pith. sign in

arxiv: 2604.15254 · v1 · submitted 2026-04-16 · 🧮 math.FA

Convexity and concavity in Banach lattices

Enrique Garc\'ia-S\'anchez This is my paper

Pith reviewed 2026-05-10 09:24 UTC · model grok-4.3

classification 🧮 math.FA
keywords Banach lattices(p,q)-convexityp-convexificationfactorization theoremsrenormingslattice normsoperator factorization
0
0 comments X

The pith

Convexity and concavity in Banach lattices connect through (p,q)-properties that can be improved by p-convexification renormings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to give a detailed self-contained course that revisits the classical notions of (p,q)-convexity, (p,q)-concavity, and upper and lower p-estimates from a modern viewpoint. It integrates these with more recent developments and explains how p-convexification and p-concavification produce renormings that sharpen the relevant constants. A reader would care because the same techniques also yield a unified account of factorization theorems for the associated operators and their use in representing convex or concave Banach lattices.

Core claim

These notes revisit from a modern perspective the classical notions of (p,q)-convexity, (p,q)-concavity and upper and lower p-estimates, 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, and 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.

What carries the argument

The p-convexification and p-concavification techniques that construct renormings improving convexity and concavity constants while preserving the lattice structure.

Load-bearing premise

The reader already possesses sufficient background in Banach space theory and lattice norms to follow the detailed proofs.

What would settle it

An explicit Banach lattice in which the p-convexification procedure fails to produce a renorming with strictly better convexity constant than claimed.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper is a detailed, self-contained set of expository notes on convexity and concavity in Banach lattices. It revisits the classical notions of (p,q)-convexity, (p,q)-concavity, upper and lower p-estimates and their interrelations; explains p-convexification and p-concavification renorming techniques; and provides comprehensive expositions of the factorization theorems of Krivine, Maurey-Nikishin, Pietsch and Pisier together with their applications to the representation of convex and concave Banach lattices.

Significance. If the detailed proofs and relations are faithfully reproduced, the notes constitute a valuable modern reference that integrates recent developments while remaining accessible to beginners. The self-contained treatment of the standard factorization results and the explicit construction of renormings are particular strengths that could make this a standard resource in Banach-lattice theory.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'integrating more recent developments in the area' would benefit from naming one or two concrete recent references or results so that readers immediately see what is new in the exposition.
  2. [Introduction] The manuscript repeatedly refers to 'the main relations' and 'well-known results' without a dedicated roadmap subsection; adding a short diagram or table summarizing the implication lattice among (p,q)-convexity, p-estimates and factorization would improve navigability for beginners.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and encouraging report, which recognizes the notes as a self-contained modern reference integrating classical and recent developments in Banach lattice theory. The recommendation for minor revision is appreciated. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; expository reproduction of classical results

full rationale

The manuscript is explicitly framed as self-contained introductory notes that revisit and exposit standard definitions, relations, and theorems on (p,q)-convexity/concavity, p-estimates, renormings, and factorization results (Krivine, Maurey-Nikishin, Pietsch, Pisier) from the existing Banach-lattice literature. No new theorems, quantitative predictions, or derivations are asserted; every load-bearing step consists of faithful reproduction of prior, independently established facts with no internal fitting, self-referential prediction, or reduction of a claimed result to its own inputs by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is an expository review relying entirely on standard mathematical frameworks from functional analysis without introducing new free parameters, ad-hoc axioms, or invented entities.

axioms (1)
  • standard math Standard axioms and properties of Banach lattices, norms, and operator theory as established in prior literature.
    The notes build upon classical definitions and theorems without re-deriving foundational results.

pith-pipeline@v0.9.0 · 5445 in / 1121 out tokens · 60819 ms · 2026-05-10T09:24:33.125862+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Albiac, N

    F. Albiac, N. J. Kalton, Topics in Banach Space Theory , Second Edition, Springer, 2016

    work page 2016

  2. [2]

    Byrd, Factoring operators satisfying p-estimates, Trans

    S. Byrd, Factoring operators satisfying p-estimates, Trans. Amer. Math. Soc. 310 (1988), 567–582

    work page 1988

  3. [3]

    Cohn, Measure Theory, Second Edition, Birkh¨ auser, New York, 2013

    D.L. Cohn, Measure Theory, Second Edition, Birkh¨ auser, New York, 2013

    work page 2013

  4. [4]

    Creekmore, Type and cotype in Lorentz Lpq spaces

    J. Creekmore, Type and cotype in Lorentz Lpq spaces. Indag. Math. (Proceedings) 84 (1981), 145–152

    work page 1981

  5. [5]

    Diestel, H

    J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators , Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995

    work page 1995

  6. [6]

    Enflo, Banach spaces which can be given an equivalent uniformly convex norm

    P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm . Israel J. Math. 13 (1972), 281–288

    work page 1972

  7. [7]

    Fabian, P

    M. Fabian, P. Habala, P. H´ ajek, V. Montesinos, V. Zizler,Banach Space Theory. The Basis for Linear and Nonlinear Analysis. Springer, New York, 2011

    work page 2011

  8. [8]

    Garc´ ıa-S´ anchez, D

    E. Garc´ ıa-S´ anchez, D. H. Leung, M. A. Taylor, P. Tradacete,Banach lattices with upper p-estimates: free and injective objects. Math. Ann. 391 (2025), 3363–3398

    work page 2025

  9. [9]

