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

TL;DR

This paper introduces new weighted H"older-Zygmund spaces using wavelet transforms and provides an elementary proof of the continuity theorem for wavelet transforms on specific function spaces.

Contribution

It presents a novel class of weighted H"older-Zygmund spaces with regularly varying weights and analyzes them through wavelet transforms and Littlewood-Paley pairs.

Findings

Elementary proof of wavelet transform continuity

Definition of new weighted H"older-Zygmund spaces

Analysis of these spaces via wavelet transforms

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 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.