Dichotomies for Lorentz spaces

Szymon Głąb; Filip Strobin; Chan Yang

Open Mathematics, Tome 11 (2013), p. 1228-1242 / Harvested from The Polish Digital Mathematics Library

Résumé

Assume that L p,q, $L^{p_{1}, q_{1}}, . . ., L^{p_{n}, q_{n}}$ are Lorentz spaces. This article studies the question: what is the size of the set $E = {(f_{1}, . . ., f_{n}) \in L^{p_{1,} q_{1}} \times \dots \times L^{p_{n}, q_{n}} : f_{1} \dots f_{n} \in L^{p, q}}$ . We prove the following dichotomy: either $E = L^{p_{1}, q_{1}} \times \dots \times L^{p_{n}, q_{n}}$ or E is σ-porous in $L^{p_{1}, q_{1}} \times \dots \times L^{p_{n}, q_{n}}$ , provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E = L^{p_{1}, q_{1}} \times \dots \times L^{p_{n}, q_{n}}$ or E is meager. This is a generalization of the results for classical L p spaces.

Publié le : 2013-01-01

Zbl 1273.46017

EUDML-ID : urn:eudml:doc:269740

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     author = {Szymon G\l \k ab and Filip Strobin and Chan Yang},
     title = {Dichotomies for Lorentz spaces},
     journal = {Open Mathematics},
     volume = {11},
     year = {2013},
     pages = {1228-1242},
     zbl = {1273.46017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0241-9}
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Szymon Głąb; Filip Strobin; Chan Yang. Dichotomies for Lorentz spaces. Open Mathematics, Tome 11 (2013) pp. 1228-1242. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0241-9/

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