Stabilizing isomorphisms from ${\ell }_p({\ell }_2)$ into $L_p[0,1]$

Abstract

Let $1<p\ne 2<\infty$, $\epsilon >0$ and let $T:{\ell }_p({\ell }_2)\stackrel{into}{\to }{L}_p[0,1]$ 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.