Multilinear commutators of parabolic Calderón-Zygmund operators on generalized weighted variable parabolic Morrey spaces. (English) Zbl 1513.42030
Summary: We established the boundedness of multilinear commutators of parabolic Calderón-Zygmund operators \(T_{\mathbf{b}}\) on generalized weighted variable exponent parabolic Morrey spaces \(M_{w}^{\mathrm{p}(\cdot),\varphi}\) with the weight function \(w\) belonging to Muckenhoupt’s class \(A_{\mathrm{p}(\cdot)}(\mathbb{R}^n)\). When \(\mathbf{b}=(b_1,\dots,b_m)\), \(b_i \in BMO(\mathbb{R}^n)\), \(i=1,\dots,m\) and \(w \in A_{\mathrm{p}(\cdot)}(\mathbb{R}^n)\), the sufficient conditions on the pair \((\varphi_1, \varphi_2)\) which ensure the boundedness of the operator \(T_{\mathbf{b}}\) from \(M_{w}^{\mathrm{p}(\cdot),\varphi_1}\) to \(M_{w}^{\mathrm{p}(\cdot),\varphi_2}\) are found.
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B35 | Function spaces arising in harmonic analysis |
Keywords:
parabolic Calderón-Zygmund operator; generalized weighted variable exponent parabolic Morrey space; multilinear commutator; \(BMO\)References:
| [1] | Akbulut, A., Badalov, X.A., Hasanov, J.J., Serbetci, A.: p(x)-admissible sublinear sin-gular operators in the generalized variable exponent Morrey spaces, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 36 (1), Mathematics, 10-17 (2016). · Zbl 1513.42031 |
| [2] | Akbulut, A., Burenkov, V.I., Guliyev, V.S.: Anisotropic fractional maximal commuta-tors with BM O functions on anisotropic morrey-type spaces, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 40 (4), Mathematics, 1332 (2020). |
| [3] | Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable ex-ponent Morrey spaces, Georgian Math. J. 15 (2), 195-208 (2008). · Zbl 1263.42002 |
| [4] | Badalov, X.A.: Maximal and singular integral operators and their commutators in the vanishing generalized weighted Morrey spaces with variable exponent, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 38 (1), Mathematics, 30-42 (2018). · Zbl 1400.42020 |
| [5] | Bandaliyev, R.A., Guliyev, V.S.: Embedding theorems between variable-exponent Mor-rey spaces, Math. Notes 106 (3-4), 488-500 (2019). · Zbl 1444.46023 |
| [6] | Besov, O.V., Il’in, V.P., Lizorkin, P.I.: The L p -estimates of a certain class of non-isotropically singular integrals, (Russian) Dokl. Akad. Nauk SSSR, 169, 1250-1253 (1966). · Zbl 0158.12903 |
| [7] | Cohen, J.: A sharp estimate for a multilinear singular integral on R n , Indiana Univ. Math. J. 30, 693-702 (1981). · Zbl 0596.42004 |
| [8] | Cohen J., Gosselin, J.: On multilinear singular integral operators on R n , Studia Math. 72, 199-223 (1982). · Zbl 0427.42005 |
| [9] | Cohen, J., Gosselin, J.: A BMO estimate for multilinear singular integral operators, Illinois J. Math. 30, 445-465 (1986). · Zbl 0619.42012 |
| [10] | Coifman, R., Meyer, Y.: Wavelets, Calderón-Zygmund and multilinear operators, Cam-bridge Studies in Advanced Math. (Cambridge: Cambridge University Press) vol. 48 (1997). · Zbl 0916.42023 |
| [11] | Coifman, R.R., Weiss, G.: Factorization theorems for Hardy spaces in several vari-ables, Ann. of Math. 103 (3), 611-635 (1976). · Zbl 0326.32011 |
| [12] | Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function, Rend Mat. 7, 273-279 (1987). · Zbl 0717.42023 |
| [13] | Chiarenza, F., Frasca, M., Longo, P.: W 2,p -solvability of Dirichlet problem for nondi-vergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (2), 841-853 (1993). · Zbl 0818.35023 |
| [14] | Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces, Foundations and Harmonic Analysis, 1st edn. Birkhauser, Basel, (2013). · Zbl 1268.46002 |
| [15] | Ding, Y., Lu, S.Z.: Weighted boundedness for a class of rough multilinear operators, Acta Math. Sinica 17, 517-526 (2001). · Zbl 0985.42011 |
| [16] | Fabes, E.B., Rivère, N.: Singular integrals with mixed homogeneity, Studia Math. 27, 19-38 (1966). · Zbl 0161.32403 |
| [17] | Di Fazio, G., Ragusa, M.A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Func. Anal. 112, 241-256 (1993). · Zbl 0822.35036 |
| [18] | Ekincioglu, I., Keskin, C., Serbetci, A. : Multilinear commutators of Calderón-Zygmund operator on generalized variable exponent Morrey spaces, Positivity 25 (4), 1551-1567 (2021). · Zbl 1479.42038 |
| [19] | Guliyev, V.S.: Integral operators on function spaces on the homogeneous groups and on domains in R n , Doctoral dissertation, Moscow, Mat. Inst. Steklov, 329 pp. (1994) (in Russian). |
