Lipschitz p-convex and q-concave maps

Transcription

1 Lipschitz p-convex and q-concave maps J. Alejandro Chávez-Domínguez Instituto de Ciencias Matemáticas, CSIC-UAM-UCM-UC3M, Madrid and Department of Mathematics, University of Texas at Austin Conference on Geometric Functional Analysis and its Applications Université de Franche-Comté, Besançon October 27 th, 2014 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

2 Ribe s program Theorem (Ribe) If two Banach spaces are uniformly homeomorphic, they are crudely finitely representable in each other. Ribe s program (as described by Mendel and Naor) The search for purely metric reformulations of basic linear concepts and invariants from the local theory of Banach spaces. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

3 Ribe s program Some examples of successes: 1 Superreflexivity (Bourgain, 1986). 2 Equivalent norm with modulus of convexity of power type p (Lee/Naor/Peres, 2009; Mendel/Naor, 2008). 3 Rademacher cotype (Mendel/Naor, 2008). 4 Rademacher type (Mendel/Naor, 2007). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

4 Ribe s program for maps The local theory of Banach spaces is not only concerned with the spaces but also with the morphisms between them, and moreover there is a rich interplay between the properties of the spaces and those of the morphisms. Nonlinear characterizations of linear properties for maps: 1 p-summing maps (Farmer/Johnson, 2009). 2 Factorization through a subspace of L p (Johnson/Maurey/Schechtman, 2009). 3 p-nuclear operators (Chen/Zheng, 2012). 4 Modulus of convexity of power type p up to a renorming of the domain (C). 5 Rademacher cotype (C). 6 Rademacher type (C). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

5 Banach lattices Banach lattice = Banach space + partial order compatible with the norm and the algebraic structure (think of a Banach space of functions or sequences, like L p [0, 1] or c 0 ). The important conditions: Both addition and multiplication by positive scalars preserve the order. For every x and y there exist a least upper bound x y and a greatest lower bound x y. x y whenever x y, where x = x ( x). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

6 The beauty of Banach lattices Paraphrasing Lindenstrauss and Tzafriri: this additional ingredient makes the theory of Banach lattices in some regards simpler, cleaner and more complete than the theory for general Banach spaces. For Banach lattices, Rademacher type, Rademacher cotype and having an equivalent uniformly p-convex norm are intimately related to the notions of convexity and concavity. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

7 Connections to other properties J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

8 The Krivine functional calculus for lattices In a space of functions, say C(K) for concreteness, we can consider expressions like ( n f j p) 1/p, f1,..., f n C(K). j=1 Krivine developed a functional calculus that allows us to make sense of this expression in any Banach lattice. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

9 Linear p-convex maps Consider 1 p. A linear map T : X E from a Banach space X to a Banach lattice E is called p-convex if there exists a constant M < such that for all v 1,..., v n X ( n Tv j p) 1/p ( n ) 1/p, M v j p X if 1 p < E j=1 j=1 or n Tv j M max v j 1 j n X, if p =. E j=1 The smallest such constant M is denoted M (p) (T). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

10 Linear p-convex maps Consider 1 p. A linear map T : X E from a Banach space X to a Banach lattice E is called p-convex if there exists a constant M < such that for all v 1,..., v n X ( n Tv j p) 1/p ( n ) 1/p, M v j p X if 1 p < E j=1 j=1 or n Tv j M max v j 1 j n X, if p =. E j=1 The smallest such constant M is denoted M (p) (T). IDEA: T induces a bounded map l p (X) E(l p ). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

11 Linear q-concave maps A linear map S : E Y from a Banach lattice E to a Banach space Y is called q-concave if there exists a constant M < such that for all v 1,..., v n E, ( n ) 1/q ( n Sv j q Y M v j q) 1/q, if 1 q < E j=1 j=1 or max Sv j 1 j n Y n v j, if q =. E j=1 The smallest such constant M is denoted M (q) (S). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

