Composition and multiplication operators between Orlicz function spaces
- Tadeusz Chawziuk^{1},
- Yunan Cui^{2}Email author,
- Yousef Estaremi^{3},
- Henryk Hudzik^{1} and
- Radosław Kaczmarek^{1}
https://doi.org/10.1186/s13660-016-0972-9
© Chawziuk et al. 2016
Received: 11 August 2015
Accepted: 14 January 2016
Published: 11 February 2016
Abstract
Composition operators and multiplication operators between two Orlicz function spaces are investigated. First, necessary and sufficient conditions for their continuity are presented in several forms. It is shown that, in general, the Radon-Nikodým derivative \(\frac{d(\mu\circ\tau^{-1})}{d\mu}(s)\) need not belong to \(L^{\infty}(\Omega)\) to guarantee the continuity of the composition operator \(c_{\tau}x(t)=x(\tau(t))\) from \(L^{\Phi}(\Omega)\) into \(L^{\Psi}(\Omega)\). Next, the problem of compactness of these operators is considered. We apply a compactness criterion in Orlicz spaces which involves compactness with respect to the topology of local convergence in measure and equi-absolute continuity in norm of all the elements of the set under consideration. In connection with this, we state some sufficient conditions for equi-absolute continuity of the composition operator \(c_{\tau}\) and the multiplication operator \(M_{w}\) from one Orlicz space into another. Also the problem of necessary conditions is discussed.
Keywords
MSC
1 Introduction
Since the early 1930s composition operators have been a subject of study of many mathematicians or physicists. At the beginning they were used to solve problems in mathematical physics and classical mechanics [1, 2], or to study ergodic transformations [3, 4]. Up until now many Ph.D. theses have been defended (for instance: Boyd [5], Gupta [6], Ridge [7], Schwartz [8], Singh [9], Swanton [10], Veluchamy [11]), numerous books published [12, 13], and innumerable papers printed on the composition operator or the weighted composition operator in various function spaces, e.g., \(L^{p}\) spaces ([13–23], and others), Orlicz spaces ([24–27], and others), Musielak-Orlicz spaces [28, 29], Musielak-Orlicz spaces of Bochner type ([30] or [31]), Orlicz-Lorentz spaces [32–37], Hilbert spaces [38–40] and many other types of spaces (for instance: [41–43]). The multiplication operator has also been a subject of research of many mathematicians (see for instance: [44–49]). For more details as regards the historical background we refer the reader to [13].
Let \(L^{0}(\Omega)=L^{0}(\Omega,\Sigma,\mu)\) be the space of all (abstract classes of) Σ-measurable functions from Ω into \(\mathbb {R}\) with respect to the equivalence relation: \(x\sim y\) if and only if \(x(t)=y(t)\) for μ-a.e. \(t\in\Omega\).
It is obvious that \(c_{\tau}x\in L^{0}(\Omega)\) and \(M_{w} x \in L^{0}(\Omega )\) if \(x\in L^{0}(\Omega)\).
Remark 1.1
We do not assume, if not specifically stated otherwise, that the mapping τ is a surjection, i.e., \(\tau(\Omega)=\Omega\).
A function \(\Phi:\mathbb{R}\rightarrow\mathbb{R}_{+}=[0,\infty)\) is said to be an Orlicz function if Φ is convex, even, continuous, vanishing only at 0.
The complementary function in the sense of Young to an Orlicz function Φ is defined to be the function \(\Phi^{*}:[0,\infty)\rightarrow [0,\infty]\) such that \(\Phi^{*}(u)=\sup_{v>0} \{uv-\Phi(v)\}\).
Throughout the paper, we will make use of Ishii’s theorem from [56].
Theorem 1.1
For any \(A\in\Sigma\), by \(I_{\Phi}(x,A)\) we mean the value of the modular \(I_{\Phi}\) at x in the Orlicz space \(L^{\Phi}(A)\) generated by the Orlicz function Φ over the measure space \((A,\Sigma\cap A,\mu|_{A})\). In the case when \(A=\Omega\), we will write \(I_{\Phi}(x)\) instead of \(I_{\Phi}(x,\Omega)\).
