Let $M$ be a closed subspace of $L\_p(0,1)$, $1<p<\infty$, $p\neq 2$.
Suppose that M contains copies of $\ell\_p^n$ uniformly.


Does $M$ contain a copy of $\ell\_p$?


The result is true for $p=1$, since subspaces of $L\_1(0,1)$ containing no copies of $\ell\_1$ are superreflexive.



Finite representability of $\ell_p$ in subspaces of $L_p(0,1)$
Yes. There is probably a quick proof using ultraproducts but I am not sure. Here is a very rough sketch of a proof.


$1\le p<2$ case. Let $\varepsilon\_k\searrow 0$, and suppose $(x^n\_i)\_{i=1}^n\in X$ are $(1+\varepsilon\_k)$-equivalent to the unit vector basis of $\ell\_p^n$'s. We will use a theorem of Dor that functions that are equivalent to $\ell\_p^n$'s are essentially disjoint. Inductively, pick an infinite sequence $(y\_i)$ which is equivalent to the unit vector basis of $\ell\_p$ as follows. Let $y\_1=x^1\_1$. By Dor's result for all $n$, $(x^n\_i)$ are essentially disjoint, say they live on disjoint sets $I^n\_i$'s. For a large $n\_1$, $y\_1$ restricted to some (in fact, many)) $I^{n\_1}\_i$ has small norm. Pick $y\_2=x^{n\_1}\_i$. Then find a large $n\_2$ so that both $y\_1$ and $y\_2$ have smaller norm on some $I^{n\_2}\_i$, and continue this process. We also want our choices be an unconditional sequence. So a bit more care is needed to pick $y\_i$'s incorporating gliding hump argument so that they are equivalent to a block basis of the Haar basis: Suppose $y\_1$ is finitely supported. Pick a very large $n\_1$ so that still a very large subset of $(x^{n^1}\_i)$ have essentially the same coordinates on the support of $y\_1$. Then take $y\_2$ be a difference of two while also ensuring the process above. Then the resulting sequence $(y\_i)$ is unconditional and they are large on disjoint subsets, then you can use a square function estimate to show the upper-$p$ estimate. The lower-$p$ estimate is automatic since $p<2$.


For $p>2$, you can use Kadec-Pelczynski's argument to get a sequence of peak functions. For any $\delta>0$, consider the set
$M(\delta)=\{x\in L\_p: m(\{t:|x(t)|>\delta\})\ge \delta\}$. If a sequence is unconditional and belong to some $M(\delta)$, then it is isomorphic to $\ell\_2$. Using this we can pick a sequence $(y\_i)$ of peak functions equivalent to a block basis of the Haar basis as above from the collection $(x^n\_i)\_{i=1}^n\in X$ which are $(1+\varepsilon\_k)$-equivalent to the unit vector basis of $\ell\_p^n$'s.


*Dor, Leonard E.*, [**On projections in (L\_1)**](https://doi.org/10.2307/1971039), Ann. Math. (2) 102, 463-474 (1975). [ZBL0314.46027](https://zbmath.org/?q=an:0314.46027).


