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MaureyRosenthal domination for abstract Banach lattices
Journal of Inequalities and Applications volume 2013, Article number: 213 (2013)
Abstract
We extend the MaureyRosenthal theorem on integral domination and factorization of pconcave operators from a pconvex Banach function space through {L}^{p}spaces for the case of operators on abstract pconvex Banach lattices satisfying some essential lattice requirements  mainly order density of its order continuous part  that are shown to be necessary. We prove that these geometric properties can be characterized by means of an integral inequality giving a domination of the pointwise evaluation of the operator for a suitable weight also in the case of abstract Banach lattices. We obtain in this way what in a sense can be considered the most general factorization theorem of operators through {L}^{p}spaces. In order to do this, we prove a new representation theorem for abstract pconvex Banach lattices with the Fatou property as spaces of pintegrable functions with respect to a vector measure.
MSC:46G10, 46E30, 46B42.
1 Introduction
The so called MaureyRosenthal theorem on domination and factorization of operators through {L}^{p}spaces provides a large set of tools for the analysis of operators on Banach spaces. Essentially, this result provides, for a Banach lattice F of a particular class, an equivalence between the pconcavity of a Banach space valued operator T:F\to E and the fact that it satisfies a pointwise integral inequality involving the norms of the evaluations \parallel T(x)\parallel, x\in F. In recent years, this (family of) theorem(s) has been studied widely by several authors, and nowadays we have a clear methodology for giving a unified version of this technique. Following the seminal ideas in [1] that allowed the understanding by means of the basic principle of a lot of similar arguments found in the mathematical literature since the sixties, several generalizations and applications have been done (see, for instance, [2–5]).
The version of this theorem that is normally used depends strongly on two facts that are deeply connected with the essential nature of the Banach function spaces.

(1)
The (topological) dual X{(\mu )}^{\ast} of an order continuous Banach function space X(\mu ) (when defined over a σadditive measure μ) is again a Banach function space (the Köthe dual or associate space X{(\mu )}^{\prime}), i.e., X{(\mu )}^{\ast}=X{(\mu )}^{\prime}. The second space is defined as integrals of some class of functions.

(2)
If X(\mu ) is a pconvex Banach function space, its p th power is again a Banach function space.
In this paper we show a general version of the MaureyRosenthal theorem that is obtained by relaxing these requirements as much as possible. In order to do this, we provide a function space representation of a class of Banach lattices satisfying the necessary requirements, which allows to perform the tandem ‘Banach lattices/order notions’ versus ‘function spaces/vector measures’ to provide the support for the factorization through an {L}^{p}space. Therefore, the key of the arguments that prove this type of factorization is in fact the structure of the space from which the operator is defined. This is the reason why our main tool is given by the representation of Banach lattices by means of spaces {L}^{p}(m) and {L}_{w}^{p}(m) of pintegrable functions with respect to a vector measure on a δring. This allows, for instance, to prove factorization theorems through {\ell}^{p}(\mathrm{\Gamma})spaces of an uncountable set Γ.
From the point of view of the general representation of Banach lattices as function spaces, the main goal of this paper is to get a representation theorem for pconvex Banach lattices with the (σ)Fatou property as general as possible by using vector measures defined on a δring to complete the picture when the existence of a weak unit is not assumed. Section 3 is devoted to this. This becomes the main tool for proving, in Section 4, our general MaureyRosenthal theorem, closing in this way the question of how far this kind of arguments can be extended: examples and counterexamples are also given.
2 Preliminaries
As the reader will notice soon, the setting that is needed for proving the main result of this paper (Theorem 4) and allows to prove the most general version of the MaureyRosenthal theorem (Corollary 6) is unusually technical and contains some notions and arguments that are not standard in the framework of the Banach lattices. This is the reason why we include a long section of preliminaries collecting all the results that are needed.
2.1 Banach lattices
We mainly use the terminology and the notation of [6] and [7]. Let F be a real Banach lattice. An ideal \tilde{F} of F is a closed subspace of F satisfying that if y\in F with y\le x for some x\in \tilde{F}, then y\in \tilde{F}. An ideal \tilde{F} in F is said to be order dense in F if, for every 0\le x\in F, there exists an upwards directed system ({x}_{\tau})\subset \tilde{F} such that 0\le {x}_{\tau}\uparrow x, and is said to be super order dense if this condition holds for increasing sequences. We say that F is order continuous if, for every ({x}_{\tau})\subset F with {x}_{\tau}\downarrow 0, it follows that \parallel {x}_{\tau}\parallel \downarrow 0 and F is σorder continuous if this is the behavior on sequences. We denote by {F}_{an} the order continuous part of F, that is, the largest order continuous ideal in F. Similarly, the σorder continuous part of F is the largest σorder continuous ideal in F and is denoted by {F}_{a}. Of course {F}_{an}\subset {F}_{a}. We say that F has the Fatou property if, for every upwards directed system 0\le {x}_{\tau}\uparrow in F such that sup\parallel {x}_{\tau}\parallel <\mathrm{\infty}, it follows that there exists x=sup{x}_{\tau} in F and, moreover, \parallel x\parallel =sup\parallel {x}_{\tau}\parallel. If this condition holds for increasing sequences, we say that F has the σFatou property. Remark that under these conditions, {F}_{an}={F}_{a} (see, for instance, [[7], Theorems 113.1, 103.6]).
Let 1\le p<\mathrm{\infty}. We say that F is pconvex if there exists a constant M>0 such that
for all n and {x}_{1},\dots ,{x}_{n}\in F. The smallest constant satisfying the previous inequality is called the pconvexity constant of F and is denoted by {\mathbf{M}}^{(p)}(F). Similarly, a linear operator T:F\to E, where F is a Banach lattice and E is an arbitrary Banach space, is said to be pconcave if there exists a constant M<\mathrm{\infty} such that
for every choice of vectors {({x}_{j})}_{j=1}^{n}, n\in \mathbb{N} in F. The smallest possible value of M is denoted by {\mathbf{M}}_{(p)}(T).