2 Continuity of the composition operator \(c_{\tau}\) from \(L^{\Phi}\) into \(L^{\Psi}\) and from \(L^{\Phi}\) onto \(L^{\Psi}\)
We are interested in finding necessary and sufficient conditions for the continuity of the composition operator \(c_{\tau}\) from the Orlicz space \(L^{\Phi}(\Omega)=L^{\Phi}(\Omega,\Sigma,\mu)\) equipped with the Luxemburg norm into the Orlicz space \(L^{\Psi}(\Omega)=L^{\Psi}(\Omega ,\Sigma,\mu)\) endowed with the corresponding Luxemburg norm. The following fact will be very helpful.
Fact 2.1
For an arbitrary function \(x\in L^{0}(\Omega)\), we have \(c_{\tau}x\in L^{\Psi}(\Omega)\) if and only if \(x\in L^{\Psi}_{h}(\tau(\Omega))\), where \(L^{\Psi}_{h}(\tau(\Omega))=L^{\Psi}_{h}(\tau(\Omega),\Sigma\cap\tau(\Omega), \mu|_{\Sigma\cap\tau(\Omega)})\) is a weighted Orlicz space with the weight function \(h(s)=\frac{d\mu\circ\tau^{-1}}{d\mu}(s)\).
Proof
Theorem 2.1
Moreover, if \(\mu(\Omega\backslash\tau(\Omega))=0\), then condition (1) is necessary for the continuity of the composition operator \(c_{\tau}\) from \(L^{\Phi}(\Omega)\) into \(L^{\Psi}(\Omega)\).
Proof
We will show that if condition (1) is satisfied then \(c_{\tau}x\in L^{\Psi}(\Omega)\) whenever \(x\in L^{\Phi}(\Omega)\), i.e., \(x\in L^{\Psi}_{h}(\tau(\Omega))\) whenever \(x\in L^{\Phi}(\Omega)\) (see Fact 2.1).
Now assume that \(\mu(\Omega\backslash\tau(\Omega))=0\). If the inclusion in the assumption of the theorem fails to hold, then there exists a function x belonging to \(L^{\Phi}(\tau(\Omega))=L^{\Phi}(\Omega)\) but not belonging to \(L^{\Psi}_{h}(\tau(\Omega))\). In virtue of Fact 2.1, we obtain \(x\in L^{\Phi}(\Omega)\) and, simultaneously, \(c_{\tau}x \notin L^{\Psi}(\Omega)\), hence \(c_{\tau}\) does not even act from \(L^{\Phi}(\Omega)\) into \(L^{\Psi}(\Omega)\). □
The preceding theorem can be formulated in a different language, which in some situations might be more useful.
Theorem 2.2
Proof
Remark 2.1
Notice that if \(b(\chi):=\sup\{u\geq0\colon\chi(u)<\infty\}<\infty\) then condition (2) implies that \(\Vert h\Vert _{L^{\infty}(\tau(\Omega))}\leq b(\chi)<\infty\), that is, the Radon-Nikodým derivative \(h=\frac{d\mu\circ\tau^{-1}}{\,}d\mu\) is essentially bounded. Moreover, it is easy to see that the integral (2) can be finite for some \(h\notin L^{\infty}(\tau(\Omega))\) if and only if the function χ has only finite values (for the case when \(\Phi=\Psi\) see [50]; the proof in the case when \(\Phi\neq\Psi\) is similar).
The next theorem states a necessary and sufficient condition in order that χ is such a function.
Theorem 2.3
The function \(\chi=(\Phi\circ K\Psi^{-1})^{*}\), with \(K>1\), assumes only finite values (i.e., \(b(\chi)=\infty\)) if and only if \(\liminf_{t\rightarrow\infty}\frac{\Phi(Kt)}{\Psi(t)}=\infty\).
Proof
Now we show that if the function χ from Theorem 2.2 assumes only finite values, i.e., \(b(\chi)=\infty\), then it may happen that the composition operator \(c_{\tau}\) from \(L^{\Phi}(\Omega)\) into \(L^{\Psi}(\Omega )\) is continuous despite the fact that \(h=\frac{d\mu\circ\tau^{-1}}{d\mu }\notin L^{\infty}(\tau(\Omega))\).
