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<p>Given 1 &lt; <i>p</i> &lt; ∞, <i>p</i> ≠ 2, we show that the <i>T</i>1 theorem for the Hilbert transform fails for <i>L</i><sup><i>p</i></sup>, despite holding for <i>p</i> = 2. More precisely, we construct a pair of locally finite positive Borel measures (<i>σ</i>, <i>ω</i>) that satisfy the two-tailed <i>A</i><sub><i>p</i></sub> condition, and satisfy both of the <i>L</i><sup><i>p</i></sup>-testing conditions for the Hilbert transform <i>H</i>, yet <i>H</i><sub><i>σ</i></sub>: <i>L</i><sup><i>p</i></sup>(<i>σ</i>) ↛ <i>L</i><sup><i>p</i></sup>(<i>ω</i>). In the opposite direction, the <i>T</i>1 theorem for the Hilbert transform for <i>p</i> = 2 was proved a decade ago in the two part paper [LaSaShUr3] and [Lac].</p>

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The T1 theorem for the Hilbert transform fails when p ≠ 2

  • 期刊论文

  • 发表日期：

    2025年7月13日

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  • Michel Alexis,
  • Jose Luis Luna-Garcia,
  • Eric T. Sawyer,
  • Ignacio Uriarte-Tuero

摘要

Given 1 < p < ∞, p ≠ 2, we show that the T1 theorem for the Hilbert transform fails for Lp, despite holding for p = 2. More precisely, we construct a pair of locally finite positive Borel measures (σ, ω) that satisfy the two-tailed Ap condition, and satisfy both of the Lp-testing conditions for the Hilbert transform H, yet Hσ: Lp(σ) ↛ Lp(ω). In the opposite direction, the T1 theorem for the Hilbert transform for p = 2 was proved a decade ago in the two part paper [LaSaShUr3] and [Lac].