Let F, \tilde{F} be Banach lattices and let T:F\to \tilde{F} be a linear operator. If Tx\ge 0 whenever 0\le x\in F, the operator T is said to be positive. Every positive linear operator between Banach lattices is always continuous (see [[8], p.2] or [[9], Theorem 4.3]) and, in particular, every inclusion F\subset \tilde{F} of Banach lattices with the same order is continuous. We say that T is an order isomorphism if it is onetoone, onto and satisfies T(x\wedge y)=Tx\wedge Ty for all x,y\in F. If, moreover, {\parallel Tx\parallel}_{\tilde{F}}={\parallel x\parallel}_{F} for all x\in F, we will say that T is an order isometry. We say that F and \tilde{F} are order isomorphic (order isometrics) if there is an order isomorphism (isometry) T:F\to \tilde{F}.
Let (\mathrm{\Omega},\mathrm{\Sigma},\mu ) be a measure space (without assumptions of finiteness on μ). As usual, a property holds μalmost everywhere (briefly, μa.e.) if it holds except on a μnull set. We denote by {L}^{0}(\mu ) the space of all measurable real functions on Ω, where functions which are equal μa.e. are identified. The space {L}^{0}(\mu ) is an Archimedean vector lattice when endowed with the μa.e. pointwise order. By a Banach function space (briefly, B.f.s.) related to μ, we mean a Banach space E\subset {L}^{0}(\mu ) satisfying that for every f\in {L}^{0}(\mu ), we have f\in E whenever f\le g with g\in E and, moreover, {\parallel f\parallel}_{E}\le {\parallel g\parallel}_{E}. Every B.f.s. is a Banach lattice with the μa.e. pointwise order, in which convergence in norm of a sequence implies μa.e. convergence for some subsequence.
2.2 Integration with respect to vector measures on δrings
The spaces {L}^{1}(\nu ) and {L}_{w}^{1}(\nu ) of integrable and weakly integrable functions with respect to a vector measure defined on a σalgebra and with values in a Banach space E have been studied in depth by many authors and their behavior is now well understood (see [10], [[11], Chapter 3] and the references therein). However, this setting is not rich enough for our analysis, since vector measures on σalgebras can only be used for representing Banach lattices which have a weak unit, and our needs require to work in a more general context. In [12], there is an analysis of the space {L}^{1}(\nu ) with ν defined on a δring and a detailed study of the lattice behavior of the corresponding space {L}_{w}^{1}(\nu ) can be found in [13]. The results in both papers give evidence of how large the difference can be between the δring and σalgebra cases and justify the use of the general framework of δrings in this paper. More information on the integration of vector measures on δrings and its applications can be found in [14–17] and [[18], pp.2223].
For the pconvexification of these spaces  the spaces {L}^{p}(\nu ) and {L}_{w}^{p}(\nu ) of pintegrable functions  the fundamental results that are needed in this paper are also known. When the vector measure ν is defined on a σalgebra, all the relevant (geometric, lattice, topological) properties of the spaces {L}^{p}(\nu ) of pintegrable functions and {L}_{w}^{p}(\nu ) of weakly pintegrable functions with 1\le p<\mathrm{\infty} can be found in [19] and [11, 19, 20]. For the δring case, the study of the main lattice properties of the spaces {L}^{p}(\nu ) and {L}_{w}^{p}(\nu ) is developed in [21], where the general case 0<p<\mathrm{\infty} is also considered (although for 0<p<1 these spaces are not necessarily Banach spaces, completeness is proved, but under a quasinorm, and the analogous definitions for Banach lattices are considered).
The theory of integration with respect to a vector measure defined on a δring is due to Lewis [22] and Masani and Niemi [23, 24] (see also [12, 13] and [21]). This integration theory extends the classical one for vector measures defined on σalgebras. Let ℛ be a δring of subsets of an abstract set Ω, that is, a ring closed under countable intersections. We denote by {\mathcal{R}}^{\mathrm{loc}} the σalgebra given by the subsets A\subset \mathrm{\Omega} such that A\cap B\in \mathcal{R} for all B\in \mathcal{R}. Clearly, if ℛ is a σalgebra, then {\mathcal{R}}^{\mathrm{loc}}=\mathcal{R}. We write \mathcal{M}({\mathcal{R}}^{\mathrm{loc}}) for the space of measurable real functions on (\mathrm{\Omega},{\mathcal{R}}^{\mathrm{loc}}) and \mathcal{S}(\mathcal{R}) for the space of simple functions with support in ℛ (or ℛsimple functions).
Let \lambda :\mathcal{R}\to \mathbb{R} be a countably additive measure, that is, for every sequence ({A}_{n}) of pairwise disjoint sets in ℛ with \bigcup {A}_{n} in ℛ, the sum \sum \lambda ({A}_{n}) converges to \lambda (\bigcup {A}_{n}). The variation of λ is the countably additive measure \lambda :{\mathcal{R}}^{\mathrm{loc}}\to [0,\mathrm{\infty}] defined by
We have that \lambda (A)<\mathrm{\infty} for every A\in \mathcal{R}. The space {L}^{1}(\lambda ) of integrable functions with respect to λ is defined as the space {L}^{1}(\lambda ) with the usual norm. Every ℛsimple function \phi ={\sum}_{i=1}^{n}{\alpha}_{i}{\chi}_{{A}_{i}} is in {L}^{1}(\lambda ), where the integral of φ with respect to λ is defined by \int \phi \phantom{\rule{0.2em}{0ex}}d\lambda ={\sum}_{i=1}^{n}{\alpha}_{i}\lambda ({A}_{i}). Furthermore, the space \mathcal{S}(\mathcal{R}) is dense in {L}^{1}(\lambda ). For every f\in {L}^{1}(\lambda ), the integral of f with respect to λ is defined, as usual, as \int f\phantom{\rule{0.2em}{0ex}}d\lambda =lim\int {\phi}_{n}\phantom{\rule{0.2em}{0ex}}d\lambda for any sequence ({\phi}_{n})\subset \mathcal{S}(\mathcal{R}) converging to f in {L}^{1}(\lambda ).
