Abstract
The Orlicz spaces \(X^{\varPhi }\) associated to a quasi-Banach function space X are defined by replacing the role of the space \(L^1\) by X in the classical construction of Orlicz spaces. Given a vector measure m, we can apply this construction to the spaces \(L^1_w(m),\) \(L^1(m)\) and \(L^1(\Vert m\Vert )\) of integrable functions (in the weak, strong and Choquet sense, respectively) in order to obtain the known Orlicz spaces \(L^{\varPhi }_w(m)\) and \(L^{\varPhi }(m)\) and the new ones \(L^{\varPhi }(\Vert m\Vert ).\) Therefore, we are providing a framework where dealing with different kind of Orlicz spaces in a unified way. Some applications to complex interpolation are also given.
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1 Introduction
The Banach lattice \(L^1(m)\) of integrable functions with respect to a vector measure m (defined on a \(\sigma \)-algebra of sets and with values in a Banach space) has been systematically studied during the last 30 years and it has proved to be a efficient tool to describe the optimal domain of operators between Banach function spaces (see [18] and the references therein). The Orlicz spaces \(L^{\varPhi }(m)\) and \(L^{\varPhi }_w(m)\) associated to m were introduced in [8] and they have recently shown in [5] their utility in order to characterize compactness in \(L^1(m).\)
On the other hand, the quasi-Banach lattice \(L^1(\Vert m\Vert )\) of integrable functions (in the Choquet sense) with respect to the semivariation of m was introduced in [9]. Some properties of this space and their corresponding \(L^p(\Vert m\Vert )\) with \(p>1\) have been obtained, but in order to achieve compactness results in \(L^1(\Vert m\Vert )\) we would need to dispose of certain Orlicz spaces related to \(L^1(\Vert m\Vert ).\)
In [10] some generalized Orlicz spaces \(X_{\varPhi }\) have been obtained by replacing the role of the space \(L^1\) by a Banach function space X in the classical construction of Orlicz spaces. Moreover, the spaces X they consider are allways supposed to possess the \(\sigma \)-Fatou property. However, these Orlicz spaces do not cover our situation since:
-
the space \(L^1(\Vert m\Vert )\) is only a quasi-Banach function space, and
-
in most of the time \(L^1(m)\) lacks the \(\sigma \)-Fatou property.
Thus, the purpose of this work is to provide a construction of certain Orlicz spaces \(X^{\varPhi }\) valid for the case of X being an arbitrary quasi-Banach function space (in general without the \(\sigma \)-Fatou property), with the underlying idea that it can be applied simultaneously to the spaces \(L^1(\Vert m\Vert )\) and \(L^1(m)\) among others. In a subsequent paper [6] we shall employ these Orlicz spaces \(L^{1}(\Vert m\Vert )^{\varPhi }\) and their main properties here derived in order to study compactness in \(L^1(\Vert m\Vert ).\)
The organization of the paper goes as follows: Section 2 contains the preliminaries which we will need later. Section 3 contains a discussion of completeness in the quasi-normed context without any additional hypothesis on \(\sigma \)-Fatou property. Section 4 is devoted to introduce the Orlicz spaces \(X^{\varPhi }\) associated to a quasi-Banach function space X and obtain their main properties. In Sect. 5, we show that the construction of the previous section allows to capture the Orlicz spaces associated to a vector measure and we take advantage of its generality to introduce the Orlicz spaces associated to its semivariation. Finally, in Sect. 6 we present some applications of this theory to compute their complex interpolation spaces.
2 Preliminaries
Throughout this paper, we shall always assume that \(\varOmega \) is a nonempty set, \(\varSigma \) is a \(\sigma \)-algebra of subsets of \(\varOmega ,\) \(\mu \) is a finite positive measure defined on \(\varSigma \) and \(L^0(\mu )\) is the space of (\(\mu \)-a.e. equivalence classes of) measurable functions \(f:\varOmega \rightarrow {\mathbb {R}}\) equipped with the topology of convergence in measure.
Recall that a quasi-normed space is any real vector space X equipped with a quasi-norm, that is, a function \(\Vert \cdot \Vert _{X} : X\rightarrow [0,\infty )\) which satisfies the following axioms:
-
(Q1)
\(\Vert x\Vert _{X}=0\) if and only if \(x=0.\)
-
(Q2)
\(\Vert \alpha x\Vert _{X} = |\alpha | \Vert x\Vert _{X},\) for \(\alpha \in {\mathbb {R}}\) and \(x\in X.\)
-
(Q3)
There exists \(K\ge 1\) such that \(\left\| x_1+x_2\right\| _{X} \le K\left( \left\| x_1\right\| _{X} + \left\| x_2\right\| _{X}\right) ,\) for all \(x_1,x_2\in X.\)
The constant K in (Q3) is called a quasi-triangle constant of X. Of course if we can take \(K=1,\) then \(\Vert \cdot \Vert _{X}\) is a norm and X is a normed space. A quasi-normed function space over \(\mu \) is any quasi-normed space X satisfying the following properties:
-
(a)
X is an ideal in \(L^0(\mu )\) and a quasi-normed lattice with respect to the \(\mu \)-a.e. order, that is, if \(f\in L^0(\mu ),\) \(g\in X\) and \(|f|\le |g|\) \(\mu \)-a.e., then \(f\in X\) and \(\Vert f\Vert _{X} \le \Vert g\Vert _{X}.\)
-
(b)
The characteristic function of \(\varOmega ,\) \(\chi _{\varOmega },\) belongs to X.
If, in addition, the quasi-norm \(\Vert \cdot \Vert _{X}\) happens to be a norm, then X is called a normed function space. Note that, with this definition, any quasi-normed function space over \(\mu \) is continuously embedded into \(L^0(\mu ),\) as it is proved in [18, Proposition 2.2].
Remark 1
Many of the results that we will present in this paper are true if we assume that the measure space \(\left( \varOmega ,\varSigma ,\mu \right) \) is \(\sigma \)-finite. In this case, the previous condition (b) must be replaced by
-
(b’)
The characteristic functions \(\chi _{A}\) belong to X for all \(A\in \varSigma \) such that \(\mu (A)<\infty .\)
Nevertheless we prefer to present the results in the finite case for clarity and simplicity in the proofs.
We say that a quasi-normed function space X has the \(\sigma \)-Fatou property if for any positive increasing sequence \((f_n)_n\) in X with \(\displaystyle \sup _{n}\Vert f_n\Vert _{X}<\infty \) and converging pointwise \(\mu \)-a.e. to a function f, then \(f\in X\) and \(\displaystyle \Vert f\Vert _{X} = \sup _{n}\Vert f_n\Vert _{X}.\) And a quasi-normed function space X is said to be \(\sigma \)-order continuous if for any positive increasing sequence \((f_n)_n\) in X converging pointwise \(\mu \)-a.e. to a function \(f\in X,\) then \(\left\| f-f_n\right\| _{X}\rightarrow 0.\)
A complete quasi-normed function space is called a quasi-Banach function space (briefly q-B.f.s.). If, in addition, the quasi-norm happens to be a norm, then X is called a Banach function space (briefly B.f.s.). It is known that if a quasi-normed function space has the \(\sigma \)-Fatou property, then it is complete and hence a q-B.f.s. (see [18, Proposition 2.35]) and that inclusions between q-B.f.s. are automatically continuous (see [18, Lemma 2.7]).
Given a countably additive vector measure \(m:\varSigma \rightarrow Y\) with values in a real Banach space Y, there are several ways of constructing q-B.f.s. of integrable functions. Let us recall them briefly. The semivariation of m is the finite subadditive set function defined on \(\varSigma \) by
where \(\left| \langle m, y^*\rangle \right| \) denotes the variation of the scalar measure \(\langle m, y^*\rangle :\varSigma \rightarrow {\mathbb {R}}\) given by \(\langle m, y^*\rangle (A):= \langle m(A), y^*\rangle \) for all \(A\in \varSigma ,\) and \(B_{Y^*}\) is the unit ball of \(Y^*,\) the dual of Y. A set \(A\in \varSigma \) is called m-null if \(\Vert m\Vert (A)=0.\) A measure \(\mu := \left| \langle m, y^*\rangle \right| ,\) where \(y^*\in B_{Y^*},\) that is equivalent to m (in the sense that \(\Vert m\Vert (A)\rightarrow 0\) if and only if \(\mu (A)\rightarrow 0\)) is called a Rybakov control measure for m. Such a measure always exists (see [7, Theorem 2, p.268]). Let \(L^0(m)\) be the space of (m-a.e. equivalence classes of) measurable functions \(f:\varOmega \rightarrow {\mathbb {R}}.\) Thus, \(L^0(m)\) and \(L^0(\mu )\) are just the same whenever \(\mu \) is a Rybakov control measure for m.