Example 2.1
Theorem 2.4
- (i)
\(\exists_{K>1} \mathop{\exists_{A\in\Sigma\cap \Omega}}\limits_{\mu(A)=0} \exists_{g\in L^{1}_{+}(\Omega)} \forall_{s\in\Omega\backslash A} \forall_{u\geq0} \Psi(u)h(s)\leq\Phi(Ku)+g(s)\);
- (ii)
\(\exists_{K>1} \mathop{\exists_{A\in\Sigma\cap \Omega}}\limits_{\mu(A)=0} \exists_{p\in L^{1}_{+}(\Omega)} \forall_{s\in\Omega\backslash A} \forall_{u\geq0} \Phi(u)\leq\Psi(Ku)h(s)+p(s)\),
Proof
Obviously, the condition \(\mu(\Omega\backslash\tau(\Omega))=0\) is necessary for \(c_{\tau}(L^{\Phi}(\Omega))=L^{\Psi}(\Omega)\), so in the further part of the proof we assume this condition holds. Therefore, by Fact 2.1, we know that \(c_{\tau}\) acts from \(L^{\Phi}(\Omega)\) onto \(L^{\Psi}(\Omega)\) if and only if \(L^{\Phi}(\Omega)=L^{\Psi}_{h}(\Omega)\). Equivalently, this holds if and only if we have two inclusions: \(L^{\Phi}(\Omega) \subset L^{\Psi}_{h}(\Omega)\) and \(L^{\Psi}_{h}(\Omega)\subset L^{\Phi}(\Omega)\). The first inclusion holds if and only if condition (i) is satisfied and the reverse inclusion holds if and only if condition (ii) is satisfied (see [51] and [56]), and this finishes the proof. □
It is interesting and profitable to observe that the preceding theorem can be written in the following form.
Theorem 2.5
- (1)
\(\int_{\Omega} \chi(h(s) )\,d\mu(s)< \infty\);
- (2)
\(\int_{\Omega} h(s)q (\frac{1}{h(s)} )\,d\mu(s)<\infty\),
Proof
In the proof of Theorem 2.2 we already showed that condition (i) from Theorem 2.4 is equivalent to condition (1). So, the proof will be finished if we show that condition (2) is equivalent to condition (ii) from Theorem 2.4.
3 Continuity of the multiplication operator \(M_{w}\) from \(L^{\Phi}\) into \(L^{\Psi}\) and from \(L^{\Phi}\) onto \(L^{\Psi}\)
We will state criteria in order that \(M_{w}\) map \(L^{\Phi}(\Omega, \Sigma ,\mu)\) into \(L^{\Psi}(\Omega, \Sigma,\mu)\), where Φ and Ψ are distinct Orlicz functions. Note that \(M_{w} x \in L^{\Psi}(\Omega, \Sigma ,\mu)\) means that there is \(\lambda>0\) such that \(I_{\Psi}(\lambda w(t)x(t) )=\int_{\Omega} \Psi(\lambda w(t)x(t) )\,d\mu (t)<\infty\). This is equivalent to the fact that \(x\in L^{\Psi_{w}} (\Omega, \Sigma,\mu)\), where \(L^{\Psi_{w}} (\Omega, \Sigma,\mu)\) is a Musielak-Orlicz space generated by the Musielak-Orlicz function \(\Psi_{w} (t,u):=\Psi(w(t)u )\). Let us begin with the following.
Theorem 3.1
The multiplication operator \(M_{w}\) maps \(L^{\Phi}(\Omega, \Sigma,\mu)\) into \(L^{\Psi}(\Omega, \Sigma,\mu)\) if and only if \(\int_{\Omega}\chi_{K} (t,1)\,d\mu(t)<\infty\) for some \(K>1\), where \(\chi_{K} (t,u)\) is, for fixed \(t\in\Omega\), the function complementary in the sense of Young to the function \(\Phi\circ\frac{K}{w(t)} \Psi^{-1}\) with respect to u.
