Polynomials as Lipschitz maps on the Veronese cone

TL;DR

This paper constructs a special metric space related to Banach spaces where homogeneous polynomials factor through Lipschitz maps, and uses this to connect polynomial summing properties with Lipschitz maps, generalizing known linear results.

Contribution

It introduces a new metric space framework for homogeneous polynomials on Banach spaces, linking polynomial properties to Lipschitz map properties, extending linear operator theorems.

Findings

Homogeneous polynomials factor through Lipschitz maps on the constructed space.

The metric on the space is norm-independent up to a constant.

Lipschitz q-summing polynomials correspond to Lipschitz q-summing maps.

Abstract

Given a Banach space $X$ and $d\in \mathbb{N}$, we construct a metric space $\mathbb{V}_X^d$ with the property that every $d$-homogeneous polynomial defined on $X$ factors through a Lipschitz map on it. We prove that the metric on $\mathbb{V}_X^d$ is independent (up to a constant) of the norm of the tensor space in which it is embedded. We apply this fact to prove that a homogeneous polynomial is Lipschitz $q$-summing as a polynomial if and only if its associated Lipschitz map is Lipschitz $q$-summing. This result generalizes the already known theorem for linear operators