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Maximal Averages on the Affine Group $G_n$ and applications

Ji Li, Chun-Yen Shen, Chaojie Wen

TL;DR

This work analyzes maximal operators on the general affine group $G_n = \mathbb{R}^n \rtimes \mathbb{R}_+$, identifying three fundamental averaging motions (translations, dilations with a modular weight, and geodesics) and proving sharp $L^p$ bounds and endpoint behavior. Translation and fixed-geodesic averages enjoy Euclidean-like or hyperbolic-regularity, yielding $L^p$ boundedness for all $p>1$ (and strong type $(1,1)$ for geodesics), while dilation averages require a modular correction to attain $L^p$ boundedness but fail at the endpoint $p=1$. The paper also shows a probabilistic interpretation via Brownian motion drift and discrete random walks, linking modular weights to drift compensation and boundary convergence. These results illuminate how non-unimodularity and negative curvature shape maximal inequalities on solvable groups, with implications for harmonic analysis on non-compact spaces and stochastic processes on hyperbolic geometries.

Abstract

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.

Maximal Averages on the Affine Group $G_n$ and applications

TL;DR

This work analyzes maximal operators on the general affine group , identifying three fundamental averaging motions (translations, dilations with a modular weight, and geodesics) and proving sharp bounds and endpoint behavior. Translation and fixed-geodesic averages enjoy Euclidean-like or hyperbolic-regularity, yielding boundedness for all (and strong type for geodesics), while dilation averages require a modular correction to attain boundedness but fail at the endpoint . The paper also shows a probabilistic interpretation via Brownian motion drift and discrete random walks, linking modular weights to drift compensation and boundary convergence. These results illuminate how non-unimodularity and negative curvature shape maximal inequalities on solvable groups, with implications for harmonic analysis on non-compact spaces and stochastic processes on hyperbolic geometries.

Abstract

The general affine group sits at the intersection of harmonic analysis on solvable groups and the geometry of negatively curved symmetric spaces. In this work, we characterize the -behavior of maximal operators associated with the fundamental motions of . We establish a sharp dichotomy: while translation and geodesic averages exhibit Euclidean-like or improved regularity (yielding 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 boundedness for , but we establish a fundamental failure at the endpoint . Specifically, we prove that dilation maximal operators and those associated with expansive random walks fail the weak-type 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.
Paper Structure (25 sections, 13 theorems, 125 equations)

This paper contains 25 sections, 13 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Group Structure
  4. Visualization of Motion
  5. Haar Measure and Modular Function
  6. Translation Averages
  7. Dilation Averages
  8. The failure of standard averaging via Lebesgue measure
  9. The adjusted averaging operator
  10. Fixed Geodesic Averages
  11. Sharpness of Estimates and Endpoint Behavior
  12. Translation Bounds
  13. Dilation Bounds and the $p \to 1$ Blow-up
  14. Geodesic Decay Rates
  15. Directional vs. Global Maximal Operators
  16. ...and 10 more sections

Key Result

Theorem 1

The translation maximal operator $\mathcal{M}_{\mathrm{trans}}$ is bounded on $L^p(G_n)$ for all $1 < p \le \infty$ and is of weak-type $(1,1)$.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition : Translation maximal operator
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 17 more