12 Linear q-concave maps A linear map S : E Y from a Banach lattice E to a Banach space Y is called q-concave if there exists a constant M < such that for all v 1,..., v n E, ( n ) 1/q ( n Sv j q Y M v j q) 1/q, if 1 q < E j=1 j=1 or max Sv j 1 j n Y n v j, if q =. E j=1 The smallest such constant M is denoted M (q) (S). IDEA: S induces a bounded map E(l q ) l q (Y). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

13 A factorization theorem (Nikishin/Maurey) Let E be a Banach space, (Ω, Σ, µ) a σ-finite measure space, 1 p < q <, T : E L p (µ) a linear map and 0 < C <. TFAE: (a) There exists a density function h on Ω and a linear map S : E L q (hdµ) with S C such that E S L q (hdµ) T i q,p L p (µ) j L p (hdµ) ( n ) 1/q (b) For every x 1,..., x n in E, Tx j q j=1 Lp(µ) ( n ) 1/q C x j q. j=1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

14 Factorization through L p Theorem (Krivine) Let E, F be Banach spaces and L a Banach lattice. Suppose that T : E L is p-convex and S : L F is p-concave. Then the operator ST can be factored through an L p (µ) space. Moreover, we may arrange to have ST = S 1 T 1 with T 1 : E L p (µ), S 1 : L p (µ) F, T 1 M (p) (T) and S 1 M (p) (S). E T L S F T 1 L p (µ) S 1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

15 Goals I did not try yet to obtain a fully nonlinear characterization of convexity/concavity. I started by working on a partially nonlinear situation, with maps between metric spaces and Banach lattices. I was aiming for nonlinear factorization theorems. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

16 Lipschitz p-convex maps Let 1 p. Let X be a metric space and E a Banach lattice. A Lipschitz map T : X E is called Lipschitz p-convex if there exists a constant C 0 for any x j, x j X and λ j 0, ( n ) 1/p ( λ j Tx j Tx j p n ) 1/p C λ j d(x j, x j) p, E j=1 (with the obvious adjustment if p = ). The smallest such constant C is called the Lipschitz p-convexity constant of T and is denoted by M (p) Lip (T). j=1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

17 Remark It would be nice to apply the usual Farmer/Johnson/Mendel/Schechtman argument and obtain that in the definition of Lipschitz p-convexity it suffices to consider λ j = 1, i.e. ( n ) 1/p ( Tx j Tx j p n ) 1/p C d(x j, x j) p. E j=1 j=1 That is indeed the case if the lattice has certain continuity properties, but let s avoid the technicalities. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

18 Nonlinear Maurey/Nikishin factorization (C) Let X be a metric space, (Ω, Σ, µ) a σ-finite measure space, 1 p < q <, T : X L p (µ) a Lipschitz map and 0 < C <. TFAE: (a) There exists a density function h on Ω and a Lipschitz map S : X L q (hdµ) with Lip(S) C such that X S L q (hdµ) T i q,p L p (µ) j L p (hdµ) (b) For every x j, x j X and λ j 0, ( n ) 1/q λ j Tx j Tx j q j=1 Lp(µ) ( n ) 1/q C λ j d(x j, x j) q. j=1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

19 Molecules and the Lipschitz-free space A molecule on a metric space X is m : X R such that m(x) = 0. Those of the form x X with x, x X are called atoms. m xx := χ {x} χ {x } The Lipschitz-free space of X is (the completion of) the space of molecules with the norm { n n } m F (X) := inf a j d(x j, x j) : m = a j m xj x j. F (X) = Lip 0 (X). j=1 j=1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

20 Universal property of Lipschitz-free spaces Theorem (Arens/Eells) Let X be a metric space with a designated point 0 X. The map ι : x m x0 is an isometric embedding of X into F (X). Moreover, for any Banach space E and any Lipschitz map T : X E with T(0) = 0 there is a unique linear map T : F (X) E such that T ι = T. Furthermore, T = Lip(T). X ι F (X) T T E J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

