# Lattice Lipschitz superposition operators on Banach function spaces

Roger Arnau∗, Jose M. Calabuig, Ezgi Erdoğan,
Enrique A. Sánchez Pérez  Ezgi Erdoğan. Department of Mathematics, Faculty of
Sciences, University of Marmara, 34722, Kadıköy, Istanbul, Turkey.
<EMAIL_ADDRESS>Roger Arnau∗, Jose M. Calabuig, Enrique A. Sánchez
Pérez
∗Corresponding author. Instituto Universitario de Matemática Pura y Aplicada,
Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia,
Spain<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We analyse and characterise the notion of lattice Lipschitz operator (a class
of superposition operators, diagonal Lipschitz maps) when defined between
Banach function spaces. After showing some general results, we restrict our
attention to the case of those Lipschitz operators which are representable by
pointwise composition with a strongly measurable function. Mimicking the
classical definition and characterizations of (linear) multiplication
operators between Banach function spaces, we show that under certain
conditions the requirement for a diagonal Lipschitz operator to be well-
defined between two such spaces $X(\mu)$ and $Y(\mu)$ is that it can be
represented by a strongly measurable function which belongs to the Bochner
space $\mathcal{M}(X,Y)\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}.$ Here,
$\mathcal{M}(X,Y)$ is the space of multiplication operators between $X(\mu)$
and $Y(\mu),$ and $Lip_{0}(\mathbb{R})$ is the space of real-valued Lipschitz
maps with real variable that are equal to $0$ in $0.$ This opens the door to a
better understanding of these maps, as well as finding the relation of these
operators to some normed tensor products and other classes of maps.

###### Key words and phrases:

Lipschitz; Operator; lattice Lipschitz, Banach function space, diagonal map

###### 2010 Mathematics Subject Classification:

Primary: 47J10, 46E30, 26A16

The first author was supported by a contract of the Programa de Ayudas de
Investigación y Desarrollo (PAID-01-21), Universitat Politècnica de València.
This publication is part of the R&D&I project/grant PID2022-138342NB-I00
funded by MCIN/ AEI/10.13039/501100011033/ (Spain).

##  1\. Introduction and notation

Lipschitz functions, which appeared as a tool in mathematical analysis, have
found relevant applications in the field of differential equations from the
beginning of the 20th century. Today, they have become a fundamental tool also
in many fields of applied mathematics and artificial intelligence. Recently,
more attention has returned to this topic from pure analysis, in a stream of
research that tries to extend linear and multilinear issues and methods to the
field of Lipschitz functions [4, 15, 29, 23]: summability of operators [10,
33, 35], ideals of Lipschitz operators [1, 2, 14, 16, 31], lattice geometric
properties of Lipschitz maps [11, 17] and many others are topics of current
interest to the mathematical community.

Following this trend, we are interested in the Lipschitz version of diagonal
maps (which appeared in the Euclidean space context) or multiplication
operators (when working in the setting of function spaces). As can be easily
seen this diagonal property coincides with a pointwise domination requirement,
for which an order in the space is needed; since we are interested in the
setting of the Banach function spaces, the requirement on the order can be
easily formulated to provide the (equivalent) notion of lattice Lipschitz
operator. No explanation is needed when we state that diagonalization of
operators between finite dimensional spaces, and also multiplication
operators, apply in a vast number of pure and applied contexts, and extensions
of these genuine linear concepts to other classes of operators are fully
justified [24, 25]. Thus, our motivation for this research is twofold. On the
one hand, we want to study Lipschitz diagonal operators to provide tools to
extend the fundamental factorization theorems for operators between function
spaces, which are now at the core of functional analysis ([20], see the last
section of the article). On the other hand, guided by some recent applications
of Lipschitz functions in artificial intelligence algorithms (mainly in
reinforcement learning [9, 13, 22] but also in some separation techniques with
applications in clustering procedures [34]), we want to increase the
mathematical tools for this applied research.

The family of functions we are interested in are a particular case of what are
called superposition operators [6, 5]. Since the eighties of the last century
there have been some specific investigations on different properties of
Lipschitz-type operators that are also superposition operators. In fact, some
relevant characterizations of locally Lipschitz superposition operators
involving inequalities such as the ones we analyze here can be found in some
papers on the subject (see for example [6, Th. 1]). However, although these
inequalities are interesting for analyzing the properties of Lipschitz
superposition operators, in this article we face the problem of the functional
representation of these maps. In other words, in this paper we are interested
in finding some general conditions to ensure that a given lattice Lipschitz
operator can be represented as a strongly measurable vector function belonging
to some special Köthe-Bochner space, by using a specific definition of how a
vector function acts on a scalar function belonging to a Banach function
space.

Thus, we will devote the second part of the article to analyze how to relate
lattice Lipschitz operators and strongly measurable vector valued functions.
In the first part, we will present the main results on the structure of
lattice Lipschitz maps and diagonal Lipschitz operators on Banach function
spaces. These spaces are the natural extension of classical Lebesgue spaces
(but also of Euclidean spaces), and include, for example, Lorentz spaces of
sequences and functions, as well as Orlicz spaces. The direct translation of
the notion of lattice Lipschitz operator, which has originally been done in
Euclidean spaces, gives a rather vague concept, which we show could include
some “pathological” maps, which appear essentially due to the fact that the
space of real-valued Lipschitz operators $Lip_{0}(\mathbb{R})$ is not
separable. To ensure a good behavior of the maps involved we will focus on the
case of superposition operators that allow a representation as a pointwise
composition with a strongly measurable function.

Starting from the description of the notion of “diagonal” operator between
Banach function spaces in the linear case (Section 2), we will show some
general properties, examples and counterexamples (Section 3), and, in the
following Section 4 we will focus the attention on lattice Lipschitz operators
that allow representations by strongly measurable functions, showing our main
representation theorems. Finally, in Section 5 we show some applications, in
connection with some classical normed tensor products and operator ideals.

Let us write the direct translation of the notion of lattice Lipschitz
operator that has been originally given in the framework of Euclidean spaces
([8, Definition 1]), to the setting of the Banach lattices. In the context of
the paper [8], we say that a Lipschitz operator $T:L\to V$ on the Euclidean
space $L$ is lattice Lipschitz if there is a constant $K>0$ such that for
every pair of elements $x,y\in L,$

$\big{|}T(x)-T(y)\big{|}\leq K\big{|}x-y\big{|}.$

Other notions related to this definition can be found in the literature on
maps on Banach lattices (see for example [18, Ch.3], [26]).

Before getting into the subject, let us recall some basic definitions and
results from the theory of Banach function spaces and general functional
analysis. Throughout the paper $(\Omega,\Sigma,\mu)$ will be a finite measure
space. Every representation of a simple function
$f=\sum_{i=1}^{n}\lambda_{i}\chi_{A_{i}}$ is assumed to be disjoint, that is
$A_{i}\cap A_{j}=\emptyset$ for $i\neq j.$ We assume that all the measures in
the current manuscript are complete, meaning that every subset of a $\mu$-null
(measurable) set is again measurable.

Given a Banach space $E$, let us review some basic definitions about
measurability of functions from $\Omega$ to $X$. A simple function is a
function as $\psi=\sum_{i=1}^{n}x_{i}\chi_{A_{i}},$ for $x_{1},...,x_{n}\in E$
and disjoint $A_{1},...,A_{n}\in\Sigma.$ A vector valued function
$\Psi:\Omega\to E$ is said to be weakly measurable if the real function given
by the composition $x^{*}\circ\Psi$ is measurable for every $x^{*}\in E^{*}.$
$\Phi$ is strongly measurable if it equals almost everywhere the limit of a
sequence of simple functions, i.e.,

$\lim_{n}\|\Psi(w)-\psi_{n}(w)\|_{X}=0\quad\text{ for almost every }\quad
w\in\Omega.$

A function $\Psi:\Omega\to E$ is essentially separably valued if there is a
$\mu-$null set $A\in\Sigma$ such that the restricted range
$\Psi(\Omega\setminus A)$ is a separable set. Pettis measurability theorem
establishes that a vector valued function $\Psi$ is strongly measurable if and
only if it is weakly measurable and essentially separably valued. In
particular, if $E$ is separable we have that weak and strong measurability
coincide.

If $(\Omega,\Sigma,\mu)$ is a finite measure space, a Banach function space
$X(\mu)$ is an ideal in $L^{0}(\mu)$ with the $\mu-$a.e. order that is a
Banach space, satisfying that for all $f,g\in X(\mu)$ such that $|f|\leq|g|$
we have $\|f\|_{X(\mu)}\leq\|g\|_{X(\mu)}.$ It is a Banach lattice, and we
always assume that $\chi_{\Omega}\in X(\mu)$ (and so this function is an order
unit). We say that a Banach function space $X(\mu)$ has the Fatou property if
for every increasing sequence $(f_{n})_{n}\in X(\mu)$ that is norm bounded and
converging $\mu-$a.e. to a measurable function $f,$ we have that $f\in X(\mu)$
too.

Recall now some notions and notation on Köthe-Bochner spaces of vector valued
integrable functions. If $E$ is a Banach space and $X(\mu)$ is a Banach
function space over the finite measure space $(\Omega,\Sigma,\mu),$ the Köthe-
Bochner space $X(\mu,E)$ is the Banach space of all the strongly measurable
functions $\Psi:\Omega\to E$ such that the norm function
$w\mapsto\|\Psi(w)\|_{E}$ belongs to $X(\mu).$ The norm in this space is given
in this case by

$\|\Psi\|_{X(\mu,E)}=\Big{\|}\,\|\Psi(w)\|_{E}\,\Big{\|}_{X(\mu)}.$

Particular cases of these spaces are the Bochner spaces of $p-$integrable
functions for $1\leq p<\infty,$ that are given when $X(\mu)=L^{p}(\mu).$ In
this case, the space $L^{p}(\mu,E)$ can be represented as the completion of
the tensor product $L^{p}(\mu)\widehat{\otimes}_{\Delta_{p}}E,$ where
$\Delta_{p}$ is the norm for the tensor product inherited from this
identification.

The space $\mathcal{M}(X(\mu),Y(\mu))$ is the space of multiplication
operators from $X(\mu)$ to $Y(\mu)$ with the usual operator norm (see [12,
28]). We identify each of these operators with the function that defines it,
although we sometimes use the notation $M_{g}$ for the operator defined by the
function $g$ to distinguish it from the function $g$ itself. If the measure
$\mu$ is fixed in the context, we will write $\mathcal{M}(X,Y)$ for this space
for simplicity.

We refer to [18] for main issues on Lipschitz operators, to [27, 32] for
definitions and results on Banach function spaces, to [19, 20] for general
issues on factorization in spaces of linear operators and normed tensor
products, and to [21] for (vector-valued) strongly measurable functions.

