The Ionescu–Wainger multiplier theorem establishes good $L^p$ bounds for Fourier multiplier operators localized to major arcs; it has become an indispensible tool in discrete harmonic analysis. We give a simplified proof of this theorem with more explicit constants (removing logarithmic losses that were present in previous versions of the theorem), and give a more general variant involving adelic Fourier multipliers. We also establish a closely related adelic sampling theorem that shows that ${\ell }^p({\mathbb{Z}}^d)$ norms of functions with Fourier transform supported on major arcs are comparable to the $L^p({\mathbb{A}}_{\mathbb{Z}}^d)$ norm of their adelic counterparts.

中文翻译：

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IONESCU-WAINGER乘子定理和ADELES

Ionescu-Wainger乘法定理建立了良好的 ${\mathrm{大号}}^p$局部于主要弧的傅立叶乘法器算子的边界；它已成为离散谐波分析中必不可少的工具。我们用更明确的常数（消除了该定理的先前版本中存在的对数损失）为该定理提供了简化的证明，并给出了包含阿迪克傅立叶乘法器的更一般的变体。我们还建立了密切相关的adelic采样定理，该定理表明${\ell }^p（{\mathbb{ž}}^d）$ 在主要弧上支持傅立叶变换的函数范数与 ${\mathrm{大号}}^p（{\mathbb{一种}}_{\mathbb{ž}}^d）$ 他们尊贵的同行的规范。