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On Stević-Sharma operator from the Zygmund space to the Bloch-Orlicz space
Advances in Difference Equations volume 2015, Article number: 228 (2015)
Abstract
Let \({\mathbb{D}}\) be the open unit disk in the complex plane \({\mathbb{C}}\), φ an analytic self-map of \({\mathbb{D}}\) and \(H({\mathbb{D}})\) the space of all analytic functions on \({\mathbb{D}}\). In order to unify the products of composition, multiplication, and differentiation operators, Stević and Sharma introduced the following so-called Stević-Sharma operator: \(T_{\psi_{1},\psi_{2},\varphi }f(z)=\psi_{1}(z)f(\varphi (z))+\psi_{2}(z)f'(\varphi (z))\), \(f\in H({\mathbb{D}})\), where \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). Here we characterize the boundedness and compactness of the operator \(T_{\psi_{1},\psi_{2},\varphi }\) from the Zygmund space to the Bloch-Orlicz space.
1 Introduction
Let \({\mathbb{D}}=\{z\in {\mathbb{C}}:|z|<1\}\) be the open unit disk in the complex plane \({\mathbb{C}}\) and \(H({\mathbb{D}})\) the class of all analytic functions on \({\mathbb{D}}\). Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi\in H({\mathbb{D}})\). The weighted composition operator \(W_{\varphi ,\psi}\) on \(H({\mathbb{D}})\) is defined by
If \(\psi\equiv1\), it becomes the composition operator, usually denoted by \(C_{\varphi }\). If \(\varphi (z)=z\), it becomes the multiplication operator, usually denoted by \(M_{\psi}\). Hence, since \(W_{\varphi ,\psi}=M_{\psi}C_{\varphi }\), it is a product-type operator. A standard problem is to provide function theoretic characterizations when φ and ψ induce a bounded or compact weighted composition operator (see, e.g., [1–5] and the references therein).
A systematic study of other product-type operators started by Stević et al. since the publication of papers [6] and [7]. Before that there were a few papers in the topic, e.g., [8]. The differentiation operator on \(H({\mathbb{D}})\) is defined by
The next two product-type operators \(D C_{\varphi }\) and \(C_{\varphi }D\), attracted some attention first (see, e.g., [9–12] and the references therein). The publication of [7] attracted some attention in product-type operators involving integral-type ones (see, e.g., [13–17] and the references therein). Since that time there has been a great interest in various product-type operators on spaces of holomorphic functions. For example, the six product-type operators from Bergman spaces to Bloch type spaces
were studied by Sharma in [18]. The next product-type operators \(W_{\varphi ,\psi}D\) and \(DW_{\varphi ,\psi}\), which were considered in [19] and [20], are included in (1) as the first and sixth operators, respectively. For some other product-type operators, see, e.g., [14, 21–29] and the references therein.
In order to treat operators in (1) in a unified manner, Stević and Sharma introduced the following so-called Stević-Sharma operator:
For example, in [30] and [31] the operator was studied on the weighted Bergman space.
By using Stević-Sharma operator all six possible products of composition, multiplication, and differentiation operators can be obtained. More specifically we have
Furthermore, by using this operator all possible difference operators of product-type operators in (1) can also be obtained. For example
etc., where \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). In this paper we characterize the boundedness and compactness of the Stević-Sharma operator from the Zygmund space to the Bloch-Orlicz space. As the applications of our main results, readers can obtain some characterizations for the boundedness and compactness for all six product-type operators in (1), as well as above mentioned differences operators from the Zygmund space to the Bloch-Orlicz space.
Now we present the needed spaces and some facts. For \(\alpha >0\), the weighted Zygmund space \(\mathcal{Z}_{\alpha }\) consists of all \(f\in H({\mathbb{D}})\) such that
It is a Banach space with the norm
When \(\alpha =1\), this space is the Zygmund space and is denoted by \(\mathcal{Z}\) [32]. From Zygmund’s theorem (see Theorem 5.3 in [33]), we know that \(f\in \mathcal{Z}\) if and only if f is continuous on \(\overline{{\mathbb{D}}}\) and
For some results on Zygmund-type spaces and some concrete operators on them, see, for example, [15, 23, 32] and the references therein.