##  2\. Main reference: the linear case

Let $(\Omega,\Sigma,\mu)$ be a finite measure space, and $X(\mu)$ and $Y(\mu)$
two Banach function spaces. Consider the space of measurable functions
$\mathcal{M}(X(\mu),Y(\mu))$, that define multiplication operators
$M_{h}:X(\mu)\to Y(\mu)$ given by $M_{h}(f)(w)=h(w)\cdot f(w),$ $f\in X(\mu).$
Recall that, depending on $X(\mu)$ and $Y(\mu),$ that space could only contain
the $0$ function. We will see that the adaptation of this notion to the
Lipschitz case fits the concept of lattice Lipschitz operator, that can be
more specified for the concrete case of operators between Banach function
spaces.

######  Definition 2.1.

We say that a (non-linear) operator between Banach function spaces
$T:X(\mu)\to Y(\mu)$ is lattice Lipschitz if there exists a real-valued non-
negative function $K(\cdot)$ such that for every pair of functions $f,g\in
X(\mu)$ and $w\in\Omega,$

$\big{|}T(f)(w)-T(g)(w)\big{|}\leq K(w)\,\big{|}f(w)-g(w)\big{|}\quad\mu-a.e.$

In case $K(\cdot)$ is $\mu$-measurable, the condition can be expressed as
$|T(f)-T(g)|\leq K|f-g|$ with the pointwise order on $L^{0}(\mu)$. Extending
the classical terminology for constants appearing in Lipschitz inequalities,
the function $K$ will be called a bound or associated function for $T$. We
will call also lattice Lipschitz operators to maps $T:X(\mu)\to L^{0}(\mu)$.

We start by a basic property, which gives a relation between lattice Lipschitz
operators and the linear ones. A related result, but given in a different
framework, can be found in [7, Lemma 1.1] and the comments after it.

######  Proposition 2.2.

Let $(\Omega,\Sigma,\mu)$ be a measure space and $X(\mu),Y(\mu)$ be two Banach
function spaces (or, eventually, $Y(\mu)=L^{0}(\mu)$). Consider a lattice
Lispchitz $T:X(\mu)\to Y(\mu)$ operator that satisfies that $T(0)=0.$ Then,
for every $f,g\in X(\mu)$ and disjoint sets $A,B\in\Sigma$,

  * (i)

$T(f\chi_{A})=T(f)\chi_{A}$ $\mu-$a.e., and so

  * (ii)

$T(f\chi_{A}+g\chi_{B})=T(f\chi_{A})+T(g\chi_{B})$ $\mu-$a.e.

###### Proof.

Let $f$ and $A$ be as in the statement. Since $f\chi_{A}\in X(\mu)$, there
exists a null set $N$ such that

$|T(f)-T(f\chi_{A})|(w)\leq K(w)\cdot|f-f\chi_{A}|(w),\quad
w\in\Omega\setminus N,$

and another null set $N^{\prime}$ that $|T(f\chi_{A})|(w)\leq
K(w)|f\chi_{A}|(w)$ for each $w\in\Omega\setminus N^{\prime}$. Let
$w\in\Omega\setminus(N\cup N^{\prime})$. If $w\not\in A$,

$|T(f)\chi_{A}-T(f\chi_{A})|(w)=|0-T(f\chi_{A})(w)|\leq
K(w)\cdot|f(w)|\chi_{A}(w)=0.$

If $w\in A$,

$\displaystyle|T(f)\chi_{A}-T(f\chi_{A})|(w)$
$\displaystyle=|T(f)(w)-T(f\chi_{A})(w)|$ $\displaystyle\leq
K(w)|f(w)-(f\chi_{A})(w)|=0.$

Then, $T(f)\chi_{A}=T(f\chi_{A})$ $\mu-$a.e. This gives (i). For (ii), just
note that using (i) we get $\mu-$a.e.

$\displaystyle T(f\chi_{A}+g\chi_{B})$
$\displaystyle=T\big{(}(f\chi_{A}+g\chi_{B})(\chi_{A}+\chi_{B})\big{)}=T(f\chi_{A}+g\chi_{B})(\chi_{A}+\chi_{B})$
$\displaystyle=T(f\chi_{A}+g\chi_{B})\chi_{A}+T(f\chi_{A}+g\chi_{B})\chi_{B}$
$\displaystyle=T\big{(}(f\chi_{A}+g\chi_{B})\chi_{A}\big{)}+T\big{(}(f\chi_{A}+g\chi_{B})\chi_{B}\big{)}=T(f\chi_{A})+T(g\chi_{B}).$

∎

We start by giving the linear reference, that is, the characterization of the
lattice Lipschitz maps for the simplest case when the map is linear. Of course
it is essentially known, but it will help to find what we can expect for the
non-linear case.

######  Proposition 2.3.

Let $T:X(\mu)\to Y(\mu)$ be a linear continuous map. Then it is a lattice
Lipschitz operator if and only if there is a function
$h\in\mathcal{M}(X(\mu),Y(\mu))$ such that $T=M_{h}.$ In this case,
$K(w)=|h(w)|$ is a bound function for $T$.

###### Proof.

A well-known Radon-Nikodym Theorem argument proves the result; let us write
the detailed proof for the aim of completeness, since some of the ideas in it
will be used for the non-linear case.

Suppose first that $T$ is lattice Lipschitz. Assume without loss of generality
that the measure $\mu$ is finite. Since $T$ is linear, $T(0)=0$ and we can use
the lattice Lipschitz order inequality for single functions, and not
necessarily for differences of functions. All the inequalities below are
defined for concrete functions of the equivalence classes of the functions
involved and has to be understood $\mu-$a.e., even if this is not explicitly
written. Consider the set function $\nu:\Sigma\to\mathbb{R}$ given by
$A\mapsto\nu(A):=\int_{\Omega}T(\chi_{A})d\mu.$ Note that
$T(\chi_{\Omega})=h\in Y(\mu)\subset L^{1}(\mu),$ since $\mu$ is finite, and
so $\nu(A)=\int_{A}T(\chi_{\Omega})$ for all $A\in\Sigma$ by Proposition 2.2.

1) As $h$ is a $\mu$-measurable function, we assume that is non-negative.
Otherwise consider it as a difference of two positive functions and repeat the
reasoning for the difference of two integrals in the above equation. Let $f\in
X(\mu)$ be a simple function and take $B\in\Sigma.$ Again by Proposition 2.2
and the linearity of $T$, we get

$\int_{B}T(f)d\mu=\int_{\Omega}T(f)\chi_{B}d\mu=\int_{\Omega}T(f\chi_{B})d\mu=\int_{\Omega}h\,f\chi_{B}\,d\mu=\int_{B}h\,f\,d\mu.$

As a consequence, $T(f)=h\cdot f$

2) Fix now $f\in X(\mu)$ and assume without loss of generality that $f\geq 0$.
Take an increasing sequence of simple function such that $f_{n}\uparrow f$
pointwise and observe that, by the non negativity of $h$, $T(f_{n})=h\cdot
f_{n}\uparrow h\cdot f=T(f_{n})$. Then, by the Monotone Convergence Theorem
and the previous paragraph we get that

$\int_{B}T(f)\,d\mu=\lim_{n}\int_{B}T(f_{n})\,d\mu=\lim_{n}\int_{B}f_{n}\,d\nu=\int_{B}f\,d\nu,$

for any $B\in\Sigma$. Then, $T(f)=h\cdot f=M_{h}(f)$ Moreover, since it is
defined for every $f\in X(\mu),$ we get that $h\in\mathcal{M}(X(\mu),Y(\mu)).$

For the converse implication, given a multiplication operator
$M_{h}\in\mathcal{M}(X(\mu),Y(\mu)),$ we only need to consider the following
direct inequalities, for $f,g\in X(\mu);$

$\big{|}M_{h}(f)-M_{h}(g)\big{|}(w)=|M_{h}(f-g)|(w)=|h(w)(f-g)(w)|\leq|h(w)|\,\big{|}f(w)-g(w)\big{|},$

and so the result holds for $K(w)=|h(w)|$ $\mu-$a.e.

∎

##  3\. Pointwise diagonal Lipschitz operators

Following the idea that suggests the case of multiplication operators, we
establish in this section what would be the Lipschitz version of a
multiplication operator. Let us show the arguments to find it. Before starting
to explain our ideas, let us recall some fundamental definitions and results.

We will find the fundamental functions to represent this class of operators as
Bochner integrable functions. Consider the vector space of all the real-valued
functions of one real variable $\mathcal{RV}(\mathbb{R}).$ Let
$\mathcal{F}(\mathbb{R})\subset\mathcal{RV}(\mathbb{R})$ be a Banach space and
take a vector valued function $\Phi:\Omega\to\mathcal{F}(\mathbb{R})$ (for
example a strongly measurable function). Let $Z$ be a Banach space of (classes
of a.e. equal) real functions; we are mainly (but not only) thinking on
$Z=Y(\mu),$ a Banach function space over $\mu.$ We want to show that we can
define an operator $T:X(\mu)\to Z$ using $\Phi$ and when it satisfies
Definition 2.1.

######  Example 3.1.

Consider $\mathcal{F}(\mathbb{R})=Lip(\mathbb{R}),$ the space of Lipschitz
real functions of one real variable. Consider the constant function
$\Phi:\Omega\to Lip(\mathbb{R})$ given by $\Phi(w)(v):=\frac{1}{1+|v|}$ for
all $w\in\Omega.$ Let $X(\mu)$ be any Banach function space in a finite
measure space and define $T:X(\mu)\to X(\mu)$ as

$T(f)(w)=\big{(}\Phi(w)\circ
f\big{)}(w)=\Phi(w)\big{(}f(w)\big{)}=\frac{1}{1+|f(w)|},\quad f\in
X(\mu),\,\,\,w\in\Omega\,\,\,\mu-a.e.$

Note that, for all $f,g\in X(\mu)$ and $w\in\Omega,$

$\displaystyle\big{|}T(f)(w)-T(g)(w)\big{|}$
$\displaystyle=\Big{|}\frac{1}{1+|f(w)|}-\frac{1}{1+|g(w)|}\Big{|}=\Big{|}\frac{|g(w)|-|f(w)|}{(1+|f(w)|)(1+|g(w)|)}\Big{|}$
$\displaystyle\leq\frac{\big{|}g(w)-f(w)\big{|}}{(1+|f(w)|)(1+|g(w)|)}\leq\big{|}g(w)-f(w)\big{|}.$

Therefore, it is lattice Lipschitz, and $K(w)$ can be defined to be the
constant function $\mathbf{1}.$ The function $g$ in Remark LABEL:Tsub is in
this case the constant function $\mathbf{1},$ since $T(0)=\mathbf{1}.$

######  Remark 3.2.

Let us show first that this covers the linear case explained in Proposition
2.3. If $\mathcal{F}(\mathbb{R})=L(\mathbb{R})\subseteq Lip_{0}(\mathbb{R}),$
the space of linear real functions of one real variable, of course this space
can be identified with the space of the real numbers $\mathbb{R}$ by means of
the map $\mathbb{R}\ni r\mapsto f_{r}(\cdot)=r\cdot(\cdot)\in L(\mathbb{R}).$

Take now a function $h\in\mathcal{M}(X(\mu),Y(\mu))$ and consider the Bochner
integrable function

$\displaystyle\Phi_{h}:\Omega$ $\displaystyle\to L(\mathbb{R})$ $\displaystyle
w$ $\displaystyle\mapsto\Phi_{h}(w)(\cdot)=f_{h(w)}(\cdot)=h(w)\cdot(\cdot).$

That is, it takes at each $w$ the value provided by the multiplication of the
real number $h(w)$ and the variable. Then the operator can be given by the
composition

$T(f)(w)=\big{(}\Phi_{h}(w)\circ f\big{)}(w)=h(w)\cdot f(w)=M_{h}(f)(w),\quad
f\in X(\mu),$

that is well-defined $\mu-$a.e. Recall that, by Proposition 2.3, the class of
operators above that are also linear coincides with the class of linear
lattice Lipschitz operators, that is, those satisfying that for all $f\in
X(\mu),\ \big{|}T(f)(w)\big{|}\leq K(w)\,|f(w)|,$ for a certain function
$K(w)$ belonging to $\mathcal{M}(X(\mu),Y(\mu)).$

Let us restrict our attention to vector valued functions for the case that
$\mathcal{F}(\mathbb{R})=Lip_{0}(\mathbb{R})$ with its Lipschitz constant as
norm. The following results illustrate the construction of lattice Lipschitz
operators from a vector-valued function, as in the cases above. The conditions
of the map $\Phi:\Omega\to Lip_{0}(\mathbb{R})$ in terms of measurability
needed for having a well defined lattice Lipschitz operator will be discussed
hereafter.

