Post on 23-Jan-2023
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR
OPERATORS
DONGYANG CHEN AND BENTUO ZHENG
Abstract. In this note we introduce strongly Lipschitz p-integral operators, stronglyLipschitz p-nuclear operators and Lipschitz p-nuclear operators. It is shown that for alinear operator, the Lipschitz p-nuclear norm is the same as its usual p-nuclear normunder certain conditions. We also prove that the Lipschitz 2-dominated operators and thestrongly Lipschitz 2-integral operators are the same with equal norms. Finally, we showthat the Lipschitz p-integral norm of a Lipschitz map from a finite metric space into aBanach space is the same as its Lipschitz p-nuclear norm.
1. Introduction
The linear theory of p-summing operators goes back to the work of Grothendieck in [G].
And the most remarkable result can date its beginning in the paper of Pietsch [P1]. A.
Pietsch defined the class of p-summing operators and proved some good properties that
made it appear like an interesting class. Among the main results that appear in [P1], we
can find the Domination/Factorization Theorem, ideal, inclusion and composition properties
and good relations with other classes of linear operators, such as the nuclear and Hilbert-
Schmidt operators.
At the beginning of the 80’s, mainly due to [P2], the idea of generalizing the theory of
ideals of linear operators to the multilinear (and polynomial) setting appeared.
Recently Farmer and Johnson [FJ] introduced the notion of Lipschitz p-summing oper-
ators and the notion of Lipschitz p-integral operators and proved a nonlinear version of
the Pietsch domination theorem. In [CZh1], the authors proved a nonlinear version of
Grothendieck’s theorem for Lipschitz p-summing operators. However there is no known
nonlinear analogue of p-nuclear operators.
In this paper, we introduce natural notions of Lipschitz p-nuclear operators and strongly
Lipschitz p-nuclear operators. We also introduce notions of Lipschitz p-dominated operators
and strongly Lipschitz p-integral operators.
Date: Version: January 15, 2012.Dongyang Chen’s research is supported in part by National Natural Science Foundation of China (Grant
Nos. 10526034,10701063); Dongyang Chen is a participant in the NSF Workshop in Analysis and Probability,Texas A&M University.Bentuo Zheng’s Research is supported in part by NSF grant DMS-0800061 and NSF DMS-1068838.
1
2 DONGYANG CHEN AND BENTUO ZHENG
In Section 2, it is shown that for a bounded linear operator T from a separable Banach
space into a dual space, the Lipschitz p-nuclear norm of T coincides with its p-nuclear norm.
So the notion of Lipschitz p-nuclear operators is really a generalization of p-nuclear operators
in this setting. In the same section, we also prove a factorization theorem for strongly
Lipschitz p-nuclear operators and the result is a nonlinear analogue of the factorization
theorem for p-nuclear operators.
In Section 3, we give a characterization of Lipschitz p-dominated operators in terms of
domination and factorization. As a consequence we show that a Lipschitz map is Lipschitz
2-dominated if and only if it is strongly Lipschitz 2-integral. Moreover the Lipschitz 2-
dominated norm is the same as the strongly Lipschitz 2-integral norm.
In the last section, we provide some applications to Lipschitz maps from finite metric
spaces to Banach spaces. To be more precise we show that a Lipschitz p-integral operator
T from a finite metric space into a Banach space is automatically Lipschitz p-nuclear and
the Lipschitz p-integral norm equals to the Lipschitz p-nuclear norm. We also give some
estimates of the norms in terms of the Lipschitz constant of T and the cardinality of the
finite metric space. We believe that the natural notions introduced together with the results
obtained are pioneering and provide foundational contribution to the theory of nonlinear
p-integral and nonlinear p-nuclear operators.
We assume readers are familiar with linear p-summing, p-integral and p-nuclear operators.
Standard notations and related results for linear operators can be found in [DJT],[P] and
[T].
2. Nonlinear p-nuclear operators
Suppose that 1 ≤ p <∞ and that u : X → Y is a linear operator between Banach spaces.
We say that u is p-summing if there is a constant c ≥ 0 such that regardless of the natural
number n and regardless of the choice of x1, ..., xn in X we have( n∑i=1
‖u(xi)‖p)1/p
≤ c · sup
( n∑i=1
|x∗(xi)|p)1/p
: x∗ ∈ BX∗,
where BX∗ is the unit ball of X∗. The least constant c for which this inequality always
holds is denoted by πp(u). We shall write Πp(X,Y ) for the set of all p-summing operators
from X to Y .
