Sharp estimates for conditionally centred moments and for compact operators 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 $L^p([0,1])$, $1<p<\infty$.

References in corpus (1)

• Real and complex operator norms