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

TL;DR

This paper derives precise bounds for the moments of conditionally centered random variables and applies these results to determine the optimal constant in the compact approximation property of $L^p$ spaces.

Contribution

It provides sharp estimates for conditionally centered moments and identifies the best constant for the compact approximation property in $L^p$ spaces.

Findings

Sharp bounds for moments of conditionally centered variables

Optimal constant in the compact approximation property of $L^p$ spaces

Enhanced understanding of operator behavior on $L^p$ spaces

Abstract

Let $(\Omega, \mathcal{F}, \mathbf{P})$ be a probability space, $\xi$ be a random variable on $(\Omega, \mathcal{F}, \mathbf{P})$, $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $\mathbf{E}^\mathcal{G} = \mathbf{ E}(\cdot | \mathcal{G})$ be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of $\xi - \mathbf{E}^\mathcal{G}\xi$ in terms of the moments of $\xi$. This allows us to find the optimal constant in the bounded compact approximation property of $L^p([0, 1])$, $1 < p < \infty$.