Proof
Theorem 3.2
- (i)
\(\int_{\Omega}\chi_{K} (t,1)\,d\mu(t)< \infty\) for some \(K>1\), where \(\chi_{K} (t,u)\) is the function defined in Theorem 3.1;
- (ii)
\(\int_{\Omega}U_{K} (t,1)\,d\mu(t)< \infty\) for some \(K>1\), where \(U_{K} (t,u)\) is, for fixed \(t\in\Omega\), the function complementary in the sense of Young, with respect to u, to the function \((\Psi\circ Kw(\cdot)\Phi^{-1} ) (t,u)=\Psi(Kw(t)\Phi^{-1}(u) )\).
Proof
4 Compactness of the composition operator \(c_{\tau}\) from one Orlicz space into another
We begin with some notions that will be useful in the following. Let \((\Omega,\Sigma, \mu)\) be a non-atomic, complete and σ-finite measure space. We say that functions in a set A contained in the Musielak-Orlicz space \(L^{\Phi}(\Omega)\) have equi-absolutely continuous norms if for any real number \(\varepsilon>0\) there exist a set \(B_{\varepsilon}\in\Sigma\) with \(\mu(B_{\varepsilon})<\infty\) and a real number \(\delta=\delta(\varepsilon)>0\) such that for any function \(x\in A\) we have \(\|x \chi_{\Omega\setminus B_{\varepsilon}}\|_{\Phi}<\varepsilon \) and \(\|x\chi_{B}\|_{\Phi}<\varepsilon\) whenever \(B\in\Sigma\cap B_{\varepsilon}\) and \(\mu(B)< \delta\).
Let \(L^{\Phi}(\Omega)=L^{\Phi}(\Omega, \Sigma, \mu)\) and \(L^{\Psi}(\Omega )=L^{\Psi}(\Omega, \Sigma, \mu)\) be distinct Orlicz spaces. We say that the operator \(T: L^{\Phi}(\Omega) \rightarrow L^{\Psi}(\Omega)\) is equi-absolutely continuous if for any bounded set \(A \subset L^{\Phi}(\Omega)\) all functions of the set \(T (A)\subset L^{\Psi}(\Omega)\) have equi-absolutely continuous norms.
We will make use of the following theorem which gives necessary and sufficient conditions for the relative compactness of a set of functions in a Musielak-Orlicz space.
Theorem 4.1
(Theorem 1.2 in [57])
Let \((\Omega,\Sigma,\mu)\) be a non-atomic σ-finite measure space and let φ be a Musielak-Orlicz function. If the functions in a set \(A\subset L^{\varphi}(\Omega)\) all have equi-absolutely continuous norms and A is relatively compact with respect to local convergence in measure, then A is relatively compact in \(E^{\varphi}(\Omega)\), the subspace of absolutely continuous functions in \(L^{\varphi}(\Omega)\).
Conversely, if a set \(A\subset E^{\varphi}(\Omega)\) is relatively compact, then all the functions in A have equi-absolutely continuous norms and A is relatively compact with respect to local convergence in measure.
In the proof of the forthcoming theorem we will need the following.
Lemma 4.1
(Lemma 8.3 in [51])
The following theorem will be of great importance in proving necessary and sufficient conditions for the compactness of the composition operator \(c_{\tau}: L^{\Phi}(\Omega) \rightarrow L^{\Psi}(\Omega)\).
Theorem 4.2
Condition (8) is necessary for the equi-absolute continuity of \(c_{\tau}\) if \(\mu(\Omega)<\infty\).
Proof
Sufficiency. First we prove that for any \(\varepsilon>0\) there exists a set \(D\in\Sigma\) with \(\mu(\Omega\setminus D)<\infty\) such that all the functions in the set \(\{c_{\tau}x: x\in S(L^{\Phi})\}\), where \(S(L^{\Phi})\) is the unit sphere of \(L^{\Phi}\), satisfy the condition \(\|(c_{\tau}x)\chi_{D}\|_{\Psi}<\varepsilon\).
Let \(\bar{r}_{n} (s)=\widetilde{r}_{n} (s) \chi_{\Omega_{n}} (s)\). Since \(\bar {r}_{n}\) are bounded measurable functions and \(h\in L^{1} (\operatorname {supp}\bar {r}_{n})\), we get \(\bar{r}_{n} \in E^{\Psi}_{h} (\Omega)\) with \(\|\bar{r}_{n}\| _{\Psi,h} =1\) and \(\bar{r}_{n} \in L^{\Phi}(\Omega)\) for any \(n\in\mathbb {N}\cup\{0\}\).
