Unions et intersections d’espaces $L^p$ invariantes par translation ou convolution

Bertrandias, Jean-Paul; Datry, Christian; Dupuis, Christian

Annales de l'Institut Fourier, Tome 28 (1978) no. 2, pp. 53-84.

Étude des propriétés des unions et intersections d’espaces $L^p\left(s\right)$ relatifs à un ensemble $S$ de mesures positives sur un groupe commutatif localement compact lorsque $S$ est invariant par translation ou stable par convolution.

Dans des cas particuliers, on retrouve les propriétés d’espaces étudiés par A. Beurling et par B. Koremblium.

On étudie aussi les espaces ${\ell }^p(L^{p^{\prime }})$ formés des fonctions appartenant localement à $L^{p^{\prime }}$ et qui ont un comportement ${\ell }^p$ à l’infini.

This paper is concerned with properties of unions and intersections of $L^p\left(s\right)$ spaces where $s$ belongs to a set $S$ of positive measures on a locally compact abelian group and where $S$ is translation invariant or convolution invariant.

In special cases, we find again properties of spaces studied by A. Beurling and by B. Koremblium.

We also study the spaces ${\ell }^p(L^{p^{\prime }})$ of functions belonging locally to $L^{p^{\prime }}$ and with ${\ell }^p$ behaviour at infinity.

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