Consider for the moment $\Phi:\Omega\to Lip_{0}(\mathbb{R})$ a strongly
mesurable function, as an almost everywhere limit of a sequence of simple
functions defined by $\psi(w)=\sum_{i=1}^{n}\chi_{A_{i}}(w)\phi_{i}.$ As for
the notation, and to facilitate the understanding of the arguments we will
also write these functions as $\psi(w)=\sum_{i=1}^{n}\phi_{i}\chi_{A_{i}}(w).$
It is relevant to mention that such a function $\psi$ defines an operator
$T_{\psi}:X(\mu)\to Y(\mu)$ acting as a superposition operator, that is, by
the natural formula

$T_{\psi}(f)(w)=\sum_{i=1}^{n}\chi_{A_{i}}(w)\phi_{i}\big{(}f(w)\big{)},\quad
w\in\Omega$ (1)

whenever for each $i=1,...,n,$ the functions $w\mapsto\phi_{i}(f(w))$ belong
to $Y(\mu).$ This formula has to be understood as defined $\mu-$a.e.; indeed,
if $f$ and $g$ are measurable functions that are equal but in a $\mu-$null set
$A\in\Sigma$ and its equivalence class belongs to $X(\mu)$, we clearly have
that $T_{\psi}(f)(w)=T_{\psi}(g)(w)$ for $w\in\Omega\setminus A,$ and so
$T_{\psi}(f)\in Y(\mu).$ In other words, the map $T_{\psi}$ is defined
independently of the representative we take for the function $f.$

######  Lemma 3.3.

Using the notation above, every simple function
$\psi=\sum_{i=1}^{n}\phi_{i}\chi_{A_{i}}\in X(\mu,Lip_{0}(\mathbb{R}))$ gives
a lattice Lipschitz operator $T_{\psi}:X(\mu)\to X(\mu)$ with bound function

$K_{T_{\psi}}=\sum_{i=1}^{n}\chi_{A_{i}}Lip(\phi_{i})\in L^{\infty}(\mu).$

###### Proof.

Note first that the composition of a measurable function $f$ with a Lipschitz
(and so continuous) map $\phi\in Lip_{0}(\mathbb{R})$ gives also a measurable
function, and

$|\phi\circ f(w)|=|\phi\circ f(w)-\phi\circ 0(w)|=|\phi(f(w))-\phi(0)|\leq
Lip(\phi)\,|f(w)|,$

where the inequality is defined $\mu-$a.e., and so $\phi\circ f\in X(\mu)$ too
due to the ideal property of the Banach function spaces.

Let us compute the value of the Lipschitz bound function for the operator
$T_{\psi}:X(\mu)\to X(\mu)$ associated to a simple function
$\psi=\sum_{i=1}^{n}\phi_{i}\chi_{A_{i}},$ $\phi_{i}\in Lip_{0}(\mathbb{R}).$
For every $f,g\in X(\mu),$ we have

$\displaystyle|T_{\psi}(f)(w)-T_{\psi}(g)(w)|$
$\displaystyle=\left|\sum_{i=1}^{n}\chi_{A_{i}}(w)\cdot\big{(}\phi_{i}(f(w))-\phi_{i}(g(w))\big{)}\right|$
$\displaystyle\leq\sum_{i=1}^{n}\chi_{A_{i}}(w)\big{|}\phi_{i}(f(w))-\phi_{i}(g(w))\big{|}$
$\displaystyle\leq\sum_{i=1}^{n}\chi_{A_{i}}(w)Lip(\phi_{i})\big{|}f(w)-g(w)\big{|}$
$\displaystyle=\left(\sum_{i=1}^{n}\chi_{A_{i}}(w)Lip(\phi_{i})\right)\cdot\big{|}f(w)-g(w)\big{|}.$

∎

Note that the next result gives a constructive procedure for finding well-
defined superposition operators as limits of simple functions. This is not the
usual way of introducing such superposition maps ([6, 5]), and opens the door
to a new point of view for understanding superposition operators as vector-
valued integrable functions.

######  Proposition 3.4.

Let $X(\mu)$ and $Y(\mu)$ be Banach function spaces and let $\Phi:\Omega\to
Lip_{0}(\mathbb{R})$ be a strongly measurable function. Then

  * (i)

the composition (superposition) formula

$T_{\Phi}(f)(w)=\big{(}\Phi(w)\circ f\big{)}(w),\quad w\in\Omega$

gives a well-defined map $T_{\Phi}:X(\mu)\to L^{0}(\mu),$ and

  * (ii)

if $(\psi_{n})_{n}$ is a sequence of simple functions converging $\mu-$a.e. to
$\Phi$ with bound functions $K_{T_{\psi_{n}}}$ such that

$\sup_{n}K_{T_{\psi_{n}}}\in\mathcal{M}(X(\mu),Y(\mu)),$

then $T_{\Phi}$ gives a lattice Lipschitz map $T_{\Phi}:X(\mu)\to Y(\mu).$

###### Proof.

(i) We only need to prove that the composition with a certain function $f$ is
well-defined as a representative of a $\mu-$a.e. class of functions of
$Y(\mu).$ Since $\Phi$ is strongly measurable, we get that there is a
$\mu-$null set $B$ and a sequence of simple functions $\psi_{n}$ such that for
every $w\in\Omega\setminus B$

$\Phi(w)=\lim_{n}\psi_{n}(w).$

Since $\psi_{n}(w)$ are simple functions with values in $Lip_{0}(\mathbb{R}),$
by Lemma 3.3 the maps $T_{\psi_{n}}:X(\mu)\to X(\mu)\subset L^{0}(\mu)$ are
well-defined. We have that for $0\in X(\mu)$ and every $n,$ $\psi_{n}(w)(0)=0$
for all $w\in\Omega,$ and so $\Phi(w)(0)=0$ $\mu-$a.e. Thus, for every $f\in
X(\mu)$ and almost all $w\in\Omega,$

$\displaystyle\big{|}\Phi(w)(f(w))-\psi_{n}(w)(f(w))\big{|}$
$\displaystyle=\big{|}\Phi(w)(f(w))-\psi_{n}(w)(f(w))-\Phi(w)(0)-\psi_{n}(w)(0)\big{|}$
$\displaystyle\leq Lip\big{(}\Phi(w)-\psi_{n}(w)\big{)}\,|f(w)-0|\to_{n}0.$

Therefore, $\lim_{n}\psi_{n}(w)(f(w))=\Phi(w)(f(w))$ $\mu-$a.e., and so the
operator $T_{\Phi}$ is defined at least pointwise for all $f\in X(\mu),$
giving a measurable function for each such $f$ that is an a.e. pointwise limit
of functions in $X(\mu).$

(ii) For a.e. every $w\in\Omega$ and $n\in\mathbb{N},$

$\displaystyle|\Phi(w)(f(w))|$
$\displaystyle\leq\big{|}\Phi(w)(f(w))-\psi_{n}(w)(f(w))\big{|}+|\psi_{n}(w)(f(w))|$
$\displaystyle\leq
Lip\big{(}\Phi(w)-\psi_{n}(w)\big{)}\,|f(w)|+\sup_{n}K_{T_{\psi_{n}}}(w)\cdot|f(w)|.$

Since $Lip\big{(}\Phi(w)-\psi_{n}(w)\big{)}\to_{n}0$ and
$\sup_{n}K_{T_{\psi_{n}}}(w)\cdot|f(w)|\in Y(\mu)$ by hypothesis, we have that
$|\Phi(w)(f(w))|$ is a measurable function a.e. bounded by a function in
$Y(\mu).$ By the ideal property of the Banach function space $Y(\mu),$ we get
that the operator is well-defined.

On the other hand, if $f,g\in X(\mu)$ we have that for a.e. $w,$

$\displaystyle|\Phi(w)$ $\displaystyle(f(w))-\Phi(w)(g(w))|$
$\displaystyle\leq\big{|}\Phi(w)(f(w))-\psi_{n}(w)(f(w))-\Phi(w)(g(w))+\psi_{n}(w)(g(w))\big{|}$
$\displaystyle\qquad+|\psi_{n}(w)(f(w))-\psi_{n}(w)(g(w))|$ $\displaystyle\leq
Lip\big{(}\Phi(w)-\psi_{n}(w)\big{)}\cdot|f(w)-g(w)|+\sup_{n}K_{T_{\psi_{n}}}(w)\cdot|f(w)-g(w)|,$

and since this holds for every $n,$ we obtain

$|T_{\Phi}(f(w))-T_{\Phi}(g(w))|\leq\sup_{n}K_{T_{\psi_{n}}}(w)\cdot|f(w)-g(w)|$

and so $T_{\Phi}$ is a lattice Lipschitz operator for which we can get a bound
function that defines a multiplication operator from $X(\mu)$ to $Y(\mu),$
since

$K_{\Phi}\leq\sup_{n}K_{T_{\psi_{n}}}\in\mathcal{M}(X(\mu),Y(\mu)).$

∎

The operator $T(f)(w)=\Phi(w)(f(w))$ used in Example 3.1 and Proposition 3.4
will be called superposition operator an will be denoted by
$T_{\Phi}(f)=\Phi\circ f$ by abuse of notation.

It is relevant to mention that we can define lattice Lipschitz operators for
which it is not possible to provide a representation by means of a vector-
valued function $\Phi.$ We show an example of this situation below.

