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Abstract

Abstract. We present a new proof of the Beurling-Malliavin theorem, often called the "multiplier theorem", concerning the existence of a real-valued function on R with spectrum in a given (small) interval and with a given small majorant of the modulus. This proof pertains entirely to real analysis. It only involves elementary facts about the Hilbert transformation; neither complex variable methods nor potential theory is exploited. The heart of the proof is Theorem 2, which treats preservation of the Lipschitz property under the Hilbert transformation. We also include a short survey of earlier proofs of the Beurling-Malliavin theorem and its generalizations to model (coinvariant) subspaces of the Hardy space H 2 (R)."...Yet the seventh proof of that exists, reliable beyond doubt! And it will be shown you in a while." [Bu, p. 453] §0. Introduction 0.1. Our purpose in this (mainly expository) paper is to present a new, relatively simple and self-contained proof of the Beurling-Malliavin multiplier theorem. Sometimes it is called "The first Beurling-Malliavin theorem", and we shall use the symbol BM1 to refer to it. Theorem BM2, lying beyond the scope of this paper, will be touched upon in Subsection 3. (X.Y means Subsection Y of §X).We shall reduce Theorem BM1 to a certain problem about the Hilbert transformation on the real line. The only new result in the present paper is Theorem 2 in Subsection 2.1, proved by F. L. Nazarov. The reduction of Theorem BM1 to that statement is contained in [HMII] and, implicitly,in [Koo2, p. 502]. Here we reproduce this reduction along the lines of [HMII], but with great simplifications. In particular, our presentation is free of the so-called model subspaces of the Hardy space H 2 (R), which were the main topic of [HMI] and [HMII] (we shall talk about them separately in Subsection 3.5). We wanted to give a focused and generally accessible proof of Theorem BM1, avoiding anywhere near important references to other sources. Our paper is written for nonexperts. We only assume the knowledge of elementary properties of the Fourier transformation on R. For "good" functions f , the Fourier transformation is defined by the formulâ f (ξ) =