# New results on MS-Lipschitz summing operators

Maatougui Belaala, Athmane Ferradi and Khalil Saadi  Laboratory of Functional
Analysis and Geometry of Spaces
University of M’sila
E-mail:
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###### Abstract.

This paper focuses on the study of MS-Lipschitz $p$-summing operators, which
were initially defined by the authors in [14]. Our objective is to establish
relationships between a Lipchitz mapping $T:X\rightarrow Y,$ where $X$ and $Y$
are pointed metric spaces, and its linearizations $\widehat{T},$
$\widetilde{T}$ and $T^{\\#}$ for certain notion of summability. Moreover, we
extend our investigation by introducing new definitions in the category of
Lipschitz mappings defined on metric spaces, known as MS-strongly Lipschitz
$p$-summing and MS-Lipschitz $p$-nuclear. We provide several results and
characterizations for these new concepts.

††2020 Mathematics Subject Classification. Primary 47B10, 46B28, 47L20. Key
words and phrases. Strictly Lipschitz $p$-summing; MS-Lipschitz $p$-summing;
Lipschitz $p$-summing operators; MS-Lipschitz $p$-nuclear; $p$-summing
operators; strongly $p$-summing operators; Pietsch factorization;
factorization theorems.

##  1\. Introduction and preliminaries

The theory of $p$-summing operators has undergone several stages of
development. It originated in the 1950s with Grothendieck’s pioneering work
[8], where he introduced the concept of $1$-summing operators. In 1967 [11],
Pietsch made a significant contribution by defining $p$-summing operators for
all positive values of $p$. His most notable result in this theory is the
Pietsch Factorization Theorem, which provides an integral characterization.
Since then, this theory has witnessed substantial advancements. Researchers
have expanded this theory in various directions, including the sublinear,
multilinear, and more recently, the Lipschitz case. Let $Y$ be a pointed
metric space and $G$ be a Banach space. We denote by $Lip_{0}\left(Y,G\right)$
the Banach space of all Lipschitz functions (or Lipschitz operators)
$T:Y\rightarrow G$ such that $T(0)=0$. This space is equipped with pointwise
addition and the Lipschitz norm. If $Y$ is equal to $\mathbb{K}$, we simply
denoted by $Lip_{0}(Y)\left(=Y^{\\#}\right)$. We consider the evaluation
functionals $\delta_{x}\in Lip_{0}(Y)^{\ast}$ for $x\in Y$ such that
$\delta_{x}\left(s\right)=s\left(x\right)$ for $s\in Lip_{0}(Y)$. The
Lipschitz-free space $\mathcal{F}\left(Y\right)$ is the closed space generated
by these evaluation functionals. See [6, 7, 10, 17, 18] for more details about
free Banach spaces. We have $Y^{\\#}=\mathcal{F}\left(Y\right)^{\ast}$ holds
isometrically. Every Lipschitz mapping $T:Y\rightarrow G$ induces a unique
linear operator $\widehat{T}:\mathcal{F}\left(Y\right)\longrightarrow G$ such
that

$\widehat{T}\circ\delta_{Y}=T,$

where $\delta_{Y}:Y\rightarrow\mathcal{F}\left(Y\right)$ is the canonical
embedding so that $\delta_{Y}\left(x\right)=\delta_{x}$ for $x\in Y.$ In this
case, the identification

$Lip_{0}\left(Y,G\right)=\mathcal{B}\left(\mathcal{F}\left(Y\right),G\right),$
(1.1)

holds isometrically. It is well-known that if a Lipschitz mapping
$T:Y\rightarrow G$ is Lipschitz $p$-summing, its linearization $\widehat{T}$
is not necessarily $p$-summing. However, in [14, Theorem 3.5.], the author
introduced the concept of strictly Lipschitz $p$-summing, which provides a
well-established relation between $T$ and its linearization $\widehat{T}$ for
the concept of $p$-summing. Now, let $Y$ and $W$ be two pointed metric spaces.
For every Lipschitz operator $T:Y\rightarrow W,$ we can associate another
linear operator $\widetilde{T}$ such that the following diagram commutes

$\begin{array}[]{ccc}Y&\underrightarrow{T}&W\\\
\downarrow\delta_{Y}&&\downarrow\delta_{W}\\\
\mathcal{F}\left(Y\right)&\underrightarrow{\widetilde{T}}&\mathcal{F}\left(W\right)\end{array}$

i.e., $\widetilde{T}\circ\delta_{Y}=\delta_{W}\circ T$. The Lipschitz adjoint
map $T^{\\#}:W^{\\#}\rightarrow Y^{\\#}$ of $T$ is defined as follows

$T^{\\#}\left(f\right)\left(x\right)=f\left(T\left(x\right)\right),\text{ for
every }f\in W^{\\#}\text{ and }x\in Y.$

Note that (as shown in [7, p. 124]), it is straightforward to verify that

$\widetilde{T}^{\ast}=T^{\\#},$ (1.2)

where $T^{\\#}:W^{\\#}\rightarrow Y^{\\#}$ is the linear map called the
Lipschitz adjoint, which is defined by $T^{\\#}\left(f\right)=f\circ T.$ Our
objective in this paper is to establish relationships between a Lipschitz
mapping $T:Y\rightarrow W$, where $Y$ and $W$ are metric spaces, and its
linearizations $\widetilde{T}$ and $T^{\\#}$. We focus on a specific notion of
summability and examine how the properties of $T$ are reflected in its
linearizations. In a previous work [15], the authors introduced the concept of
M-strictly Lipschitz $p$-summing operators, also referred to as MS-Lipschitz
$p$-summing, defined on pointed metric spaces and established a significant
connection between $T$ and its linearization $\widetilde{T}$. Moreover, we
will extend this idea by introducing a new definitions, called MS-strongly
Lipschitz $p$-summing and MS-Lipschitz $p$-nuclear, in the category of
Lipschitz mappings defined on metric spaces. The connection between $T$ and
its linearizations, $\widetilde{T}$ and $T^{\\#}$, will be established for
these new concepts.

The structure of the paper is as follows:

Section 1 provides a brief review of the standard notations that will be
employed throughout the paper. Section 2 is dedicated to the examination of
various characterizations of MS-Lipschitz $p$-summing operators defined
between pointed metric spaces. In Section 3, we introduce the definition of
MS-strongly Lipschitz $p$-summing and MS-Lipschitz $p$-nuclear operators.
These operators exhibit remarkable properties, particularly their associations
with linearization operators. Furthermore, we present a Pietsch Domination
Theorem that applies specifically to these classes of operators.

