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 satisfies a $\log$-H\"older 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.
"Variable Exponent Lebesgue Spaces on Metric Spaces: The Hardy-Littlewood Maximal Operator." Real Anal. Exchange 30 (1) 87 - 104, 2004-2005.