Lipschitz stable sequence classes: an approach to Rademacher type and cotype of Lipschitz functions

Abstract

In this paper, we extend sequence-class methods from linear and multilinear theory to the Lipschitz setting, highlighting the substantial differences that arise from the lack of linearity. First, we establish a general criterion for lifting a Lipschitz mapping between Banach spaces to a Lipschitz operator between sequence spaces, and we use it to define the class ${\hat{I}}_{(Z,Y)}^{{\mathrm{Lip}}_0}$ of $(Z,Y)$-summing Lipschitz functions, where $Z$ and $Y$ are sequence classes. We then introduce the notion of Lipschitz stable sequence class and show that $[{\hat{I}}_{(Z,Y)}^{{\mathrm{Lip}}_0},\ddot{I}{€}_{(Z,Y)}^{\mathrm{Lip}}]$ is a Banach Lipschitz ideal whenever the sequence classes satisfy this property. As applications, we present Rademacher type and cotype for Lipschitz functions and identify them with $(Z,Y)$-summing Lipschitz spaces for concrete choices of sequence classes. We prove that the type case forms a Banach Lipschitz ideal, whereas the cotype case does not, and we analyse composition and maximality for the type ideals.

23 pages