Let’s review some essential notations and terminology that will be used in the
following discussion. In this paper, we adopt the following conventions: The
letters $E$, $F$, and $G$ represent Banach spaces. The letters $Y$, $W$, and
$Z$ denote metric spaces with a distinguished point (pointed metric spaces),
which we denote as 0. Given a metric space $Y$ and a Banach space $G$, we
represent their Lipschitz tensor product as $Y\boxtimes G$. This tensor
product is the vector space spanned by the linear functional
$\delta_{\left(x,y\right)}\boxtimes h$ on $Lip_{0}\left(Y,G^{\ast}\right)$,
where $G^{\ast}$ denotes the topological dual of $G$. The functional is
defined as follows: For any $s\in Lip_{0}\left(Y,G^{\ast}\right)$,

$\delta_{\left(x,y\right)}\boxtimes h\left(s\right)=\left\langle
s\left(x\right)-s\left(y\right),h\right\rangle.$

When $\mathfrak{n}\in\mathcal{F}\left(Y\right)$ is expressed as
$\mathfrak{n}=\sum_{j=1}^{m}\delta_{\left(x_{j},y_{j}\right)},$ we have the
following for every $s\in Y^{\\#}$

$\left\langle\mathfrak{n},s\right\rangle=\sum_{j=1}^{m}s(x_{j})-s(y_{j}).$

For a more detailed understanding of the properties of the space $Y\boxtimes
G$, we refer to [1]. Now, let’s consider a Lipschitz operator $T:Y\rightarrow
W$ between pointed metric spaces, and
$z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes f_{l}\in
Y\boxtimes W^{\\#}.$ The action of $T$ on $z$ is given by

$\left|\left\langle
T,z\right\rangle\right|=\left|\sum_{l=1}^{k}f_{l}\left(T\left(x_{l}\right)\right)-f_{l}\left(T\left(y_{l}\right)\right)\right|$

Let $G$ be a Banach space. In the following definitions, $B_{G}$ represents
the closed unit ball of $G$, and $G^{\ast}$ denotes its (topological) dual.
For $1\leq p\leq\infty$ and $m\in\mathbb{N}^{\ast}$, we define two Banach
spaces as follows:

\- $\ell_{p}^{m}\left(G\right):$ This space consists of all sequences
$\left(x_{j}\right)_{j=1}^{m}$ in $G$ with the norm given by

$\left\|\left(x_{j}\right)_{j}\right\|_{\ell_{p}^{m}\left(G\right)}=(\sum_{j=1}^{m}\left\|x_{j}\right\|^{p})^{\frac{1}{p}}.$

\- $\ell_{p}^{m,w}\left(G\right):$ This space comprises all sequences
$\left(x_{j}\right)_{j=1}^{m}$ in $G,$ with the norm defined as

$\left\|\left(x_{j}\right)_{j}\right\|_{\ell_{p}^{m,w}\left(G\right)}=\underset{x^{\ast}\in
B_{G^{\ast}}}{\sup}(\sum_{j=1}^{m}\left|\left\langle
x_{j},x^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}.$

If $G$ is equal to $\mathbb{K}$, we can simplify the notation as
$\ell_{p}^{m}$ and $\ell_{p}^{m,w}.$ In particular, let
$\left(\mathfrak{n}_{j}\right)_{j=1}^{m_{1}}\in\mathcal{F}\left(Y\right)$ be
such that
$\mathfrak{n}_{j}=\sum_{i=1}^{m_{2}}\delta_{\left(x_{j}^{i},y_{j}^{i}\right)}$
for $1\leq j\leq m_{1}.$ In this case, we have

$\displaystyle\left\|\left(\mathfrak{n}_{j}\right)_{j}\right\|_{\ell_{p}^{m,w}\left(\mathcal{F}\left(Y\right)\right)}$
$\displaystyle=$ $\displaystyle\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\left\langle\mathfrak{n}_{j},s\right\rangle\right|^{p})^{\frac{1}{p}}$
$\displaystyle=$ $\displaystyle\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\sum_{i=1}^{m_{2}}s(x_{j}^{i})-s(y_{j}^{i})\right|^{p})^{\frac{1}{p}}.$

Let us review the following concepts:

\- The linear operator $u:F\rightarrow G$ is said to be $p$-summing if there
exists a positive constant $K$ such that, for any sequence
$\left(x_{j}\right)_{j=1}^{m}$ belonging to $F$, the following inequality
holds

$\left\|\left(u\left(x_{j}\right)\right)_{j=1}^{m}\right\|_{\ell_{p}^{m}\left(G\right)}\leq
K\left\|\left(x_{j}\right)_{j=1}^{m}\right\|_{\ell_{p}^{m,w}\left(F\right)}.$
(1.3)

The class of $p$-summing linear operators from the Banach space $F$ into $G$,
denoted as $\Pi_{p}(F,G)$, forms a Banach space itself when equipped with the
norm $\pi_{p}(u)$. This norm is defined as the smallest constant $K$ for which
the inequality (1.3) holds.

\- The linear operator $u:F\rightarrow G$ is said to be (Cohen) strongly
$p$-summing if there exists a positive constant $K$ such that, for any
sequence $\left(x_{j}\right)_{j=1}^{m}$ belonging to $F$ and any
$\left(y_{j}^{\ast}\right)_{j=1}^{m}$ belonging to $G^{\ast}$, the following
inequality holds

$\left\|\left(\left\langle
u\left(x_{j},y_{j}^{\ast}\right)\right\rangle\right)_{j=1}^{m}\right\|_{\ell_{1}^{m}}\leq
K\left\|\left(x_{j}\right)_{j=1}^{m}\right\|_{\ell_{p}^{m}\left(F\right)}\left\|\left(y_{j}^{\ast}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m,w}\left(G^{\ast}\right)}.$
(1.4)

The class of strongly $p$-summing operators from the Banach space $F$ into
$G$, denoted as $\mathcal{D}_{p}(F,G)$, forms a Banach space itself when
equipped with the norm $d_{p}(u)$. This norm is defined as the smallest
constant $K$ for which the inequality (1.4) holds.

\- The linear operator $u:F\rightarrow G$ is said to be (Cohen) $p$-nuclear if
there exists a positive constant $K$ such that, for any sequence
$\left(x_{j}\right)_{j=1}^{m}$ belonging to $F$ and any
$\left(y_{j}^{\ast}\right)_{j=1}^{m}$ belonging to $G^{\ast}$, the following
inequality holds

$\left\|\left(\left\langle
u\left(x_{j},y_{j}^{\ast}\right)\right\rangle\right)_{j=1}^{m}\right\|_{\ell_{1}^{m}}\leq
K\left\|\left(x_{j}\right)_{j=1}^{m}\right\|_{\ell_{p}^{m,w}\left(F\right)}\left\|\left(y_{j}^{\ast}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m,w}\left(G^{\ast}\right)}.$
(1.5)

The class of strongly $p$-summing operators from the Banach space $F$ into
$G$, denoted as $\mathcal{N}_{p}(F,G)$, forms a Banach space itself when
equipped with the norm $n_{p}(u)$. This norm is defined as the smallest
constant $K$ for which the inequality (1.5) holds.

