Differentiability of the Evolution Map and Mackey Continuity
aa r X i v : . [ m a t h . F A ] S e p Differentiability of the Evolution Mapand Mackey Continuity
Maximilian Hanusch ∗ Institut f¨ur MathematikUniversit¨at PaderbornWarburger Straße 10033098 PaderbornGermanySeptember 6, 2019
Abstract
We solve the differentiability problem for the evolution map in Milnor’s infinite dimensionalsetting. We first show that the evolution map of each C k -semiregular Lie group G (for k ∈ N ⊔ { lip , ∞} ) admits a particular kind of sequentially continuity – called Mackey k-continuity.We then prove that this continuity property is strong enough to ensure differentiability of theevolution map. In particular, this drops any continuity presumptions made in this context sofar. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, whichmakes it particular among the continuity conditions traditionally considered. As an applicationof the introduced notions, we discuss the strong Trotter property in the sequentially-, and theMackey continuous context. We furthermore conclude that if the Lie algebra of G is a Fr´echetspace, then G is C k -semiregular (for k ∈ N ⊔ {∞} ) if and only if G is C k -regular. Contents ∗ [email protected] Auxiliary Results 12
In 1983 Milnor introduced his regularity concept [15] as a tool to extend proofs of fundamentalLie theoretical facts to infinite dimensions. Specifically, he adapted (and weakened) the regularityconcept introduced in 1982 by Omori, Maeda, Yoshioka and Kobayashi for Fr´echet Lie groups[20] to such Lie groups that are modeled over complete Hausdorff locally convex vector spaces.Then, he used this notion to prove the integrability of Lie algebra homomorphisms to Lie grouphomomorphisms under certain regularity and connectedness presumptions. In this paper, we workin the slightly more general setting introduced by Gl¨ockner in [2] – specifically meaning that anycompleteness presumption on the modeling space is dropped. Roughly speaking, regularity is concerned with definedness, continuity, and smoothness of theevolution map (product integral) – a notion that naturally generalizes the concept of the Riemannintegral for curves in locally convex vector spaces, to infinite dimensional Lie groups (Lie algebravalued curves are thus integrated to Lie group elements). For instance, the exponential map ofa Lie group is the restriction of the evolution map to constant curves; and, given a principalfibre bundle, then holonomies are evolutions of such Lie algebra valued curves that are pairingsof smooth connections with derivatives of curves in the base manifold of the bundle. Althoughindividual arguments show that the generic infinite dimensional Lie group is C ∞ -regular or stronger,only recently general regularity criteria had been found [3, 7, 16]. Differentiability of the evolutionmap (hence, of the exponential map) is one of the key components of the regularity problem.In [3,7], this issue had been discussed in the standard topological context – implicitly meaning that Confer also [16, 17] for an introduction to this area. To prevent confusion, we additionally remark that Milnor’sdefinition of an infinite dimensional manifold M involves the requirement that M is a regular topological space, i.e.,fulfills the separation axioms T , T . Deviating from that, in [2], only the T property of M is explicitly presumed –This, however, makes no difference in the Lie group case, because topological groups are automatically T . C k -topology was presumed. In this paper, we solvethe differentiability problem in full generality, as we drop any continuity presumption made in thiscontext so far. The results obtained in particular imply that if the Lie group is modeled over aFr´echet space, with evolution map defined on all C k -curves (the Lie group is C k -semiregular), thenthe evolution map is automatically smooth w.r.t. to the C k -topology (the Lie group is C k -regular).We furthermore generalize the results obtained in [4, 8] concerning the strong Trotter property byweakening the continuity presumptions made there.More specifically, let G denote an infinite dimensional Lie group as defined in [2] that is modeledover the Hausdorff locally convex vector space E . We let g denote the Lie algebra of G ; as well asd q R g the differential of the right translation R g : G ∋ h h · g by g ∈ G , at the point q ∈ G . Wefurthermore define (right logarithmic derivative) C ([0 , , g ) ∋ δ r ( µ ) := d µ R µ − ( ˙ µ ) ∀ µ ∈ C ([0 , , G )as well as D := { δ r ( µ ) | µ ∈ C ([0 , , G ) } and C ∗ ([0 , , G ) := { µ ∈ C ([0 , , G ) | µ (0) = e } . Theevolution maps are given by Evol : D ∋ δ r ( µ ) µ · µ − (0) ∈ C ∗ ([0 , , G )evol : D ∋ φ Evol( φ )(1) ∈ G as well asEvol k := Evol | D k and evol k := evol | D k , with D k := D ∩ C k ([0 , , g ) for each k ∈ N ⊔ { lip , ∞ , c } . We say that G is C k -semiregularif C k ([0 , , g ) ⊆ D holds; hence, if each φ ∈ C k ([0 , , g ) admits a (necessarily unique) solution µ ∈ C ∗ ([0 , , G ) to the differential equation δ r ( µ ) = φ . It was shown in [3] (cf. Theorem E in [3])that if G is C k -semiregular for k ∈ N ⊔ {∞} , then Evol k (thus, evol k ) is smooth if and only if evol k is of class C . Then, it was proven in [7] (cf. Theorem 4 in [7]) that evol k is of class C if and onlyif it is continuous, with g Mackey complete for k ∈ N ≥ ⊔ { lip , ∞} (as well as integral complete for k = 0). All these statement have been established in the standard topological context – specificallymeaning that evol k (and Evol k ) was presumed to be continuous w.r.t. the C k -topology. In thispaper, we more generally show that, cf. (the more comprehensive) Theorem 2 in Sect. 6.2.1 Theorem A.