We recall that a linear operator u : X → Y between Banach spaces is p-nuclear(1 ≤ p ≤∞)if there are two bounded linear operators a : `p → Y and b : X → `∞ and λ ∈ `p such
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 3
that the following diagram commutes:
Xu−−→ Yyb xa
`∞Mλ−−−→ `p
whereMλ : `∞ → `p is the diagonal operator defined as follows: Mλ((ξn)n) = (λnξn)n, (ξn)n ∈`∞. Then ‖Mλ‖ = πp(Mλ) = ιp(Mλ) = ‖λ‖p. The collection of all p-nuclear operators be-
tween X and Y is denoted by Np(X,Y ). For u ∈ Np(X,Y ), we define the p-nuclear norm
νp(u) of u to be the infimum of ‖a‖ · ‖Mλ‖ · ‖b‖,the infimum being taken over all factoriza-
tions as above. It is well-known (see,for example, Proposition 8.11 in [T]) that u : X → Y
is p-nuclear(1 ≤ p ≤ ∞) if and only if u can be written in the form u =∑
j x∗j ⊗ yj , where
(x∗j )j in X∗ and (yj)j in Y satisfy Np((x∗j )j , (yj)j) <∞. Here
Np((x∗j )j , (yj)j) = (
∑j
‖x∗j‖)(supj‖yj‖), p = 1
Np((x∗j )j , (yj)j) = (
∑j
‖x∗j‖p)1p supy∗∈BY ∗
(∑j
| < y∗, yj > |p∗)
1p∗ , 1 < p <∞, 1
p+
1
p∗= 1
Np((x∗j )j , (yj)j) = (sup
j‖x∗j‖) sup
y∗∈BY ∗
∑j
| < y∗, yj > |, p =∞.
Moreover, νp(u) = inf Np((x∗j )j , (yj)j), the infimum being taken over all such representations
as above.
Let X be a metric space and Y be a Banach space. We say that a Lipschitz mapping
T : X → Y is called Lipschitz p-nuclear if there are two Lipschitz mappings A : `p → Y
and B : X → `∞ and λ ∈ `p such that the following diagram commutes:
XT−−−→ YyB xA
`∞Mλ−−−→ `p
We set νLp (T ) := inf Lip(A) · ‖Mλ‖ ·Lip(B), the infimum being taken over all factorizations
as above. The collection of all Lipschitz p-nuclear operators from X to Y is denoted by
NLp (X,Y ).
Next theorem shows that under certain conditions, the Lipschitz p-nuclear norm of a
Lipschitz mapping coincides with its p-nuclear norm. Differentiation technique is used in the
proof. For definitions of Gateaux differentiability, ω∗-Gateaux differentiability, Gauss null
sets and Aronszajn null sets, we refer readers to the book of Benyamini and Lindenstrauss
[BL].
4 DONGYANG CHEN AND BENTUO ZHENG
Theorem 2.1. Let T be a bounded linear operator from a separable Banach space X into
a dual space Y . Then νLp (T ) = νp(T )(1 ≤ p <∞).
Proof. We use the method of [FJ] and [JMS]. Consider a typical Lipschitz p-nuclear factor-
ization:
XT−−−→ YyB xA
`∞Mλ−−−→ `p
It follows from Theorem 7.9 ([BL]) that B : X → `∞ is ω∗-Gateaux differentiable outside a
Gauss null set. Take such x0 ∈ X. Then
D∗B(x0)(x) = ω∗ − limt→0
B(x0 + tx)−B(x0)
t
exists for all x ∈ X and is a bounded linear operator in x. Since X is separable, it
follows from Theorem 6.42 ([BL]) that MλB : X → `p is Gateaux differentiable outside an
Aronszajn null set. It is well-known (see, Proposition 6.25, Proposition 6.27 and Theorem
6.32 of [BL]) that Gauss null sets coincide with Aronszajn null sets. Since Mλ : `∞ → `p is
weak*-to-weak continuous, we have
MλD∗B(x0) = DMλB(x0), x0 ∈ X \G,G is a Gauss null set in X.
By translation, we may assume that x0 = 0 and B(0) = 0, that is, MλD∗B(0) = DMλB(0).
Next we show that B can be replaced by D∗B(0) by constructing a Lipschitz map B :
`p → Y such that
T = BMλD∗B(0) and Lip(B) ≤ Lip(A).
Indeed, define, for each n,
Bn : `p → Y by ξ 7→ nA(ξ/n) , for all ξ ∈ `p.
Then Lip(Bn) = Lip(A) and, for all x ∈ X, we have
‖Tx−BnMλD∗B(0)(x)‖ ≤ Lip(A)‖nMλB
(xn
)−DMλB(0)(x)‖ → 0 (n→∞).
Since Bn(ξ)∞n=1 is bounded in Y and Y is a dual space, the weak* limit of Bn(ξ) through
some fixed free ultrafilter U of natural numbers exists for all ξ ∈ `p. Then, we set
B(ξ) = ω∗ − limUBn(ξ) for all ξ ∈ `p.
Thus
T = BMλD∗B(0) and Lip(B) ≤ Lip(A).