Let E be a Banach space and let \nu :\mathcal{R}\to E be a vector measure, that is, for every sequence ({A}_{n}) of pairwise disjoint sets in ℛ with \bigcup {A}_{n}\in \mathcal{R}, the sum \sum \nu ({A}_{n}) converges to \nu (\bigcup {A}_{n}) in E. The semivariation of ν is the map \parallel \nu \parallel :{\mathcal{R}}^{\mathrm{loc}}\to [0,\mathrm{\infty}] given by \parallel \nu \parallel (A)=sup\{{x}^{\ast}\nu (A):{x}^{\ast}\in {B}_{{E}^{\ast}}\} for all A\in {\mathcal{R}}^{\mathrm{loc}}, where {x}^{\ast}\nu  is the variation of the measure {x}^{\ast}\nu :\mathcal{R}\to \mathbb{R} given by {x}^{\ast}\nu (A)=\u3008\nu (A),{x}^{\ast}\u3009, {E}^{\ast} denotes the topological dual of E and {B}_{{E}^{\ast}} the unit ball of {E}^{\ast}. The semivariation of ν is monotone increasing, countably subadditive, finite on ℛ and satisfies \frac{1}{2}\parallel \nu \parallel (A)\le sup\{{\parallel \nu (B)\parallel}_{X}:B\in \mathcal{R}\cap {2}^{A}\}\le \parallel \nu \parallel (A) for all A\in {\mathcal{R}}^{\mathrm{loc}}. Thus, the vector measure ν is bounded (i.e., its range is a bounded set in E) if and only if \parallel \nu \parallel (\mathrm{\Omega})<\mathrm{\infty}. A set A\in {\mathcal{R}}^{\mathrm{loc}} is νnull if \parallel \nu \parallel (A)=0, or equivalently, \nu (B)=0 for all B\in \mathcal{R}\cap {2}^{A} and a property holds νalmost everywhere (briefly, νa.e.) if it holds as usual except on a νnull set.
A function f\in \mathcal{M}({\mathcal{R}}^{\mathrm{loc}}) is said to be weakly integrable with respect to ν if f\in {L}^{1}({x}^{\ast}\nu ) for all {x}^{\ast}\in {E}^{\ast}, or equivalently, if {\parallel f\parallel}_{\nu}<\mathrm{\infty}, where
We denote by {L}_{w}^{1}(\nu ) the space of all weakly integrable functions with respect to ν, where functions which are equal νa.e. are identified, which is a Banach space when endowed with the norm {\parallel \cdot \parallel}_{\nu}. A function f\in {L}_{w}^{1}(\nu ) is integrable with respect to ν if, for each A\in {\mathcal{R}}^{\mathrm{loc}}, there exists a vector denoted by {\int}_{A}f\phantom{\rule{0.2em}{0ex}}d\nu \in E, satisfying the barycentric formula {x}^{\ast}({\int}_{A}f\phantom{\rule{0.2em}{0ex}}d\nu )={\int}_{A}f\phantom{\rule{0.2em}{0ex}}d{x}^{\ast}\nu for all {x}^{\ast}\in {E}^{\ast}. We write \int f\phantom{\rule{0.2em}{0ex}}d\nu for {\int}_{\mathrm{\Omega}}f\phantom{\rule{0.2em}{0ex}}d\nu. We denote by {L}^{1}(\nu ) the space of all integrable functions with respect to ν. Then, since {L}^{1}(\nu ) is a closed subspace of {L}_{w}^{1}(\nu ), it is a Banach space with the norm {\parallel \cdot \parallel}_{\nu}. Moreover, \mathcal{S}(\mathcal{R}) is dense in {L}^{1}(\nu ), where for every ℛsimple function \phi ={\sum}_{i=1}^{n}{\alpha}_{i}{\chi}_{{A}_{i}}, we have that \int \phi \phantom{\rule{0.2em}{0ex}}d\nu ={\sum}_{i=1}^{n}{\alpha}_{i}\nu ({A}_{i}). The equality {L}_{w}^{1}(\nu )={L}^{1}(\nu ) holds whenever the space E does not contain a copy of {c}_{0} (see [[22], Theorem 5.1]).
We will identify {L}^{0}(\nu )={L}^{0}(\mu ) and say B.f.s. related to ν for B.f.s. related to μ, for any measure \mu :{\mathcal{R}}^{\mathrm{loc}}\to [0,\mathrm{\infty}] with the same null sets as ν (the existence of such a measure is guaranteed in [[25], Theorem 3.2]). Therefore, {L}^{1}(\nu ) and {L}_{w}^{1}(\nu ) are both B.f.s.’ related to ν. The space {L}^{1}(\nu ) is always order continuous and {L}_{w}^{1}(\nu ) have the σFatou property. Furthermore, {L}^{1}(\nu ) is always order dense in {L}_{w}^{1}(\nu ) (actually, in {L}^{0}(\nu )) and {({L}_{w}^{1}(\nu ))}_{a}={L}^{1}(\nu ). We denote by {[{L}^{1}(\nu )]}_{\sigma \text{F}} the minimal B.f.s. related to ν, with the σFatou property and containing {L}^{1}(\nu ). It can be described as
The integration operator {I}_{\nu}:{L}^{1}(\nu )\to E given by {I}_{\nu}(f)=\int f\phantom{\rule{0.2em}{0ex}}d\nu is linear and continuous with \parallel {I}_{\nu}(f)\parallel \le {\parallel f\parallel}_{\nu}. A vector measure \nu :\mathcal{R}\to E with values in a Banach lattice E is positive if \nu (A)\ge 0 for all A\in \mathcal{R}.
