Sharp estimates for conditionally centred moments and for compact operators on $L^p$ spaces

Abstract

Let (Ω, F , P) be a probability space, ξ be a random variable on (Ω, F , P), G be a sub-σ-algebra of F , and let E G = E(•|G) be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of ξ − E G ξ in terms of the moments of ξ. This allows us to find the optimal constant in the bounded compact approximation property of L p ([0, 1]), 1 < p < ∞.