# Finite representability of $\ell_p$ in subspaces of $L_p(0,1)$

Let $$M$$ be a closed subspace of $$L_p(0,1)$$, $$1, $$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.

$$1\le p<2$$ 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.
• Are you assuming that the subspace contains almost isometric copies of $\ell^n_p$ for all $n$? For $p=1$ this is equivalent to containing copies of $\ell^n_p$, but I think there is no equivalence for $p>1$. Commented Apr 2, 2023 at 15:41
• By Krivine's theorem (finite dimensiona version) you can pass to further blocks of those $\ell_p^n$'s to get almost isometric copies. See, for instance, Maurey's Handbook article for the Krivine's theorem. Commented Apr 2, 2023 at 17:28
• @BunyaminSari: You can simplify a bit and not use Dor's theorem or Aldous/Kriviine-Maurey. Assume $p<2$. Justify that there is a normalized sequence that has big disjoint pieces. By passing to a subsequence of differences, you can assume the sequence is weakly null, and hence, by passing to another subsequence, is unconditional. You get the lower $p$-estimate from the diagonal principle and the upper $p$-estimate from type $p$. Commented Apr 11, 2023 at 20:14