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A summability principle and applications

Nacib Albuquerque, Gustavo Araújo, Lisiane Rezende, Joedson Santos

TL;DR

This work develops an anisotropic inclusion principle for the class of $\Lambda$-summing multilinear operators, unifying the isotropic absolute summing and the multiple summing theories through a block-structured $\Lambda$-summing framework. The authors prove a general inclusion theorem for block summing classes, then derive concrete applications to Hardy–Littlewood-type inequalities and a Grothendieck-type coincidence in anisotropic settings. The results yield improved multilinear Hardy–Littlewood estimates with explicit anisotropic exponents and provide new sufficient conditions for when multilinear summing classes coincide with the full operator space. Collectively, the paper extends classical summability theory, offering sharper bounds and a cohesive, block-based perspective on multilinear operators in Banach spaces.

Abstract

This paper investigates summability principles for multilinear summing operators. The main result presents a novel inclusion theorem for a class of summing operators, which generalizes several classical results. As applications, we derive improved estimates for Hardy--Littlewood inequalities on multilinear forms and prove a Grothendieck--type coincidence result in anisotropic settings.

A summability principle and applications

TL;DR

This work develops an anisotropic inclusion principle for the class of -summing multilinear operators, unifying the isotropic absolute summing and the multiple summing theories through a block-structured -summing framework. The authors prove a general inclusion theorem for block summing classes, then derive concrete applications to Hardy–Littlewood-type inequalities and a Grothendieck-type coincidence in anisotropic settings. The results yield improved multilinear Hardy–Littlewood estimates with explicit anisotropic exponents and provide new sufficient conditions for when multilinear summing classes coincide with the full operator space. Collectively, the paper extends classical summability theory, offering sharper bounds and a cohesive, block-based perspective on multilinear operators in Banach spaces.

Abstract

This paper investigates summability principles for multilinear summing operators. The main result presents a novel inclusion theorem for a class of summing operators, which generalizes several classical results. As applications, we derive improved estimates for Hardy--Littlewood inequalities on multilinear forms and prove a Grothendieck--type coincidence result in anisotropic settings.

Paper Structure

This paper contains 8 sections, 9 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Key Concepts
  3. Multilinear summing operators: classical approach
  4. Multilinear summing operators: unified approach
  5. Inclusion theorem for block summing classes
  6. Applications
  7. Hardy--Littlewood block variants inequalities
  8. A new Grothendieck inclusion result

Key Result

Proposition 2.2

Let $E_{1}, \dots, E_{m}, F$ be Banach spaces and let $m,d$ be positive integers with $1\leq d\leq m$, and $(\mathbf{r},\mathbf{p}) := (r_{1}, \ldots, r_{d}; p_{1}, \ldots, p_{m}) \in \lbrack 1,\infty)^{d+m}$. Let also $\mathcal{I}_d = \{I_{1}, \dots, I_{d}\}$ be a partition of $\{1,\ldots,m\}$ and

Theorems & Definitions (14)

  • Example 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1: Hardy--Littlewood/Dimant--Sevilla-Peris dimanthardy
  • Theorem 4.2
  • Theorem 4.3
  • proof
  • Corollary 4.4
  • ...and 4 more