New classes of weighted Hölder-Zygmund spaces and the wavelet transform

Abstract

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces ${\mathcal{S}}_0({\mathbb{R}}^n)$ and $\mathcal{S}({\mathbb{H}}^{n+1})$. We then introduce and study a new class of weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.

18 pages