Skip to main content
SpringerLink
Log in

On the regularity of the one-sided Hardy-Littlewood maximal functions

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators \(\mathcal{M}^+\) and \(\mathcal{M}^-\). More precisely, we prove that \(\mathcal{M}^+\) and \(\mathcal{M}^-\) map W 1,p(ℝ) → W 1,p(ℝ) with 1 < p < 1, boundedly and continuously. In addition, we show that the discrete versions M + and M − map BV(ℤ) → BV(ℤ) boundedly and map l 1(ℤ) → BV(ℤ) continuously. Specially, we obtain the sharp variation inequalities of M + and M −, that is

$$Var\left( {{M^ + }\left( f \right)} \right) \leqslant Var\left( f \right)andVar\left( {{M^ - }\left( f \right)} \right) \leqslant Var\left( f \right)$$

if f ∈ BV(ℤ), where Var(f) is the total variation of f on ℤ and BV(ℤ) is the set of all functions f: ℤ → ℝ satisfying Var(f) < 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. M. Aldaz, J. Pérez Lázaro: Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Am. Math. Soc. 359 (2007), 2443–2461.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Bober, E. Carneiro, K. Hughes, L. B. Pierce: On a discrete version of Tanaka’s theorem for maximal functions. Proc. Am. Math. Soc. 140 (2012), 1669–1680.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. P. Calderón: Ergodic theory and translation invariant operators. Proc. Natl. Acad. Sci. USA 59 (1968), 349–353.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Carneiro, K. Hughes: On the endpoint regularity of discrete maximal operators. Math. Res. Lett. 19 (2012), 1245–1262.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Carneiro, D. Moreira: On the regularity of maximal operators. Proc. Am. Math. Soc. 136 (2008), 4395–4404.

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Dunford, J. Schwartz: Convergence almost everywhere of operator averages. Proc. Natl. Acad. Sci. USA 41 (1955), 229–231.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Hajłasz, J. Onninen: On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 29 (2004), 167–176.

    MathSciNet  MATH  Google Scholar 

  8. G. H. Hardy, J. E. Littlewood: A maximal theorem with function-theoretic applications. Acta Math. 54 (1930), 81–116.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Kinnunen: The Hardy-Littlewood maximal function of a Sobolev function. Isr. J. Math. 100 (1997), 117–124.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Kinnunen, P. Lindqvist: The derivative of the maximal function. J. Reine Angew. Math. 503 (1998), 161–167.

    MathSciNet  MATH  Google Scholar 

  11. J. Kinnunen, E. Saksman: Regularity of the fractional maximal function. Bull. Lond. Math. Soc. 35 (2003), 529–535.

    Article  MathSciNet  MATH  Google Scholar 

  12. O. Kurka: On the variation of the Hardy-Littlewood maximal function. Ann. Acad. Sci. Fenn. Math. 40 (2015), 109–133.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Liu, T. Chen, H. Wu: A note on the endpoint regularity of the Hardy-Littlewood maximal functions. Bull. Aust. Math. Soc. 94 (2016), 121–130.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Liu, H. Wu: On the regularity of the multisublinear maximal functions. Can. Math. Bull. 58 (2015), 808–817.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Luiro: Continuity of the maximal operator in Sobolev spaces. Proc. Am. Math. Soc. 135 (2007), 243–251.

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Luiro: On the regularity of the Hardy-Littlewood maximal operator on subdomains of Rn. Proc. Edinb. Math. Soc. II. 53 (2010), 211–237.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Sawyer: Weighted inequalities for the one-sided Hardy-Littlewood maximal function. Trans. Am. Math. Soc. 297 (1986), 53–61.

    Article  MathSciNet  MATH  Google Scholar 

  18. E. M. Stein, R. Shakarchi: Real Analysis. Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis 3, Princeton University Press, Princeton, 2005.

    Google Scholar 

  19. H. Tanaka: A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function. Bull. Aust. Math. Soc. 65 (2002), 253–258.

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Temur: On regularity of the discrete Hardy-Littlewood maximal function. Available at ArXiv:1303.3993v1 [math. CA].

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, No. 579 Qianwangang Road Economic and Technical Development Zone, Qingdao, Shandong, 266590, P.R. China

    Feng Liu

  2. School of Sciences, Nanchang Institute of Technology, No. 289 Tianxiang Road, Nanchang, Jiangxi, 330099, P.R. China

    Suzhen Mao

Authors
  1. Feng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suzhen Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Liu.

Additional information

The first author was supported by NNSF of China (No. 11526122), Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (No. 2015RCJJ053), Research Award Fund for Outstanding Young Scientists of Shandong Province (No. BS2015SF012) and Support Program for Outstanding Young Scientific and Technological Top-notch Talents of College of Mathematics and Systems Science (No. Sxy2016K01).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, F., Mao, S. On the regularity of the one-sided Hardy-Littlewood maximal functions. Czech Math J 67, 219–234 (2017). https://doi.org/10.21136/CMJ.2017.0475-15

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2017.0475-15

Keywords

MSC 2010

Search

Navigation