    Garc´ ıa-S´ anchez, D.H

    E. Garc´ ıa-S´ anchez, D.H. Leung, M. A. Taylor, P. Tradacete,Banach lattices with upper p-estimates: renorming and factorization . Arxiv preprint

    work page

  10. [10]

    Grafakos, Classical Fourier Analysis, Volume 249 of Graduate Texts in Mathematics

    L. Grafakos, Classical Fourier Analysis, Volume 249 of Graduate Texts in Mathematics. Springer, New York, 2014

    work page 2014

  11. [11]

    Johnson, B

    W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric Structures in Banach Spaces, Volume 19 of Memoirs of the American Mathematical Society. Providence–Rhode Island, 1979

    work page 1979

  12. [12]

    J. L. Krivine, Th´ eor` emes de factorisation dans les espaces r´ eticules. S´ eminaire Maurey-Schwartz (1973- 1974), Expos´ e no. 22 et 23, 22 p.http://www.numdam.org/item/SAF_1973-1974____A24_0/ 70 E. GARC ´IA-S´ANCHEZ

    work page 1973

  13. [13]

    Kwapie´ n,Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients

    S. Kwapie´ n,Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Studia Math. 44 (1972), 583–595

    work page 1972

  14. [14]

    D. H. Leung, Isomorphic classification of atomic weak Lp spaces. In: N. Kalton, E. Saab, S. Montgomery- Smith (eds.), Interaction between functional analysis, harmonic analysis, and probability. Proceedings of a conference held at the University of Missouri, Columbia, MO, USA, May 29-June 3, 1994 . New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math....

    work page 1994

  15. [15]

    Lindenstrauss, L

    J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II . Springer–Verlag, 1979

    work page 1979

  16. [16]

    H. P. Lotz, Extensions and liftings of positive linear mappings on Banach lattices . Trans. Amer. Math. Soc. 211 (1975), 85–100

    work page 1975

  17. [17]

    Maurey, Type et cotype dans les espaces munis de structures locales inconditionnelles

    B. Maurey, Type et cotype dans les espaces munis de structures locales inconditionnelles . S´ eminaire Maurey-Schwartz (1973-1974), Expos´ e no. 24 et 25, 25 p. http://www.numdam.org/item/SAF_ 1973-1974____A25_0/

    work page 1973

  18. [18]

    Maurey, Th´ eor` emes de Factorisation Pour Les Op´ erateurs Lin´ eaires ` a Valeurs Dans Les EspacesLp

    B. Maurey, Th´ eor` emes de Factorisation Pour Les Op´ erateurs Lin´ eaires ` a Valeurs Dans Les EspacesLp. Ast´ erisque,11 (1974) (French)

    work page 1974

  19. [19]

    Meyer-Nieberg, Banach Lattices

    P. Meyer-Nieberg, Banach Lattices. Springer, Berlin, 1991

    work page 1991

  20. [20]

    Milman, On some criteria for the regularity of spaces of type (B)

    D. Milman, On some criteria for the regularity of spaces of type (B). C. R. (Doklady) Acad. Sci. U.R.S.S 20 (1938), 243–246

    work page 1938

  21. [21]

    S. J. Montgomery-Smith, The pp in Pisier’s factorization theorem. In: M¨ uller PFX, Schachermayer W, eds. Geometry of Banach Spaces: Proceedings of the Conference Held in Strobl, Austria 1989 . London Math. Soc. Lecture Note Ser. Cambridge University Press; 1991, 231–236

    work page 1989

  22. [22]

    E. M. Nikishin, Resonance theorems and superlinear operators. Russ. Math. Surv. 25 (6) (1970), 125– 187

    work page 1970

  23. [23]

    B. J. Pettis, A proof that every uniformly convex space is reflexive . Duke Math. J. 5 (1939), 249–253

    work page 1939

  24. [24]

    Pisier, Factorization of operators through Lp,8 or Lp,1 and non-commutative generalizations

    G. Pisier, Factorization of operators through Lp,8 or Lp,1 and non-commutative generalizations. Math. Ann. 276 (1986), 105–136

    work page 1986

  25. [25]

    Raynaud, P

    Y. Raynaud, P. Tradacete, Interpolation of Banach lattices and factorization of p-convex and q-concave operators. Integr. Equ. Oper. Theory 66 (2010), 79—112

    work page 2010

  26. [26]

    Reisner, Operators which factor through convex Banach lattices

    S. Reisner, Operators which factor through convex Banach lattices . Canad. J. Math. 32 (1980), 1482– 1500

    work page 1980

  27. [27]

    H. U. Schwarz, Banach Lattices and Operators. Teubner Verlag, Leipzig, 1984

    work page 1984

  28. [28]

    Tomczak-Jaegermann, Banach–Mazur distances and finite-dimensional operator ideals

    N. Tomczak-Jaegermann, Banach–Mazur distances and finite-dimensional operator ideals . Longman Scientific & Technical, New York, 1989. Instituto de Ciencias Matem ´aticas (CSIC-UAM-UC3M-UCM), Consejo Superior de Inves- tigaciones Cient´ıficas, C/ Nicol´as Cabrera, 13–15, Campus de Cantoblanco UAM, 28049 Madrid, Spain. Email address: enrique.garcia@icmat.es

    work page 1989