Asymptotic smoothness and universality in Banach spaces

TL;DR

The paper investigates the complexity and universality properties of certain classes of Banach spaces related to asymptotic smoothness, establishing their Borel structure and constructing universal families with optimality results.

Contribution

It demonstrates the Borel nature of classes A_p and N_p, constructs universal families, and proves the non-existence of universal spaces for these classes.

Findings

Classes A_p and N_p are Borel in the space of separable Banach spaces.

Constructed small families that are both injectively and surjectively universal for these classes.

Proved that no single universal space exists for these classes.

Abstract

For $1<p\leqslant \infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\textsf{A}_p$ and $\textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by proving in particular that none of these classes admits a universal space.