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 5
Finally, since B|MλD∗B(0)(X) is linear and Y is 1-complemented in Y ∗∗, Theorem 7.2 ([BL])
yields that there is a linear operator A : `p → Y such that
A∣∣∣MλD
∗B(0)(X)
= B∣∣∣MλD
∗B(0)(X)
= T and ‖A‖ ≤ Lip(B).
Therefore,
T = AMλD∗B(0) and νp(T ) ≤ νLp (T ).
So,
νp(T ) = νLp (T ).
Let M be a metric space. M# is the space of all real-valued Lipschitz functions under the
(semi)-norm Lip(·). We follow the usual convention of considering M as a pointed metric
space by designating a special point 0 ∈M and identifying M# with the Lipschitz functions
on M that are zero at 0. With this convention (M#, Lip(·)) is a Banach space and the unit
ball BM# of M# is a compact Hausdorff space in the topology of pointwise convergence on
M .
A Lipschitz mapping T : X → Y from a pointed metric space X to a Banach space Y is
called strongly Lipschitz p-nuclear if T can be written in the form T =∑
j fj ⊗ yj , where
(fj)j in X# and (yj)j in Y satisfy NLp ((fj)j , (yj)j) <∞. Here
NLp ((fj)j , (yj)j) = (
∑j
Lip(fj))(supj‖yj‖), p = 1
NLp ((fj)j , (yj)j) = (
∑j
Lip(fj)p)
1p supy∗∈BY ∗
(∑j
| < y∗, yj > |p∗)
1p∗ , 1 < p <∞
NLp ((fj)j , (yj)j) = (sup
jLip(fj)) sup
y∗∈BY ∗
∑j
| < y∗, yj > |, limj
Lip(fj) = 0, p =∞.
The strongly Lipschitz p-nuclear norm is defined by sνLp (T ) := inf NLp ((fj)j , (yj)j), where
the infimum being taken over all the strongly Lipschitz p-nuclear representations of T .
Our next result is a factorization theorem for strongly Lipschitz p-nuclear operators.
Theorem 2.2. Let 1 ≤ p ≤ ∞ and let X be a pointed metric space and Y be a Banach
space. A Lipschitz mapping T : X → Y is strongly Lipschitz p-nuclear if and only if T has
a factorization T = AMλB such that the following diagram commutes:
XT−−−→ YyB xA
`∞Mλ−−−→ `p (c0, p =∞)
6 DONGYANG CHEN AND BENTUO ZHENG
where B is a Lipschitz mapping from X to `∞ with B(0) = 0, Mλ ∈ L(`∞, `p) (L(`∞, c0), p =
∞) a diagonal operator and A ∈ L(`p, Y ) (L(c0, Y ), p =∞). Moreover, sνLp (T ) = inf ‖A‖ ·‖Mλ‖ · Lip(B), where the infimum is taken over all the above factorizations.
Proof. Assume that T is strongly Lipschitz p-nuclear and let T =∑
j fj ⊗ yj , with (fj)j in
X# and (yj)j in Y be a strongly Lipschitz p-nuclear representation of T . Let
B : X → `∞, x 7→ (fj(x)
Lip(fj))j
Mλ : `∞ → `p(c0, p =∞), (tj)j 7→ (Lip(fj)tj)j , λ = (Lip(fj))j
A : `p(c0, p =∞)→ Y, (sj)j 7→∑j
sjyj .
Then B is a Lipschitz mapping from X to `∞ with
B(0) = 0,Lip(B) ≤ 1, and ‖Mλ‖ = (∑j
Lip(fj)p)
1p (1 ≤ p <∞).
For p = 1,
‖A((sj)j)‖ = supy∗∈BY ∗
| < y∗,∑j
sjyj > | ≤ (∑j
|sj |)(supj‖yj‖).
Thus
‖A‖ ≤ supj‖yj‖ for p = 1.
For 1 < p <∞,
‖A((sj)j)‖ = supy∗∈BY ∗
| < y∗,∑j
sjyj > | ≤ supy∗∈BY ∗
(∑j
|sj |p)1p (∑j
| < y∗, yj > |p∗)
1p∗ .
Thus
‖A‖ ≤ supy∗∈BY ∗
(∑j
| < y∗, yj > |p∗)
1p∗ for 1 < p <∞.
For p =∞,
‖Mλ‖ = supj
Lip(fj), ‖A‖ ≤ supy∗∈BY ∗
∑j
| < y∗, yj > |.
Thus,
T = AMλB and inf ‖A‖‖Mλ‖Lip(B) ≤ sνLp (T ).
Conversely, if Mλ =∑
j δjej ⊗ ej and λ = (δj)j ∈ `p (c0, p =∞), then for x ∈ X,
T (x) = AMλB(x) =∑j
δj < B(x), ej > A(ej).
