Abstract
Let T be a Calderón-Zygmund operator of type \(\omega \) with \(\omega (t)\) being nondecreasing and satisfying a kind of Dini’s type condition and let \(T_{\vec {b}}\) be the multilinear commutators of T with \(BMO^m\) functions. In this paper, we study the boundedness of the operators T and \(T_{\vec {b}}\) on generalized weighted variable exponent Morrey spaces \(M^{p(\cdot ),\varphi }(w)\) with the weight function w belonging to variable Muckenhoupt’s class \(A_{p(\cdot )}({{\mathbb {R}}^n})\). We find the sufficient conditions on the pair \((\varphi _1,\varphi _2)\) with \(\vec {b} \in BMO^m({{\mathbb {R}}^n})\) which ensures the boundedness of the operators T and \(T_{\vec {b}}\) from \(M^{p(\cdot ),\varphi _1}(w)\) to \(M^{p(\cdot ),\varphi _2}(w)\).
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References
Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15(2), 195–208 (2008)
Akbulut, A., Badalov, X.A., Hasanov, J.J., Serbetci, A.: \(p(x)\)-admissible sublinear singular operators in the generalized variable exponent Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Mat. Sci. Math. 36(1), 10–17 (2016)
Chiarenza, F., Frasca, M., Longo, P.: \(W^{2, p}\)-solvability of Dirichlet problem for nondivergence ellipic equations with VMO coefficients. Trans. Amer. Math. Soc. 336, 841–853 (1993)
Cruz-Uribe, D., Fiorenza, A., Martell, J., Perez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces, Foundations and Harmonic Analysis, 1st edn. Birkhauser, Basel (2013)
Fan, X.: Variable exponent Morrey and Campanato spaces. Nonlinear Anal. 72(11), 4148–4161 (2010)
Fazio, G.. Di., Hakim, D.I., Sawano, Y.: Elliptic equations with discontinuous coefficients in generalized Morrey spaces. Eur. J. Math. 3(3), 728–762 (2017)
Diening, L., Harjulehto, P., Hästö, P., Ružička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer-Verlag, Berlin (2011)
Duoandikoetxea, J., Rosenthal, M.: Boundedness of operators on certain weighted Morrey spaces beyond the Muckenhoupt range. Potential Anal. 53, 1255–1268 (2020)
Duoandikoetxea, J., Rosenthal, M.: Muckenhoupt-type conditions on weighted Morrey spaces. J. Fourier Anal. Appl. 27(32), 1–33 (2021)
Ekincioglu, I., Keskin, C., Serbetci, A.: Multilinear commutators of Calderón-Zygmund operator on generalized variable exponent Morrey spaces. Positivity (2021). https://doi.org/10.1007/s11117-021-00828-3
Guliyev, V.S.: Integral operators on function spaces on the homogeneous groups and on domains in \({{\mathbb{R}}^n}\). (in Russian) Doctoral dissertation, Moscow, Mat. Inst. Steklov, pp. 329 (1994)
Guliyev, V.S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. Art. ID 503948, pp. 20 (2009)
Guliyev, V.S.: Generalized weighted Morrey spaces and higher order commutators of sublinear operators. Eurasian Math. J. 3(3), 33–61 (2012)
Guliyev, V.S.: Generalized local Morrey spaces and fractional integral operators with rough kernel. J. Math. Sci. (N. .Y) 193(2), 211–227 (2013)
Guliyev, V.S., Hasanov, J., Samko, S.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107(2), 285–304 (2010)
Guliyev, V.S., Samko, S.G.: Maximal, potential and singular operators in the generalized variable exponent Morrey spaces on unbounded sets. J. Math. Sci. (N. Y.) 193(2), 228–248 (2013)
Guliyev, V.S., Hasanov, J.J., Badalov, X.A.: Maximal and singular integral operators and their commutators on generalized weighted Morrey spaces with variable exponent. Math. Ineq. Appl. 21(1), 41–61 (2018)
Ismayilova, A.F.: Calderón-Zygmund operators with kernels of Dini’s type and their multilinear commutators on generalized Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. 41(4), 1–12 (2021)
Hasanov, J.J., Aliyev, S.S., Seymur, S., Guliyev, Y.Y.: Commutators of potential and singular integral operators in generalized variable exponent Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. 35(4), 75–83 (2015)
Hästö, P., Diening, L.: Muckenhoupt weights in variable exponent spaces. http://www.helsinki.fi/pharjule/varsob/publications.shtml
