Maximal Averages on the Affine Group $G_n$ and applications

The general affine group $G_n$ sits at the intersection of harmonic analysis on solvable groups and the geometry of negatively curved symmetric spaces. In this work, we characterize the $L^p$-behavior of maximal operators associated with the fundamental motions of $G_n$. We establish a sharp dichotomy: while translation and geodesic averages exhibit Euclidean-like or improved regularity (yielding $L^1$ boundedness for the latter), dilation averages are governed by the group's non-unimodularity. We prove that dilation averages require a modular-weighted correction to achieve $L^p$ boundedness for $p > 1$, but we establish a fundamental failure at the endpoint $p=1$. Specifically, we prove that dilation maximal operators and those associated with expansive random walks fail the weak-type $(1,1)$ estimate due to an exponential drift-to-volume mismatch. These results connect analytic maximal inequalities to the transience of Brownian motion, demonstrating that modular weights are necessary to compensate for the stochastic drift in the upper half-space.