LIPSCHITZ p-integral OPERATORS AND LIPSCHITZ p-nuclear OPERATORS. 1. Introduction

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1 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS DONGYANG CHEN AND BENUO ZHENG Abstract. In this note we introduce strongly Lipschitz p-integral operators, strongly Lipschitz p-nuclear operators and Lipschitz p-nuclear operators. It is shown that for a linear operator, the Lipschitz p-nuclear norm is the same as its usual p-nuclear norm under certain conditions. We also prove that the Lipschitz 2-dominated operators and the strongly Lipschitz 2-integral operators are the same with equal norms. Finally, we show that the Lipschitz p-integral norm of a Lipschitz map from a finite metric space into a Banach space is the same as its Lipschitz p-nuclear norm. 1. Introduction he 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 heorem, 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 operators and the notion of Lipschitz p-integral operators and proved a nonlinear version of the Pietsch domination theorem. Grothendieck s theorem for Lipschitz p-summing operators. nonlinear analogue of p-nuclear operators. In [CZh1], the authors proved a nonlinear version of However there is no known 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, Dongyang Chen s research is supported in part by National Natural Science Foundation of China (Grant Nos , ); Dongyang Chen is a participant in the NSF Workshop in Analysis and Probability, exas A&M University. Bentuo Zheng s Research is supported in part by NSF grant DMS and NSF DMS

2 2 DONGYANG CHEN AND BENUO ZHENG In Section 2, it is shown that for a bounded linear operator from a separable Banach space into a dual space, the Lipschitz p-nuclear norm of 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. o be more precise we show that a Lipschitz p-integral operator 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 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 [DJ],[P] and []. 2. Nonlinear p-nuclear operators Suppose that 1 p < and that u : 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 x 1,..., x n in we have where B ( n i=1 ) 1/p {( n u(x i ) p c sup i=1 x (x i ) p ) 1/p : x B is the unit ball of. he least constant c for which this inequality always holds is denoted by π p (u). We shall write Π p (, Y ) for the set of all p-summing operators from to Y. We recall that a linear operator u : Y between Banach spaces is p-nuclear(1 p )if there are two bounded linear operators a : l p Y and b : l and λ l p such },

3 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 3 that the following diagram commutes: b l u M λ where M λ : l l p is the diagonal operator defined as follows: M λ ((ξ n ) n ) = (λ n ξ n ) n, (ξ n ) n l. hen M λ = π p (M λ ) = ι p (M λ ) = λ p. he collection of all p-nuclear operators between and Y is denoted by N p (, Y ). For u N p (, Y ), we define the p-nuclear norm ν p (u) of u to be the infimum of a M λ b,the infimum being taken over all factorizations as above. It is well-known (see,for example, Proposition 8.11 in []) that u : Y is p-nuclear(1 p ) if and only if u can be written in the form u = x y, where Y a (x ) in and (y ) in Y satisfy N p ((x ), (y ) ) <. Here lp N p ((x ), (y ) ) = ( x )(sup y ), p = 1 N p ((x ), (y ) ) = ( x p ) 1 p N p ((x ), (y ) ) = (sup sup y B Y ( x ) < y, y > p ) 1 p, 1 , p =. Moreover, ν p (u) = inf N p ((x ), (y ) ), the infimum being taken over all such representations as above. Let be a metric space and Y be a Banach space. We say that a Lipschitz mapping : Y is called Lipschitz p-nuclear if there are two Lipschitz mappings A : l p Y and B : l and λ l p such that the following diagram commutes: B l M λ Y A lp We set νp L ( ) := inf Lip(A) M λ Lip(B), the infimum being taken over all factorizations as above. he collection of all Lipschitz p-nuclear operators from to Y is denoted by Np L (, 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 Gâteaux differentiability, ω -Gâteaux differentiability, Gauss null sets and Aronszan null sets, we refer readers to the book of Benyamini and Lindenstrauss [BL].

