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Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 2109–2132 | Cite as

Necessary and sufficient conditions for the bounds of the commutator of a Littlewood-Paley operator with fractional differentiation

  • Xiongtao Wu
  • Yanping ChenEmail author
  • Liwei Wang
  • Wenyu Tao
Article
  • 44 Downloads

Abstract

For \(b\in L_{\mathrm{loc}}({\mathbb {R}}^n)\) and \(0<\alpha <1\), we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely
$$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg |^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$
Here, we obtain the necessary and sufficient conditions for the function b to guarantee that \(g_{\Omega ,\alpha ;b}\) is a bounded operator on \(L^2({\mathbb {R}}^n)\). More precisely, if \(\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}\) and \(b\in I_{\alpha }(BMO)\), then \(g_{\Omega ,\alpha ;b}\) is bounded on \(L^2({\mathbb {R}}^n)\). Conversely, if \(g_{\Omega ,\alpha ;b}\) is bounded on \(L^2({\mathbb {R}}^n)\), then \(b \in Lip_\alpha ({\mathbb {R}}^n)\) for \(0<\alpha < 1\).

Keywords

Commutator Littlewood-Paley operator Rough kernel BMO Sobolev spaces 

Mathematics Subject Classification

42B20 42B25 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare no conflicts of interest.

