Geodesic
A Dixmier-Malliavin Theorem for Lie Groupoids
Journal of Lie theory, Tome 32 (2022) no. 3, pp. 879-898
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A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. The analytical heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from functions vanishing to given order on an invariant submanifold of the unit space.
Classification : 22A22, 58H05
Mots-clés : Dixmier-Malliavin, Lie groupoid, convolution 
@article{JLT_2022_32_3_JLT_2022_32_3_a13,
     author = {M. D. Francis},
     title = {A {Dixmier-Malliavin} {Theorem} for {Lie} {Groupoids}},
     journal = {Journal of Lie theory},
     pages = {879--898},
     year = {2022},
     volume = {32},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JLT_2022_32_3_JLT_2022_32_3_a13/}
}
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M. D. Francis. A Dixmier-Malliavin Theorem for Lie Groupoids. Journal of Lie theory, Tome 32 (2022) no. 3, pp. 879-898. http://geodesic.mathdoc.fr/item/JLT_2022_32_3_JLT_2022_32_3_a13/