######  Example 3.5.

Consider the Lebesgue measure space $([0,1],\Sigma,\mu)$ and take a pointwise
bounded non-measurable function $h:[0,1]\to\mathbb{R}$ that is positive and
with pointwise bound $K>0$ (for example, the function given by 1 plus the
characteristic function of a non-measurable set). Consider the space of
essentially bounded measurable functions $X(\mu)=L^{\infty}(\mu)$ and consider
the multiplication operator $M_{h}:X(\mu)\to\mathcal{F}(\mathbb{R}),$ where
$M_{h}(f)=h\cdot f,$ and $\mathcal{F}(\mathbb{R})$ is the space of all classes
of real functions with real variable that are equal $\mu-$a.e. Note that this
map is well-defined when defined from a Banach function space $X(\mu)$ over
$\mu.$

This means that we can consider $M_{h}$ defined from $X(\mu)$ to the space
$\mathcal{F}(\mathbb{R})_{\mu}$ of all the equivalence classes of functions
that are equal $\mu-$a.e. A Banach lattice structure can be given for a
subspace $Z$ of this space by dividing every function in it by $h;$ for
example, the (classes of) functions that satisfy that
$\mathcal{F}(\mathbb{R})_{\mu}\ni v\mapsto v/h\in X(\mu)$ defines a Banach
lattice with $\mu-$a.e. order and the norm given by this map. $M_{h}:X(\mu)\to
Z$ is clearly well-defined. On the other hand, for every $w$ and every pair of
measurable functions $f,g,$

$|M_{h}(f)(w)-M_{h}(g)(w)|=h(w)\cdot|f(w)-g(w)|\leq K\,|f(w)-g(w)|$

and so it is a lattice Lipschitz map. The vector valued function $\Phi$ that
represents the lattice Lipschitz operator is $w\mapsto h(w)(\cdot),$ where the
real number $h(w)$ is considered as a constant that defines a Lipschitz map in
$Lip_{0}(\mathbb{R}),$ which cannot be, by construction, measurable: in other
case, the scalar function $w\mapsto\|\Phi(w)\|_{Lip(\mathbb{R})}=h(w)$ would
be measurable.

Let us explain an example in which a non-measurable Lipschitz valued function
is constructed using a typical cardinality argument. In order to do it, let us
recall that the spaces of Lipschitz functions ares not separable in general,
as a consequence of the fact that $\mathbb{R}$ is not totally bounded.

######  Remark 3.6.

The space of Lipschitz bounded funrction
$(Lip_{b}(\mathbb{R}),\|\cdot\|_{\infty})$ and the space of Lipschitz
functions that vanish at 0 $(Lip_{0}(\mathbb{R}),Lip(\cdot))$ are not
separable. To see this, consider the set of natural numbers $\mathbb{N}$ as a
subset of $\mathbb{R}.$ For every subset $S\subset\mathbb{N}\setminus\\{0\\},$
define the function $f_{S}:\mathbb{R}\to\mathbb{R}$ given by

$r\mapsto\min\Big{\\{}\frac{1}{4},\min\\{|r-s|:s\in
S\cup\\{0\\}\\}\Big{\\}}=\min\Big{\\{}\frac{1}{4},|r|,d(r,S)\\}\Big{\\}},$

which are $1$-Lispchitz functions bounded by $1/4$.

It is immediate to notice that $\|f_{S}-f_{D}\|_{\infty}\geq 1/4,$ which
implies that these functions are separated, what, together with the fact that
there are as many functions as subsets of $\mathbb{N}\setminus\\{0\\},$ gives
that the family $\mathcal{H}:=\\{f_{S}:S\subset\mathbb{N}\\}$ cannot the
approximated by any countable family of functions. So,
$(Lip_{b}(\mathbb{R}),\|\cdot\|_{\infty})$ is not separable.

Note also that the same construction can be used to prove that
$(Lip_{0}(\mathbb{R}),Lip(\cdot))$ is not separable. Indeed, for all the
functions $f_{S}$ we have that $f_{S}(0)=0$ and $Lip(f_{S}-f_{D})=1$ for all
$S,D\subset\mathbb{N}\setminus\\{0\\}$ such that $S\neq D.$

Let us show some examples of non-strongly measurable functions that still
define lattice Lipschitz operators. We will show examples in (non
$\sigma-$finite) discrete measure spaces as well as in finite measure spaces
(Lebesgue measure).

######  Example 3.7.

Consider the measure space
$(B_{Lip_{b}(\mathbb{R})},\mathcal{P}(B_{Lip_{b}(\mathbb{R})}),c)$, where $c$
is the counting (extended) measure. Since $Lip_{b}(\mathbb{R})$ is not
separable, take a non-countable set $N$ in it that cannot be approximated by
any countable set (for example, as in Remark 3.6).

Define the vector valued function $\Phi:B_{Lip_{b}(\mathbb{R})}\to
B_{Lip_{b}(\mathbb{R})}$ by $\Phi=id\cdot\chi_{N}$ that is,

$\Phi(\tau)=\sum_{h\in
N}h\cdot\chi_{\\{h\\}}(\tau)=\left\\{\begin{array}[]{ll}\tau&\mathrm{if\
}\tau\in N\\\ 0&\mathrm{if\ }\tau\notin N\,.\end{array}\right.$

It cannot be strongly measurable, since if there is a sequence
$(s_{n})_{n=1}^{\infty}$ of simple functions such that
$s_{n}=\sum_{i=1}^{m_{n}}\eta_{n}^{i}\chi_{A_{n}^{i}}\to\Phi$ pointwise, then
for all $\tau\in N$, $s_{n}(\tau)=\eta_{n}^{i_{n}}\to\Phi(\tau)=\tau$.

Note, however, that we can define a lattice Lipschitz operator

$T_{\Phi}:\ell^{\infty}(B_{Lip_{b}(\mathbb{R})})\to\ell^{\infty}(B_{Lip_{b}(\mathbb{R})})$

by pointwise composition, as we have done in the representation lemma 3.4.
Indeed,

$T_{\Phi}\big{(}(r_{\tau})_{\tau\in
B_{Lip_{b}(\mathbb{R})}}\big{)}=(\Phi(\tau)(r_{\tau}))_{\tau\in
B_{Lip_{b}(\mathbb{R})}},$

Therefore, for any $r,t\in\ell^{\infty}(B_{Lip_{b}(\mathbb{R})})$ and $\eta\in
B_{Lip_{b}(\mathbb{R})}$,

$\Big{|}T_{\Phi}(r)(\eta)-T_{\Phi}(t)(\eta)\Big{|}=\Big{|}\Phi(\eta)(r(\eta))-\Phi(\eta)(t(\eta))\Big{|}\leq
Lip(\Phi(\eta))\big{|}r(\eta)-t(\eta)\big{|}.$

Thus, $T_{\Phi}$ is lattice Lipschitz, with bound function
$K=Lip(\Phi(\cdot))=\chi_{N}.$ Note that the same construction can be done for
$Lip_{0}(\mathbb{R})$ instead of $Lip_{b}(\mathbb{R}).$

This example allows a general version, taking into account Pettis
measurability theorem. Note first that the only condition required in the
example above with the counting measure is the existence of a bijection
relating any element of the measure space with an element of the set
$\mathcal{H}$ defined by the functions $f_{S}.$ If $(\Omega,\Sigma,\mu)$ is a
measure space, any function $\Phi$ that satisfies that $\Phi(\Omega\setminus
A)\cap\mathcal{H}$ is uncountable for any $\mu-$null set $A\in\Sigma$ gives a
similar example. For instance, consider the measure space
$(\mathcal{P}(\mathbb{N}\setminus\\{0\\}),\mathcal{P}\big{(}\mathcal{P}(\mathbb{N}\setminus\\{0\\})\big{)},\nu),$
where $\nu$ is given by

$\nu(\\{S\\})=\sum_{n\in
S}\frac{1}{2^{n}},\quad\text{and}\quad\nu(\Lambda)=\sum_{S\in\Lambda}\nu(\\{S\\}).$

The function $\Phi:\mathcal{P}(\mathbb{N}\setminus\\{0\\})\to
Lip_{0}(\mathbb{R})$ given by $\Phi(S)=f_{S}$ is well-defined, and obviously
non-strongly measurable, due to the non-countable nature of
$\mathcal{P}(\mathbb{N}\setminus\\{0\\}).$

Next results shows where it could be ensured that the range of $T$ is a space
of measurable functions in therms of $\Phi$.

######  Lemma 3.8.

Let $(\Omega,\Sigma,\mu)$ be a measure space and $T:X(\mu)\to L^{0}(\mu)$ a
lattice Lipschitz operator. Then, there exists
$\Phi:\mathbb{R}\times\Omega\to\mathbb{R}$ such that

$\Phi(\cdot)(\lambda)=T(\lambda\chi_{\Omega}),\quad\lambda\in\mathbb{R},$

(that is, $\Phi(\cdot)(\lambda)$ is a particular representative of
$T(\lambda\chi_{\Omega})$), and for any $w\in\Omega,$
$\Phi(w):\mathbb{R}\to\mathbb{R}$ is a $K(w)$-Lipschitz function.

###### Proof.

Denote by $\mathfrak{M}$ the set of all real $\mu$-measurable functions
defined in all $\Omega$ without the usual $\mu$-almost everywhere equivalence.
That is, all possible representatives of classes of functions in $L^{0}(\mu)$.
We claim that there exists $\Psi:\mathbb{Q}\to\mathfrak{M}$ such that
$\Psi(\lambda)$ is a representative of $T(\lambda\chi_{\Omega})$ defined in
all $\Omega$ and for any $\lambda,\xi\in\mathbb{Q}$

$|\Psi(\lambda)(w)-\Psi(\xi)(w)|\leq K(w)|\lambda-\xi|\quad\text{for all}\quad
w\in\Omega.$ (2)

That is, for any $w\in\Omega$, $\Psi(\cdot)(w):\mathbb{Q}\to\mathbb{R}$ is a
$K(w)$-Lipschitz function.

Let $\mathbb{Q}=(\lambda_{n})_{n=1}^{\infty}$, and choose $\Psi(\lambda_{1})$
to be any representative of $T(\lambda_{1}\chi_{\Omega})$ defined in all
$\Omega$. Assume now that $\Psi$ is defined in $\\{\lambda_{n}\\}_{n=1}^{m-1}$
satisfying the given property. Choose $\sigma:\Omega\to\mathbb{R}$ to be any
representative of $T(\lambda_{m}\chi_{\Omega})$. By hypothesis, there exists
$N_{m}^{n}$ null sets such that

$|\Psi(\lambda_{n})(w)-\sigma(w)|\leq|\lambda_{n}\chi_{\Omega}(w)-\lambda_{m}\chi_{\Omega}(w)|=K(w)|\lambda_{n}-\lambda_{m}|$

for any $w\not\in N_{m}^{n}$, $n\leq m$. Let $N_{m}=N_{m}^{1}\cup
N_{m}^{2}\cup\cdots\cup N_{m}^{n-1}$ and define
$\Psi(\lambda_{m}):\Omega\to\mathbb{R}$ as

$\Psi(\lambda_{m})(w)=\begin{cases}\sigma(w)&w\not\in N_{m}\\\
\Psi(\lambda_{1})(w)&w\in N_{m}^{1}\\\ \Psi(\lambda_{2})(w)&w\in
N_{m}^{2}\setminus N_{m}^{1}\\\ \quad\vdots\\\ \Psi(\lambda_{m-1})(w)&w\in
N_{m}^{m-1}\setminus(N_{m}^{1}\cup\cdots\cup N_{m}^{m-2})\\\ \end{cases}.$

Clearly, (2) is satisfied for any $\lambda_{m},\lambda_{n}$ in all $\Omega$,
also for elements in $N_{m}$. Proceeding inductively, $\Psi$ is correctly
defined in $\mathbb{Q}$.

