THE IONESCU–WAINGER MULTIPLIER THEOREM AND THE ADELES

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.