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SHARP INEQUALITIES FOR THE VARIATION OF THE DISCRETE MAXIMAL FUNCTION

Part of: Difference and functional equations Linear function spaces and their duals Difference equations Real functions Harmonic analysis in several variables

Published online by Cambridge University Press:  04 November 2016

JOSÉ MADRID
JOSÉ MADRID*
Affiliation:
IMPA—Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil email [email protected]
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Abstract

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In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].

Keywords

Sobolev spacesdiscrete maximal operatorsbounded variation

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 26A45: Functions of bounded variation, generalizations 39A12: Discrete version of topics in analysis 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E39: Sobolev (and similar kinds of) spaces of functions of discrete variables
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 94 - 107
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 uncentered Hardy–Littlewood maximal function’, J. Geom. Anal. 22 (2012), 132–167.CrossRefGoogle 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(5) (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.CrossRefGoogle Scholar
Carneiro, E., Finder, R. and Sousa, M., ‘On the variation of maximal operators of convolution type II’, Preprint, 2015, http://arxiv.org/abs/1512.02715v1.Google Scholar
Carneiro, E. and Hughes, K., ‘On the endpoint regularity of discrete maximal operators’, Math. Res. Lett. 19(6) (2012), 1245–1262.Google Scholar
Carneiro, E. and Madrid, J., ‘Derivative bounds for fractional maximal functions’, Trans. Amer. Math. Soc., to appear, http://dx.doi.org/10.1090/tran/6844.CrossRefGoogle Scholar
Carneiro, E. and Moreira, D., ‘On the regularity of maximal operators’, Proc. Amer. Math. Soc. 136(12) (2008), 4395–4404.CrossRefGoogle 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 Malý, 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(1) (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(4) (2003), 529–535.Google Scholar
Kurka, O., ‘On the variation of the Hardy–Littlewood maximal function’, Ann. Acad. Sci. Fenn. Math. 40 (2015), 109–133.CrossRefGoogle Scholar
Lang, S., Algebraic Number Theory, 2nd edn (Springer, New York, 1994).CrossRefGoogle Scholar
Luiro, H., ‘Continuity of the maximal operator in Sobolev spaces’, Proc. Amer. Math. Soc. 135(1) (2007), 243–251.CrossRefGoogle Scholar
Steinerberger, S., ‘A rigidity phenomenon for the Hardy–Littlewood maximal function’, Studia Math. 229(3) (2015), 263–278.Google Scholar
Tanaka, H., ‘A remark on the derivative of the one-dimensional Hardy–Littlewood maximal function’, Bull. Aust. Math. Soc. 65(2) (2002), 253–258.Google Scholar
Temur, F., ‘On regularity of the discrete Hardy–Littlewood maximal function’, Preprint, 2013, http://arxiv.org/abs/1303.3993.Google Scholar