Define $\Phi(w)(\lambda)=\Psi(\lambda)(w)$ in $\Omega\times\mathbb{Q}$. Then,
given any $w\in\Omega$, $\Phi(w)\\!\\!\restriction_{\mathbb{Q}}$ is a
$K(w)$-Lipschitz function, so it can be extended to $\mathbb{R}$ keeping the
Lipschitz constant, for example following McShane formula, see [30]. For
$\lambda\in\mathbb{R}\setminus\mathbb{Q}$, consider
$\tilde{T}(\lambda\chi_{\Omega})(w)$ a representative of
$T(\lambda\chi_{\Omega})(w)$. By the continuity of $\Phi(w)$, if
$\mathbb{Q}\ni\lambda_{n}\to\lambda$,

$\displaystyle|\tilde{T}(\lambda\chi_{\Omega})(w)-\Phi(w)(\lambda)|$
$\displaystyle=\lim_{n\to\infty}|\tilde{T}(\lambda\chi_{\Omega})(w)-\Phi(w)(\lambda_{n})|$
$\displaystyle\leq\lim_{n\to\infty}K(w)|\lambda-\lambda_{n}|=0,$

for almost every $w\in\Omega$. Then,
$\Phi(\cdot)(\lambda):\Omega\to\mathbb{R}$ belongs to $\mathfrak{M}$ and is a
representative of $T(\lambda\chi_{\Omega})$. ∎

Let us recall the construction of a predual of $Lip_{0}(M)$ for any pointed
metric space $M$. For each $x\in M$, consider the evaluation functional
$\delta_{x}:Lip_{0}(M)\to\mathbb{R}$ defined as $\delta_{x}(f)=f(x)$ as an
element in $\big{(}Lip_{0}(M)\big{)}^{*}$. Define the so called Lipschitz-free
space $\mathfrak{F}(M)=\overline{\text{span}}\\{\delta_{x}:\ x\in
M\\}\subseteq\big{(}Lip_{0}(M)\big{)}^{*}$ using the dual norm which, in fact,
satisfies $\|\delta_{x}-\delta_{y}\|=d(x,y)$ for $x,y\in M$. Then,
$\big{(}\mathfrak{F}(M)\big{)}^{*}=Lip_{0}(M)$. This construction is explained
in detail in [18], where other constructions of a predual of $Lip_{0}(M)$,
such as the Arens-Eells space $\AE(M)$, are also studied.

######  Theorem 3.9.

Let $(\Omega,\Sigma,\mu)$ a measure space and $X(\mu)$ a Banach function
space. The operator $T:X(\mu)\to L^{0}(\mu)$ is lattice Lipschitz if and only
if there exists $\Phi:\Omega\to Lip_{0}(\mathbb{R})$ such that
$T(f)(w)=\Phi(w)(f(w))$ (for almost any $w\in\Omega$) and
$p\circ\Phi:\Omega\to\mathbb{R}$ is $\mu$-measurable for any
$p\in\mathfrak{F}(\mathbb{R})$.

###### Proof.

Assume that $T$ is lattice Lispchitz. Define $\Phi(w)(\lambda)$ as the
concrete representative of $T(\lambda\chi_{\Omega})$ evaluated on $w$ given by
Lemma 3.8. For any $\lambda\in\mathbb{R}$,
$\delta_{\lambda}\circ\Phi=\Phi(\cdot)(\lambda)$ is measurable since is a
representative of $T(\lambda\chi_{\Omega})$. By taking linear combinations and
limits of function in $L^{0}(\mu)$, we obtain that $p\circ\Phi$ is
$\mu$-measurable for any $p\in\mathfrak{F}(\mathbb{R})$.

For the reciprocal, let us see first that $T(f)(w)=\Phi(w)(f(w))$ is well
defined from $X(\mu)$ to $L^{0}(\mu)$. Let
$f=\sum_{i=1}^{n}\lambda_{i}\chi_{A_{i}}\in X(\mu)$ a simple function. Then,

$T(f)(w)=\Phi(w)\left(\sum_{i=1}^{n}\lambda_{i}\chi_{A_{i}}(w)\right)=\sum_{i=1}^{n}\Phi(w)(\lambda_{i})\chi_{A_{i}}(w)\in
L^{0}(\mu),$

since
$w\mapsto\Phi(w)(\lambda_{i})=\langle\Phi(w),\delta_{\lambda_{i}}\rangle$ is
$\mu$-measurable for all $i$.

Since every limit of simple functions is measurable, we have that $T(f)\in
L^{0}(\mu)$ for all $f\in X(\mu).$ Moreover, if $f=g$ $\mu$-a.e. in $\Omega$,
clearly $T(f)(w)=\Phi(w)(f(w))=\Phi(w)(g(w))=T(f)(w)$ $\mu$-a.e., so $T$ is
well defined. To see that $T$ satisfies the lattice Lipschitz inequality, let
$f,g\in X$. For any choose of the representatives,

$|T(f)(w)-T(g)(w)|=|\Phi(w)(f(w))-\Phi(w)(g(w))|\leq Lip\
\Phi(w)\cdot|f(w)-g(w)|,$

so we can select $K(w)=Lip\ \Phi(w)$. ∎

######  Corollary 3.10.

If $\Phi:\Omega\to Lip_{0}(\mathbb{R})$ is weakly $\mu$-measurable (or
strongly $\mu$-measurable) and $X(\mu)$ is any Banach function space, then
$T:X\to L^{0}(\mu)$ defined as $T(f)(w)=\Phi(w)(f(w))$ is a lattice Lipschitz
operator.

###### Proof.

Assume that $\Phi$ is weakly measurable (since strong measurability implies
it). Given any $p\in\mathfrak{F}(\mathbb{R})$, as
$p\in\big{(}\mathfrak{F}(\mathbb{R})\big{)}^{**}=Lip_{0}(\mathbb{R})^{*}$,
$p\circ\Phi$ is $\mu$-measurable, so by Theorem 3.9, $T$ is a lattice
Lipschitz operator from $X$ to $L^{0}(\mu)$. ∎

######  Example 3.11.

Consider the Lebesgue measurable space $([0,1],\Sigma,\mu)$. Each real number
in $[0,1]$ can be identified with a subset of natural numbers via a binary
representation of its digits. Since that representation is not always unique,
choose the one that is eventually 0 rather that 1. Define $\Phi$ as

$\displaystyle\Phi:[0,1]$ $\displaystyle\to Lip_{0}(\mathbb{R})$
$\displaystyle w=\sum_{i=1}^{\infty}c_{i}\frac{1}{2^{i}}$
$\displaystyle\mapsto\left(\lambda\mapsto\min\left\\{\frac{1}{2},|\lambda|,d(\lambda,\\{i\in\mathbb{N}:c_{i}=0\\})\right\\}\right).$

Let us see that $T:L^{1}(\mu)\to L^{1}(\mu)$ defined by
$T(f)(w)=\Phi(w)(f(w))$ is a well defined lattice Lispchitz operator using
Theorem 3.9. Let $\lambda\in\mathbb{R}$ and consider
$\delta_{\lambda}\in\mathfrak{F}(\mathbb{R})$, we claim that
$w\mapsto\langle\Phi(w),\delta_{\lambda}\rangle=\Phi(w)(\lambda)$ is
$\mu$-measurable. Consider, for simplicity, the function

$\varphi_{\lambda}(w)=\min\left(\left\\{\frac{1}{2}\right\\}\cup\big{\\{}|\lambda-i|:\
c_{i}=0,\ i\in\mathbb{N}\big{\\}}\right),$

where $w=\sum_{i=1}^{\infty}c_{i}\frac{1}{2^{i}}$ is the binary representation
of $w$ as before. Observe that
$\Phi(w)(\lambda)=\min\\{|\lambda|,\varphi_{\lambda}(w)\\}$, so it is enough
to see that $\varphi_{\lambda}:\Omega\to[0,\frac{1}{2}]\subseteq\mathbb{R}$ is
$\mu$-measurable. Let $0<s<\frac{1}{2}$, and consider

$\varphi_{\lambda}^{-1}\big{(}]-\infty,s[\big{)}=\left\\{w=\sum_{i=1}^{\infty}c_{i}\frac{1}{2^{i}}\in\Omega:\
\\{0,1\\}\,\reflectbox{$\in$}\,c_{i}\not\to 1,\ \exists c_{i_{0}}=0:|\lambda-
i_{0}|<s\right\\}.$

If $d(\lambda,\mathbb{N})\geq s$,
$\varphi_{\lambda}^{-1}\big{(}]-\infty,s[\big{)}=\emptyset\in\Sigma$.
Alternatively, pick $i_{0}$ to be the nearest natural number to $\lambda$, so

$\displaystyle\varphi_{\lambda}^{-1}\big{(}]-\infty,s[\big{)}$
$\displaystyle=\left\\{w=\sum_{i=1}^{\infty}c_{i}\dfrac{1}{2^{i}}\in\Omega:\
\\{0,1\\}\,\reflectbox{$\in$}\,c_{i}\not\to 1,\ c_{i_{0}}=0\right\\}$
$\displaystyle=\bigcup_{j=0}^{2^{i_{0}-1}-1}\left[\frac{2j}{2^{i_{0}}},\frac{2j+1}{2^{i_{0}}}\right[\in\Sigma,$

which proves that $\varphi_{\lambda}$ is a Lebesgue measurable function. This
situation, together with the fact that $Lip\ \Phi(w)$ is bounded by 1, shows
that $T$ maps $L^{1}(\mu)$ to $L^{1}(\mu)$.

If $w,\tau\in[0,1]$ are different, $Lip(\Phi(w)-\Phi(\tau))=1$ (reasoning as
in Remark 3.6). Then, $\Phi(\Omega)$ is not separable, so $\Phi$ cannot be
approximated by simple functions.

Observe that any Banach function space $X(\mu)$ can be considered instead of
$L^{1}(\mu)$, having the same result.

##  4\. Lattice Lipschitz operators that are representable by strongly
measurable functions

We now show a representation theorem for lattice Lipschitz operators on spaces
of Banach functions in terms of a vector-valued measurable function with
values in $Lip_{0}(\mathbb{R}).$ We will see that these operators can in
general be written in terms of pointwise diagonal Lipschitz maps, and
sometimes a representation using a strongly measurable function is also
available. The main technical problem throughout this section is the handling
of the $\mu-$null sets appearing in Lipschitz-type inequalities, which have to
be properly understood: a $\mu-$a.e. property sometimes concerns the terms
involved in different ways. Essentially, these inequalities concern both the
real numbers and the elements of $\Omega,$ and exchanging the roles of these
variables is a delicate move.

