  For each ordinal $\xi$ and each $1<p<\infty$, we offer a natural, ismorphic
characterization of those spaces and operators which admit an equivalent
$\xi$-$p$-asymptotically uniformly smooth norm. We also introduce the notion of
$\xi$-asymptotically uniformly flat norms and provide an isomorphic
characterization of those spaces and operators which admit an equivalent
$\xi$-asymptotically uniformly flat norm.
  Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem
regarding the $\xi$-asymptotic smoothness of $C(K)$ in terms of the
Cantor-Bendixson index of $K$. We also prove that for all ordinals, both the
isomorphic properties and isometric properties we study pass from Banach spaces
to their injective tensor products.
  We study the classes of $\xi$-$p$-asymptotically uniformly smooth,
$\xi$-$p$-asymptotically uniformly smoothable, $\xi$-asymptotically uniformly
flat, and $\xi$-asymptotically uniformly flattenable operators. We show that
these classes are either a Banach ideal or a right Banach ideal when assigned
an appropriate ideal norm.