Given 0<p<\mathrm{\infty}, the p th power space of {L}_{w}^{1}(\nu ) is defined as {L}_{w}^{p}(\nu )=\{f\in {L}^{0}(\nu ):{f}^{p}\in {L}_{w}^{1}(\nu )\} and a function in {L}_{w}^{p}(\nu ) will be called weakly pintegrable with respect to ν. Similarly, the p th power space of {L}^{1}(\nu ) is defined as {L}^{p}(\nu )=\{f\in {L}^{0}(\nu ):{f}^{p}\in {L}^{1}(\nu )\}, and a function in {L}^{p}(\nu ) will be called pintegrable with respect to ν. Both spaces are B.f.s.’ related to ν for p\ge 1 and qB.f.s.’ for p<1 when the lattice (quasi)norm given by {\parallel f\parallel}_{p,\nu}={\parallel {f}^{p}\parallel}_{\nu}^{\frac{1}{p}}, f\in {L}_{w}^{p}(\nu ), is considered. The space {L}^{p}(\nu ) is order continuous and the space {L}_{w}^{p}(\nu ) has the σFatou property. Moreover, \mathcal{S}(\mathcal{R}) is dense in {L}^{p}(\nu ); {({L}_{w}^{p}(\nu ))}_{a}={L}^{p}(\nu ); {L}^{p}(\nu ) is order dense in {L}_{w}^{p}(\nu ) (also in {L}^{0}(\nu )) and if {L}_{w}^{1}(\nu ) has the Fatou property, so does {L}_{w}^{p}(\nu ). Related to the convexity behavior of these spaces, both are pconvex with pconvexity constant {\mathbf{M}}^{(p)}({L}_{w}^{p}(\nu ))={\mathbf{M}}^{(p)}({L}^{p}(\nu ))=1.
3 Representing pconvex Banach lattices
Representation of Banach lattices as spaces of integrable functions with respect to a vector measure is nowadays a wellknown useful technique. Depending on the fact that either the lattice contains a weak unit or not, either vector measures on σalgebras or on δrings must be used. Curbera proved in [[26], Theorem 8] that an order continuous Banach lattice F with a weak unit is always order isometric to a space {L}^{1}(\nu ), where ν is defined on a σalgebra. If the existence of a weak unit is not assumed, the result remains true but for ν defined on a δring. This was first stated in [[18], pp.2223] with an outlined proof and later in [[27], Theorem 5] with a full detailed proof. Thinking now about the space {L}_{w}^{1}(\nu ), Curbera and Ricker showed in [[28], Theorem 2.5] that every Banach lattice F with the σFatou property, having a weak unit which belongs to the σorder continuous part {F}_{a} of F, is order isometric to a space {L}_{w}^{1}(\nu ) with ν defined on a σalgebra. Again, the corresponding result in the case when F has not a weak unit can be established by using a vector measure defined on a δring as Delgado and Juan proved in [[27], Theorem 10]. In this case, every Banach lattice F with the Fatou property, and such that its σorder continuous part is an order dense subset in F, can be represented as a space {L}_{w}^{1}(\nu ) for some vector measure ν defined on a δring. Furthermore, a representation theorem for the class of σFatou Banach lattices F with the σorder continuous part as a super order dense ideal in F, using again vector measures on δrings, is established in [[13], Proposition 6.1]. In this case, F is order isometric to the σFatou completion of {L}^{1}(\nu ).
Similar results are known for representing Banach lattices with convexity properties, and in this case, the spaces of pintegrable functions with respect to vector measures play a fundamental role. For 1<p<\mathrm{\infty}, if F is an order continuous pconvex Banach lattice, then F is order isomorphic to an {L}^{p}space with ν defined on a δring [[21], Theorem 10]. When there exists also a weak unit in F, the Banach lattice can be represented with an {L}^{p}space but with ν defined on a σalgebra (see [[20], Proposition 2.4]). On the other hand, if E is a pconvex Banach lattice with the σFatou property and has a weak unit belonging to {E}_{a}, then E is order isomorphic to a space {L}_{w}^{p}(\nu ) with ν on a σalgebra [29]. The aim of this section is to get a representation theorem for pconvex Banach lattices with the (σ)Fatou property as general as possible by using vector measures defined on a δring to complete the picture when the existence of a weak unit is not assumed. The starting point to prove this result is the corresponding representation theorem for the case p=1. In the proof of this theorem, the vector measure which allows to establish the order isometry has a special behavior (the socalled ℛdecomposability), under which the space {L}_{w}^{1}(\nu ) has the Fatou property (see [[27], Proposition 8] and [[13], Theorem 5.8 and Section 6]). We recall here the corresponding definition as follows.
A vector measure ν is said to be ℛdecomposable if we can write \mathrm{\Omega}=({\bigcup}_{\alpha \in \mathrm{\Delta}}{\mathrm{\Omega}}_{\alpha})\cup N, where N\in {\mathcal{R}}^{\mathrm{loc}} is a νnull set and \{{\mathrm{\Omega}}_{\alpha}:\alpha \in \mathrm{\Delta}\} is a family of pairwise disjoint sets in ℛ satisfying that

(i)
if {A}_{\alpha}\in \mathcal{R}\cap {2}^{{\mathrm{\Omega}}_{\alpha}} for all \alpha \in \mathrm{\Delta}, then {\bigcup}_{\alpha \in \mathrm{\Delta}}{A}_{\alpha}\in {\mathcal{R}}^{\mathrm{loc}}, and

(ii)
for each {x}^{\ast}\in {X}^{\ast}, if {Z}_{\alpha}\in \mathcal{R}\cap {2}^{{\mathrm{\Omega}}_{\alpha}} is {x}^{\ast}\nu null for all \alpha \in \mathrm{\Delta}, then {\bigcup}_{\alpha \in \mathrm{\Delta}}{Z}_{\alpha} is {x}^{\ast}\nu null.
(N can be taken to be disjoint with {\bigcup}_{\alpha \in \mathrm{\Delta}}{\mathrm{\Omega}}_{\alpha}).
Note that each pconvex Banach lattice F can be renormed equivalently in a way that F with the new norm and the same order is a pconvex Banach lattice with pconvexity constant equal to 1 (see [[8], Proposition 1.d.8]).
Theorem 1 Let p>1 and let F be a pconvex Banach lattice with pconvexity constant equal to 1, having the Fatou property and such that its σorder continuous part {F}_{a} is an order dense subset. Then there exists a vector measure ν on a δring and with values in {F}_{a} such that F and {L}_{w}^{p}(\nu ) are order isometric.