A measurable function \(f:\varOmega \rightarrow {\mathbb {R}}\) is called weakly integrable (with respect to m) if f is integrable with respect to \(|\langle m, y^*\rangle |\) for all \(y^*\in Y^*.\) A weakly integrable function f is said to be integrable (with respect to m) if, for each \(A\in \varSigma \) there exists an element (necessarily unique) \(\displaystyle \int _A f \, dm \in Y,\) satisfying
Given a measurable function \(f:\varOmega \rightarrow {\mathbb {R}},\) we shall also consider its distribution function (with respect to the semivariation of the vector measure m)
where \(\left[ |f|>t\right] := \left\{ w\in \varOmega : |f(w)|>t \right\} .\) This distribution function is bounded, non-increasing and right-continuous.
Let \(L^1_w(m)\) be the space of all (m-a.e. equivalence classes of) weakly integrable functions, \(L^1(m)\) the space of all (m-a.e equivalence classes of) integrable functions and \(L^1(\Vert m\Vert )\) the space of all (m-a.e. equivalence classes of) measurable functions f such that its distribution function \(\Vert m\Vert _{f}\) is Lebesgue integrable in \((0,\infty ).\) Letting \(\mu \) be any Rybakov control measure for m, we have that \(L^1_w(m)\) becomes a B.f.s. over \(\mu \) with the \(\sigma \)-Fatou property when endowed with the norm
Moreover, \(L^1(m)\) is a closed \(\sigma \)-order continuous ideal of \(L^1_w(m).\) In fact, it is the closure of \({\mathscr {S}}(\varSigma ),\) the space of simple functions supported on \(\varSigma .\) Thus, \(L^1(m)\) is a \(\sigma \)-order continuous B.f.s. over \(\mu \) endowed with same norm (see [18, Theorem 3.7] and [18, p.138])). It is worth noting that space \(L^1(m)\) does not generally have the \(\sigma \)-Fatou property. In fact, if \(L^1(m) \ne L^1_w(m),\) then \(L^1(m)\) does not have the \(\sigma \)-Fatou property. See [2] for details.
On the other hand, \(L^1(\Vert m\Vert )\) equipped with the quasi-norm
is a q-B.f.s. over \(\mu \) with the \(\sigma \)-Fatou property (see [4, Proposition 3.1]) and it is also \(\sigma \)-order continuous (see [4, Proposition 3.6]). We will denote by \(L^{\infty }(m)\) the B.f.s. of all (m-a.e. equivalence classes of) essentially bounded functions equipped with the essential sup-norm.
3 Completeness of quasi-normed lattices
In this section we present several characterizations of completeness which will be needed later. We begin by recalling one of them valid for general quasi-normed spaces (see [10, Theorem 1.1]).
Theorem 1
Let X be a quasi-normed space with a quasi-triangle constant K. The following conditions are equivalent:
-
(i)
X is complete.
-
(ii)
For every sequence \((x_n)_n\subseteq X\) such that \(\displaystyle \sum _{n=1}^{\infty }K^n\Vert x_n\Vert _{X}<\infty \) we have \(\displaystyle \sum _{n=1}^{\infty } x_n \in X.\) In this case, the inequality \(\displaystyle \left\| \sum _{n=1}^{\infty } x_n \right\| _{X} \le K \sum _{n=1}^{\infty } K^n \Vert x_n\Vert _{X}\) holds.
The next result is a version of Amemiya’s Theorem ( [10, Theorem 2, p.290]) for quasi-normed lattices.
Theorem 2
Let X be a quasi-normed lattice. The following conditions are equivalent:
-
(i)
X is complete.
-
(ii)
For any positive increasing Cauchy sequence \((x_n)_n\) in X there exists \(\displaystyle \sup _{n} x_n \in X.\)
Proof
(i) \(\Rightarrow \) (ii) is evident because the limit of increasing convergent sequences in a quasi-normed lattice is always its supremum.
(ii) \(\Rightarrow \) (i) Let \((x_n)_n\) be a positive increasing Cauchy sequence in X. It is sufficient to prove that \((x_n)_n\) is convergent in X for X being complete (see, for example [1, Theorem 16.1]). By hypothesis, there exists \(x:=\displaystyle \sup _{n} x_n \in X.\) We have to prove that \((x_n)_n\) converges to x and for this it is enough the convergence of a subsequence of \((x_n)_n.\) So, let us take a subsequence of \((x_n)_n,\) that we still denote by \((x_n)_n,\) such that \(\Vert x_{n+1}-x_n\Vert _{X}\le \displaystyle \frac{1}{K^n n^3},\) for all \(n\in {\mathbb {N}}\) where K is a quasi-triangle constant of X. Thus, the sequence \(y_n:=\displaystyle \sum _{i=1}^{n} i (x_{i+1}-x_i)\) is positive, increasing and Cauchy. Indeed, given \(m>n,\) we have
Applying (ii) again, we deduce that there exists \(y:=\displaystyle \sup _{n} y_n\in X.\) Moreover, given \(n\in {\mathbb {N}},\) we have
Therefore, \(0\le x-x_n \le \displaystyle \frac{1}{n}y\) and hence \(\Vert x-x_n\Vert _{X}\le \displaystyle \frac{1}{n}\Vert y\Vert _{X}\rightarrow 0.\) \(\square \)
Applying Theorem 2 to the sequence of partial sums of a given sequence, we see that completeness in quasi-normed lattices can still be characterized by a Riesz-Fischer type property.
Corollary 1
Let X be a quasi-normed lattice with a quasi-triangle constant K. The following conditions are equivalent:
-
(i)
X is complete.
-
(ii)
For every positive sequence \((x_n)_n\subseteq X\) such that \(\displaystyle \sum _{n=1}^{\infty }K^n\Vert x_n\Vert _{X}<\infty \) there exists \(\displaystyle \sup _{n} \sum _{i=1}^{n} x_i \in X.\)
4 Orlicz spaces \(X^{\varPhi }\)
In this section we introduce the Orlicz spaces \(X^{\varPhi }\) associated to a quasi-Banach function space X and a Young function \(\varPhi \) and obtain their main properties.
