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Eurasian Math. J., 2012, Volume 3, Number 3, Pages 11–32 (Mi emj94)  

This article is cited in 31 scientific papers (total in 31 papers)

Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I

V. I. Burenkov

Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, Kazakhstan

Abstract: The survey is aimed at providing detailed information about recent results in the problem of the boundedness in general Morrey-type spaces of various important operators of real analysis, namely of the maximal operator, fractional maximal operator, Riesz potential, singular integral operator, Hardy operator. The main focus is on the results which contain, for a certain range of the numerical parameters, necessary and sufficient conditions on the functional parameters characterizing general Morrey-type spaces, ensuring the boundedness of the aforementioned operators from one general Morrey-type space to another one. The major part of the survey is dedicated to the results obtained by the author jointly with his co-authores A. Gogatishvili, M. L. Goldman, H. V. Guliyev, V. S. Guliyev, P. Jain, R. Mustafaev, E. D. Nursultanov, R. Oinarov, A. Serbetci, T. V. Tararykova. Part I of the survey contains discussion of the definition and basic properties of the local and global general Morrey-type spaces, of embedding theorems, and of the boundedness properties of the maximal operator. Part II of the survey will contain discussion of boundedness properties of the fractional maximal operator, Riesz potential, singular integral operator, commutators of singular integral operator, Hardy operator. It will also contain discussion of interpolation theorems, of methods of proofs and of open problems.

Keywords and phrases: local and global Morrey-type spaces, embedding operator, maximal operator, fractional maximal operator, Riesz potential, singular integral operator, Hardy operator, interpolation.

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MSC: 42B20, 42B25, 42B35, 46E30, 47B38
Received: 15.07.2011
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Citation: V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I”, Eurasian Math. J., 3:3 (2012), 11–32

Citation in format AMSBIB
\Bibitem{Bur12}
\by V.~I.~Burenkov
\paper Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces.~I
\jour Eurasian Math. J.
\yr 2012
\vol 3
\issue 3
\pages 11--32
\mathnet{http://mi.mathnet.ru/emj94}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3024128}
\zmath{https://zbmath.org/?q=an:1270.42014}


