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Riesz potential in the local Morrey–Lorentz spaces and some applications

  • Vagif S. Guliyev , Abdulhamit Kucukaslan , Canay Aykol and Ayhan Serbetci EMAIL logo

Abstract

In this paper, the necessary and sufficient conditions are found for the boundedness of the Riesz potential Iα in the local Morrey–Lorentz spaces Mp,q;λloc(n). This result is applied to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups on the local Morrey–Lorentz spaces Mp,q;λloc(n).

MSC 2010: 42B20; 42B35; 47G10

Funding statement: The research of V. S. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1.

Acknowledgements

The authors would like to express their gratitude to the referees for their valuable comments and suggestions.

References

[1] J. Alvarez, J. Lakey and M. Guzmán-Partida, Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math. 51 (2000), no. 1, 1–47. Search in Google Scholar

[2] K. F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26. 10.4064/sm-72-1-9-26Search in Google Scholar

[3] C. Aykol, V. S. Guliyev, A. Kucukaslan and A. Serbetci, The boundedness of Hilbert transform in the local Morrey–Lorentz spaces, Integral Transforms Spec. Funct. 27 (2016), no. 4, 318–330. 10.1080/10652469.2015.1121483Search in Google Scholar

[4] C. Aykol, V. S. Guliyev and A. Serbetci, Boundedness of the maximal operator in the local Morrey–Lorentz spaces, J. Inequal. Appl. 2013 (2013), Paper No. 346. 10.1186/1029-242X-2013-346Search in Google Scholar

[5] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988. Search in Google Scholar

[6] V. I. Burenkov, H. V. Guliyev and V. S. Guliyev, Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces, J. Comput. Appl. Math. 208 (2007), no. 1, 280–301. 10.1016/j.cam.2006.10.085Search in Google Scholar

[7] V. I. Burenkov and V. S. Guliyev, Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal. 30 (2009), no. 3, 211–249. 10.1007/s11118-008-9113-5Search in Google Scholar

[8] V. I. Burenkov, V. S. Guliyev, A. Serbetci and T. V. Tararykova, Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces, Eurasian Math. J. 1 (2010), no. 1, 32–53. 10.1134/S1064562408050025Search in Google Scholar

[9] A.-P. Calderón, Spaces between L1 and L and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. 10.4064/sm-26-3-301-304Search in Google Scholar

[10] F. Chiarenza and M. Frasca, Morrey spaces and Hardy–Littlewood maximal function, Rend. Mat. Appl. (7) 7 (1987), no. 3–4, 273–279. Search in Google Scholar

[11] G. Di Fazio and M. A. Ragusa, Commutators and Morrey spaces, Boll. Unione Mat. Ital. A (7) 5 (1991), no. 3, 323–332. Search in Google Scholar

[12] V. S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in n, (in Russian), Doctor’s degree dissertation, Steklov Mathematical Institute, Moscow, 1994. Search in Google Scholar

[13] V. S. Guliyev, Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications (in Russian), Elm, Baku, 1999. Search in Google Scholar

[14] V. S. Guliyev, C. Aykol, A. Kucukaslan and A. Serbetci, Maximal operator and Calderon–Zygmund operators in local Morrey–Lorentz spaces, Integral Transforms Spec. Funct. 27 (2016), no. 11, 866–877. 10.1080/10652469.2016.1227329Search in Google Scholar

[15] V. S. Guliyev, A. Serbetci and I. Ekincioglu, Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces, J. Math. Anal. Appl. 336 (2007), no. 1, 425–437. 10.1016/j.jmaa.2007.02.080Search in Google Scholar

[16] K.-P. Ho, Sobolev–Jawerth embedding of Triebel–Lizorkin–Morrey–Lorentz spaces and fractional integral operator on Hardy type spaces, Math. Nachr. 287 (2014), no. 14–15, 1674–1686. 10.1002/mana.201300217Search in Google Scholar

[17] S. Lu, Y. Ding and D. Yan, Singular Integrals and Related Topics, World Scientific, Hackensack, 2007. 10.1142/6428Search in Google Scholar

[18] G. Mingione, Gradient estimates below the duality exponent, Math. Ann. 346 (2010), no. 3, 571–627. 10.1007/s00208-009-0411-zSearch in Google Scholar

[19] R. O’Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129–142. Search in Google Scholar

[20] M. A. Ragusa, Embeddings for Morrey–Lorentz spaces, J. Optim. Theory Appl. 154 (2012), no. 2, 491–499. 10.1007/s10957-012-0012-ySearch in Google Scholar

[21] N. Samko, Weighted Hardy and singular operators in Morrey spaces, J. Math. Anal. Appl. 350 (2009), no. 1, 56–72. 10.1016/j.jmaa.2008.09.021Search in Google Scholar

[22] N. Samko, Weighted Hardy and potential operators in Morrey spaces, J. Funct. Spaces Appl. 2012 (2012), Article ID 678171. 10.1155/2012/678171Search in Google Scholar

[23] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), no. 2, 145–158. 10.4064/sm-96-2-145-158Search in Google Scholar

[24] E. M. Stein, On the functions of Littlewood–Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430–466. 10.1090/S0002-9947-1958-0112932-2Search in Google Scholar

[25] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar

Received: 2016-06-17
Revised: 2016-10-10
Accepted: 2017-01-12
Published Online: 2018-10-30
Published in Print: 2020-12-01

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