Recall that a Young function is any function \(\varPhi :[0,\infty )\rightarrow [0,\infty )\) which is strictly increasing, continuous, convex, \(\varPhi (0)=0\) and \(\displaystyle \lim _{t\rightarrow \infty }\varPhi (t)=\infty .\) A Young function \(\varPhi \) satisfies the following useful inequalities (which we shall use without explicit mention) for all \(t\ge 0\):
In particular, from the second of the previous inequalities it follows that for all \(t_0>0\) there exists \(C>0\) such that \(\varPhi (t)\ge C t\) for all \(t\ge t_0.\) For a given \(t_0>0,\) just take \(\displaystyle C:=\frac{\varPhi (t_0)}{t_0}>0\) and observe that \(\displaystyle \varPhi (t) = \varPhi \left( t_0\frac{t}{t_0}\right) \ge \frac{t}{t_0}\varPhi (t_0) = C t\) for all \(t\ge t_0.\)
Moreover, it is easy to prove using the convexity of \(\varPhi \) that
for all \(N\in {\mathbb {N}},\) \(\alpha \ge 1\) and \(t_1,\dots ,t_N\ge 0.\)
A Young function \(\varPhi \) has the \(\varDelta _2\)-property, written \(\varPhi \in \varDelta _2,\) if there exists a constant \(C>1\) such that \(\varPhi (2t)\le C\varPhi (t)\) for all \(t\ge 0.\) Equivalently, \(\varPhi \in \varDelta _2\) if for any \(c>1\) there exists \(C>1\) such that \(\varPhi (ct)\le C\varPhi (t),\) for all \(t\ge 0.\)
Definition 1
Let \(\varPhi \) be a Young function. Given a quasi-normed function space X over \(\mu ,\) the corresponding (generalized) Orlicz class \(\widetilde{X}^{\varPhi }\) is defined as the following set of (\(\mu \)-a.e. equivalence classes of) measurable functions:
Proposition 1
Let \(\varPhi \) be a Young function and X be a quasi-normed function space over \(\mu .\) Then, \(\widetilde{X}^{\varPhi }\) is a solid convex set in \(L^0(\mu ).\) Moreover, \(\widetilde{X}^{\varPhi } \subseteq X.\)
Proof
Let \(f,g\in \widetilde{X}^{\varPhi }\) and \(0\le \alpha \le 1.\) According to the convexity and monotonicity properties of \(\varPhi \) we have \(\varPhi (|\alpha f + (1-\alpha )g|) \le \alpha \varPhi (|f|) + (1-\alpha )\varPhi (|g|) \in X.\) The ideal property of X yields \(\varPhi (|\alpha f + (1-\alpha )g|)\in X\) which means that \(\begin{aligned} \alpha f + (1-\alpha )g \in \widetilde{X}^{\varPhi } \end{aligned}\) and proves the convexity of \(\widetilde{X}^{\varPhi }.\) Clearly, \(\widetilde{X}^{\varPhi }\) is solid, since \(|h|\le |f|\) implies that \(\varPhi (|h|)\le \varPhi (|f|)\in X,\) for any \(h\in L^0(\mu ).\) Moreover, since \(\varPhi \) is a convex function, there exists \(C>0\) such that \(\varPhi (t)\ge C t,\) for all \(t > 1.\) Thus, for all \(f\in \widetilde{X}^{\varPhi },\)
which gives \(f\in X.\) \(\square \)
Definition 2
Let \(\varPhi \) be a Young function. Given a quasi-normed function space X over \(\mu ,\) the corresponding (generalized) Orlicz space \(X^{\varPhi }\) is defined as the following set of (\(\mu \)-a.e. equivalence classes of) measurable functions:
Proposition 2
Let \(\varPhi \) be a Young function and X be a quasi-normed function space over \(\mu .\) Then, \(X^{\varPhi }\) is a linear space, an ideal in \(L^0(\mu )\) and \(\widetilde{X}^{\varPhi } \subseteq X^{\varPhi }\subseteq X.\)
Proof
Let \(f,g\in X^{\varPhi }\) and \(\alpha \in {\mathbb {R}}.\) Then, there exist \(c_1, c_2>0\) such that \(\displaystyle \frac{|f|}{c_1}, \displaystyle \frac{|g|}{c_2}\in \widetilde{X}^{\varPhi }.\) Setting \(c:=\max \{c_1,c_2\}\) and using the convexity of \(\widetilde{X}^{\varPhi }\) we have
and hence \(\displaystyle \frac{|f+g|}{2c} \in \widetilde{X}^{\varPhi }\) since \(\widetilde{X}^{\varPhi }\) is solid, which proves that \(f+g\in X^{\varPhi }.\) Note that this also implies that \(nf\in X^{\varPhi }\) for any \(n\in {\mathbb {N}}.\) Taking \(n_0\in {\mathbb {N}}\) such that \(|\alpha |\le n_0,\) it follows that there exists \(c_0>0\) such that \(\displaystyle \frac{|\alpha f|}{c_0}\le \frac{n_0|f|}{c_0} \in \widetilde{X}^{\varPhi },\) which yields \(\displaystyle \frac{|\alpha f|}{c_0}\in \widetilde{X}^{\varPhi }\) and so \(\alpha f\in X^{\varPhi }.\)
It is evident that \(\widetilde{X}^{\varPhi } \subseteq X^{\varPhi }\) and \(X^{\varPhi }\) inherits the ideal property from \(\widetilde{X}^{\varPhi },\) since \(|h|\le |f|\) implies that \(\displaystyle \frac{|h|}{c_1} \le \frac{|f|}{c_1}\in \widetilde{X}^{\varPhi }\) for any \(h\in L^0(\mu ).\) Moreover, taking into account Proposition 1, we have \(\displaystyle \frac{|f|}{c_1}\in \widetilde{X}^{\varPhi } \subseteq X\) and so \(f\in X\) which proves that \(X^{\varPhi } \subseteq X.\) \(\square \)
Definition 3
Let \(\varPhi \) be a Young function and X be a quasi-normed function space over \(\mu .\) Given \(f\in X^{\varPhi },\) we define
The functional \(\Vert \cdot \Vert _{X^{\varPhi }}\) in \(X^{\varPhi }\) is called the Luxemburg quasi-norm.
Proposition 3
Let \(\varPhi \) be a Young function and X be a quasi-normed function space (respectively, normed function space) over \(\mu .\) Then, \(\Vert \cdot \Vert _{X^{\varPhi }}\) is a quasi-norm (respectively, norm) in \(X^{\varPhi }.\) Moreover, \(X^{\varPhi }\) equipped with the Luxemburg quasi-norm, is a quasi-normed (respectively, normed) function space over \(\mu .\)
Proof
First, note that \(\Vert \cdot \Vert _{X^{\varPhi }}:X^{\varPhi }\rightarrow [0,\infty ).\) Given \(f\in X^{\varPhi },\) there exists \(c>0\) such that \(\varPhi \displaystyle \left( \frac{|f|}{c}\right) \in X.\) Let \(M:=\displaystyle \left\| \varPhi \left( \frac{|f|}{c}\right) \right\| _X<\infty .\) On the one hand, if \(M\le 1\) then \(\Vert f\Vert _{X^{\varPhi }}\le c < \infty .\) On the other hand, if \(M>1\) then \(\displaystyle \varPhi \left( \frac{|f|}{Mc}\right) \le \frac{1}{M}\varPhi \left( \frac{|f|}{c}\right) \in X\) and so \(\displaystyle \left\| \varPhi \left( \frac{|f|}{Mc}\right) \right\| _X \le \frac{1}{M} \left\| \varPhi \left( \frac{|f|}{c}\right) \right\| =1,\) which implies that \(\Vert f\Vert _{X^{\varPhi }}\le Mc < \infty .\)
If \(f=0,\) then \(\displaystyle \left\| \varPhi \left( \frac{|f|}{c}\right) \right\| _X=0\le 1\) for all \(c>0\) and so \(\Vert f\Vert _{X^{\varPhi }}=0.\) Now, suppose that \(\Vert f\Vert _{X^{\varPhi }}=0\) and that \(\mu \left( \left[ f\ne 0\right] \right) >0,\) that is, \(\displaystyle \left\| \varPhi \left( \frac{|f|}{c}\right) \right\| _X\le 1\) for all \(c>0\) and there exist \(\varepsilon >0\) and \(A\in \varSigma \) such that \(\mu (A)>0\) and \(|f|\chi _A \ge \varepsilon \chi _A.\) Given \(c>0,\) we have \(\displaystyle \varPhi \left( \frac{\varepsilon }{c}\right) \chi _A \le \varPhi \left( \frac{|f|\chi _A}{c}\right) \le \varPhi \left( \frac{|f|}{c}\right) .\) Therefore,
and keeping in mind that \(\displaystyle \lim _{t\rightarrow \infty }\varPhi (t)=\infty ,\) we can take \(c>0\) such that
which yields a contradiction.
On the other hand, given \(f\in X^{\varPhi }\) and \(\lambda \in {\mathbb {R}},\) it is clear that
Now, let \(f, g\in X^{\varPhi }\) and take \(K\ge 1\) as in (Q3). Given \(a, b>0\) such that \(\displaystyle \left\| \varPhi \left( \frac{|f|}{a}\right) \right\| _{X}\le 1\) and \(\displaystyle \left\| \varPhi \left( \frac{|g|}{b}\right) \right\| _{X}\le 1,\) we have
Hence, \(\displaystyle \left\| \varPhi \left( \frac{|f+g|}{K(a+b)}\right) \right\| _{X} \le \frac{a}{(a+b)} \left\| \varPhi \left( \frac{|f|}{a}\right) \right\| _{X} + \frac{b}{(a+b)} \left\| \varPhi \left( \frac{|g|}{b}\right) \right\| _{X} \le 1\) which implies that \(\Vert f+g\Vert _{X^{\varPhi }} \le K(a+b).\) By the arbitrariness of a and b we deduce that \(\Vert f+g\Vert _{X^{\varPhi }} \le K(\Vert f\Vert _{X^{\varPhi }}+\Vert g\Vert _{X^{\varPhi }}).\)
Thus, we have proved that \(\Vert \cdot \Vert _{X^{\varPhi }}\) is a quasi-norm in \(X^{\varPhi }\) with the same quasi-triangle constant as the one of the quasi-norm of X. Moreover, we have already proved that \(X^{\varPhi }\) equipped with the Luxemburg quasi-norm is a quasi-normed space and an ideal in \(L^0(\mu ).\) It is also clear that the Luxemburg quasi-norm is a lattice quasi-norm: \(|f|\le |g|\) implies that \(\displaystyle \varPhi \left( \frac{|f|}{k}\right) \le \varPhi \left( \frac{|g|}{k}\right) \) for all \(k>0\) and this guarantees that \(\Vert f\Vert _{X^{\varPhi }} \le \Vert g\Vert _{X^{\varPhi }}.\) In addition, \(\chi _{\varOmega }\in X^{\varPhi },\) since \(\displaystyle \varPhi \left( \frac{|\chi _{\varOmega }|}{c}\right) = \varPhi \left( \frac{1}{c}\right) \chi _{\varOmega }\in X,\) for all \(c>0,\) and hence \(X^{\varPhi }\) is in fact a quasi-normed function space. \(\square \)
Remark 2
The inclusion of \(X^{\varPhi }\subseteq X\) is continuous provided X and \(X^{\varPhi }\) be q-B.f.s. We will see in Theorem 3 that the completeness is transferred from X to \(X^{\varPhi }.\)
Once we have checked that \(X^{\varPhi }\) is quasi-normed function space, it is immediate that \(L^{\infty }(\mu )\) is contained in \(X^{\varPhi }\) and this inclusion is continuous with norm \(\Vert \chi _{\varOmega }\Vert _{X^{\varPhi }}.\) The next result establishes the relation between the norm of this inclusion and the norm \(\Vert \chi _{\varOmega }\Vert _{X}\) of the continuous inclusion of \(L^{\infty }(\mu )\) into X.