##  2\. M-strictly Lipschitz p-summing operators

Consider a pointed metric space $Y$ and let $G$ be a Banach space. The concept
of Lipschitz tensor product, denoted by $Y\boxtimes G$, was introduced by
Cabrera-Padilla et al. [1]. An element $z$ in $Y\boxtimes G$ can be
represented as $z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes
h_{l}$ and can be viewed as a linear functional on
$Lip_{0}\left(Y,G^{\ast}\right)$. The action of this linear functional is
defined by

$\left\langle
z,s\right\rangle=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes
h_{l}\left(s\right)=\sum_{l=1}^{k}\left(s\left(x_{l}\right)-s\left(y_{l}\right)\right)h_{l}\text{
for every }s\in Lip_{0}\left(Y,G^{\ast}\right).$

The relationship between $Y\boxtimes G$ and $\mathcal{F}\left(Y\right)\otimes
G$ is straightforward, where $Y\boxtimes G$ is a vector subspace of
$\mathcal{F}\left(Y\right)\otimes G$. Given an element $z\in Y\boxtimes G$, we
can define the set $A_{z}$ as the set of all representations of $z$ in
$\mathcal{F}\left(Y\right)\otimes G$, that is,

$A_{z}:=\left\\{\left(\left(\mathfrak{n}_{j}\right)_{j=1}^{m},\left(h_{j}\right)_{j=1}^{m}\right):m\in\mathbb{N},\text{
}\mathfrak{n}_{j}\in\mathcal{F}\left(Y\right),h_{j}\in
G:z=\sum_{j=1}^{m}\mathfrak{n}_{j}\otimes h_{j}\right\\}.$ (2.1)

Let’s consider $\beta$ as a tensor norm defined on Banach spaces. Based on
[14, Theorem 3.1], it has been established that there exists a corresponding
Lipschitz cross-norm, denoted as $\beta^{L},$ defined on $Y\boxtimes G$ by:

$\beta^{L}(\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes
h_{l})=\beta(\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\otimes h_{l}),$
(2.2)

with $\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\otimes h_{l}\ $is an
element of $\mathcal{F}\left(Y\right)\otimes G$. Before presenting the
following definition, it is necessary to recall the norms of Chevet-Saphar
$d_{p}$ and $g_{p}$ [2, 12, 16], which are defined on two Banach spaces $F$
and $G$

$d_{p}\left(z\right)=\inf\left\\{\left\|\left(x_{j}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m,w}\left(F\right)}\left\|\left(y_{j}\right)_{j=1}^{m}\right\|_{\ell_{p}^{m}\left(G\right)}\right\\}\text{
},$

and

$g_{p}\left(z\right)=\inf\left\\{\left\|\left(x_{j}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m}\left(F\right)}\left\|\left(y_{j}\right)_{j=1}^{m}\right\|_{\ell_{p}^{m,w}\left(G\right)}\right\\},$

here the $\inf$ is taken over all representations of $z$ in the form of
$z=\sum_{j=1}^{m}x_{j}\otimes y_{j}\in F\otimes G.$ It is worth noting that we
can utilize the Chevet-Saphar norms to provide equivalent definitions for
(1.3) and (1.4) (see [12, p. 140]). Specifically, the linear operator
$u:F\rightarrow G$ is said to be $p$-summing if there exists a constant $K>0$
such that for any $z=\sum_{j=1}^{m}x_{j}\otimes y_{j}^{\ast}\in F\otimes
G^{\ast},$ the following inequality holds:

$\left|\left\langle u,z\right\rangle\right|=\left|\sum_{j=1}^{m}\left\langle
u\left(x_{j}\right),y_{j}^{\ast}\right\rangle\right|\leq
Kd_{p}\left(z\right).$

If we replace $d_{p}\left(u\right)$ with $g_{p}\left(u\right)$, we obtain the
definition of strongly $p$-summing. In [14], the Lipschitz cross-norm
$d_{p}^{L}$ is defined as follows:

$d_{p}^{L}\left(z\right)=d_{p}(\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\otimes
t_{l})=\inf_{\left(\left(\mathfrak{n}_{j}\right)_{j=1}^{m},\left(h_{j}\right)_{j=1}^{m}\right)\in
A_{z}}\left\\{\left\|\left(\mathfrak{n}_{j}\right)_{j=1}^{m}\right\|_{l_{p}^{m,w}\left(\mathcal{F}\left(Y\right)\right)}\left\|\left(h_{j}\right)_{j=1}^{m}\right\|_{l_{p^{\ast}}^{m}\left(G\right)}\right\\}.$

Most definitions of summability for Lipschitz mappings are typically defined
from a metric space to a Banach space. However, in the following definition of
MS-Lipschitz $p$-summing, we consider Lipschitz operators defined on metric
spaces. This new perspective allows us to establish a meaningful relationship
between $T$ and its linearization $\widetilde{T}$. Before proceeding, let us
recall the following definition introduced in [14].

Definition 2.1. [14] Consider $1\leq p\leq\infty,$ $Y$ be a pointed metric
space and $G$ be a Banach space. The Lipschitz operator $T:Y\rightarrow G$ is
considered to be strictly Lipschitz $p$-summing if there exists a constant
$K>0$ such that for every
$z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes h_{l}^{\ast}\in
Y\boxtimes G^{\ast}$ we have

$\left|\left\langle T,z\right\rangle\right|\leq Kd_{p}^{L}(z).$ (2.3)

Building upon the aforementioned idea, we have introduced the concept of MS-
Lipschitz $p$-summing operators [15]. In contrast to considering elements from
the dual space of $G$, we now focus on elements from its Lipschitz space
$G^{\\#}$.

Definition 2.2. [15] For $1\leq p\leq\infty$ and $Y,W$ being two pointed
metric spaces, a Lipschitz operator $T:Y\rightarrow W$ is considered to be MS-
Lipschitz $p$-summing if there exists a constant $K>0$ such that for every
$z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes f_{l}\in
Y\boxtimes W^{\\#}$ we have

$\left|\left\langle T,z\right\rangle\right|\leq Kd_{p}^{L}(z).$ (2.4)

The set of all MS-Lipschitz $p$-summing operators from $Y$ into $W$ is denoted
by $\Pi_{p}^{MSL}\left(Y,W\right)$. It represents the collection of operators
that satisfy the MS-Lipschitz $p$-summing property. The constant
$\pi_{p}^{MSL}\left(T\right)$ corresponds to the smallest value of $K$ that
satisfies inequality (2.4) for a given operator $T$. It is important to note
that if $W$ is a Banach space, the set $\Pi_{p}^{MSL}\left(Y,W\right)$ does
not possess the structure of a vector space.

Proposition 2.3. Let $1\leq p\leq\infty.$Every MS-Lipschitz $p$-summing
operator from a pointed metric space $Y$ into a Banach space $G$ is strictly
Lipschitz $p$-summing.

