A simple and consistent definition of homogeneous Besov spaces on stratified Lie groups

TL;DR

This paper defines homogeneous Besov spaces on stratified Lie groups using Littlewood-Paley decomposition, proves their independence from specific decompositions, and characterizes them via wavelet transforms, ensuring consistency with previous spectral calculus approaches.

Contribution

It introduces a unified, decomposition-independent definition of homogeneous Besov spaces on stratified Lie groups and provides wavelet-based characterizations.

Findings

Spaces are independent of the decomposition used.

Homogeneous Besov spaces are consistent with spectral calculus methods.

Wavelet transforms effectively characterize these spaces.

Abstract

We introduce a general definition of homogeneous Besov spaces on a stratified Lie group $G$, based on a Littlewood-Paley-type decomposition of Schwartz functions with all moments vanishing. We show that under mild and intuitive conditions the spaces thus defined are independent of the decomposition employed. A corollary of this is that previously constructed versions of homogeneous Besov spaces on $G$, relying on the spectral calculus of a sub-Laplacian of the group, are consistent, i.e., independent of the choice of sub-Laplacian. We further prove characterizations of homogeneous Besov spaces using continuous wavelet transforms, with a large variety of analysing wavelets to choose from.