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

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