Nonlinear generalized sections and vector bundle homomorphisms in Colombeau spaces of generalized functions

Abstract

Key words Colombeau algebras, generalized sections, generalized vector bundle homomorphisms, point values MSC (2010) 26E15, 46F30, 46T30We define and characterize spaces of manifold-valued generalized functions and generalized vector bundle homomorphisms in the setting of the full diffeomorphism-invariant vector-valued Colombeau algebra. Furthermore, we establish point value characterizations for these spaces.

Full and Special Colombeau Algebras

“…Moreover, if one knows two pϕ ε , εq-local generalized numbers or points to be moderate their equivalence can be tested for without resorting to derivatives (cf. [28]): Proof. This can be seen by a straightforward application of the chain rule [18, 3.18, p. 33], similarly to [19,28].…”

“…Moreover, if one knows two pϕ ε , εq-local generalized numbers or points to be moderate their equivalence can be tested for without resorting to derivatives (cf. [28]): Proposition 29. A generalized point or generalized number which is pϕ ε , εqlocal is moderate or negligible if and only if the respective tests of Definitions 25 and 26 hold uniformly for ϕ and ψ 1 , .…”

“…Then by Theorem 14 there exist m P N, ϕ P SpΩq, K Ď Ω compact, a sequence pε k q k with ε k ă 1{k and a sequence px k q k in K such that |Rpϕq ε k px k q| ą ε m k . As in [28] we can choose a compactly supported generalized point X 0 P C 8 pSKpΩq, Ωq which is moderate in the sense of Definition 26 (i) and such that X 0 pϕ ε k q " x k for infinitely many k. Defining X P C 8 pSKpΩq I , Ω I q by Xpϕq ε :" X 0 pϕ ε q, we see that X is ϕ ε -local and RpXqpϕq ε k " Rpϕq ε k px k q, hence RpXq is not negligible. This means that (ii) entails r R " 0.…”

“…Moreover, if one knows two (ϕ ε , ε)-local generalized numbers or points to be moderate their equivalence can be tested for without resorting to derivatives (cf. [31]). Proof.…”

“…k . As in [31] we can choose a compactly supported generalized point X 0 ∈ C ∞ (SK(Ω), Ω) which is moderate in the sense of Definition 26 (i) and such that X 0 (ϕ ε k ) = x k for infinitely many k. Defining X ∈ C ∞ (SK(Ω) I , Ω I ) by X(ϕ) ε := X 0 (ϕ ε ), we see that X is ϕ ε -local and R(X)(ϕ) ε k = R(ϕ) ε k (x k ), hence R(X) is not negligible. This means that (ii) entails R = 0.…”