Ho, Kwok-Pun.: Singular integral operators, John-Nirenberg inequalities and Tribel-Lizorkin type spaces on weighted Lebesgue spaces with variable exponents. Rev. Un. Mat. Argentina 57(1), 85–101 (2016)
Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282(2), 219–231 (2009)
Kopaliani, T.S.: Infimal convolution and Muckenhoupt \(A_{p(\cdot )}\) condition in variable \(L^p\) spaces. Arch. Math. 89(2), 185–192 (2007)
Kovacik, O., Rakosnik, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41, 592–618 (1991)
Lin, Y.: Strongly singular Calderón-Zygmund operator and commutator on Morrey type spaces. Acta Math. Sin. (Engl. Ser.) 23(11), 2097–2110 (2007)
Lin, Y., Lu, Sh.: Strongly singular Calderón-Zygmund operators and their commutators. Jordan J. Math. Stat. 1(1), 31–49 (2008)
Liu, Z.G., Lu, S.Z.: Endpoint estimates for commutators of Calderón-Zygmund type operators. Kodai Math. J. 25(1), 79–88 (2002)
Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220, 1222–1264 (2009)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938)
Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis (Sendai, 1990), S. Igari, Ed, ICM 90 Satellite Conference Proceedings, Springer, Tokyo, Japan, 183-189 (1991)
Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)
Nakamura, S.: Generalized weighted Morrey spaces and classical operators. Math. Nachr. 289(17–18), 2235–2262 (2016)
Nakamura, S., Sawano, Y.: The singular integral operator and its commutator on weighted Morrey spaces. Collect. Math. 68, 145–174 (2017)
Nakamura, S., Sawano, Y., Tanaka, H.: Weighted local Morrey spaces. Ann. Acad. Sci. Fenn. Math. 45(1), 67–93 (2020)
Peng, L.Z.: Generalized Calderón-Zygmund operators and their weighted norm inequalities. Adv. Math. 14, 97–115 (1985)
Pérez, C., Trujillo-González, R.: Sharp weighted estimates for multilinear commutators. J. London Math. Soc. 65, 672–692 (2002)
Persson, L.E., Samko, N., Wall, P.: Calderón-Zygmund type singular operators in weighted generalized Morrey spaces. J. Fourier Anal. Appl. 22(2), 413–426 (2016)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker Inc., New York (1991)
Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350, 56–72 (2009)
Samko, N.: On two-weight estimates for the maximal operator in local Morrey spaces. Int. J. Math. 25(11), 1450099 (2014)
Sawano, Y.: A thought on generalized Morrey spaces. J. Indones. Math. Soc. 25(3), 210–281 (2019)
Tan, J., Liu, Z., Zhao, J.: On some multilinear commutators in variable Lebesgue spaces. J. Math. Inequal. 11(3), 715–734 (2017)
Wang, H.: Boundedness of \(\theta \)-type Calderón-Zygmund operators and commutators in the generalized weighted Morrey spaces. J. Funct. Spaces Art. ID 1309348, pp. 18 (2016)
Zhang, P., Sun, J.: Commutators of multilinear Calderón-Zygmund operators with kernels of Dini’s type and applications. J. Math. Inequal. 13(4), 1071–1093 (2019)
Yabuta, K.: Generalizations of Calderón-Zygmund operators. Studia Math. 82, 17–31 (1985)
Ye, X.F.: Some estimates for multilinear commutators on the weighted Morrey spaces. Math. Sci. (Springer) 6, Art. 33, pp. 6 (2012)
Yang, D.Y., Meng, Y.: Boundedness of commutators in Morrey spaces over nonhomogeneous spaces. Beijing Shifan Daxue Xuebao 41(5), 449–454 (2005)
Acknowledgements
The author would like to express their gratitude to the referees for his very valuable comments and suggestions. The research of author was partially supported by grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455), by Grant of 1st Azerbaijan-Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/2017-21/01/1-M-08) and by the RUDN University Strategic Academic Leadership Program.
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Guliyev, V.S. Calderón-Zygmund operators with kernels of Dini’s type on generalized weighted variable exponent Morrey spaces. Positivity 25, 1771–1788 (2021). https://doi.org/10.1007/s11117-021-00846-1
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DOI: https://doi.org/10.1007/s11117-021-00846-1