Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces

TL;DR

This paper extends classical results on the coarse Lipschitz structure from reflexive to quasi-reflexive Banach spaces, revealing embedding limitations based on asymptotic convexity and smoothness properties.

Contribution

It generalizes known coarse Lipschitz structure results to quasi-reflexive spaces and establishes new embedding restrictions between spaces with different asymptotic geometries.

Findings

q-asymptotically uniformly convex spaces do not embed into p-asymptotically uniformly smooth quasi-reflexive spaces for q<p

Extension of classical results to quasi-reflexive setting

New non-embedding results based on asymptotic properties

Abstract

In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for $1\le q<p$, a $q$-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a $p$-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga.