Dynamics Seminar

We consider the Hardy-Littlewood maximal operator associated with ball averages on discrete groups, with balls defined by invariant metrics associated with a variety of length functions. Under natural assumptions on the rough radial structure of the group under consideration, we establish a weak-type maximal inequality for the Hardy-Littlewood operator. These assumptions are related to a coarse radial median inequality, to almost exact polynomial-exponential growth of balls, and to the rough radial rapid decay property. These concepts will all be defined and explained during the talk.

We give a variety of examples where the rough radial structure assumptions hold, including any lattice in a connected semisimple Lie group with finite center, with respect to the invariant Riemannian distance on symmetric space restricted to an orbit of the lattice.

For non-elementary word-hyperbolic groups we establish that the Hardy-Littlewood maximal operator with respect to balls defined by a word metric satisfies the weak-type (1,1)-maximal inequality, which is the optimal result.

This is joint work with Koji Fujiwara, Kyoto University.