Let fj = δj < B(·), ej >. Then
fj ∈ X# and T =∑j
fj ⊗A(ej).
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 7
For p = 1, ∑j
Lip(fj) ≤ Lip(B)‖Mλ‖ and supj‖A(ej)‖ ≤ ‖A‖
For 1 < p <∞,
(∑j
Lip(fj)p)
1p ≤ Lip(B)‖Mλ‖
and
supy∗∈BY ∗
(∑j
| < y∗, A(ej) > |p∗)
1p∗ = sup
y∗∈BY ∗(∑j
| < A∗y∗, ej > |p∗)
1p∗
= supy∗∈BY ∗
‖A∗y∗‖
= ‖A‖.
For p =∞,
‖Mλ‖ = supj|δj |, sup
jLip(fj) ≤ Lip(B) sup
j|δj |
and
limj
Lip(fj) = 0.
Moreover,
supy∗∈BY ∗
∑j
| < y∗, A(ej) > | = supy∗∈BY ∗
∑j
| < A∗y∗, ej > |
= supy∗∈BY ∗
‖A∗y∗‖
= ‖A‖.
Thus, for 1 ≤ p ≤ ∞,
NLp ((fj)j , (yj)j) ≤ ‖A‖‖Mλ‖Lip(B).
This implies that
sνLp (T ) ≤ inf ‖A‖‖Mλ‖Lip(B).
Remark 2.1. It follows from Theorem 2.2 that strongly Lipschitz p-nuclear operators are
Lipschitz p-nuclear (1 ≤ p <∞).
Using the result of Theorem 2.2 and a similar argument as in Theorem 2.1, we can show
that for a linear operator, the strongly Lipschitz p-nuclear norm is the same as its usual
p-nuclear norm if the domain is separable.
Theorem 2.3. Let T be a bounded linear operator from a separable Banach space X into
a Banach space Y . Then sνLp (T ) = νp(T ) (1 ≤ p <∞).
8 DONGYANG CHEN AND BENTUO ZHENG
3. Nonlinear p-integral operators
We begin this section with the definitions of Lipschitz p-summing operators and Lipschitz
p-integral operators [FJ] and some known results. Recall that a Lipschitz map T from a
metric space (X, dX) into a metric space (Y, dY ) is said to be Lipschitz p-summing (1 ≤ p <∞) if there is a constant C > 0 such that regardless of the natural number n and regardless
of the choices of xini=1 and yini=1 in X, we have( n∑i=1
dY (Txi, Tyi)p
)1/p
≤ C supf∈B
X#
( n∑i=1
|f(xi)− f(yi)|p)1/p
,
the least constant C for which the above inequality holds is denoted by πLp (T ). We use
πLp (X,Y ) to denote the set of all Lipschitz p-summing mappings from X into Y . Farmer
and Johnson proved in [FJ] a nonlinear Pietsch domination theorem: a map T : X → Y
between two metric spaces X and Y is Lipschitz p-summing if and only if there exist a
constant C > 0 and a regular Borel probability measure µ on BX# such that
‖Tx− Ty‖p ≤ Cp∫BX#
|f(x)− f(y)|pdµ(f), ∀x, y ∈ X
in this case, πLp (T ) = inf C,the infimum being taken over all possible C’s and µ’s.
Let X be a metric space and Y be a pointed metric space. We say that a Lipschitz
mapping T : X → Y is Lipschitz p-integral (1 ≤ p ≤ ∞) if there are a probability measure
space (Ω,Σ, µ) and two Lipschitz mappings A : Lp(µ)→ (Y #)∗ and B : X → L∞(µ) giving
rise to the following commutative diagram:
XT−−−→ Y
KY−−−−→ (Y #)∗yB xAL∞(µ)
ip−−−−−−−−→ Lp(µ)
where KY : Y → (Y #)∗ is the canonical evaluation map. We define the Lipschitz p-integral
norm ιLp (T ) of T to be the infimum of Lip(A) Lip(B), the infimum being taken over all such
factorizations. The collection of all Lipschitz p-integral operators from X to Y is denoted
by ILp (X,Y ). When Y is a normed space, we can replace KY by the canonical embedding
kY of Y into Y ∗∗ in the above definition without changing the Lipschitz p-integral norm
because Y ∗∗ is norm-one complemented in (Y #)∗ ([L]). An easy check shows that ILp (X,Y )
is a Banach space whenever X and Y are Banach spaces. Since the Banach ideal [Ip, ιp] of
p-integral operators (1 ≤ p ≤ ∞) is maximal, the Lipschitz p-integral norm is the same as
the usual p-integral norm for a linear operator.