4 4 DONGYANG CHEN AND BENUO ZHENG heorem 2.1. Let be a bounded linear operator from a separable Banach space into a dual space Y. hen ν L p ( ) = ν p ( )(1 p < ). Proof. We use the method of [FJ] and [JMS]. Consider a typical Lipschitz p-nuclear factorization: B l M λ It follows from heorem 7.9 ([BL]) that B : l is ω -Gâteaux differentiable outside a Gauss null set. ake such x 0. hen Y A D B(x 0 )(x) = ω lim t 0 B(x 0 + tx) B(x 0 ) t exists for all x and is a bounded linear operator in x. lp Since is separable, it follows from heorem 6.42 ([BL]) that M λ B : l p is Gâteaux differentiable outside an Aronszan null set. It is well-known (see, Proposition 6.25, Proposition 6.27 and heorem 6.32 of [BL]) that Gauss null sets coincide with Aronszan null sets. Since M λ : l l p is weak*-to-weak continuous, we have M λ D B(x 0 ) = D Mλ B(x 0 ), x 0 \ G, G is a Gauss null set in. By translation, we may assume that x 0 = 0 and B(0) = 0, that is, M λ D B (0) = D M λ B(0). Next we show that B can be replaced by D B (0) by constructing a Lipschitz map B : l p Y such that = BM λ D B(0) and Lip( B) Lip(A). Indeed, define, for each n, B n : l p Y by ξ na(ξ/n), for all ξ l p. hen Lip(B n ) = Lip(A) and, for all x, we have ( x x B n M λ DB(0)(x) Lip(A) nm λ B D Mλ B(0)(x) 0 n) (n ). Since {B n (ξ)} n=1 is bounded in Y and Y is a dual space, the weak* limit of B n(ξ) through some fixed free ultrafilter U of natural numbers exists for all ξ l p. hen, we set B(ξ) = ω lim B n (ξ) for all ξ l p. U hus = BM λ DB(0) and Lip( B) Lip(A).

5 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 5 Finally, since B Mλ D B (0)() is linear and Y is 1-complemented in Y, heorem 7.2 ([BL]) yields that there is a linear operator Ã : l p Y such that Ã = B = and Ã Lip( B). (0)() Mλ DB (0)() herefore, Mλ D B = ÃM λd B(0) and ν p ( ) ν L p ( ). So, ν p ( ) = ν L p ( ). 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 B M # of M # is a compact Hausdorff space in the topology of pointwise convergence on M. A Lipschitz mapping : Y from a pointed metric space to a Banach space Y is called strongly Lipschitz p-nuclear if can be written in the form = f y, where (f ) in # and (y ) in Y satisfy Np L ((f ), (y ) ) <. Here N L p ((f ), (y ) ) = ( Lip(f ))(sup y ), p = 1 Np L ((f ), (y ) ) = ( Lip(f ) p ) 1 p sup y B Y ( < y, y > p ) 1 p, 1 , lim Lip(f ) = 0, p =. y B Y he strongly Lipschitz p-nuclear norm is defined by sν L p ( ) := inf N L p ((f ), (y ) ), where the infimum being taken over all the strongly Lipschitz p-nuclear representations of. Our next result is a factorization theorem for strongly Lipschitz p-nuclear operators. heorem 2.2. Let 1 p and let be a pointed metric space and Y be a Banach space. A Lipschitz mapping : Y is strongly Lipschitz p-nuclear if and only if has a factorization = AM λ B such that the following diagram commutes: B l Y A M λ lp (c 0, p = )