Proof. Let $T:X\rightarrow E$ be a MS-Lipschitz $p$-summing operator. Let
$x_{l},y_{l}\in Y$ and $h_{k}^{\ast}\in E^{\ast}$ $\left(1\leq l\leq
k\right),$ then

$\displaystyle\left|\left\langle T,z\right\rangle\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{l=1}^{k}h_{l}^{\ast}\left(T\left(x_{l}\right)\right)-h_{l}^{\ast}\left(T\left(y_{l}\right)\right)\right|$
$\displaystyle\leq$ $\displaystyle Kd_{p}^{L}(z),$

where $z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes
h_{k}^{\ast}\in Y\boxtimes G^{\\#}.$ So, as $Y\boxtimes G^{\ast}\subset
Y\boxtimes G^{\\#},$ the definition of $d_{p}^{L}(z)$ on $Y\boxtimes G^{\\#}$
is smaller than on $Y\boxtimes G^{\ast},$ consequently, the condition
$\left(2.3\right)$ is verified.$\qquad\blacksquare$

Remark 2.4. In the case where $F$ and $G$ are Banach spaces, it is well-known
that the definitions of strictly Lipschitz $p$-summing, Lipschitz $p$-summing,
and $p$-summing coincide for linear operators from $F$ to $G$ (see [14,
Proposition 3.8]). Furthermore, in our specific case, the definition of MS-
Lipschitz $p$-summing implies $p$-summing; however, the converse is not true,
as illustrated in the following example: Consider the identity operator
$id_{F}:F\rightarrow F$. It can be easily demonstrated that
$\widetilde{id_{F}}=id_{\mathcal{F}\left(F\right)}$, indicating that the
following diagram is commutative

$\begin{array}[]{ccc}F&\underrightarrow{id_{F}}&F\\\
\downarrow\delta_{F}&&\downarrow\delta_{F}\\\
\mathcal{F}\left(F\right)&\underrightarrow{id_{\mathcal{F}\left(F\right)}}&\mathcal{F}\left(F\right)\end{array}$

If $F$ is a finite-dimensional space, then $id_{F}$ is indeed $p$-summing, and
consequently, it is also strictly Lipschitz $p$-summing. However,
$id_{\mathcal{F}\left(F\right)}$ cannot be $p$-summing since
$\mathcal{F}\left(F\right)$ is not finite-dimensional. Therefore, $id_{F}$ is
not MS-Lipschitz $p$-summing.

The following statement presents the main result of this section.

Theorem 2.5. Consider $1\leq p\leq\infty.$ Let $Y$ and $W$ be two pointed
metric spaces. Let $T:X\rightarrow Y$ be a Lipschitz operator. The following
properties are equivalent.

1) $T$ is MS-Lipschitz $p$-summing.

2)
$\widetilde{T}:\mathcal{F}\left(Y\right)\rightarrow\mathcal{F}\left(W\right)$
is $p$-summing.

3) $\delta_{Y}\circ T:Y\rightarrow\mathcal{F}\left(W\right)$ is strictly
Lipschitz $p$-summing.

4) There is a constant $K>0$ such that for every
$\left(x_{j}^{i}\right)_{j=1}^{m_{1}},\left(y_{j}^{i}\right)_{j=1}^{m_{1}}$ in
$Y;\left(1\leq i\leq m_{2}\right)$ and $m_{1},m_{2}\in\mathbb{N}^{\ast}$, we
have

$(\sum_{j=1}^{m_{1}}\left\|\sum_{i=1}^{m_{2}}\delta_{W}\circ
T(x_{j}^{i})-\delta_{W}\circ T(y_{j}^{i})\right\|^{p})^{\frac{1}{p}}\leq
K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\sum_{i=1}^{m_{2}}s(x_{j}^{i})-s(y_{j}^{i})\right|^{p})^{\frac{1}{p}}.$
(2.5)

Proof. $1)\Leftrightarrow 2)$ See [15, Proposition 2.4.].

$2)\Rightarrow 3):$ Suppose that $\widetilde{T}$ is $p$-summing. Then

$\displaystyle\sum_{j=1}^{m_{1}}(\left\|\sum_{i=1}^{m_{2}}\delta_{W}\circ
T(x_{j}^{i})-\delta_{W}\circ T(y_{j}^{i})\right\|^{p})^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle(\sum_{j=1}^{m_{1}}\left\|\widetilde{T}\left(\mathfrak{n}_{j}\right)\right\|^{p})^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle\pi_{p}\left(\widetilde{T}\right)\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|s\left(\mathfrak{n}_{j}\right)\right|^{p})^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle\pi_{p}\left(\widetilde{T}\right)\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\sum_{i=1}^{m_{2}}s(x_{j}^{i})-s(y_{j}^{i})\right|^{p})^{\frac{1}{p}}$

$3)\Rightarrow 2):$ We have

$\displaystyle(\sum_{j=1}^{m_{1}}\left\|\widetilde{T}\left(\mathfrak{n}_{j}\right)\right\|^{p})^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle(\sum_{j=1}^{m_{1}}\left\|\sum_{i=1}^{m_{2}}\delta_{W}\circ
T\left(x_{j}^{i}\right)-\delta_{W}\circ
T\left(y_{j}^{i}\right)\right\|^{p})^{\frac{1}{p}}$ $\displaystyle\leq$
$\displaystyle K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\sum_{i=1}^{m_{2}}s(x_{j}^{i})-s(y_{j}^{i})\right|^{p})^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|s\left(\mathfrak{n}_{j}\right)\right|^{p})^{\frac{1}{p}}.$

As a consequence, $\widetilde{T}$ is $p$-summing, and by applying the result
in [4, Theorem 2.12], we can obtain the desired result.

$3)\Leftrightarrow 4)$ It is immediate.$\qquad\blacksquare$

By setting $m_{2}=1$ in formula (2.5) and considering the isometric property
of $\delta_{W}$, we arrive at the precise formulation of Lipschitz $p$-summing
as originally defined by Farmer [5], indeed

$\displaystyle(\sum_{j=1}^{m_{1}}\left\|\delta_{W}\circ
T(x_{j})-\delta_{W}\circ T(y_{j})\right\|^{p})^{\frac{1}{p}}$ $\displaystyle=$
$\displaystyle(\sum_{j=1}^{m_{1}}d\left(T(x_{j}),T(y_{j})\right)^{p})^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|s(x_{j})-s(y_{j})\right|^{p})^{\frac{1}{p}}.$

Corollary 2.6. Consider a Lipschitz operator $T:Y\rightarrow W$ between
pointed metric spaces. The following properties are equivalent.

1) $T$ is MS-Lipschitz $p$-summing.

2) The Lipschitz adjoint $T^{\\#}:Y^{\\#}\rightarrow X^{\\#}$ is strongly
$p^{\ast}$-summing.

Proof. According to (1.2), the dual operator of $\widetilde{T}$ is $T^{\\#}$.
To establish the equivalence between the given properties, we can make use of
the result mentioned in [3, Theorem 2.2.2]. $\blacksquare$

Proposition 2.7. Consider a Lipschitz operator $T:Y\rightarrow W$ between
pointed metric spaces such that $Y$ or $W$ is finite, then $T$ is MS-Lipschitz
$p$-summing.

Proof. Suppose $Y$ is a finite metric space. According to [17, Example 2.3.6],
we know that the space $\mathcal{F}\left(Y\right)$ is finite-dimensional.
Therefore, the linearization
$\widetilde{T}:\mathcal{F}\left(Y\right)\rightarrow\mathcal{F}\left(W\right)$
is $p$-summing, and consequently, $T:Y\rightarrow W$ is MS-Lipschitz
$p$-summing. $\blacksquare$

The Pietsch domination theorem is an intriguing characterization that is
satisfied by the class of MS-Lipschitz $p$-summing operators.

Theorem 2.8. Consider a Lipschitz operator $T:Y\rightarrow W$ between pointed
metric spaces. We have the following equivalent properties.

1) $T$ is MS-Lipschitz $p$-summing.

