Maximal Averages on the Affine Group G n and Applications

Let be equipped with the left Haar measure . We study maximal averages associated with three basic motions on : horizontal translations, vertical dilations, and fixed hyperbolic geodesics in the upper half‐space model. The translation maximal operator is the Euclidean Hardy–Littlewood maximal operator on each horizontal slice. The Haar‐compatible dilation maximal operator is of weak type and bounded on for , but it is not strongly bounded on . By contrast, the unweighted Lebesgue dilation average is unbounded on every finite and is not of weak type . For fixed hyperbolic geodesic averages, the large‐time part is strongly bounded on because of modular exponential decay. The small‐time part is a finite‐type parabolic maximal problem. Using the corresponding local finite‐type endpoint estimate for the geodesic slice, we prove in weak Orlicz form, together with the strong bounds for . We also show that the strong endpoint fails. Finally, we record a discrete random‐walk maximal inequality whose sufficient condition is expressed through the modular drift , where is the probability measure defining the right random walk.