Abstract
Recently we have introduced a product-type operator and studied it on some spaces of analytic functions on the unit disc. Here we start investigating the operator on the space of analytic functions on the upper half-plane. We characterize the boundedness and compactness of the operator between Hardy and α-Bloch spaces on the domain.
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1 Introduction
Let \({\mathbb {D}}\) be the unit disc in the complex plane \({\mathbb{C}}\), \(\Pi^{+}=\{z \in \mathbb{C} : \Im z > 0\}\) the upper half-plane in \({\mathbb {C}}\), and \(\widehat{\Pi ^{+}}=\overline{\Pi^{+}} \cup \{\infty \}\). Let Ω be a domain in \({\mathbb {C}}\). We denote by \(H(\Omega )\) the space of all analytic functions on Ω and by \(S(\Omega )\) the class of all analytic self-maps of Ω.
For \(0< p<\infty \), the Hardy space of \(\Pi^{+}\), denoted by \(H^{p}(\Pi^{+})\), consists of all \(f\in H(\Pi^{+})\) such that
For \(1 \leq p < \infty \), \(H^{p}(\Pi^{+})\) is a Banach space.
Let \(\alpha >0\). The α-Bloch or Bloch-type space on \(\Pi^{+}\), denoted by \(\mathcal{B}^{\alpha }(\Pi^{+})\), consists of all \(f \in H(\Pi^{+})\) such that
With the norm (1), the α-Bloch space is a Banach space. For Bloch-type spaces on various domains and operators on them, see, for example, [1–13] and the references therein. For a natural extension of Bloch-type spaces and for Zygmud-type spaces, see [14–16].
For \(\varphi \in S(\Omega )\), the composition operator \(C_{\varphi }\) is the linear operator defined by
For \(\psi \in H(\Omega )\), the multiplication operator \(M_{\psi }\) is defined on \(H(\Omega )\) by
The product of these two operators
is the so-called weighted composition operator.
By D we denote the differentiation operator, that is,
These concrete operators, along with some integral-type ones, are among those considerably studied, during the last five decades, on spaces of analytic functions on various domains in \({\mathbb {C}}\) or domains in the complex-vector space \({\mathbb {C}}^{n}\). Majority of the papers on the operators are devoted to investigating them on spaces of analytic functions on \({\mathbb {D}}\). Much less papers consider the operators on spaces of analytic functions on other domains, including the upper half-plane. Even for some popular operators, such as the composition ones, up to the end of the previous century, there are no many papers on popular spaces such as \(H^{p}(\Pi^{+})\) (see, e.g., [17–20] and the references therein). Hence, any new result on the spaces and operators on \(\Pi^{+}\) is of some interest. Regarding some operators on the mentioned spaces, let us mention that some basic results on the boundedness of composition operators from \(H^{p}(\Pi^{+})\) to the classical Bloch space \(\mathcal{B}(\Pi^{+})\) can be found in note [21]. For related investigations of composition or weighted composition operators on some other spaces, see [14, 15, 22–25]. Let us mention that the behavior of composition operators on spaces of analytic functions in \(\Pi^{+}\) is considerably different from the behavior of composition operators on spaces of analytic functions in \(\mathbb{D}\). For example, every analytic self-map of \(\mathbb{D}\) induces a bounded composition operator on the corresponding Hardy space, which is not always the case on the space \(H^{p}(\Pi^{+})\) [17, 18].
From 1968 to 2005, experts more or less studied theoretic properties of only operators (2)–(5) and integral-type ones on spaces of analytic functions in terms of their symbols. The only product-type operator among (2)–(5) is the weighted composition operator. Since 2005, some experts started studying some other product-type operators. The first product-type operators different from weighted composition ones that attracted some attention were the products of composition and differentiation operators (see, e.g., [7, 11, 13, 26–29] and the references therein). Some generalizations of the products of composition and differentiation operators, containing iterated differentiation, have appeared soon after them (see [12, 30–33]). Around 2008, Li and Stević have initiated studying products of integral and composition operators, including in some cases the differentiation operator, on spaces of analytic functions on \({\mathbb {D}}\) (see, e.g., [34]), which have been considerably studied recently (see, e.g., [3, 35–40]). For some other results and related product-type operators, see, for example, [4–6, 35, 41–46].
