Lipschitz ( q , p ) -Summing Maps from C ( K ) -Spaces to Metric Spaces

Variants of the notion of ( q , p ) -summing operator are introduced in the setting of Lipschitz mappings acting between metric spaces. Some classes of these operators from C ( K ) -spaces to metric spaces are studied. An integral domination estimate is proved for a class of the mentioned Lipschitz ( q , p ) -summing maps. It is shown that under some conditions this domination is equivalent to ( q , 1 ) -summability of these Lipschitz maps. As an application, we recover Pisier’s result, which provides this equivalence in the setting of the linear operators.


Introduction
The formal study of the class of absolutely p-summing linear operators began in 1967 with the seminal work of Pietsch [10], and they have become an important tool for the study of summability in Banach spaces. The notion of (q, p)-summing operator due to Mityagin and Pełczyński [9] completes the previous one, and is also fundamental The smallest such constant C is the Lipschitz (q, p)-summing constant of and is denoted by π L q, p ( ). The space of all Lipschitz (q, p)-summing maps from M into N is denoted by L q, p (M, N ). In the case p = q, we recover the space L p (M, N ) of Lipschitz p-summing maps introduced by Farmer and Johnson in [6]. Note that directly from the definition, it is clear that the Lipschitz (q, p)-summing norm has the ideal property: whenever the composition makes sense. Similarly as in [6] a natural question of general nature appears: What results about (q, p)-summing operators have analogs for Lipschitz (q, p)-summing maps in the setting of metric spaces ? We point out that in the theory of linear operators involving Banach lattices, two of the most important classes of such operators are the (q, p)-convex and (q, p)concave ones (see [5, pp. 326-345]). Pisier's famous theorem for (q, 1)-summing linear operators from C(K )-spaces to Banach spaces plays a key role in the linear theory of (q, p)-concave operators, which is deeply connected with the linear theory of (q, p)-summing operators (see [5, pp. 326-345]). This motivates to study nonlinear counterparts of these two concepts, considering Lipschitz maps from Banach lattices to metric spaces. Let E be a Banach lattice and (N , d N ) a metric space. Given 1 ≤ p ≤ q < ∞, a Lipschitz map : E → N is called Lipschitz (q, p)-concave if, for any choice of finitely many elements x 1 , y 1 , . . . , x n , y n in E, one has n k=1 d N ( x k , y k ) q 1/q ≤ C n k=1 |x k − y k | p 1/ p E .
The least constant C that satisfies the above estimate is denoted by K L q, p ( ). The relevant paper [6] started with the analysis of Lipschitz p-summing mappings, showing that under certain mild requirements, an integral domination is equivalent to the natural version of p-summing operators for Lipschitz maps. In the current paper, we study some classes of (q, 1)-summing-(q, 1)-concave-type inequalities for Lipschitz maps on C(K )-spaces. This is inspired by the above-mentioned Pisier result, which allows to reduce the questions about (q, p)-summing operators from C(K )-spaces to the canonical embedding of C(K ) into Lorentz spaces L q,1 (μ). To be more precise, Pisier's theorem states that if a linear operator T : C(K ) → Y is (q, 1)summing, where Y is a Banach space, then there exist a Radon probability measure μ ∈ M(K ) and a constant γ ≤ q 1/q such that Here we let L q,1 (μ) denote the Lorentz space over (K , μ) equipped with the norm where f * denotes the non-decreasing rearrangement of | f |.
The primary purpose of this paper is to establish a general approach for understanding concavity inequalities involving different exponents for a class of Lipschitz mappings acting from C(K )-spaces to metric spaces. In technical terms, this means to find a natural version of Pisier's theorem for some class of Lipschitz mappings. Our main reference is Pisier's proof from the mentioned paper [13].

Lipschitz (q, p)-Summing Maps from C(K )-Space
Let (M, d M ) and (N , d N ) be metric spaces and let W ⊂ B M # be a compact subset with respect to the topology of the pointwise convergence on M.
