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Eurasian Math. J., 2011, Volume 2, Number 1, Pages 52–80 (Mi emj42)  

This article is cited in 21 scientific papers (total in 21 papers)

On boundedness of the Hardy operator in Morrey-type spaces

V. I. Burenkova, P. Jainb, T. V. Tararykovaa

a Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, Kazakhstan
b Department of Mathematics, Deshbandhu College, University of Delhi, New Delhi, India

Abstract: In this paper we study the boundedness of the Hardy operator $H_\alpha$ in local and global Morrey-type spaces $LM_{p\theta,w(\cdot)}$, $GM_{p\theta,w(\cdot)}$ respectively, characterized by numerical parameters $p,\theta$ and a functional parameter $w$. We reduce this problem to the problem of a continuous embedding of one local Morrey-type space to another one. This allows obtaining, for all admissible values of the numerical parameters $\alpha,p_1,p_2,\theta_1,\theta_2$, sufficient conditions on the functional parameters $w_1$ and $w_2$ ensuring the boundedness of $H_\alpha$ from $LM_{p_1\theta_1,w_1(\cdot)}$ to $LM_{p_2\theta_2,w_2(\cdot)}$ and from $GM_{p_1\theta_1,w_1(\cdot)}$ to $GM_{p_2\theta_2,w_2(\cdot)}$. Moreover, for a certain range of the numerical parameters and under certain a priori assumptions on $w_1$ and $w_2$ these sufficient conditions coincide with the necessary ones.

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

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MSC: 47B38, 46E30
Received: 14.01.2011

Citation: V. I. Burenkov, P. Jain, T. V. Tararykova, “On boundedness of the Hardy operator in Morrey-type spaces”, Eurasian Math. J., 2:1 (2011), 52–80

Citation in format AMSBIB
\by V.~I.~Burenkov, P.~Jain, T.~V.~Tararykova
\paper On boundedness of the Hardy operator in Morrey-type spaces
\jour Eurasian Math. J.
\yr 2011
\vol 2
\issue 1
\pages 52--80

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    This publication is cited in the following articles:
    1. 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  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. 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
    4. Burenkov V.I., Oinarov R., “Necessary and Sufficient Conditions For Boundedness of the Hardy-Type Operator From a Weighted Lebesgue Space to a Morrey-Type Space”, Math. Inequal. Appl., 16:1 (2013), 1–19  crossref  mathscinet  zmath  isi  scopus
    5. 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
    6. Zhao F., Lu Sh., “the Best Bound For N-Dimensional Fractional Hardy Operators”, Math. Inequal. Appl., 18:1 (2015), 233–240  crossref  mathscinet  zmath  isi  scopus
    7. 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
    8. D. V. Prokhorov, V. D. Stepanov, “Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces”, Sb. Math., 207:8 (2016), 1159–1186  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. 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
    10. V. I. Burenkov, N. A. Kydyrmina, “Sobolev embedding theorem for the Sobolev-Morrey spaces”, Bull. Karaganda Univ-Math., 83:3 (2016), 32–40  mathscinet  isi
    11. V. I. Burenkov, T. V. Tararykova, “Young’s inequality for convolutions in Morrey-type spaces”, Eurasian Math. J., 7:2 (2016), 92–99  mathnet
    12. E. I. Berezhnoi, “A discrete version of local Morrey spaces”, Izv. Math., 81:1 (2017), 1–28  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. 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
    14. 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
    15. V. S. Guliyev, S. G. Hasanov, Y. Sawano, “Decompositions of Local Morrey-Type Spaces”, Positivity, 21:3 (2017), 1223–1252  crossref  isi
    16. N. Samko, “Commutators With Coefficients in Cmo of Weighted Hardy Operators in Generalized Local Morrey Spaces”, Mediterr. J. Math., 14:2 (2017)  crossref  isi
    17. 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 Conference Proceedings, 1880, eds. T. Kalmenov, M. Sadybekov, Amer Inst Physics, 2017, UNSP 030001  crossref  isi
    18. H Amjad, A. Mudassar, “Weak and Strong Type Estimates For the Commutators of Hausdorff Operators”, Math. Inequal. Appl., 20:1 (2017), 49–56  crossref  isi
    19. Ts. Batbold, Y. Sawano, “Sharp Bounds For M-Linear Hilbert-Type Operators on the Weighted Morrey Spaces”, Math. Inequal. Appl., 20:1 (2017), 263–283  crossref  isi
    20. 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
    21. Hakim D.I., “Complex Interpolation of Predual Spaces of General Local Morrey-Type Spaces”, Banach J. Math. Anal., 12:3 (2018), 541–571  crossref  mathscinet  zmath  isi
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