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Banach lattices with upper $p$-estimates: Renorming and factorization

Enrique García-Sánchez, Denny H. Leung, Mitchell A. Taylor, Pedro Tradacete

TL;DR

The paper develops a comprehensive theory for Banach lattices with upper $p$-estimates, modeling them via Lorentz spaces $L_{p, ty}$ and $L_{p,1}$. It introduces a robust abstract factorization framework for $(p, ty)$-convex and $(q,1)$-concave settings, including duality, minimal/maximal factorizations, and Calderón-type interpolation to handle simultaneous convexity/concavity. It proves representation and embedding results for upper $p$-estimate lattices, analyzes push-outs and universality, and studies operator ideals factoring through convex/concave lattices, establishing perfectness and diagonal-factor models. The work highlights fundamental differences from classical $p$-convexity, including isometric vs. isomorphic distinctions and renorming limitations, and proposes candidate universal objects for the upper $p$-estimate regime. Overall, it extends the Reisner–MN–RT paradigm to upper $p$-estimates, offering new tools for renorming, factorization, and universal constructions in Banach lattices.

Abstract

The notions of $p$-convexity and concavity are fundamental tools for studying Banach lattices, as they partition the class of Banach lattices into a scale of spaces with $L_p$-like properties. Upper and lower $p$-estimates provide a refinement of this scale, modeled by the Lorentz spaces $L_{p,\infty}$ and $L_{p,1}$, respectively. In this article, we provide a comprehensive treatment of Banach lattices with upper $p$-estimates. In particular, we show that many well-known theorems about $p$-convex Banach lattices have analogues in the upper $p$-estimate setting, including the ability to represent all such spaces inside of infinity sums of model spaces, to canonically factor the convex operators and identify their associated operator ideals, as well as to give a precise description of the free objects and push-outs. Proving these results is far from straightforward and will require the development of a variety of new tools that avoid convexification and concavification procedures. In fact, we will identify many fundamental differences between the theories of $p$-convexity and upper $p$-estimates, particularly with regards to isometric problems and renormings.

Banach lattices with upper $p$-estimates: Renorming and factorization

TL;DR

The paper develops a comprehensive theory for Banach lattices with upper -estimates, modeling them via Lorentz spaces and . It introduces a robust abstract factorization framework for -convex and -concave settings, including duality, minimal/maximal factorizations, and Calderón-type interpolation to handle simultaneous convexity/concavity. It proves representation and embedding results for upper -estimate lattices, analyzes push-outs and universality, and studies operator ideals factoring through convex/concave lattices, establishing perfectness and diagonal-factor models. The work highlights fundamental differences from classical -convexity, including isometric vs. isomorphic distinctions and renorming limitations, and proposes candidate universal objects for the upper -estimate regime. Overall, it extends the Reisner–MN–RT paradigm to upper -estimates, offering new tools for renorming, factorization, and universal constructions in Banach lattices.

Abstract

The notions of -convexity and concavity are fundamental tools for studying Banach lattices, as they partition the class of Banach lattices into a scale of spaces with -like properties. Upper and lower -estimates provide a refinement of this scale, modeled by the Lorentz spaces and , respectively. In this article, we provide a comprehensive treatment of Banach lattices with upper -estimates. In particular, we show that many well-known theorems about -convex Banach lattices have analogues in the upper -estimate setting, including the ability to represent all such spaces inside of infinity sums of model spaces, to canonically factor the convex operators and identify their associated operator ideals, as well as to give a precise description of the free objects and push-outs. Proving these results is far from straightforward and will require the development of a variety of new tools that avoid convexification and concavification procedures. In fact, we will identify many fundamental differences between the theories of -convexity and upper -estimates, particularly with regards to isometric problems and renormings.
Paper Structure (19 sections, 58 theorems, 172 equations)

This paper contains 19 sections, 58 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Content of the paper
  3. An overview of -convexity and upper -estimates
  4. Banach lattices with upper -estimates vs. sublattices of
  5. A finite dimensional counterexample
  6. The renormings of weak-
  7. Factorization of convex and concave operators
  8. Geometry of factorization
  9. Duality of convex sets
  10. Convexity and concavity
  11. Minimal convex factorization
  12. Maximal concave factorization
  13. Duality of factorization
  14. Factorization of a simultaneously convex and concave operator
  15. Factoring a -convex and -concave operator
  16. ...and 4 more sections

Key Result

Theorem 2.3

Let $1\leq r<p<\infty$, $E$ a Banach space, $T:E\rightarrow L_r(\mu)$ a linear operator and $C>0$. The following are equivalent:

Theorems & Definitions (119)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Maurey's Factorization Theorem
  • Theorem 2.4: Pisier's Factorization Theorem
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Corollary 2.8
  • proof
  • Remark 2.9
  • ...and 109 more