On subspaces of Orlicz spaces spanned by independent copies of a mean zero function
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| Format: | Preprint |
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2024
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| _version_ | 1866917729478377472 |
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| author | Astashkin, Sergey V. |
| author_facet | Astashkin, Sergey V. |
| contents | 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 $\ell_ψ$. 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 $ψ$ satisfy the inequality: $α_ψ^0>β_M^\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_14870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On subspaces of Orlicz spaces spanned by independent copies of a mean zero function Astashkin, Sergey V. Functional Analysis 46B03, 46B09, 46A45 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 $\ell_ψ$. 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 $ψ$ satisfy the inequality: $α_ψ^0>β_M^\infty$. |
| title | On subspaces of Orlicz spaces spanned by independent copies of a mean zero function |
| topic | Functional Analysis 46B03, 46B09, 46A45 |
| url | https://arxiv.org/abs/2407.14870 |