Proof The hypothesis on F gives an ℛdecomposable vector measure {\nu}_{1} on a δring ℛ and an order isometry \phi :F\to {L}_{w}^{1}({\nu}_{1}) (see Theorem 10 in [27] and Section 6 in [13]). Remark that {L}_{w}^{1}({\nu}_{1}) is then pconvex with pconvexity constant equal to 1 and, consequently, the space {L}_{w}^{1/p}({\nu}_{1}) is a B.f.s., that is, its quasinorm is actually a norm (see [[13], Proposition 6]). Moreover, {L}_{w}^{1/p}({\nu}_{1}) has the Fatou property and {L}^{1/p}({\nu}_{1}) is order dense in {L}_{w}^{1/p}({\nu}_{1}) (see the comments before Section 4 in [21]). Take now the vector measure {\nu}_{2}:\mathcal{R}\to {L}^{1/p}({\nu}_{1}) defined by {\nu}_{2}(A)={\chi}_{A}, A\in \mathcal{R} for which the integration operator {I}_{{\nu}_{2}}:{L}^{1}({\nu}_{2})\to {L}^{1/p}({\nu}_{1}) is the identity map and {L}^{1}({\nu}_{2})={L}^{1/p}({\nu}_{1}) with equal norms (see the proof of Theorem 10 in [21]). It can be checked easily that {\nu}_{2} is also ℛdecomposable since {\nu}_{1} is; in order to see this, note that the vector measures {\nu}_{1} and {\nu}_{2} have the same null sets, and follow the construction of ℛ in Section 3 in [27]. Hence, {L}_{w}^{1}({\nu}_{2}) has the Fatou property (see [[13], Theorem 5.8]).
We claim now that {L}_{w}^{1}({\nu}_{2})={L}_{w}^{1/p}({\nu}_{1}) with equal norms. For showing this, take 0\le f\in {L}_{w}^{1}({\nu}_{2}). Since {L}^{1}({\nu}_{2}) is order dense in {L}^{0}({\nu}_{2}) (see Remark 4.3 in [13]), there exists an upwards directed system {({f}_{\tau})}_{\tau} in {L}^{1}({\nu}_{2}) such that 0\le {f}_{\tau}\uparrow f in {L}^{0}({\nu}_{2}). Then 0\le {f}_{\tau}\uparrow in {L}_{w}^{1/p}({\nu}_{1}) and sup{\parallel {f}_{\tau}\parallel}_{\frac{1}{p},{\nu}_{1}}=sup{\parallel {f}_{\tau}\parallel}_{{\nu}_{2}}\le {\parallel f\parallel}_{{\nu}_{2}}. Therefore, the Fatou property of {L}_{w}^{1/p}({\nu}_{1}) gives h\in {L}_{w}^{1/p}({\nu}_{1}) such that {\parallel h\parallel}_{\frac{1}{p},{\nu}_{1}}={sup}_{\tau}{\parallel {f}_{\tau}\parallel}_{\frac{1}{p},{\nu}_{1}}. Since for each τ we have that {f}_{\tau}\le h {\nu}_{1}a.e. or, equivalently, {\nu}_{2}a.e., then f\le h and so f\in {L}_{w}^{1/p}({\nu}_{1}). On the other hand, {f}_{\tau}\le f {\nu}_{2}a.e. (i.e., {\nu}_{1}a.e.) for all τ and thus h\le f. Therefore, {\parallel f\parallel}_{\frac{1}{p},{\nu}_{1}}={\parallel h\parallel}_{\frac{1}{p},{\nu}_{1}}={sup}_{\tau}{\parallel {f}_{\tau}\parallel}_{\frac{1}{p},{\nu}_{1}}={sup}_{\tau}{\parallel {f}_{\tau}\parallel}_{{\nu}_{2}} due to the Fatou property of {L}_{w}^{1}({\nu}_{2}) as 0\le {f}_{\tau}\uparrow f also in {L}_{w}^{1}({\nu}_{2}). By taking positive and negative parts for a general f\in {L}_{w}^{1}({\nu}_{2}), we have that {L}_{w}^{1}({\nu}_{2})\subset {L}_{w}^{1/p}({\nu}_{1}) with equal norms.
The converse inclusion can be proved by using the same arguments. Therefore, the equality {L}_{w}^{1}({\nu}_{2})={L}_{w}^{1/p}({\nu}_{1}) holds with equal norms. Consequently, {L}_{w}^{p}({\nu}_{2})={L}_{w}^{1}({\nu}_{1}) with equal norms, and hence E and {L}_{w}^{p}({\nu}_{2}) are order isometric. □
Remark 2 A proof based on similar arguments to those in the previous theorem allows us to represent pconvex Banach lattices (with pconvexity constant equal to 1) having the σFatou property and such that {E}_{a} is super order dense in E. In this case, E is order isometric to {[{L}^{p}(\nu )]}_{\sigma \text{F}} for some vector measure ν defined on a δring. This result generalizes [[29], Theorem 4] where E has a weak unit in {E}_{a}. Remark that our proof differs from the one given in [[29], Theorem 4].
4 The MaureyRosenthal theorem for abstract Banach lattices
In this section we prove the main result of this paper. As we said in the introduction, our aim is to explore the limits of the arguments that allow to prove the factorization theorem regarding the structure of the Banach lattice where the operator is defined.
We say that an operator T:G\to E from a closed subspace G of a Banach lattice F with {F}_{an}\subset G\subset F on a Banach space E is Fatou if it satisfies that for every upwards directed system 0\le {x}_{\tau}\uparrow x, with {x}_{\tau}\in {F}_{an} for every τ and x\in G, we have that {lim}_{\tau}\parallel T({x}_{\tau})\parallel =\parallel T(x)\parallel. Continuous operators between two Banach lattices F and \tilde{F}, when restricted to G={F}_{an}, are clearly Fatou. Also, positive order continuous operators from a Banach lattice F into a Fatou Banach lattice \tilde{F}, when restricted to a closed subspace G of F with {F}_{an}\subset G\subset F, are so (for the definition of order continuous operator, see, for instance, [9]). Obviously, there are examples of operators which are not Fatou, as we show in the next example.