Lemma 1
Let \(\varPhi \) be a Young function and X be a quasi-normed function space over \(\mu .\)
-
(i)
For all \(A\in \varSigma \) with \(\mu (A)>0,\) \(\Vert \chi _A\Vert _{X^{\varPhi }} = \displaystyle \frac{1}{\varPhi ^{-1}\left( \frac{1}{\Vert \chi _A\Vert _X} \right) }.\)
-
(ii)
For all \(f\in L^{\infty }(\mu ),\) \(\Vert f\Vert _{X^{\varPhi }} \le \displaystyle \frac{\Vert f\Vert _{L^{\infty }(\mu )}}{\varPhi ^{-1}\left( \frac{1}{\Vert \chi _{\varOmega }\Vert _X} \right) }.\)
Proof
(i) Write \(\alpha :=\displaystyle \frac{1}{\varPhi ^{-1}\left( \frac{1}{\Vert \chi _A\Vert _X} \right) }.\) On the one hand,
and so \(\Vert \chi _A\Vert _{X^{\varPhi }}\le \alpha .\) On the other hand, given \(k>0\) such that \(\displaystyle \frac{\chi _A}{k}\in \widetilde{X}^{\varPhi }\) with \(\displaystyle \left\| \varPhi \left( \frac{\chi _A}{k}\right) \right\| _{X}\le 1,\) we have \(\displaystyle \varPhi \left( \frac{1}{k}\right) \Vert \chi _A\Vert _{X}\le 1,\) that is, \(\displaystyle \varPhi \left( \frac{1}{k}\right) \le \frac{1}{\Vert \chi _A\Vert _{X}}\) or, equivalently, \(\displaystyle \frac{1}{k} \le \varPhi ^{-1}\left( \frac{1}{\Vert \chi _A\Vert _{X}}\right) ,\) which finally leads to \(\alpha \le k\) and so \(\alpha \le \Vert \chi _A\Vert _{X^{\varPhi }}.\)
(ii) Since \(|f|\le \Vert f\Vert _{L^{\infty }(\mu )}\chi _{\varOmega },\) for any \(f\in L^{\infty }(\mu ),\) we have \(\begin{aligned} \Vert f\Vert _{X^{\varPhi }} \le \Vert f\Vert _{L^{\infty }(\mu )} \Vert \chi _{\varOmega }\Vert _{X^{\varPhi }} \end{aligned}\) and the result follows applying (i) to \(\chi _{\varOmega }.\) \(\square \)
The following two results explore the close relationship between the quantities \(\Vert f\Vert _{X^{\varPhi }}\) and \(\Vert \varPhi (|f|)\Vert _X.\) This entails interesting consequences on boundedness in \(X^{\varPhi },\) allowing us to obtain a sufficient condition and a necessary condition for it.
Lemma 2
Let \(\varPhi \) be a Young function, X be a quasi-normed function space over \(\mu \) and \(H\subset L^0(\mu ).\)
-
(i)
If \(f\in \widetilde{X}^{\varPhi },\) then \(\Vert f\Vert _{X^{\varPhi }} \le \max \{1, \Vert \varPhi (|f|)\Vert _X \}.\)
-
(ii)
If \(\{ \varPhi (|h|) : h\in H \}\) is bounded in X, then H is bounded in \(X^{\varPhi }.\)
Proof
(i) On the one hand, \(\Vert \varPhi (|f|)\Vert _X\le 1\) directly implies that
On the other hand, if \(\Vert \varPhi (|f|)\Vert _X\ge 1,\) then \(\begin{aligned} \displaystyle \varPhi \left( \frac{|f|}{\Vert \varPhi (|f|)\Vert _X}\right) \le \displaystyle \frac{1}{\Vert \varPhi (|f|)\Vert _X} \varPhi (|f|)\in X \end{aligned}\) and hence \(\displaystyle \varPhi \left( \frac{|f|}{\Vert \varPhi (|f|)\Vert _X}\right) \in X\) with \(\displaystyle \left\| \varPhi \left( \frac{|f|}{\Vert \varPhi (|f|)\Vert _X}\right) \right\| _{X} \le 1.\) This also leads to \(\Vert f\Vert _{X^{\varPhi }} \le \Vert \varPhi (|f|)\Vert _X = \max \{1, \Vert \varPhi (|f|)\Vert _X \}.\)
(ii) If \(\Vert \varPhi (|h|)\Vert _X \le M<\infty ,\) for all \(h\in H,\) according to (i) we have that \(\Vert h\Vert _{X^{\varPhi }} \le \max \{1, \Vert \varPhi (|h|)\Vert _X \}\le \max \{1,M\}<\infty ,\) for all \(h\in H.\) \(\square \)
Lemma 3
Let \(\varPhi \) be a Young function, X be a quasi-normed function space over \(\mu \) and \(f\in X^{\varPhi }.\)
-
(i)
If \(\Vert f\Vert _{X^{\varPhi }}<1,\) then \(f\in \widetilde{X}^{\varPhi }\) with \(\Vert \varPhi (|f|)\Vert _X \le \Vert f\Vert _{X^{\varPhi }}.\)
-
(ii)
If \(\Vert f\Vert _{X^{\varPhi }}>1\) and \(f\in \widetilde{X}^{\varPhi },\) then \(\Vert \varPhi (|f|)\Vert _X \ge \Vert f\Vert _{X^{\varPhi }}.\)
-
(iii)
If \(H\subseteq X^{\varPhi }\) is bounded, then there exists a Young function \(\Psi \) such that the set \(\{ \Psi (|h|) : h\in H \}\) is bounded in X.
Proof
(i) Given \(0<k<1\) such that \(\displaystyle \frac{|f|}{k}\in \widetilde{X}^{\varPhi }\) with \(\displaystyle \left\| \varPhi \left( \frac{|f|}{k}\right) \right\| _{X}\le 1,\) we have
Therefore, \(\varPhi (|f|)\in X\) with \(\Vert \varPhi (|f|)\Vert _X \le k \displaystyle \left\| \varPhi \left( \frac{|f|}{k}\right) \right\| _{X}\le k\) and keeping in mind that \(\Vert f\Vert _{X^{\varPhi }}<1,\) we obtain
(ii) Let \(0<\varepsilon <\Vert f\Vert _{X^{\varPhi }}-1\) and observe that \(\displaystyle \left\| \varPhi \left( \frac{|f|}{\Vert f\Vert _{X^{\varPhi }}-\varepsilon }\right) \right\| _{X} > 1.\) Thus,
and letting \(\varepsilon \rightarrow 0,\) it follows that \(\Vert \varPhi (|f|)\Vert _X \ge \Vert f\Vert _{X^{\varPhi }}.\)
(iii) Take \(M>0\) such that \(\Vert h\Vert _{X^{\varPhi }}<M,\) for all \(h\in H.\) Since \(\displaystyle \left\| \frac{h}{M}\right\| _{X^{\varPhi }}< 1,\) for all \(h\in H,\) (i) guarantees that \(\displaystyle \varPhi \left( \frac{|h|}{M}\right) \in X\) with \(\displaystyle \left\| \varPhi \left( \frac{|h|}{M}\right) \right\| _{X}\le \left\| \frac{h}{M}\right\| _{X^{\varPhi }}<1,\) for all \(h\in H.\) Defining \(\Psi (t):=\varPhi \displaystyle \left( \frac{t}{M}\right) ,\) for all \(t\ge 0,\) we produce a Young function such that \(\{ \Psi (|h|) : h\in H \}\) is bounded in X. \(\square \)
We are now in a position to establish the remarkable fact that Orlicz spaces \(X^{\varPhi }\) are always complete for any q-B.f.s. X. It is worth pointing out that standard proofs in the Banach setting require the \(\sigma \)-Fatou property of X to obtain the \(\sigma \)-Fatou property of \(X^{\varPhi }\) (see the next Theorem 4) and as a byproduct, the completeness of this last space. However, as we have said before, there are many complete spaces without the \(\sigma \)-Fatou property, to which it is not possible to apply Theorem 4. Herein lies the importance of the result that we will show next about completeness of \(X^\varPhi .\)
Theorem 3
Let \(\varPhi \) a Young function and X be a q-B.f.s. over \(\mu .\) Then, \(X^{\varPhi }\) is complete (and hence it is a q-B.f.s. over \(\mu \)).
