Abstract
The equivalence of the Haar system in a rearrangement invariant space \( X \) on \( [0,1] \) and a sequence of pairwise disjoint functions in some Lorentz space is known to imply that \( X=L_{2}[0,1] \) up to the equivalence of norms. We show that the same holds for the class of uniform disjointly homogeneous rearrangement invariant spaces and obtain a few consequences for the properties of isomorphic embeddings of such spaces. In particular, the \( L_{p}[0,1] \) space with \( 1<p<\infty \) is the only uniform \( p \)-disjointly homogeneous rearrangement invariant space on \( [0,1] \) with nontrivial Boyd indices which has two rearrangement invariant representations on the half-axis \( (0,\infty) \).
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Funding
The work was supported by the Russian Science Foundation (Grant no. 23–71–30001) at Lomonosov Moscow State University.
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Translated from Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 3, pp. 435–445. https://doi.org/10.33048/smzh.2024.65.301
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Astashkin, S.V. On Isomorphic Embeddings in the Class of Disjointly Homogeneous Rearrangement Invariant Spaces. Sib Math J 65, 505–513 (2024). https://doi.org/10.1134/S0037446624030017
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DOI: https://doi.org/10.1134/S0037446624030017
Keywords
- isomorphism
- rearrangement invariant space
- Orlicz space
- Lorentz space
- disjoint functions
- disjointly homogeneous space
- \( p \)-disjointly homogeneous space