2) There exist a constant $K>0$, a Radon probability $\mu$ on $B_{Y^{\\#}}$
such that for every
$\left(x^{i}\right)_{i=1}^{m},\left(y^{i}\right)_{i=1}^{m}\subset Y,$ we have

$\left\|\sum_{i=1}^{m}\delta_{W}\circ T(x^{i})-\delta_{W}\circ
T(y^{i})\right\|\leq
K(\int_{B_{Y^{\\#}}}\left|\sum_{i=1}^{m}s\left(x^{i}\right)-s\left(y^{i}\right)\right|^{p}d\mu\left(s\right))^{\frac{1}{p}}.$
(2.6)

In this case, we have

$\pi_{p}^{MSL}\left(T\right)=\inf\left\\{K:\text{ {verifying}
{(2.6)}}\right\\}.$

Proof. $1)\Rightarrow 2):$ Since $T$ is MS-Lipschitz $p$-summing, then
$\widetilde{T}:\mathcal{F}\left(Y\right)\rightarrow\mathcal{F}\left(W\right)$
is $p$-summing. By Pietsch domination theorem for $p$-summing linear operator
[4, Theorem 2.12], we have

$\displaystyle\left\|\widetilde{T}\left(\sum_{i=1}^{m}\delta_{\left(x^{i},y^{i}\right)}\right)\right\|$
$\displaystyle\leq$
$\displaystyle\pi_{p}\left(\widetilde{T}\right)(\int_{B_{Y^{\\#}}}\left|\left\langle\sum_{i=1}^{m}\delta_{\left(x^{i},y^{i}\right)},s\right\rangle\right|^{p}d\mu\left(s\right))^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\pi_{p}\left(\widetilde{T}\right)(\int_{B_{Y^{\\#}}}\left|\sum_{i=1}^{m}s\left(x^{i}\right)-s\left(y^{i}\right)\right|^{p}d\mu\left(s\right))^{\frac{1}{p}}.$

On the other hand,

$\displaystyle\left\|\widetilde{T}\left(\sum_{i=1}^{m}\delta_{\left(x^{i},y^{i}\right)}\right)\right\|$
$\displaystyle=$
$\displaystyle\left\|\sum_{i=1}^{m}\widetilde{T}\left(\delta_{\left(x^{i},y^{i}\right)}\right)\right\|$
$\displaystyle=$
$\displaystyle\left\|\sum_{i=1}^{m}\widetilde{T}\circ\delta_{Y}\left(x^{i}\right)-\widetilde{T}\circ\delta_{Y}\left(y^{i}\right)\right\|$
$\displaystyle=$ $\displaystyle\left\|\sum_{j=1}^{n}\delta_{W}\circ
T(x^{i})-\delta_{W}\circ T(y^{i})\right\|$

Therefore, we have obtained the desired result.

$2)\Rightarrow 1):$ Similarly, we can apply the same argument. $\blacksquare$

We conclude this section with a result concerning Lipschitz operators that
have a finite image. The following Lemma establishes a relationship between
the free space of $T\left(Y\right)$ and the image
$\widetilde{T}\left(\mathcal{F}\left(Y\right)\right)$.

Lemma 2.9. Let $X$ and $Y$ be two pointed metric spaces. Consider a Lipschitz
operator $T:Y\rightarrow W$ such that $T\left(Y\right)$ is a closed subset of
$W$. Then, we have the following

$\widetilde{T}\left(\mathcal{F}\left(Y\right)\right)=\mathcal{F}\left(T\left(Y\right)\right).$

Proof. By [17, Theorem 2.2.6], we have

$\displaystyle\mathcal{F}\left(T\left(Y\right)\right)$ $\displaystyle=$
$\displaystyle\overline{span}\left\\{\delta_{T\left(x\right)}:x\in Y\right\\}$
$\displaystyle=$
$\displaystyle\overline{span}\left\\{\delta_{W}\left(T\left(x\right)\right):x\in
Y\right\\}$ $\displaystyle=$
$\displaystyle\overline{span}\left\\{\widetilde{T}\left(\delta_{x}\right):x\in
Y\right\\}$ $\displaystyle=$
$\displaystyle\widetilde{T}\left(\mathcal{F}\left(Y\right)\right).\qquad\blacksquare$

Corollary 2.10. Let $Y$ and $W$ be two pointed metric spaces. Suppose that
$T:Y\rightarrow W$ is a Lipschitz operator such that $T\left(Y\right)$ is a
finite set. Then, the linearization $\widetilde{T}$ has finite rank.
Consequently, we can conclude that every finite Lipschitz operator is MS-
Lipschitz $p$-summing.

##  3\. Cohen MS-Lipschitz $p$-nuclear operators

In [3], Cohen introduced the concepts of strongly $p$-summing and $p$-nuclear
operators in the category of linear operators. Since then, many authors have
explored and extended these notions in various directions, including
multilinear, sublinear, and Lipschitz cases. We will further extend these
concepts using a similar approach to the one presented in the previous
section.

Definition 3.1. Consider a Lipschitz operator $T:Y\rightarrow W$ between
pointed metric spaces. For $1\leq p\leq\infty,$ $T$ is said to be MS-strongly
Lipschitz $p$-summing if there exists a constant $K>0$ such that the following
condition holds for any
$z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes f_{l}\in
Y\boxtimes W^{\\#}$

$\left|\left\langle T,z\right\rangle\right|\leq Kg_{p}^{L}(z).$ (3.1)

We denote the set of all MS-strongly Lipschitz $p$-summing operators from $Y$
into $W$ as $\mathcal{D}_{p}^{MSL}\left(Y,W\right),$ and
$d_{p}^{MSL}\left(T\right)$ represents the smallest constant $K$ that
satisfies (3.1). If $W$ is a Banach space, it’s important to note that
$\mathcal{D}_{p}^{MSL}\left(Y,W\right)$ does not possess the structure of a
vector space.

Theorem 3.2. Consider $1\leq p\leq\infty$ and a Lipschitz operator
$T:Y\rightarrow W$ between pointed metric spaces. The following properties are
equivalent.

1) $T$ is MS-strongly Lipschitz $p$-summing.

2) There exists a constant $K>0$ such that for every
$\left(x_{j}\right)_{j=1}^{m},\left(y_{j}\right)_{j=1}^{m}$ in $Y$ and
$\left(f_{j}\right)_{j=1}^{m}$in $W^{\\#};$ ($m\in\mathbb{N}^{\ast}$),

$\left|\sum_{j=1}^{m}f_{j}\left(T(x_{j})\right)-f_{j}\left(T(y_{j})\right)\right|\leq
K(\sum_{j=1}^{m}d\left(x_{j},y_{j}\right)^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m,w}\left(W^{\\#}\right)}.$
(3.2)

3) $\delta_{W}\circ T:Y\rightarrow\mathcal{F}\left(W\right)$ is strongly
Lipschitz $p$-summing.

4)
$\widetilde{T}:\mathcal{F}\left(Y\right)\rightarrow\mathcal{F}\left(W\right)$
is strongly $p$-summing.

Proof.

