The Trouvé group for spaces of test functions

Abstract

The Trouvé group ${\mathcal{G}}_{\mathcal{A}}$ from image analysis consists of the flows at a fixed time of all time-dependent vectors fields of a given regularity $\mathcal{A}({\mathbb{R}}^d,{\mathbb{R}}^d)$. For a multitude of regularity classes $\mathcal{A}$, we prove that the Trouvé group ${\mathcal{G}}_{\mathcal{A}}$ coincides with the connected component of the identity of the group of orientation preserving diffeomorphims of ${\mathbb{R}}^d$ which differ from the identity by a mapping of class $\mathcal{A}$. We thus conclude that ${\mathcal{G}}_{\mathcal{A}}$ has a natural regular Lie group structure. In many cases we show that the mapping which takes a time-dependent vector field to its flow is continuous. As a consequence we obtain that the scale of Bergman spaces on the polystrip with variable width is stable under solving ordinary differential equations.

23 pages

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Cited by in corpus (1)

• Manifolds of mappings for continuum mechanics