Geodesic
Compactness of Hardy Operators Involving Suprema
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 1, pp. 219-252.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We study compactness properties of Hardy operators involving suprema on weighted Banach function spaces. We first characterize the compactness of abstract operators assumed to have their range in the class of non-negative monotone functions. We then define a category of pairs of weighted Banach function spaces for which a suitable Muckenhoupt-type condition implies the boundedness of Hardy operators involving suprema, and prove a criterion for the compactness of these operators between such a couple of spaces. Finally, we characterize the compactness of these operators on weighted Lebesgue spaces including those which do not belong to the above-mentioned category. 
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Pernecká, Eva; Pick, Luboš. Compactness of Hardy Operators Involving Suprema. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 1, pp. 219-252. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_1_a9/

[1] C. Bennett - R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics Vol. 129, Academic Press, Boston 1988. | MR | Zbl

[2] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull., 21 (1978), 405-408. | DOI | MR | Zbl

[3] A. Cianchi - R. Kerman - B. Opic - L. Pick, A sharp rearrangement inequality for fractional maximal operator, Studia Math., 138 (2000), 277-284. | fulltext EuDML | MR | Zbl

[4] M. Cwikel - E. Pustylnik, Weak type interpolation near ``endpoint'' spaces, J. Funct. Anal., 171 (1999), 235-277. | DOI | MR | Zbl

[5] R. Ya. Doktorskii, Reiteration relations of the real interpolation method, Soviet Math. Dokl., 44 (1992), 665-669. | MR

[6] D. E. Edmunds - P. Gurka - L. Pick, Compactness of Hardy type integral operators in weighted Banach function spaces, Studia Math., 109 (1994), 73-90. | fulltext EuDML | MR | Zbl

[7] W. D. Evans - B. Opic, Real interpolation with logarithmic functors and reiteration, Canad. J. Math., 52 (2000), 920-960. | DOI | MR | Zbl

[8] A. Gogatishvili - B. Opic - L. Pick, Weighted inequalities for Hardy-type operators involving suprema, Collect. Math., 57, 3 (2006), 227-255. | fulltext EuDML | MR | Zbl

[9] A. Gogatishvili - L. Pick, Discretization and anti-discretization of rearrangement-invariant norms, Publ. Mat., 47 (2003), 311-358. | fulltext EuDML | DOI | MR | Zbl

[10] R. Kerman - L. Pick, Optimal Sobolev imbeddings, Forum Math., 18, 4 (2006), 535-570. | DOI | MR | Zbl

[11] W. A. J. Luxemburg - A. C. Zaanen, Compactness of integral operators in Banach function spaces, Math. Ann., 149 (1963), 150-180. | fulltext EuDML | DOI | MR | Zbl

[12] V. G. Maz'Ya, Einbettungssätze für Sobolewsche Räume. Teil 1 (Embedding Theorems for Sobolev Spaces. Part 1), Teubner-Texte zur Mathematik, Leipzig 1979. | MR

[13] V. G. Maz'Ya, Sobolev Spaces, Russian Edition Leningrad. Univ., Leningrad 1985, English Translation Springer-Verlag, Berlin-Heidelberg 1985. | DOI | MR

[14] B. Muckenhoupt, Hardy's inequality with weights, Studia Math., 44 (1972), 31-38. | fulltext EuDML | DOI | MR | Zbl

[15] L. Pick, Supremum operators and optimal Sobolev inequalities Function Spaces, Differential Operators and Nonlinear Analysis Vol. 4. Proceedings of the Spring School held in Syöte, June 1999, V. Mustonen and J. Rákosník (Eds.), Mathematical Institute of the Academy of Sciences of the Czech Republic, Praha, 2000, 207-219. | MR

[16] L. Pick, Optimal Sobolev Embeddings, Rudolph-Lipschitz-Vorlesungsreihe no. 43, Rheinische Friedrich-Wilhelms-Universität Bonn, 2002, x+144.

[17] E. Pustylnik, Optimal interpolation in spaces of Lorentz-Zygmund type, J. Anal. Math., 79 (1999), 113-157. | DOI | MR | Zbl

[18] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145-158. | fulltext EuDML | DOI | MR | Zbl