Example 3 Consider {\ell}^{\mathrm{\infty}}(\mathrm{\Gamma}), where Γ is an uncountable set. Take its (σ)order continuous part {c}_{0}(\mathrm{\Gamma}) and a continuous operator S:{c}_{0}(\mathrm{\Gamma})\to {\ell}^{\mathrm{\infty}}(\mathrm{\Gamma}) with \parallel S\parallel =\frac{1}{2}. Use the HahnBanach theorem to find a functional \varphi \in {({\ell}^{\mathrm{\infty}}(\mathrm{\Gamma}))}^{\ast} such that \varphi (x)=0 for all x\in {c}_{0}(\mathrm{\Gamma}) and \varphi ({\chi}_{\mathrm{\Gamma}})=1, and define the operator T:{c}_{0}(\mathrm{\Gamma})+span\{{\chi}_{\mathrm{\Gamma}}\}\to {\ell}^{\mathrm{\infty}}(\mathrm{\Gamma}) given by T(\cdot )=S({P}_{{c}_{0}(\mathrm{\Gamma})}\cdot )+\varphi (\cdot ){\chi}_{\mathrm{\Gamma}}, where {P}_{{c}_{0}(\mathrm{\Gamma})} is the projection of every element of the (direct) sum {c}_{0}(\mathrm{\Gamma})+span\{{\chi}_{\mathrm{\Gamma}}\} endowed with the max norm on {c}_{0}(\mathrm{\Gamma}). To see that T is not Fatou, just consider x={\chi}_{\mathrm{\Gamma}}, and for every finite set N of Γ, define {x}_{N} as the characteristic function {\chi}_{N}. Take the increasing net \{{x}_{N}:N\text{finite},N\subset \mathrm{\Gamma}\} (order is given by inclusion of the indexes), and notice that for all N, \parallel T({x}_{N})\parallel =\parallel S({x}_{N})\parallel \le \frac{1}{2} and \parallel T(x)\parallel =\parallel {\chi}_{\mathrm{\Gamma}}\parallel =1. We will come back to this example in Remark 7(2).
Theorem 4 Let 1\le p<\mathrm{\infty} and consider a positive vector measure \nu :\mathcal{R}\to F on a δring with values in a Banach lattice, and a closed subspace of measurable functions Y such that {L}^{p}(\nu )\subset Y\subset {L}_{w}^{p}(\nu ). Let T:Y\to E be a Fatou operator, where E is a Banach space. Then the following statements are equivalent.

(1)
T{}_{{L}^{p}(\nu )} is pconcave.

(2)
There exists a measure \eta :{\mathcal{R}}^{\mathrm{loc}}\to {\mathbb{R}}^{+} that is absolutely continuous with respect to ν and such that
\parallel T(f)\parallel \le {(\int {f}^{p}\phantom{\rule{0.2em}{0ex}}d\eta )}^{1/p}<+\mathrm{\infty},\phantom{\rule{1em}{0ex}}f\in Y. 
(3)
There are a scalar measure \eta :{\mathcal{R}}^{\mathrm{loc}}\to {\mathbb{R}}^{+} and a continuous operator S:E\to {L}^{p}(\eta ) such that the following diagram commutes:
where [i] is the inclusion/quotient map given by [i](f)=[f]  the equivalence class of f with respect to η.
Proof Let us prove \text{(1)}\to \text{(2)}. Consider first the restriction {T}_{0} of T to the space {L}^{p}(\nu ). A standard separation procedure gives the existence of a positive element {\varphi}_{0}:={M}_{(p)}(T){\varphi}_{1} of {({L}^{1}(\nu ))}^{\ast} (where {\varphi}_{1}\in {B}_{{L}^{1}{(\nu )}^{\ast}}) satisfying that
To see this, since {({L}^{p}(\nu ))}^{\frac{1}{p}}={L}^{1}(\nu ), it is enough to apply KyFan’s lemma to the concave family of convex continuous functions \psi :({B}_{{L}^{1}{(\nu )}^{\ast}},{\tau}_{w\ast})\to \mathbb{R} defined as
for each finite set {f}_{1},\dots ,{f}_{n}\in {L}^{p}(\nu ) (see, for instance, this technique in [1]  see Corollary 5 and the proof of Theorem 1  or the proof of Theorem 4.1 in [3]).
Let us show now that we can identify {\varphi}_{0} with a measure on the measurable space (\mathrm{\Omega},{\mathcal{R}}^{\mathrm{loc}}). Since the space {L}^{1}(\nu ) is order continuous and ν is positive, the inequalities
give easily that the set function \eta :\mathcal{R}\to {\mathbb{R}}^{+} defined by \eta (A)={\varphi}_{0}({\chi}_{A}) is countably additive, and so it defines a measure with the domain in the semiring ℛ. Using the Carathéodory extension procedure (see, for instance, Section 9.4 in [30]), we get that η can be extended to the σalgebra of measurable sets with respect to the outer measure defined by {\eta}^{\ast}.
So, we only need to show the following claim: each set in {\mathcal{R}}^{\mathrm{loc}} is ηmeasurable. To see that, we have to prove that for a set B\in {\mathcal{R}}^{\mathrm{loc}}, \eta (A)={\eta}^{\ast}(A\cap B)+{\eta}^{\ast}(A\cap {B}^{c}) for each A\in \mathcal{R} (see, for instance, Lemma 9.26 in [30]). Notice that fixed A\in \mathcal{R}, A\cap B and A\cap {B}^{c} are also in ℛ due to the definition of {\mathcal{R}}^{\mathrm{loc}}, so we really have to prove that {\varphi}_{0}({\chi}_{A})={\varphi}_{0}({\chi}_{A\cap B})+{\varphi}_{0}({\chi}_{A\cap {B}^{c}}) for every A\in \mathcal{R}. Finally, since {\chi}_{A}, {\chi}_{A\cap B} and {\chi}_{A\cap {B}^{c}} belong to {L}^{1}(\nu ), the linearity of {\varphi}_{0} proves the claim.