$1)\Rightarrow 2):$ Let $T$ be a MS-strongly Lipschitz $p$-summing operator.
Let $\left(x_{j}\right)_{j=1}^{m},\left(y_{j}\right)_{j=1}^{m}$ in $Y$ and
$\left(f_{j}\right)_{j=1}^{m}\subset W^{\\#}.$ We get

$\left|\left\langle
T,z\right\rangle\right|=\left|\sum_{j=1}^{m}f_{j}\left(T(x_{j})\right)-f_{j}\left(T(y_{j})\right)\right|\leq
d_{p}^{MSL}\left(T\right)g_{p}^{L}(z),$

where $z=\sum_{j=1}^{m}\delta_{\left(x_{j},y_{j}\right)}\boxtimes f_{j}\in
Y\boxtimes W^{\\#}.$ Then

$\displaystyle\left|\left\langle T,z\right\rangle\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{j=1}^{m}f_{j}\left(T(x_{j})\right)-f_{j}\left(T(y_{j})\right)\right|$
$\displaystyle\leq$ $\displaystyle
d_{p}^{MSL}\left(T\right)(\sum_{j=1}^{m}\left\|\delta_{\left(x_{j},y_{j}\right)}\right\|^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m,w}\left(W^{\\#}\right)}$
$\displaystyle\leq$ $\displaystyle
d_{p}^{MSL}\left(T\right)(\sum_{j=1}^{m}d\left(x_{j},y_{j}\right)^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m,w}\left(W^{\\#}\right)}$

$2)\Rightarrow 3):$ We will show that $\delta_{W}\circ
T:Y\rightarrow\mathcal{F}\left(W\right)$ is strongly Lipschitz $p$-summing.
Let $\left(x_{j}\right)_{j=1}^{m},\left(y_{j}\right)_{j=1}^{m}$ in $Y$ and
$\left(f_{j}\right)_{j=1}^{m}\subset
W^{\\#}\left(=\mathcal{F}\left(W\right)^{\ast}\right).$ Then

$\displaystyle\left|\sum_{j=1}^{m}\left\langle\delta_{W}\circ
T(x_{j})-\delta_{W}\circ T(y_{j}),f_{j}\right\rangle\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{j=1}^{m}f_{j}\left(\delta_{W}\circ
T(x_{j})\right)-f_{j}\left(\delta_{W}\circ T(y_{j})\right)\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{j=1}^{m}f_{j}\left(T(x_{j})\right)-f_{j}\left(T(y_{j})\right)\right|$
$\displaystyle\leq$ $\displaystyle
K(\sum_{j=1}^{m}d\left(x_{j},y_{j}\right)^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m}\right\|_{\ell_{p^{\ast}}^{m,w}\left(W^{\\#}\right)}$

Then $\delta_{Y}\circ T$ is strongly Lipschitz $p$-summing.

$3)\Rightarrow 4):$ We know that

$\widehat{\delta_{W}\circ T}=\widetilde{T}.$

Furthermore, according to [13, Proposition 3.1], the linearization
$\widehat{\delta_{W}\circ T}$ is strongly $p$-summing, which implies that
$\widetilde{T}$ is also strongly $p$-summing.

$4)\Rightarrow 1):$ Let
$z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes f_{l}\in
Y\boxtimes W^{\\#}.$ Assuming that $\widetilde{T}$ is strongly $p$-summing, we
can deduce from [12, Proposition 6.12] that $\widetilde{T}$ satisfies the
following

$\left|\sum_{l=1}^{k}\left\langle\widetilde{T}\left(\mathfrak{n}_{l}\right),f_{l}\right\rangle\right|\leq
Kg_{p}(\sum_{l=1}^{k}\mathfrak{n}_{l}\otimes f_{l}),$

If we put $\mathfrak{n}_{l}=\delta_{\left(x_{l},y_{l}\right)}$ for $1\leq
l\leq k,$ we find

$\displaystyle\left|\sum_{l=1}^{l}\left\langle\widetilde{T}\left(\delta_{\left(x_{l},y_{l}\right)}\right),f_{l}\right\rangle\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{k=1}^{l}f_{l}\left(T\left(x_{l}\right)\right)-f_{l}\left(T\left(y_{l}\right)\right)\right|$
$\displaystyle\leq$ $\displaystyle Kg_{p}(z)=Kg_{p}^{L}(z).\qquad\blacksquare$

Using the same reasoning as in Corollary 2.6, we can establish the following
result.

Corollary 3.3. Consider $1\leq p\leq\infty$ and a Lipschitz operator
$T:Y\rightarrow W$ between pointed metric spaces. The following properties are
equivalent.

1) $T$ is MS-strongly Lipschitz $p$-summing.

2) The Lipschitz adjoint $T^{\\#}:W^{\\#}\rightarrow Y^{\\#}$ is
$p^{\ast}$-summing.

The following integral characterization is an adaptation of the linear case.
To prove it, we rely on the fact that $T^{\\#}$ is $p^{\ast}$-summing or
$\widetilde{T}$ is strongly $p$-summing.

Theorem 3.4. Consider $1\leq p\leq\infty$ and a Lipschitz operator
$T:Y\rightarrow W$ between pointed metric spaces. The following properties are
equivalent.

1) $T$ is MS-strongly Lipschitz $p$-summing.

2) There exist a constant $K>0$ and a Radon probability $\mu$ on
$B_{Lip_{0}\left(W\right)^{\ast}}$ such that for every $x,y\in Y$ and $f\in
W^{\\#},$ we have

$\left|f\left(T(x)\right)-f\left(T(y)\right)\right|\leq
Kd\left(x,y\right)\left(\int_{B_{Lip_{0}\left(W\right)^{\ast}}}\left|\left\langle
f,\mathfrak{n}\right\rangle\right|^{p^{\ast}}d\mu\left(\mathfrak{n}\right)\right)^{\frac{1}{p^{\ast}}}.$

Let us now recall the definition of the tensor norm $w_{p}$ on the product of
two Banach spaces $F\otimes G$, which has been studied in [12, p. 180]. Let
$p\in\left[1,\infty\right]\ $we have

$w_{p}\left(z\right)=\inf\left\\{\left\|\left(x_{j}\right)_{j=1}^{m}\right\|_{l_{p}^{m,w}\left(F\right)}\left\|\left(y_{j}\right)_{j=1}^{m}\right\|_{l_{p^{\ast}}^{m,w}\left(G\right)}\right\\},$

where the infimum is taken over all representations of $z$ of the form
$z=\sum_{j=1}^{m}x_{j}\otimes y_{j}\in F\otimes G.$

Definition 3.5. Consider a Lipschitz operator $T:Y\rightarrow W$ between
pointed metric spaces. For $1\leq p\leq\infty,$ $T$ is said to be MS-Lipschitz
$p$-nuclear if there is a constant $K>0$ such that such that for every
$z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes f_{l}\in
Y\boxtimes W^{\\#}$ we have

$\left|\left\langle T,z\right\rangle\right|\leq Kw_{p}^{L}(z),$ (3.3)

We define $\mathcal{N}_{p}^{MSL}\left(Y,W\right)$ as the set of all MS-
Lipschitz $p$-nuclear operators from $Y$ into $W$. Additionally,
$n_{p}^{MSL}\left(T\right)$ represents the smallest constant $K$ that
satisfies (3.3). If $W$ is a Banach space, it is important to reiterate that
$\mathcal{N}_{p}^{MSL}\left(Y,W\right)$ does not possess the structure of a
vector space.

