$B-$ Riezs potential in $B-$ local Morrey-Lorentz spaces

Abstract

In this paper, the Riesz potential (B−Riesz potential) which are generated by the Laplace-Bessel differential operator will be studied. We obtain the necessary and sufficient conditions for the boundedness of the B−Riesz potential $I_{\gamma }^{\alpha }$ in the B-local Morrey-Lorentz spaces $M_{p,q,\lambda,\gamma }^{loc}(\mathbb{R}_{k,+}^{n})$ with the use of the rearrangement inequalities and boundedness of the Hardy operators $H_{\upsilon }^{\beta }$ and $\mathcal{H}_{\upsilon}^{\beta }$ with power weights.

Keywords

• Morrey spaces
• Lorentz spaces
• local Morrey-Lorentz spaces
• Riesz potential

References

1. Aykol, C., Guliyev, V. S., Küçükaslan, A., Şerbetçi, A., The boundedness of Hilbert transform in the local Morrey-Lorentz spaces, Integral Transforms Spec. Funct., 27(4) (2016), 318–330. https://doi.org/10.1080/10652469.2015.1121483
2. Aykol, C., Guliyev, V. S., Şerbetçi, A., Boundedness of the maximal operator in the local Morrey-Lorentz spaces, J. Inequal. Appl., 2013(1) (2013), 1–11. https://doi.org/10.1186/1029-242X-2013-346
3. Aykol, C., Kaya, E., B−maximal operators, B−singular integral operators and B−Riesz potentials in variable exponent Lorentz spaces, Filomat, 37(17) (2023), 5765–5774. https://doi.org/10.2298/FIL2317765A
4. Aykol, C., Şerbetçi, A., On the boundedness of fractional B−maximal operators in the Lorentz spaces $L_{p,q,\gamma }(\mathbb{R}_{+}^{n})$, An. St. Univ. Ovidius Constanta, 17(2) (2009), 27–38.
5. Bennett, C., Sharpley, R., Interpolation of Operators, Academic Press, Boston, 1988.
6. Burenkov, V. I., Guliyev, H. V., Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces, Stud. Math., 163(2) (2004), 157–176.
7. Guliyev, V. S., On maximal function and fractional integral, associated with the Bessel differential operator, Math. Inequal. Appl., 6(2) (2003), 317–330. dx.doi.org/10.7153/mia-06-30
8. Guliyev, V. S., Sobolev theorems for anisotropic Riesz-Bessel potentials on Morrey-Bessel spaces, Dokl. Akad. Nauk, 367(2) (1999), 155–156.

9. Guliyev, V. S., Sobolev theorems for the Riesz B−potentials, Doklady Mathematics, 358 (1998), 450–451.
10. Guliyev, V. S., Şerbetçi, A., Ekincioğlu, I., On boundedness of the generalized B−potential integral operators in the Lorentz spaces, Integral Transforms Spec. Funct., 18(12) (2007), 885–895. https://doi.org/10.1080/10652460701510980
11. Guliyev, V. S., Şeerbetçi, A., Ekincioğlu, I., Necessary and sufficient conditions for the boundedness of rough B−fractional integral operators in the Lorentz spaces, J. Math. Anal. Appl., 336 (2007), 425–437. https://doi.org/10.1016/j.jmaa.2007.02.080
12. Guliyev, V. S., Safarov, Z. V., Şerbetçi, A., On the rearrangement estimates and the boundedness of the generalized fractional integrals associated with the Laplace-Bessel differential operator, Acta Math. Hung., 119 (2008), 201–217. https://doi.org/10.1007/s10474-007-6107-5
13. Guliyev, V. S., Küçükaslan A., Aykol, C., Şerbetçi, A., Riesz potential in the local Morrey-Lorentz Spaces and some applications, Georgian Math. J., 27(4) (2020), 557–567. https://doi.org/10.1515/gmj-2018-0065
14. Guliyev, V. S., Mustafayev, R. C., Integral operators of potential type in spaces of homogeneous type, Dokl. Akad. Nauk, Ross. Akad. Nauk, 354(6) (1997), 730–732.
15. Guliyev, V. S., Mustafayev, R. C., Fractional integrals in spaces of functions defined on spaces of homogeneous type, Anal. Math., 24(3) (1998), 181–200
16. Klyuchantsev, M. I., Singular integrals generated by the generalized shift operator I, Sibirsk. Math. Zh., 11(4) (1970), 810–821. https://doi.org/10.1007/BF00969676
17. Levitan, B. M., Bessel function expansions in series and Fourier integrals, Uspekhi Mat. Nauk., 6 2(42) (1951), 102–143.
18. Lorentz, G. G., Some new function spaces, Ann. Math. 51(1) (1950), 37–55. https://doi.org/10.2307/1969496
19. Lorentz, G. G., On the theory of spaces $\Lambda$, Pac. J. Math. 1(3) (1951), 411–429.
20. Peetre, J., Nouvelles proprietes d’espaces d’interpolation, C. R. Acad. Sci., Paris 256 (1963), 1424–1426.
21. Samko, N., Weighted Hardy and potential operators in Morrey spaces, J. Funct. Spaces, 2012 (2012), Article ID 678171. https://doi.org/10.1155/2012/678171
22. Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971.