| [20] | Guliyev, V.S.: Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications, Baku, Elm. 332 pp. (1999) (Russian). |
| [21] | Guliyev, V.S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. Art. ID 503948, 20 pp. (2009). · Zbl 1193.42082 |
| [22] | Guliyev, V.S., Hasanov, J.J., Samko, S.G.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces, Math. Scand. 107 (2), 285-304 (2010). · Zbl 1213.42077 |
| [23] | Guliyev, V.S.: Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J. 3 (3), 33-61 (2012). · Zbl 1271.42019 |
| [24] | Guliyev, V.S., Samko, S.G.: Maximal, potential and singular operators in the gener-alized variable exponent Morrey spaces on unbounded sets, Problems in mathematical analysis. J. Math. Sci. (N. Y.) 193 (2), 228-248 (2013). · Zbl 1278.42030 |
| [25] | Guliyev, V.S.: Generalized local Morrey spaces and fractional integral operators with rough kernel, Problems in mathematical analysis. No. 71. J. Math. Sci. (N. Y.) 193 (2), 211-227 (2013). · Zbl 1277.42021 |
| [26] | Guliyev, V.S., Alizadeh, F.Ch.: Multilinear commutators of Calderón-Zygmund opera-tor on generalized weighted Morrey spaces, J. Funct. Spaces 2014, Art. ID 710542, 9 pp. · Zbl 1472.42018 |
| [27] | Guliyev, V.S., Omarova, M.N.: Multilinear singular and fractional integral operators on generalized weighted Morrey spaces, · Zbl 1316.42016 |
| [28] | 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). · Zbl 1384.42013 |
| [29] | Guliyev, V.S., Omarova, M.N., Softova, L.: The Dirichlet problem in a class of gener-alized weighted Morrey spaces, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 45 (2), 270-285 (2019). · Zbl 1441.35114 |
| [30] | Guliyev, V.S.: Calderón-Zygmund operators with kernels of Dini’s type on generalized weighted variable exponent Morrey spaces, Positivity (2021). · Zbl 1482.42024 |
| [31] | https://doi.org/10.1007/s11117-021-00846-1 · doi:10.1007/s11117-021-00846-1 |
| [32] | Guliyev, V.S., Ismayilova, A.F.: Calderón-Zygmund operators with kernels of Dini’s type and their multilinear commutators on generalized weighted Morrey spaces, TWMS J. Pure Appl. Math. 12 (2), 1-13 (2021). · Zbl 1500.42006 |
| [33] | 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. 41 (4), Mathematics, 1-12 (2021). |
| [34] | Hamzayev, V.H.: Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized weighted Morrey spaces, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 38 (1), Mathematics, 79-94 (2018). · Zbl 1400.42011 |
| [35] | Hästö, P., Diening, L.: Muckenhoupt weights in variable exponent spaces, preprint, http://www.helsinki.fi/pharjule/varsob/publications.shtml. |
| [36] | Kwok-Pun, Ho.: 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). · Zbl 1343.42017 |
| [37] | Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2), 219-231 (2009). · Zbl 1160.42008 |
| [38] | Kopaliani, T.S.: Infimal convolution and Muckenhoupt A p(•) condition in variable L p spaces, Arch. Math. 89 (2), 185-192 (2007). · Zbl 1181.42015 |
| [39] | Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maxi-mal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220, 1222-1264 (2009). · Zbl 1160.42009 |
| [40] | Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43, 126-166 (1938). · JFM 64.0460.02 |
| [41] | Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (Sendai, 1990), S. Igari, Ed, ICM 90 Satellite Conference Proceed-ings, pp. 183-189, Springer, Tokyo, Japan, 1991. · Zbl 0771.42007 |
| [42] | Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr. 166, 95-103 (1994). · Zbl 0837.42008 |
| [43] | Pérez, C., Trujillo-González, R.: Sharp weighted estimates for multilinear commuta-tors, J. London Math. Soc. 65, 672-692 (2002). · Zbl 1012.42008 |
| [44] | Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces, Marcel Dekker Inc., New York (1991). · Zbl 0724.46032 |
| [45] | Sawano, Y.: A thought on generalized Morrey spaces, J. Indones. Math. Soc. 25 (3), 210-281 (2019). · Zbl 1451.26032 |
| [46] | Tan, J., Liu, Z., Zhao, J.: On some multilinear commutators in variable Lebesgue spaces, J. Math. Inequal. 11 (3), 715-734 (2017). · Zbl 1375.42033 |
| [47] | Xiao, Feng Ye: Some estimates for multilinear commutators on the weighted Morrey spaces, Math. Sci. (Springer) 6, Art. 33, 6 pp. (2012). · Zbl 1271.42027 |
| [48] | Yang, Do., Meng, Y.: Boundedness of commutators in Morrey spaces over nonhomoge-neous spaces, Beijing Shifan Daxue Xuebao 41 (5), 449-454 (2005). · Zbl 1098.42011 |
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