Compactness of Hardy Operators Involving Suprema
Bollettino dell'Unione Matematica Italiana (2013)
- Volume: 6, Issue: 1, page 219-252
- ISSN: 0392-4041
Access Full Article
top
Abstract
topHow to cite
topPernecká, Eva, and Pick, Luboš. "Compactness of Hardy Operators Involving Suprema." Bollettino dell'Unione Matematica Italiana 6.1 (2013): 219-252. <http://eudml.org/doc/294057>.
@article{Pernecká2013,
abstract = {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.},
author = {Pernecká, Eva, Pick, Luboš},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {219-252},
publisher = {Unione Matematica Italiana},
title = {Compactness of Hardy Operators Involving Suprema},
url = {http://eudml.org/doc/294057},
volume = {6},
year = {2013},
}
TY - JOUR
AU - Pernecká, Eva
AU - Pick, Luboš
TI - Compactness of Hardy Operators Involving Suprema
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/2//
PB - Unione Matematica Italiana
VL - 6
IS - 1
SP - 219
EP - 252
AB - 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.
LA - eng
UR - http://eudml.org/doc/294057
ER -
References
top- BENNETT, C. - SHARPLEY, R., Interpolation of Operators, Pure and Applied Mathematics Vol. 129, Academic Press, Boston1988. Zbl0647.46057MR928802
- BRADLEY, J. S., Hardy inequalities with mixed norms, Canad. Math. Bull., 21 (1978), 405-408. Zbl0402.26006MR523580DOI10.4153/CMB-1978-071-7
- CIANCHI, A. - KERMAN, R. - OPIC, B. - PICK, L., A sharp rearrangement inequality for fractional maximal operator, Studia Math., 138 (2000), 277-284. Zbl0968.42014MR1758860
- CWIKEL, M. - PUSTYLNIK, E., Weak type interpolation near ``endpoint'' spaces, J. Funct. Anal., 171 (1999), 235-277. Zbl0978.46008MR1745635DOI10.1006/jfan.1999.3502
- DOKTORSKII, R. YA., Reiteration relations of the real interpolation method, Soviet Math. Dokl., 44 (1992), 665-669. MR1153547
- EDMUNDS, D. E. - GURKA, P. - PICK, L., Compactness of Hardy type integral operators in weighted Banach function spaces, Studia Math., 109 (1994), 73-90. Zbl0821.46036MR1267713
- EVANS, W. D. - OPIC, B., Real interpolation with logarithmic functors and reiteration, Canad. J. Math., 52 (2000), 920-960. Zbl0981.46058MR1782334DOI10.4153/CJM-2000-039-2
- GOGATISHVILI, A. - OPIC, B. - PICK, L., Weighted inequalities for Hardy-type operators involving suprema, Collect. Math., 57, 3 (2006), 227-255. Zbl1116.47041MR2264321
- GOGATISHVILI, A. - PICK, L., Discretization and anti-discretization of rearrangement-invariant norms, Publ. Mat., 47 (2003), 311-358. Zbl1066.46023MR2006487DOI10.5565/PUBLMAT_47203_02
- KERMAN, R. - PICK, L., Optimal Sobolev imbeddings, Forum Math., 18, 4 (2006), 535-570. Zbl1120.46018MR2254384DOI10.1515/FORUM.2006.028
- LUXEMBURG, W. A. J. - ZAANEN, A. C., Compactness of integral operators in Banach function spaces, Math. Ann., 149 (1963), 150-180. Zbl0106.30804MR145374DOI10.1007/BF01349240
- MAZ'YA, V. G., Einbettungssätze für Sobolewsche Räume. Teil 1 (Embedding Theorems for Sobolev Spaces. Part 1), Teubner-Texte zur Mathematik, Leipzig1979. MR571891
- MAZ'YA, V. G., Sobolev Spaces, Russian Edition Leningrad. Univ., Leningrad1985, English Translation Springer-Verlag, Berlin-Heidelberg1985. MR817985DOI10.1007/978-3-662-09922-3
- MUCKENHOUPT, B., Hardy's inequality with weights, Studia Math., 44 (1972), 31-38. Zbl0236.26015MR311856DOI10.4064/sm-44-1-31-38
- PICK, L., 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. MR1755311
- PICK, L., Optimal Sobolev Embeddings, Rudolph-Lipschitz-Vorlesungsreihe no. 43, Rheinische Friedrich-Wilhelms-Universität Bonn, 2002, x+144.
- PUSTYLNIK, E., Optimal interpolation in spaces of Lorentz-Zygmund type, J. Anal. Math., 79 (1999), 113-157. Zbl0987.46035MR1749309DOI10.1007/BF02788238
- SAWYER, E., Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145-158. Zbl0705.42014MR1052631DOI10.4064/sm-96-2-145-158
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.