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Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on the generalized weighted Morrey spaces

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Abstract

The goal of this paper is to characterize the local sharp estimate \((I_{\rho } f)^{\#}(x) \le C \, M_{\rho } f(x)\) and by using this inequality to get necessary and sufficient conditions on the triple functions \((\varphi , \rho , \omega )\) which satisfy the equivalence of norms of the generalized fractional integral operator \(I_{\rho }\) and the generalized fractional maximal operator \(M_{\rho }\) on the generalized weighted Morrey spaces \({\mathcal {M}}_{p,\varphi }({\mathbb {R}}^{n},\omega )\) and generalized weighted central Morrey spaces \(\dot{{\mathcal {M}}}_{p,\varphi }({{\mathbb {R}}}^n,\omega )\), when \(\omega \in A_{\infty }\)-Muckenhoupt class.

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Acknowledgements

The author would like to express his gratitude to Prof. Dr. Amiran Gogatishvili for his very valuable comments and suggestions. The author would like to express his gratitude to the referees for their (his/her) very valuable comments and suggestions. The research of Abdulhamit Kucukaslan was supported by the grant of The Scientific and Technological Research Council of Turkey, Grant TUBITAK-1059B191600675.

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Authors and Affiliations

  1. Institute of Mathematics of Czech Academy of Sciences, 115 67, Prague, Czech Republic

    Abdulhamit Kucukaslan

  2. School of Applied Sciences, Pamukkale University, 20680, Denizli, Turkey

    Abdulhamit Kucukaslan

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Correspondence to Abdulhamit Kucukaslan.

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Communicated by Sorina Barza.

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Kucukaslan, A. Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on the generalized weighted Morrey spaces. Ann. Funct. Anal. 11, 1007–1026 (2020). https://doi.org/10.1007/s43034-020-00066-w

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  • DOI: https://doi.org/10.1007/s43034-020-00066-w

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