Remark also that η is absolutely continuous with respect to ν, that is, if B\in {\mathcal{R}}^{\mathrm{loc}} is νnull, then B is also ηnull. To prove this fact, it suffices to fix A\in \mathcal{R} with A\subset B and check that \eta (A)=0, but \eta (A)={\varphi}_{0}({\chi}_{A})\le \parallel {\varphi}_{0}\parallel \cdot {\parallel {\chi}_{A}\parallel}_{{L}^{1}(\nu )}=\parallel {\varphi}_{0}\parallel \cdot \parallel \nu \parallel (A)=0 since B is νnull. Consequently, a (νa.e.) class in {L}^{0}(\nu ) represented by f can be considered then a (ηa.e.) class in {L}^{0}(\eta ) represented by the same f. We will call again η the restriction of {\eta}^{\ast} to {\mathcal{R}}^{\mathrm{loc}}.
Each ℛsimple function is of course ηintegrable  recall that η comes from a functional of {({L}^{1}(\nu ))}^{\ast}  and then the density of this set in {L}^{p}(\nu ) gives that the inequality involving {\varphi}_{0} that was proved using Ky Fan’s lemma can be rewritten as
since for such functions, \int {f}^{p}\phantom{\rule{0.2em}{0ex}}d\eta =\u3008{f}^{p},{\varphi}_{0}\u3009. Notice also that for all f\in {L}^{p}(\nu ), we have that \int {f}^{p}\phantom{\rule{0.2em}{0ex}}d\eta <+\mathrm{\infty}. For this aim, take 0\le f\in {L}^{p}(\nu ) and an increasing sequence ({f}_{n}) of ℛsimple functions such that 0\le {f}_{n}\uparrow f in the νa.e. order and in the norm of {L}^{p}(\nu ). Then 0\le {f}_{n}\uparrow in {L}^{p}(\eta ). Moreover, for every n,
so {sup}_{n}{\parallel {f}_{n}\parallel}_{{L}^{p}(\eta )}<+\mathrm{\infty}, and consequently f\in {L}^{p}(\eta ) with {sup}_{n}{\parallel {f}_{n}\parallel}_{{L}^{p}(\eta )}={\parallel f\parallel}_{{L}^{p}(\eta )} due to the σFatou property of the space {L}^{p}(\eta ). The same occurs for a general f\in {L}^{p}(\eta ) following a standard procedure with f={f}^{+}{f}^{}. This allows to write a factorization of {T}_{0} through the space {L}^{p}(\eta ).
Now we prove that this is enough for getting an extension of {T}_{0} that coincides with T using the same construction that proves Proposition 2.7 in [31]. In the proof, the operator considered is the integration map. As this is not our case, this is the reason why we write here a detailed proof. Take an element 0\le f\in Y. Then there is an upwards directed system ({f}_{\tau})\subset {L}^{p}(\nu ) with 0\le {f}_{\tau}\uparrow f νa.e. due to the order density of {L}^{p}(\nu ) in {L}_{w}^{p}(\nu ). Moreover, following the same arguments as above, {f}_{\tau}\uparrow in {L}^{p}(\eta ) with {sup}_{\tau}{\parallel {f}_{\tau}\parallel}_{{L}^{p}(\eta )}<+\mathrm{\infty}. Finally, since the space {L}^{p}(\eta ) is order continuous and has the σFatou property, Theorem 113.4 in [7] yields that {L}^{p}(\eta ) has the Fatou property, and so f\in {L}^{p}(\eta ). Now the Fatou property of the operator gives also that {lim}_{\tau}\parallel T({f}_{\tau})\parallel =\parallel T(f)\parallel. Consequently, since for every τ,
we have that
the last equality due again to the Fatou property of {L}^{p}(\eta ). Therefore,
holds also for 0\le f\in Y. Again, a standard argument  decomposing a general f\in Y as {f}^{+}{f}^{}, 0\le {f}^{+},{f}^{}  gives the result for all f\in Y.
For \text{(2)}\to \text{(3)}, consider the inclusion/quotient map [i]:Y\to {L}^{p}(\eta ), given by [i](f):=[f], that is well defined by the absolute continuity of η with respect to ν. Notice that the ℛsimple functions are dense in {L}^{p}(\eta ), and so the range [i](Y) is obviously dense in {L}^{p}(\eta ), and we can define an operator S:{L}^{p}(\eta )\to E by S(f):=T(f) for each ℛsimple function f\in Y (note that T(f)=T({f}^{\prime}) if f={f}^{\prime} ηa.e.), and then for all functions in {L}^{p}(\eta ) by density.
The implication \text{(3)}\to \text{(1)} can be proved just by considering the following inequalities:
which hold for every finite family {({f}_{i})}_{i=1}^{n}\subset Y and n\in \mathbb{N}, where we use the fact that {L}^{p}(\eta ) is a pconvex space with pconvexity constant equal to 1. □
Example 5 Consider 1\le p<q<\mathrm{\infty} and the Fatou operator T:{\ell}^{q}(\mathrm{\Gamma})\to E, where Γ is uncountable. Remark that {\ell}^{q}(\mathrm{\Gamma}) is clearly qconvex and so {\ell}^{q}(\mathrm{\Gamma}) is pconvex (see Proposition 1.d.5 in [8]). Since the space is order continuous, Theorem 10 in [21] gives the existence of a vector measure on a δring \nu :\mathcal{R}\to {\ell}^{\frac{q}{p}}(\mathrm{\Gamma}) such that {\ell}^{q}(\mathrm{\Gamma}) is order isomorphic to {L}^{p}(\nu ). Furthermore, since {\ell}^{\frac{q}{p}}(\mathrm{\Gamma}) does not have a copy of {c}_{0}, {L}^{1}(\nu )={L}_{w}^{1}(\nu ) and so {L}^{p}(\nu )={L}_{w}^{p}(\nu ) also. Notice that then the operator T can be considered as a Fatou operator from {L}_{w}^{p}(\nu ) into E. Now, if T is pconcave, Theorem 4 assures that there is a factorization through a space {L}^{p}(\eta ). This natural and easy MaureyRosenthal type theorem, which is a direct consequence of Theorem 4, can only be proved using the abstract setting of the vector measure representation of pconvex Banach lattices that has been shown in the previous section. Remark that {\ell}^{q}(\mathrm{\Gamma}) cannot be written as an {L}^{p}(\beta ) for a vector measure β defined on a σalgebra since the space has not a weak unit.