In the case where M is a Banach space and W = B M * , then is said to be Lipschitz (q, p)-B M * -summing (or simply Lipschitz (q, p)-summing by abuse of notation if the context is clearly fixed). That is, such mapping satisfies for all choices of finite sets of vectors an inequality as for a given constant C. The smallest such constant is denoted by π ( * ) q, p ( ). We list some easily verified properties, which allow to generate examples of Lipschitz (q, p)-summing maps. Let X , Y be Banach spaces and M, N be metric spaces. Then the following statements are true: summing if and only if it is (q, p)-summing in the classical sense. • If S : X → Y is a linear operator and : Y → M is a Lipschitz (q, p)-B Y *summing map, then S : X → M is a Lipschitz (q, p)-B X * -summing map with • If X is rearrangement invariant space on the Lebesgue measure space [0, 1] (or in general on an atomless probability measure space (K , μ), K a compact Hausdorff space) with the fundamental function φ X (t) = t 1/q for all t ∈ [0, 1], then the inclusion map id : Recall that a Banach space X has (Rademacher) cotype r ∈ [2, ∞] if there is a constant C > 0 such that no matter how we select finitely many vectors x 1 , . . . , Here, as usual {r k } ∞ k=1 denotes the sequence of Rademacher functions on [0, 1] given by r k (t) = sign(sin 2 k π t) for all t ∈ (0, 1).
Worth noticing that a linear operator T : X → Y acting between Banach spaces is (q, p)-summing in the classical sense if and only if : X → R + given by x := T x Y for all x ∈ X is a Lipschitz (q, p)-B X * -summing map. In particular, if (X , · ) is a Banach space, then the norm · : X → R + is a Lipschitz (q, 1)-B X * -summing map if and only if the identity id X is (q, 1)-summing. Combining this fact with [5, Corollary 16.8], we conclude that if 2 < q < ∞ and (E, · ) is a Banach lattice, then the Lipschitz map · : L → R + is (q, 1)-B X * -summing if and only if E has Rademacher cotype q.
Let us fix our attention on the case of maps acting in C(K )-spaces, which is the main concern of the present work. Clearly, : C(K ) → (N , d N ) is Lipschitz (q, p)-B C(K ) * -summing map whenever there is a positive constant γ such that, for any choice of finitely many functions f 1 , g 1 , . . . , f n , g n in C(K ) one has Thus is Lipschitz (q, p)-B C(K ) * -summing if and only if is Lipschitz (q, p)concave, and the constants π ( * ) q, p and K L q, p coincide. As it was mentioned in the Introduction, the aim of this paper is to find variants of Pisier's theorem for some class of Lipschitz (q, 1)-B C(K ) * -summing maps from C(K )-spaces to metric spaces. It should be pointed out that the transition is not at all direct. We devote the last part of the present section to explain what are the main obstacles that are found to extend Pisier's result to the Lipschitz case.
First notice that the analysis of the proof of Pisier's result (see for example Equation (5) in the proof of Theorem 10.8 in [5]) shows that for every q ∈ [1, ∞) there is a constant γ (q) ≤ q such that the following statements about a linear operator T from C(K ) to a Banach space Y are equivalent: (iii) There exists a probability measure μ ∈ M(K ) such that In fact, Pisier's factorization theorem for linear operators can be written as the equivalence among these statements. Indeed, in the linear case, the implications from (ii) and (iii) to (i) are direct; we will see that this is not the case for Lipschitz maps. For a linear operator, the implication (ii)⇒(i) is only a consequence of a straightforward calculation, whereas proving (i)⇒(ii) requires an elaborate argument; the statements (ii) and (iii) are obviously equivalent. In the case of Lipschitz maps, the implication (i)⇒(ii) is not true in general, as the following lemma shows. Lemma 2.1 Let X be an infinite-dimensional Banach space and suppose K is an infinite compact Hausdorff space. Then, for every q ∈ [1, ∞) there exists a Lipschitz (q, 1)-concave map : C(K ) → X , for which there is no probability Borel measure μ on K such that for some C > 0 one has Proof We use Benyamini and Sternfeld result [2], which states that for any infinitedimensional Banach space X there exists a Lipschitz retraction R from the unit ball B X onto the unit sphere S X . It is a known fact that the map P : X → B X given by is a Lipschitz projection with Lip(P) ≤ 2. Then := R • P is a Lipschitz projection from X onto S X . We claim that if S : C(K ) → X is any map and := • S, then there is no probability Borel measure μ on K such that for some C > 0 and q ∈ [1, ∞), one has Indeed, since C(K ) is an infinite-dimensional space, it contains an isomorphic copy of c 0 . This implies that there exists a weakly null sequence ( f n ) in C(K ) such that f n ∞ = 1 for each n ∈ N. In particular, for every t ∈ K , one has f n (t) → 0 as n → ∞. By Lebesgue's Dominated Convergence Theorem, it follows that for any probability Borel measure μ on K , f n X = 1 for each n ∈ N and so the claim is proved. Taking any linear (q, 1)-concave operator S : C(K ) → X , we conclude that = • S is a Lipschitz (q, 1)-concave map too. This completes the proof.
