Embeddings of Orlicz–Lorentz spaces into 𝐿₁

Prochno, J.

doi:10.1090/spmj/1638

Embeddings of Orlicz–Lorentz spaces into $L_1$
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by J. Prochno

St. Petersburg Math. J. 32 (2021), 59-70

DOI: https://doi.org/10.1090/spmj/1638

Published electronically: January 11, 2021

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Abstract:

It is shown that the Orlicz–Lorentz spaces $\ell ^n_{M,a}$, $n\in \mathbb {N}$, with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\| \cdot \|_{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $\mathrm {d}^n(a,p)$. The approach is based on combinatorial averaging techniques, and a new result of independent interest is proved, which relates suitable averages with Orlicz–Lorentz norms.

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