21 Universal property of Lipschitz-free spaces Theorem (Arens/Eells) Let X be a metric space with a designated point 0 X. The map ι : x m x0 is an isometric embedding of X into F (X). Moreover, for any Banach space E and any Lipschitz map T : X E with T(0) = 0 there is a unique linear map T : F (X) E such that T ι = T. Furthermore, T = Lip(T). X ι F (X) T T E NOTE: To evaluate the norm of the linear extension T, it suffices to look at the images of atoms. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

22 Relating the linear and nonlinear theorems Let X be a metric space, (Ω, Σ, µ) a σ-finite measure space, 1 p < q <, T : X L p (µ) a Lipschitz map. L q (hdµ) S X i q,p T L p (µ) j L p (hdµ) J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

23 Relating the linear and nonlinear theorems Let X be a metric space, (Ω, Σ, µ) a σ-finite measure space, 1 p < q <, T : X L p (µ) a Lipschitz map. F (X) ι Ŝ L q (hdµ) S T X i q,p T L p (µ) j L p (hdµ) J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

24 A consequence of the linear theorem Let X be a metric space, (Ω, Σ, µ) a σ-finite measure space, 1 p < q <, T : X L p (µ) a Lipschitz map and 0 < C <. TFAE: (a) There exists a density function h on Ω and a Lipschitz map S : X L q (hdµ) with Lip(S) C such that X S L q (hdµ) T i q,p L p (µ) j L p (hdµ) (b) For every m 1,..., m n F (X), ( n ) 1/q ( ˆTm j q n ) 1/q C m j q F (X), where j=1 Lp(µ) j=1 ˆT : F (X) L p (µ) is the linearization of T. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

25 Putting everything together Corollary (C) Let X be a metric space, (Ω, Σ, µ) a σ-finite measure space, 1 p < q <, T : X L p (µ) a Lipschitz map and 0 < C <. TFAE: (a) For every m 1,..., m n F (X), ( n ) 1/q ˆTm j q j=1 Lp(µ) ( n ) 1/q C m j q F (X), j=1 where ˆT : F (X) L p (µ) is the linearization of T. (b) For every x j, x j X and λ j 0, ( n ) 1/q λ j Tx j Tx j q j=1 Lp(µ) ( n ) 1/q C λ j d(x j, x j) q. j=1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

26 General equivalence Theorem (C) Let X be a metric space and E a Banach lattice. A Lipschitz map T : X E is Lipschitz p-convex if and only if ˆT : F (X) E is p-convex. Moreover, in this case the p-convexity constants are the same. X ι F (X) T T E J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

27 General equivalence Theorem (C) Let X be a metric space and E a Banach lattice. A Lipschitz map T : X E is Lipschitz p-convex if and only if ˆT : F (X) E is p-convex. Moreover, in this case the p-convexity constants are the same. X ι F (X) T T E The if part is trivial: p-convexity of ˆT clearly implies Lipschitz p-convexity of T with no increment in the constant, since m xx F (X) = d(x, x ) and ˆTm xx = Tx Tx. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

28 The proof Now suppose that T is Lipschitz p-convex. The strategy of the proof will be to show that T : E F (X) = Lip 0 (X) is p -concave. Let ϕ j E be arbitrary. For any x j, x j X with x j x j we have ( ϕ j, Tx j Tx j p ) 1/p d(x j, x j ) j = sup j α j p 1 j α j ϕ j, Tx j Tx j d(x j, x j ). Using Hölder s inequality for lattices, the latter is bounded by (( ) 1/p )(( sup ϕ j α j p j p 1 j j ( ) 1/p ϕ j p sup j L j α j p 1 Txj α j p Tx j p d(x j, x j )p ( j ) 1/p ) α j p Tx j Tx j p d(x j, x j )p ) 1/p E J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

