On subspaces of Orlicz spaces, generated by independent copies of a mean zero function

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Abstract

We study subspaces of Orlicz spaces $L_M$ generated by independent copies $f_k$, $k=1,2,…$, of functions $f\in L_M$, $\int_0^1 f(t) dt=0$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions, guaranteeing that the unit ball of such subspace consists of functions with equicontinuous norms in $L_M$, are found. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. We prove that there is a wide class of Orlicz spaces $L_M$ (containing $L^p$-spaces, $1\le p< 2$), for which each of these properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy the inequality: $\alpha_\psi^0>\beta_M^\infty$.

About the authors

Sergei Vladimirovich Astashkin

Samara National Research University; Lomonosov Moscow State University; Moscow Center for Fundamental and Applied Mathematics; Bahçesehir University

Email: astash@ssau.ru
ORCID iD: 0000-0002-8239-5661
Doctor of physico-mathematical sciences, Professor

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