Stabilizing isomorphisms from ℓ_p (ℓ₂) into l_p [0, 1]

Let 1 < p ≠ 2 < ∞, ε > 0 and let T be an isomorphism from ℓp(ℓ2) into Lp[0, 1]. Then there is a subspace Y ⊂ ℓp(ℓ2), (1+ ε)-isomorphic to ℓp(ℓ2) such that T|Y is an (1+ ε)-isomorphism and T (Y) is Kp-complemented in Lp [0, 1], with Kp depending only on p. Moreover, Kp ≤ (1 + ε)γp if p > 2 and Kp ≤ (1 + ε)γp/(p−1) if 1 < p < 2, where γr is the Lr norm of a standard Gaussian variable.