Boundedness of Toeplitz operators related to singular integral operators
Izvestiya. Mathematics , Tome 82 (2018) no. 6, pp. 1225-1238.

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We establish that Toeplitz-type operators related to singular integral operators with variable Calderón–Zygmund kernels are bounded on weighted Morrey spaces. To do this, we prove weighted inequalities for the sharp maximal functions of such operators.
Keywords: Toeplitz-type operator, singular integral operator, sharp maximal function, Morrey space, weighted $\mathrm{BMO}$, weighted Lipschitz function.
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Ya. Tan; L. Liu. Boundedness of Toeplitz operators related to singular integral operators. Izvestiya. Mathematics , Tome 82 (2018) no. 6, pp. 1225-1238. http://geodesic.mathdoc.fr/item/IM2_2018_82_6_a5/

[1] R. R. Coifman, R. Rochberg, G. Weiss, “Factorization theorems for Hardy spaces in several variables”, Ann. of Math. (2), 103:3 (1976), 611–635 | DOI | MR | Zbl

[2] S. Janson, “Mean oscillation and commutators of singular integral operators”, Ark. Math., 16:1-2 (1978), 263–270 | DOI | MR | Zbl

[3] M. Paluszynski, “Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss”, Indiana Univ. Math. J., 44:1 (1995), 1–17 | DOI | MR | Zbl

[4] G. Di Fazio, M. A. Ragusa, “Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients”, J. Funct. Anal., 112:2 (1993), 241–256 | DOI | MR | Zbl

[5] D. K. Palagachev, L. G. Softova, “Singular integral operators, Morrey spaces and fine regularity of solutions to PDE's”, Potential Anal., 20:3 (2004), 237–263 | DOI | MR | Zbl

[6] J. García-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud., 116, North-Holland Publishing Co., Amsterdam, 1985, x+604 pp. | MR | Zbl

[7] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., 43, Monographs in Harmonic Analysis, III, Princeton Univ. Press, Princeton, NJ, 1993, xiv+695 pp. | MR | Zbl

[8] C. Pérez, “Endpoint estimates for commutators of singular integral operators”, J. Funct. Anal., 128:1 (1995), 163–185 | DOI | MR | Zbl

[9] C. Pérez, R. Trujillo-González, “Sharp weighted estimates for multilinear commutators”, J. London Math. Soc. (2), 65:3 (2002), 672–692 | DOI | MR | Zbl

[10] S. Bloom, “A commutator theorem and weighted $\mathrm{BMO}$”, Trans. Amer. Math. Soc., 292:1 (1985), 103–122 | DOI | MR | Zbl

[11] Bei Hu, Jiajun Gu, “Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions”, J. Math. Anal. Appl., 340:1 (2008), 598–605 | DOI | MR | Zbl

[12] S. G. Krantz, Song-Ying Li, “Boundedness and compactness of integral operators on spaces of homogeneous type and applications. I”, J. Math. Anal. Appl., 258:2 (2001), 629–641 | DOI | MR | Zbl

[13] Yan Lin, Shanzhen Lu, “Toeplitz operators related to strongly singular Calderón–Zygmund operators”, Sci. China Ser. A, 49:8 (2006), 1048–1064 | DOI | MR | Zbl

[14] Lanzhe Liu, “Sharp maximal function inequalities and boundedness for Toeplitz type operator related to general fractional integral operators”, Banach J. Math. Anal., 7:1 (2013), 142–159 | DOI | MR | Zbl

[15] Shanzhen Lu, Huixia Mo, “Toeplitz-type operators on Lebesgue spaces”, Acta Math. Sci. Ser. B (Engl. Ed.), 29:1 (2009), 140–150 | DOI | MR | Zbl

[16] A. P. Calderón, A. Zygmund, “On singular integrals with variable kernels”, Appl. Anal., 7:3 (1978), 221–238 | DOI | MR | Zbl

[17] Lanzhe Liu, “The continuity for multilinear singular integral operators with variable Calderón–Zygmund kernel on Hardy and Herz spaces”, Sib. elektron. matem. izv., 2 (2005), 156–166 | MR | Zbl

[18] Lanzhe Liu, “A good $\lambda$ estimate for multilinear singular integral operators with variable Calderón–Zygmund kernel”, Kragujevac J. Math., 27 (2005), 19–30 | MR | Zbl

[19] Lanzhe Liu, “Weighted estimates of multilinear singular integral operators with variable Calderón–Zygmund kernel for the extreme cases”, Vietnam J. Math., 34:1 (2006), 51–61 | MR | Zbl

[20] Hong Xu, Lanzhe Liu, “Weighted boundedness for multilinear singular integral operator with variable Calderón–Zygmund kernel”, Afr. Diaspora J. Math., 6:1 (2008), 1–12 | MR | Zbl

[21] Shanzhen Lu, Dachun Yang, Zusheng Zhou, “Oscillatory singular integral operators with variable Calderón–Zygmund kernels”, Southeast Asian Bull. Math., 23:3 (1999), 457–470 | MR | Zbl

[22] C. B. Morrey, Jr., “On the solutions of quasi-linear elliptic partial differential equations”, Trans. Amer. Math. Soc., 43:1 (1938), 126–166 | DOI | MR | Zbl

[23] G. Di Fazio, M. A. Ragusa, “Commutators and Morrey spaces”, Boll. Un. Mat. Ital. A (7), 5:3 (1991), 323–332 | MR | Zbl

[24] Lanzhe Liu, “Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators”, Acta Math. Sci. Ser. B (Engl. Ed.), 25:1 (2005), 89–94 | DOI | MR | Zbl

[25] T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces”, Harmonic analysis (Sendai, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, 183–189 | MR | Zbl

[26] Y. Komori, S. Shirai, “Weighted Morrey spaces and a singular integral operator”, Math. Nachr., 282:2 (2009), 219–231 | DOI | MR | Zbl

[27] J. García-Cuerva, “Weighted $H^p$ spaces”, Dissertationes Math. (Rozprawy Mat.), 162 (1979), 63 pp. | MR | Zbl

[28] J. Peetre, “On the theory of $\mathscr L_{p,\lambda}$-spaces”, J. Funct. Anal., 4 (1969), 71–87 | DOI | MR | Zbl