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

Eugene Shargorodsky and Teo Sharia

Mathematische Nachrichten, 2023, vol. 296, issue 1, 368-381

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

Date: 2023
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