Variable Exponent Lebesgue Spaces on Metric Spaces: The Hardy-Littlewood Maximal Operator

In this article we introduce variable exponent Lebesgue spaces on metric measure spaces and consider a central tool in geometric analysis: the Hardy-Littlewood maximal operator. We show that the maximal operator is bounded provided the variable exponent satises a log-Hölder type estimate. This condition is known to be essentially sharp in real Euclidean space, however, we show that this is not so in metric spaces.