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Preprints, Working Papers, ... Year : 2023

Harmonic analysis on the boundary of hyperbolic groups

Abstract

In this paper we show that a Möbius-structure $\mathcal{M}$ of dimension $Q$ has a minimal Ahlfors-David constant. This shows that a Möbius space is uniformly $Q$-Ahlfors-David regular. In summary, many classical theorems of harmonic analysis on $\mathbb{R}^n$ admit a Möbius-invariant formulation in the context of Möbius-geometry. We use this observation to show that the Knapp-Stein operator $$ (I_d^\alpha u_d)(x) = \int \frac{u_d(y)}{d(x,y)^{Q - \alpha}} \, d\mu_d(y), \quad\quad (\, 0 < \alpha < \frac{Q}{2}\,) $$ is a continuous operator on the weighted $L^2$-space $L^2((\frac{d'}{d})^{\alpha} d\mu_d)$, with a norm independent of $d$ and $d'$. From here we construct a Sobolev space $\mathcal{H}^{-\alpha}_d$ on $s$-densities for a given $s$ as a function of $\alpha$. We would like to say that the construction is topologically independent of the metric $d$. In this paper we prove that the norms on a large class of functions are comparable. The work is inspired by a paper by Astengo, Cowling, and Di Blasio, who construct uniformly bounded representations for simple Lie groups of rank $1$. We formulate the problem in a much more general framework of groups acting on Möbius structures. In particular, all hyperbolic groups.
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Dates and versions

hal-03963498 , version 1 (30-01-2023)
hal-03963498 , version 2 (31-01-2023)
hal-03963498 , version 3 (11-02-2023)
hal-03963498 , version 4 (21-02-2023)

Identifiers

  • HAL Id : hal-03963498 , version 4

Cite

Georg Alexander Gruetzner. Harmonic analysis on the boundary of hyperbolic groups. 2023. ⟨hal-03963498v4⟩
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