We also remark that a mapping T : C(K ) → M satisfying the integral domination provided by (iii) of Pisier's equivalent statements is not necessarily Lipschitz, as we will show below, while being (q, 1)-concave obviously implies being Lipschitz. Let us present an example showing that in general the metric variant of condition (iii) does not imply that the map is Lipschitz. Therefore, in particular, it is not (q, 1)-concave. It must be said that a direct computation shows that (iii) implies that the (q, 1)-concavity inequality is satisfied with a uniform constant whenever the functions f k , The problem, however, arises when this requirement is not imposed on the functions.
Fix 0 < r < 1 and let μ be a probability regular Borel measure on a compact Hausdorff space K . Then the space L r (μ) of Lebesgue r -integrable functions is a complete metric space endowed with the distance d r given by Letting q := 1/r , we conclude by Jensen's inequality that for all f , g ∈ C(K ), In particular, this shows that the inclusion map j : is not a Lipschitz map. Otherwise, there would exist a constant γ > 0 such that, for every λ > 0 and all f , g ∈ C(K ) Taking g = 0 and f = 1, we get a contradiction for λ → 0.
However, we point out that a positive result can also be obtained regarding the implication (iii)⇒(i), as the next lemma shows. The key point is to consider a sort of "homogenization" of the composition of d and T instead of the composition. In fact, if T is (sub-)homogeneous (see the definition in the next section), the result below holds for ρ = d.
Suppose that there exists a probability measure μ ∈ M(K ) and a constant γ > 0 such that, for some q ∈ [1, ∞), one has Then, for every metric ρ on M which satisfies In particular, it is a Lipschitz map which satisfies the above integral domination formula, with the same q too.
Proof The proof is given by a straightforward calculation. Take This shows that : The assertion on the integral domination is also direct.

Pisier's Result for Lipschitz Maps
In this section, we prove our main result about Lipschitz (q, 1)-concave maps and integral dominations, and we show some consequences. Given any metric space (M, d), we start by introducing some general concepts and metric notations having roots in the Hahn-Banach Theorem, in the setting of normed spaces.
A basic example of a metric attaining set is In the case of M := C(K )-space, we will need more specific definitions.
there is an element ϕ f ,g ∈ W such that the following two conditions are satisfied We will consider a special family of Lipschitz maps from C(K )-spaces to a metric space (M, d).
• As usual, a Lipschitz map : and λ ≥ 0. We will need also a weaker version of this notion; a Lipschitz map and in addition, for all f 1 , f 2 , g 1 , g 2 ∈ C(K ), we have We provide examples of Lipschitz maps from C(K ) space into metric spaces which are sub-homogeneous or metric sublinear. • Let X be a normed space and (E, ≤) a normed vector lattice. Assume that p : X → E is a sublinear map, that is, for all x, y ∈ X , λ ∈ R, p satisfies the following properties: Combining with the estimate ( * ), we conclude that if additionally p : To show a concrete example of sublinear map, we may take X = E := L q (R n ) with 1 < q ≤ ∞ and p := M, where M is the Hardy-Littlewood maximal operator defined for all f ∈ L 1 loc (R n ) by where B(x, r ) := {y ∈ R n ; y − x < r } is the open ball in R n equipped with the Euclidean norm · , while |B(x, r )| is the Lebesgue measure of the ball B(x, r ). It is well known that M is bounded in L q (R n ) for all q ∈ (1, ∞]. As explained above, we get that := M • T : C(K ) → L q (R n ) is a sub-homogeneous Lipschitz map for any bounded linear operator T : • Direct examples of sub-homogeneous and metric sublinear Banach-space-valued maps are the affine maps, that is, Lipschitz maps that are defined as an addition of a fixed point and a linear operator. More precisely, let X be a Banach space, T : C(K ) → X a bounded linear operator, and let x 0 ∈ X . Then it is obvious that the affine map : C(K ) → X defined by is both a sub-homogeneous and metric sublinear Lipschitz map with Lip( ) = T . • Let (X , · ) be a normed space, and let Y be an arbitrary nonempty set. Given a one-to-one map ϕ : X → Y , we define on M := ϕ(X ) a metric d induced by ϕ and given by d(x, y) Given a bounded linear operator T : C(K ) → X , we define a map : It is clear that is a sub-homogeneous and metric sublinear Lipschitz map with Lip = T . • It appears a natural question to ask whether there is a suitable characterization of Lipschitz metric sublinear maps from C(K )-space to any metric space. Below we provide such a characterization, and we will show that the mentioned question is closely related with a specific construction of metrics d in the space C(K ) such that the identity i : In what follows, for simplicity of presentation, we assume that (M, d) is a metric space and : C(K ) → M a Lipschitz map. We define the functional ρ : We are ready to state the following result.
