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

Abstract

Let (Ω,  , 𝐏) be a probability space, 𝜉 be a random variable on (Ω,  , 𝐏),  be a sub-𝜎-algebra of  , and let 𝐄  = 𝐄(⋅|) be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of 𝜉 − 𝐄  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 < ∞.