Construction of a certain superharmonic majorant
Annales de l'Institut Fourier, Tome 44 (1994) no. 3, p. 729-766
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 fonctionF(z):1π-|𝔍z||z-t|2f(t)dt-Al|𝔍z|,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 laquelle1π-yf(t)(x-t)2+y2dt=αy;on a alors - (Y α (x)/(1+x 2 ))dx<.
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 functionF(z):1π-|𝔍z||z-t|2f(t)dt-Al|𝔍z|,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 making1π-yf(t)(x-t)2+y2dt=αyby Y α (x); then - (Y α (x)/(1+x 2 ))dx<.
@article{AIF_1994__44_3_729_0,
     author = {Koosis, Paul},
     title = {Construction of a certain superharmonic majorant},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {3},
     year = {1994},
     pages = {729-766},
     doi = {10.5802/aif.1416},
     zbl = {0812.31001},
     mrnumber = {96j:31002},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_3_729_0}
}
Koosis, Paul. Construction of a certain superharmonic majorant. Annales de l'Institut Fourier, Tome 44 (1994) no. 3, pp. 729-766. doi : 10.5802/aif.1416. http://www.numdam.org/item/AIF_1994__44_3_729_0/

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