Previous |  Up |  Next

Article

Keywords:
generalized weighted Morrey space; sublinear operator; commutator; BMO space; maximal operator; Calderón-Zygmund operator
Summary:
In this paper, the boundedness of a large class of sublinear commutator operators $T_{b}$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }(w)$ with the weight function $w$ belonging to Muckenhoupt's class $A_{p}$ is studied. When $1<p<\infty $ and $b \in {\rm BMO}$, sufficient conditions on the pair $(\varphi _1,\varphi _2)$ which ensure the boundedness of the operator $T_{b}$ from $M_{p,\varphi _1}(w)$ to $M_{p,\varphi _2}(w)$ are found. In all cases the conditions for the boundedness of $T_{b}$ are given in terms of Zygmund-type integral inequalities on $(\varphi _1,\varphi _2)$, which do not require any assumption on monotonicity of $\varphi _1(x,r)$, $\varphi _2(x,r)$ in $r$. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.
References:
[1] Burenkov, V. I., Gogatishvili, A., Guliyev, V. S., Mustafayev, R. C.: Boundedness of the fractional maximal operator in local Morrey-type spaces. Complex Var. Elliptic Equ. 55 739-758 (2010). MR 2674862 | Zbl 1207.42015
[2] Burenkov, V. I., Guliyev, H. V., Guliyev, V. S.: Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces. J. Comput. Appl. Math. 208 280-301 (2007). DOI 10.1016/j.cam.2006.10.085 | MR 2347750 | Zbl 1134.46014
[3] Burenkov, V. I., Guliyev, V. S.: Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces. Potential Anal. 30 211-249 (2009). DOI 10.1007/s11118-008-9113-5 | MR 2480959 | Zbl 1171.42003
[4] Chiarenza, F., Frasca, M., Longo, P.: Interior $W^{2,p}$ estimates for non-divergence elliptic equations with discontinuous coefficients. Ric. Mat. 40 149-168 (1991). MR 1191890
[5] Chiarenza, F., Frasca, M., Longo, P.: $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336 841-853 (1993). MR 1088476
[6] Coifman, R. R., Meyer, Y.: Beyond pseudodifferential operators. Asterisque 57 Société Mathématique de France, Paris (1978), French. MR 0518170
[7] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 611-635 (1976). DOI 10.2307/1970954 | MR 0412721 | Zbl 0326.32011
[8] Fazio, G. Di, Ragusa, M. A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112 241-256 (1993). DOI 10.1006/jfan.1993.1032 | MR 1213138 | Zbl 0822.35036
[9] Ding, Y., Yang, D., Zhou, Z.: Boundedness of sublinear operators and commutators on $L^{p,\omega}(\mathbb{R}^n)$. Yokohama Math. J. 46 15-27 (1998). MR 1670757
[10] Garcí{a}-Cuerva, J., Francia, J. L. Rubio de: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116. Mathematical Notes 104 North-Holland, Amsterdam (1985). MR 0807149
[11] Grafakos, L.: Classical and Modern Fourier Analysis. Pearson/Prentice Hall Upper Saddle River (2004). MR 2449250 | Zbl 1148.42001
[12] Guliyev, V. S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009 Article ID 503948, 20 pages (2009). MR 2579556 | Zbl 1193.42082
[13] Guliyev, V. S.: Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications Baku (1996).
[14] Guliyev, V. S.: Integral Operators on Function Spaces on the Homogeneous Groups and on Domains in $\mathbb{R}^n$. Doctoral Degree Dissertation. Mat. Inst. Steklov Moskva (1994), Russian.
