# ANNALES DE L'INSTITUT FOURIER

Some remarks on convolution equations
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

 $\text{cv.}\phantom{\rule{4pt}{0ex}}\text{sing.}\phantom{\rule{4pt}{0ex}}\text{supp.}\phantom{\rule{0.166667em}{0ex}}S*T=\phantom{\rule{0.166667em}{0ex}}\text{cv.}\phantom{\rule{4pt}{0ex}}\text{sing.}\phantom{\rule{4pt}{0ex}}\text{supp.}\phantom{\rule{0.166667em}{0ex}}S+\phantom{\rule{0.166667em}{0ex}}\text{cv.}\phantom{\rule{4pt}{0ex}}\text{sing.}\phantom{\rule{4pt}{0ex}}\text{supp.}\phantom{\rule{0.166667em}{0ex}}T$

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

 $\text{cv.}\phantom{\rule{4pt}{0ex}}\text{sing.}\phantom{\rule{4pt}{0ex}}\text{supp.}\phantom{\rule{0.166667em}{0ex}}S*T=\phantom{\rule{0.166667em}{0ex}}\text{cv.}\phantom{\rule{4pt}{0ex}}\text{sing.}\phantom{\rule{4pt}{0ex}}\text{supp.}\phantom{\rule{0.166667em}{0ex}}S+\phantom{\rule{0.166667em}{0ex}}\text{cv.}\phantom{\rule{4pt}{0ex}}\text{sing.}\phantom{\rule{4pt}{0ex}}\text{supp.}\phantom{\rule{0.166667em}{0ex}}T$

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

@article{AIF_1973__23_1_55_0,
author = {Berenstein, C. A. and Dostal, M. A.},
title = {Some remarks on convolution equations},
journal = {Annales de l'Institut Fourier},
pages = {55--73},
publisher = {Imprimerie Louis-Jean},
volume = {23},
number = {1},
year = {1973},
doi = {10.5802/aif.444},
zbl = {0241.46039},
mrnumber = {49 \#5822},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_1973__23_1_55_0/}
}
Berenstein, C. A.; Dostal, M. A. Some remarks on convolution equations. Annales de l'Institut Fourier, Tome 23 (1973) no. 1, pp. 55-73. doi : 10.5802/aif.444. https://aif.centre-mersenne.org/item/AIF_1973__23_1_55_0/

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