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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 \(p, q> 1\) .

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A Sufficient Condition for Haar Multipliers in Triebel-Lizorkin Spaces

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  • 发表日期：

    2025年

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  • Gustavo Garrigós,
  • Andreas Seeger,
  • Tino Ullrich

摘要

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 \(p, q> 1\) .