To treat product-type operators consisting of exactly one composition, multiplication and differentiation operator in a unified manner, we have recently introduced a generalized operator and studied it on the weighted Bergman spaces (see [45] and [46]). See also papers [5, 42] on the operator on some other spaces of functions defined by
where \(\psi_{1}, \psi_{2} \in H(\mathbb{D})\) and \(\varphi \in S( \mathbb{D})\).
It is clear that operator (6) includes the composition, multiplication, differentiation, weighted composition, weighted differentiation composition, and many other concrete operators, including those in [6, 41, 43], which are obtained by some concrete choices of the symbols \(\psi_{1}\), \(\psi_{2}\), and φ. This is one of the reasons why the operator is of some importance for investigation.
So far the operator has not been considered between spaces of analytic functions on the upper half-plane. Here we start investigating the operator on such spaces by characterizing the boundedness and compactness of the operator between the Hardy and α-Bloch spaces on the domain, that is, of \(T_{\psi_{1} , \psi_{2} , \varphi }:H^{p}(\Pi ^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\). We provide a complete characterization of the compactness. The paper can be regarded as a continuation of our investigations in [14, 15, 22–25].
Throughout this paper, constants are denoted by C; they are positive and not necessarily the same at each occurrence. The notation \(A \asymp B\) means that \(B \lesssim A \lesssim B\), where \(A \lesssim B\) means that there is a positive constant C such that \(A \leq C B\).
2 Auxiliary results
First, we quote a point evaluation lemma, which is a folklore result (see, e.g., [15, Lemma 3]).
Lemma 1
Let \(p\in (0,\infty )\) and \(n\in {\mathbb {N}}_{0}\). Then
for some positive constant \(C=C(p,n)\) independent of f.
The following lemma can be found in [14, Lemma 2.1].
Lemma 2
Let \(p\in [1,\infty )\), \(k\in {\mathbb {N}}_{0}\), \(\zeta \in \Pi^{+}\), and
Then
To deal with the compactness of the operator \(T_{\psi_{1},\psi_{2}, \varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\), in the lemma that follows, we give its characterization, which is typical for concrete operators between spaces of analytic functions. The thesis of Schwartz [47] is one of the first sources that presents such a characterization for the case of a concrete operator, more precisely, for a composition operator. It is interesting that the proof of our present lemma is more complicated than those of the corresponding lemmas that we have had so for (for example, the lemma in [24]). Hence we will present a detailed proof of the lemma.
We say that a sequence \((f_{n})_{n\in {\mathbb {N}}}\) in \(H^{p}(\Pi^{+})\) converges weakly to zero if it is norm bounded in \(H^{p}(\Pi^{+})\) and converges to zero on compacts of \(\Pi^{+}\).
Lemma 3
Let \(p\ge 1\), \(\alpha \in (0,+\infty )\), \(\psi_{1},\psi_{2}\in H(\Pi^{+})\), and \(\varphi \in S(\Pi^{+})\). Then \(T_{\psi_{1},\psi_{2},\varphi }:H ^{p}(\Pi^{+})\rightarrow \mathcal{B}^{\alpha }( \Pi^{+})\) is compact if and only if for any sequence \((f_{n})_{n\in {\mathbb {N}}}\) weakly convergent to zero, we have
Proof
Let the operator be compact. Suppose that \((f_{n})_{n \in {\mathbb {N}}}\) weakly converges to zero. Then there are a subsequence \((f_{n_{k}})_{k\in {\mathbb {N}}}\) and \(g\in \mathcal{B}^{\alpha }(\Pi^{+})\) such that
Let \(K\subset \Pi^{+}\) be compact. Then
From this and from (1) we have
and
Let \(A\subset \Pi^{+}\) and \(A_{\varepsilon }=\{z\in \Pi^{+}:d(z,K)\le \varepsilon \}\), where \(\varepsilon >0\). Note that if A is compact, then for each \(\varepsilon < d _{K}\), the set \(A_{\varepsilon }\) is also a compact set as bounded and closed.