[15] Guliyev, V. S., Aliyev, S. S., Karaman, T.: Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces. Abstr. Appl. Anal. 2011 Article ID 356041, 18 pages (2011). MR 2819766 | Zbl 1228.42017
[16] 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. 107 285-304 (2010). MR 2735097 | Zbl 1213.42077
[17] Hörmander, L.: Pseudo-differential operators and hypoelliptic equations. Proc. Sympos. Pure Math. 10, Chicago, Ill., 1966 American Mathematical Society Providence 138-183 (1967). MR 0383152 | Zbl 0167.09603
[18] Karaman, T., Guliyev, V. S., Serbetci, A.: Boundedness of sublinear operators generated by Calderón-Zygmund operators on generalized weighted Morrey spaces. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) LX f.1, 18 pages (2014). MR 3252469
[19] Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282 219-231 (2009). DOI 10.1002/mana.200610733 | MR 2493512 | Zbl 1160.42008
[20] Lin, Y.: Strongly singular Calderón-Zygmund operator and commutator on Morrey type spaces. Acta Math. Sin., Engl. Ser. 23 2097-2110 (2007). DOI 10.1007/s10114-007-0974-0 | MR 2359125 | Zbl 1131.42014
[21] Lin, Y., Lu, S.: Strongly singular Calderón-Zygmund operators and their commutators. Jordan Journal of Mathematics and Statistics 1 31-49 (2008). Zbl 1279.42018
[22] Liu, L.: Weighted weak type estimates for commutators of Littlewood-Paley operator. Jap. J. Math., New Ser. 29 1-13 (2003). MR 1986863 | Zbl 1046.42013
[23] Liu, L., Lu, S.: Weighted weak type inequalities for maximal commutators of BochnerRiesz operator. Hokkaido Math. J. 32 85-99 (2003). DOI 10.14492/hokmj/1350652427 | MR 1962028
[24] Liu, Y., Chen, D.: The boundedness of maximal Bochner-Riesz operator and maximal commutator on Morrey type spaces. Anal. Theory. Appl. 24 321-329 (2008). DOI 10.1007/s10496-008-0321-z | MR 2471861 | Zbl 1199.42105
[25] Lu, S., Ding, Y., Yan, D.: Singular Integrals and Related Topics. World Scientific Publishing Hackensack (2007). MR 2354214 | Zbl 1124.42011
[26] Lu, G., Lu, S., Yang, D.: Singular integrals and commutators on homogeneous groups. Anal. Math. 28 103-134 (2002). DOI 10.1023/A:1016568918973 | MR 1918254 | Zbl 1026.43007
[27] Miller, N.: Weighted Sobolev spaces and pseudodifferential operators with smooth symbols. Trans. Am. Math. Soc. 269 91-109 (1982). DOI 10.1090/S0002-9947-1982-0637030-4 | MR 0637030 | Zbl 0482.35082
[28] Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis. Proceedings of a conference in Sendai, Japan, 1990 S. Igari Springer Tokyo 183-189 (1991). MR 1261439 | Zbl 0771.42007
[29] Jr., C. B. Morrey: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43 126-166 (1938). DOI 10.1090/S0002-9947-1938-1501936-8 | MR 1501936 | Zbl 0018.40501
[30] Muckenhoupt, B., Wheeden, R. L.: Weighted bounded mean oscillation and the Hilbert transform. Stud. Math. 54 221-237 (1976). MR 0399741 | Zbl 0318.26014
[31] Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166 95-103 (1994). DOI 10.1002/mana.19941660108 | MR 1273325 | Zbl 0837.42008
[32] Peetre, J.: On the theory of $L_{p,\lambda}$ spaces. J. Funct. Anal. 4 71-87 (1969). DOI 10.1016/0022-1236(69)90022-6 | MR 0241965
[33] Polidoro, S., Ragusa, M. A.: Hölder regularity for solutions of ultraparabolic equations in divergence form. Potential Anal. 14 341-350 (2001). DOI 10.1023/A:1011261019736 | MR 1825690 | Zbl 0980.35081
[34] Sawano, Y.: Generalized Morrey spaces for non-doubling measures. NoDEA, Nonlinear Differ. Equ. Appl. 15 413-425 (2008). DOI 10.1007/s00030-008-6032-5 | MR 2465971 | Zbl 1173.42317
[35] Shi, X., Sun, Q.: Weighted norm inequalities for Bochner-Riesz operators and singular integral operators. Proc. Am. Math. Soc. 116 665-673 (1992). DOI 10.1090/S0002-9939-1992-1136237-1 | MR 1136237 | Zbl 0786.42006
[36] Sjölin, P.: Convergence almost everywhere of certain singular integrals and multiple Fourier series. Ark. Mat. 9 (1971), 65-90. DOI 10.1007/BF02383638 | MR 0336222 | Zbl 0212.41703
[37] Soria, F., Weiss, G.: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43 187-204 (1994). DOI 10.1512/iumj.1994.43.43009 | MR 1275458 | Zbl 0803.42004
[38] Stein, E. M.: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series 43. Monographs in Harmonic Analysis III Princeton University Press, Princeton (1993). MR 1232192 | Zbl 0821.42001
[39] Stein, E. M.: On the functions of Littlewood-Paley, Lusin and Marcinkiewicz. Trans. Am. Math. Soc. 88 430-466 (1958). DOI 10.1090/S0002-9947-1958-0112932-2 | MR 0112932
[40] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30 Princeton University Press, Princeton (1970). MR 0290095 | Zbl 0207.13501
[41] Taylor, M. E.: Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics 100 Birkhäuser, Boston (1991). MR 1121019 | Zbl 0746.35062
[42] Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Pure and Applied Mathematics 123 Academic Press, Orlando (1986). MR 0869816 | Zbl 0621.42001
[43] Torchinsky, A., Wang, S.: A note on the Marcinkiewicz integral. Colloq. Math. 60/61 235-243 (1990). MR 1096373 | Zbl 0731.42019
[44] Vargas, A. M.: Weighted weak type $(1,1)$ bounds for rough operators. J. Lond. Math. Soc., II. Ser. 54 297-310 (1996). DOI 10.1112/jlms/54.2.297 | MR 1405057 | Zbl 0884.42011
Partner of
EuDML logo