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A REMARK ON THE REGULARITY OF THE DISCRETE MAXIMAL OPERATOR

Part of: Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  01 December 2016

FENG LIU
FENG LIU*
Affiliation:
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, PR China email liufeng860314@163.com
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Abstract

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We study the regularity properties of several classes of discrete maximal operators acting on $\text{BV}(\mathbb{Z})$ functions or $\ell ^{1}(\mathbb{Z})$ functions. We establish sharp bounds and continuity for the derivative of these discrete maximal functions, in both the centred and uncentred versions. As an immediate consequence, we obtain sharp bounds and continuity for the discrete fractional maximal operators from $\ell ^{1}(\mathbb{Z})$ to $\text{BV}(\mathbb{Z})$ .

Keywords

discrete maximal operatorfractional maximal operatorbounded variationcontinuity

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 108 - 120
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Aldaz, J. M., Colzani, L. and Pérez Lázaro, J., ‘Optimal bounds on the modulus of continuity of the uncentred Hardy–Littlewood maximal function’, J. Geom. Anal. 22 (2012), 132–167.Google Scholar
Aldaz, J. M. and Pérez Lázaro, J., ‘Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities’, Trans. Amer. Math. Soc. 359 (2007), 2443–2461.CrossRefGoogle Scholar
Bober, J., Carneiro, E., Hughes, K. and Pierce, L. B., ‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc. 140 (2012), 1669–1680.Google Scholar
Brezis, H. and Lieb, E., ‘A relation between pointwise convergence of functions and convergence of functionals’, Proc. Amer. Math. Soc. 88 (1983), 486–490.CrossRefGoogle Scholar
Carneiro, E. and Hughes, K., ‘On the endpoint regularity of discrete maximal operators’, Math. Res. Lett. 19 (2012), 1245–1262.CrossRefGoogle Scholar
Carneiro, E. and Madrid, J., ‘Derivative bounds for fractional maximal functions’, Trans. Amer. Math. Soc., to appear, Preprint, http://arxiv:1510.02965v2.Google Scholar
Carneiro, E. and Moreira, D., ‘On the regularity of maximal operators’, Proc. Amer. Math. Soc. 136 (2008), 4395–4404.Google Scholar
Carneiro, E. and Svaiter, B. F., ‘On the variation of maximal operators of convolution type’, J. Funct. Anal. 265 (2013), 837–865.Google Scholar
Hajłasz, P. and Maly, J., ‘On approximate differentiability of the maximal function’, Proc. Amer. Math. Soc. 138 (2010), 165–174.CrossRefGoogle Scholar
Hajłasz, P. and Onninen, J., ‘On boundedness of maximal functions in Sobolev spaces’, Ann. Acad. Sci. Fenn. Math. 29 (2004), 167–176.Google Scholar
Kinnunen, J., ‘The Hardy–Littlewood maximal function of a Sobolev function’, Israel J. Math. 100 (1997), 117–124.Google Scholar
Kinnunen, J. and Lindqvist, P., ‘The derivative of the maximal function’, J. reine angew. Math. 503 (1998), 161–167.Google Scholar
Kinnunen, J. and Saksman, E., ‘Regularity of the fractional maximal function’, Bull. Lond. Math. Soc. 35 (2003), 529–535.CrossRefGoogle Scholar
Kurka, O., ‘On the variation of the Hardy–Littlewood maximal function’, Ann. Acad. Sci. Fenn. Math. 40 (2015), 109–133.CrossRefGoogle Scholar
Liu, F., Chen, T. and Wu, H., ‘A note on the end-point regularity of the Hardy–Littlewood maximal functions’, Bull. Aust. Math. Soc. 94 (2016), 121–130.CrossRefGoogle Scholar
Liu, F. and Wu, H., ‘On the regularity of the multisublinear maximal functions’, Canad. Math. Bull. 58 (2015), 808–817.Google Scholar
Luiro, H., ‘Continuity of the maximal operator in Sobolev spaces’, Proc. Amer. Math. Soc. 135 (2007), 243–251.Google Scholar
Luiro, H., ‘On the regularity of the Hardy–Littlewood maximal operator on subdomains of ℝ n ’, Proc. Edinb. Math. Soc. (2) 53 (2010), 211–237.Google Scholar
Madrid, J., ‘Sharp inequalities for the variation of the discrete maximal function’, Bull. Aust. Math. Soc. , to appear. DOI: https://doi.org./10.1017/S0004972716000903.Google Scholar
Tanaka, H., ‘A remark on the derivative of the one-dimensional Hardy–Littlewood maximal function’, Bull. Aust. Math. Soc. 65 (2002), 253–258.Google Scholar
Temur, F., ‘On regularity of the discrete Hardy–Littlewood maximal function’, Preprint, 2013, arXiv:1303.3993v1.Google Scholar