Let us first prove the more general result, which does not explicitly require
any measurability property of the fundamental function $\Phi_{T}.$ Instead,
the function is assumed to act only on a special class of subsets of functions
of $X(\mu),$ which are those of the type $f=\lambda\chi_{\Omega}$ for
$\lambda\in\mathbb{R}.$We show that the lattice Lipschitz operators are
disjoint, imitating the linear case and focusing on their “diagonal” nature.

######  Proposition 4.1.

Let $\Phi:\Omega\to Lip_{0}(\mathbb{R})$ be a strongly measurable function
such that $\Phi\in\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))$. Consider the
corresponding operator $T_{\Phi}:X(\mu)\to L_{0}(\mu)$, given by the pointwise
composition $T_{\Phi}(f)(w)=\Phi(w)(f(w))$. Then, $T_{\Phi}$ is

  * i)

a well-defined operator $T_{\Phi}:X(\mu)\to Y(\mu)$,

  * ii)

Lipschitz continous, and

  * iii)

lattice Lipschitz with the best bound function appearing in the lattice
Lipschitz inequality $K(\cdot)=Lip(\Phi(\cdot));$ that is, if there is any
other measurable function $h$ such that the lattice Lipschitz inequality holds
for $h(\cdot)=K(\cdot),$ then $Lip(\Phi(\cdot))\leq h(\cdot)$ $\mu-$a.e.

Moreover, if there is another function $\Psi:\Omega\to Lip_{0}(\mathbb{R})$
such that $\Phi=\Psi$ $\mu-$a.e., then $T_{\Phi}=T_{\Psi}$ and the bound
functions $w\mapsto Lip(\Phi(w))$ and $w\mapsto Lip(\Psi(w))$ are $\mu-$a.e.
equal.

###### Proof.

Given $f\in X(\mu)$, by Corollary 3.10, $T(f)$ is a $\mu$-measurable function.
As $\Phi(\Omega)\subseteq Lip_{0}(\mathbb{R})$, $T(0)=0$ and so for almost any
$w\in\Omega$, $|T(f)(w)|\leq Lip(\Phi(w))\cdot|f(w)|$. Then, since
$Lip(\Phi(\cdot))\in\mathcal{M}(X(\mu),Y(\mu))$,
$Lip(\Phi(\cdot))\cdot|f(w)|\in Y(\mu)$ and, by the ideal property of
$Y(\mu)$, $T(f)\in Y(\mu)$. Moreover, given $f,g\in X(\mu)$

$\|T(f)-T(g)\|_{Y(\mu)}\leq\|Lip(\Phi(\cdot))\|_{\mathcal{M}(X(\mu),Y(\mu))}\cdot\|f-g\|_{X(\mu)}.$

Consider now a pair of functions $f,g\in X(\mu)$ and $w\in\Omega$,

$|T(f)(w)-T(g)(w)|=|\Phi(w)(f(w))-\Phi(w)(g(w))|\leq
Lip(\Phi(w))\,|f(w)-g(w)|$

and so $T_{\Phi}$ is lattice Lipschitz with bound function
$K(w)=Lip(\Phi(w))$. Note that $\Phi(w)$ (and so $Lip(\Phi(w))$) is defined
$\mu-$a.e.

We claim now that for every $w\in\Omega$ and every $\varepsilon>0$ there are
functions $f_{1},f_{2}\in X(\mu)$ such that

$|T_{\Phi}(f_{1})(w)-T_{\Phi}(f_{2})(w)|>=(Lip(\Phi(w))-\varepsilon)\,{|f_{1}-f_{2}|(w)}.$

Note that this implies that $Lip(\Phi(w))$ is the best constant that can be
written in this inequality. Indeed, fix $w\in\Omega.$ Consider
$\Phi(w)=\varphi_{w}\in Lip_{0}(\mathbb{R}).$ By definition of the Lipschitz
constant, there are two different real numbers $r_{1}$ and $r_{2}$ such that

$\frac{|\varphi_{w}(r_{1})-\varphi_{w}(r_{2})|}{|r_{1}-r_{2}|}>Lip(\varphi_{w})-\varepsilon.$

Consider the measurable functions $\hat{f}_{1}=r_{1}\chi_{\Omega}$ and
$\hat{f}_{2}=r_{2}\chi_{\Omega}.$ Then

$|(\Phi\circ\hat{f}_{1})(w)-(\Phi\circ\hat{f}_{2})(w)|=|\varphi_{w}(r_{1})-\varphi_{w}(r_{2})|>(Lip(\varphi_{w})-\varepsilon)\,{|r_{1}-r_{2}|}.$

Thus, since this can be done for all $r_{1},r_{2}\in\mathbb{R}$ for each
$\varepsilon>0$ and $w$ there are functions $f_{1},f_{2}\in X(\mu)$ having as
representatives the defined above functions $\hat{f}_{1}$ and $\hat{f}_{2}$ in
such a way that

$|T_{\Phi}(f_{1})(w)-T_{\Phi}(f_{2})(w)|>(Lip(\Phi(w))-\varepsilon)\,{|r_{1}-r_{2}|}=(Lip(\Phi(w))-\varepsilon)\,{|f_{1}-f_{2}|(w)}.$

Since this holds for all $\varepsilon,$ this means that the contant
$Lip(\Phi(w))$ cannot be improved for this fixed $w.$

Suppose now that there is another function $h\leq Lip(\Phi(\cdot))$ such that
there is a non-null set $A$ satisfying
$h\,\chi_{A}<Lip(\Phi(\cdot))\,\chi_{A}$ and for all $f_{1},f_{2}\in X(\mu),$

$|\Phi(f_{1})(w)-T_{\Phi}(f_{2})(w)|\leq h(w)\,{|f_{1}(w)-f_{2}(w)|},\quad
w\in A.$

By the claim, this means that for every $r_{1},r_{2}\in\mathbb{R},$

$|\varphi_{w}(r_{1})-\varphi_{w}(r_{2})|\leq
h(w)|r_{1}-r_{2}|<Lip(\Phi(w))|r_{1}-r_{2}|,$

for all $w$ belonging to a non-null set, what contradicts the definition of
$Lip(\Phi(w)).$

Finally, suppose that the measurable functions $\Phi$ and $\Psi$ on
$Lip(\mathbb{R})$ are equal $\mu-$a.e. and let $f\in X(\mu).$ Clearly, for two
representatives $\hat{f}_{1}$ and $\hat{f}_{2}$ of $f$ we get
$\Phi\circ\hat{f}_{1}=\Psi\circ\hat{f}_{2}$ $\mu-$a.e. and so we clearly have
that $T_{\Phi}$ and $T_{\Psi}$ are equal as operators from $X(\mu)$ to
$Y(\mu).$ The associated bound functions are equal $\mu-$a.e. as a consequence
of the definition of the Köthe-Bochner spaces: the measurable functions
$w\mapsto\|\Phi(w)\|_{Lip_{0}(\mathbb{R})}=Lip(\Phi(w))$ and
$w\mapsto\|\Psi(w)\|_{Lip_{0}(\mathbb{R})}=Lip(\Psi(w))$ are equal $\mu-$a.e.
∎

Observe that, in contrast to this previous case, a general lattice Lipschitz
operator can be non-Lipschitz. For example, let $([0,1],\Sigma,\mu)$ be the
Legesgue measure space and consider $X(\mu)=Y(\mu)=L^{1}(\mu)$, so
$\mathcal{M}(X(\mu),Y(\mu))=L^{\infty}(\mu)$. Define $T:L^{1}(\mu)\to
L^{1}(\mu)$ as $T(f)=\inf\\{f^{2},1/\sqrt{\cdot}\\}$. Then,
$|T(f)(w)-T(g)(w)|\leq 2/\sqrt{w}\cdot|f(w)-g(w)|$, but considering
$f_{n}=n\chi_{[0,1/n^{4}]}$ and the zero function, it can be shown that $T$ is
not Lipschitz continous.

Moreover, if $T:X(\mu)\to Y(\mu)$ is a lattice Lipschitz operator such that
$T(f)(w)=\Phi(w)\big{(}f(w)\big{)}$ for a strongly measurable function
$\Phi:\Omega\to Lip_{0}(\mathbb{R})$, there exists a sequence of simple
vector-valued functions $\Phi_{n}:\Omega\to Lip_{0}(\mathbb{R})$ such that
$Lip(\Phi(w)-\Phi_{n}(w))\to 0$ for almost any $w\in\Omega$. But that
convergence has no relation with the norm in $Y(\mu)$, so it should not be
possible to conclude strong claims about the continuity of $T$ without more
hypothesis, as in the previous proposition. In fact, in more “pathological”
cases as the ones provided by $L^{\infty}(\mu)$, easy examples of lattice
Lispchitz non continuous operators can be found. For example, $T:L^{1}(\mu)\to
L^{\infty}(\mu)$ defined as $T(f)=\sup\big{\\{}\inf\\{f,1\\},-1\big{\\}}$.
There, $\Phi$ is a constant function, so it is strongly measurable. This
example does not contradict Theorem 1 of [6], since that result requires, in
our context, $Lip\ \Phi(w)$ to belong to $\mathcal{M}(X(\mu),Y(\mu))$ which is
only composed of the null function in this example.

The result above motivates the following definition, that will provide some
representation results for the class of lattice Lipschitz operators with
representing functions $\Phi$ satisfying certain properties.

######  Definition 4.2.

We say that a map $T:X(\mu)\to Y(\mu)$ is a strongly lattice Lipschitz
operator if it can be written as a pointwise composition $T(f)=\Phi\circ f$
for all $f\in X(\mu)$ with a strongly measurable function $\Phi:\Omega\to
Lip_{0}(\mathbb{R})$ belonging to the Köthe-Bochner space
$\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R})).$ We will write
$SLL(X(\mu),Y(\mu))$ for the space of all such operators endowed with the norm

$\|T_{\Phi}\|_{SLL(X(\mu),Y(\mu))}:=\big{\|}Lip(\Phi(\cdot))\big{\|}_{\mathcal{M}(X,Y)}.$

Note that, in particular, $T(0)=0.$

For example, given any Banach function space $X(\mu)$ since
$\mathcal{M}(X(\mu),X(\mu))=L^{\infty}(\mu)$ (see [28, Th.1]) we have that
$SLL(X(\mu),X(\mu))$ is the set of $T_{\Phi}:X(\mu)\to X(\mu)$ for which
$\Phi:\Omega\to Lip_{0}(\mathbb{R})$ is a strongly measurable function such
that $\text{ess}\sup Lip(\Phi(\cdot))$ is finite.

Next result gives a characterization of strongly lattice Lipschitz operators.
We will need the requirement that $X(\mu)\subseteq Y(\mu);$ it is satisfied
each time $\mu$ a finite measure and $\chi_{\Omega}\in\mathcal{M}(X,Y)$. For
example, let $\mu$ be the Lebesgue measure in $[0,1]$, $X(\mu)=L^{p}[0,1]$ and
$Y(\mu)=L^{q}[0,1]$ for $1\leq p,q\leq\infty.$ If $p<q$, as $L^{p}[0,1]\supset
L^{q}[0,1]$, we have that $SLL(L^{p}[0,1],L^{q}[0,1])=\\{0\\}$ (see for
example [12, Ex.2.1]). But if $p>q$, $L^{p}[0,1]\subset L^{q}[0,1]$, and is it
well known that $\mathcal{M}(L^{p}[0,1],L^{q}[0,1])=L^{r}[0,1],$ where
$1/p+1/r=1/q,$ so $SLL(L^{p}[0,1],L^{q}[0,1])=\\{T_{\Phi}:\Phi\in
L^{r}(\mu,Lip_{0}(\mathbb{R}))\\}$. Thus, for the $L^{p}(\mu)$ spaces of
finite measures, the condition $X(\mu)\subseteq Y(\mu)$ is always satisfied
when there are more strong lattice Lispchitz operators than the null one.