Recently, the Bloch-Orlicz space was introduced in [4] by Ramos Fernández. More precisely, let Ψ be a strictly increasing convex function such that \(\Psi(0)=0\). From these conditions it follows that \(\lim_{t\to+\infty}\Psi(t)=+\infty\). The Bloch-Orlicz space associated with the function Ψ, denoted by \(\mathcal{B}^{\Psi}\), is the class of all \(f\in H({\mathbb{D}})\) such that
for some \(\lambda>0\) depending on f. The Minkowski functional
defines a seminorm for \(\mathcal{B}^{\Psi}\), where
Moreover, \(\mathcal{B}^{\Psi}\) is a Banach space with the norm
In fact, Ramos Fernández in [4] proved that \(\mathcal{B}^{\Psi}\) is isometrically equal to \(\mu_{\Psi}\)-Bloch space, where
Thus, for \(f\in \mathcal{B}^{\Psi}\) it follows that
This equivalent norm is useful to us for the study of operator \(T_{\psi_{1},\psi_{2},\varphi }\) from the Zygmund space to the Bloch-Orlicz space. It is obvious to see that if \(\Psi(t)=t^{p}\) with \(p>0\), then the space \(\mathcal{B}^{\Psi}\) coincides with the weighted Bloch space \(\mathcal{B}^{\alpha }\), where \(\alpha =1/p\). Also, if \(\Psi(t)=t\log(1+t)\), then \(\mathcal{B}^{\Psi}\) coincides with the Log-Bloch space (see [34]). For the generalization of the Log-Bloch spaces, see, for example, [35, 36].
Let X and Y be Banach spaces. It is said that a linear operator \(L:X\to Y \) is bounded if there exists a positive constant K such that
for all \(f\in X\). The operator \(L:X\rightarrow Y\) is said to be compact if it maps bounded sets into relatively compact sets. It is well known that the norm of operator \(L:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is defined by
and written by \(\|L\|\).
Throughout this paper, a positive constant C may differ from one occurrence to the other. The notation \(a\lesssim b\) means that there exists a positive constant C such that \(a\leq Cb\). When \(a\lesssim b\) and \(b\lesssim a\), we write \(a\simeq b\).
2 Main results and proofs
In order to characterize the compactness, we need the following result, which is proved in a standard way [5]. So, the proof is omitted.
Lemma 1
Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). Then the bounded operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact if and only if for every bounded sequence \(\{f_{j}\}_{j\in {\mathbb{N}}}\) in \(\mathcal{Z}\) such that \(f_{j}\to0\) uniformly on every compact subset of \({\mathbb{D}}\) as \(j\to\infty\), it follows that
We state the following useful result whose first estimate was essentially proved in [37], while the second essentially follows from the point evaluation estimate for the Bloch functions (see, e.g., [38]). See also [2].
Lemma 2
For each \(f\in \mathcal{Z}\) and \(z\in {\mathbb{D}}\), it follows that
The following lemma was proved in [37], Lemma 2.5.
Lemma 3
Let \(\{f_{j}\}_{j\in {\mathbb{N}}}\) be a bounded sequence in \(\mathcal{Z}\) which uniformly converges to zero on compact subsets of \({\mathbb{D}}\) as \(j\to\infty\). Then
For \(w\in {\mathbb{D}}\) and \(1/2<|w|<1\), we define the function
By using this function, the test functions in the Zygmund space can be obtained as follows:
From [9] we have the next result on the functions \(g_{w}\) and \(h_{w}\).
Lemma 4
Let \(w\in {\mathbb{D}}\) and \(1/2<|w|<1\). Then
Moreover,
Now we characterize the boundedness of the operator \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\).
Theorem 1
Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). Then the following statements are equivalent.
-
(i)
The operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded.