Suppose that $W$ is a Banach space. By restricting the previous definition to
the linear forms of $W^{\ast}$, we obtain the definition of (Cohen) Lipschitz
$p$-nuclear operators as introduced in [9]. Indeed, let
$z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes a_{l}^{\ast}\in
Y\boxtimes W^{\ast}$ we have

$\displaystyle\left|\left\langle T,z\right\rangle\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{l=1}^{k}a_{l}^{\ast}\left(T\left(x_{l}\right)\right)-a_{l}^{\ast}\left(T\left(y_{l}\right)\right)\right|=\left|\sum_{l=1}^{k}\left\langle
T\left(x_{l}\right)-T\left(y_{l}\right),a_{l}^{\ast}\right\rangle\right|$
(3.1) $\displaystyle\leq$ $\displaystyle
Kw_{p}^{L}(z)=Kw_{p}(\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\otimes
a_{l}^{\ast})$ $\displaystyle\leq$ $\displaystyle
K\left\|\left(\delta_{\left(x_{l},y_{l}\right)}\right)_{l=1}^{k}\right\|_{\ell_{p}^{k,w}\left(\mathcal{F}\left(Y\right)\right)}\left\|\left(a_{l}^{\ast}\right)_{l=1}^{k}\right\|_{\ell_{p^{\ast}}^{k,w}\left(W^{\ast}\right)}$
(3.2) $\displaystyle\leq$ $\displaystyle K\sup_{f\in
Y^{\\#}}(\sum_{l=1}^{k}\left|f(x_{l})-f(y_{l})\right|^{p})^{\frac{1}{p}}\sup_{\left\|a\right\|_{W}=1}(\sum_{l=1}^{k}\left|\left\langle
a_{l}^{\ast},a\right\rangle\right|^{p^{\ast}})^{\frac{1}{p^{\ast}}},$ (3.3)

Consequently, it follows that every MS-Lipschitz $p$-nuclear operator is also
Lipschitz $p$-nuclear.

Theorem 3.6. Consider $1\leq p\leq\infty$ and a Lipschitz operator
$T:Y\rightarrow W$ between pointed metric spaces. The following properties are
equivalent.

1) $T$ is MS-Lipschitz $p$-nuclear.

2) There is a constant $K>0$ such that for every
$\left(x_{j}^{i}\right)_{j=1}^{m_{1}},\left(y_{j}^{i}\right)_{j=1}^{m_{1}}$ in
$Y,$ $\left(f_{j}\right)_{j=1}^{m_{1}}\subset W^{\\#};\left(1\leq i\leq
m_{2}\right)$ and $m_{1},m_{2}\in\mathbb{N}^{\ast}$, we have

$\left|\sum_{j=1}^{m_{1}}\sum_{i=1}^{m_{2}}f_{j}\left(T(x_{j}^{i})\right)-f_{j}\left(T(y_{j}^{i})\right)\right|\leq
K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\sum_{i=1}^{m_{2}}s(x_{j}^{i})-s(y_{j}^{i})\right|^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m_{1}}\right\|_{\ell_{p^{\ast}}^{m_{1},w}\left(W^{\\#}\right)}.$
(3.4)

3)
$\widetilde{T}:\mathcal{F}\left(Y\right)\rightarrow\mathcal{F}\left(W\right)$
is $p$-nuclear.

Proof.

$1)\Rightarrow 2):$ Let $T$ be a MS-Lipschitz $p$-nuclear operator. Let
$\left(x_{j}^{i}\right)_{j=1}^{m_{1}},\left(y_{j}^{i}\right)_{j=1}^{m_{1}}$ in
$Y$ and $\left(f_{j}\right)_{j=1}^{m_{1}}$ in $W^{\\#};\left(1\leq i\leq
m_{2}\right)$, we have

$\left|\sum_{j=1}^{m_{1}}\sum_{i=1}^{m_{2}}f_{j}\left(T(x_{j}^{i})\right)-f_{j}\left(T(y_{j}^{i})\right)\right|\leq
Kg_{p}^{L}(z),$

where
$z=\sum_{j=1}^{m_{1}}\sum_{i=1}^{m_{2}}\delta_{\left(x_{j}^{i},y_{j}^{i}\right)}\boxtimes
f_{j}\in Y\boxtimes W^{\\#}.$ Then

$\displaystyle\left|\sum_{j=1}^{m_{1}}\sum_{i=1}^{m_{2}}f_{j}\left(T(x_{j}^{i})\right)-f_{j}\left(T(y_{j}^{i})\right)\right|$
$\displaystyle\leq$ $\displaystyle n_{p}^{MSL}\left(T\right)\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|s(\sum_{i=1}^{m_{2}}\delta_{\left(x_{j}^{i},y_{j}^{i}\right)})\right|^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m_{1}}\right\|_{\ell_{p^{\ast}}^{m_{1},w}\left(W^{\\#}\right)}$
$\displaystyle\leq$ $\displaystyle n_{p}^{MSL}\left(T\right)\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\sum_{i=1}^{m_{2}}s(x_{j}^{i})-s(y_{j}^{i})\right|^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m_{1}}\right\|_{\ell_{p^{\ast}}^{m_{1},w}\left(W^{\\#}\right)}.$

$2)\Rightarrow 3):$ Let
$\left(\mathfrak{n}_{j}\right)_{j=1}^{m_{1}}\subset\mathcal{F}\left(Y\right)$
and $\left(f_{j}\right)_{j=1}^{m_{1}}\subset W^{\\#}$ such that

$\mathfrak{m}_{j}=\sum_{i=1}^{m_{2}}\delta_{\left(x_{j}^{i},y_{j}^{i}\right)}\in\mathcal{F}\left(Y\right),\text{
}\left(1\leq j\leq m_{1}\right).$

Then

$\displaystyle(\sum_{j=1}^{m_{1}}\left|\left\langle\widetilde{T}\left(\mathfrak{n}_{j}\right),f_{j}\right\rangle\right|^{p})^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle(\sum_{j=1}^{m_{1}}\left|\left\langle\sum_{i=1}^{m_{2}}\left(\delta_{W}\circ
T\left(x_{j}^{i}\right)-\delta_{W}\circ
T\left(x_{j}^{i}\right)\right),f_{j}\right\rangle\right|^{p})^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{1}}(\left|\sum_{i=1}^{m_{2}}f_{j}\left(T(x_{j}^{i})\right)-f_{j}\left(T(y_{j}^{i})\right)\right|^{p})^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|\sum_{i=1}^{m_{2}}s(x_{j}^{i})-s(y_{j}^{i})\right|^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m_{1}}\right\|_{\ell_{p^{\ast}}^{m_{1},w}\left(W^{\\#}\right)}$
$\displaystyle\leq$ $\displaystyle K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|s(\sum_{i=1}^{m_{2}}\delta_{(x_{j}^{i},y_{j}^{i})})\right|^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m_{1}}\right\|_{\ell_{p^{\ast}}^{m_{1},w}\left(W^{\\#}\right)}$
$\displaystyle\leq$ $\displaystyle K\sup_{s\in
Y^{\\#}}(\sum_{j=1}^{m_{1}}\left|s\left(\mathfrak{n}_{j}\right)\right|^{p})^{\frac{1}{p}}\left\|\left(f_{j}\right)_{j=1}^{m_{1}}\right\|_{\ell_{p^{\ast}}^{m_{1},w}\left(W^{\\#}\right)}$

Then $\widetilde{T}$ is $p$-nuclear.

