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 p (Z d ) norms of functions with Fourier transform supported on major arcs are comparable to the L p (A d Z ) norm of their adelic counterparts. §1. Introduction. This paper will be concerned with the L p theory of Fourier multiplier operators on various locally compact abelian groups, such as Z d , R d , andIn order to treat these groups in a unified fashion, we adopt the following abstract harmonic analysis notation. Definition 1.1 (Pontryagin duality). An LCA group is a locally compact abelian group G = (G, +) equipped with a Haar measure μ G . A Pontryagin dual of an LCA group G is an LCA group G * = (G * , +) with a Haar measure μ G * and a continuous bihomomorphism (x, ξ ) → x • ξ (which we call a pairing) from G × G * to the unit circle T = R/Z, such that the Fourier transformwhere e : T → C is the standard character e(θ ) := e 2πiθ , extends to a unitary map from L 2 (G) to L 2 (G * ); in particular, we have the Plancherel identityfor all f ∈ L 2 (G), as well as the inversion formulaIf ⊂ G * is measurable, we say that f ∈ L 2 (G) is Fourier supported in if F G f vanishes outside of (modulo null sets). The space of such functions will be denoted L 2 (G) .If m ∈ L ∞ (G * ), we define the associated Fourier multiplier operator T m : L 2 (G) → L 2 (G) by the formulaWe refer to m as the symbol of T m .

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