29 The proof The Lipschitz p-convexity of T allows us to bound this by ( ) 1/p ( ϕ j p M (p) Lip j (T) sup α j p d(x j, x ) j )p 1/p E j α j p 1 d(x j j, x j )p ( = M (p) Lip (T) ) 1/p ϕ j p Therefore, ( (ˆT ϕ j )(x j) (ˆT ϕ j )(x j ) p ) 1/p d(x j, x j ) M (p) j Lip (T) j E ( ) 1/p ϕ j p so taking the supremum over all pairs x j, x j X with x j x j we conclude ( ) ˆT ϕ 1/p ( p j M (p) Lip Lip (T) ) 1/p ϕ j p. j j E J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48 j. E,

30 The proof Since the ϕ j L were arbitrary, this means that ˆT : L Lip 0 (X) is p -concave with M (p )(ˆT ) M (p) Lip (T), and by duality ˆT : F (X) L is p-convex with M (p) (ˆT) M (p) Lip (T). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

31 Remarks For a moment, one could think that in particular we have a result in the spirit of the Godefroy/Kalton theorem for the BAP, that is, for a Banach lattice E E is p-convex F (E) is p-convex. However, what we do have is id E : E E is p-convex id E : F (E) E is p-convex. Because of the role played by duality, it seems unlikely that these ideas could be used to prove a more similar result for other classes of operators obtained by replacing the expression ( j x j p ) 1/p by other homogeneous functions given by the Krivine functional calculus for Banach lattices. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

32 Linear p-concave maps A linear operator S : L E from a Banach lattice L to a Banach space E is called p-concave if there exists a constant M < such that for all v 1,..., v n L ( n ) 1/p ( n Sv j p E M v j p) 1/p, if 1 p < L j=1 j=1 or max Sv j 1 j n E n v j, if p =. L j=1 The smallest such constant M is denoted M (p) (T). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

33 Lipschitz p-concave maps Let X be a metric space and L a Banach lattice. A Lipschitz map T : L X is called Lipschitz p-concave if there exists a constant C 0 such that for any v j, v j L, ( n ) 1/p ( d(tv j, Tv j) p n ) 1/p C v j v j p. L j=1 The smallest such constant C is the Lipschitz p-concavity constant of T and is denoted by M Lip (p) (T). NOTE: In the case of Lipschitz concavity we can always add constants to the inequality. j=1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

34 Factorization through L p Theorem (Krivine) Let E, F be Banach spaces and L a Banach lattice. Suppose that T : E L is p-convex and S : L F is p-concave. Then the operator ST can be factored through an L p (µ) space. Moreover, we may arrange to have ST = S 1 T 1 with T 1 : E L p (µ), S 1 : L p (µ) F, T 1 M (p) (T) and S 1 M (p) (S). E T L S F T 1 L p (µ) S 1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

35 Can we get something nonlinear easily? Suppose that T : X L is Lipschitz p-convex and S : L Y is Lipschitz p-concave. X T L S Y J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

36 Can we get something nonlinear easily? Suppose that T : X L is Lipschitz p-convex and S : L Y is Lipschitz p-concave. X T L S Y ι X F (X) T ι Y F (Y) J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

37 Can we get something nonlinear easily? Suppose that T : X L is Lipschitz p-convex and S : L Y is Lipschitz p-concave. X T L S Y ι X F (X) T ι Y F (Y) The map ι Y S is not linear! J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

38 Nonlinear factorization through L p Theorem (C) Let X, Y be metric spaces with Y complete and L a Banach lattice. Suppose that T : X L is Lipschitz p-convex and S : L Y is Lipschitz p-concave. Then the operator ST can be factorized through an L p (µ) space. Moreover, we may arrange to have ST = S 1 T 1 with T 1 : X L p (µ), S 1 : L p (µ) Y, Lip(T 1 ) M (p) Lip Lip(S 1 ) M Lip (p) (S). X T L S Y (T) and T 1 L p (µ) S 1 J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

39 Magic? So far we have obtained nice results, but it s not quite clear why they work. The situation will be greatly clarified thanks to the factorization theory of Raynaud and Tradacete. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

