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

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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 $L^p([0, 1])$, $1 < p < \infty$.
Original languageEnglish
Pages (from-to)368-381
Number of pages14
JournalMathematische Nachrichten
Volume296
Issue number1
Early online date18 Dec 2022
Publication statusPublished - 29 Jan 2023

Keywords

  • math.PR
  • math.FA
  • 60E15, 47A30, 46B20, 46B28, 46E30, 47B07

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