Asymptotic structure and coarse Lipschitz geometry of Banach spaces

Abstract: In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly smoothness and convexity, and several distinct Banach-Saks-like properties. Among other results, we characterize the Banach spaces which are either coarsely or uniformly homeomorphic to $T^{p_1}\oplus \ldots \oplus T^{p_n}$, where each $T^{p_j}$ denotes the $p_j$-convexification of the Tsirelson space, for $p_1,\ldots,p_n\in (1,\ldots, \infty)$, and $2\not\in{p_1,\ldots ,p_n}$. We obtain applications to the coarse Lipschitz geometry of the $p$-convexifications of the Schlumprecht space, and some hereditarily indecomposable Banach spaces. We also obtain some new results on the linear theory of Banach spaces.