6 6 DONGYANG CHEN AND BENUO ZHENG where B is a Lipschitz mapping from to l with B(0) = 0, M λ L(l, l p ) (L(l, c 0 ), p = ) a diagonal operator and A L(l p, Y ) (L(c 0, Y ), p = ). Moreover, sν L p ( ) = inf A M λ Lip(B), where the infimum is taken over all the above factorizations. Proof. Assume that is strongly Lipschitz p-nuclear and let = f y, with (f ) in # and (y ) in Y be a strongly Lipschitz p-nuclear representation of. Let B : l, x ( f (x) Lip(f ) ) M λ : l l p (c 0, p = ), (t ) (Lip(f )t ), λ = (Lip(f )) A : l p (c 0, p = ) Y, (s ) s y. hen B is a Lipschitz mapping from to l with B(0) = 0, Lip(B) 1, and M λ = ( Lip(f ) p ) 1 p (1 p < ). For p = 1, hus For 1 ( A sup y for p = 1. s y > sup y B Y ( s )(sup y ). s p ) 1 p ( < y, y > p ) 1 p. hus For p =, hus, A sup y B Y ( < y, y > p ) 1 p for 1 . y B Y = AM λ B and inf A M λ Lip(B) sν L p ( ). Conversely, if M λ = δ e e and λ = (δ ) l p (c 0, p = ), then for x, (x) = AM λ B(x) = δ < B(x), e > A(e ). Let f = δ < B( ), e >. hen f # and = f A(e ).

7 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 7 For p = 1, For 1 p ) 1 p = sup y B Y ( < A y, e > p ) 1 p = sup A y y B Y = A. For p =, and M λ = sup δ, sup Lip(f ) Lip(B) sup δ lim Lip(f ) = 0. Moreover, sup y B Y < y, A(e ) > = sup y B Y = sup A y y B Y = A. < A y, e > hus, for 1 p, N L p ((f ), (y ) ) A M λ Lip(B). his implies that sν L p ( ) inf A M λ Lip(B). Remark 2.1. It follows from heorem 2.2 that strongly Lipschitz p-nuclear operators are Lipschitz p-nuclear (1 p < ). Using the result of heorem 2.2 and a similar argument as in heorem 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. heorem 2.3. Let be a bounded linear operator from a separable Banach space into a Banach space Y. hen sν L p ( ) = ν p ( ) (1 p < ).

8 8 DONGYANG CHEN AND BENUO 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 from a metric space (, d ) into a metric space (Y, d Y ) 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 {x i } n i=1 and {y i} n i=1 in, we have ( n ) 1/p ( n ) 1/p d Y ( x i, y i ) p C sup f(x i ) f(y i ) p, i=1 f B # i=1 the least constant C for which the above inequality holds is denoted by π L p ( ). We use π L p (, Y ) to denote the set of all Lipschitz p-summing mappings from into Y. Farmer and Johnson proved in [FJ] a nonlinear Pietsch domination theorem: a map : Y between two metric spaces and Y is Lipschitz p-summing if and only if there exist a constant C > 0 and a regular Borel probability measure µ on B # such that x y p C p f(x) f(y) p dµ(f), x, y B # in this case, π L p ( ) = inf C,the infimum being taken over all possible C s and µ s. Let be a metric space and Y be a pointed metric space. We say that a Lipschitz mapping : Y is Lipschitz p-integral (1 p ) if there are a probability measure space (Ω, Σ, µ) and two Lipschitz mappings A : L p (µ) (Y # ) and B : L (µ) giving rise to the following commutative diagram: B L (µ) Y K Y (Y # ) A i p L p (µ) where K Y : Y (Y # ) is the canonical evaluation map. We define the Lipschitz p-integral norm ι L p ( ) of to be the infimum of Lip(A) Lip(B), the infimum being taken over all such factorizations. he collection of all Lipschitz p-integral operators from to Y is denoted by Ip L (, Y ). When Y is a normed space, we can replace K Y by the canonical embedding k Y 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 Ip L (, Y ) is a Banach space whenever and Y are Banach spaces. Since the Banach ideal [I p, ι p ] 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.

