Estimation of the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces

Abstract

For each ordinal $\alpha< \omega_1$, we prove the existence of a separable, reflexive Banach space with a basis and Szlenk index $\omega^{\alpha+1}$ which is universal for the class of separable, reflexive Banach spaces $X$ such that the Szlenk indices $Sz(X), Sz(X^*)$ do not exceed $\omega^\alpha$.