Asymptotic properties of Banach spaces and coarse quotient maps

Zhang, Sheng

doi:10.1090/proc/14097

Asymptotic properties of Banach spaces and coarse quotient maps
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Proc. Amer. Math. Soc. 146 (2018), 4723-4734 Request permission

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.

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