Construction of a certain superharmonic majorant

Koosis, Paul

Koosis, Paul

Annales de l'Institut Fourier, Tome 44 (1994), p. 729-766 / Harvested from Numdam

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Résumé
Abstract

Soit $f (t) \geq 0$ une fonction définie sur $ℝ$ telle que $\int_{- \infty}^{\infty} (f (t) / (1 + t^{2})) d t < \infty$ et que $| f (t) - f (t^{'}) | \leq l | t - t^{'} |$ ; on montre comment obtenir une majorante surharmonique (finie ) sur $ℂ$ de la fonction

F (z) : \frac{1}{π} \int_{- \infty}^{\infty} \frac{| 𝔍 z |}{| z - t |^{2}} f (t) d t - A l | 𝔍 z |,

$A$ étant une (grande) constante absolue. On en tire des démonstrations assez constructives des deux théorèmes principaux du multiplicateur dûs à Beurling et à Malliavin. Le procédé repose sur une version du lemme suivant qui remonte très probablement à Beurling : étant donné une fonction $f (t)$ bornée inférieurement par une quantité $> 0$ et telle que $\int_{- \infty}^{\infty} (f (t) / (1 + t^{2}) d t < \infty$ , fixons une constante $α > 0$ et, pour chaque $x \in ℝ$ , désignons par $Y_{α} (x)$ l’unique valeur $> 0$ de $y$ pour laquelle

\frac{1}{π} \int_{- \infty}^{\infty} \frac{y f (t)}{(x - t)^{2} + y^{2}} d t = α y;

on a alors $\int_{- \infty}^{\infty} (Y_{α} (x) / (1 + x^{2})) d x < \infty$ .

Given a function $f (t) \geq 0$ on $ℝ$ with $\int_{- \infty}^{\infty} (f (t) / (1 + t^{2})) d t < \infty$ and $| f (t) - f (t^{'}) | \leq l | t - t^{'} |$ , a procedure is exhibited for obtaining on $ℂ$ a (finite) superharmonic majorant of the function

F (z) : \frac{1}{π} \int_{- \infty}^{\infty} \frac{| 𝔍 z |}{| z - t |^{2}} f (t) d t - A l | 𝔍 z |,

where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f (t)$ , positive and bounded away from 0 on $ℝ$ , is such that $\int_{- \infty}^{\infty} (f (t) / (1 + t^{2}) d t < \infty$ and denote, for any constant $α > 0$ and each $x \in ℝ$ , the unique value $> 0$ of $y$ making

\frac{1}{π} \int_{- \infty}^{\infty} \frac{y f (t)}{(x - t)^{2} + y^{2}} d t = α y

by $Y_{α} (x)$ ; then $\int_{- \infty}^{\infty} (Y_{α} (x) / (1 + x^{2})) d x < \infty$ .

Publié le : 1994-01-01

MR 96j:31002 | Zbl 0812.31001

DOI : https://doi.org/10.5802/aif.1416

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     author = {Koosis, Paul},
     title = {Construction of a certain superharmonic majorant},
     journal = {Annales de l'Institut Fourier},
     volume = {44},
     year = {1994},
     pages = {729-766},
     doi = {10.5802/aif.1416},
     mrnumber = {96j:31002},
     zbl = {0812.31001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1994__44_3_729_0}
}

Koosis, Paul. Construction of a certain superharmonic majorant. Annales de l'Institut Fourier, Tome 44 (1994) pp. 729-766. doi : 10.5802/aif.1416. http://gdmltest.u-ga.fr/item/AIF_1994__44_3_729_0/

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