A sufficient condition for Haar multipliers in Triebel-Lizorkin spaces

TL;DR

This paper establishes a sufficient condition based on variation norms for the boundedness of Haar multiplier operators on Triebel-Lizorkin spaces, extending understanding beyond unconditional cases.

Contribution

It provides a new optimal sufficient condition for Haar multipliers in Triebel-Lizorkin spaces when the Haar system is not unconditional.

Findings

Derived a sufficient condition involving variation norms for boundedness.

Extended results to Triebel-Lizorkin spaces with non-unconditional Haar systems.

Identified optimality of the condition in certain parameter ranges.

Abstract

We consider Haar multiplier operators $T_m$ acting on Sobolev spaces, and more generally Triebel-Lizorkin spaces $F^s_{p,q}(\mathbb{R})$, for indices in which the Haar system is not unconditional. When $m$ depends only on the Haar frequency, we give a sufficient condition for the boundedness of $T_m$ in $F^s_{p,q}$, in terms of the variation norms $\|m\|_{V_u}$, which is optimal in $u$ (up to endpoints) when $p, q> 1$.