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Rosenthal’s inequalities: ${\Delta }$-norms and quasi-Banach symmetric sequence spaces

Volume 255 / 2020

Yong Jiao, Fedor Sukochev, Guangheng Xie, Dmitriy Zanin Studia Mathematica 255 (2020), 55-81 MSC: Primary 46E30; Secondary 60G50. DOI: 10.4064/sm190503-28-10 Published online: 27 May 2020

Abstract

Let $X$ be a symmetric quasi-Banach function space with the Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq 0}\subset X$ is a sequence of independent random variables. We present a necessary and sufficient condition on $X$ such that the quantity $$ \Bigl \|\, \Bigl \|\sum _{k=0}^nf_ke_k\Bigr \|_{E}\, \Bigr \|_X $$ admits an equivalent characterization in terms of disjoint copies of $(f_k)_{k=0}^n$ for every $n\ge 0$; in particular, we obtain a deterministic description of $$ \Big \|\, \Big \|\sum _{k=0}^nf_ke_k\Big \|_{\ell _q}\, \Big \|_{L_p} $$ for all $0 \lt p,q \lt \infty ,$ which is the ultimate form of Rosenthal’s inequality. We also consider the case of a $\Delta $-normed symmetric function space $X$, defined via an Orlicz function $\Phi $ satisfying the $\Delta _2$-condition. That is, we provide a formula for “$E$-valued $\Phi $-moments” $\mathbb {E}(\Phi (\|(f_k)_{k\geq 0} \|_E))$, in terms of the sum of disjoint copies of $f_k, k\geq 0.$

Authors

  • Yong JiaoSchool of Mathematics and Statistics
    Central South University
    Changsha 410075, People’s Republic of China
    e-mail
  • Fedor SukochevSchool of Mathematics and Statistics
    University of NSW
    Sydney, 2052, Australia
    e-mail
  • Guangheng XieSchool of Mathematics and Statistics
    Central South University
    Changsha 410075, People’s Republic of China
    e-mail
  • Dmitriy ZaninSchool of Mathematics and Statistics
    University of NSW
    Sydney, 2052, Australia
    e-mail

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