Fractional cartesian products of sets

Blei, Ron C.

Annales de l'Institut Fourier, Tome 29 (1979) no. 2, pp. 79-105.

Soit $E$ un sous-ensemble du dual $\Gamma$ d’un groupe compact $G$. On dit que $E$ est exactement $p$-Sidon (resp. exactement non-$p$-Sidon) quand $(*)C_E(G{)}^{\wedge }\subset {\ell }^r$ si et seulement si $r\in [p,\infty ]$ (resp. $r\in (p,\infty )$). On dit que $E$ est exactement ${\Lambda }^{\beta }$ (resp. exactement non-${\Lambda }^{\beta }$) quand $E$ vérifie $(**)$toute $f\in L_E^2\left(G\right)$ est telle que, quel que soit $\lambda >0$,

$${\int }_G\exp \left(\lambda \right|f{|}^{2/\alpha }<\infty$$

si et seulement si $\alpha \in [\beta ,\infty )$ (resp. $\alpha \in (\beta ,\infty )$).

Dans ce travail, pour chaque $p\in [1,2)$ et $\beta \in [1,\infty )$, on construit des ensembles qui sont exactement $p$-Sidon, exactement non-$p$-Sidon, exactement ${\Lambda }^{\beta }$ et exactement non-${\Lambda }^{\beta }$.

Let $E$ be a subset of a discrete abelian group whose compact dual is $G$. $E$ is exactly $p$-Sidon (respectively, exactly non-$p$-Sidon) when $(*)C_E(G{)}^{\wedge }\subset {\ell }^r$ holds if and only if $r\in [p,\infty ]$ (respectively, $r\in (p,\infty )$). $E$ is said to be exactly ${\Lambda }^{\beta }$ (respectively, exactly non-${\Lambda }^{\beta }$) if $E$ has the property ${\displaystyle (**)\text{every}f\in L_E^2\left(G\right)\text{satisfies}{\int }_G\exp \left(\lambda \right|f{|}^{2/\alpha }<\infty ,\text{for}\text{all}\lambda >0,}$ if and only if $\alpha \in [\beta ,\infty )$ (respectively, $\alpha \in (\beta ,\infty )$).

In this paper, for every $p\in [1,2)$ and $\beta \in [1,\infty )$, we display sets which are exactly $p$-Sidon, exactly non-$p$-Sidon, exactly ${\Lambda }^{\beta }$ and exactly non-${\Lambda }^{\beta }$.

[1] R. C. Blei, Multidimensional extensions of the Grothendieck inequality (to appear in Arkiv für Mathematik). | Zbl

[2] A. Bonami, Étude des coefficients de Fourier des fonctions de Lp(G), Ann. Inst. Fourier, Grenoble, 20 (1970), 335-402. | Numdam | MR | Zbl

[3] M. Bozejko and T. Pytlik, Some types of lacunary Fourier series, Colloq. Math., 25 (1972), 117-124. | MR | Zbl

[4] R. E. Edwards and K. A. Ross, p-Sidon sets, J. of Functional Anal., 15 (1974), 404-427. | MR | Zbl

[5] A. Figà-Talamanca, An example in the theory of lacunary Fourier series, Boll. Unione Mat. Ital., 3 (1970), 375-378. | MR | Zbl

[6] G. W. Johnson and G. S. Woodward, On p-Sidon sets, Indiana Univ. Math. J., 24 (1974), 161-167. | MR | Zbl

[7] J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quartely J. Math., 1 (1930), 164-174. | JFM

[8] G. Pisier, Ensembles de Sidon et processus gaussiens (preprint). | Zbl

[9] W. Rudin, Trigonometric series with gaps, J. Math. Mechanics, 9 (1960), 203-227. | MR | Zbl

[10] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1967.