Fourier coefficients of continuous functions and a class of multipliers

Kislyakov, Serguei V.

doi:10.5802/aif.1138

Kislyakov, Serguei V.

Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 147-183.

Résumé
Abstract

Soit $x$ une fonction bornée sur $Z$ ; on définit le multiplicateur avec un symbole $x$ (noté par $M_{x}$ ) par $(M_{x} f) \hat{} = x \hat{f}$ , $f \in L^{2} (T)$ . On étudie des conditions sur $x$ qui garantissent “l’inégalité interpolationnelle” $∥ M_{x} f ∥_{L^{p}} \leq C ∥ f ∥_{L^{1}}^{α} ∥ M_{x} f ∥_{L^{q}}^{1 - α}$ (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 ${| \hat{f} (n) |}_{n \in E}$ 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 $(M_{x} f) \hat{} = x \hat{f}$ , $f \in L^{2} (T)$ . We give some conditions on $x$ ensuring the “interpolation inequality” $∥ M_{x} f ∥_{L^{p}} \leq C ∥ f ∥_{L^{1}}^{α} ∥ M_{x} f ∥_{L^{q}}^{1 - α}$ (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 $E \subset Z$ every positive sequence in $ℓ^{2} (E)$ can be majorized by the sequence ${$ $| \hat{f} (n) |}_{n \in E}$ for some continuous funtion $f$ with spectrum in $E$ .

MR Zbl

DOI : 10.5802/aif.1138

@article{AIF_1988__38_2_147_0,
     author = {Kislyakov, Serguei V.},
     title = {Fourier coefficients of continuous functions and a class of multipliers},
     journal = {Annales de l'Institut Fourier},
     pages = {147--183},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {2},
     year = {1988},
     doi = {10.5802/aif.1138},
     mrnumber = {89j:42004},
     zbl = {0607.42004},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1138/}
}

TY  - JOUR
AU  - Kislyakov, Serguei V.
TI  - Fourier coefficients of continuous functions and a class of multipliers
JO  - Annales de l'Institut Fourier
PY  - 1988
SP  - 147
EP  - 183
VL  - 38
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.1138/
DO  - 10.5802/aif.1138
LA  - en
ID  - AIF_1988__38_2_147_0
ER  -

%0 Journal Article
%A Kislyakov, Serguei V.
%T Fourier coefficients of continuous functions and a class of multipliers
%J Annales de l'Institut Fourier
%D 1988
%P 147-183
%V 38
%N 2
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.1138/
%R 10.5802/aif.1138
%G en
%F AIF_1988__38_2_147_0

Kislyakov, Serguei V. Fourier coefficients of continuous functions and a class of multipliers. Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 147-183. doi : 10.5802/aif.1138. https://www.numdam.org/articles/10.5802/aif.1138/

Bibliographie
Cité par

[1] J. Bourgain, Bilinear forms on Hé and bounded bianalytic functions, Trans. Amer. Math. Soc., 286, N° 1 (1984), 313-337. | MR | Zbl

[2] R. R. Coifman, G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, N° 4 (1977), 569-645. | MR | Zbl

[3] K. De Leeuw, Y. Katznelson, J.-P. Kahane, Sur les coefficients de Fourier des fonctions continues, C. R. Acad. Sci. Paris, Sér. A, 285, N° 16 (1977), 1001-1003. | MR | Zbl

[4] J. García-Cuerva, J. L. Rubio De Francia, Weighted norm inequalities and related topics, North Holland, Amsterdam, New York, Oxford, 1985. | MR | Zbl

[5] S. V. Hruščëv (S. V. Khrushtchëv), S. A. Vinogradov, Free interpolation in the space of uniformly convergent Taylor series, Lecture Notes Math., 864, Springer, Berlin, 1981, 171-213. | MR | Zbl

[6] S. V. Kislyakov, On reflexive subspaces of the space C*A, Funktsionalnyi Anal. i ego Prilozhen., 13, No 1 (1979), 21-30 (Russian). | Zbl

[7] S. V. Kislyakov, Fourier coefficients of boundary values of functions analytic in the disc and in the bidisc, Trudy Matem. Inst. im. V. A. Steklova, 155 (1981), 77-94 (Russian). | MR | Zbl

[8] S. V. Kislyakov, A substitute for the weak type (1, 1) inequality for multiple Riesz projections, Linear and Complex Analysis Problem Book, Lecture Notes Math., 1043, Springer, Berlin, 1984, 322-324.

[9] B. Maurey, Nouveaux théorèmes de Nikishin (suite et fin), Séminaire Maurey-Schwartz, 1973-1974, Exposé No V, École Polytechnique, Paris, 1974. | Numdam | Zbl

[10] W. Rudin, Trigonometric series with gaps, J. Math. Mech., 9, N° 2 (1960), 203-227. | MR | Zbl

[11] S. Sawyer, Maximal inequalities of weak type, Ann. Math., 84, N° 1 (1966), 157-173. | MR | Zbl

[12] S. Sidon, Einige Sätze und Fragestellungen über Fourier-Koeffizienten, Math. Z., 34, N° 4 (1932), 477-480. | Zbl

[13] P. Sjögren, P. Sjölin, Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets, Ann. Inst. Fourier, 31, n° 1 (1981), 157-175. | Numdam | MR | Zbl

[14] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970. | MR | Zbl

[15] S. V. Vinogradov, A strengthening of the Kolmogorov theorem on conjugate function and interpolation properties of uniformly convergent power series, Trudy Matem. Inst. im V. A. Steklova, 155 (1981), 7-40 (Russian). | MR | Zbl

[16] A. Zygmund, Trigonometric series, vol. I, II, Cambridge at the University Press, 1959. | Zbl

Cité par Sources :