Polynomials as Lipschitz maps on the Veronese cone

2024/12/13 by Maite Fernández-Unzueta, Fernández-Unzueta, Maite
Mathematics · #46B28 #46T99 #47H60 #47L22 #51F99 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2412.10527

openalex publication_date 2024/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a Banach space X and d∈ ℕ, we construct a metric space \mathbbV_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 \mathbbV_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

Related

• Lipschitz trivial values of polynomial mappings
• Lineability of summing sets of homogeneous polynomials
• The Segre cone of Banach spaces and multilinear operators
• Approximation by Lipschitz, analytic maps on certain Banach spaces
• Orbits of homogeneous polynomials on Banach spaces