Small Subspaces of L_p

TL;DR

This paper characterizes small subspaces of L_p for 2<p<∞, showing they either embed into a sum of ℓ_p and ℓ_2 or contain a subspace isomorphic to ℓ_p(ℓ_2), solving longstanding problems from the 1970s.

Contribution

It provides an intrinsic characterization of subspace embeddings in L_p using weakly null trees and the infinite asymptotic game, advancing understanding of their structure.

Findings

Subspaces of L_p either embed into ℓ_p ⊕ ℓ_2 or contain ℓ_p(ℓ_2)

Characterization via weakly null trees and asymptotic game

Solves problems from the 1970s about small subspaces of L_p

Abstract

We prove that if $X$ is a subspace of $L_p$ $(2<p<\infty)$, then either $X$ embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace $Y,$ which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic characterization of when $X$ embeds into $\ell_p \oplus \ell_2$ in terms of weakly null trees in $X$ or, equivalently, in terms of the "infinite asymptotic game" played in $X$. This solves problems concerning small subspaces of $L_p$ originating in the 1970's. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000's.