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.

💡 Research Summary

This paper investigates three natural families of maximal averaging operators on the general affine group (G_n=\mathbb{R}^n\rtimes\mathbb{R}+), a non‑unimodular solvable Lie group that can be identified with the upper half‑space (\mathbb{R}^n\times\mathbb{R}+) equipped with the Poincaré metric, i.e. the real hyperbolic space (\mathbb{H}^{n+1}). The authors focus on the (L^p) mapping properties of maximal operators associated with (i) horizontal translations, (ii) vertical dilations, and (iii) hyperbolic geodesics. Their main contribution is a sharp dichotomy: translation and geodesic averages behave essentially like Euclidean or even better, while dilation averages are dramatically affected by the non‑unimodularity of the group and require a modular weight to be bounded.

Group preliminaries.
The group law is ((x,y)(x’,y’)=(x+yx’,yy’)) with left Haar measure (d\mu(x,y)=dx,dy,y^{n+1}) and modular function (\Delta(x,y)=y^{-n}). The right Haar measure is (d\mu_R(x,y)=dx,dy,y^{-1}). The factor (y^{-n}) measures how right translation by ((0,y)) expands the left Haar measure by (y^n). This non‑unimodular feature is the source of the difficulties for dilation averages.

1. Horizontal translation maximal operator (M_{\mathrm{trans}}).
Defined by \