  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.