Proof
Let \((h_n)_n\) be a positive increasing Cauchy sequence in \(X^{\varPhi }\) and take \(K\ge 1\) as in (Q3). Then, we can choose a subsequence of \((h_n)_n,\) that we denote by \((f_n)_n,\) such that \(\Vert f_{n+1}-f_{n}\Vert _{X^{\varPhi }} < \displaystyle \frac{1}{2^{2n} K^{2n}},\) for all \(n\in {\mathbb {N}}.\) Thus,
for all \(n\in {\mathbb {N}},\) and by Lemma 3 it follows that
which proves that \(\displaystyle \sum _{n=1}^{\infty } K^n \left\| \varPhi \left( 2^{n} K^{n} \left( f_{n+1}-f_{n}\right) \right) \right\| _{X} \le \sum _{n=1}^{\infty } \frac{1}{2^{n}} < \infty . \) The completeness of X ensures that the function \(f:=\displaystyle \sum _{n=1}^{\infty } \varPhi \left( 2^{n} K^{n} \left( f_{n+1}-f_{n}\right) \right) \in X,\) by Theorem 1. Note that \(f\in L^0(\mu )\) and the convergence of that series is also \(\mu \)-a.e, since X is continuously included in \(L^0(\mu ).\) Given \(N\in {\mathbb {N}},\) let \(g_N:=\displaystyle \sum _{n=1}^{N} (f_{n+1}-f_{n})\) and denote by \(g:=\displaystyle \sup _{N} g_N \) pointwise \(\mu \)-a.e. Applying (1) with \(\alpha :=K,\) it follows that for all \(N\in {\mathbb {N}},\)
Therefore, \(0\le g_N\le \varPhi ^{-1}(f)\in L^0(\mu )\) for all \(N\in {\mathbb {N}}\) and so \(g\in L^0(\mu )\) with \(0\le g\le \varPhi ^{-1}(f) \in X^{\varPhi },\) which guarantees that \(g\in X^{\varPhi }.\) But
for all \(N\in {\mathbb {N}}\) and so there also exists \(\displaystyle \sup _{n} f_n = g + f_1 \in X^{\varPhi }.\) Since \((f_n)_n\) is a subsequence of the original increasing sequence \((h_n)_n,\) the supremum of the whole sequence must exists and be the same as the supremum of the subsequence. By applying Amemiya’s Theorem 2 we conclude that \(X^{\varPhi }\) is complete. \(\square \)
If the q-B.f.s. X has the \(\sigma \)-Fatou property, then we can improve a little more our knowledge about \(X^{\varPhi }\) as the following proposition makes evident.
Theorem 4
Let \(\varPhi \) be a Young function and X be a q-B.f.s. over \(\mu \) with the \(\sigma \)-Fatou property.
-
(i)
If \(0\ne f\in X^{\varPhi }\) then \(\displaystyle \frac{|f|}{\Vert f\Vert _{X^{\varPhi }}}\in \widetilde{X}^{\varPhi }\) with \(\displaystyle \left\| \varPhi \left( \frac{|f|}{\Vert f\Vert _{X^{\varPhi }}}\right) \right\| _{X}\le 1.\)
-
(ii)
If \(f\in X^{\varPhi }\) with \(\Vert f\Vert _{X^{\varPhi }}\le 1\) then \(f\in \widetilde{X}^{\varPhi }\) with \(\Vert \varPhi (|f|)\Vert _X \le \Vert f\Vert _{X^{\varPhi }}.\)
-
(iii)
\(X^{\varPhi }\) also has the \(\sigma \)-Fatou property.
Proof
(i) Take a sequence \((k_n)_n\) such that \(k_n \downarrow \Vert f\Vert _{X^{\varPhi }}\) and \(\displaystyle \left\| \varPhi \left( \frac{|f|}{k_n}\right) \right\| _{X}\le 1,\) for all \(n\in {\mathbb {N}}.\) Then, \(\displaystyle \frac{|f|}{k_n}\uparrow \frac{|f|}{\Vert f\Vert _{X^{\varPhi }}}\) and so \(\displaystyle \varPhi \left( \frac{|f|}{k_n}\right) \uparrow \varPhi \left( \frac{|f|}{\Vert f\Vert _{X^{\varPhi }}}\right) ,\) since \(\varPhi \) is continuous and increasing. The \(\sigma \)-Fatou property of X guarantees that \(\displaystyle \varPhi \left( \frac{|f|}{\Vert f\Vert _{X^{\varPhi }}}\right) \in X\) and
(ii) According to (i) and the inequality
we deduce that \(\varPhi (|f|)\in X\) and \(\displaystyle \Vert \varPhi (|f|)\Vert _X \le \Vert f\Vert _{X^{\varPhi }} \left\| \varPhi \left( \frac{|f|}{\Vert f\Vert _{X^{\varPhi }}}\right) \right\| _{X}\le \Vert f\Vert _{X^{\varPhi }}. \)
(iii) Let \((f_n)_n\) in \(X^{\varPhi }\) with \(0\le f_n\uparrow f\) \(\mu \)-a.e. and \(M:=\displaystyle \sup _n \Vert f_n\Vert _{X^{\varPhi }}<\infty .\) Then, \(\displaystyle \varPhi \left( \frac{f_n}{M}\right) \uparrow \varPhi \left( \frac{f}{M}\right) \) \(\mu \)-a.e. and \(\displaystyle \left\| \frac{f_n}{M}\right\| _{X^{\varPhi }}\le 1\) for all \(n\in {\mathbb {N}}.\) Applying (ii), we deduce that \(\displaystyle \varPhi \left( \frac{f_n}{M}\right) \in X\) with \(\displaystyle \left\| \varPhi \left( \frac{f_n}{M}\right) \right\| _{X}\le 1\) for all \(n\in {\mathbb {N}}\) and using the \(\sigma \)-Fatou property of X, it follows that \(\displaystyle \varPhi \left( \frac{f}{M}\right) \in X\) with \(\begin{aligned} \left\| \varPhi \left( \frac{f}{M}\right) \right\| _{X} = \sup _n \left\| \varPhi \left( \frac{f_n}{M}\right) \right\| _{X}\le 1. \end{aligned}\) This implies that \(f\in X^{\varPhi }\) with \(\Vert f\Vert _{X^{\varPhi }}\le M\) and we also have \(M\le \Vert f\Vert _{X^{\varPhi }},\) since \(f_n\le f\in X^{\varPhi }.\) Thus, \(\Vert f\Vert _{X^{\varPhi }}=M,\) which proves that \(X^{\varPhi }\) has the \(\sigma \)-Fatou property. \(\square \)
The relation between the Orlicz class and its corresponding Orlicz space is greatly simplified when the Young function has the \(\varDelta _2\)-property. In addition, this has far-reaching consequences on convergence in \(X^{\varPhi }\) as we state in the next result.
Theorem 5
Let X be a quasi-normed function space over \(\mu \) and \(\varPhi \in \varDelta _2.\)
-
(i)
The Orlicz space and the Orlicz class coincide: \(X^{\varPhi } = \widetilde{X}^{\varPhi }.\)
-
(ii)
\(\Vert f_n\Vert _{X^{\varPhi }}\rightarrow 0\) if and only if \(\Vert \varPhi (|f_n|)\Vert _{X}\rightarrow 0,\) for all \((f_n)_n\subseteq X^{\varPhi }.\)
-
(iii)
If X is \(\sigma \)-order continuous, then \(X^{\varPhi }\) is also \(\sigma \)-order continuous.
