On uniform boundedness of dyadic averaging operators in spaces of Hardy-Sobolev type

TL;DR

This paper provides an elementary proof of the uniform boundedness of dyadic averaging operators in Hardy-Sobolev and Triebel-Lizorkin spaces, crucial for establishing Haar basis properties, using wavelet characterizations.

Contribution

It offers a new, simpler proof of boundedness results that were previously established, leveraging wavelet characterizations of the function spaces.

Findings

Proves uniform boundedness of dyadic averaging operators in Hardy-Sobolev and Triebel-Lizorkin spaces

Uses wavelet characterizations to simplify the proof

Supports Schauder basis properties of Haar systems in these spaces

Abstract

We give an alternative proof of recent results by the authors on uniform boundedness of dyadic averaging operators in (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type. This result served as the main tool to establish Schauder basis properties of suitable enumerations of the univariate Haar system in the mentioned spaces. The rather elementary proof here is based on characterizations of the respective spaces in terms of orthogonal compactly supported Daubechies wavelets.