Suppose that G is C k -semiregular for k ∈ N ⊔ { lip , ∞} . Then, evol k is differentiable • for k = 0 if and only if g is integral complete. • for k ∈ N ≥ ⊔ {∞} if and only if g is Mackey complete.In this case, evol k is differentiable, with d φ evol k ( ψ ) = d e L R φ (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) ∀ φ, ψ ∈ C k ([ r, r ′ ] , g ) . This theorem will be derived from significantly more fundamental results established in this paper:Let Ξ : U → V ⊆ E be a fixed chart around e , and P the system of continuous seminorms on E . Apair ( φ, ψ ) ∈ C ([0 , , g ) × C ([0 , , g ) is said to be • admissible if φ + ( − δ, δ ) · ψ ⊆ D holds for some δ > • regular if it is admissible, withlim ∞ h → Ξ (cid:0) Evol( φ ) − · Evol( φ + h · ψ ) (cid:1) = 0 . The C k -topology is recalled in Sect. 2.2.4; and, the evolution maps are defined below. Here, C lip ([0 , , g ) denotes the set of Lipschitz curves, and C c ([0 , , g ) denotes the set of constant curves. Notably, this formula is well known from the finite dimensional context (cf., e.g., the proof of (1.13.4) Propositionin [1]), and also for regular Lie groups in the convenient setting [11]. ∞ h → α = β for α : ( − δ, ⊔ (0 , δ ) × [0 , → E with δ > β : [0 , → E iflim h → sup { p ( α ( h, t ) − β ( t )) | t ∈ [0 , } = 0 ∀ p ∈ P holds, where p : E → R ≥ denotes the continuous extension of the seminorm p ∈ P to the completion E of E . Then, the first result we want to mention is, cf. Proposition 3 in Sect. 6.2 Proposition B.
Suppose that ( φ, ψ ) is admissible.1) The pair ( φ, ψ ) is regular if and only if we have lim ∞ h → /h · Ξ (cid:0) Evol( φ ) − · Evol( φ + h · ψ ) (cid:1) = R • r (d e Ξ ◦ Ad Evol( φ )( s ) − )( ψ ( s )) d s ∈ E.
2) If ( φ, ψ ) is regular, then ( − δ, δ ) ∋ h evol( φ + h · ψ ) ∈ G is differentiable at h = 0 (for δ > suitably small) if and only if R Ad Evol( φ )( s ) − ( ψ ( s )) d s ∈ g holds. In this case, we have dd h (cid:12)(cid:12) h =0 evol( φ + h · ψ ) = d e L evol( φ ) (cid:0) R Ad Evol( φ )( s ) − ( ψ ( s )) d s (cid:1) . Evidently, each ( φ, ψ ) ∈ C k ([0 , , g ) × C k ([0 , , g ) is admissible if and only if G is C k -semiregular.In Sect. 4, we furthermore prove that, cf. Theorem 1 in Sect. 4 Theorem C. If G is C k -semiregular for k ∈ N ⊔ { lip , ∞} , then G is Mackey k -continuous. Here, Mackey k-continuity is a specific kind of sequentially continuity (cf. Sect. 3.3) that, in par-ticular, implies that each admissible ( φ, ψ ) ∈ C k ([0 , , g ) × C k ([0 , , g ) is regular (cf. Lemma 15in Sect. 3.3) – Theorem A thus follows immediately from Proposition B and Theorem C. We willconclude from Theorem C and Theorem 4 in [7] that, cf. Corollary 7 in Sect. 7 Corollary D.
Suppose that g is a Fr´echet space; and let k ∈ N ⊔ {∞} be fixed. Then, G is C k -regular if and only if G is C k -semiregular. Now, Proposition B is actually a consequence of a more fundamental differentiability result (Propo-sition 2 in Sect. 6) that we will also use to generalize Theorem 5 in [7]. Specifically, we will provethat, cf. Theorem 3 in Sect. 6.3
Theorem E.
