Geodesic
On a characteristic of strongly embedded subspaces in symmetric spaces
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 2, pp. 25-32

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It is shown that the presence of a lower $p$-estimate with constant $1$ in the symmetric space $E$ is sufficient for the condition of equivalence of convergence in norm and in measure on the subspace $H$ of the space $E$ to be satisfied if and only if the numerical characteristic $\eta_ {E}(H) 1. $ The last criterion is also valid for symmetric spaces “close” to $L_ {1},$ more precisely, for which an analog of the Dunford–Pettis criterion of weak compactness is valid. In particular, it is shown that spaces “close” to $L_ {1},$ have the binary property: the characteristic $\eta_{E}(H)$ takes only two values, $0$ and $1$. This gives an example of binary Orlicz spaces different from the spaces $L_{p}$.
Keywords: rearrangement invariant space, Orlicz space, Orlicz norm, lower $p$-estimate with constant one, strongly embedded subspace, equivalent norms, convergence in measure.
Mots-clés : Luxemburg norm 
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     author = {S. I. Strakhov},
     title = {On a characteristic of strongly embedded subspaces in symmetric spaces},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {25--32},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2021_27_2_a2/}
}
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S. I. Strakhov. On a characteristic of strongly embedded subspaces in symmetric spaces. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 2, pp. 25-32. http://geodesic.mathdoc.fr/item/VSGU_2021_27_2_a2/