We establish wavelet characterizations of homogeneous Besov spaces on
stratified Lie groups, both in terms of continuous and discrete wavelet
systems. The associated transform is an analogue of the $\phi$-transform
introduced by Frazier and Jawerth for the characterization of function spaces
in the Euclidean setting.

We first introduce a notion of homogeneous Besov space $\dot{B}_{p,q}^s$ in
terms of a Calder\'on-type decomposition into bandlimited tempered
distributions, in analogy to the well-known characterization of the Euclidean
case. For the context of stratified Lie groups, frequency bands are defined
with reference to the spectral measure of a sub-Laplacian of the group. The
convolution kernels involved in this decomposition are oscillatory Schwartz
functions, i.e., wavelets. We then prove that the homogeneous Besov spaces can
also be characterized in terms of the heat semigroup; in particular, their
definition is to a certain extent independent of the precise choice of wavelet
in the initial definition. Finally, we turn to the discretization of the
wavelet criterion, and establish the existence of universal wavelet frames and
associated atomic decomposition formulas for all homogeneous Besov spaces
${\dot B}_{p,q}^{s}$, with $1 \le p,q < \infty$ and $s \in \mathbb{R}$.