Besov and Triebel-Lizorkin spaces on homogeneous groups

TL;DR

This work provides a unified theory of Besov spaces $\dot{\mathbf{B}}^{\sigma}_{p,q}(N)$ and Triebel-Lizorkin spaces $\dot{\mathbf{F}}^{\sigma}_{p,q}(N)$ on arbitrary homogeneous groups $N$, defined via Littlewood-Paley-type decompositions independent of any fixed operator. It establishes continuous maximal characterizations, wavelet-transform and coorbit-space descriptions, and robust molecular decompositions, enabling operator-boundedness results within this framework. The theory recovers classical spaces such as Hardy spaces, homogeneous Sobolev spaces, and Lipschitz spaces on stratified groups as specific instances, while providing new maximal-characterizations for $p=\infty$ cases. The results offer a cohesive link between harmonic analysis on homogeneous groups and function-space theory, with implications for spectral multipliers, coorbit theory, and boundedness of operators in non-Euclidean settings.

Abstract

This paper develops a theory of Besov spaces $\dot{\mathbf{B}}^σ_{p,q} (N)$ and Triebel-Lizorkin spaces $\dot{\mathbf{F}}^σ_{p,q} (N)$ on an arbitrary homogeneous group $N$ for the full range of parameters $p, q \in (0, \infty]$ and $σ\in \mathbb{R}$. Among others, it is shown that these spaces are independent of the choice of the Littlewood-Paley decomposition and that they admit characterizations in terms of continuous maximal functions and molecular frame decompositions. The defined spaces include as special cases various classical function spaces, such as Hardy spaces on homogeneous groups and homogeneous Sobolev spaces and Lipschitz spaces associated to sub-Laplacians on stratified groups.

Theorems & Definitions (72)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3: Left Taylor inequality
• Lemma 3.1
• proof
• Definition 3.2
• Remark 3.3
• Proposition 3.4
• proof
• Lemma 3.5