Universality, complexity and asymptotically uniformly smooth Banach spaces

TL;DR

This paper constructs a Banach space that is universal for all separable Banach spaces with a $p$-asymptotically uniformly smooth norm and proves the complexity of this class.

Contribution

It introduces a universal Banach space for the class of $p$-asymptotically uniformly smooth spaces and establishes the class as analytic complete.

Findings

Existence of a universal Banach space for the class of $p$-asymptotically uniformly smooth spaces.

The class of such spaces is shown to be analytic complete.

Extension of previous results for the case $p=\infty$.

Abstract

For $1<p\le \infty$, we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent $p$-asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by Kalton, Werner and Kurka in the case $p=\infty$.