Type and cotype with respect to arbitrary orthonormal systems

Let Φ = (φ_k)_{k ∈ N} be an orthonormal system on some σ-finite measure space (Ω, p). We study the notion of cotype with respect to Φ for an operator T between two Banach spaces X and Y, defined by c_Φ(T): = inf c such that where (g_k)(k ∈ N) is the sequence of independent and normalized gaussian variables. It is shown that this Φ-cotype coincides with the usual notion of cotype 2 iff c_Φ(I_{lⁿ ∞} ∼ uniformly in n iff there is a positive η > 0 such that for all n ∈ ℕ one can find an orthonormal Ψ = (ψ_l)ⁿ₁ ⊂ span (φ_k|k ∈ ℕ) and a sequence of disjoint measurable sets (A_l)ⁿ₁ ⊂ Ω with ∫_Al|ψ_l|²dp ≥ η for all l = 1,., n. A similar result holds for the type situation. The study of type and cotype with respect to orthonormal systems of a given length provides the appropriate approach to this result. We intend to give a quite complete picture for orthonormal systems in measure space with few atoms.