Corollary 6 Let p\ge 1, E be a Banach space and F be a pconvex Banach lattice with the Fatou property. Suppose that {F}_{a} is order dense in F. Consider a closed subspace G such that {F}_{a}\subset G\subset F. Then the following statements are equivalent for a Fatou operator T:G\to E.

(1)
T{}_{{F}_{a}} is pconcave.

(2)
There exist a vector measure ν on a δring \mathcal{R}\to {F}_{a}, a scalar measure \eta :{\mathcal{R}}^{\mathrm{loc}}\to {\mathbb{R}}^{+} that is absolutely continuous with respect to ν and an order isomorphism on the range \phi :F\to {L}_{w}^{p}(\nu ) such that
\parallel T(x)\parallel \le {(\int {\phi (x)}^{p}\phantom{\rule{0.2em}{0ex}}d\eta )}^{1/p}<+\mathrm{\infty},\phantom{\rule{1em}{0ex}}x\in G. 
(3)
There are a vector measure on a δring \nu :\mathcal{R}\to {F}_{a}, a scalar measure \eta :{\mathcal{R}}^{\mathrm{loc}}\to {\mathbb{R}}^{+}, an order isomorphism on the range \phi :F\to {L}_{w}^{p}(\nu ) and a continuous operator S:{L}^{p}(\eta )\to E such that the following diagram commutes:
Proof First, remark that {F}_{a}={F}_{an} due to the Fatou property of F. Let us show \text{(1)}\to \text{(2)}. The existence of the vector measure on a δring and the order isomorphism \phi :F\to {L}_{w}^{p}(\nu ) are direct consequences of Theorem 1. So, we can apply \text{(1)}\to \text{(2)} in Theorem 4 to the subspace \phi (G) that satisfies {L}^{p}(\nu )\subset \phi (G)\subset {L}_{w}^{p}(\nu ). Note that T\circ {\phi}^{1}{}_{{L}^{p}(\nu )} is also pconcave. This gives the measure \eta :\mathcal{R}\to {\mathbb{R}}^{+} satisfying that
If f=\phi (x) with x\in G, we obtain the inequality in (2).
The implication \text{(2)}\to \text{(3)} is given by \text{(2)}\to \text{(3)} in Theorem 4 composing also with φ. Finally, \text{(3)}\to \text{(1)} is given directly by Theorem 4. □
Remark 7 To finish the paper, let us report that the main requirements in Theorem 4  density of the σorder continuous part together with T being a Fatou operator  are necessary, which means that in some sense this result is optimal.

(1)
A simple counterexample that proves that the order density of the σorder continuous part is necessary is a functional \varphi \ge 0 in {({L}^{\mathrm{\infty}}[0,1])}^{\ast} not belonging to {L}^{1}[0,1]. The σorder continuous part of {L}^{\mathrm{\infty}}[0,1] is trivial, the operator ϕ is obviously pconcave and {L}^{\mathrm{\infty}}[0,1] is pconvex for every 1\le p. Since {({L}^{\mathrm{\infty}}[0,1])}_{[p]}={L}^{\mathrm{\infty}}[0,1], the factorization would mean that \varphi \in {L}^{1}[0,1] just using the domination by the measure and the RadonNikodým theorem, which gives a contradiction. Notice that ϕ is trivially a Fatou operator.

(2)
A bit more elaborated example is the following. Consider {\ell}^{\mathrm{\infty}}(\mathrm{\Gamma}), where Γ is an uncountable set, and let a functional \varphi :{\ell}^{\mathrm{\infty}}(\mathrm{\Gamma})\to \mathbb{R} with ϕ not belonging to {\ell}^{1}(\mathrm{\Gamma})={({c}_{0}(\mathrm{\Gamma}))}^{\ast}=ca(\mathrm{\Gamma}). Recall that {({\ell}^{\mathrm{\infty}}(\mathrm{\Gamma}))}^{\ast}=ca(\mathrm{\Gamma})\oplus pa(\mathrm{\Gamma}), where ca(\mathrm{\Gamma}) is the space of the countably additive measures and pa(\mathrm{\Gamma}) is the space of the purely finite additive measures (see, for example, Corollary 13.11, Section 9.10 and Section 9.11 in [30]). Let {P}_{1} be the projection of {({\ell}^{\mathrm{\infty}}(\mathrm{\Gamma}))}^{\ast} on ca(\mathrm{\Gamma}). We have that {\varphi}_{1}:=\varphi {P}_{1}(\varphi )\ne 0. Since {\varphi}_{1} is not countably additive, there is a sequence {({A}_{i})}_{i=1}^{\mathrm{\infty}} of measurable sets such that {\sum}_{i=1}^{\mathrm{\infty}}{\varphi}_{1}({\chi}_{{A}_{i}})\ne {\varphi}_{1}({\chi}_{{\bigcup}_{i=1}^{\mathrm{\infty}}{A}_{i}}). Let us define now T:{c}_{0}(\mathrm{\Gamma})+span\{{\chi}_{\mathrm{\Gamma}};{\chi}_{{A}_{i}},i\in \mathbb{N}\}\to \mathbb{R} given by T(\cdot ):={\varphi}_{1}(\cdot ). The operator T is clearly pconcave, but T is not Fatou, which can be proved using an argument similar to the one given in Example 3. Moreover, T cannot be bounded by an integral; otherwise, {\varphi}_{1} would be countably additive.
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The authors are supported by grants MTM201123164 and MTM201236740C0202 of the Ministerio de Economía y Competitividad (Spain).
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Juan, M.A., Sánchez Pérez, E.A. MaureyRosenthal domination for abstract Banach lattices. J Inequal Appl 2013, 213 (2013). https://doi.org/10.1186/1029242X2013213
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DOI: https://doi.org/10.1186/1029242X2013213
Keywords
 integral inequality
 Banach lattice
 pconvexity
 pconcavity