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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board Member: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, J. / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

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Riesz type potential operators in generalized grand Morrey spaces

1Department of Mathematical Analysis, I. Javakhishvili Tbilisi State University, 2 University St., Tbilisi 0186, Georgia; and International Black Sea University, 3 Agmashenebeli Ave., Tbilisi 0131, Georgia

2Department of Mathematical Analysis, I. Javakhishvili Tbilisi State University, 2 University St., Tbilisi 0186, Georgia; Department of Mathematics, Georgian Technical University, 77 Kostava St., Tbilisi 0175, Georgia; and Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town, Lahore, Pakistan

3Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7a no. 43–82 Ed. Carlos Ortiz 604, Bogotá, Colombia; and Instituto Superior Técnico, Departamento de Matemática, Centro CEAF, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Citation Information: Georgian Mathematical Journal. Volume 20, Issue 1, Pages 43–64, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: 10.1515/gmj-2013-0009, March 2013

Publication History

Received:
2012-04-30
Revised:
2012-07-26
Accepted:
2012-09-04
Published Online:
2013-03-06

Abstract.

We introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy–Littlewood maximal operator as well as the boundedness of Calderón–Zygmund operators. The boundedness of Riesz type potential operators is also obtained in the framework of homogeneous and also in the nonhomogeneous cases in generalized grand Morrey spaces.

Keywords: Morrey spaces; Hardy–Littlewood maximal operator; Calderón–Zygmund operator; potentials

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