9 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 9 Next we introduce the notions of Lipschitz p-dominated operators and strongly Lipschitz p-integral operators. A Lipschitz mapping : Y between Banach spaces and Y is Lipschitz p-dominated(1 p < )if there exist a Banach space Z and a L Π p (, Z) such that x y Lx Ly, x, y he collection of all Lipschitz p-dominated operators between and Y is denoted by D L p (, Y ). For D L p (, Y ), we set d L p ( ) to be the infimum of π p (L), the infimum being taken over all the above Z and L. We say that a Lipschitz mapping : Y between Banach spaces and Y is strongly Lipschitz p-integral (1 p ) if there are a probability measure space (Ω, Σ, µ) and a Lipschitz mapping A : L p (µ) Y and a bounded linear operator B : L (µ) giving rise to the following commutative diagram: B L (µ) Y J Y Y A i p L p (µ) We define the strongly Lipschitz p-integral norm sι L p ( ) of 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 [DJ]). For convenience of readers, we include the proof here. hroughout the rest of this section, we assume that and Y are Banach spaces. heorem 3.1. Let 1 p. A Lipschitz mapping : Y is strongly Lipschitz p- integral if and only if every time we take a weak -compact norming subset K of B, there exist a regular Borel probability measure µ on K and a Lipschitz map : L p (µ) Y such that the following diagram commutes: i C(K) Y J Y Y p L p (µ), where p, i are the canonical maps. In this case, sι L p ( ) = inf Lip( ), where the infimum is taken over all possible µ s and s.

10 10 DONGYANG CHEN AND BENUO ZHENG Proof. We write = inflip( ), where the infimum is taking over all for which the diagram in the statement of the theorem commutes. Fix a weak -compact norming subset K of B. hen there exist a regular Borel probability measure µ on K and a Lipschitz map : L p (µ) Y such that J Y = p i : i C(K) p L p (µ) Y. Note that p = i p : C(K) L (µ) ip L p (µ). Let B = i and A =. hen J Y = Ai p B : B L (µ) ip L p (µ) A Y. his implies that sι L p ( ) B Lip(A) Lip( ) and hence sι L p ( ). Conversely, fix a weak -compact norming subset K of B and a typical factorization J Y = Ai p B : B L (µ) ip L p (µ) A Y. Since L (µ) is inective, there is a bounded linear operator B : C(K) L (µ) such that Bi = B and B = B. Applying i p B : C(K) Lp (µ) to Corollary 2.15 in [DJ], we obtain a regular Borel probability measure ν on K and a bounded linear operator C : L p (ν) L p (µ) such that We set = AC and then hus i p B = Cp : C(K) p L p (ν) C L p (µ) and C = π p (i p B). J Y = Ai p B = A(i p B)i = A(C p )i = p i. Lip( ) Lip(A) C = Lip(A)π p (i p B) Lip(A) B = Lip(A) B. his implies that sι L p ( ).

11 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 11 he following result characterizes Lipschitz p-dominated operators. heorem 3.2. For a Lipschitz mapping : Y the following are equivalent: (i) D L p (, Y ); (ii) there exists a constant C > 0 such that for any {x i } n i=1 and {y i} n i=1 in, we have ( n ) 1/p ( n ) 1/p x i y i p C x, x i y i p ; i=1 sup x B (iii) there exist a constant C > 0 and a probability measure µ on B i=1 such that for all x, y, we have x y C( B x, x y p dµ(x )) 1/p. In that case d L p ( ) = inf{c : C as in (ii)} = inf{c : C, µ as in (iii)}. Proof. (i) (ii). Choose any L π p (, Z) such that x y Lx Ly, x, y. hen, for any finite sequence {x i } n i=1 and {y i} n i=1 in, we have ( n ) 1/p ( n ) 1/p x i y i p Lx i Ly i p hus, hat is, i=1 (ii) (iii). Fix x, y. Define i=1 π p (L) sup x B ( n ) 1/p x, x i y i p. i=1 inf{c : C as in (ii)} π p (L). inf{c : C as in (ii)} d L p ( ). g {x,y} : B R by x x y p C p x, x y p. Let Q be the positive convex hull of {g {x,y} : x, y }, that is, Q = { i λ i g {xi,y i } : i λ i = 1, λ i > 0} and let P = {F C(B, R) : F (x ) > 0, x B }. hen 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 B and a real number c so that g dµ c < F dµ, g Q, F P. B B

