Fractional cartesian products of sets

Blei, Ron C. 

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

Abstract (VO)

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-${\Lambda }^{\beta }$.

Résumé

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-${\Lambda }^{\beta }$.

Détail

DOI : 10.5802/aif.744