On the other hand, we have
for all \(f\in H(\Pi^{+})\) and \(z\in \Pi^{+}\).
Let \(A\subset \Pi^{+}\) and
Note that if A is compact, then \(A(i)\) is also compact.
Hence from (11) we easily get
for each \(z\in K\).
From (9), (10), and (12) it follows that
From (8) and (13) it follows that
on each compact \(K\in \Pi^{+}\).
Further, note that
for each compact K. The compactness of the set \(\varphi (K)\) also implies that
on each compact \(K\in \Pi^{+}\).
From this, since
it follows that
on each compact \(K\in \Pi^{+}\).
From (14) and (15) we obtain \(g(z)=0\) for every \(z\in \Pi^{+}\), since each \(z\in \Pi^{+}\) lies in a compact subset of \(\Pi^{+}\).
Using the fact in (8), it follows that
Such a procedure can be applied to any subsequence of \((f_{n})_{n \in {\mathbb {N}}}\), from which it follows that (7) holds.
Now assume that, for any sequence \((f_{n})_{n\in {\mathbb {N}}}\) weakly convergent to zero, we have \(\lim_{n\rightarrow \infty } \Vert T_{\psi_{1}, \psi_{2},\varphi } f_{n} \Vert _{\mathcal{B}^{\alpha }(\Pi^{+})}=0\). Let \((\widehat{f}_{n})_{n\in {\mathbb {N}}}\) be a sequence of functions such that \(M:=\sup_{n\in {\mathbb {N}}} \Vert \widehat{f}_{n} \Vert _{H^{p}(\Pi^{+})}<+\infty \). By Lemma 1 the sequence is uniformly bounded on compacts of \(\Pi^{+}\), and consequently normal. Hence there are a subsequence \((\widehat{f}_{n _{k}})_{k\in {\mathbb {N}}}\) and \(\widehat{f}\in H(\Pi^{+})\) such that
on each compact \(K\in \Pi^{+}\). The Fatou lemma along with (17) implies \(\Vert \widehat{f} \Vert _{H^{p}(\Pi^{+})} \leq M\). Hence, the sequence \((\widehat{f}_{n_{k}}-\widehat{f})_{k\in {\mathbb {N}}}\) weakly converges to zero, and consequently
from which the compactness of the operator \(T_{\psi_{1},\psi_{2}, \varphi }:H^{p}(\Pi^{+})\rightarrow \mathcal{B}^{\alpha }( \Pi^{+})\) follows. □
3 Boundedness and compactness of \(T_{\psi_{1},\psi_{2},\varphi}:H^{p}(\Pi^{+})\to\mathcal{B}^{\alpha}(\Pi^{+})\)
In this section, we characterize the boundedness and compactness of the operator \(T_{\psi_{1} , \psi_{2} , \varphi }:H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\). We also give upper and lower bounds for the norm of \(T_{\psi_{1} , \psi_{2} , \varphi }\) acting between these spaces.
Theorem 4
Let \(1\leq p <\infty \), \(\alpha >0\), \(\psi_{1}, \psi_{2} \in H(\Pi ^{+})\) and \(\varphi\in S(\Pi^{+})\). Then \(T_{\psi_{1},\psi_{2},\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{ \alpha }(\Pi^{+})\) is bounded if and only if the following conditions are satisfied:
Moreover,
Proof
First, suppose that conditions (i)–(iii) hold. Then by Lemma 1 we have
We also have
Combining (19) and (20), we have
from which it follows that
Conversely, suppose that \(T_{\psi_{1},\psi_{2},\varphi } : H^{p}(\Pi ^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\) is bounded. Consider the family of functions
where \(w\in \Pi^{+}\).
Since the functions \(f_{w}\) are linear combinations of the functions in Lemma 2 (for \(k=2,3,4\)), from this by using the lemma it follows that
We also have
from which with some simple calculation we obtain
Since \(T_{\psi_{1},\psi_{2},\varphi }: H^{p}(\Pi^{+})\to \mathcal{B} ^{\alpha }(\Pi^{+})\) is bounded, we have
for every \(w\in \Pi^{+}\).
