Convexity and concavity in Banach lattices

TL;DR

This paper provides a comprehensive, modern introduction to convexity and concavity in Banach lattices, revisiting classical notions and integrating recent developments and factorization results.

Contribution

It offers a detailed exposition of $(p,q)$-convexity and concavity, $p$-convexification techniques, and factorization theorems, enhancing understanding and applications in Banach lattice theory.

Findings

Revisits classical convexity and concavity notions with modern perspective

Details $p$-convexification and $p$-concavification techniques for Banach lattices

Provides comprehensive factorization results for convex and concave operators

Abstract

These notes are a detailed, self-contained introductory course on convexity and concavity in Banach lattices, suitable for both experts and beginners. We revisit, from a modern perspective, the classical notions of $(p,q)$-convexity, $(p,q)$-concavity and upper and lower $p$-estimates, and the main relations between these properties, integrating more recent developments in the area. We explain in full detail the $p$-convexification and $p$-concavification techniques and how they can be used to build renormings of Banach lattices that improve the convexity and concavity constants. We also provide a comprehensive exposition of the main factorization results for $(p,q)$-convex and $(p,q)$-concave operators, including well-known results from Krivine, Maurey--Nikishin, Pietsch and Pisier, and their applications to the representation of convex and concave Banach lattices.