Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 503-522
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.
Copyright © 2009 by Academia Scientiarum Fennica