$3)\Rightarrow 1):$ Now, we suppose that $\widetilde{T}$ is $p$-nuclear. Let
$x_{l},y_{l}\in Y$ and $f_{l}\in W^{\\#}$ $\left(1\leq l\leq k\right)$ we have

$\displaystyle\left|\sum_{l=1}^{k}f_{l}\left(T(x_{l})\right)-f_{l}\left(T(y_{l})\right)\right|$
$\displaystyle=$ $\displaystyle\left|\sum_{l=1}^{k}\left\langle\delta_{W}\circ
T\left(x_{l}\right)-\delta_{W}\circ
T\left(y_{l}\right),f_{l}\right\rangle\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{l=1}^{k}\left\langle\widehat{T}\left(\delta_{\left(x_{l},y_{l}\right)}\right),f_{l}\right\rangle\right|$
$\displaystyle\leq$ $\displaystyle
n_{p}\left(\widetilde{T}\right)w_{p}(z)=n_{p}\left(\widetilde{T}\right)w_{p}^{L}(z),$

where $z=\sum_{l=1}^{k}\delta_{\left(x_{l},y_{l}\right)}\boxtimes f_{l}\in
Y\boxtimes W^{\\#}.$ Finally, $T$ is MS-Lipschitz $p$-nuclear and we have

$n_{p}^{MSL}\left(T\right)\leq
n_{p}\left(\widetilde{T}\right).\qquad\blacksquare$

By utilizing the result presented in [3, Theorem 2.2.4], we can establish the
following relationship between $T$ and its Lipschitz adjoint $T^{\\#}$.

Corollary 3.7. Consider $1\leq p\leq\infty$ and a Lipschitz operator
$T:Y\rightarrow W$ between pointed metric spaces. The following properties are
equivalent.

1) $T$ is MS-Lipschitz $p$-nuclear.

2) The Lipschitz adjoint $T^{\\#}:W^{\\#}\rightarrow Y^{\\#}$ is
$p^{\ast}$-nuclear.

The following integral characterization is an adaptation of the linear case.
Its proof will be omitted.

Theorem 3.8. Consider $1\leq p\leq\infty$ and a Lipschitz operator
$T:Y\rightarrow W$ between pointed metric spaces. The following properties are
equivalent.

1) $T$ is MS-Lipschitz $p$-nuclear.

2) There exist a constant $K>0,$ a Radon probability $\mu$ on $B_{Y^{\\#}}$
and $\eta\in B_{Lip_{0}\left(W\right)^{\ast}}$ such that for every
$\left(x^{i}\right)_{i=1}^{m},\left(y^{i}\right)_{i=1}^{m}$ in $Y$ and $f\in
W^{\\#},$ we have

$\displaystyle\left|\sum_{i=1}^{m}f\left(T(x^{i})\right)-f\left(T(x^{i})\right)\right|$
$\displaystyle\leq$ $\displaystyle
K(\int_{B_{Y^{\\#}}}\left|\sum_{i=1}^{m}s(x^{i})-s(y^{i})\right|^{p}d\mu\left(s\right))^{\frac{1}{p}}\times$
$\displaystyle(\int_{B_{Lip_{0}\left(W\right)^{\ast}}}\left|\left\langle
s,\mathfrak{n}\right\rangle\right|^{p^{\ast}}d\eta\left(\mathfrak{n}\right))^{\frac{1}{p^{\ast}}}.$

Theorem 3.9. Let $Y,W$ and $Z$ be three pointed metric spaces. Let
$u:Y\rightarrow W$ be an MS-Lipschitz $p$-summing operator and $v:W\rightarrow
Z$ be an MS-strongly Lipschitz $p$-summing operator. Then the composition
$T=v\circ u$ is MS-Lipschitz $p$-nuclear.

Proof. By [7, p. 124], we have

$\widetilde{T}=\widetilde{v}\circ\widetilde{u}.$

According to a result due to Cohen [3], the linear operator $\widetilde{u}$
being $p$-summing and $\widetilde{v}$ being strongly $p$-summing imply that
$\widetilde{T}$ is $p$-nuclear. Consequently, $T$ is also MS-Lipschitz
$p$-nuclear. $\blacksquare$

Remark 3.10.

In the linear case, the converse of the previous statement is true. However,
in our case, it is unknown whether every MS-Lipschitz $p$-nuclear operator can
be expressed as the product of an MS-Lipschitz $p$-summing operator and an MS-
strongly $p$-summing operator.

Declarations

Conflict of interest. The authors declare that they have no conflicts of
interest.

## References

  * [1] M. G. Cabrera-Padilla, J. A. Chávez-Domínguez, A. Jiménez-Vargas and M. Villegas-Vallecillos, Lipschitz tensor product, Khayyam J. Math. 1 (2015), no. 2, 185–218.
  * [2] S. Chevet, Sur certains produits tensoriels topologiques d’espaces de Banach, Z. Wahrscheinlichkeitstheorie verw. Geb. 11, (1969) 120-138.
  * [3] J. S. Cohen, Absolutely $p$-summing, $p$-nuclear operators and their conjugates, Math. Ann. 201 (1973) 177-200.
  * [4] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge University Press, 1995.
  * [5] J. D. Farmer and W. B. Johnson, Lipschitz p-summing Operators, Proc. Amer. Math. Soc. 137, no.9, (2009) 2989-2995.
  * [6] J. Flood, Free Topological Vector Spaces, Dissertationes Math. (Rozprawy Mat.) 221, 1984.
  * [7] G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Mathematica 159 (1) (2003) 121-141.
  * [8] A. Grothendieck, Resume de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8, 1–79 (1953)
  * [9] L. Mezrag and A. Tallab, On Lipschitz $\tau$(p)-summing operators, Colloq.Math. 147 (1) (2017), 95-114.
  * [10] V.G. Pestov, Free Banach spaces and representation of topological groups, Funct. Anal. Appl. 20 (1986) 70-72.
  * [11] A. Pietsch, Absolute p-summierende Abbildungen in normierten Raumen. Stud. Math 28, 333–353 (1967)
  * [12] R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Science & Business Media, 15 janv. (2002).
  * [13] K. Saadi, Some properties of Lipschitz strongly p-summing operators, J. Math. Anal. Appl. 423 (2015), no. 2, 1410-1426.
  * [14] K. Saadi, On the composition ideals of Lipschitz mappings, Banach J. Math. Anal. (2017) .
  * [15] K. Saadi and M. Belaala, Further results on strictly Lipschitz summing operators. Moroccan Journal of Pure and Applied Analysis, Volume 8 (2022) - Issue 2 (May 2022).
  * [16] P. Saphar, Applications à puissance nucléaire et applications de Hilbert-Schmidt dans les espaces de Banach, Annales scientifiques de L’E.N.S. $3^{e}$ série, tome 83, 2 (1966) 113-151.
  * [17] R. J. A. Varea, Geometry and structure of Lipschitz-free spaces and their biduals, Ph.D. dissertation, Universitat Politècnica De València, October 2020.
  * [18] N. Weaver, Lipschitz Algebras, World Scientific, Singapore, New Jersey, London, Hong Kong, 1999.