Moreover, if ρ is not a metric, the results above remain true if we consider the quotient metric space C(K )/ρ induced by ρ instead of (C(K ), ρ).
Proof It is easy to verify that ρ is a semimetric, so we skip the proof. For (a), fix f 1 , f 2 , g 1 , g 2 ∈ C(K ) and let ε > 0. From the definition of ρ, it follows that we can find in C(K ) representations of Then, by Since ε was arbitrary, it follows that the map i : C(K ) → (C(K ), ρ) is metric sublinear.
The proof of (b) is obvious. For (c), assume that : C(K ) → X is metric sublinear. Clearly, for any representation in This implies that is, is Lipschitz with Lipschitz constant equal to 1. The same computations but argued in the opposite direction prove the converse implication. Indeed, take f 1 , f 2 , g 1 , g 2 ∈ C(K ). Then, for f = f 1 + f 2 and g = g 1 + g 2 , we have f − g = ( f 1 − g 1 ) + ( f 2 − g 2 ) and so, by our hypothesis that : (C(K ), ρ) → X is Lipschitz with Lipschitz constant equal to 1, we get as required. The verification of the last statement of the result is straightforward.
We conclude our discussion with the following remark. It can be easily seen that if a Lischitz map : C(K ) → X is (q, p)-concave and ρ defined in Lemma 3.1 is a metric, then the identity map i : C(K ) → (C(K ), ρ) is also (q, p)-concave. Indeed, if there exists C > 0 such that for all f 1 , g 1 , . . . , f n , g n ∈ C(K ), then one has This obvious observation, in combination with the below-proven version of Pisier's theorem, seems to have an interesting application. In fact, starting from a (q, p)concave sub-homogeneous Lipschitz map, we obtain a method to provide a subhomogeneous metric sublinear identity map that is also (q, p)-concave. The integral domination given by the following results for this map can also be translated in terms of the original Lipschitz map.
We are now ready to prove a version of Pisier's theorem for some class of Lipschitz (q, p)-summing maps from a C(K )-space to a metric space (M, d). For simplicity, we present the result when K L q, p (T ) = 1. The result can be formulated in the case when K L q, p (T ) is not equal to one, introducing a new metric d/K L q, p (T ) in the space M and modifying the assumptions about the metric attaining set.

