Extending Whitney's extension theorem: nonlinear function spaces

Abstract

This article shows that there is a continuous extension operator for compactly-supported smooth sections of vector bundles on possibly noncompact smooth manifolds, where the closed set to which sections are restricted satisfy a mild restriction on possible boundary cusps. These function spaces are locally convex but in general not Fréchet and so one cannot use the existing theory for Fréchet spaces of smooth functions. Further, in the global, nonlinear case of all smooth functions between possibly noncompact smooth manifolds, we prove that the analogous restriction map between locally convex manifolds of smooth functions is a submersion in the sense of admitting continuous local splittings on charts, under the assumption the closed set is either a submanifold with corners, or is locally convex with rough boundary and compact. In all these results the spaces of functions are spaces of smooth functions, and not spaces of Whitney jets.

Smoothing operators for vector-valued functions and extension operators

“…Due to condition (b), a result of Frerick [17] provides a continuous right inverse for the natural map C ∞ (R d , R) → E(R). Combining these facts, one concludes (see [52,Corollary 3.5]):…”

“…Given d ∈ N, let • 2 be the euclidean norm on R d and B r (x) := {y ∈ R d : y − x 2 < r} be the open ball around x ∈ R d of radius r > 0. The following definition from [52] (which is formulated there for closed subsets of metric spaces) 3 combines conditions from [7] and [17]. (a) R has no narrow fjords, viz., for each x ∈ R there exists n ∈ N, a compact neighbourhood K of x in R and C > 0 such that all y, z ∈ K can be joined by a rectifyable curve γ lying inside R 0 except perhaps for finitely many points, and the path length L(γ) satisfies L(γ) ≤ C n z − y 2 .…”

Lie groupoids of mappings taking values in a Lie groupoid

“…In this appendix, we present a construction of the canonical manifold structure on the spaces C ℓ (K, M ) for K a compact manifold (possibly with rough boundary), ℓ ∈ N 0 ∪ {∞}, and M a (possibly infinite-dimensional) smooth manifold which admits a local addition (in the sense recalled in Definition A.7). Constructions of the manifold structure are well known in special cases; see [10,17,20,24] (for ℓ = ∞ and K a manifold with corners), [34] (for ℓ = ∞ and K with rough boundary), and [41] (for K without boundary, finite ℓ and M of finite dimension). In all approaches mentioned (with the exception of [34]), the construction hinges on a version of the so-called Ω-Lemma.…”

“…Constructions of the manifold structure are well known in special cases; see [10,17,20,24] (for ℓ = ∞ and K a manifold with corners), [34] (for ℓ = ∞ and K with rough boundary), and [41] (for K without boundary, finite ℓ and M of finite dimension). In all approaches mentioned (with the exception of [34]), the construction hinges on a version of the so-called Ω-Lemma. This result is not currently available for manifolds with rough boundary 16 .…”

“…This result is not currently available for manifolds with rough boundary 16 . However, as suggested by the work of Michor and carried out in [34,Section 5], one can circumvent this problem for compact source manifolds by using exponential laws for C k,ℓ -functions (as provided in [1]). We work out this approach here and mention that the authors are not aware of a published source for the results if ℓ < ∞, in the current generality.…”