Comparing gaussian and Rademacher cotype for operators on the space of continuous functions
Volume 118 / 1996
Studia Mathematica 118 (1996), 101-115
DOI: 10.4064/sm-118-2-101-115
Abstract
We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if $(∑_k ((∥Tx_k∥_F)/(√log(k+1)))^q)^{1/q} ≤ c ∥ ∑_k ɛ_{k} x_{k} ∥_{L_{2}(C(K))}$, for all sequences $(x_k)_{k∈ℕ} ⊂ C(K)$ with $(∥Tx_k∥)_{k=1}^n$ decreasing. (2) T is of Rademacher cotype q if and only if $(∑_k (∥Tx_k∥_{F} √((log(k+1))^q) )^{1/q} ≤ c ∥∑_k g_{k}x_{k}∥_{L_2(C(K))}$, for all sequences $(x_k)_{k∈ℕ} ⊂ C(K)$ with $(∥Tx_k∥)_{k=1}^n$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.
