The Trouvé group for spaces of test functions

Abstract

The Trouvé group G A from image analysis consists of the flows at a fixed time of all time-dependent vectors fields of a given regularity A(R d , R d ). For a multitude of regularity classes A, we prove that the Trouvé group G A coincides with the connected component of the identity of the group of orientation preserving diffeomorphims of R d which differ from the identity by a mapping of class A. We thus conclude that G A has a natural regular Lie group structure. In many cases we show that the mapping which takes a timedependent 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. Theorem 2.4 ([4, Theorem 2.2]). A function f : I → E is measurable by seminorm if and only if f is weakly measurable and for each p ∈ P there exists a nullset N p ⊆ I such that f (I \ N p ) is separable. For a simple function f : I → E, the integral over a measurable set J ⊆ I is clearly defined as J f (t) dt := y∈E λ(f −1 ({y}) ∩ J) · y, where λ is the Lebesgue measure on I. A function f : I → E which is measurable by seminorm is called integrable by seminorm if ∀p ∈ P, ∀n ∈ N : p • (f p n − f ) ∈ L 1 (I), ∀J ⊆ I measurable ∃F J ∈ E ∀p ∈ P : p F J − J f p n (t) dt n→∞ −→ 0;in this case F J =: J f (t) dt. For a complete locally convex space E, integrability of p • f for each p ∈ P already implies integrability by seminorm. If E is a Banach space, then clearly integrability by seminorm coincides with Bochner integrability and the respective integrals coincide. It is easily seen that for each p ∈ P,

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“…(Cleary, D [M] (R n ) is nontrivial only in the non-quasianalytic setting.) Note that various aspects of these global classes have recently been studied in [11,12,16]. We have continuous (with respect to the natural locally convex topologies) inclusions…”

Manifolds of mappings for continuum mechanics

“…Trouvé groups. For the following see [103], [108], [85]. Trouvé groups are useful for introducing topological metrics on certain groups of diffeomorphism on R d starting from a suitable reproducing kernel Hilbert space of vector fields without using any Lie algebra structure; see 8.12 below.…”

“…)) is a suitable vector space of time dependent vector fields. It seems that for a wide class of spaces A the Trouvé group G A is independent of the choice of F A if the latter contains the piecewise smooth curves and is contained in the curves which are integrable by seminorms; a precise statement is still lacking, but see [85], [84], [82], and citations therein.…”

“…• For B, W ∞,p , Schwartz functions S, C ∞ c , and many classes of Denjoy-Carleman functions, where G A is always a regular Lie group; see [85].…”