-229 (2009)
Stević, S: On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces. Nonlinear Anal. 71(12), 6323-6342 (2009)
Stević, S: Products of composition and differentiation operators on the weighted Bergman space. Bull. Belg. Math. Soc. Simon Stevin 16, 623-635 (2009)
Stević, S: Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Sib. Math. J. 50(4), 726-736 (2009)
Stević, S: Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211(1), 222-233 (2009)
Stević, S: Weighted differentiation composition operators from \(H^{\infty}\) and Bloch spaces to nth weighted-type spaces on the unit disk. Appl. Math. Comput. 216, 3634-3641 (2010)
Stević, S, Sharma, AK, Bhat, A: Products of multiplication composition and differentiation operators on weighted Bergman spaces. Appl. Math. Comput. 217(20), 8115-8125 (2011)
Stević, S, Ueki, SI: Integral-type operators acting between weighted-type spaces on the unit ball. Appl. Math. Comput. 215, 2464-2471 (2009)
Zhu, X: Weighted composition operators from area Nevanlinna spaces into Bloch spaces. Appl. Math. Comput. 215, 4340-4346 (2010)
Acknowledgements
This work is supported by the NNSF of China (Nos. 11201127; 11271112) and IRTSTHN (14IRTSTHN023).
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Li, H., Guo, Z. On a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces. J Inequal Appl 2015, 132 (2015). https://doi.org/10.1186/s13660-015-0658-8
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DOI: https://doi.org/10.1186/s13660-015-0658-8