On summability of multilinear operators and applications

TL;DR

This paper unifies and extends a broad family of inequalities related to multilinear operators, providing a deeper theoretical framework with practical applications, including improved inequalities in recent research.

Contribution

It introduces a unified formulation of inequalities for multilinear operators, creating new inequalities and improving existing results with a novel approach.

Findings

Unified a large family of inequalities in a single framework

Developed new inequalities previously unseen by earlier methods

Enhanced existing inequalities, demonstrating practical applicability

Abstract

This paper has two clear motivations: a technical and a practical. The technical motivation unifies in a single and crystal clear formulation a huge family of inequalities that have been produced separately in the last 90 years in different contexts. But we do not just join inequalities; our method also create a family of inequalities invisible by previous approaches. The practical motivation is to show that our deeper approach has strength to attack various problems. We provide new applications of our family of inequalities, continuing the recent work by Maia et al., that, by using our main theorem, substantially improved an inequality of Carando et al. which seemed impossible to be achieved by their original method.