Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T23:11:10.618Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Regularity of the Multisublinear Maximal Functions

Published online by Cambridge University Press:  20 November 2018

Feng Liu  and
Huoxiong Wu
Feng Liu
Affiliation:
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, P. R. China e-mail: [email protected]
Huoxiong Wu
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, P. R. China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

Keywords

42B2546E35regularitymultisublinear maximal operatorSobolev spacespartial derivativequasicontinuity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 808 - 817
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Aldaz, J. M. and J. Pérez Lâzaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Amer. Math. Soc. 359(2007), no. 5, 2443–2461. http://dx.doi.Org/10.1090/S0002-9947-06-04347-9 Google Scholar
[2] Carneiro, E. and Moreira, D., On the regularity of maximal operators. Proc. Amer. Math. Soc. 136(2008), no. 12, 4395–4404. http://dx.doi.Org/10.1090/S0002-9939-08-09515-4 Google Scholar
[3] Fédérer, H. and Ziemer, W., The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana Univ. Math. J. 22(1972/73), 139–158. http://dx.doi.Org/10.1512/iumj.1973.22.22013 Google Scholar
[4] Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order. Second éd., Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.Google Scholar
[5] Hajlasz, P. and Onninen, J., On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 29(2004), no. 1,167-176.Google Scholar
[6] Kinnunen, J., The Hardy-Littlewood maximal function of a Sobolev function. Israel J. Math. 100(1997), 117–124. http://dx.doi.Org/10.1007/BF02773636 Google Scholar
[7] Kinnunen, J. and Lindqvist, P., The derivative of the maximal function. J. Reine. Angew. Math. 503(1998), 161–167.Google Scholar
[8] Kinnunen, J. and Saksman, E., Regularity of the fractional maximal function. Bull. London Math. Soc. 35(2003), no. 4, 529–535. http://dx.doi.Org/10.1112/S0024609303002017 Google Scholar
[9] Kurka, O., On the variation of the Hardy-Littlewood maximal function. arxiv:1210.0496v1Google Scholar
[10] Luiro, H., Continuity of the maximal operator in Sobolev spaces. Proc. Amer. Math. Soc. 135(2007), no. 1, 243–251. http://dx.doi.Org/10.1090/S0002-9939-06-08455-3 Google Scholar
[11] Tanaka, H., A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function. Bull. Austral. Math. Soc. 65(2002), no. 2, 253–258. http://dx.doi.Org/10.1017/S0004972700020293 Google Scholar