40 Factoring weakly compact operators Easy way to construct a weakly compact operator: go through a reflexive space. X Y linear Z reflexive linear J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

41 Factoring weakly compact operators Easy way to construct a weakly compact operator: go through a reflexive space. X w-compact Y linear Z reflexive linear J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

42 Factoring weakly compact operators Easy way to construct a weakly compact operator: go through a reflexive space. X w-compact Y linear Z reflexive linear Theorem (Davis-Figiel-Johnson-Pelczynski) This is the only way. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

43 Factoring p-convex maps Easy way to construct a p-convex linear map: go through a p-convex Banach lattice. X L W p convex J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

44 Factoring p-convex maps Easy way to construct a p-convex linear map: go through a p-convex Banach lattice. X p convex L linear W p convex positive J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

45 Factoring p-convex maps Easy way to construct a p-convex linear map: go through a p-convex Banach lattice. X p convex L linear W p convex positive Theorem (Raynaud/Tradacete) This is the only way. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

46 Factoring p-convex maps Theorem (Raynaud/Tradacete) Let L be a Banach lattice, E a Banach space and 1 p. A linear operator T : E L is p-convex if and only if there exist a p-convex Banach lattice W, a positive operator (in fact, an injective interval preserving lattice homomorphism) ψ : W L and another linear operator R : E W such that T = ψr. E T L R W ψ Moreover, M (p) (T) = inf R M (p) (I W ) ψ. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

47 Factorization characterization of Lipschitz p-convexity Theorem (C) Let L be a Banach lattice, X a metric space and 1 p. A Lipschitz map T : X L is Lipschitz p-convex if and only if there exist a p-convex Banach lattice W, a positive operator (in fact, an injective interval preserving lattice homomorphism) ψ : W L and another Lipschitz map R : E W such that T = ψr. X T L R W ψ Moreover, M (p) Lip (T) = inf Lip(R) M(p) (I W ) ψ. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

48 Factorization characterization of Lipschitz p-convexity Theorem (C) Let L be a Banach lattice, X a metric space and 1 p. A Lipschitz map T : X L is Lipschitz p-convex if and only if there exist a p-convex Banach lattice W, a positive operator (in fact, an injective interval preserving lattice homomorphism) ψ : W L and another Lipschitz map R : E W such that T = ψr. X T L R W ψ Moreover, M (p) Lip (T) = inf Lip(R) M(p) (I W ) ψ. NOTE: This theorem implies the linear one. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

49 A proof without duality Suppose that T : X L is Lipschitz p-convex. X R T W ψ L J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

50 A proof without duality Suppose that T : X L is Lipschitz p-convex. F (X) X T L ι R R W ψ J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

51 A proof without duality Suppose that T : X L is Lipschitz p-convex. F (X) X T L ι R R Hence T = ψ R is p-convex and with the same constant. W ψ J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

52 Factorization characterization of q-concavity Theorem (Reisner; Raynaud/Tradacete) Let L be a Banach lattice, E a Banach space and 1 q. A linear operator T : L E is q-concave if and only if there exist a q-concave Banach lattice V, a positive operator φ : L V (in fact, a lattice homomorphism with dense image), and another operator S : V E such that T = Sψ. Moreover, M (q) (T) = inf φ M (q) (I V ) S. L φ T V S E J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

53 Factorization characterization of Lipschitz q-concavity Theorem (C) Let L be a Banach lattice, X a complete metric space and 1 q. A Lipschitz map T : L X is Lipschitz q-concave if and only if there exist a q-concave Banach lattice V, a positive operator φ : L V (in fact, a lattice homomorphism with dense image), and another Lipschitz map S : V X such that T = Sφ. L T X φ V S Moreover, M Lip (q) (T) = inf φ M (q)(i V ) Lip(S). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

