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Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces

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The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

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References

  1. D. R. Adams, “A note on Riesz potentials,” Duke Math. J. 42, 765–778 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. S. Balakishiyev, V. S. Guliyev, F. Gürbüz, and A. Serbetci, “Sublinear operators with rough kernel generated by Calderon-Zygmund operators and their commutators on generalized local Morrey spaces,” J. Inequal. Appl. 61, doi:10.1186/s13660-015-0582-y (2015).

  3. L. Caffarelli, “Elliptic second order equations,” Rend. Semin.Math. Fis. Milano 58, 253–284 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Chiarenza and M. Frasca, “Morrey spaces and Hardy–Littlewood maximal function,” Rend. Mat. 7, 273–279 (1987).

    MathSciNet  MATH  Google Scholar 

  5. F. Chiarenza, M. Frasca, and P. Longo, “Interior W2, p-estimates for nondivergence elliptic equations with discontinuous coefficients,” Ricerche Mat. 40 (1), 149–168 (1991).

    MathSciNet  MATH  Google Scholar 

  6. F. Chiarenza, M. Frasca, and P. Longo, “W2, p-solvability of Dirichlet problem for nondivergence elliptic equations with VMO coefficients,” Trans. Amer.Math. Soc. 336 (2), 841–853 (1993).

    MathSciNet  MATH  Google Scholar 

  7. R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Ann. ofMath. 103 (3), 611–636 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Ding, D. C. Yang, and Z. Zhou, “Boundedness of sublinear operators and commutators on L p, (Rn),” Yokohama Math. J. 46, 15–27 (1998).

    MathSciNet  MATH  Google Scholar 

  9. G. Di Fazio and M. A. Ragusa, “Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients,” J. Funct. Anal. 112 (2), 241–256 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Di Fazio, D. K. Palagachev, and M. A. Ragusa, “Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients,” J. Funct. Anal. 166, 179–196 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. S. Guliyev, “Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces,” J. Inequal. Appl., Art. ID 503948 (2009).

    Google Scholar 

  12. V. S. Guliyev, S. S. Aliyev, T. Karaman, and P. S. Shukurov, “Boundedness of sublinear operators and commutators on generalized Morrey Space,” Integ. Equ. Oper. Theory. 71 (3), 327–3555 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Gürbüz, Boundedness of Some Potential-Type Sublinear Operators and Their Commutators with Rough Kernels on Generalized Local Morrey Spaces, Ph.D. thesis (Ankara University, Ankara, Turkey, 2015) [in Turkish].

    Google Scholar 

  14. F. Gürbüz, “Parabolic sublinear operators with rough kernel generated by parabolic Calderón–Zygmund operators and parabolic local Campanato space estimates for their commutators on the parabolic generalized localMorrey spaces,” OpenMath. 14, 300–323 (2016).

    MATH  Google Scholar 

  15. F. Gürbüz, “Parabolic sublinear operators with rough kernel generated by parabolic fractional integral operators and parabolic local Campanato space estimates for their commutators on the parabolic generalized localMorrey spaces,” Adv.Math. (China), in press (2017).

    Google Scholar 

  16. S. Janson, “Mean oscillation and commutators of singular integral operators,” Ark. Mat. 16, 263–270 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  17. F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Pure Appl.Math. 14, 415–426 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Karaman, Boundedness of Some Classes of Sublinear Operators on Generalized Weighted Morrey Spaces and Some Applications, Ph.D. thesis (Ankara University, Ankara, Turkey, 2012) [in Turkish].

    Google Scholar 

  19. G. Lu, S. Z. Lu, and D. C. Yang, “Singular integrals and commutators on homogeneous groups,” Anal. Math. 28, 103–134 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  20. S.Z. Lu, Y. Ding, and D. Y. Yan, Singular Integrals and Related Topics, (WorldScientificPubl., Singapore, 2006).

    MATH  Google Scholar 

  21. S. Z. Lu, “Some results and problems on commutators,” Front.Math. China 6, 821–833 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Mazzucato, “Besov–Morrey spaces: functions space theory and applications to nonlinear PDE,” Trans. Amer.Math. Soc. 355, 1297–1364 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  23. T. Mizuhara, Boundedness of Some Classical Operators on Generalized Morrey Spaces, Harmonic Analysis, S. Igari (editor), in ICM 90 Satellite Proceedings, Springer-Verlag (Tokyo, 1991).

  24. C. B. Morrey, “On the solutions of quasi-linear elliptic partial differential equations,” Trans. Amer. Math. Soc. 43, 126–166 (1938).

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Muckenhoupt and R. L. Wheeden, “Weighted norm inequalities for singular and fractional integrals,” Trans. Amer.Math. Soc. 161, 249–258 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  26. E. Nakai, “Hardy–Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces,” Math. Nachr. 166, 95–103 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  27. D. K. Palagachev and L. G. Softova, “Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s,” Potential Anal. 20, 237–263 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Paluszynski, “Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg, and Weiss,” Indiana Univ. Math. J. 44, 1–17 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Peetre, “On the theory of Mp,” J. Funct. Anal. 4, 71–87 (1969).

    Article  MATH  Google Scholar 

  30. A. Ruiz and L. Vega, “On local regularity of Schrödinger equations,” Int. Math. Res. Not. 1, 13–27 (1993).

    Article  MATH  Google Scholar 

  31. S. G. Shi, Z. W. Fu, and F. Y. Zhao, “Estimates for operators on weighted Morrey spaces and their applications to nondivergence elliptic equations,” J. Inequal. Appl. 2013, 390 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  32. S. G. Shi and S. Z. Lu, “A characterization of Campanato space via commutator of fractional integral,” J. Math. Anal. Appl. 419, 123–137 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  33. L. G. Softova, “Singular integrals and commutators in generalized Morrey spaces,” Acta Math. Sin. (Engl. Ser.) 22, 757–766 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  34. F. Soria and G. Weiss, “A remark on singular integrals and power weights,” Indiana Univ. Math. J. 43, 187–204 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  35. E. M. Stein, “On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz,” Trans. Amer. Math. Soc. 88, 430–466 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  36. E. M. Stein, Singular Integrals and Differentiability of Functions (PrincetonUniversity Press, Princeton, NJ, 1970).

    MATH  Google Scholar 

  37. E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals (Princeton University Press, Princeton, NJ, 1993).

    MATH  Google Scholar 

  38. A. Torchinsky, Real Variable Methods in Harmonic Analysis Pure and Applied Math. 123 (Academic Press, New York, 1986).

    Google Scholar 

  39. A. Torchinsky and S. Wang, “A note on the Marcinkiewicz integral,” Colloq. Math. 60/61, 235–243 (1990).

    MathSciNet  MATH  Google Scholar 

  40. R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis of Pure and Applied Mathematics (Marcel Dekker, New York, NY, 1977), Vol. 43.

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  1. Department of Mathematics, Ankara University, Ankara, Turkey

    F. Gürbüz

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  1. F. Gürbüz
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Correspondence to F. Gürbüz.

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Gürbüz, F. Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces. Math Notes 101, 429–442 (2017). https://doi.org/10.1134/S0001434617030051

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