[Paper Review] Lipschitz $p$-convex and $q$-concave maps

Javier Alejandro Chávez‐Domínguez|arXiv (Cornell University)|Jun 24, 2014

Advanced Banach Space Theory7 references3 citations

TL;DR

This paper introduces Lipschitz $p$-convex and $q$-concave maps as nonlinear analogues of classical $p$-convex and $q$-concave linear maps between Banach lattices. It establishes nonlinear factorization theorems through $L_p$ spaces, proves that a Lipschitz map is $p$-convex if and only if its linearization is $p$-convex, and characterizes these maps via factorizations through $p$-convex and $q$-concave Banach lattices, extending results of Raynaud and Tradacete to the Lipschitz setting.

ABSTRACT

The notions of $p$-convexity and $q$-concavity are mostly known because of their importance as a tool in the study of isomorphic properties of Banach lattices, but they also play a role in several results involving linear maps between Banach spaces and Banach lattices. In this paper we introduce Lipschitz versions of these concepts, dealing with maps between metric spaces and Banach lattices, and start by proving nonlinear versions of two well-known factorization theorems through $L_p$ spaces due to Maurey/Nikishin and Krivine. We also show that a Lipschitz map from a metric space into a Banach lattice is Lipschitz $p$-convex if and only if its linearization is $p$-convex. Furthermore, we elucidate why there is such a close relationship between the linear and nonlinear concepts by proving characterizations of Lipschitz $p$-convex and Lipschitz $q$-concave maps in terms of factorizations through $p$-convex and $q$-concave Banach lattices, respectively, in the spirit of the work of Raynaud and Tradacete.

Motivation & Objective

• To develop a nonlinear theory of $p$-convex and $q$-concave maps between metric spaces and Banach lattices.
• To extend classical linear factorization theorems of Maurey/Nikishin and Krivine to the Lipschitz setting.
• To establish a precise correspondence between Lipschitz $p$-convexity and the $p$-convexity of the linearization of a map.
• To characterize Lipschitz $p$-convex and $q$-concave maps via factorizations through $p$-convex and $q$-concave Banach lattices, respectively.
• To explore the feasibility of a Lipschitz version of the Raynaud–Tradacete factorization result for maps that are both $p$-convex and $q$-concave.

Proposed method

• Introduces the concept of Lipschitz $p$-convex and $q$-concave maps using norm inequalities involving finite sums of absolute values raised to powers $p$ and $q$, respectively.
• Uses the Arens–Eells space ${\mathscr{F}}(X)$ to linearize Lipschitz maps from a metric space $X$ to a Banach lattice $E$, enabling the study of their linearization properties.
• Proves that a Lipschitz map $T$ is $p$-convex if and only if its linearization $\tilde{T}$ is $p$-convex, establishing a fundamental bridge between linear and nonlinear theories.
• Applies nonlinear factorization theorems through $L_p$ spaces, generalizing the Maurey/Nikishin and Krivine factorization results to the Lipschitz context.
• Characterizes Lipschitz $p$-convex and $q$-concave maps via factorizations through $p$-convex and $q$-concave Banach lattices, respectively, in the spirit of Raynaud and Tradacete.
• Proposes a conjectural Lipschitz version of the Raynaud–Tradacete factorization, suggesting a factorization through intermediate spaces with slightly adjusted convexity and concavity exponents.

Experimental results

Research questions

• RQ1 Can the classical Maurey/Nikishin factorization theorem be extended to Lipschitz maps between metric spaces and Banach lattices?
• RQ2 Is a Lipschitz map $T$ from a metric space to a Banach lattice $p$-convex if and only if its linearization $\tilde{T}$ is $p$-convex?
• RQ3 Can the composition of a Lipschitz $p$-convex map followed by a Lipschitz $q$-concave map be factored through an $L_p$ space, generalizing Krivine’s theorem?
• RQ4 Can Lipschitz $p$-convex and $q$-concave maps be characterized by factorizations through $p$-convex and $q$-concave Banach lattices, respectively?
• RQ5 Is there a Lipschitz analogue of the Raynaud–Tradacete factorization result that allows factorization of maps that are both $p$-convex and $q$-concave through spaces with adjusted exponents $p_0 < p$ and $q_0 > q$?

Key findings

• A Lipschitz map $T$ from a metric space to a Banach lattice is Lipschitz $p$-convex if and only if its linearization $\tilde{T}$ is $p$-convex, establishing a precise correspondence between the linear and nonlinear settings.
• The paper proves a nonlinear version of the Maurey/Nikishin factorization theorem, showing that a Lipschitz $p$-convex map factors through an $L_p$ space with constant depending on the $p$-convexity constant.
• The composition of a Lipschitz $p$-convex map followed by a Lipschitz $q$-concave map factors through an $L_p$ space, generalizing Krivine’s theorem to the Lipschitz category.
• Lipschitz $p$-convex and $q$-concave maps are characterized via factorizations through $p$-convex and $q$-concave Banach lattices, respectively, extending the work of Raynaud and Tradacete to the nonlinear setting.
• The formal inclusion $i_{p,q}:L_p(0,1)\to L_q(0,1)$ for $1<q<p<\infty$ is shown to be Lipschitz $q$-concave and Lipschitz $p$-convex but does not factor as a composition of a Lipschitz $q$-concave and a Lipschitz $p$-convex map, indicating limitations of naïve factorization schemes.
• A conjectural Lipschitz version of the Raynaud–Tradacete factorization is proposed, suggesting that a map both $p$-convex and $q$-concave may factor through spaces with adjusted exponents $p_0 < p$ and $q_0 > q$, though such a result remains open due to the incompatibility of complex interpolation with Lipschitz maps.