Geodesic
Extension of distributions and representation by derivatives of continuous functions.
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 7 (1996) no. 1, pp. 31-40

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

It is proved that any Banach valued distribution on a bounded set can be extended to all of \( \mathbb{R}^{d} \) if and only if it is a derivative of a uniformly continuous function. A similar result is given for distributions on an unbounded set. An example shows that this does not extend to Frechet valued distributions. This relies on the fact that a Banach valued distribution is locally a derivative of a uniformly continuous function. For sake of completeness, a global representation of a Banach valued distribution by derivatives of functions with compact supports is given. 
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Lemoine, Jérôme; Simon, Jacques. Extension of distributions and representation by derivatives of continuous functions.. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 7 (1996) no. 1, pp. 31-40. http://geodesic.mathdoc.fr/item/RLIN_1996_9_7_1_a2/