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

TL;DR

This paper investigates subspaces of Orlicz spaces generated by independent zero-mean functions, characterizing their structure and embedding properties through dilations and Matuszewska-Orlicz indices.

Contribution

It provides a characterization of subspaces spanned by independent copies in Orlicz spaces and links their properties to the indices of the Orlicz functions.

Findings

Subspaces are isomorphic to Orlicz sequence spaces.

Conditions for strong embedding based on dilations are established.

Characterization of properties using Matuszewska-Orlicz indices.

Abstract

We study subspaces of Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,\dots$, of a function $f\in L_M$, $\int_0^1 f(t)\,dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. 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 a subspace consists of functions with equicontinuous norms in $L_M$, are found. In particular, 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 the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy the inequality: $\alpha_\psi^0>\beta_M^\infty$.