By Gilles Pisier

Inspired via a query of Vincent Lafforgue, the writer experiences the Banach areas X pleasant the next estate: there's a functionality \varepsilon\to \Delta_X(\varepsilon) tending to 0 with \varepsilon>0 such that each operator T\colon \ L_2\to L_2 with \|T\|\le \varepsilon that's at the same time contractive (i.e., of norm \le 1) on L_1 and on L_\infty needs to be of norm \le \Delta_X(\varepsilon) on L_2(X). the writer exhibits that \Delta_X(\varepsilon) \in O(\varepsilon^\alpha) for a few \alpha>0 if X is isomorphic to a quotient of a subspace of an ultraproduct of \theta-Hilbertian areas for a few \theta>0 (see Corollary 6.7), the place \theta-Hilbertian is intended in a touch extra basic experience than within the author's previous paper (1979)

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1. Any θ-Hilbertian space is a quotient of a subspace (or equivalently is a subquotient) of an arcwise θ-Hilbertian space. 2. 8 are equivalent to (i)’ X is C-isomorphic to a quotient of a subspace (or equivalently to a subquotient) of a θ-Hilbertian space. 6 also holds with “θ-Hilbertian” instead of “arcwise θ-Hilbertian” everywhere. CHAPTER 8 Fourier and Schur multipliers Let G be a locally compact Abelian group. Let M (G) be the classical Banach space of complex (Radon) measures on G equipped with the total variation norm: μ M (G) = |μ|(G).

Is in the unit ball of Sp (Y ) (resp. 3) ˜ v(s, t) = ξ(s), η˜(t) . Let S ⊂ Lp (μ) be any subspace supplementary to Sp (recall that we assume Sp ﬁnite dimensional). Clearly ξ ∈ Lp ⊗ Y can be written ξ = ξ1 + ξ2 with ξ1 ∈ Sp ⊗ Y and ξ2 ∈ S ⊗ Y . 4) ξ2 (s), η(t) = 0. Let Z ⊂ Y be the closed span in Y of all elements of the form x(s)ξ2 (s)dμ(s) with x a scalar valued function in Lp . Note that ξ2 is Z-valued. 4) z, η(t) = 0 for any z in Z, so that η deﬁnes an element of Lp (Z ⊥ ) with η ≤ 1. Let q1 : Y → Y /Z be the quotient map.

Consider a measurable family { z } of norms on Cn indexed by z ∈ ∂D. By measurable, we mean that z → x z is measurable for any x in Cn . 1) ∀z ∈ ∂D ∀x ∈ Cn k1 (z) x ≤ x z ≤ k2 (z) x , where x denotes the Euclidean norm of x (here any ﬁxed norm would do just as well), sometimes denoted also below by x n2 . Let X(z) = (Cn , z ). Following [15] we say that {X(z) | z ∈ ∂D} is a compatible family of Banach spaces. When this holds for constant functions k1 > 0 and k2 > 0, we will say that the family is strongly compatible.