Asymptotic properties of Banach spaces and coarse quotient maps

TL;DR

This paper establishes relationships between the asymptotic geometric properties of Banach spaces under coarse quotient maps, linking moduli of smoothness and convexity through quantitative bounds.

Contribution

It provides new bounds connecting the asymptotic moduli of Banach spaces when related by coarse quotient maps, extending understanding of their geometric structure.

Findings

Bound on the $eta$-modulus of $X$ via the asymptotic uniform smoothness of $Y

Bound on the asymptotic uniform convexity of $X$ when the map is a coarse homeomorphism

Quantitative relationships between asymptotic properties under coarse quotient mappings

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 ($\beta$)-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.