12 12 DONGYANG CHEN AND BENUO 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 g {x,y}, we have x y p C p B x, x y p dµ 0. Moreover, inf{c : C, µ as in (iii)} inf{c : C as in (ii)}. (iii) (i). Define L : L p (B, µ) by x (Lx)(x ) = x, x, x B. hen x y C Lx Ly x, y and π p (L) 1. hus, d L p ( ) π p (C L) C. his implies d L p ( ) inf{c : C, µ as in(iii)}. Remark 3.1. In the above theorem, B can be replaced by any weak*-compact norming subset of B. heorem 3.2 clearly implies that a Lipschitz p-dominated operator is Lipschitz p-summing. In the nonlinear Pietsch domination theorem by Farmer and Johnson [FJ], B # is used for supremum and integral. However the geometric structure of # 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 B for the supremum and integral instead of B #. heorem 3.3. Let : Y be a Lipschitz mapping and let K be a weak* compact norming subset of B. he following are equivalent: (i) Dp L (, Y ); (ii) there exist a regular Borel probability measure µ on K, a closed subspace p of L p (µ) and a Lipschitz map ˆ : p Y such that (a) p i () p ; (b) ˆ p i =. In

13 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 13 other words, the following diagram commutes: i Y i () C(K) p i () p p L p (µ); (iii) there exist a regular Borel probability measure µ on K and a Lipschitz map L p (µ) l (B Y ) such that the following diagram commutes: i C(K) Y i Y l (B Y ) p L p (µ); (iv) there exist a probability space (Ω, Σ, µ) and a Lipschitz mapping ũ : L p (µ) l (B Y ) and a linear mapping v : L (µ) such that the the following diagram commutes v L (µ) Y i Y l (B Y ) ũ i p L p (µ). In addition, we may choose µ and ˆ in (ii) or µ and in (iii) so that Lip( ˆ ) = Lip( ) = d L p ( ); in (iv) we may arrange that v = 1 and Lip(ũ) = d L p ( ). Proof. (i) (ii): It follows from heorem 3.2 that there is a regular Borel probability measure µ on K such that for all x, y, we have x y d L p ( )( x, x y p dµ(x )) 1/p. Define a Lipschitz mapping u : p i () Y by u( p i (x)) = (x), x. hen B Lip(u) d L p ( ). We let p be the norm closure of p i () in L p (µ) and ˆ be the extension of u to p. hen For the converse, Lip( ˆ ) d L p ( ) and ˆ p i =. d L p ( ) = d L p ( ˆ p i ) Lip( ˆ ).

14 14 DONGYANG CHEN AND BENUO ZHENG hus d L p ( ) = Lip( ˆ ). (ii) (iii): Let µ, p and ˆ be as in (ii). Since l (B Y ) is an absolute 1-Lipschitz retract, there is a Lipschitz extension of i Y ˆ to Lp (µ) with Lip( ) = Lip(i Y ˆ ). hen is the required Lipschitz mapping. (iii) (iv): Let µ and be as in (iii). Note that Let u = i. hen Let v = Moreover, and hus p = i p : C(K) L (µ) ip L p (µ). i Y = i p u. u u and ũ = u. his implies that i Y = ũi p v. Lip(ũ) Lip( ) = d L p ( ) d L p ( ) = d L p (i Y ) = d L p (ũi p v) Lip(ũ)π p (i p v) Lip(ũ). v = 1 and Lip(ũ) = d L p ( ). (iv) (i): It follows from heorem 3.2 and the fact that i p v is p-summing and i Y isometric embedding. is an heorem 3.3 yields the following corollaries immediately. Corollary 3.4. Let K be a compact Hausdorff space. A Lipschitz mapping : 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 : L p (µ) Y such that p =. Moreover, we may arrange that Lip( ) = d L p ( ).