Proof
(i) Given \(f\in X^{\varPhi },\) there exists \(c>0\) such that \(\displaystyle \varPhi \left( \frac{|f|}{c}\right) \in X.\) If \(c\le 1,\) then
and if \(c>1,\) then there exist \(C>1\) such that \(\varPhi (ct)\le C\varPhi (t)\) for all \(t\ge 0\) by the \(\varDelta _2\)-property of \(\varPhi .\) Therefore, \(\displaystyle \varPhi (|f|) = \varPhi \left( c \ \frac{|f|}{c}\right) \le C \ \varPhi \left( \frac{|f|}{c}\right) \in X. \) In any case, it follows that \(\varPhi (|f|)\in X,\) which means that \(f\in \widetilde{X}^{\varPhi }.\)
(ii) If \(\Vert f_n\Vert _{X^{\varPhi }}\rightarrow 0,\) then \(\Vert \varPhi (|f_n|)\Vert _{X}\rightarrow 0\) as a consequence of Lemma 3 (i). Suppose now that \(\Vert f_n\Vert _{X^{\varPhi }}\) does not converges to 0. Then, there exists \(\varepsilon >0\) and a subsequence \((f_{n_k})_k\) of \((f_n)_n\) such that \(\Vert f_{n_k}\Vert _{X^{\varPhi }}>\varepsilon \) for all \(k\in {\mathbb {N}}.\) We can assume that \(\varepsilon <1\) and that \((f_{n_k})_k\) is the whole \((f_n)_n\) without loss of generality. Since \(\varPhi \in \varDelta _2\) and \(\displaystyle \frac{1}{\varepsilon }>1,\) there exist \(C>1\) such that \(\displaystyle \varPhi \left( \frac{|f_n|}{\varepsilon }\right) \le C \varPhi (|f_n|).\) By (i), we deduce that \(\displaystyle \varPhi \left( \frac{|f_n|}{\varepsilon }\right) \in X\) and hence \(\displaystyle \left\| \varPhi \left( \frac{|f_n|}{\varepsilon }\right) \right\| _{X} > 1.\) Thus, \(\begin{aligned} \Vert \varPhi (|f_n|)\Vert _{X} \ge \displaystyle \frac{1}{C} \displaystyle \left\| \varPhi \left( \frac{|f_n|}{\varepsilon }\right) \right\| _{X}> \frac{1}{C} >0, \end{aligned}\) which means that \(\Vert \varPhi (|f_n|)\Vert _{X}\) does not converges to 0.
(iii) Let \((f_n)_n\) and f in \(X^{\varPhi }\) such that \(0\le f_n\uparrow f\) \(\mu \)-a.e. Then, \(\varPhi \left( f-f_n\right) \downarrow 0\) \(\mu \)-a.e. Since X is \(\sigma \)-order continuous, it follows that \(\Vert \varPhi \left( f-f_n\right) \Vert _{X}\rightarrow 0\) and by (ii) this implies that \(\Vert f-f_n\Vert _{X^{\varPhi }}\rightarrow 0,\) which gives the \(\sigma \)-order continuity of \(X^{\varPhi }.\) \(\square \)
5 Application: Orlicz spaces associated to a vector measure
First of all observe that classical Orlicz spaces \(L^{\varPhi }(\mu )\) with respect to a positive finite measure \(\mu \) are obtained applying the construction \(X^{\varPhi }\) of section 4 to the B.f.s. \(X=L^1(\mu ),\) that is, \(L^{\varPhi }(\mu ) = L^1(\mu )^{\varPhi }\) equipped with the norm \(\Vert \cdot \Vert _{L^{\varPhi }(\mu )} := \Vert \cdot \Vert _{L^1(\mu )^{\varPhi }}.\) Using these classical Orlicz spaces, the Orlicz spaces \(L^{\varPhi }_w(m)\) and \(L^{\varPhi }(m)\) with respect to a vector measure \(m:\varSigma \rightarrow Y\) were introduced in [8] in the following way:
equipped with the norm
and \(L^{\varPhi }(m)\) is the closure of simple functions \({\mathscr {S}}(\varSigma )\) in \(L^{\varPhi }_w(m).\) The next result establishes that these Orlicz spaces \(L^{\varPhi }_w(m)\) and \(L^{\varPhi }(m)\) can be obtained as generalized Orlicz spaces \(X^{\varPhi }\) by taking X to be \(L^1_w(m)\) and \(L^1(m),\) respectively.
Proposition 4
Let \(\varPhi \) be a Young function and \(m:\varSigma \rightarrow Y\) a vector measure.
-
(i)
\(L^{\varPhi }_w(m) = L^1_w(m)^{\varPhi }\) and \(\Vert f\Vert _{L^{\varPhi }_w(m)} = \Vert f\Vert _{L^1_w(m)^{\varPhi }},\) for all \(f\in L^{\varPhi }_w(m).\)
-
(ii)
\(L^{\varPhi }(m) \subseteq L^1(m)^{\varPhi }\) and if \(\varPhi \in \varDelta _2,\) then \(L^{\varPhi }(m) = L^1(m)^{\varPhi }.\)
Proof
(i) Suppose that \(f\in L^1_w(m)^{\varPhi }\) and let \(k>0\) such that \(\displaystyle \varPhi \left( \frac{|f|}{k}\right) \in L^1_w(m)\) with \(\displaystyle \left\| \varPhi \left( \frac{|f|}{k}\right) \right\| _{L^1_w(m)}\le 1.\) Given \(y^*\in B_{Y^*}\) we have \(\displaystyle \varPhi \left( \frac{|f|}{k}\right) \in L^1(|\langle m, y^*\rangle |)\) with
This implies that \(f\in L^{\varPhi }(|\langle m, y^*\rangle |)\) with \(\Vert f\Vert _{L^{\varPhi }(|\langle m, y^*\rangle |)}\le k.\) Hence, \(f\in L^{\varPhi }_w(m)\) with \(\Vert f\Vert _{L^{\varPhi }_w(m)} \le \Vert f\Vert _{L^1_w(m)^{\varPhi }}.\)
Reciprocally, suppose now that \(f\in L^{\varPhi }_w(m),\) write \(M:=\Vert f\Vert _{L^{\varPhi }_w(m)}\) and let \(y^*\in B_{Y^*}.\) Since \(f\in L^{\varPhi }(|\langle m, y^*\rangle |)\) and \(\Vert f\Vert _{L^{\varPhi }(|\langle m, y^*\rangle |)} \le M,\) we have that \(\displaystyle \frac{f}{M}\in L^{\varPhi }(|\langle m, y^*\rangle |)\) with \(\displaystyle \left\| \frac{f}{M}\right\| _{L^{\varPhi }(|\langle m, y^*\rangle |)} \le 1.\) Applying Theorem 4 (ii) to the space \(X=L^{1}(|\langle m, y^*\rangle |),\) it follows that \(\displaystyle \varPhi \left( \frac{|f|}{M}\right) \in L^1(|\langle m, y^*\rangle |)\) with \(\begin{aligned} \left\| \varPhi \left( \frac{|f|}{M}\right) \right\| _{L^1(|\langle m, y^*\rangle |)} \le \displaystyle \left\| \frac{f}{M}\right\| _{L^{\varPhi }(|\langle m, y^*\rangle |)} \le 1. \end{aligned}\) Then, the arbitrariness of \(y^*\in B_{Y^*}\) guarantees that \(\displaystyle \varPhi \left( \frac{|f|}{M}\right) \in L^1_w(m)\) with \(\displaystyle \left\| \varPhi \left( \frac{|f|}{M}\right) \right\| _{L^1_w(m)} \le 1\) and hence \(f\in L^1_w(m)^{\varPhi }\) with \(\Vert f\Vert _{L^1_w(m)^{\varPhi }} \le M.\)
(ii) Since \(L^1(m)^{\varPhi }\) is a B.f.s., simple functions \({\mathscr {S}}(\varSigma ) \subseteq L^1(m)^{\varPhi }\) and \(L^1(m)^{\varPhi }\) is a closed subspace of \(L^1_w(m)^{\varPhi }.\) Thus, taking in account (i), we deduce that \(L^{\varPhi }(m) \subseteq L^1(m)^{\varPhi }.\) If in addition \(\varPhi \in \varDelta _2,\) we have
where the first equality is due to Theorem 5 (i) applied to \(X=L^1(m)\) and the second one can be found in [8, Proposition 4.4]. \(\square \)
The Orlicz spaces \(L^{\varPhi }(m)\) have been recently employed in [5] to locate the compact subsets of \(L^{1}(m).\) Motivated by the idea of studying compactness in \(L^{1}(\Vert m\Vert )\) (see [6] for details), we introduce the Orlicz spaces \(L^{\varPhi }(\Vert m\Vert )\) as the Orlicz spaces \(X^{\varPhi }\) associated to the q-B.f.s. \(X = L^1(\Vert m\Vert ).\) For further reference, we collect together all the information that our general theory provide about these new Orlicz spaces.