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 9
Next we introduce the notions of Lipschitz p-dominated operators and strongly Lipschitz
p-integral operators. A Lipschitz mapping T : X → Y between Banach spaces X and Y
is Lipschitz p-dominated(1 ≤ p < ∞)if there exist a Banach space Z and a L ∈ Πp(X,Z)
such that
‖Tx− Ty‖ ≤ ‖Lx− Ly‖,∀ x, y ∈ X
The collection of all Lipschitz p-dominated operators between X and Y is denoted by
DLp (X,Y ). For T ∈ DLp (X,Y ), we set dLp (T ) to be the infimum of πp(L), the infimum
being taken over all the above Z and L. We say that a Lipschitz mapping T : X → Y
between Banach spaces X and Y is strongly Lipschitz p-integral (1 ≤ p ≤ ∞) if there are
a probability measure space (Ω,Σ, µ) and a Lipschitz mapping A : Lp(µ) → Y ∗∗ and a
bounded linear operator B : X → L∞(µ) giving rise to the following commutative diagram:
XT−−−→ Y
JY−−−→ Y ∗∗yB xAL∞(µ)
ip−−−−−−−−→ Lp(µ)
We define the strongly Lipschitz p-integral norm sιLp (T ) of T to be the infimum of Lip(A) ·‖B‖, the infimum being taken over all such factorizations. In this note, we prove that
the Lipschitz 2-dominated operators and the strongly Lipschitz 2-integral operators are the
same with equal norms. Moreover, it is proved that the Lipschitz p-dominated operators
and the strongly Lipschitz p-integral operators are also the same with equal norms when
the domain space is C(K)(K is a compact Hausdorff space).
We give a characterization of strongly Lipschitz p-integral operators whose proof is very
similar to the proof in the linear setting (see e.g.page 98 in [DJT]). For convenience of
readers, we include the proof here.
Throughout the rest of this section, we assume that X and Y are Banach spaces.
Theorem 3.1. Let 1 ≤ p ≤ ∞. A Lipschitz mapping T : X → Y is strongly Lipschitz p-
integral if and only if every time we take a weak∗-compact norming subset K of BX∗, there
exist a regular Borel probability measure µ on K and a Lipschitz map T : Lp(µ)→ Y ∗∗ such
that the following diagram commutes:
XT−−−→ Y
JY−−−→ Y ∗∗yiX xTC(K)
jp−−−−−−−−→ Lp(µ),
where jp, iX are the canonical maps. In this case, sιLp (T ) = inf Lip(T ), where the infimum
is taken over all possible µ’s and T ’s.
10 DONGYANG CHEN AND BENTUO ZHENG
Proof. We write 4 = infLip(T ), where the infimum is taking over all T for which the
diagram in the statement of the theorem commutes. Fix a weak∗-compact norming subset
K of BX∗ . Then there exist a regular Borel probability measure µ on K and a Lipschitz
map T : Lp(µ)→ Y ∗∗ such that
JY T = T jpiX : XiX−→ C(K)
jp−→ Lp(µ)T−→ Y ∗∗.
Note that
jp = ipj∞ : C(K)j∞−−→ L∞(µ)
ip−→ Lp(µ).
Let B = j∞iX and A = T . Then
JY T = AipB : XB−→ L∞(µ)
ip−→ Lp(µ)A−→ Y ∗∗.
This implies that
sιLp (T ) ≤ ‖B‖Lip(A) ≤ Lip(T )
and hence sιLp (T ) ≤ 4.
Conversely, fix a weak∗-compact norming subset K of BX∗ and a typical factorization
JY T = AipB : XB−→ L∞(µ)
ip−→ Lp(µ)A−→ Y ∗∗.
Since L∞(µ) is injective, there is a bounded linear operator B : C(K)→ L∞(µ) such that
BiX = B and ‖B‖ = ‖B‖.
Applying ipB : C(K) → Lp(µ) to Corollary 2.15 in [DJT], we obtain a regular Borel
probability measure ν on K and a bounded linear operator C : Lp(ν)→ Lp(µ) such that
ipB = Cjp : C(K)jp−→ Lp(ν)
C−→ Lp(µ) and ‖C‖ = πp(ipB).
We set T = AC and then
JY T = AipB = A(ipB)iX = A(Cjp)iX = T jpiX .
Thus
4 ≤ Lip(T ) ≤ Lip(A)‖C‖
= Lip(A)πp(ipB)
≤ Lip(A)‖B‖
= Lip(A)‖B‖.
This implies that
4 ≤ sιLp (T ).
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 11
The following result characterizes Lipschitz p-dominated operators.