15 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 15 Corollary 3.5. A Lipschitz mapping : Y is Lipschitz 2-dominated if and only if there exist a regular probability measure µ on B and a Lipschitz mapping : L 2 (µ) Y such that the following diagram commutes: i C(B ) Y 2 L2 (µ). Moreover, we may arrange that Lip( ) = d L 2 ( ). heorem 3.6. If : Y is strongly Lipschitz p-integral, then it is Lipschitz p- dominated with d L p ( ) sι L p ( ). Proof. Given a typical strongly Lipschitz p-integral factorization: B L (µ) Y J Y Y A i p L p (µ). Since i p is p-summing and B is linear, it follows from the ideal property of Lipschitz p- dominated operators that J Y is Lipschitz p-dominated. Hence is also Lipschitz p- dominated and d L p ( ) = d L p (J Y ) = d L p (Ai p B) Lip(A) B. his implies d L p ( ) sι L p ( ). Combining the previous results, we have the following two immediate corollaries. Corollary 3.7. he 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 : C(K) Y is strongly Lipschitz p-integral with equality of norms.

16 16 DONGYANG CHEN AND BENUO ZHENG 4. Applications to Lipschitz mappings from a finite metric space into a Banach space his section deals with the relationship between Lipschitz p-integral and Lipschitz p- nuclear operators from a finite metric space into a Banach space. heorem 4.1. Let be a finite metric space and Y be a Banach space. hen, for any 1 p < and any mapping : Y, we have (a) ν L p ( ) = ι L p ( ); (b) ν L p ( ) = ι L p ( ) C (log ) 2 Lip( ), where C is an absolute constant. Proof. (a) Consider a typical p-integral factorization J Y : B L (µ) ip L p (µ) A Y. Fix ε > 0. hen, there exists a finite dimensional subspace E of L (µ), of dimension N say, together with an isomorphism ν : E l N such that ν ν ε and B() E. Moreover, i p (E) is contained in a (1 + ε)-complemented subspace F of L p (µ). Let J : F L p (µ) be the corresponding canonical embedding and P : L p (µ) F the corresponding proection with P 1 + ε. he principle of local reflexivity guarantees the existence of a linear operator W : span{ai p B()} Y such that hus, Consider W 1 + ε and W span{aipb()} J Y (Y ) = id JY (Y ). J Y = W J Y. ũ = P i p E ν 1 : l N F. hen ũ is p-summing and hence p-integral and π p (ũ) = ι p (ũ) = ν p (ũ) P ι p (i p ) ν 1 (1 + ε) ν 1. Note that hen J Y = W J Y = W Ai p B = W AJũνB.

17 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 17 νp L ( ) = νp L (J Y ) = νp L (W AJũνB) Lip(W AJ)ν p (ũ) Lip(νB) (1 + ε) 3 Lip(A) Lip(B). hus ν L p ( ) = ι L p ( ). (b) It follows from Farmer-Johnson Factorization heorem ([FJ])that i Y ũ i () C(K) p i () p p L p (µ), where = ũ p i and π L p ( ) = Lip(ũ). It follows from heorem 1 of [JLS] that there exists û : L p (µ) Y such that where C 1 is an absolute constant. Note that û p = ũ and Lip(û) (C 1 log ) Lip(ũ), p = i p i : C(K) L (µ) ip L p (µ). Let b = i. hen = ũ p i = ûi p b. hus νp L ( ) = ι L p ( ) Lip(b) Lip(û) (C 1 log )πp L ( ). Finally, as was mentioned in [FJ], J.Bourgain([B]) really proved that πp L (id ) π1 L (id ) (C 2 log ), where C 2 is an absolute constant. It follows from the ideal property of Lipschitz p-summing operators that πp L ( ) = πp L ( id ) (C 2 log ) Lip( ).