Definition 4
Let \(\varPhi \) be a Young function and \(m:\varSigma \rightarrow Y\) a vector measure. We define the Orlicz spaces associated to the semivariation of m as \(L^{\varPhi }(\Vert m\Vert ):=L^1(\Vert m\Vert )^{\varPhi }\) equipped with \(\Vert f\Vert _{L^{\varPhi }(\Vert m\Vert )}:= \Vert f\Vert _{L^1(\Vert m\Vert )^{\varPhi }},\) for all \(f\in L^{\varPhi }(\Vert m\Vert ).\)
Corollary 2
Let \(\varPhi \) be a Young function, \(m:\varSigma \rightarrow Y\) a vector measure and \(\mu \) any Rybakov control measure for m. Then,
-
(i)
\(L^{\varPhi }(\Vert m\Vert )\) is a q-B.f.s. over \(\mu \) with the \(\sigma \)-Fatou property.
-
(ii)
If \(\varPhi \in \varDelta _2,\) then \(L^{\varPhi }(\Vert m\Vert )\) is \(\sigma \)-order continuous.
-
(iii)
\(L^{\varPhi }(\Vert m\Vert ) \subseteq L^1(\Vert m\Vert )\) with continuous inclusion.
Proof
Apply Theorems 3, 4 and 5 to the q-B.f.s \(X=L^1(\Vert m\Vert ).\) See also Proposition 2 and Remark 2. \(\square \)
Corollary 3
Let \(\varPhi \) be a Young function, \(m:\varSigma \rightarrow Y\) a vector measure, \(f\in L^{\varPhi }(\Vert m\Vert )\) and \(H\subseteq L^0(m).\)
-
(i)
If \(\varPhi (|f|)\in L^1(\Vert m\Vert ),\) then \(\Vert f\Vert _{L^{\varPhi }(\Vert m\Vert )} \le \max \{1, \Vert \varPhi (|f|)\Vert _{L^1(\Vert m\Vert )} \}.\)
-
(ii)
If \(\Vert f\Vert _{L^{\varPhi }(\Vert m\Vert )}\le 1,\) then \(\varPhi (|f|)\in L^1(\Vert m\Vert )\) and \(\Vert \varPhi (|f|)\Vert _{L^1(\Vert m\Vert )} \le \Vert f\Vert _{L^{\varPhi }(\Vert m\Vert )}.\)
-
(iii)
If \(\Vert f\Vert _{L^{\varPhi }(\Vert m\Vert )}>1\) and \(\varPhi (|f|)\in L^1(\Vert m\Vert ),\) then \(\Vert \varPhi (|f|)\Vert _{L^1(\Vert m\Vert )} \ge \Vert f\Vert _{L^{\varPhi }(\Vert m\Vert )}.\)
-
(iv)
If \(\{\varPhi (|h|) : h\in H \}\) is bounded in \(L^{1}(\Vert m\Vert ),\) then H is bounded in \(L^{\varPhi }(\Vert m\Vert ).\)
-
(v)
If H is bounded in \(L^{\varPhi }(\Vert m\Vert ),\) then there exists a Young function \(\Psi \) such that \(\{\Psi (|h|) : h\in H \}\) is bounded in \(L^{1}(\Vert m\Vert ).\)
Proof
Particularize Lemmas 2 and 3 to \(X=L^1(\Vert m\Vert ).\) Note that, in fact, we can use (ii) of Theorem 4. \(\square \)
Corollary 4
Let \(\varPhi \in \varDelta _2,\) \(m:\varSigma \rightarrow Y\) a vector measure and \((f_n)_n\subseteq L^{\varPhi }(\Vert m\Vert ).\)
-
(i)
\(L^{\varPhi }(\Vert m\Vert ) = \{ f\in L^0(m) : \varPhi (|f|)\in L^1(\Vert m\Vert ) \}.\)
-
(ii)
\(\Vert f_n\Vert _{L^{\varPhi }(\Vert m\Vert )}\rightarrow 0\) if and only if \(\Vert \varPhi (|f_n|)\Vert _{L^1(\Vert m\Vert )}\rightarrow 0.\)
Proof
Apply Theorem 5 to the space \(X=L^1(\Vert m\Vert ).\) \(\square \)
6 Application: interpolation of Orlicz spaces
In this section all the q-B.f.s. will be supposed to be complex. This means that \(L^0(\mu )\) will be assumed to be in fact the space of all (\(\mu \)-a.e. equivalence classes of) \({\mathbb {C}}\)-valued measurable functions on \(\varOmega .\) Recall that a complex q-B.f.s X over \(\mu \) is the complexification of the real q-B.f.s. \(X_{{\mathbb {R}}}:=X\cap L^0_{{\mathbb {R}}}(\mu ),\) where \(L^0_{{\mathbb {R}}}(\mu )\) is the space of all (\(\mu \)-a.e. equivalence classes of) \({\mathbb {R}}\)-valued measurable functions on \(\varOmega \) (see [18, p.24] for more details) and this allows to extend all the real q-B.f.s. defined above to complex q-B.f.s. following a standard argument.
The complex method of interpolation, \([X_0,X_1]_{\theta }\) with \(0<\theta <1,\) for pairs \((X_0,X_1)\) of quasi-Banach spaces was introduced in [10] as a natural extension of Calderón’s original definition for Banach spaces. It relies on a theory of analytic functions with values in quasi-Banach spaces which was developed in [10] and [10]. It is important to note that there is no analogue of the Maximum Modulus Principle for general quasi-Banach spaces, but there is a wide subclass of quasi-Banach spaces called analytically convex (A-convex) in which that principle does hold. For a q-B.f.s. X it can be proved that analytical convexity is equivalent to lattice convexity (L-convexity), i.e., there exists \(0<\varepsilon <1\) so that if \(f\in X\) and \(0\le f_i\le f,\) \(i=1,\dots ,n,\) satisfy \(\displaystyle \frac{f_1+\cdots + f_n}{n}\ge (1-\varepsilon )f,\) then \(\displaystyle \max _{1\le i \le n}\Vert f_i\Vert _{X} \ge \varepsilon \Vert f\Vert _{X}\) (see [10, Theorem 4.4]). This is also equivalent to X be s-convex for some \(s>0\) (see [10, Theorem 2.2]). We recall that X is called s-convex if there exists \(C\ge 1\) such that
for all \(n\in {\mathbb {N}}\) and \(f_1, \dots , f_n\in X.\) Observe that, X is s-convex if and only if its s-th power \(X_{[s]}\) is 1-convex, where the s-th power \(X_{[s]}\) of a q-B.f.s. X over \(\mu \) (for any \(0<s<\infty \)) is the q-B.f.s. \(X_{[s]}:=\displaystyle \left\{ f\in L^0(\mu ) : |f|^{\frac{1}{s}}\in X \right\} \) equipped with the quasi-norm \(\Vert f\Vert _{X_{[s]}} = \displaystyle \left\| |f|^{\frac{1}{s}} \right\| _{X}^{s},\) for all \(f\in X_{[s]}\) (see [18, Proposition 2.22]).
The following result provides a condition under which the L-convexity of X can be transferred to its Orlicz space \(X^{\varPhi }.\) When X possesses the \(\sigma \)-Fatou property, this can be derived from [10, Proposition 3.3], but we make apparent that this property can be dropped. Recall that a function \(\psi \) on the semiaxis \([0,\infty )\) is said to be quasiconcave if \(\psi (0)=0,\) \(\psi (t)\) is positive and increasing for \(t>0\) and \(\displaystyle \frac{\psi (t)}{t}\) is decreasing for \(t>0.\) Observe that a quasiconcave function \(\psi \) satisfies the following inequalities for all \(t\ge 0\):
Theorem 6
If X is an L-convex q-B.f.s. and \(\varPhi \in \varDelta _2,\) then \(X^{\varPhi }\) is L-convex.