Theorem 3.2. For a Lipschitz mapping T : X → Y the following are equivalent:
(i) T ∈ DLp (X,Y );
(ii) there exists a constant C > 0 such that for any xini=1 and yini=1 in X, we have( n∑i=1
‖Txi − Tyi‖p)1/p
≤ C supx∈BX∗
( n∑i=1
|〈x∗, xi − yi〉|p)1/p
;
(iii) there exist a constant C > 0 and a probability measure µ on BX∗ such that for
all x, y ∈ X, we have ‖Tx − Ty‖ ≤ C(∫BX∗|〈x∗, x − y〉|p dµ(x∗))1/p. In that case
dLp (T ) = infC : C as in (ii) = infC : C, µ as in (iii).
Proof. (i) ⇒ (ii). Choose any L ∈ πp(X,Z) such that
‖Tx− Ty‖ ≤ ‖Lx− Ly‖,∀ x, y ∈ X.
Then, for any finite sequence xini=1 and yini=1 in X, we have( n∑i=1
‖Txi − Tyi‖p)1/p
≤( n∑i=1
‖Lxi − Lyi‖p)1/p
≤ πp(L) supx∈BX∗
( n∑i=1
|〈x∗, xi − yi〉|p)1/p
.
Thus,
infC : C as in (ii) ≤ πp(L).
That is,
infC : C as in (ii) ≤ dLp (T ).
(ii) ⇒ (iii). Fix x, y ∈ X. Define
gx,y : BX∗ → R by x∗ 7→ ‖Tx− Ty‖p − Cp|〈x∗, x− y〉|p.
Let Q be the positive convex hull of gx,y : x, y ∈ X, that is,
Q = ∑i
λigxi,yi :∑i
λi = 1, λi > 0
and let
P = F ∈ C(BX∗ ,R) : F (x∗) > 0,∀ x∗ ∈ BX∗.
Then it follows from (ii) and the approximation that P ∩ Q = ∅. It follows from the
separation theorem and the Riesz representation theorem that there is a finite signed Borel
regular measure µ on BX∗ and a real number c so that∫BX∗
g dµ ≤ c <∫BX∗
F dµ,∀ g ∈ Q,∀ F ∈ P.
12 DONGYANG CHEN AND BENTUO ZHENG
Since 0 ∈ Q and all positive constants are in P , we see that c = 0 and µ is a positive
measure , which we may assume by rescaling is a probability measure. In particular, for
gx,y, we have
‖Tx− Ty‖p − Cp∫BX∗|〈x∗, x− y〉|p dµ ≤ 0.
Moreover,
infC : C, µ as in (iii) ≤ infC : C as in (ii).
(iii) ⇒ (i). Define
L : X → Lp(BX∗ , µ) by x 7→ (Lx)(x∗) = 〈x∗, x〉, ∀ x∗ ∈ BX∗ .
Then
‖Tx− Ty‖ ≤ C · ‖Lx− Ly‖ ∀ x, y ∈ X
and
πp(L) ≤ 1.
Thus,
dLp (T ) ≤ πp(C · L) ≤ C.
This implies
dLp (T ) ≤ infC : C, µ as in(iii).
Remark 3.1. In the above theorem, BX∗ can be replaced by any weak*-compact norming
subset of BX∗ . Theorem 3.2 clearly implies that a Lipschitz p-dominated operator is Lip-
schitz p-summing. In the nonlinear Pietsch domination theorem by Farmer and Johnson
[FJ], BX# is used for supremum and integral. However the geometric structure of X# is
poorly understood and in most cases, it is very hard to handle. So the importance of the
above theorem is that in (ii) and (iii), we use BX∗ for the supremum and integral instead
of BX# .
Theorem 3.3. Let T : X → Y be a Lipschitz mapping and let K be a weak* compact
norming subset of BX∗. The following are equivalent:
(i) T ∈ DLp (X,Y );
(ii) there exist a regular Borel probability measure µ on K, a closed subspace Xp of Lp(µ)
and a Lipschitz map T : Xp → Y such that (a) jpiX(X) ⊆ Xp; (b) T jpiX = T . In
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 13
other words, the following diagram commutes:
XT−−−→ YyiX xT
iX(X)jp|iX (X)−−−−−→ Xp
∩ ∩C(K)
jp−−−→ Lp(µ);
(iii) there exist a regular Borel probability measure µ on K and a Lipschitz map T ·Lp(µ)→ `∞(BY ∗) such that the following diagram commutes:
XT−−−→ Y
iY−−−→ `∞(BY ∗)yiX xTC(K)
jp−−−−−−−−→ Lp(µ);
(iv) there exist a probability space (Ω,Σ, µ) and a Lipschitz mapping u : Lp(µ) →`∞(BY ∗) and a linear mapping v : X → L∞(µ) such that the the following dia-
gram commutes
XT−−−→ Y
iY−−−→ `∞(BY ∗)yv xuL∞(µ)
ip−−−−−−−−→ Lp(µ).
In addition, we may choose µ and T in (ii) or µ and T in (iii) so that Lip(T ) =
Lip(T ) = dLp (T ); in (iv) we may arrange that ‖v‖ = 1 and Lip(u) = dLp (T ).
