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Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces


Authors: Yoshihiro Sawano, Satoko Sugano and Hitoshi Tanaka
Journal: Trans. Amer. Math. Soc. 363 (2011), 6481-6503
MSC (2010): Primary 42B35; Secondary 42B25
DOI: https://doi.org/10.1090/S0002-9947-2011-05294-3
Published electronically: July 26, 2011
MathSciNet review: 2833565
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Abstract: The action of the generalized fractional integral operators and the generalized fractional maximal operators is investigated in the framework of Morrey spaces. A typical property of the functions which belongs to Morrey spaces under pointwise multiplication by the generalized fractional integral operators and the generalized fractional maximal operators is established. The boundedness property of the fractional integral operators on the predual of Morrey spaces is shown as well. A counterexample concerning the Fefferman-Phong inequality is given by the use of the characteristic function of the Cantor set.


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Additional Information

Yoshihiro Sawano
Affiliation: Department of Mathematics, Kyoto University, Kitasirakawa, Sakyoku, Kyoto, 606-8502, Japan
Email: yosihiro@math.kyoto-u.ac.jp

Satoko Sugano
Affiliation: Department of Mathematics, Kobe City College of Technology, 8-3 Gakuen-higashi- machi, Nishi-ku, Kobe 651-2194, Japan
Email: sugano@kobe-kosen.ac.jp

Hitoshi Tanaka
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
Email: htanaka@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2011-05294-3
Keywords: Generalized fractional integral operator, generalized fractional maximal operator, Morrey space, Olsen’s inequality, block space.
Received by editor(s): July 16, 2008
Received by editor(s) in revised form: December 22, 2009
Published electronically: July 26, 2011
Additional Notes: The third author was supported by the Global COE program at the Graduate School of Mathematical Sciences at the University of Tokyo, and was also supported by the Fūjyukai Foundation.
Article copyright: © Copyright 2011 American Mathematical Society

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