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Lorentz norm inequalities for the Hardy operator involving suprema
Author:
Dmitry V. Prokhorov
Journal:
Proc. Amer. Math. Soc. 140 (2012), 1585-1592
MSC (2010):
Primary 26D15; Secondary 47G10
Posted:
January 4, 2012
MathSciNet review:
2869142
Full-text PDF
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References |
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Additional Information
Abstract: The weighted Lorentz norm inequalities for the Hardy operator involving suprema are characterized.
- 1.
Kenneth
F. Andersen and Benjamin
Muckenhoupt, Weighted weak type Hardy inequalities with
applications to Hilbert transforms and maximal functions, Studia Math.
72 (1982), no. 1, 9–26. MR 665888
(83k:42018)
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Bennett and Robert
Sharpley, Interpolation of operators, Pure and Applied
Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988. MR 928802
(89e:46001)
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María
J. Carro and Javier
Soria, Weighted Lorentz spaces and the Hardy operator, J.
Funct. Anal. 112 (1993), no. 2, 480–494. MR 1213148
(94f:42025), http://dx.doi.org/10.1006/jfan.1993.1042
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Huann
Ming Chung, Richard
A. Hunt, and Douglas
S. Kurtz, The Hardy-Littlewood maximal function on
𝐿(𝑝,𝑞) spaces with weights, Indiana Univ.
Math. J. 31 (1982), no. 1, 109–120. MR 642621
(83b:42021), http://dx.doi.org/10.1512/iumj.1982.31.31012
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David
E. Edmunds, Petr
Gurka, and Luboš
Pick, Compactness of Hardy-type integral operators in weighted
Banach function spaces, Studia Math. 109 (1994),
no. 1, 73–90. MR 1267713
(95c:47033)
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Amiran
Gogatishvili, Bohumír
Opic, and Luboš
Pick, Weighted inequalities for Hardy-type operators involving
suprema, Collect. Math. 57 (2006), no. 3,
227–255. MR 2264321
(2007g:26019)
- 7.
Amiran
Gogatishvili and Luboš
Pick, A reduction theorem for supremum operators, J. Comput.
Appl. Math. 208 (2007), no. 1, 270–279. MR 2347749
(2009a:26013), http://dx.doi.org/10.1016/j.cam.2006.10.048
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M.
L. Gol′dman, H.
P. Heinig, and V.
D. Stepanov, On the principle of duality in Lorentz spaces,
Canad. J. Math. 48 (1996), no. 5, 959–979. MR 1414066
(97h:42008), http://dx.doi.org/10.4153/CJM-1996-050-3
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Elena
Lomakina and Vladimir
Stepanov, On the compactness and approximation numbers of
Hardy-type integral operators in Lorentz spaces, J. London Math. Soc.
(2) 53 (1996), no. 2, 369–382. MR 1373067
(97f:47031), http://dx.doi.org/10.1112/jlms/53.2.369
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Dmitry
V. Prokhorov, Inequalities for Riemann-Liouville operator involving
suprema, Collect. Math. 61 (2010), no. 3,
263–276. MR 2732371
(2011j:26026), http://dx.doi.org/10.1007/BF03191232
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Eric
Sawyer, Weighted Lebesgue and Lorentz norm
inequalities for the Hardy operator, Trans.
Amer. Math. Soc. 281 (1984), no. 1, 329–337. MR 719673
(85f:26013), http://dx.doi.org/10.1090/S0002-9947-1984-0719673-4
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Vladimir
D. Stepanov, Weighted norm inequalities for integral operators and
related topics, Nonlinear analysis, function spaces and applications,
Vol. 5 (Prague, 1994), Prometheus, Prague, 1994, pp. 139–175.
MR
1322312 (96m:26019)
- 1.
- K. F. Andersen and B. Muckenhoupt.
Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions.
Stud. Math., 72:9-26, 1982. MR 665888 (83k:42018)
- 2.
- C. Bennett and R. Sharpley.
Interpolation of Operators.
Pure and Applied Mathematics 129, Academic Press, Inc., Boston, MA, 1988. MR 928802 (89e:46001)
- 3.
- M. J. Carro and J. Soria.
Weighted Lorentz spaces and the Hardy operator.
J. Funct. Anal., 112(2):480-494, 1993. MR 1213148 (94f:42025)
- 4.
- H.-M. Chung, R. A. Hunt, and D. S. Kurtz.
The Hardy-Littlewood maximal function on  spaces with weights.
Indiana Univ. Math. J., 31:109-120, 1982. MR 642621 (83b:42021)
- 5.
- D. E. Edmunds, P. Gurka, and L. Pick.
Compactness of Hardy-type integral operators in weighted Banach function spaces.
Stud. Math., 109(1):73-90, 1994. MR 1267713 (95c:47033)
- 6.
- A. Gogatishvili, B. Opic, and L. Pick.
Weighted inequalities for Hardy-type operators involving suprema.
Collect. Math., 57(3):227-255, 2006. MR 2264321 (2007g:26019)
- 7.
- A. Gogatishvili and L. Pick.
A reduction theorem for supremum operators.
J. Comput. Appl. Math., 208(1):270-279, 2007. MR 2347749 (2009a:26013)
- 8.
- M. Gol
 dman, H. Heinig, and V. Stepanov.
On the principle of duality in Lorentz spaces.
Can. J. Math., 48(5):959-979, 1996. MR 1414066 (97h:42008)
- 9.
- E. Lomakina and V. Stepanov.
On the compactness and approximation numbers of Hardy-type integral operators in Lorentz spaces.
J. Lond. Math. Soc., II. Ser., 53(2):369-382, 1996. MR 1373067 (97f:47031)
- 10.
- D. V. Prokhorov.
Inequalities for Riemann-Liouville operator involving suprema.
Collect. Math., 61(3):263-276, 2010. MR 2732371 (2011j:26026)
- 11.
- E. Sawyer.
Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator.
Trans. Amer. Math. Soc., 281:329-337, 1984. MR 719673 (85f:26013)
- 12.
- V. D. Stepanov.
Weighted norm inequalities for integral operators and related topics.
Krbec, Miroslav (ed.) et al., Nonlinear analysis, function spaces and applications. Vol. 5. Proceedings of the spring school held in Prague, May 23-28, 1994. Prague: Prometheus Publishing House. 139-175, 1994. MR 1322312 (96m:26019)
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Additional Information
Dmitry V. Prokhorov
Affiliation:
Computing Centre of the Far Eastern Branch of the Russian Academy of Sciences, Kim Yu Chen 65, Khabarovsk 680000, Russia
Email:
prohorov@as.khb.ru
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-10976-1
PII:
S 0002-9939(2012)10976-1
Keywords:
Weighted Lorentz norm inequalities,
Hardy operator involving suprema
Received by editor(s):
August 12, 2010
Received by editor(s) in revised form:
November 1, 2010
Posted:
January 4, 2012
Communicated by:
Richard Rochberg
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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