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arXiv · 2008.05066

The Ionescu--Wainger multiplier theorem and the adeles

Terence Tao
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Abstract

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 $\ell^p(\Z^d)$ norms of functions with Fourier transform supported on major arcs are comparable to the $L^p(\A_\Z^d)$ norm of their adelic counterparts.

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Terence Tao. 2021-04-25. The Ionescu--Wainger multiplier theorem and the adeles. https://arxiv.org/abs/2008.05066

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