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Weak Type (1,1) Behavior for the Maximal Operator with L 1-Dini Kernel

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Abstract

Let M Ω be the maximal operator with homogeneous kernel Ω. In the present paper, we show that if Ω satisfies the L 1-Dini condition on 𝕊n−1, then the following weak type (1,1) behaviors

$\lim\limits _{\lambda \rightarrow 0_{+}}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})=\frac {1}{n} \|\Omega \|_{1} \|f\|_{1},$
$\sup\limits_{\lambda >0}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})\lesssim {\bigg ((\log n)\|\Omega \|_{1}+{\int }_{0}^{1/n}\frac {\tilde {\omega }_{1}(\delta )}{\delta }d\delta \bigg )}\|f\|_{1}$

hold for the maximal operator M Ω and \(f\in L^{1}(\mathbb {R}^{n})\), here \(\tilde {\omega }_{1}\) denotes the L 1 integral modulus of continuity of Ω defined by translation in \(\mathbb {R}^{n}\).

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References

  1. Aldaz, J.M.: The weak type (1,1) bounds for the maximal function associated to cubes grow to infinity with the dimension. Ann. Math. 173, 1013–1023 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubrun, G.: Maximal inequality for high-dimensional cubes. Confluentes Math. 1(2), 169–179 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Calderón, A., Weiss, M., Zygmund, A.: On singular integrals, vol. 10, pp. 56–73. Amer. Math. Soc, Providence (1967)

  4. Calderón, A., Zygmund, A.: On singular integrals. Amer. J. Math. 78, 289–309 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  5. Christ, M., Rubio de Francia, J.-L.: Weak type (1,1) bounds for rough operators II. Invent. Math. 93, 225–237 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Davis, B.: On the weak type (1,1) inequality for conjugate functions. Proc. Amer. Math. Soc. 44, 307–311 (1974)

    MathSciNet  MATH  Google Scholar 

  7. Ding, Y.: Weak type bounds for a class of rough operators with power weights. Proc. Amer. Math. Soc. 125, 2939–2942 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, Y., Lai, X.: L 1-Dini conditions and limiting behavior of weak type estimates for singular integrals. to appear in Rev. Mat. Iberoam. arXiv:1508.07519

  9. Ding, Y., Lu, S.: The \(L^{p_{1}}\times L^{p_{2}}\times \cdot \cdot \times L^{p_{k}}\) boundedness for some rough operators. J. Math. Anal. Appl. 203, 166–186 (1996)

  10. Ding, Y., Lu, S.: Weighted norm inequalities for fractional integral operator with rough kernel. Canad. J. Math. 50, 29–39 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding, Y., Lu, S.: Higher order commutators for a class of rough operators. Ark. Mat. 37, 33–44 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ding, Y., Lu, S.: Homogeneous fractional integrals on Hardy spaces Tohoku. Math. J. 52, 153–162 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Duoandikoetxea, J.: Weighted norm inequalities for homogeneous singular integrals. Trans. Amer. Math. Soc. 336, 869–880 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Duoandikoetxea, J., Rubio de Francia, J.-L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fefferman, R.: A theory of entropy in Fourier analysis. Adv. Math. 30(3), 171–201 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grafakos, L., Stefanov, A.: Convolution Calderón-Zygmund singular integral operators with rough kernels. In: Analysis of Divergence: Control and Management of Divergent Processes, vol. 119–143. Birkhauser, Boston-Basel-Berlin (1999)

  17. Iakovlev, A.S., Strömberg, J.-O.: Lower bounds for the weak type (1,1) estimate for the maximal function associated to cubes in high dimensions. Math. Res. Lett. 20 (5), 907–918 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iwaniec, T., Martin, G.: Riesz transforms and related singular integrals. J. Reine Angew. Math. 473, 25–57 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Janakiraman, P.: Weak-type estimates for singular integrals and the Riesz transform. Indiana. Univ. Math. J. 53, 533–555 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Janakiraman P.: Limiting weak-type behavior for singular integral and maximal operators. Trans. Amer. Math. Soc. 5, 1937–1952 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lu, S., Ding, Y., Yan, D.: Singular Integrals and Related Topics. World Scientific Publishing Co. Pte. Ltd, Hackensack (2007)

    Book  MATH  Google Scholar 

  22. Melas, A.: The best constant for the centered Hardy-Littlewood maximal inequality. Ann. Math. 157, 647–688 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Pichorides, S.: On the best values of the constants arising in the theorems of M. Riesz, Zygmund and Kolmogorov. Studia Math. 44, 165–179 (1972)

    MathSciNet  MATH  Google Scholar 

  25. Rudin, W.: Real and Complax Analysis, 3rd edn. McGraw-Hill Publishing Company, New Delhi (1987)

    Google Scholar 

  26. Seeger, A.: Singular integral operators with rough convolution kernels. J. Amer. Math. Soc. 9, 95–105 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stein, E.M.: The development of square functions in the work of A. Zygmund. Bull. Amer. Math. Soc. (N.S.) 7(2), 359–376 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stein, E.M.: Harmonic analysis: Real-variable methods, orthogonality and oscillatory integrals. In: Princeton Mathematical Series, 43 Monographs in Harmonic Analysis III. Princeton University Press, Princeton (1993)

  29. Stein, E.M., Strömberg, J.-O.: Behavior of maximal functions in \(\mathbb {R}^{n}\) for large n. Ark. Mat. 21, 259–269 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Yong Ding & Xudong Lai

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  1. Yong Ding
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  2. Xudong Lai
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Correspondence to Xudong Lai.

Additional information

The work is supported by NSFC (No.11371057, No.11471033, No.11571160), SRFDP (No.20130003110003) and the Fundamental Research Funds for the Central Universities (No.2014KJJCA10).

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Ding, Y., Lai, X. Weak Type (1,1) Behavior for the Maximal Operator with L 1-Dini Kernel. Potential Anal 47, 169–187 (2017). https://doi.org/10.1007/s11118-017-9612-3

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