Construction of a certain superharmonic majorant
Annales de l'Institut Fourier, Volume 44 (1994) no. 3, pp. 729-766.

Given a function f(t)0 on with - (f(t)/(1+t 2 ))dt< and |f(t)-f(t )|l|t-t |, a procedure is exhibited for obtaining on a (finite) superharmonic majorant of the function

F ( z ) : 1 π - | 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 , is such that - (f(t)/(1+t 2 )dt< and denote, for any constant α>0 and each x, the unique value >0 of y making

1 π - y f ( t ) ( x - t ) 2 + y 2 d t = α y

by Y α (x); then - (Y α (x)/(1+x 2 ))dx<.

Soit f(t)0 une fonction définie sur telle que - (f(t)/(1+t 2 ))dt< et que |f(t)-f(t )|l|t-t |; on montre comment obtenir une majorante surharmonique (finie ) sur de la fonction

F ( z ) : 1 π - | 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 - (f(t)/(1+t 2 )dt<, fixons une constante α>0 et, pour chaque x, désignons par Y α (x) l’unique valeur >0 de y pour laquelle

1 π - y f ( t ) ( x - t ) 2 + y 2 d t = α y ;

on a alors - (Y α (x)/(1+x 2 ))dx<.

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     title = {Construction of a certain superharmonic majorant},
     journal = {Annales de l'Institut Fourier},
     pages = {729--766},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
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     year = {1994},
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Koosis, Paul. Construction of a certain superharmonic majorant. Annales de l'Institut Fourier, Volume 44 (1994) no. 3, pp. 729-766. doi : 10.5802/aif.1416. http://archive.numdam.org/articles/10.5802/aif.1416/

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