Skip to main content
SpringerLink
Log in

A Note on Endpoint Sobolev Regularity of a Class of Bilinear Maximal Functions

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

We consider the endpoint Sobolev regularity of a class of bilinear restricted maximal function

$$\begin{aligned}&{\mathfrak {M}}_a(\vec {f})(x)\\&\quad :=\sup \limits _{x\in B(z,a|x|)}\frac{1}{|B(z,a|x|)|^2}\int _{B(z,a|x|)}|f_1(y)|dy\int _{B(z,a|x|)}|f_2(y)|dy. \end{aligned}$$

We show that the map \({\mathfrak {M}}_a: W^{1,1}({\mathbb {R}}^n) \times W^{1,1}({\mathbb {R}}^n)\rightarrow W^{1,1/2}({\mathbb {R}}^n)\) is bounded if \(a\in (0,1/2)\) and not bounded if \(a\in [1/2, +\infty )\).

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Aldaz, J.M., Colzani, L., Pérez Lázaro, J.: Optimal bounds on the modulus of continuity of the uncentered Hardy–Littlewood maximal function. J. Geom. Anal. 22, 132–167 (2012)

    Article  MathSciNet  Google Scholar 

  2. Aldaz, J.M., Pérez Lázaro, J.: Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Am. Math. Soc. 359(5), 2443–2461 (2007)

    Article  MathSciNet  Google Scholar 

  3. Carneiro, E., Moreira, D.: On the regularity of maximal operators. Proc. Am. Math. Soc. 136(12), 4395–4404 (2008)

    Article  MathSciNet  Google Scholar 

  4. Carneiro, E., Madrid, J.: Derivative bounds for fractional maximal functions. Trans. Am. Math. Soc. 369(6), 4063–4092 (2017)

    Article  MathSciNet  Google Scholar 

  5. Carneiro, E., Mardid, J., Pierce, L.B.: Endpoint Sobolev and BV continuity for maximal operators. J. Funct. Anal. 273(10), 3262–3294 (2017)

    Article  MathSciNet  Google Scholar 

  6. Carneiro, E., Svaiter, B.F.: On the variation of maximal operators of convolution type. J. Funct. Anal. 265, 837–865 (2013)

    Article  MathSciNet  Google Scholar 

  7. Hajłasz, P., Maly, J.: On approximate differentiability of the maximal function. Proc. Am. Math. Soc. 138(1), 165–174 (2010)

    Article  MathSciNet  Google Scholar 

  8. Hajłasz, P., Onninen, J.: On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 29(1), 167–176 (2004)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Kinnunen, J., Saksman, E.: Regularity of the fractional maximal function. Bull. Lond. Math. Soc. 35(4), 529–535 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Liu, F.: Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequal. Appl. 21(1), 25–40 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Liu, F., Chen, T., Wu, H.: A note on the endpoint regularity of the Hardy–Littlewood maximal functions. Bull. Aust. Math. Soc. 94(1), 121–130 (2016)

    Article  MathSciNet  Google Scholar 

  15. Liu, F., Mao, S.: On the regularity of the one-sided Hardy–Littlewood maximal functions. Czechoslov. Math. J. 67(142), 219–234 (2017)

    Article  MathSciNet  Google Scholar 

  16. Liu, F., Wu, H.: On the regularity of the multisublinear maximal functions. Can. Math. Bull. 58(4), 808–817 (2015)

    Article  MathSciNet  Google Scholar 

  17. Liu, F., Wu, H.: Endpoint regularity of multisublinear fractional maximal functions. Can. Math. Bull. 60(3), 586–603 (2017)

    Article  MathSciNet  Google Scholar 

  18. Liu, F., Wu, H.: On the regularity of maximal operators supported by submanifolds. J. Math. Anal. Appl. 453, 144–158 (2017)

    Article  MathSciNet  Google Scholar 

  19. Luiro, H.: Continuity of the maixmal operator in Sobolev spaces. Proc. Am. Math. Soc. 135(1), 243–251 (2007)

    Article  MathSciNet  Google Scholar 

  20. Luiro, H.: On the regularity of the Hardy–Littlewood maximal operator on subdomains of \({\mathbb{R}}^n\). Proc. Edinb. Math. Soc. 53(1), 211–237 (2010)

    Article  MathSciNet  Google Scholar 

  21. Luiro, H.: The variation of the maximal function of a radial function. Arkiv Mat. 56(1), 147–161 (2018)

    Article  MathSciNet  Google Scholar 

  22. Madrid, J.: Sharp inequalities for the variation of the discrete maximal function. Bull. Aust. Math. Soc. 95, 94–107 (2017)

    Article  MathSciNet  Google Scholar 

  23. Tanaka, H.: A remark on the derivative of the one-dimensional Hardy–Littlewood maximal function. Bull. Aust. Math. Soc. 65(2), 253–258 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, People’s Republic of China

    Feng Liu

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

    Qingying Xue

  3. Research Center for Mathematics and Data Science, Kwansei Gakuin University, Gakuen 2-1, Sanda, 669-1337, Japan

    Kôzô Yabuta

Authors
  1. Feng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingying Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kôzô Yabuta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported partly by NSFC (Grant No. 11701333) and SP-OYSTTT-CMSS (Grant No. Sxy2016K01). The second author was supported partly by NNSF of China (Nos. 11671039, 11871101) and NSFC-DFG (No. 11761131002).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, F., Xue, Q. & Yabuta, K. A Note on Endpoint Sobolev Regularity of a Class of Bilinear Maximal Functions. Results Math 75, 13 (2020). https://doi.org/10.1007/s00025-019-1139-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-1139-z

Keywords

Mathematics Subject Classification

Search

Navigation