54 Nonlinear factorization through L p revisited Lemma (Raynaud/Tradacete) If W, V are quasi-banach lattices with W p-convex and V p-concave, then every lattice homomorphism h : W V factors trough some L p (µ), and the factors are lattice homomorphisms. The nonlinear Krivine theorem is an easy consequence of the lemma and our characterizations: if T 1 : X L is Lipschitz p-convex and T 2 : L Y is Lipschitz p-concave (with Y complete), X T 1 L T 2 Y R W ψ φ V S J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

55 Nonlinear factorization through L p revisited Lemma (Raynaud/Tradacete) If W, V are quasi-banach lattices with W p-convex and V p-concave, then every lattice homomorphism h : W V factors trough some L p (µ), and the factors are lattice homomorphisms. The nonlinear Krivine theorem is an easy consequence of the lemma and our characterizations: if T 1 : X L is Lipschitz p-convex and T 2 : L Y is Lipschitz p-concave (with Y complete), X T 1 L T 2 Y R W ψ φ V S L p (µ) J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

56 Nonlinear factorization through L p L q Corollary (C) Now if T 1 : X L is Lipschitz p-convex and T 2 : L Y is Lipschitz q-concave, with Y complete and 1 p < q, then T 2 T 1 Lipschitz factors through a canonical inclusion i p,q : L p (µ) L q (µ). X R L p (µ) T 1 L T 2 Y S i p,q L q (µ) J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

57 More related results Proposition Let X, Y be metric spaces with Y complete and E a Banach lattice, and 1 p, q. Suppose that T : X E is Lipschitz p-convex and S : E Y is Lipschitz q-concave. Then for every θ (0, 1), ST factors through a Banach lattice U θ that is Corollary p p(1 θ)+θ -convex and q 1 θ -concave. Let E be a Banach lattice, 1 p, q, and assume that T : E E is both Lipschitz p-convex and Lipschitz q-concave. Then for each θ (0, 1), T 2 p factors through a p(1 θ)+θ -convex and q 1 θ -concave Banach lattice. In particular, if p > 1 and q < then T factors through a super reflexive Banach lattice. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

58 Lipschitz factorization implies linear factorization If a linear map T : X Y between Banach spaces can be factored as a Lipschitz p-convex map followed by a Lipschitz q-convex one, is there a factorization where the factor maps are in addition linear? J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

59 Lipschitz factorization implies linear factorization If a linear map T : X Y between Banach spaces can be factored as a Lipschitz p-convex map followed by a Lipschitz q-convex one, is there a factorization where the factor maps are in addition linear? Theorem (C) Let T : X Y be a linear map between a Banach space X and a dual Banach space Y, and assume that T admits a factorization T = T 2 T 1 where T 1 is Lipschitz p-convex and T 2 is Lipschitz q-concave, with 1 q < p <. Then there is also a factorization T = τ 2 τ 1 where τ 1 is p-convex and τ 2 is q-concave, and moreover M (p) (τ 1 ) M (p) Lip (T 1) and M (q) (τ 2 ) M Lip (q) (T 2). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

60 Lipschitz factorization implies linear factorization If a linear map T : X Y between Banach spaces can be factored as a Lipschitz p-convex map followed by a Lipschitz q-convex one, is there a factorization where the factor maps are in addition linear? Theorem (C) Let T : X Y be a linear map between a Banach space X and a dual Banach space Y, and assume that T admits a factorization T = T 2 T 1 where T 1 is Lipschitz p-convex and T 2 is Lipschitz q-concave, with 1 q < p <. Then there is also a factorization T = τ 2 τ 1 where τ 1 is p-convex and τ 2 is q-concave, and moreover M (p) (τ 1 ) M (p) Lip (T 1) and M (q) (τ 2 ) M Lip (q) (T 2). Remark: When q = 2 the theorem holds for a general Banach space Y, because in a Hilbert space all subspaces are 1-complemented. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

