Abstract
We study four asymptotic smoothness properties of Banach spaces, denoted T p , A p , N p \mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p , and P p \mathsf {P}_p . We complete their description by proving the missing renorming characterization for A p \mathsf {A}_p . We show that asymptotic uniform flattenability (property T ∞ \mathsf {T}_\infty ) and summable Szlenk index (property A ∞ \mathsf {A}_\infty ) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of A p \mathsf {A}_p and N p \mathsf {N}_p in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.
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