Stabilizing isomorphisms from $\ell_{p}(\ell_{2})$ into $L_p[0,1]$

Abstract

Let $1< p\neq 2<\infty$, $\mathcal{E}>0$ and let $T$ be an isomorphism from $\ell_p(\ell_2)$ into $L_p[0,1]$. Then there is a subspace $Y\subset \ell_p(\ell_2)$, $(1+\mathcal{E})$-isomorphic to $\ell_p(\ell_2)$ such that $T_{|Y}$ is an $(1+\mathcal{E})$-isomorphism and $T\left(Y\right)$ is $K_p$-complemented in $L_{p}\left[0,1\right]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\mathcal{E})\gamma_p$ if $p>2$ and $K_p\le (1+\mathcal{e})\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable.