The Ionescu--Wainger multiplier theorem and the adeles

2020/08/12 by Terence Tao, Tao, Terence · 1 citation
Computer Science · Mathematics · #42B15 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2008.05066

openalex publication_date 2020/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℓ^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.

Citations

• On the Product of the Primes

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• Pointwise convergence of polynomial multiple ergodic averages along the primes

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