Thus for each \(\zeta \in \Pi^{+}\), we have
Since \(\zeta \in \Pi^{+}\) is arbitrary, we have that
Now, consider the family of functions
Since the functions \(g_{w}\) are also linear combinations of the functions in Lemma 2, we have \(L_{2}:=\sup_{w\in \Pi^{+}} \Vert g_{w} \Vert _{H^{p}(\Pi^{+})} \le 1\).
We also have
from which it follows that
Since \(T_{\psi_{1},\psi_{2},\varphi }: H^{p}(\Pi^{+})\to \mathcal{B} ^{\alpha }(\Pi^{+})\) is bounded, for each \(\zeta \in \Pi^{+}\), we have
Since \(\zeta \in \Pi^{+}\) is arbitrary, we have that
Now, consider the family of functions
Once again proceeding as before, we can show that \(\sup_{w\in \Pi^{+}} \Vert h_{w} \Vert _{H^{p}(\Pi^{+})} \lesssim 1\).
We have
Thus
Therefore by the boundedness of \(T_{\psi_{1},\psi_{2},\varphi }: H ^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\), for each \(\zeta \in \Pi^{+}\), we have
and consequently
Combining (24), (25), and (26), we have
finishing the proof of the theorem. □
Corollary 5
Let \(1\le p<\infty\), \(\alpha >0\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }: H^{p}(\Pi^{+}) \to \mathcal{B}^{\alpha }(\Pi^{+})\) is bounded if and only if
Moreover,
Corollary 6
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(M_{\psi } C_{\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Corollary 7
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi } M_{\psi } : H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Corollary 8
Let \(1\le p<\infty\), \(\alpha >0\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }D: H^{p}(\Pi^{+}) \to \mathcal{B}^{\alpha }(\Pi^{+})\) is bounded if and only if
Moreover,
Corollary 9
Let \(1\le p<\infty\), \(\alpha >0\) and \(\varphi \in S(\Pi^{+})\). Then \(DC_{\varphi }: H^{p}(\Pi^{+}) \to \mathcal{B}^{\alpha }(\Pi^{+})\) is bounded if and only if
Moreover,
Corollary 10
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(M_{\psi }C_{\varphi }D: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Corollary 11
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }M_{\psi }D: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Corollary 12
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(M_{\psi }D C_{\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Corollary 13
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(D M_{\psi }C_{\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Corollary 14
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }D M_{\psi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Corollary 15
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(D C_{\varphi }M_{\psi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is bounded if and only if
Moreover,
Recall that for a function f defined in \(\Pi^{+}\), \(\lim_{z \to \partial \widehat{\Pi ^{+}}}f(z)=0\) if and only if for every \(\varepsilon >0\), there is a compact set \(K\subset \Pi^{+}\) such that \(\vert f(z) \vert <\varepsilon \) for \(z\in \Pi^{+} \setminus K\).
The following result characterizes the compactness of the operator \(T_{\psi_{1},\psi_{2},\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{ \alpha }(\Pi^{+})\).
Theorem 16
Let \(1 \leq p < \infty \), \(\alpha >0\), \(\psi_{1},\psi_{2} \in H(\Pi^{+})\), and \(\varphi \in S(\Pi^{+})\). Then \(T_{\psi_{1},\psi_{2},\varphi }: H ^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\) is compact if and only if it is bounded and the following conditions are satisfied:
Proof
Assume that the operator is bounded and conditions (i)–(iii) hold. Then by Theorem 4 the quantities \(M_{j}\), \(j=\overline{1,3}\), are finite, whereas from conditions (i)–(iii) we have that, for each \(\varepsilon >0\), there exists a compact set K in \(\Pi^{+}\) such that
provided that \(\varphi (z) \in \Pi^{+} \setminus K\).
Let \((f_{n})_{n\in {\mathbb {N}}}\) be a sequence in \(H^{p}(\Pi^{+})\) weakly convergent to zero. Then, by inequalities (35)–(37) and Lemma 1, for every \(z \in \Pi^{+}\) with \(\varphi (z) \in \Pi^{+} \setminus K\), we have
Since K is compact, \(\varphi (K)\) is also compact. Hence
Choose \(n_{0} \in \mathbb{N}\) such that
and
for all \(n \geq n_{0}\).
