ON THE BOUNDEDNESS OF THE RIESZ POTENTIAL AND ITS COMMUTATOR’S IN THE GLOBAL MORREY TYPE SPACES WITH VARIABLE EXPONENTS
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.05Keywords:
Morrie-type global spaces, variable exponent, Riess potential, Riesz potential commutator, boundary operatorAbstract
The paper considers the global Morrey-type spaces GMp(.),θ(.),w(.)(Ω) with variable exponents p(.), θ(.), where Ω ⊂ Rn is an unbounded domain. The questions of boundedness of the Riesz potential and its commutator in these spaces are investigated. We give the conditions for variable exponents (p1(.), p2(.)), (θ1(.), θ2(.)) and on the functions (w1(.), w2(.)) under which the Riesz potential I α , will be bounded from GMp1(.),θ1(.),w1(.)(Ω) to GMp2(.),θ2(.),w2(.)(Ω). The same conditions are obtained for the boundedness of the commutator of the Riesz potential in these spaces. In the case when the exponents p, θ constant numbers, the questions of boundedness of the Riesz potential and its commutator in global Morrey spaces were previously studied by other authors. There are also well-known results on the boundedness of the Riesz potential in global Morrey-type spaces with variable exponents, when the domain Ω ⊂ Rn is bounded.
References
[2] V. Burenkov, V.Guliyev. Necessary and sufficient conditions for boundedness of the Riesz potential in the local Morreytype spaces // Potential Anal. – 2009. – Vol.31, – Pp.1-39.
[3] V. Burenkov, V.Guliyev. Necessary and sufficient conditions for boundedness of the Riesz potential in the local Morreytype spaces // Doklady Ross.Akad.Nauk.Matematika. – 2007. – Vol.412, No.5, – Pp. 585-589 (in Russian).English transl. in Acad.Sci.Dokl.Math.– 2007. – Vol.76.
[4] V. Burenkov. Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces // Eurasian. Math. J. – 2012. – Vol.3, No.3, – Pp. 11-32.
[5] V. Burenkov. Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces // Eurasian. Math. J. – 2013. – Vol.4, No.1, – Pp. 21-45.
[6] L.Diening, P.Harjulehto, P.Hasto, M.Ruzicka. Lebesgue and Sobolev spaces with variable exponents // Monograpgh. – 2010. – Pp. 1-493.
[7] L.Diening Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(.) and Wk,p(.) . // Math.Nachr. – 2004. – Vol.268, – Pp.31-43.
[8] A. Almeida,J.Hasanov S.Samko. Maximal and potential operators in variable exponent Morrey spaces // Georgian. Math. J. – 2008. – Vol.15, No.2, – Pp. 195-208.
[9] V. Guliyev,S.Samko, J.Hasanov Boundedness of the maximal, potential type and singular integral operators in the generalized variable exponent Morrey spaces. // Math.Scand. – 2010. – Vol.107, – Pp. 285-304.
[10] V. Guliyev,S.Samko, J.Hasanov. Boundedness of maximal, potential type,and singular integral operators in the generalized variable exponent Morrey type spaces // J.Math.Sciences. – 2010. – Vol.170, No. 4. – Pp. 423-442.
[11] V. Guliyev,S.Samko. Maximal, potential, and singular operators in the generalized variable exponent Morrey spaces on unbounded sets // J.Math.Sciences. – 2013. – Vol.193, No. 2. – Pp. 228-247.
[12] V. Guliyev, J.Hasanov, X.Badalov. Commutators of Riesz potential in the vanishing generalized Morrey spaces with variable exponent // J.Math.Sciences. – 2019. – Vol.22, No. 1. – Pp. 331-351.
[13] D. Edmunds, V.Kokilasvili, A.Meskhi. On the boundedness and compactness of weighted Hardy operators in spaces Lp(x) // Georgian.Math.J. – 2005. – Vol.12, No. 1. – Pp. 27-44