-
(ii)
The functions \(\psi_{1}\), \(\psi_{2}\), and φ satisfy the following conditions:
$$\begin{aligned}& M_{1}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z)\bigl| \psi_{1}'(z)\bigr|< \infty,\\& M_{2}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi_{1}(z)\varphi '(z)+\psi _{2}'(z) \bigr| \log\frac{e}{1-|\varphi (z)|^{2}}< \infty, \end{aligned}$$and
$$M_{3}:=\sup_{z\in {\mathbb{D}}}\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}}< \infty. $$
Moreover, if the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is nonzero and bounded, then
Proof
(i) ⇒ (ii). Suppose that \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded. For a fixed \(w\in {\mathbb{D}}\) and \(|\varphi (w)|>1/2\), let \(f(z)=h_{\varphi (w)}(z)-c_{1}+c_{2}\), where
Then by Lemma 4
By using the boundedness of \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi }\) to the function f, we have
from which we get
From (4) it follows that
Let \(h_{0}(z)\equiv1\in \mathcal{Z}\). Then by the boundedness of \(T_{\psi _{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\), we obtain
Considering \(h_{1}(z)=z\in \mathcal{Z}\), by the boundedness of \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) we have
From (6), (7), the boundedness of φ, and the triangle inequality, we obtain
Considering \(h_{2}(z)=z^{2}\in \mathcal{Z}\), we have
From (6), (8), (9), the boundedness of \(\varphi ^{2}\), and the triangle inequality, we get
Then from (10) we have
From (5) and (11) we finally have \(M_{3}<\infty\).
Now we prove that \(M_{2}<\infty\). For a fixed \(w\in {\mathbb{D}}\) and \(|\varphi (w)|>1/2\), let \(g(z)=g_{\varphi (w)}(z)-c_{1}\). Then
By using the boundedness of \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\), we have
From (4), (12), and the triangle inequality, it follows that
and then
From (8), we obtain
Hence, from (14) and (15) we have \(M_{2}<\infty\).
(ii) ⇒ (i). By Lemma 2, for all \(f\in \mathcal{Z}\) we have
It is clear that
Hence from (16) and (17) it follows that \(T_{\psi _{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded.
Suppose that the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is nonzero and bounded. Then from the proof of (i) ⇒ (ii) it is not hard to see that
Since the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is nonzero, we have \(\|T_{\psi_{1},\psi_{2},\varphi }\|>0\). From this we can find a positive constant C such that \(1\leq C\|T_{\psi_{1},\psi_{2},\varphi }\|\), which means that
Then combing (18) and (19) gives
It is clear from (16) and (17) that
Hence from (20) and (21) the asymptotic expression of \(\| T_{\psi_{1},\psi_{2},\varphi }\|\) follows. The proof is finished. □
Next we characterize the compactness of operator \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\).
Theorem 2
Let φ be an analytic self-map of \({\mathbb{D}}\) and \(\psi_{1},\psi_{2}\in H({\mathbb{D}})\). Then the following statements are equivalent.
-
(i)
The operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact.
