The classical Hardy-Littlewood maximal inequality was proved originally for the real line by Hardy and Littlewood (1930), and for Euclidean spaces by Wiener (1939). A key property utilized in the proofs is the fact that the volume growth of the balls is polynomial in the radius. This was generalized by Calderon (1953) who introduced the volume doubling property, which became the cornerstone of a theory of maximal inequalities for doubling metric measure spaces.

But what do you do when the volume growth of the balls in the space under consideration is actually exponential ? We will begin by describing the scope of this problem and the (rather short) history of some of the main results obtained. We will then describe some recent results which establish weak-type maximal inequalities for the Hardy-Littlewood operator. The arguments are based on general geometric and growth conditions only, and we also describe many examples satisfying these conditions. We will conclude by formulating some concrete open problems.