Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic smoothness in Banach spaces, three space properties and
  applications   We study four asymptotic smoothness properties of Banach spaces, denoted
$\textsf{T}_p,\textsf{A}_p, \textsf{N}_p$ and $\textsf{P}_p$. We complete their
description by proving the missing renorming theorem for $\textsf{A}_p$. We
prove that asymptotic uniform flattenability (property $\textsf{T}_\infty$) and
summable Szlenk index (property $\textsf{A}_\infty$) 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 $\textsf{A}_p$ and $\textsf{N}_p$ in terms of
equivalent renormings, new coarse Lipschitz rigidity results for these classes.

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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
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The categories of this paper are: adap-org