-
(ii)
The functions \(\psi_{1}\), \(\psi_{2}\), and φ satisfy the following conditions:
$$\begin{aligned}& M_{1}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi'_{1}(z) \bigr|< \infty, \\& L_{1}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z) \bigl| \psi_{1}(z)\varphi '(z)+\psi _{2}'(z) \bigr|< \infty, \\& L_{2}:=\sup_{z\in {\mathbb{D}}}\mu_{\Psi}(z)\bigl| \psi_{2}(z)\bigr|\bigl|\varphi '(z)\bigr|< \infty , \\& \lim_{|\varphi (z)|\to1^{-}}\mu_{\Psi}(z) \bigl|\psi_{1}(z) \varphi '(z)+\psi _{2}'(z) \bigr|\log \frac{e}{1-|\varphi (z)|^{2}}=0, \end{aligned}$$and
$$\lim_{|\varphi (z)|\to1^{-}}\frac{\mu_{\Psi}(z)|\psi_{2}(z)||\varphi '(z)|}{1-|\varphi (z)|^{2}}=0. $$
Proof
(i) ⇒ (ii). Suppose that (i) holds. Then it is clear that the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded. In the proof of Theorem 1, we have shown that \(M_{1}<\infty\), \(L_{1}<\infty\) and \(L_{2}<\infty\). Consider a sequence \(\{\varphi (z_{i})\}_{i\in {\mathbb{N}}}\) in \({\mathbb{D}}\) such that \(|\varphi (z_{i})|\to1^{-}\) as \(i\to\infty\). If such a sequence does not exist, then the last two conditions (ii) obviously hold. We may suppose, without loss of generality, that \(|\varphi (z_{i})|>1/2\) for all \(i\in {\mathbb{N}}\). Using this sequence, we define the function sequence
Then from a calculation we see that \(\sup_{i\in {\mathbb{N}}}\|f_{i}\|_{\mathcal{Z}}\leq C\) and \(f_{i}\to0\) uniformly on every compact subset of \({\mathbb{D}}\) as \(i\to\infty\). So by Lemma 1
Moreover, we have
Hence we get
From this, Lemmas 1 and 3, and since \(M_{1}\) is finite, we obtain
On the other hand, take the sequence \(g_{i}(z)=g_{\varphi (z_{i})}(z)-c_{i}\), \(i\in {\mathbb{N}}\), where \(c_{i}=g_{\varphi (z_{i})}(\varphi (z_{i}))\). Then \(\sup_{i\in {\mathbb{N}}}\|g_{i}\|_{\mathcal{Z}}\leq C\),
Hence we have
By the compactness \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\), Lemma 1 and (22), we get
(ii) ⇒ (i). We first prove that \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded. We observe that the conditions in (ii) imply that for every \(\varepsilon>0\), there is an \(\eta\in(0,1)\), such that for any \(z\in K=\{z\in {\mathbb{D}}:|\varphi (z)|>\eta\}\)
and
From the fact \(L_{1}<\infty\) and (23), we obtain
From the fact \(L_{2}<\infty\) and (24), we also obtain
Hence from Theorem 1 it follows that the operator \(T_{\psi_{1},\psi _{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is bounded.
In order to prove that the operator \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact, by Lemma 1 we just need to prove that, if \(\{f_{i}\}_{i\in {\mathbb{N}}}\) is a sequence in \(\mathcal{Z}\) such that \(\sup_{i\in {\mathbb{N}}}\|f_{i}\|_{\mathcal{Z}}\leq M\) and \(f_{i}\to0\) uniformly on any compact subset of \({\mathbb{D}}\) as \(i\to\infty\), then
For such a chosen ε and η, by using (23), (24), and Lemma 2 we have
Since \(f_{i}\to\) uniformly on compact subsets of \({\mathbb{D}}\) as \(i\to\infty\) implies that for each \(k\in {\mathbb{N}}\), \(f_{i}^{(k)}\to0\) uniformly on compact subsets of \({\mathbb{D}}\) as \(i\to\infty\), from (25) and Lemma 3 we get
It is clear that
Hence from (27) and Lemma 1, we see that \(T_{\psi_{1},\psi_{2},\varphi }:\mathcal{Z}\to \mathcal{B}^{\Psi}\) is compact. The proof is finished. □
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Acknowledgements
The author would like to thank the anonymous referee very much for providing valuable suggestions for the improvement of this paper. This work was supported by the National Natural Science Foundation of China (No.11201323), the Key Fund Project of Sichuan Provincial Department of Education (No. 15ZA0221), the Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2013QZJ01) and the Cultivation Project of Sichuan University of Science and Engineering (No. 2015PY04).
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Jiang, Zj. On Stević-Sharma operator from the Zygmund space to the Bloch-Orlicz space. Adv Differ Equ 2015, 228 (2015). https://doi.org/10.1186/s13662-015-0567-7
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DOI: https://doi.org/10.1186/s13662-015-0567-7
MSC
- 47B38
- 47B33
- 47B37
Keywords
- Zygmund space
- Bloch-Orlicz space
- Stević-Sharma operator
- boundedness
- compactness