Some remarks on convolution equations

Berenstein, C. A.; Dostal, M. A.

Annales de l'Institut Fourier, Tome 23 (1973) no. 1, pp. 55-73.

Par voie d’une description de la topologie des espaces ${\mathbf{E}}^{\prime }\left(\Omega \right)$ ($\Omega$ ouvert convexe dans $R^n$) via la transformation de Fourier, c’est-à-dire leurs structures analytiques uniformes, on arrive à une formule qui décrit l’enveloppe convexe du support singulier d’une distribution $T$, $T\in {\mathbf{E}}^{\prime }$. On donne des applications à une classe des distributions qui satisfont à l’égalité

$$\text{cv.}\text{sing.}\text{supp.}S*T=\text{cv.}\text{sing.}\text{supp.}S+\text{cv.}\text{sing.}\text{supp.}T$$

pour toutes $S\in {\mathbf{E}}^{\prime }$.

Using a description of the topology of the spaces ${\mathbf{E}}^{\prime }\left(\Omega \right)$ ($\Omega$ open convex subset of $R^n$) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution $T$, $T\in {\mathbf{E}}^{\prime }$. We give applications to a class of distributions $T$ satisfying

$$\text{cv.}\text{sing.}\text{supp.}S*T=\text{cv.}\text{sing.}\text{supp.}S+\text{cv.}\text{sing.}\text{supp.}T$$

for all $S\in {\mathbf{E}}^{\prime }$.

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