Dominated bilinear forms and 2-homogeneous polynomials

TL;DR

This paper explores the relationship between the cotype of Banach spaces and the domination properties of 2-homogeneous polynomials, establishing conditions under which such polynomials are not r-dominated.

Contribution

It establishes a connection between the cotype of Banach spaces and the parameters r for which all 2-homogeneous polynomials are r-dominated, providing new insights into polynomial domination.

Findings

If cotX > 2, then for every 1 <= r < (cotX)*, there exists a non-r-dominated 2-homogeneous polynomial on X.

The paper links geometric properties of Banach spaces to polynomial domination behavior.

Provides conditions under which 2-homogeneous polynomials fail to be r-dominated.

Abstract

The main goal of this note is to establish a connection between the cotype of the Banach space X and the parameters r for which every 2-homogeneous polynomial on X is r-dominated. Let cotX be the infimum of the cotypes assumed by X and (cotX)* be its conjugate. The main result of this note asserts that if cotX > 2, then for every 1<= r < (cotX)* there exists a non-r-dominated 2-homogeneous polynomial on X.