We study continuity envelopes of function spaces $B_{p,q}^s(ℝⁿ,w)$ and $F_{p,q}^s(ℝⁿ,w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

We give a characterization of the Hölder-Zygmund spaces ${𝒞}^{\sigma }\left(G\right)$ ($0<\sigma <\infty$) on a stratified Lie group $G$ in terms of Littlewood-Paley type decompositions, in analogy to the well-known characterization of the Euclidean case. Such decompositions are defined via the spectral measure of a sub-Laplacian on $G$, in place of the Fourier transform in the classical setting. Our approach mainly relies on almost orthogonality estimates and can be used to study other function spaces such as Besov and Triebel-Lizorkin spaces...

The paper deals with local means and wavelet bases in weighted and unweighted function spaces of type $B_{pq}^s$ and $F_{pq}^s$ on ℝⁿ and on ⁿ.