dongyang chen and bentuo zheng - Department of Computer ...

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 =∞.

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].

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:


`∞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).

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 |, 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:


`∞Mλ−−−→ `p (c0, p =∞)

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 | ≤ supy∗∈BY ∗

(∑j

|sj |p)1p (∑j

| < y∗, yj > |p∗)

1p∗ .

Thus

‖A‖ ≤ supy∗∈BY ∗

(∑j

| < y∗, yj > |p∗)

1p∗ for 1 |.

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).

For p = 1, ∑j

Lip(fj) ≤ Lip(B)‖Mλ‖ and supj‖A(ej)‖ ≤ ‖A‖

For 1 |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 <∞).

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.

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.


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 ).

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.


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


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 ).


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 ).

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.

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


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 ).

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.


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.

References

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[BL] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Vol.1, Amer. Math.Soc. Colloq. Pub., Vol.48, Amer. Math. Soc., Providence, RI, 2000.

[CZh1] D. Chen and B. Zheng, Remarks on Lipschitz p-summing operators, Proc. Amer. Math. Soc.139(2011), no.8,2891-2898.

[DJT] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge Studies in Adv.Math., Vol.43, Cambridge Univ. Press, Cambridge, 1995.

[FJ] J.D. Farmer and W.B. Johnson, Lipschitz p-summing operators, Proc. Amer. Math. Soc. 137 (2009),no.9, 2989–2995.

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[JLS] W.B.Johnson, J.Lindenstrauss and G.Schechtman, Extensions of Lipschitz maps into Banachspaces, Israel J.Math.54(1986),no.2, 129-138.

[JMS] W.B. Johnson, B. Maurey and G. Schechtman, Nonlinear factorization of linear operators, Bull.London Math. Soc. 41 (2009), 663–668.

[L] J.Lindenstrauss, On non-linear projections in Banach spaces, Michigan J.Math.11(1964)268-287.[MN] M.Mendel and A.Naor, Some applications of Ball’s extension theorem, Proc. Amer. Math. Soc. 134

(2006), no.9, 2577–2584.[P] A.Pietsch, Operator ideals, VEB Deutscher Verlag, Berlin; North-Holland, Amsterdam.[P1] A. Pietsch, Absolut p-summierende Abbildungen in normierten Raumen, Studia Math. 28 (1967),

333-353.[P2] A. Pietsch, Ideals of multilinear functionals (designs of a theory), Proceedings of the Second In-

ternational Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics(Leipzig), Teubner-Texte, 1983, pp. 185-99.

[T] N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, volume38 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific &Technical, Harlow, 1989.

School of Mathematical Sciences, Xiamen University, Xiamen,361005,ChinaE-mail address: [email protected]

Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240E-mail address: [email protected]

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