Haar projection numbers and failure of unconditional convergence in Sobolev spaces

TL;DR

This paper characterizes the conditions under which the Haar system forms an unconditional basis in various Sobolev and Triebel-Lizorkin spaces, revealing limitations of unconditional convergence related to Haar projection numbers.

Contribution

It precisely determines the range of Sobolev spaces where Haar is unconditional and extends the analysis to Triebel-Lizorkin spaces, providing bounds on projection norms.

Findings

Haar system is unconditional in certain Sobolev spaces for 1<p<∞

Projection norms depend on properties of Haar frequency sets

Unconditional convergence fails outside specific Sobolev space ranges

Abstract

For $1<p<\infty$ we determine the precise range of $L_p$ Sobolev spaces for which the Haar system is an unconditional basis. We also consider the natural extensions to Triebel-Lizorkin spaces and prove upper and lower bounds for norms of projection operators depending on properties of the Haar frequency set.