18 18 DONGYANG CHEN AND BENUO ZHENG his implies that ν L p ( ) = ι L p ( ) C 1 C 2 (log ) 2 Lip( ). he following result gives a better estimate on the constant of heorem 4.1 when the range space is a 2-uniformly convex Banach space and the cardinality of is relatively small compared to p. heorem 4.2. Let be a finite metric space and Y be a 2-uniformly convex Banach space. hen, for any 2 p < and for any mapping : Y, we have νp L ( ) 24K 2 (Y ) p 1 Lip( )( ) 1/p where K 2 (Y ) is the 2-uniform convexity constant of Y. Proof. Suppose that = {x 1,..., x n }. Consider the Fréchet map F : l n (n = ) given by F(x) = (d(x, x i )) n i=1. hen F is an isometric embedding. Define g : Id p F() Y by Id p F(x) (x), where Id p : l n l n p and Id p : l n p l n are the identity mappings. Note that Id p is p-nuclear with ν p (Id p ) = ι p (Id p ) = π p (Id p ) = n 1/p. It follows from heorem 1.1 ([MN]) that there exists g : l n p Y such that Lip( g) 24K 2 (Y ) p 1 Lip(g) 24K 2 (Y ) p 1 Lip( ), and g = g. Id p F() hen = g Id p F, and ν L p ( ) Lip( g)ν p (Id p ) 24K 2 (Y ) p 1 Lip( ) n 1/p.

19 LIPSCHIZ p-inegral OPERAORS AND LIPSCHIZ p-nuclear OPERAORS 19 Acknowledgements. his work is done during the first author s visit to Department of Mathematics, exas 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 [B] J.Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J.Math.52(1985),no.1-2, [BL] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Vol.1, Amer. Math. Soc. Colloq. Pub., Vol.48, Amer. Math. Soc., Providence, RI, [CZh1] D. Chen and B. Zheng, Remarks on Lipschitz p-summing operators, Proc. Amer. Math. Soc. 139(2011), no.8, [DJ] J. Diestel, H. Jarchow and A. onge, Absolutely summing operators, Cambridge Studies in Adv. Math., Vol.43, Cambridge Univ. Press, Cambridge, [FJ] J.D. Farmer and W.B. Johnson, Lipschitz p-summing operators, Proc. Amer. Math. Soc. 137 (2009), no.9, [G] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. Sao Paulo 8 (1956), [JLS] W.B.Johnson, J.Lindenstrauss and G.Schechtman, Extensions of Lipschitz maps into Banach spaces, Israel J.Math.54(1986),no.2, [JMS] W.B. Johnson, B. Maurey and G. Schechtman, Nonlinear factorization of linear operators, Bull. London Math. Soc. 41 (2009), [L] J.Lindenstrauss, On non-linear proections in Banach spaces, Michigan J.Math.11(1964) [MN] M.Mendel and A.Naor, Some applications of Ball s extension theorem, Proc. Amer. Math. Soc. 134 (2006), no.9, [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), [P2] A. Pietsch, Ideals of multilinear functionals (designs of a theory), Proceedings of the Second International Conference on Operator Algebras, Ideals, and their Applications in heoretical Physics (Leipzig), eubner-exte, 1983, pp [] N. omczak-jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & echnical, Harlow, School of Mathematical Sciences, iamen University, iamen,361005,china address: cdy@xmu.edu.cn Department of Mathematical Sciences, he University of Memphis, Memphis, N address: bzheng@memphis.edu

Non-linear factorization of linear operators

Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Non-linear factorization of linear operators W. B. Johnson, B. Maurey and G. Schechtman Abstract We show, in particular,