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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]
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Abstract

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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