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, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio
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Riesz type potential operators in generalized grand Morrey spaces
- Department 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
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.
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