Proof. (i) ⇒ (ii): It follows from Theorem 3.2 that there is a regular Borel probability
measure µ on K such that for all x, y ∈ X, we have
‖Tx− Ty‖ ≤ dLp (T )(
∫BX∗|〈x∗, x− y〉|p dµ(x∗))1/p.
Define a Lipschitz mapping u : jpiX(X)→ Y by u(jpiX(x)) = T (x), x ∈ X. Then
Lip(u) ≤ dLp (T ).
We let Xp be the norm closure of jpiX(X) in Lp(µ) and T be the extension of u to Xp.
Then
Lip(T ) ≤ dLp (T ) and T jXp iX = T.
For the converse,
dLp (T ) = dLp (T jXp iX) ≤ Lip(T ).
14 DONGYANG CHEN AND BENTUO ZHENG
Thus
dLp (T ) = Lip(T ).
(ii) ⇒ (iii): Let µ,Xp and T be as in (ii). Since l∞(BY ∗) is an absolute 1-Lipschitz
retract, there is a Lipschitz extension T of iY T to Lp(µ) with Lip(T ) = Lip(iY T ). Then T
is the required Lipschitz mapping.
(iii)⇒ (iv): Let µ and T be as in (iii). Note that
jp = ipj∞ : C(K)j∞−−→ L∞(µ)
ip−→ Lp(µ).
Let u = j∞iX . Then
iY T = T ipu.
Let v = u‖u‖ and u = ‖u‖T . This implies that
iY T = uipv.
Moreover,
Lip(u) ≤ Lip(T ) = dLp (T )
and
dLp (T ) = dLp (iY T )
= dLp (uipv)
≤ Lip(u)πp(ipv)
≤ Lip(u).
Thus
‖v‖ = 1 and Lip(u) = dLp (T ).
(iv)⇒ (i): It follows from Theorem 3.2 and the fact that ipv is p-summing and iY is an
isometric embedding.
Theorem 3.3 yields the following corollaries immediately.
Corollary 3.4. Let K be a compact Hausdorff space. A Lipschitz mapping T : C(K)→ Y
is Lipschitz p-dominated if and only if there exist a regular Borel probability measure µ on
K and a Lipschitz mapping T : Lp(µ)→ Y such that T jp = T . Moreover, we may arrange
that Lip(T ) = dLp (T ).
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 15
Corollary 3.5. A Lipschitz mapping T : X → Y is Lipschitz 2-dominated if and only if
there exist a regular probability measure µ on BX∗ and a Lipschitz mapping T : L2(µ)→ Y
such that the following diagram commutes:
XT−−−→ YyiX xT
C(BX∗)j2−−→ L2(µ).
Moreover, we may arrange that Lip(T ) = dL2 (T ).
Theorem 3.6. If T : X → Y is strongly Lipschitz p-integral, then it is Lipschitz p-
dominated with dLp (T ) ≤ sιLp (T ).
Proof. Given a typical strongly Lipschitz p-integral factorization:
XT−−−→ Y
JY−−−→ Y ∗∗yB xAL∞(µ)
ip−−−−−−−−→ Lp(µ).
Since ip is p-summing and B is linear, it follows from the ideal property of Lipschitz p-
dominated operators that JY T is Lipschitz p-dominated. Hence T is also Lipschitz p-
dominated and
dLp (T ) = dLp (JY T )
= dLp (AipB)
≤ Lip(A)‖B‖.
This implies
dLp (T ) ≤ sιLp (T ).
Combining the previous results, we have the following two immediate corollaries.
Corollary 3.7. The Lipschitz 2-dominated and strongly Lipschitz 2-integral operators are
the same with equality of norms.
Corollary 3.8. Let 1 ≤ p < ∞. If K is a compact Hausdorff space, then every Lipschitz
p-dominated operator T : C(K)→ Y is strongly Lipschitz p-integral with equality of norms.
16 DONGYANG CHEN AND BENTUO ZHENG
4. Applications to Lipschitz mappings from a finite metric space into a
Banach space
This section deals with the relationship between Lipschitz p-integral and Lipschitz p-
nuclear operators from a finite metric space into a Banach space.
Theorem 4.1. Let X be a finite metric space and Y be a Banach space. Then, for any
1 ≤ p <∞ and any mapping T : X → Y , we have
(a) νLp (T ) = ιLp (T );
(b) νLp (T ) = ιLp (T ) ≤ C · (log |X|)2 · Lip(T ), where C is an absolute constant.
Proof. (a) Consider a typical p-integral factorization
JY T : XB−→ L∞(µ)
ip−→ Lp(µ)A−→ Y ∗∗.