######  Remark 4.3.

Since $\Phi$ is strongly measurable we can approximate it pointwise with a
sequence of simple functions
$\Phi_{n}(w)=\sum_{i=1}^{k_{n}}\varphi^{n}_{i}\chi_{A^{n}_{i}}(w)$ but in a
$\mu-$null set $N.$ Fix a function $f\in X(\mu)$ and recall that we have
assumed that $X(\mu)\subseteq Y(\mu).$ Then for every $w\in\Omega,$

$|\Phi_{n}(w)(f(w))|=\left|\sum_{i=1}^{k_{n}}\varphi^{n}_{i}(f(w))\chi_{A^{n}_{i}}(w)\right|\leq\sum_{i=1}^{k_{n}}Lip(\varphi^{n}_{i})|f(w)|\chi_{A^{n}_{i}}(w)\in
X(\mu),$

and so $\Phi_{n}(w)(f(w))$ belongs to $Y(\mu).$ Note also that

$\sum_{i=1}^{k_{n}}Lip(\varphi^{n}_{i})|f(w)|\chi_{A^{n}_{i}}(w)=Lip(\Phi_{n}(w))\,|f(w)|$

for every $w.$ Consider now the (non-negative) functions

$h_{n}(\cdot):=\inf\big{\\{}Lip(\Phi_{k}(\cdot))\,|f(\cdot)|:k\geq
n\big{\\}}.$

Since every Banach function space is order complete, the functions $h_{n}$
belong to $Y(\mu).$ On the other hand, $Lip(\Phi(w)-\Phi_{n}(w))\to_{n}0$ for
$w\in\Omega\setminus N.$ This gives for every $w\notin N,$

$\Big{|}Lip(\Phi(w))\cdot f(w)-Lip(\Phi_{n}(w))\cdot f(w)\Big{|}\leq
Lip(\Phi(w)-\Phi_{n}(w))\cdot\big{|}f(w)\big{|}\to_{n}0,$

what implies that $h_{n}(\cdot)\to_{n}Lip(\Phi(\cdot))\,|f(\cdot)|$ $\mu-$a.e.
The problem is that, in general, a sequence of measurable functions converging
pointwise to a function in a Banach function space does not necessarily
converges in norm to this function. So we need further requirements, both in
the space and in the properties of the functions involved.

######  Theorem 4.4.

Let $X(\mu)$ and $Y(\mu)$ be Banach function spaces such that $X(\mu)\subseteq
Y(\mu)$ and $Y(\mu)$ has the Fatou property. The following facts are
equivalent for an operator $T:X(\mu)\to Y(\mu).$

  * (i)

$T$ is strongly lattice Lipschitz.

  * (ii)

There exist a strongly measurable function $\Phi:\Omega\to
Lip_{0}(\mathbb{R})$ such that $T=T_{\Phi}$ and a sequence of essentially
bounded functions $(\Phi_{n})_{n}\subset L_{\infty}(\mu,Lip_{0}(\mathbb{R})),$
$\Phi_{n}:\Omega\to Lip_{0}(\mathbb{R})$ such that

$\sup_{n}\big{\|}Lip(\Phi_{n}(\cdot))\big{\|}_{\mathcal{M}(X,Y)}<\infty$

and converges $\mu-$a.e. to $\Phi.$

Moreover, the sequence $(\Phi_{n})_{n}$ can be chosen in such a way that

$\sup_{n}\big{\|}\Phi_{n}(\cdot)\big{\|}_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}=\|\Phi\|_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}.$

###### Proof.

(i) $\Rightarrow$ (ii) If $T$ is strongly lattice Lipschitz it can be written
as $T_{\Phi}$ for a certain strongly measurable function belonging to
$\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R})).$ Also, there is a sequence of
simple functions as
$\Phi_{n}(w)=\sum_{i=1}^{k_{n}}\varphi^{n}_{i}\chi_{A^{n}_{i}}(w)$ that
converges $\mu-$a.e. to $\Phi.$ Then for every $n\in\mathbb{N}$ consider the
measurable set

$B_{n}=\\{w:Lip(\Phi_{n}(w))<Lip(\Phi(w))\\}$

and the measurable function

$\psi_{n}(w):=\Phi_{n}(w)\chi_{B_{n}}+\Phi(w)\chi_{B_{n}^{c}}.$

Clearly,

$Lip(\psi_{n}(\cdot))=Lip(\Phi_{n}(\cdot))\chi_{B_{n}}+Lip(\Phi(\cdot))\chi_{B_{n}^{c}}\leq
Lip(\Phi_{n}(\cdot))$

$\mu-$a.e. and so it is bounded $\mu-$a.e. Also, by the ideal property of the
norm of the Banach function space $\mathcal{M}(X,Y),$ we obtain the uniform
bound for the norms
$\|\psi_{n}\|_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}\leq\|\Phi\|_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}.$
Finally, it is also clear that the sequence $(\psi_{n})_{n}$ converges
$\mu-$a.e. to $\Phi,$ what gives the result.

(ii) $\Rightarrow$ (i) By hypothesis, there is a strongly measurable function
$\Phi$ representing $T$ and we have to prove that
$\Phi\in\mathcal{M}(X,Y)\big{(}\mu,Lip_{0}(\mathbb{R})\big{)},$ that is
$\|\Phi(\cdot)\|_{Lip_{0}(\mathbb{R})}=Lip(\Phi(\cdot))\in\mathcal{M}(X,Y).$

By hypothesis, we can approximate $\Phi$ $\mu-$a.e. with a sequence of simple
functions $\Phi_{n}$ that are bounded but in a $\mu-$null set $N.$ Take a
function $f\in X(\mu)$. Since $X(\mu)\subseteq Y(\mu),$ we have for
$\mu-$almost all $w$

$|\Phi_{n}(w)(f(w))|\leq
Lip(\Phi_{n}(w))|f(w)|\leq\|Lip(\Phi_{n}(w))\|_{L^{\infty}(\mu)}\,|f(w)|\in
X(\mu)\subset Y(\mu),$

and so $\Phi_{n}(w)(f(w))$ belongs to $Y(\mu).$ Take the (non-negative classes
of) functions

$h_{n}(\cdot):=\inf\big{\\{}Lip(\Phi_{k}(\cdot))\,|f(\cdot)|:k\geq
n\big{\\}}.$

Note that the infimum of every sequence of measurable functions that is
bounded by below is again measurable, and every Banach function space is order
complete, so for every $n\in\mathbb{N},$ the (equivalence class of the)
function $h_{n}$ belongs to $Y(\mu).$ Since we have
$Lip(\Phi(w)-\Phi_{n}(w))\to_{n}0$ $\mu-$a.e. we get

$\Big{|}Lip(\Phi(w))\cdot f(w)-Lip(\Phi_{n}(w))\cdot f(w)\Big{|}\leq
Lip(\Phi(w)-\Phi_{n}(w))\cdot\big{|}f(w)\big{|}\to_{n}0,$

and so $h_{n}(\cdot)\to_{n}Lip(\Phi(\cdot))\,|f(\cdot)|$ $\mu-$a.e. In
addition, the sequence $(h_{n})_{n}$ is increasing and positive. Recall that
by hypothesis $\sup_{n}\|Lip(\Phi_{n})(\cdot))\|_{M(X,Y)}<\infty,$ and so

$\displaystyle\sup_{n}\|h_{n}(\cdot)\|_{Y(\mu)}$
$\displaystyle\leq\sup_{n}\|Lip(\Phi_{n})(\cdot)\cdot|f(\cdot)|\|_{Y(\mu)}$
$\displaystyle\leq\sup_{n}\|Lip(\Phi_{n})(\cdot)\|_{\mathcal{M}(X,Y)}\cdot\|f\|_{X(\mu)}<\infty.$

Since $Y(\mu)$ has the Fatou property, we conclude that the $\mu-$a.e. limit
of the sequence defined by the functions $h_{n}$ (that is,
$Lip(\Phi(\cdot))\,|f(\cdot)|$), belongs to $Y(\mu)$ too. Since $\Phi$ is
strongly measurable, we know that $T_{\Phi}(f)$ that is given by (the
equivalence class of) $\Phi(\cdot)(f(\cdot))$ is a measurable function
belonging to $L_{0}(\mu),$ and

$|\Phi(\cdot)(f(\cdot))|\leq Lip(\Phi(\cdot))\,|f(\cdot)|.$

Since we have shown that $Lip(\Phi(\cdot))\,|f(\cdot)|\in Y(\mu),$ we obtain
by the ideal property of $Y(\mu)$ that $\Phi(\cdot)(f(\cdot))\in Y(\mu).$ Note
that all the arguments above are independent of the representative. This shows
that the operator $T_{\Phi}$ is well-defined from $X(\mu)$ to $Y(\mu).$

Note that in fact we have shown more. We have proved that for every function
$f\in X(\mu),$ the multiplication $Lip(\Phi(\cdot))\,|f(\cdot)|$ belongs to
$Y(\mu).$ Since a well-defined operator defined by the product with a
(positive) function is always continuous, we have that the multiplication
operator $M_{Lip(\Phi)}:X(\mu)\to Y(\mu)$ given by the function
$Lip(\Phi)(\cdot)$ is well-defined and, since it is positive, it is continuous
(see [3, Th.4.3]). This means that $Lip(\Phi(\cdot))\in\mathcal{M}(X,Y).$
Since $\Phi$ is strongly measurable, we get that
$\Phi\in\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R})),$ as desired.