Proof
Since \(\varPhi \in \varDelta _2,\) there exists \(s>1\) such that \(\varPhi (2t)\le s\varPhi (t)\) for all \(t\ge 0.\) From the inequality
it is easy to check that \(\displaystyle \frac{\varPhi (t)}{t^s}\) is decreasing and then \(\displaystyle \frac{\varPhi \left( t^{\frac{1}{s}}\right) }{t}\) so is. Therefore, the function \(\psi (t):=\varPhi \left( t^{\frac{1}{s}}\right) \) is quasiconcave. Take \(0<\delta <1\) such that \((1-\delta )^s=1-\varepsilon ,\) where \(\varepsilon \) is the constant from the L-convexity of X. Let \(f\in X^{\varPhi }\) and \(0\le f_i\le f,\) \(i=1,\dots ,n\) satisfying \(\displaystyle \frac{f_1+\cdots + f_n}{n}\ge (1-\delta )f.\) We can also assume that \(\Vert f\Vert _{X^{\varPhi }}=1\) without loss of generality. Note that this implies \(\Vert \varPhi (f)\Vert _{X} \ge 1.\) If we suppose, on the contrary, that \(\Vert \varPhi (f)\Vert _{X}<1\) and we take \(0<k<1\) such that \(\Vert \varPhi (f)\Vert _{X}< k^s < 1,\) then
and therefore \(\Vert f\Vert _{X^{\varPhi }}<k<1.\) Moreover, we have \(0\le \varPhi (f_i)\le \varPhi (f)\in X\) and
Thus, the L-convexity of X implies that \(\displaystyle \max _{1\le i \le n}\Vert \varPhi (f_i)\Vert _{X} \ge \varepsilon \Vert \varPhi (f)\Vert _{X} \ge \varepsilon \) and hence \(\displaystyle \max _{1\le i \le n}\Vert f_i\Vert _{X^{\varPhi }} \ge \varepsilon >\delta \) by (i) of Lemma 3. \(\square \)
The Calderón product \(X_0^{1-\theta }X_1^{\theta }\) of two q-B.f.s. \(X_0\) and \(X_1\) over \(\mu \) is the q-B.f.s. of all functions \(f\in L^0(\mu )\) such that there exist \(f_0\in B_{X_0},\) \(f_1\in B_{X_1}\) and \(\lambda >0\) for which
endowed with the quasi-norm \(\Vert f\Vert _{X_0^{1-\theta }X_1^{\theta }}= \inf \lambda ,\) where the infimum is taken over all \(\lambda \) satisfying (2). The complex method gives the result predicted by the Calderón product for nice pairs of q-B.f.s. (see [10, Theorem 3.4]).
Theorem 7
Let \(\varOmega \) be a Polish space and let \(\mu \) be a finite Borel measure on \(\varOmega .\) Let \(X_0,\) \(X_1\) be a pair of \(\sigma \)-order continuous L-convex q-B.f.s. over \(\mu .\) Then \(X_0+X_1\) is L-convex and \([X_0,X_1]_{\theta } = X_0^{1-\theta }X_1^{\theta }\) with equivalence of quasi-norms.
On the other hand, it is easy to compute the Calderón product of two Orlicz spaces associated to the same q-B.f.s:
Proposition 5
Let X be a q-B.f.s. over \(\mu ,\) \(\varPhi _0,\) \(\varPhi _1\) Young functions, \(0< \theta <1\) and \(\varPhi \) such that \(\varPhi ^{-1} := (\varPhi _{0}^{-1})^{1-\theta } (\varPhi _{1}^{-1})^{\theta }.\) Then \(\left( X^{\varPhi _0}\right) ^{1-\theta }\left( X^{\varPhi _1}\right) ^{\theta } = X^{\varPhi }.\)
Proof
Given \(f\in X^{\varPhi },\) there exists \(c>0\) such that \(h:=\displaystyle \varPhi \left( \frac{|f|}{c}\right) \in X\) and hence \(f_0:=\varPhi _0^{-1}(h)\in X^{\varPhi _0}\) and \(f_1:=\varPhi _1^{-1}(h)\in X^{\varPhi _1}.\) Taking \(\alpha :=\max \{ \Vert f_0\Vert _{X^{\varPhi _0}}, \Vert f_1\Vert _{X^{\varPhi _1}} \},\) it follows that
which yields \(f\in \left( X^{\varPhi _0}\right) ^{1-\theta }\left( X^{\varPhi _1}\right) ^{\theta }.\)
Conversely, if \(f\in \left( X^{\varPhi _0}\right) ^{1-\theta }\left( X^{\varPhi _1}\right) ^{\theta },\) then there exist \(\lambda >0,\) \(f_0\in X^{\varPhi _0}\) and \(f_1\in X^{\varPhi _1}\) such that \(|f|\le \lambda |f_0|^{1-\theta } |f_1|^{\theta }.\) This implies the existence of \(c>0\) such that \(h_0 := \displaystyle \varPhi _0\left( \frac{|f_0|}{c}\right) \in X\) and \(h_1 := \displaystyle \varPhi _1\left( \frac{|f_1|}{c}\right) \in X.\) Thus, taking \(h:=h_0+h_1\in X,\) we deduce that
and hence \(f\in X^{\varPhi }.\) \(\square \)
Combining the three previous results, we obtain conditions under which the complex method applied to Orlicz spaces associated to a q-B.f.s. over \(\mu \) keeps on producing an Orlicz space associated to the same q-B.f.s.
Corollary 5
Let \(\varOmega \) be a Polish space and let \(\mu \) be a finite Borel measure on \(\varOmega .\) Let X be an L-convex, \(\sigma \)-order continuous q-B.f.s. over \(\mu ,\) \(\varPhi _0, \varPhi _1\in \varDelta _2,\) \(0<\theta <1\) and \(\varPhi \) such that \(\varPhi ^{-1} := (\varPhi _{0}^{-1})^{1-\theta } (\varPhi _{1}^{-1})^{\theta }.\) Then, \([X^{\varPhi _0},X^{\varPhi _1}]_{\theta } = X^{\varPhi }.\)
Proof
According to Theorems 5 and 6, the hypotheses guarantee that \(X^{\varPhi _0}\) and \(X^{\varPhi _1}\) are L-convex, \(\sigma \)-order continuous q-B.f.s. Therefore, the result follows by applying Theorem 7 and Proposition 5. \(\square \)
Let us denote \(L^s(\Vert m\Vert ):=L^1(\Vert m\Vert )_{\left[ \frac{1}{s}\right] },\) for \(0<s<\infty \) and \(m:\varSigma \rightarrow Y\) a vector measure. In [4, Proposition 4.1] we proved that if \(s>1,\) then \(L^s(\Vert m\Vert )\) is r-convex for every \(r < s.\) In fact, this is true for all \(0<s<\infty \) because if \(0<s\le 1\) and \(r<s,\) then \(\displaystyle \frac{s}{r}>1\) and hence \(L^{\frac{s}{r}}(\Vert m\Vert )\) is 1-convex, that is \(L^s(\Vert m\Vert )_{[r]}\) is 1-convex, which is equivalent to \(L^s(\Vert m\Vert )\) be r-convex. This means that \(L^s(\Vert m\Vert )\) is L-convex for all \(0<s<\infty .\) In particular, \(L^1(\Vert m\Vert )\) is L-convex and we can apply Corollary 5 to it.
Corollary 6
Let \(\varOmega \) be a Polish space and let \(\mu \) be a Borel measure which is a Rybakov control measure for m. Let \(\varPhi _0, \varPhi _1\in \varDelta _2,\) \(0<\theta <1\) and \(\varPhi \) such that \(\varPhi ^{-1} := (\varPhi _{0}^{-1})^{1-\theta } (\varPhi _{1}^{-1})^{\theta }.\) Then, \([L^{\varPhi _0}(\Vert m\Vert ),L^{\varPhi _1}(\Vert m\Vert )]_{\theta } = L^{\varPhi }(\Vert m\Vert ).\)
For a similar result about complex interpolation of Orlicz type spaces \(L^{\Phi}(m)\) and \(L_w^{\Phi}(m)\) see [3, Corollary 4.2 and Theorem 4.5].
Note that, for \(p>1,\) \(\displaystyle \frac{1}{p}\)-th powers are an special case of Orlicz spaces, since \(X_{\left[ \frac{1}{p}\right] } = X^{\varPhi _{[p]}},\) where \(\varPhi _{[p]}(t)=t^p.\) If we particularize the previous Corollary to these powers, then we obtain the interpolation result below for \(L^p(\Vert m\Vert )\) spaces. In fact, this result is valid for all \(0<p_0, p_1<\infty \) due to the fact that the Calderón product commutes with powers for all indices.
Corollary 7
Let \(\varOmega \) be a Polish space and let \(\mu \) be a Borel measure which is a Rybakov control measure for m. Let \(0<\theta <1\) and \(0<p_0, p_1<\infty .\) Then \([L^{p_0}(\Vert m\Vert ),L^{p_1}(\Vert m\Vert )]_{\theta } = L^{p}(\Vert m\Vert ),\) where \(\displaystyle \frac{1}{p} = \frac{1-\theta }{p_0} + \frac{\theta }{p_1}.\)
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This research has been partially supported by La Junta de Andalucía (Spain) under the Grant FQM-133.
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del Campo, R., Fernández, A., Mayoral, F. et al. Orlicz spaces associated to a quasi-Banach function space: applications to vector measures and interpolation. Collect. Math. 72, 481–499 (2021). https://doi.org/10.1007/s13348-020-00295-1
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DOI: https://doi.org/10.1007/s13348-020-00295-1