Fix ε > 0. Then, there exists a finite dimensional subspace E of L∞(µ), of dimension N
say, together with an isomorphism ν : E → `N∞ such that
‖ν‖ ‖ν−1‖ ≤ 1 + ε and B(X) ⊆ E.
Moreover, ip(E) is contained in a (1 + ε)-complemented subspace F of Lp(µ). Let J : F →Lp(µ) be the corresponding canonical embedding and P : Lp(µ) → F the corresponding
projection with ‖P‖ ≤ 1 + ε. The principle of local reflexivity guarantees the existence of
a linear operator W : spanAipB(X) → Y such that
‖W‖ ≤ 1 + ε and W |spanAipB(X)∩JY (Y ) = idJY (Y ).
Thus,
JY T = WJY T.
Consider
u = Pip|Eν−1 : `N∞ → F.
Then u is p-summing and hence p-integral and
πp(u) = ιp(u)
= νp(u)
≤ ‖P‖ιp(ip)‖ν−1‖
≤ (1 + ε)‖ν−1‖.
Note that
JY T = WJY T = WAipB = WAJuνB.
Then
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 17
νLp (T ) = νLp (JY T )
= νLp (WAJuνB)
≤ Lip(WAJ)νp(u) Lip(νB)
≤ (1 + ε)3 Lip(A) Lip(B).
Thus
νLp (T ) = ιLp (T ).
(b) It follows from Farmer-Johnson Factorization Theorem ([FJ])that
XT−−−→ YyiX xu
iX(X)jp|iX (X)−−−−−→ Xp
∩ ∩C(K)
jp−−−→ Lp(µ),
where T = ujpiX and πLp (T ) = Lip(u). It follows from Theorem 1 of [JLS] that there exists
u : Lp(µ)→ Y such that
u|Xp = u and Lip(u) ≤ (C1 · log |X|) Lip(u),
where C1 is an absolute constant. Note that
jp = ipi∞ : C(K)j∞−−→ L∞(µ)
ip−→ Lp(µ).
Let b = j∞iX . Then
T = ujpiX = uipb.
Thus
νLp (T ) = ιLp (T )
≤ Lip(b) · Lip(u)
≤ (C1 · log |X|)πLp (T ).
Finally, as was mentioned in [FJ], J.Bourgain([B]) really proved that
πLp (idX) ≤ πL1 (idX) ≤ (C2 · log |X|),
where C2 is an absolute constant. It follows from the ideal property of Lipschitz p-summing
operators that
πLp (T ) = πLp (T · idX) ≤ (C2 · log |X|) Lip(T ).
18 DONGYANG CHEN AND BENTUO ZHENG
This implies that
νLp (T ) = ιLp (T ) ≤ C1 · C2 · (log |X|)2 Lip(T ).
The following result gives a better estimate on the constant of Theorem 4.1 when the
range space is a 2-uniformly convex Banach space and the cardinality of X is relatively
small compared to p.
Theorem 4.2. Let X be a finite metric space and Y be a 2-uniformly convex Banach space.
Then, for any 2 ≤ p <∞ and for any mapping T : X → Y , we have
νLp (T ) ≤ 24K2(Y ) ·√p− 1 · Lip(T )(|X|)1/p
where K2(Y ) is the 2-uniform convexity constant of Y .
Proof. Suppose that X = x1, ..., xn. Consider the Frechet map
F : X → `n∞(n = |X|) given by F(x) = (d(x, xi))ni=1.
Then F is an isometric embedding. Define
g : Id∞→pF(X)→ Y by Id∞→pF(x) 7→ T (x),
where Id∞→p : `n∞ → `np and Idp→∞ : `np → `n∞ are the identity mappings. Note that Id∞→p
is p-nuclear with νp(Id∞→p) = ιp(Id∞→p) = πp(Id∞→p) = n1/p. It follows from Theorem 1.1
([MN]) that there exists g : `np → Y such that
Lip(g) ≤ 24K2(Y )√p− 1 Lip(g)
≤ 24K2(Y )√p− 1 · Lip(T ),
and
g∣∣∣Id∞→p F(X)
= g.
Then
T = g Id∞→p F ,
and
νLp (T ) ≤ Lip(g)νp(Id∞→p)
≤ 24K2(Y ) ·√p− 1 · Lip(T ) · n1/p.
LIPSCHITZ p-INTEGRAL OPERATORS AND LIPSCHITZ p-NUCLEAR OPERATORS 19
Acknowledgements. This work is done during the first author’s visit to Department of
Mathematics, Texas A&M University. We are grateful to Professor W.B.Johnson,without
whose ideas and encouragement,this work would not have taken place. We would also like
to thank the referee for his valuable suggestions which made the current version of the paper
more readable.
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School of Mathematical Sciences, Xiamen University, Xiamen,361005,ChinaE-mail address: cdy@xmu.edu.cn
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240E-mail address: bzheng@memphis.edu