Theorem 3.2 Let 1 ≤ p < q < ∞, let (M, d) be a metric space and let : C(K ) → (M, d) be a Lipschitz sub-homogeneous (q, p)-concave map with K L q, p (T ) = 1. Suppose that W ⊂ C(K ) # is a τ p -compact subset that contains a C(K )-metric attaining set of subadditive (and monotone in the case p > 1) functions for the map . Then there exists a Borel measure
Proof The proof is shown in several steps: (A). Since K L q, p ( ) = 1, for each n ∈ N, we can find the least constant C n such that, no matter how we select Clearly, C n ≤ 1 and lim n C n = 1. Moreover, if for each n ∈ N we let γ (n) := 1 + 1 n , then we can choose We claim that without loss of generality, we may assume that f 1 , h 1 , . . . , f n , h n ∈ C(K ) are such that ( n k=1 | f k − h k | p ) 1/ p ∞ = 1. To see this, suppose that λ : 1). Consider functions f k and h k ∈ C(K ) given by Since the map is sub-homogeneous, it follows that Combining the above estimates yields and so this proves the claim. (B). By the C(K )-metric attaining property, for each 1 ≤ k ≤ n, there exists a subadditive function ϕ k : and In what follows, we need some properties of functions ϕ 1 , . . . , ϕ n . At first observe that by Thus, by Lip( ) ≤ π q, p ( ) = 1, for all f ∈ C(K ) one has We recall that : If in addition f k − g k ∞ ≤ 1 for each 1 ≤ k ≤ n, then the Lipschitz constant of each function ϕ k can be estimated by C n f k − h k ∞ . Indeed, In summary, for each 1 ≤ k ≤ n, the functions ϕ k are Lipschitz with norms that are uniformly bounded by C n f k − g k ∞ ≤ 1.

Let us show an upper estimate of its Lipschitz norm. For all
and so the Lipschitz norm of ϕ 0,n is less than or equal to C n . Note also that We can find a positive integer n 0 such that For each k ∈ {1, . . . , n}, we let g k : Applying the above estimate yields Since 1 = (1 − | f − g| p ) + | f − g| p , the subadditivity of the elements of W in combination with the estimate given in (C) gives We have shown that ϕ 0,n (h) ≤ h ∞ for all h ∞ ≤ 1. Hence and so, we get Combining with the estimate provided by the inequality ( * ), we obtain Now we apply the inequality 1 − |1 − x| q ≤ qx, that is true for all x ∈ [0, 1], to get (E). We represent now ϕ 0,n as an integral on W ⊂ B C(K ) # with respect to a regular Borel measure on W . For every f ∈ C(K ), we let φ f to denote the scalar function on W given by Consider φ 1 associated to 1 ∈ C(K ), and the positive Borel regular measure μ n defined on W by Here as usual, for every ϕ ∈ W , we let δ ϕ to denote the Dirac measure. Notice that μ n = μ(1) ≤ C n and, for every f ∈ B C(K ) , one has In particular, we get The hypothesis on compactness of W with respect to the topology of pointwise convergence τ p allows to consider the bounded sequence (μ n ) ∞ n=1 of all these measures as a subset of the dual space W C(W ) * . Thus, there is an accumulation point μ for this sequence with respect to the weak* topology. Using the estimate shown at the end of step (D), and taking into account that lim n γ (n) = lim n C n = lim n C n+1 = 1, we conclude that for all f , g ∈ C(K ) such that f − g ∞ < 1, one has Another straightforward limit argument shows that this inequality also holds if It is easy to see that the measure μ is positive. By the subadditivity and positivity of the elements of W , it follows that This completes the proof.
We conclude the section with the following result for the special class of subsets W of C(K ) # , which are composed of superadditive functions.
(iii) The condition (ii) holds with W containing superadditive functions.
Proof From Theorem 3.2 it follows that (i) implies (ii) with γ = q 1/q . Suppose that the condition (iii) holds. Then for all f 1 , g 1 , . . . , f n , g n ∈ C(K ) with This completes the proof.
In the following subsections, we show how our result applies in more concrete contexts. We begin by showing that we have proved a true extension, that is, our theorem gives Pisier's result when restricted to linear operators.

Pisier's Theorem for Linear Operators from C(K)-Spaces
Recall that Pisier's result can be written in the linear case as follows: Let 1 < q and 1/q + 1/q = 1 and let Y be a Banach space. A linear operator T : C(K ) → Y is (q, 1)-summing (with π q,1 (T ) = 1) if and only if there exist a positive constant C ≤ q 1/q and a probability Borel measure μ over K such that To show this result from Theorem 3.2, we at first choose a proper C(K )-metric attaining subset W ⊂ B C(K ) # . We consider the set S of linear functionals on C(K ) given by Observe that the set S is C(K )-metric attaining for an operator T . By the Hahn-Banach Theorem, given f , Clearly, for every y * ∈ Y * , one has (by T : Observe that s f ,g ∈ B C(K ) # . Indeed, if h 1 , h 2 ∈ C(K ), it follows by the metric sublinearity of that Let us denote by H the set of all these functions s f ,g with f − g ∞ ≤ 1. As it was observed, H ⊂ B C(K ) # and so we equipped it with the topology τ p of pointwise convergence, for which B C(K ) # is compact. Therefore, we can consider the closure H of H with respect to τ p as the required compact set W .