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    This publication is cited in the following articles:
    1. A. Akbulut, V. S. Guliev, Sh. A. Muradova, “On the boundedness of the anisotropic fractional maximal operator from anisotropic complementary Morrey-type spaces to anisotropic Morrey-type spaces”, Eurasian Math. J., 4:1 (2013), 7–20  mathnet  mathscinet  zmath
    2. V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. II”, Eurasian Math. J., 4:1 (2013), 21–45  mathnet  mathscinet  zmath
    3. V. I. Burenkov, D. K. Darbayeva, E. D. Nursultanov, “Description of interpolation spaces for general local Morrey-type spaces”, Eurasian Math. J., 4:1 (2013), 46–53  mathnet  mathscinet  zmath
    4. A. Gogatishvili, R. Ch. Mustafayev, “New characterization of Morrey spaces”, Eurasian Math. J., 4:1 (2013), 54–64  mathnet  mathscinet  zmath
    5. W. Sickel, “Smoothness spaces related to Morrey spaces — a survey. II”, Eurasian Math. J., 4:1 (2013), 82–124  mathnet  mathscinet  zmath
    6. T. V. Tararykova, “Comments on definitions of general local and global Morrey-type spaces”, Eurasian Math. J., 4:1 (2013), 125–134  mathnet  mathscinet  zmath
    7. Sawano Y., Shimomura T., “Sobolev Embeddings For Generalized Riesz Potentials of Functions in Morrey Spaces l-(1,l-Phi)(G) Over Nondoubling Measure Spaces”, J. Funct. Space Appl., 2013, 984259  crossref  mathscinet  zmath  isi  scopus
    8. V. I. Burenkov, E. D. Nursultanov, D. K. Chigambayeva, “Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations”, Proc. Steklov Inst. Math., 284 (2014), 97–128  mathnet  crossref  crossref  isi  elib  elib
    9. V. V. Kisil, “The real and complex techniques in harmonic analysis from the point of view of covariant transform”, Eurasian Math. J., 5:1 (2014), 95–121  mathnet
    10. Burenkov V.I., Goldman M.L., “Necessary and Sufficient Conditions For the Boundedness of the Maximal Operator From Lebesgue Spaces To Morrey-Type Spaces”, Math. Inequal. Appl., 17:2 (2014), 401–418  crossref  mathscinet  zmath  isi  elib  scopus
    11. Mustafayev R.Ch., Unver T., “Embeddings Between Weighted Local Morrey-Type Spaces and Weighted Lebesgue Spaces”, J. Math. Inequal., 9:1 (2015), 277–296  crossref  mathscinet  zmath  isi  elib  scopus
    12. A. N. Karapetyants, S. G. Samko, “Mixed Norm Bergman–Morrey-type Spaces on the Unit Disc”, Math. Notes, 100:1 (2016), 38–48  mathnet  crossref  crossref  mathscinet  isi  elib
    13. V. I. Burenkov, T. V. Tararykova, “An analog of Young's inequality for convolutions of functions for general Morrey-type spaces”, Proc. Steklov Inst. Math., 293 (2016), 107–126  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. K.-P. Ho, “Hardy-Littlewood-Polya inequalities and Hausdorff operators on block spaces”, Math. Inequal. Appl., 19:2 (2016), 697–707  crossref  mathscinet  zmath  isi  scopus
    15. N. A. Bokayev, V. I. Burenkov, D. T. Matin, “Sufficient conditions for pre-compactness of sets in the generalized Morrey spaces”, Bull. Karaganda Univ-Math., 84:4 (2016), 18–26  isi
    16. V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, v. 2, Operator Theory Advances and Applications, 249, Variable Exponent Holder, Morrey-Campanato and Grand Spaces, Springer, 2016, 569–1003  crossref  mathscinet  isi
    17. N. Bokayev, V. Burenkov, D. Matin, “On the pre-compactness of a set in the generalized Morrey spaces”, International conference on analysis and applied mathematics ICAAM 2016, AIP Conf. Proc., 1759, eds. A. Ashyralyev, A. Lukashov, Amer. Inst. Phys., 2016, 020108  crossref  isi  scopus
    18. E. Nakai, T. Sobukawa, “$B_w^u$-function spaces and their interpolation”, Tokyo J. Math., 39:2 (2016), 483–516  crossref  mathscinet  zmath  isi  scopus
    19. V. I. Burenkov, T. V. Tararykova, “Young's inequality for convolutions in Morrey-type spaces”, Eurasian Math. J., 7:2 (2016), 92–99  mathnet  mathscinet  isi
    20. E. I. Berezhnoi, “A discrete version of local Morrey spaces”, Izv. Math., 81:1 (2017), 1–28  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. A. Gogatishvili, R. Mustafayev, T. Ünver, “Embedding relations between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces”, Eurasian Math. J., 8:1 (2017), 34–49  mathnet
    22. O. G. Avsyankin, “On the Compactness of Convolution-Type Operators in Morrey Spaces”, Math. Notes, 102:4 (2017), 437–443  mathnet  crossref  crossref  mathscinet  isi  elib
    23. V. I. Burenkov, E. Liflyand, “On the boundedness of Hausdorff operators on Morrey-type spaces”, Eurasian Math. J., 8:2 (2017), 97–104  mathnet  mathscinet
    24. N. Bokayev, V. Burenkov, D. Matin, “Sufficient conditions for the pre-compactness of sets in global Morrey-type spaces”, International Conference Functional Analysis in Interdisciplinary Applications FAIA 2017, AIP Conf. Proc., 1880, eds. T. Kalmenov, M. Sadybekov, Amer. Inst. Phys., 2017, UNSP 030001  crossref  isi
    25. N. A. Bokayev, V. I. Burenkov, D. T. Matin, “On precompactness of a set in general local and global Morrey-type spaces”, Eurasian Math. J., 8:3 (2017), 109–115  mathnet  mathscinet
    26. Y. Sawano, T. Shimomura, “Boundedness of the generalized fractional integral operators on generalized Morrey spaces over metric measure spaces”, Z. Anal. ihre. Anwend., 36:2 (2017), 159–190  crossref  isi
    27. O. G. Avsyankin, “Compactness of Some Operators of Convolution Type in Generalized Morrey Spaces”, Math. Notes, 104:3 (2018), 331–338  mathnet  crossref  crossref  isi  elib
    28. Bokayev N. Matin D. Baituyakova Zh., “A Sufficient Condition For Compactness of the Commutators of Riesz Potential on Global Morrey-Type Space”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conference Proceedings, 1997, eds. Ashyralyev A. Lukashov A. Sadybekov M., Amer Inst Physics, 2018, UNSP 020008-1  crossref  isi  scopus
    29. V. I. Burenkov, D. K. Chigambayeva, E. D. Nursultanov, “Marcinkiewicz-type interpolation theorem and estimates for convolutions for Morrey-type spaces”, Eurasian Math. J., 9:2 (2018), 82–88  mathnet  mathscinet  isi
    30. N. R. Ahmedzade, Z. A. Kasumov, “On the Dirichlet problem for the Laplace equation with the boundary value in Morrey space”, Eurasian Math. J., 9:4 (2018), 9–21  mathnet  crossref
    31. Matin D.T. Baituyakova Zh.Zh. Adilkhanov A.N. Bostanov B.O., “Sufficient Conditions For the Precompactness of Sets in Local Morrey-Type Spaces”, Bull. Karaganda Univ-Math., 92:4 (2018), 54–63  crossref  isi
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