Finally, in order to prove the “moreover part” of the result, consider the
sequence $(\psi_{n})_{n}$ of vector valued functions and the associate
sequence of real valued measurable functions $(Lip(\psi_{n}(\cdot)))_{n}$ that
has been constructed in the proof of (i) $\Rightarrow$(ii). The functions
$\psi_{n}$ will play the role of the functions $\Phi_{n}$ appearing in the
last statement of the result. Since $(\psi_{n})_{n}$ converges $\mu-$a.e. to
$\Phi,$ we get that $(Lip(\psi_{n}(\cdot)))_{n}$ also converges $\mu-$a.e. to
$Lip(\Phi(\cdot)).$ Consider now the sequence of real valued functions
$(\tau_{n})_{n},$ where for each $n$

$\tau_{n}(\cdot)=\inf\big{\\{}Lip(\psi_{k})(\cdot):k\geq n\big{\\}}.$

Clearly, $\tau_{n}(\cdot)\uparrow Lip(\Phi(\cdot))$ $\mu-$a.e. and, due to the
ideal property of the space we also have
$\tau_{n}(\cdot)\in\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R})).$ Thus, we have
that

$\big{\|}\tau_{n}(\cdot)\big{\|}_{\mathcal{M}(X,Y)}\leq\big{\|}\psi_{n}(\cdot)\big{\|}_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}\leq\big{\|}\Phi(\cdot)\big{\|}_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}$

Since $Y(\mu)$ has the Fatou property we get that $\mathcal{M}(X,Y)$ has the
Fatou property too ([12, Prop. 3.3]), what gives that

$\sup_{n}\big{\|}\tau_{n}\big{\|}_{\mathcal{M}(X,Y)}=\sup_{n}\big{\|}\psi_{n}\big{\|}_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}=\big{\|}\Phi\big{\|}_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}.$

∎

##  5\. Applications: approximation formulas for strongly lattice Lipschitz
operators

In the case of strongly lattice Lipschitz operators we can obtain a good
representation in terms of tensor products (and consequently good
approximation formulas) under some mild requirements. In this section we will
show how to do so; we will also write down results for the case of
$L^{p}-$spaces of finite measure to illustrate the procedure. These results
will complement the analysis of lattice Lipschitz operators for the case of
infinite measure spaces, which are known to be defined on Euclidean spaces and
on $C(K)-$-spaces as a particular subclass of superposition operators. As we
explained in the Introduction, the good representation of these operators
playing the role of multiplication operators in the linear context is the key
to use them as useful tools in analysis and to find good applications in other
branches of applied mathematics such as Machine Learning.

Since we are working in the context of Köthe-Bochner spaces and operators
between Banach function spaces, it is natural to consider tensor product
representations for the space of strongly lattice Lipschitz operators. Each
simple tensor $t=\sum_{i=1}^{n}\varphi_{i}\otimes h_{i}\in
Lip_{0}(\mathbb{R})\otimes\mathcal{M}(X,Y)$ could be seen as an operator, in
this case by pointwise composition,

$T_{t}(f)(w)=\Big{(}\sum_{i=1}^{n}\varphi_{i}\otimes
h_{i}\Big{)}(f)(w)=\sum_{i=1}^{n}\varphi_{i}\big{(}f(w)\big{)}\cdot
h_{i}(w),\quad w\in\Omega.$

We can define a sort of canonical representation of the tensor, based on the
existence of standard representation of simple functions given by disjoint
sums of single tensors as $t=\sum_{i=1}^{n}\varphi_{i}\otimes\chi_{A_{i}},$
what gives when acts on a function $f\in X(\mu)$

$T_{t}(f)(w)=\sum_{i=1}^{n}\varphi_{i}\otimes\chi_{A_{i}}(w)(f(w))=\sum_{i=1}^{n}\varphi_{i}\big{(}f(w)\big{)}\,\chi_{A_{i}}(w),\quad
f\in X(\mu),\,\,\,w\in\Omega,$

for disjoint measurable sets $A_{1},...,A_{n}.$ Write $S_{X,Y}$ for the
subspace of the simple functions in $\mathcal{M}(X,Y)$ (which can always be
represented as above), and endow the tensor product
$Lip_{0}(\mathbb{R})\widehat{\otimes}S_{X,Y}$ with the natural norm
$\Delta_{\mathcal{M}(X,Y)}$ provided by the identification
$Lip_{0}(\mathbb{R})\widehat{\otimes}S_{X,Y}\ni t\mapsto T_{t}\in SLL(X,Y).$
Assuming that simple functions are dense in $\mathcal{M}(X,Y),$ we can obtain
that the set of tensors as above is dense in

$Lip_{0}(\mathbb{R})\widehat{\otimes}_{\Delta_{\mathcal{M}(X,Y)}}\mathcal{M}(X,Y).$

This directly gives the following

######  Corollary 5.1.

Let $X(\mu)$ and $Y(\mu)$ be Banach function spaces such that simple functions
are dense in $\mathcal{M}(X,Y)\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}.$ Then

$Lip_{0}(\mathbb{R})\widehat{\otimes}_{\Delta_{\mathcal{M}(X,Y)}}\mathcal{M}(X,Y)=\mathcal{M}(X,Y)\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}=SLL(X,Y).$

Therefore, for each strongly lattice Lipschitz operator $T$ there is a
sequence of simple tensors
$t_{n}=\sum_{i=1}^{k_{n}}\varphi^{k_{n}}\otimes\chi_{A_{i}^{n}}$ such that

$\lim_{n}\|T-\sum_{i=1}^{k_{n}}\varphi^{k_{n}}\otimes\chi_{A_{i}^{n}}\|_{\mathcal{M}(X,Y)(\mu,Lip_{0}(\mathbb{R}))}=0.$

###### Proof.

The identification of the closure of the tensor product with the space
$\mathcal{M}(X,Y)\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}$ is immediate, due to
the fact that the map
$\sum_{i=1}\varphi_{i}\otimes\chi_{A_{i}}\mapsto\sum_{i=1}\varphi_{i}\chi_{A_{i}}$
is injective, where $\sum_{i=1}\varphi_{i}\otimes\chi_{A_{i}}$ is any simple
tensor in $Lip_{0}(\mathbb{R})\otimes\mathcal{M}(X,Y)$ with $\\{A_{i}\\}_{i}$
disjoint measurable sets. Density of simple functions and coincidence of the
norm for the set of simple functions in both spaces gives the equality.

The identification of $\mathcal{M}(X,Y)\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}$
with $SLL(X,Y)$ is a consequence of requirements in Definition 4.2 and the
injectivity of the map that carries each function
$\Phi\in\mathcal{M}(X,Y)\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}$ to the operator
$T_{\Phi}$ that is defined in Proposition 4.1. The injectivity is also given
by the last statement in this proposition. This proves the result.

∎

It is well known that simple functions are dense in Bochner spaces
$L^{p}(\mu,E)$ where $E$ is any Banach space, provided that $1\leq p<\infty$
(see for example [19, S.7 .2], which are particular cases of multiplication
operator spaces such as those we are concerned with for
$E=Lip_{0}(\mathbb{R}).$ However, if $E$ is an infinite dimensional Banach
space, simple functions are not dense in $L^{\infty}(\mu,E);$ in fact, the
tensor product $L^{\infty}(\mu)\otimes E$ is not dense in $L^{\infty}(\mu,E)$
([19, Ex.4 .9]). Let us write as usual $\Delta_{p}$ for the norm on the tensor
product of $Lip_{0}(\mathbb{R})$ and $L^{p}(\mu)$ inherited from the Bochner
space $L^{p}(\mu,Lip_{0}(\mathbb{R}).$ Recall that for $\mu$ finite, $1\leq
q\leq p<\infty$ and $r$ such that $1/q=1/p+1/r,$ we have that
$L^{p}(\mu)\subseteq L^{q}(\mu)$ and
$\mathcal{M}(L^{p}(\mu),L^{q}(\mu))=L_{r}(\mu).$

######  Corollary 5.2.

Let $\mu$ be a finite measure, $1\leq q\leq p<\infty$ and $1/q=1/p+1/r.$ Then
an operator $T:L^{p}(\mu)\to L^{q}(\mu)$ is strongly lattice Lipschitz if and
only if there is a function $\Phi\in
L^{r}\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}$ such that

$T(f)(w)=\Phi(w)\circ
f(w)\quad\text{for}\,\,w\in\Omega\,\,\,\text{$\mu-$a.e}\,\,\text{ and for all
}\,\,\,\,f\in X(\mu).$

In other words, we have the isometries

$Lip_{0}(\mathbb{R})\widehat{\otimes}_{\Delta_{r}}L^{r}(\mu)=L^{r}\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}=SLL(L^{p},L^{q}).$

Using Theorem 4.4 we can easily get the next result on $L_{p}-$spaces over
finite measures. The spaces $L^{p}(\mu)$ for $1\leq p<\infty$ have the Fatou
property. In accordance with the initial purpose of the paper, next corollary
shows the similarities with the description of multiplication operators
between $L^{p}-$spaces.

######  Corollary 5.3.

Let $\mu$ be a finite measure, $1\leq q\leq p<\infty$ and $1/q=1/p+1/r.$ Then
an operator $T:L^{p}(\mu)\to L^{q}(\mu)$ is strongly lattice Lipschitz if and
only if there exists a sequence of simple functions $(\Phi_{n})_{n}\subset
L^{r}\big{(}\mu,Lip_{0}(\mathbb{R})\big{)}$ that converges pointwise
$\mu-$a.e. to the representing function
$\Phi(w)(\lambda)=T(\lambda\chi_{\Omega})(w)$ and such that

$\sup_{n}\big{\|}Lip(\Phi_{n}(\cdot))\big{\|}_{L^{r}(\mu)}<\infty.$

Although $\Delta_{p}$ is not a tensor norm, we always have that the
inequalities $\varepsilon\leq\Delta_{r}\leq\pi$ hold for tensor products as
the one above, where $\epsilon$ is the injective tensor norm and $\pi$ the
projective one (see [19, Ch.7]). For example, there is a surjection from the
tensor product $F\widehat{\otimes}_{\pi}E^{\prime}$ onto the nuclear operators
$\mathcal{N}(E,F)$ between $E$ and $F.$ On the other hand, it is also known
that the dual of the Arens-Ells space $\AE(\mathbb{R})$ is
$Lip_{0}(\mathbb{R}).$ Taking into account the inequality $\Delta_{r}\leq\pi$
we have that we can identify every nuclear operator $N:\AE(\mathbb{R})\to
L^{r}(\mu)$ with a strongly lattice Lipschitz operator $T_{N}:L^{p}(\mu)\to
L^{q}(\mu)$ through the isomorphisms written above.

Let us finish the paper with some comments on the suitable applications of the
results in pure analysis. It is well-known that multiplication operators are
the key in the contexts of the spaces of integrable functions to obtain good
diagonalization settings for operators between infinite dimensional function
spaces. For example, they allow to get some fundamental tools for the geometry
of function spaces, as the circle of ideas that permits to relate lattice
geometric properties of function spaces (p-convexity, p-concavity, Boyd
indexes,…) with factorization of operators through $L_{p}-$spaces. Some
classical results, as the famous Rosenthal Theorem, as well as the
factorization theorems of Krivine, Maurey, Nikishin, Pietsch, Pisier and many
others are in the center of, for example, summability theory for operators or
the geometry of the Banach spaces. We are then intended to search for the
generalization of the above mentioned results in the context of the Lipschitz
operators, for which it seems to be necessary to know the main properties of
the particular class of superposition operators which we call lattice
Lipschitz operators.

A lot of questions are still open regarding the results presented in this
paper. Although we have shown some representation tools, we believe that it is
possible to obtain more general results. We write below two open questions.

  * •

Which requirements are needed to assure that all lattice Lipschitz operators
are strongly lattice Lipschitz? Which are the spaces—others than the trivial
ones—for which this holds?

  * •

Are all the strongly lattice Lipschitz operators between Banach function
spaces $X(\mu)$ and $Y(\mu)$ representable by strongly measurable functions in
$\mathcal{M}(X,Y)$ with weaker requirements on $Y(\mu)$?

## Ethics declarations

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

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