POWER TYPE ξ-ASYMPTOTICALLY UNIFORMLY SMOOTH NORMS

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 xi-asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of Sz(xi)(., epsilon). We also introduce the xi-Szlenk power type and investigate ideal and factorization properties of classes associated with the xi-Szlenk power type.