We provide a new and elementary proof of the continuity theorem for the
wavelet and left-inverse wavelet transforms on the spaces S0(Rn) and S(Hn+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.Comment: 18 page