Besov and Triebel-Lizorkin spaces on homogeneous groups

TL;DR

This paper establishes a comprehensive theory of Besov and Triebel-Lizorkin spaces on homogeneous groups, including their independence from decompositions and characterizations via maximal functions and molecules.

Contribution

It introduces a unified framework for these function spaces on homogeneous groups, covering all parameter ranges and linking to classical spaces.

Findings

Spaces are independent of Littlewood-Paley decomposition

Characterizations via maximal functions and molecules are established

Includes classical spaces like Hardy, Sobolev, and Lipschitz spaces as special cases

Abstract

This paper develops a theory of Besov spaces $\dot{\mathbf{B}}^{\sigma}_{p,q} (N)$ and Triebel-Lizorkin spaces $\dot{\mathbf{F}}^{\sigma}_{p,q} (N)$ on an arbitrary homogeneous group $N$ for the full range of parameters $p, q \in (0, \infty]$ and $\sigma \in \mathbb{R}$. 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.