This is an announcement for the paper “Power type ξ-asymptotically uniformly smooth norms” by Ryan M Causeyhttp://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1.
Abstract: We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $\omega$. For every ordinal $\xi$, we characterize the operators, and therefore the Banach spaces, which admit a $\xi$ -asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of $S_{z\xi}(\cdot, \epsilon)$. We also introduce the $\xi$-Szlenk power type and investigate ideal and factorization properties of classes associated with the $\xi$-Szlenk power type.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.03666