References

  1. 1.
    Alvarez, J., Bagby, R., Kurtz, D., Pérez, C.: Weighted estimates for commutators of linear operators. Studia Math. 104, 195–209 (1993)Google Scholar
  2. 2.
    AL-Salman, A., AL-Qassem, H., Cheng, L., Pan, Y.: \(L^p\) bounds for the function of Marcinkiewicz. Math. Res. Lett. 9, 697–700 (2002)Google Scholar
  3. 3.
    AL-Salman, A., AL-Qassem, H.: A note on Marcinkiewicz integral operators. J. Math. Anal. Appl. 282, 698–710 (2003)Google Scholar
  4. 4.
    Benedek, A., Calderón, A.P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA 48, 356–365 (1962)Google Scholar
  5. 5.
    Calderón, A.P.: Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 53, 1092–1099 (1965)Google Scholar
  6. 6.
    Calderón, A.P.: Commutators, singular integrals on Lipschitz curves and application. In: Proc. Inter. Con. Math. Helsinki, pp. 85–96, p. 1980. Acad. Sci, Fennica, Helsinki (1978)Google Scholar
  7. 7.
    Calderón, A.P.: Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA 74, 1324–1327 (1977)Google Scholar
  8. 8.
    Chen, Y., Ding, Y.: Commutators of Littlewood-Paley operators. Sci. China Ser. A 52, 2493–2505 (2009)Google Scholar
  9. 9.
    Chen, Y., Ding, Y.: Necessary and sufficient conditions for the bounds of the Calderón type commutator for the Littlewood-Paley operator. Nonlinear Anal. 130, 279–297 (2016)Google Scholar
  10. 10.
    Chen, Y., Ding, Y.: Gradient estimates for commutators of square roots of elliptic operators with complex bounded measurable coefficients. J. Geom. Anal. 27, 466–491 (2017)Google Scholar
  11. 11.
    Chen, Y., Ding, Y.: \(L^p\) boundedness of the commutators of Marcinkiewicz integrals with rough kernels. Forum Math. 27, 2087–2111 (2015)Google Scholar
  12. 12.
    Chen, Y., Ding, Y., Hong, G.: Commutators with fractional differentiation and new characterizations of BMO-Sobolev spaces. Anal. PDE 9, 1497–1522 (2016)Google Scholar
  13. 13.
    Chen, Y., Ding, Y., Hofmann, S.: The commutators of the Kato square root for second order elliptic operators on \({\mathbb{R}}^n\). Acta Math. Sin. (Engl. Ser.) 32, 1121–1144 (2016)Google Scholar
  14. 14.
    Ding, Y., Fan, D., Pan, Y.: Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana Univ. Math. J. 48, 1037–1055 (1999)Google Scholar
  15. 15.
    Ding, Y., Fan, D., Pan, Y.: \(L^{p}\) boundedness of Marcinkiewicz integrals with Hardy space function kernels. Acta Math. Sin. Ser. B (English) 16, 593–600 (2000)Google Scholar
  16. 16.
    Ding, Y., Fan, D., Pan, Y.: On the \(L^p\) boundedness of Marcinkiewicz integrals. Mich. Math. J. 50, 17–26 (2002)Google Scholar
  17. 17.
    Ding, Y., Lu, S., Yabuta, K.: On commutator of Marcinkiewicz integrals with rough kernel. J. Math. Anal. Appl. 275, 60–68 (2002)Google Scholar
  18. 18.
    Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986)Google Scholar
  19. 19.
    Fan, D., Sato, S.: Weak type \((1,1)\) estimates for Marcinkiewicz integrals with rough kernels. Tohôku Math. J. 53, 265–284 (2001)Google Scholar
  20. 20.
    Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., Upper Saddle River (2004)Google Scholar
  21. 21.
    Hömander, L.: Translation invariant operators. Acta. Math. 104, 93–139 (1960)Google Scholar
  22. 22.
    Hu, G., Yan, D.: On the commutator of the Marcinkiewicz integral. J. Math. Anal. Appl. 283, 351–361 (2003)Google Scholar
  23. 23.
    Meyers, N.G.: Mean oscillation over cubes and H\(\ddot{\rm o}\)lder continuity. Proc. Am. Math. Soc. 15, 717–721 (1964)Google Scholar
  24. 24.
    Murray, M.: Commutators with fractional differentiation and BMO Sobolev spaces. Indiana Univ. Math. J. 34, 205–215 (1985)Google Scholar
  25. 25.
    Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)Google Scholar
  26. 26.
    Sakamoto, M., Yabuta, K.: Boundedness of Marcinkiewicz functions. Studia Math. 135, 103–142 (1999)Google Scholar
  27. 27.
    Sato, S.: Remarks on square functions in the Littlewood-Paley theory. Bull. Aust. Math. Soc. 58, 199–211 (1998)Google Scholar
  28. 28.
    Sato, S., Yatubta, K.: Multilinearized Littlewood-Paley operators. Scientiae Mathematicae Japonicae 6, 245–251 (2002)Google Scholar
  29. 29.
    Stein, E.M.: On the functions of Littlewood-Paley, Lusin and Marcinkiewicz. Trans. Am. Math. Soc. 88, 430–466 (1958)Google Scholar
  30. 30.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton (1970)Google Scholar
  31. 31.
    Strichartz, R.S.: Bounded mean oscillations and Sobolev spaces. Indiana Univ. Math. J. 29, 539–558 (1980)Google Scholar
  32. 32.
    Torchinsky, A., Wang, S.L.: A note on the Marcinkiewicz integral. Colloq. Math. 60(61), 235–243 (1990)Google Scholar
  33. 33.
    Walsh, T.: On the function of Marcinkiewicz. Studia Math. 44, 203–217 (1972)Google Scholar
  34. 34.
    Wu, H.: On Marcinkiewicz integral operators with rough kernels. Integral Equ. Oper. Theory 52, 285–298 (2005)Google Scholar
  35. 35.
    Xue, Q., Peng, X., Yabuta, K.: On the theory of multilinear Littlewood-Paley g-function. J. Math. Soc. Jpn. 67, 535–559 (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Xiongtao Wu
    • 1
  • Yanping Chen
    • 2
    Email author
  • Liwei Wang
    • 3
  • Wenyu Tao
    • 4
  1. 1.Department of Mathematics, School of Mathematics and StatisticsHengyang Normal UniversityHengyangChina
  2. 2.Department of Applied Mathematics, School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina
  3. 3.School of Mathematics and PhysicsAnhui Polytechnic UniversityWuhuChina
  4. 4.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina

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