  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.
