Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 175-185

ON THE CONTINUITY OF DISCRETE MAXIMAL OPERATORS IN SOBOLEV SPACES

Hannes Luiro and Juho Nuutinen

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35, FI-40014 Jyväskylä, Finland; hannes.s.luiro 'at' jyu.fi

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35, FI-40014 Jyväskylä, Finland; juho.nuutinen@jyu.fi

Abstract. We investigate the continuity of discrete maximal operators in Sobolev space W^1,p(Rⁿ). A counterexample is given as well as it is shown that the continuity follows under certain sufficient assumptions. Especially, our research verifies that for the continuity in Sobolev spaces the role of the partition of the unity used in the construction of the maximal operator is very delicate.

2010 Mathematics Subject Classification: Primary 42B25, 46E35.

Key words: Discrete maximal operator, Sobolev spaces, continuity.

Reference to this article: H. Luiro and J. Nuutinen: On the continuity of discrete maximal operators in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 39 (2014), 175-185.

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doi:10.5186/aasfm.2014.3916