Transformation de Fourier sur les espaces ${\ell }^p\left(L^p\right)$

Bertrandias, Jean-Paul; Dupuis, Christian

Annales de l'Institut Fourier, Tome 29 (1979) no. 1, pp. 189-206.

Nous étudions d’abord la transformation de Fourier sur les espaces ${\ell }^p(L^{p^{\prime }})$ qui sont formés de fonctions appartenant localement à $L^{p^{\prime }}$ et se comportant à l’infini comme des éléments de ${\ell }^p$. Si $1\le p,p^{\prime }\le 2$, les transformées de Fourier des éléments de ${\ell }^p(L^{p^{\prime }})$ appartiennent à ${\ell }^{q^{\prime }}(L^q)$. Dans les autres cas, nous donnons quelques résultats partiels.

Nous montrons ensuite que ${\ell }^2(L^1)$ est le plus grand espace vectoriel solide de fonctions mesurables sur lequel la transformation de Fourier puisse se définir par prolongement par continuité.

We study the Fourier transform on the spaces ${\ell }^p(L^{p^{\prime }})$ of functions belonging locally to $L^{p^{\prime }}$ and with ${\ell }^p$ behaviour at infinity. If $1\le p,p^{\prime }\le 2$, the Fourier transforms of all functions in ${\ell }^p(L^{p^{\prime }})$ are in ${\ell }^{q^{\prime }}(L^q)$. In the other cases, we state some partial results.

We show that the maximal solid subspace of integrable functions on which the Fourier transform can be defined by extension by continuity is the space ${\ell }^2(L^1)$.

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