Then, by the preceding and (38), for every \(z\in \Pi^{+}\) such that \(\varphi (z)\in K\), we have
Further, using (39), we have
for \(n\ge n_{0}\).
From (40) and (41) it follows that
for \(n\ge n_{0}\).
Since \(\varepsilon > 0\) is arbitrary, by Lemma 3 it follows that \(T_{\psi_{1}, \psi_{2},\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{ \alpha }(\Pi^{+})\) is compact.
Conversely, suppose that \(T_{\psi_{1}, \psi_{2},\varphi }: H^{p}(\Pi ^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\) is compact. Then, clearly, it is bounded. Let \((z_{n})_{n\in {\mathbb {N}}}\) be a sequence in \(\Pi^{+}\) such that \(\varphi (z_{n})\to \widehat{\partial \Pi^{+}}\) as \(n \to \infty \). If such a sequence does not exist, then conditions (i)–(iii) vacuously hold.
Let \(w_{n} = \varphi (z_{n})\), \(n\in {\mathbb {N}}\), and let the family of functions \((f_{n})_{n\in {\mathbb {N}}}\) be defined by
Then by using (22) and some simple estimates it is easy to see that \((f_{n})_{n\in {\mathbb {N}}}\) weakly converges to zero as \(n\to \infty \) (including the case when \(w_{n}\to\infty\) as \(n\to\infty\)). Hence
The boundedness of \(T_{\psi_{1}, \psi_{2},\varphi }: H^{p}(\Pi^{+}) \to \mathcal{B}^{\alpha }(\Pi^{+})\), together with (23), implies
from which it follows that condition (i) holds.
By considering families of functions
and
and proceeding as before, we can similarly show that
and
from which it follows that conditions (ii) and (iii) hold, respectively. □
Corollary 17
Let \(1\le p<\infty\), \(\alpha >0\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }: H^{p}(\Pi^{+}) \to \mathcal{B}^{\alpha }(\Pi^{+})\) is compact if and only if it is bounded and
Corollary 18
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(M_{\psi}C_{\varphi}: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi^{+})\) is compact if and only if it is bounded and
Corollary 19
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi } M_{\psi } : H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is compact if and only if it is bounded and
Corollary 20
Let \(1\le p<\infty\), \(\alpha >0\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }D: H^{p}(\Pi^{+}) \to \mathcal{B}^{\alpha }(\Pi^{+})\) is compact if and only if it is bounded and
Corollary 21
Let \(1\le p<\infty\), \(\alpha >0\) and \(\varphi \in S(\Pi^{+})\). Then \(DC_{\varphi }: H^{p}(\Pi^{+}) \to \mathcal{B}^{\alpha }(\Pi^{+})\) is compact if and only if it is bounded and
Corollary 22
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(M_{\psi }C_{\varphi }D: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is compact if and only if it is bounded and
Corollary 23
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }M_{\psi }D: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is compact if and only if it is bounded and
Corollary 24
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(M_{\psi }D C_{\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is compact if and only if it is bounded and
Corollary 25
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(D M_{\psi }C_{\varphi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is compact if and only if it is bounded and
Corollary 26
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(C_{\varphi }D M_{\psi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is compact if and only if it is bounded and
Corollary 27
Let \(1\le p<\infty\), \(\alpha >0\), \(\psi \in H(\Pi^{+})\) and \(\varphi \in S(\Pi^{+})\). Then \(D C_{\varphi }M_{\psi }: H^{p}(\Pi^{+})\to \mathcal{B}^{\alpha }(\Pi ^{+})\) is compact if and only if it is bounded and
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The study in the paper is a part of the investigation at the projects III 41025 and III 44006 by the Serbian Ministry of Education and Science.
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Stević, S., Sharma, A.K. On a product-type operator between Hardy and α-Bloch spaces of the upper half-plane. J Inequal Appl 2018, 273 (2018). https://doi.org/10.1186/s13660-018-1867-8
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DOI: https://doi.org/10.1186/s13660-018-1867-8