We point out that more can be said about a convenient set W in this case. Suppose that ϕ 0 ∈ B C(K ) # is the pointwise limit of a convergent net (s f γ ,g γ ) γ ∈ . For fixed h 1 , h 2 ∈ C(K ), 0 ≤ λ ≤ η and γ ∈ , we get By passing to the limit, due to the convergence with respect to τ p , we get Thus, the pointwise limit is also a sublinear map. The same calculations show that in fact the set L S of Lipschitz sublinear functionals with Lipschitz norm less than or equal to 1 is closed with respect to the topology τ p . Since (B C(K ) # , τ p ) is a compact Hausdorff space, we have that L S is also compact. This fact allows to write the following direct consequence of Theorem 3.2, that is also true if we write H instead of L S. We also use part (iii) of Corollary 3.3 for the last statement.

Proposition 3.4 Let 1 ≤ p < q < ∞, let (M, d) be a metric space, and let
: C(K ) → (M, d) be a sub-homogeneous and metric sublinear Lipschitz (q, 1)concave map with constant K L q,1 ( ) = 1. Let L S ⊂ C(K ) # be the set of sublinear Lipschitz functionals with norm less than or equal to one. Then there is a regular Borel measure μ on the compact space (L S, τ p ) such that, for all f , g ∈ C(K ) with f − g ∞ ≤ 1, one has If in addition the integral domination above holds for a subset W ⊂ L S composed of superadditive functions and is homogeneous, then the converse holds, that is, the map is Lipschitz (q, 1)-concave.

Metric Attaining Subsets of the Lipschitz Dual Space
We point out that to apply the obtained results, the key problem is finding suitable sets of functionals that achieve the metric. We show in this section a canonical way of constructing such a set under some natural conditions. Recall that if (M, d) is a metric space, then a subset S ⊂ B M # is metric attaining if, for every pair (x, y) ∈ M × M there exists a function φ x,y ∈ S such that φ x,y (x) − φ x,y (y) = d(x, y). Using such class of sets, we can define C(K )-metric attaining sets in such a way that they give a new type of domination. Let S ⊂ B M # be a metric attaining set.
Let : C(K ) → M be a Lipschitz map and suppose that the functions given by satisfy a triangular inequality. Then clearly the set where S is the closure of S in (C(K ) # , τ p ).
Observe that knowing a "good" description of the closure of S T with respect to the τ p -topology would give more precise formulas in the above domination. This is the case for example when M is a Banach space and T a linear operator, then S can be chosen to be the unit ball B M * of the dual Banach space M * .
We close the paper with some comments about relationships between classes of the introduced Lipschitz maps defined on Banach lattices. Standard computations show that every Lipschitz (q, p)-B E * -summing map from a Banach lattice E to a Banach space is Lipschitz (q, p)-concave. As we have noticed, the converse is also true when the Lipschitz map is defined on a C(K )-space. But these facts can be connected in the linear case to give a characterization of when a Lipschitz map is (q, p)-concave in terms of compositions with (q, p)-summing maps on C(K )-spaces. The following result is the Lipschitz variant of a well-known theorem that holds in the linear setting. The proof in the Lipschitz case is similar to the one for linear case (see [5,Theorem 16.5]), so we skip it. An immediate consequence of the previous result is the following corollary, which states that for certain Banach spaces and every q ∈ [1, ∞), there are only two distinct classes of non-trivial Lipschitz (q, p)-concave maps (see [5,Corollary 16.6] for the linear case). Corollary 3.7 Let 1 ≤ p < q. Suppose that the Banach space Y satisfies that every Lipschitz (q, p)-concave map from any C(K )-space to Y is Lipschitz (q, 1)-concave. Then every Lipschitz map from any Banach lattice to the Banach space Y is Lipschitz (q, p)concave if and only if it is Lipschitz (q, 1)-concave.

Declarations
Competing interests The authors declare none.
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