# The Haar System in Triebel-Lizorkin Spaces: Endpoint Results

## TL;DR

This paper characterizes the basis properties of the Haar system in Triebel-Lizorkin spaces at critical endpoint cases, completing the understanding of these properties across the entire scale.

## Contribution

It provides a comprehensive characterization of Schauder and unconditional basis properties of the Haar system in Triebel-Lizorkin spaces at endpoint parameters, extending previous results.

## Key findings

- Haar system forms a Schauder basis at specific endpoints
- Unconditional basis properties are established at critical cases
- Completes the classification of basis properties in Triebel-Lizorkin spaces

## Abstract

We characterize the Schauder and unconditional basis properties for the Haar system in the Triebel-Lizorkin spaces $F^s_{p,q}(\Bbb R^d)$, at the endpoint cases $s=1$, $s=d/p-d$ and $p=\infty$. Together with the earlier results in [10], [4], this completes the picture for such properties in the Triebel-Lizorkin scale, and complements a similar study for the Besov spaces given in [5].