Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 503-522

# MAXIMAL FUNCTIONS IN VARIABLE EXPONENT SPACES: LIMITING CASES OF THE EXPONENT

## Lars Diening, Petteri Harjulehto, Peter Hästö, Yoshihiro Mizuta and Tetsu Shimomura

Freiburg University, Section of Applied Mathematics
Eckerstrasse 1, 79104 Freiburg/Breisgau, Germany; diening 'at' mathematik.uni-freiburg.de

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; petteri.harjulehto 'at' helsinki.fi

University of Oulu, Department of Mathematical Sciences
P.O. Box 3000, FI-90014 University of Oulu, Finland; peter.hasto 'at' helsinki.fi

Hiroshima University, Division of Mathematical and Information Sciences
Higashi-Hiroshima 739-8521, Japan; mizuta 'at' mis.hiroshima-u.ac.jp

Hiroshima University, Department of Mathematics, Graduate School of Education
Higashi-Hiroshima 739-8524, Japan; tshimo 'at' hiroshima-u.ac.jp

Abstract. In this paper we study the Hardy-Littlewood maximal operator in variable exponent spaces when the exponent is not assumed to be bounded away from 1 and \infty. Within the framework of Orlicz-Musielak spaces, we characterize the function space X with the property that Mf \in Lp(.) if and only if f \in X, under the assumptions that p is log-Hölder continuous and 1 \le p- \le p+ \le \infty.

2000 Mathematics Subject Classification: Primary 42B25, 46E30.

Key words: Variable exponent Lebesgue spaces, generalized Lebesgue spaces, Orlicz-Musielak spaces, maximal functions, non-uniformly convex norms, non-doubling modulars.

Reference to this article: L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta and T. Shimomura: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34 (2009), 503-522.