Recall that: let $0<r<s<2$, then $\ell\_r$ uniformly contains a subspace isomorphic to $\ell\_s^m$, $m\ge 1$ (see [JS]).


I am wondering whether are any result for the case when $r>s>2$?


[Johnson, William B.; Schechtman, Gideon Embedding $l\_p^m$ into $l\_1^m$, Acta Math. 149 (1982), 71--85.][JS]



Banach embedding of finite dimensional spaces
For $2<r<\infty$, if $\ell\_s^n$ embeds uniformly into $\ell\_r$ for all $n$, then either $s=r$ or $s=2$. This is basically the localization to finite dimensions of the classical dichotomy theorem of Kadec and Pelczynski. The book of Albiac and Kalton is a good source for this.


