Kislyakov, Serguei V.
Fourier coefficients of continuous functions and a class of multipliers
Annales de l'institut Fourier, Tome 38 (1988) no. 2 , p. 147-183
Zbl 0607.42004 | MR 89j:42004
doi : 10.5802/aif.1138
URL stable : http://www.numdam.org/item?id=AIF_1988__38_2_147_0

Soit x une fonction bornée sur Z ; on définit le multiplicateur avec un symbole x (noté par M x ) par (M x f) ^=xf ^, fL 2 (T). On étudie des conditions sur x qui garantissent “l’inégalité interpolationnelle” M x f L p Cf L 1 α M x f L q 1-α (ici 1<p<q, α=α(p,q,x) est entre 0 et 1 et C ne dépend pas de f). Cette inégalité exprime une sorte de régularité de M x sur L 1 (T). (Pour la plupart les multiplicateurs en question ne sont pas de type faible (1,1).) On utilise ces résultats pour démontrer qu’il y a bien des sous-ensembles E de Z tels que chaque suite positive dans l 2 (E) puisse être majorée par la suite {|f ^(n)|} nE pour une fonction continue f dont le spectre soit inclus dans E.
If x is a bounded function on Z, the multiplier with symbol x (denoted by M x ) is defined by (M x f) ^=xf ^, fL 2 (T). We give some conditions on x ensuring the “interpolation inequality” M x f L p Cf L 1 α M x f L q 1-α (here 1<p<q and α=α(p,q,x) is between 0 and 1). In most cases considered M x fails to have stronger L 1 -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets EZ every positive sequence in 2 (E) can be majorized by the sequence { |f ^(n)|} nE for some continuous funtion f with spectrum in E.

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