Geodesic
Net spaces on lattices, Hardy--Littlewood type inequalities, and their converses
Eurasian mathematical journal, Tome 8 (2017) no. 3, pp. 10-27

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We introduce abstract net spaces on directed sets and prove their embedding and interpolation properties. Typical examples of interest are lattices of irreducible unitary representations of compact Lie groups and of class I representations with respect to a subgroup. As an application, we prove Hardy–Littlewood type inequalities and their converses on compact Lie groups and on compact homogeneous manifolds. 
@article{EMJ_2017_8_3_a2,
     author = {R. Akylzhanov and M. Ruzhansky},
     title = {Net spaces on lattices, {Hardy--Littlewood} type inequalities, and their converses},
     journal = {Eurasian mathematical journal},
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     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2017_8_3_a2/}
}
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R. Akylzhanov; M. Ruzhansky. Net spaces on lattices, Hardy--Littlewood type inequalities, and their converses. Eurasian mathematical journal, Tome 8 (2017) no. 3, pp. 10-27. http://geodesic.mathdoc.fr/item/EMJ_2017_8_3_a2/