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FRACTIONAL WEIGHTED SPHERICAL MEAN AND MAXIMAL INEQUALITY FOR THE WEIGHTED SPHERICAL MEAN AND ITS APPLICATION TO SINGULAR PDE

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Abstract

In this paper we establish a mean value property for the functions which is satisfied to Laplace–Bessel equation. Our results involve the generalized divergence theorem and the second Green’s identities relating the bulk with the boundary of a region on which differential Bessel operators act. Also we design a fractional weighted mean operator, study its boundedness, obtain maximal inequality for the weighted spherical mean and get its boundedness. The connection between the boundedness of the spherical maximal operator and the properties of solutions of the Euler–Poisson–Darboux equation with Bessel operators is given as an application.

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Funding

The research of V. Guliyev and I. Ekincioglu was partially supported by grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455). The research of V. Guliyev was partially supported by the RUDN University Strategic Academic Leadership Program.

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Author notes

  1. Ismail Ekincioǧlu, Vagif S. Guliyev and Elina L. Shishkina contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Faculty of Engineering and Natural Sciences, Istanbul Medeniyet University, Istanbul, Turkey

    Ismail Ekincioǧlu

  2. Institute of Applied Mathematics, Baku State University, Academic Zahid Khalilov street, 23, Baku, AZ 1148, Azerbaijan

    Vagif S. Guliyev

  3. Peoples Friendship University of Russia (RUDN University), Miklukho-Maklaya street, 6, Moscow, 117198, Russia

    Vagif S. Guliyev

  4. Voronezh State University, Universitetskaya pl. 1, Voronezh, 394018, Russia

    Elina L. Shishkina

  5. Belgorod State National Research University (“BelGU”), Pobedy Street, 85, Belgorod, 308015, Russia

    Elina L. Shishkina

Authors
  1. Ismail Ekincioǧlu
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  2. Vagif S. Guliyev
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  3. Elina L. Shishkina
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Correspondence to Elina L. Shishkina.

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Ekincioǧlu, I., Guliyev, V.S. & Shishkina, E.L. FRACTIONAL WEIGHTED SPHERICAL MEAN AND MAXIMAL INEQUALITY FOR THE WEIGHTED SPHERICAL MEAN AND ITS APPLICATION TO SINGULAR PDE. J Math Sci 266, 744–764 (2022). https://doi.org/10.1007/s10958-022-06099-x

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