61 Factorizations for just one map What can we say if a linear operator T : E F between Banach lattices is both p-convex and q-concave? Does it factor as T 2 T 1 with T 2 p-convex and T 1 q-concave? (Note that such a product is always p-convex and q-concave) φ E E q T R F F p ϕ where φ and ϕ are positive linear maps, E q is q-concave, F p is p-convex and R is a bounded linear map. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

62 Factorizations for just one map What can we say if a linear operator T : E F between Banach lattices is both p-convex and q-concave? Does it factor as T 2 T 1 with T 2 p-convex and T 1 q-concave? (Note that such a product is always p-convex and q-concave) φ E E q T R F F p ϕ where φ and ϕ are positive linear maps, E q is q-concave, F p is p-convex and R is a bounded linear map. Answer (Raynaud/Tradacete) No. Example: for 1 < q < p <, the formal inclusion L p (0, 1) L q (0, 1) does not admit such a factorization. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

63 Nonlinear implies linear again Theorem (C) Let T : E F be a linear map between Banach lattices E and F, and assume that T admits a factorization T = T 2 T 1 where T 1 is Lipschitz q-concave and T 2 is Lipschitz p-convex. Then there is also a factorization T = τ 2 τ 1 where τ 1 is q-concave and τ 2 is p-convex, and moreover M (p) (τ 2 ) M (p) Lip (T 2) and M (q) (τ 1 ) M Lip (q) (T 1). J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

64 Nonlinear implies linear again Theorem (C) Let T : E F be a linear map between Banach lattices E and F, and assume that T admits a factorization T = T 2 T 1 where T 1 is Lipschitz q-concave and T 2 is Lipschitz p-convex. Then there is also a factorization T = τ 2 τ 1 where τ 1 is q-concave and τ 2 is p-convex, and moreover M (p) (τ 2 ) M (p) Lip (T 2) and M (q) (τ 1 ) M Lip (q) (T 1). Corollary A map that is both Lipschitz p-convex and Lipschitz q-concave does not necessarily admit a factorization as a Lipschitz q-concave map followed by a Lipschitz p-convex one. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

65 Giving up a little Theorem (Raynaud/Tradacete) Suppose that a linear operator T : E F between Banach lattices is p-convex and q-concave, with 1 < p and 1 q <. Then for p every θ (0, 1), T = T 2 T 1 where T 2 is p θ = θ+(1 θ)p -convex and T 1 is q θ = q 1 θ -concave. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

66 Giving up a little Theorem (Raynaud/Tradacete) Suppose that a linear operator T : E F between Banach lattices is p-convex and q-concave, with 1 < p and 1 q <. Then for p every θ (0, 1), T = T 2 T 1 where T 2 is p θ = θ+(1 θ)p -convex and T 1 is q θ = -concave. In fact, there is a factorization q 1 θ E φ E θ T R F F θ ϕ where φ and ϕ are positive linear maps, E θ is q θ -concave, F θ is p θ -convex and R is a bounded linear map. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

67 The nonlinear situation is unclear Question Suppose that a Lipschitz map T : E F between Banach lattices is Lipschitz p-convex and Lipschitz q-concave, with 1 < p and 1 q <. Can we find 1 < p 0 < p and q < q 0 < so that there is a factorization of T as T E F φ E 0 where φ and ϕ are positive linear maps, E 0 is q 0 -concave, F 0 is p 0 -convex and R is a Lipschitz map? Moreover: given θ (0, 1), can p we have p 0 = θ+(1 θ)p and q 0 = q 1 θ? R F 0 ϕ J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48

68 Challenges 1 The proof of the linear result cannot be easily adapted to the Lipschitz context. 2 The arguments in that proof are heavily based on complex interpolation because that method works very well for lattices. 3 Complex interpolation, however, is not well suited to work with Lipschitz maps. 4 The results available require strong extra assumptions due to the fact that a Lipschitz function generally does not preserve analyticity. J.A. Chávez-Domínguez (UT Austin) Lipschitz p-convex and q-concave maps 10/27/ / 48