Maximal averages on the affine group G_n and applications

Let (Formula presented.) be equipped with the left Haar measure (Formula presented.). We study maximal averages associated with three basic motions on (Formula presented.) : 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 (Formula presented.) and bounded on (Formula presented.) for (Formula presented.), but it is not strongly bounded on (Formula presented.). By contrast, the unweighted Lebesgue dilation average is unbounded on every finite (Formula presented.) and is not of weak type (Formula presented.). For fixed hyperbolic geodesic averages, the large-time part is strongly bounded on (Formula presented.) because of modular exponential decay. The small-time part is a finite-type parabolic maximal problem. Using the corresponding local finite-type (Formula presented.) endpoint estimate for the geodesic slice, we prove (Formula presented.) in weak Orlicz form, together with the strong (Formula presented.) bounds for (Formula presented.). We also show that the strong (Formula presented.) endpoint fails. Finally, we record a discrete random-walk maximal inequality whose sufficient condition is expressed through the modular drift (Formula presented.), where (Formula presented.) is the probability measure defining the right random walk.