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Georgian Mathematical Journal

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

Editorial Board: 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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Complex interpolation of the predual of Morrey spaces over measure spaces

Victor I. Burenkov
  • S.  M. Nikol’skii Mathematical Institute, Peoples Friendship University of Russia (RUDN University), 6 Miklukho Maklay Str., 117198 Moscow, Russia; and School of Mathematics, Cardiff University, Cardiff CF24 4AG, United Kingdom
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/ Denny I. Hakim
  • Corresponding author
  • Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo, 192-0397, Japan. Current address: Department of Mathematics, Bandung Institute of Technology, Jalan Ganesha 10, Bandung 40132, Indonesia
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/ Eiichi Nakai / Yoshihiro Sawano
  • Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo, 192-0397, Japan
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/ Takuya Sobukawa / Tamara V. Tararykova
  • S. M. Nikol’skii Mathematical Institute, Peoples Friendship University of Russia (RUDN University), 6 Miklukho Maklay Str., 117198 Moscow, Russia; and School of Mathematics, Cardiff University, Cardiff CF24 4AG, United Kingdom
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Published Online: 2019-11-26 | DOI: https://doi.org/10.1515/gmj-2019-2070

Abstract

We prove that block spaces defined on n with an arbitrary Radon measure, which are known to be the preduals of Morrey spaces, are closed under the first and the second complex interpolation method. The proof of our main theorem uses the duality theorem in the complex interpolation method, the complex interpolation of certain closed subspaces of Morrey spaces, a characterization of the preduals of block spaces, and some formulas related to the Calderón product.

Keywords: Morrey spaces; block spaces; preduals of Morrey spaces; complex interpolation method

MSC 2010: 42B35; 46B70; 46B26

References

  • [1]

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

  • [2]

    J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. Google Scholar

  • [3]

    V. I. Burenkov, D. K. Darbayeva and E. D. Nursultanov, Description of interpolation spaces for general local Morrey-type spaces, Eurasian Math. J. 4 (2013), no. 1, 46–53. Google Scholar

  • [4]

    V. I. Burenkov and E. D. Nursultanov, Description of interpolation spaces for local Morrey-type spaces (in Russian), Tr. Mat. Inst. Steklova 269 (2010), 52–62; translation in Proc. Steklov Inst. Math. 269 (2010), no. 1, 46–56. Google Scholar

  • [5]

    V. I. Burenkov, E. D. Nursultanov and D. K. Chigambayeva, Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations (in Russian), Tr. Mat. Inst. Steklova 284 (2014), 105–137; translation in Proc. Steklov Inst. Math. 284 (2014), no. 1, 105–137. Google Scholar

  • [6]

    A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. CrossrefGoogle Scholar

  • [7]

    F. Cobos, J. Peetre and L. E. Persson, On the connection between real and complex interpolation of quasi-Banach spaces, Bull. Sci. Math. 122 (1998), no. 1, 17–37. CrossrefGoogle Scholar

  • [8]

    D. I. Hakim and Y. Sawano, Interpolation of generalized Morrey spaces, Rev. Mat. Complut. 29 (2016), no. 2, 295–340. Web of ScienceCrossrefGoogle Scholar

  • [9]

    D. I. Hakim and Y. Sawano, Calderón’s first and second complex interpolations of closed subspaces of Morrey spaces, J. Fourier Anal. Appl. 23 (2017), no. 5, 1195–1226. CrossrefGoogle Scholar

  • [10]

    T. Izumi, E. Sato and K. Yabuta, Remarks on a subspace of Morrey spaces, Tokyo J. Math. 37 (2014), no. 1, 185–197. CrossrefWeb of ScienceGoogle Scholar

  • [11]

    S. G. Kreĭn, Y. I. Petunīn and E. M. Semënov, Interpolation of Linear Operators, Transl. Math. Monogr. 54, American Mathematical Society, Providence, 1982. Google Scholar

  • [12]

    P. G. Lemarié-Rieusset, Multipliers and Morrey spaces, Potential Anal. 38 (2013), no. 3, 741–752. Web of ScienceCrossrefGoogle Scholar

  • [13]

    P. G. Lemarié-Rieusset, Erratum to: Multipliers and Morrey spaces [mr3034598], Potential Anal. 41 (2014), no. 4, 1359–1362. CrossrefGoogle Scholar

  • [14]

    G. J. Lozanovskiĭ, Certain Banach lattices (in Russian), Sibirsk. Mat. Ž. 10 (1969), 584–599; translation in Siberian Math. J. 10 (1969), 419–431. Google Scholar

  • [15]

    Y. Lu, D. Yang and W. Yuan, Interpolation of Morrey spaces on metric measure spaces, Canad. Math. Bull. 57 (2014), no. 3, 598–608. CrossrefGoogle Scholar

  • [16]

    E. Nakai and T. Sobukawa, Bwu-function spaces and their interpolation, Tokyo J. Math. 39 (2016), no. 2, 483–516. Web of ScienceGoogle Scholar

  • [17]

    S. Reisner, On Two Theorems of Lozanovskiĭ Concerning Intermediate Banach Lattices, Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math. 1317, Springer, Berlin (1988), 67–83. Google Scholar

  • [18]

    Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1535–1544. CrossrefGoogle Scholar

  • [19]

    Y. Sawano and H. Tanaka, Predual spaces of Morrey spaces with non-doubling measures, Tokyo J. Math. 32 (2009), no. 2, 471–486. CrossrefGoogle Scholar

  • [20]

    Y. Sawano and H. Tanaka, Fatou property of predual Morrey spaces with non-doubling measures, Int. J. Appl. Math. 27 (2014), no. 3, 283–296. Google Scholar

  • [21]

    Y. Sawano and H. Tanaka, The Fatou property of block spaces, J. Math. Sci. Univ. Tokyo 22 (2015), no. 3, 663–683. Google Scholar

  • [22]

    W. Yuan, Complex interpolation for predual spaces of Morrey-type spaces, Taiwanese J. Math. 18 (2014), no. 5, 1527–1548. Web of ScienceCrossrefGoogle Scholar

  • [23]

    W. Yuan, W. Sickel and D. Yang, Interpolation of Morrey-Campanato and related smoothness spaces, Sci. China Math. 58 (2015), no. 9, 1835–1908. CrossrefWeb of ScienceGoogle Scholar

  • [24]

    C. T. Zorko, Morrey space, Proc. Amer. Math. Soc. 98 (1986), no. 4, 586–592. CrossrefGoogle Scholar

About the article

Received: 2017-11-28

Accepted: 2018-10-18

Published Online: 2019-11-26


Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 15H03621

Award identifier / Grant number: 16K05209

Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 18-51-06005

V. I. Burenkov was supported by the “Peoples” Friendship University of Russia (RUDN University) Programme 5-100. E. Nakai and T. Sobukawa were supported by Grant-in-Aid for Scientific Research (B) (no. 15H03621), Japan Society for the Promotion of Science. Y. Sawano was supported by Grant-in-Aid for Scientific Research (C) (no. 16K05209), Japan Society for the Promotion of Science. T. V. Tararykova was supported by the Russian Foundation for Basic Research (no. 18-51-06005).


Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2070.

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