Regularity principle in sequence spaces and applications

TL;DR

This paper introduces a nonlinear regularity principle in sequence spaces that yields universal estimates, leading to new inclusion theorems for multiple summing operators and resolving classical problems like Grothendieck's theorems and Hardy--Littlewood inequalities.

Contribution

It develops a novel nonlinear regularity principle in sequence spaces and applies it to establish new inclusion theorems and solve longstanding classification problems.

Findings

Established new inclusion theorems for multiple summing operators.

Settled all Grothendieck's type (ℓ₁, ℓ₂) theorems for multilinear operators.

Solved the classification problem for admissible exponents in the anisotropic Hardy--Littlewood inequality.

Abstract

We prove a nonlinear regularity principle in sequence spaces which produces universal estimates for special series defined therein. Some consequences are obtained and, in particular, we establish new inclusion theorems for multiple summing operators. Of independent interest, we settle all Grothendieck's type $(\ell_{1},\ell_{2})$ theorems for multilinear operators. We further employ the new regularity principle to solve the classification problem concerning all pairs of admissible exponents in the anisotropic Hardy--Littlewood inequality.