The Haar system as a Schauder basis in spaces of Hardy-Sobolev type

TL;DR

This paper proves that the multivariate Haar system can serve as a Schauder basis in certain Hardy-Sobolev and Triebel-Lizorkin spaces, extending previous unconditionality results to a broader class of function spaces.

Contribution

It establishes the Haar system as a Schauder basis in Hardy-Sobolev and Triebel-Lizorkin spaces, expanding the understanding of basis properties in these function spaces.

Findings

Haar system is a Schauder basis in classical Sobolev spaces for specific parameters.

Extension of results to Hardy-Sobolev and Triebel-Lizorkin spaces in optimal parameter ranges.

Identification of conditions under which the Haar system is a conditional Schauder basis.

Abstract

We show that, for suitable enumerations, the multivariate Haar system is a Schauder basis in the classical Sobolev spaces on $\mathbb R^d$ with integrability $1<p<\infty$ and smoothness $1/p-1<s<1/p$. This complements earlier work by the last two authors on the unconditionality of the Haar system and implies that it is a {conditional} Schauder basis for a nonempty open subset of the $(1/p,s)$-diagram. The results extend to (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type in the range of parameters $\frac{d}{d+1}<p<\infty$ and $\max\{d(1/p-1),1/p-1\}<s<\min\{1,1/p\}$, which is optimal except perhaps at the end-points.