Stabilizing isomorphisms from $\ell_p(\ell_2)$ into $L_p[0,1]$

TL;DR

This paper proves that any isomorphism from the Banach space _p(_2) into L_p[0,1] can be stabilized to a nearly isometric embedding on a large subspace, which is complemented with a universal bound depending only on p.

Contribution

It establishes a stabilization result for isomorphisms from _p(_2) into L_p[0,1], ensuring the existence of a nearly isometric, complemented subspace with bounds depending solely on p.

Findings

Existence of a nearly isometric subspace within _p(_2)

Complemented subspace with universal bounds in L_p[0,1]

Bounds depend only on p and Gaussian norms

Abstract

Let $1<p\not=2<\infty$, $\epsilon>0$ and let $T:\ell_p(\ell_2)\overset{into}{\rightarrow}L_p[0,1]$ be an isomorphism. Then there is a subspace $Y\subset \ell_p(\ell_2)$ $(1+\epsilon)$-isomorphic to $\ell_p(\ell_2)$ such that: $T_{|Y}$ is an $(1+\epsilon)$-isomorphism and $T(Y)$ is $K_p$-complemented in $L_p[0,1]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\epsilon)\gamma_p$ if $p>2$ and $K_p\le (1+\epsilon)\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable.