Geodesic
On Lattice Properties of the Lorentz Spaces~$L_{p,q}$
Matematičeskie zametki, Tome 113 (2023) no. 1, pp. 11-20

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It is shown that the space $l_r$ is crudely finitely representable in the Lorentz space $L_{p,q}[0,1]$, $1$, if and only if $r=p$ or $r=q$. To the best of the author's knowledge, this is the first example of a “natural” rearrangement-invariant space $E$ on $[0,1]$ such that the set of all numbers $r$ for which $l_r$ is crudely finitely representable in $E$ is not an interval of the real line.
Keywords: finite representability, Lorentz space, rearrangement-invariant space, Banach lattice, upper (lower) estimate, ${\mathcal K}$-functional. 
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     author = {S. V. Astashkin},
     title = {On {Lattice} {Properties} of the {Lorentz} {Spaces~}$L_{p,q}$},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
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     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_113_1_a1/}
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S. V. Astashkin. On Lattice Properties of the Lorentz Spaces~$L_{p,q}$. Matematičeskie zametki, Tome 113 (2023) no. 1, pp. 11-20. http://geodesic.mathdoc.fr/item/MZM_2023_113_1_a1/