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Astashkin, S. V.

On isomorphic embeddings in the class of disjointly homogeneous rearrangement invariant spaces. (English. Russian original) Zbl 07918224

Sib. Math. J. 65, No. 3, 505-513 (2024); translation from Sib. Mat. Zh. 65, No. 3, 435-445 (2024).
Summary: 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 \((0,\infty) \).

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B25 Classical Banach spaces in the general theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47Bxx Special classes of linear operators

Keywords:

isomorphism; rearrangement invariant space; Orlicz space; Lorentz space; disjoint functions; disjointly homogeneous space; \( p \)-disjointly homogeneous space
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References:

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