On the boundedness of $B$-Riesz potential and its commutators on generalized weighted $B$-Morrey spaces

Year 2024, Volume: 53 Issue: 2, 321 - 332, 23.04.2024

İsmail Ekincioğlu , Javanshir J. Hasanov , Cansu Keskin

https://doi.org/10.15672/hujms.1221556

Abstract

In the present paper, we shall investigate a characterization for the boundedness of the $B$-Riesz potential and its commutators on the generalized weighted $B$-Morrey spaces. We also give a characterization for the generalized weighted $B$-Morrey spaces via the boundedness of the Riesz potential and its commutators generated by generalized translate operators associated with Laplace-Bessel differential operator.

Keywords

generalized translate operator, $B$-maximal operator, $B$-Riesz potential operator, commutators, generalized weighted $B$-Morrey space

Supporting Institution

TUBITAK

Project Number

119N455

Thanks

The authors would like to express their gratitude to the referees for his/her very valuable comments and suggestions.

References

• [1] D.R. Adams, A note on Riesz potentials, Duke Math. J. 42, 765-778, 1975.
• [2] A. Akbulut, I. Ekincioglu, A. Serbetci and T. Tararykova, Boundedness of the anisotropic fractional maximal operator in anisotropic local Morrey-type spaces, Eurasian Math. J. 2 (2), 5-30, 2011.
• [3] R. Ayazoglu and J.J. Hasanov, On the boundedness of B-Riesz potential in the generalized weighted B-Morrey spaces, Georgian Math. J. 23 (2), 143-155, 2016.
• [4] C. Aykol and J.J. Hasanov, On the boundedness of B-maximal commutators, commutators of B-Riesz potentials and B-singular integral operators in modified B-Morrey spaces, Acta Sci. Math. (Szeged) 86 (3-4), 521-547, 2020.
• [5] M. Bramanti and M.C. Cerutti, Commutators of singular integrals on homogeneous spaces, oll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 4, 843-884, 1996.
• [6] V.I. Burenkov and H.V. Guliyev, Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces, Studia Math. 163 (2), 157- 176, 2004.
• [7] V.I. Burenkov and V.S. Guliyev, Necessary and sufficient conditions for boundedness of the Riesz potential in the local Morrey-type spaces, Potential Anal. 31 (2), 1-39, 2009.
• [8] F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Sem. Mat. Univ. Padova 7, 273-279, 1987.
• [9] G. Di Fazio and M.A. Ragusa, Commutators and Morrey spaces, Boll. Unione Mat. Ital. Sez. A Mat. Soc. Cult. 5 (3), 323-332, 1991.
• [10] A.D. Gadjiev and I.A. Aliev, On classes of operators of potential types, generated by a generalized translate, Reports of enlarged Session of the Seminars of I.N.Vekua Inst. of Applied Mathematics Tbilisi 3 (2), 21-24, 1988 (Russian).
• [11] V.S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in $\mathbb{R}^n$, PhD thesis, Mat. Inst. Steklov, 1994.
• [12] V.S. Guliyev, Sobolev theorems for the Riesz B-potentials, Dokl. Akad. Nauk 358 (4), 450-451, 1998. (Russian)
• [13] V.S. Guliyev, Sobolev theorems for anisotropic Riesz-Bessel potentials on Morrey- Bessel spaces, Dokl. Akad. Nauk Russia 367 (2), 155-156, 1999.
• [14] V.S. Guliyev, Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J. 3 (3), 3361, 2012.
• [15] V. Guliyev, I. Ekincioglu, E. Kaya and Z. Safarov, Characterizations for the fractional maximal commutator operator in generalized Morrey spaces on Carnot group, Integral Transforms Spec. Funct. 30 (6), 453-470, 2019.
• [16] V.S. Guliyev and J.J. Hasanov, Sobolev-Morrey type inequality for Riesz potentials, associated with the Laplace-Bessel differential operator, Fract. Calc. Appl. Anal. 9 (1), 17-32, 2006.
• [17] V.S. Guliyev and J.J. Hasanov, Necessary and sufficient conditions for the boundedness of Riesz potential associated with the Laplace-Bessel differential operator in Morrey spaces, J. Math. Anal. Appl. 347, 113-122, 2008.
• [18] J.J. Hasanov, A note on anisotropic potentials, associated with the Laplace-Bessel differential operator, Oper. Matrices 2 (4), 465-481, 2008.
• [19] J.J. Hasanov, R. Ayazoglu and S. Bayrakci, B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces, Open Mathematics, 18, 715-730, 2020.
• [20] J.J. Hasanov, I. Ekincioglu and C. Keskin, A characterization for B-singular integral operator and its commutators on generalized weighted B-Morrey spaces, (accepted for publication)
• [21] I.A. Kipriyanov, Fourier-Bessel transformations and imbedding theorems, Trudy Math. Inst. Steklov 89, 130-213, 1967.
• [22] Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2), 219-231, 2009.
• [23] B.M. Levitan, Bessel function expansions in series and Fourier integrals, Uspekhi Mat. Nauk. (Russian), 6 (2), 102-143, 1951.
• [24] L.N. Lyakhov, Multipliers of the Mixed Fourier-Bessel transform, Proc. Steklov Inst. Math. 214, 234-249, 1997.
• [25] T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, 183-189, Springer- Verlag, Tokyo 1991.
• [26] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43, 126-166, 1938.
• [27] B. Muckenhoupt and E.M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118, 17-92 1965.
• [28] E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces , Math. Nachr. 166, 95-103, 1994.
• [29] E. Nakai, Generalized fractional integrals on generalized Morrey spaces , Math. Nachr. 287 (2-3), 339-351, 2014.
• [30] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivative. Theory and Applications, Gordon and Breach Sci. Publishers, 1993.
• [31] Y. Sawano, Generalized Morrey spaces for non-doubling measures, NoDEA Nonlinear Differential Equations Appl. 12 (4-5), 413-425, 2008.
• [32] Y. Sawano, S.Sugano and H. Tanaka, Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc. 363 (12), 6481-6503, 2011.
• [33] A. Serbetci and I. Ekincioglu, On Boundedness of Riesz potential generated by generalized translate operator on Ba spaces, Czech. Math. J. 54 (3), 579-589, 2004.
• [34] E.L. Shishkina and S.M. Sitnik, Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics. Elsevier, 2020.
• [35] E.M. Stein, Singular Integrals And Differentiability Properties of Functions, Princeton New Jersey, Princeton Uni. Press, 1970.
• [36] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton, NJ, Princeton Univ. Press, 1971.
• [37] K. Stempak, Almost everywhere summability of Laguerre series, Studia Math. 100 (2), 129-147, 1991.
• [38] K. Trimeche, Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comput. Harmon. Anal. 4, 97-112, 1997.