New classes of weighted H\"older-Zygmund spaces and the wavelet transform

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 H\"older-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.