Maximal Averages on the Affine Group $G_n$ and applications
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912879187329024 |
|---|---|
| author | Li, Ji Shen, Chun-Yen Wen, Chaojie |
| author_facet | Li, Ji Shen, Chun-Yen Wen, Chaojie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05127 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximal Averages on the Affine Group $G_n$ and applications Li, Ji Shen, Chun-Yen Wen, Chaojie Classical Analysis and ODEs Probability 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. |
| title | Maximal Averages on the Affine Group $G_n$ and applications |
| topic | Classical Analysis and ODEs Probability |
| url | https://arxiv.org/abs/2602.05127 |