Fourier coefficients of continuous functions and a class of multipliers

Kislyakov, Serguei V.

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

Soit $x$ une fonction bornée sur $\mathbf{Z}$ ; on définit le multiplicateur avec un symbole $x$ (noté par $M_x$) par $(M_{x}f)\widehat{}=x\widehat{f}$, $f\in L^2\left(\mathbf{T}\right)$. On étudie des conditions sur $x$ qui garantissent “l’inégalité interpolationnelle” $\parallel M_{x}f{\parallel }_{L^p}\le C\parallel f{\parallel }_{L^1}^{\alpha }\parallel M_{x}f{\parallel }_{L^q}^{1-\alpha }$ (ici $1<p<q$, $\alpha =\alpha (p,q,x)$ est entre 0 et 1 et $C$ ne dépend pas de $f$). Cette inégalité exprime une sorte de régularité de $M_x$ sur $L^1\left(\mathbf{T}\right)$. (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 $\mathbf{Z}$ tels que chaque suite positive dans $l^2\left(E\right)$ puisse être majorée par la suite $\left\{\right|\widehat{f}\left(n\right)|{\}}_{n\in E}$ pour une fonction continue $f$ dont le spectre soit inclus dans $E$.

If $x$ is a bounded function on $\mathbf{Z}$, the multiplier with symbol $x$ (denoted by $M_x)$ is defined by $(M_{x}f)\widehat{}=x\widehat{f}$, $f\in L^2\left(\mathbf{T}\right)$. We give some conditions on $x$ ensuring the “interpolation inequality” $\parallel M_{x}f{\parallel }_{L^p}\le C\parallel f{\parallel }_{L^1}^{\alpha }\parallel M_{x}f{\parallel }_{L^q}^{1-\alpha }$ (here $1<p<q$ and $\alpha =\alpha (p,q,x)$ 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 $E\subset \mathbf{Z}$ every positive sequence in ${\ell }^2\left(E\right)$ can be majorized by the sequence $\{$ $\left|\widehat{f}\left(n\right)\right|{\}}_{n\in E}$ for some continuous funtion $f$ with spectrum in $E$.

[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