Asymptotic properties of Banach spaces and coarse quotient maps

Abstract

We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space Y is a coarse quotient of a subset of a Banach space X, where the coarse quotient map is coarse Lipschitz, then the (β)-modulus of X is bounded by the modulus of asymptotic uniform smoothness of Y up to some constants. In particular, if the coarse quotient map is a coarse homeomorphism, then the modulus of asymptotic uniform convexity of X is bounded by the modulus of asymptotic uniform smoothness of Y up to some constants.

A submetric characterization of Rolewicz's property ($β$)

“…The lemma below says that co-uniformly continuous maps and co-coarsely continuous maps are "co-Lipschtiz for large distances" provided the target space is metrically convex. We refer to [1,16] for the proof. Lemma 4.2.…”