From Theorem 4.2, applying Theorem 4.1 and the definition of a compact operator, we directly get the following.
Theorem 4.3
If \((\Omega,\Sigma, \mu)\) is a non-atomic complete finite or infinite but σ-finite measure space and τ satisfies the assumption from Theorem 4.2, then the composition operator \(c_{\tau}\) from \(L^{\Phi}(\Omega)\) into \(L^{\Psi}(\Omega)\) is compact whenever the set \(c_{\tau}(S(L^{\Phi}))\) is relatively compact with respect to local convergence in measure and condition (8) from Theorem 4.2 is satisfied.
Under the assumption that \(\mu(\Omega)<\infty\), if the composition operator \(c_{\tau}\) from \(L^{\Phi}(\Omega)\) into \(E^{\Psi}(\Omega)\) is compact then the set \(c_{\tau}(S(L^{\Phi}))\) is relatively compact with respect to convergence in measure and condition (8) is satisfied.
In the case when Ω has infinite measure, we were unable to show that (8) is a necessary condition for the equi-absolute continuity of the composition operator \(c_{\tau}\). Instead, we can deduce a slightly different (and weaker) condition, as the following theorem states.
Theorem 4.4
Proof
From the preceding theorem, applying Theorem 4.1, we can deduce the following necessary condition for the compactness of the composition operator \(c_{\tau}\):
Theorem 4.5
- (1)
the set \(c_{\tau}(S(L^{\Phi}))\) is relatively compact with respect to local convergence in measure;
- (2)
the functions Φ and Ψ satisfy condition (11).
Remark 4.1
Proof of Remark 4.1
5 Compactness of the multiplication operator \(M_{w}\) from one Orlicz space into another
We state a sufficient condition for the compactness of the multiplication operator \(M_{w}:L^{\Phi}(\Omega)\rightarrow L^{\Psi}(\Omega)\).
Theorem 5.1
Proof
First we show that given \(\varepsilon>0\) there exists a set \(B\in\Sigma \) with \(\mu(B)<\infty\) such that \(\|x \chi_{\Omega\setminus B}\|_{\Psi}<\varepsilon\) for any function \(x\in S(L^{\Phi}(\Omega))\).
Next, we show that for \(\varepsilon>0\) there exists \(\delta=\delta (\varepsilon)>0\) such that for any \(D\subset B\) and any function \(x\in S(L^{\Phi})\), if \(\mu(D)<\delta\) then \(\|M_{w} x\chi_{D}\| _{\Psi}<\varepsilon\).
Remark 5.1
Let us note that in the case when \(\Phi=\Psi\), Theorem 5.1 can only hold when \(\Phi\in\Delta_{2}(\infty)\).
Applying Theorem 4.1, we directly get from Theorem 5.1 and the definition of a compact operator the following.
Theorem 5.2
If \((\Omega,\Sigma, \mu)\) is a non-atomic complete finite or infinite but σ-finite measure space, then the multiplication operator \(c_{\tau}\) form \(L^{\Phi}(\Omega)\) into \(L^{\Psi}(\Omega)\) is compact whenever the set \(M_{w}(S(L^{\Phi}))\) is relatively compact with respect to local convergence in measure and condition (12) from Theorem 5.1 is satisfied.
Theorems 5.1 and 5.2 resemble closely the sufficiency part of Theorems 4.2 and 4.3 for the composition operator. Similarly, we will formulate necessary conditions for the equi-absolute continuity of the multiplication operator: one in the case when \(\mu(\Omega)<\infty\) and the other in the case when \(\mu(\Omega)=\infty\). The respective proofs proceed along the lines of the proofs for the composition operator, and therefore will be omitted.
Theorem 5.3
Theorem 5.4
The respective necessary conditions for the compactness of the multiplication operator are analogous to the ones for the composition operator from Theorems 4.3 and 4.5.
Declarations
Acknowledgements
The second author is supported by NFS of Heilong Jiang province (A2015018).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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