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

Let $(Ω, \mathcal{F}, \mathbf{P})$ be a probability space, $ξ$ be a random variable on $(Ω, \mathcal{F}, \mathbf{P})$, $\mathcal{G}$ be a sub-$σ$-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 $ξ- \mathbf{E}^\mathcal{G}ξ$ in terms of the moments of $ξ$. This allows us to find the optimal constant in the bounded compact approximation property of $L^p([0, 1])$, $1 < p < \infty$.

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