Geodesic
Criteria for compactness in $L^p$-spaces, $p\geqslant0$
Sbornik. Mathematics, Tome 203 (2012) no. 7, pp. 1045-1064 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper puts forward new compactness criteria for spaces of summable and measurable functions on a metric space with measure satisfying the doubling condition. These criteria are formulated in terms of either local smoothness inequalities or maximal operators that measure local smoothness. Bibliography: 28 titles.
Keywords: compactness, total boundedness, space of summable functions, space of measurable functions, maximal operators, local smoothness. 
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V. G. Krotov. Criteria for compactness in $L^p$-spaces, $p\geqslant0$. Sbornik. Mathematics, Tome 203 (2012) no. 7, pp. 1045-1064. http://geodesic.mathdoc.fr/item/SM_2012_203_7_a5/

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