Suppose that G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞ , c } – additionally abelianif k = c holds. Let Φ : I × [0 , → g ( I ⊆ R open) be given with Φ( z, · ) ∈ D k for each z ∈ I . Then, lim ∞ h → /h · Ξ (cid:0) Evol(Φ( x, · )) − · Evol(Φ( x + h, · )) (cid:1) = R • r (d e Ξ ◦ Ad Evol(Φ( x, · ))( s ) )( ∂ z Φ( x, s )) d s ∈ E holds for x ∈ I , provided thata) We have ( ∂ z Φ)( x, · ) ∈ C k ([0 , , g ) .b) For each p ∈ P and s (cid:22) k , there exists L p , s ≥ , and I p , s ⊆ I open with x ∈ I p , s , such that / | h | · (cid:5) p s ∞ (Φ( x + h, · ) − Φ( x, · )) ≤ L p , s ∀ h ∈ R =0 with x + h ∈ I p , s . In particular, we have dd h (cid:12)(cid:12) h =0 evol(Φ( x + h, · )) = d e L evol(Φ( x, · )) (cid:0) R Ad Evol(Φ( x, · ))( s ) ( ∂ z Φ( x, s )) d s (cid:1) if and only if the Riemann integral on the right side exists in g . This means s = lip for k = lip, s = 0 for k = 0, 0 ≤ s ≤ k for k ∈ N , and s ∈ N for k = ∞ . The correspondingseminorms (cid:5) p s ∞ are defined in Sect. 2.1.1.
4e explicitly recall at this point that, by Theorem C, for k ∈ N ⊔ { lip , ∞} , Mackey k-continuityis automatically given if G is C k -semiregular. Finally, let exp : g ⊇ dom[exp] → G denote theexponential map of G ; and recall that a Lie group G is said to have the strong Trotter property [4, 8, 13, 19] if for each µ ∈ C ∗ ([0 , , G ) with ˙ µ (0) ∈ dom[exp], we havelim n µ ( τ /n ) n = exp( τ · ˙ µ (0)) ∀ τ ∈ [0 , ℓ ] (1)uniformly for each ℓ >
0. As already figured out in [4], the strong Trotter property implies thestrong commutator property; and, also the Trotter and the commutator property that are relevant,e.g., in representation theory of infinite dimensional Lie groups [19]. Now, Theorem I in [4] statesthat G has the strong Trotter property if G is R -regular. This was generalized in [8] to the locally µ -convex case (hence, the case where evol is C -continuous on its domain, cf. Theorem 1 in [7]).In this paper, we go a step further, as we show (cf. Proposition 1 in Sect. 5) that G has thestrong Trotter property if Evol is sequentially continuous (the precise definitions can be foundin Sect. 3.4); which is much weaker than locally µ -convexity (provided that g is not metrizable,of course). We furthermore show in Proposition 1 that (1) holds for each µ ∈ C ∗ ([0 , , G ) with˙ µ (0) ∈ dom[exp] and δ r ( µ ) ∈ C lip ([0 , , g ) if Evol is Mackey continuous, the latter condition beingeven less restrictive than sequentially continuity.This paper is organized as follows: • In Sect. 2, we provide the basic definitions; and discuss the most elementary properties of thecore mathematical objects of this paper. • In Sect. 3, we discuss the continuity notions considered in this paper. • In Sect. 4, we prove Theorem 1 (i.e., Theorem C). • In Sect. 5, we discuss the strong Trotter property in the sequentially/Mackey continuous context. • In Sect. 6, we establish the differentiability results for the evolution map. • In Sect. 7, we prove Corollary 7 (i.e., Corollary D).
In this section, we fix the notations, and discuss the properties of the product integral (evolutionmap) that we will need in the main text. The proofs of the facts mentioned but not verified in thissection, can be found, e.g., in Sect. 3 and Sect. 4 in [7].
In this paper, Manifolds and Lie groups are always understood to be in the sense of [2]; in particular,smooth, Hausdorff, and modeled over a Hausdorff locally convex vector space. If f : M → N is a C -map between the manifolds M and N , then d f : T M → T N denotes the corresponding tangentmap between their tangent manifolds – we write d x f ≡ d f ( x, · ) : T x M → T f ( x ) N for each x ∈ M .By an interval, we understand a non-empty, non-singleton connected subset D ⊆ R . The set of allcompact intervals is denoted by K = { [ r, r ′ ] ⊆ R | r < r ′ } . We furthermore let D δ := ( − δ, ⊔ (0 , δ )for each δ >
0. A curve is a continuous map γ : D → M for a manifold M and an interval D ⊆ R .If D ≡ I is open, then γ is said to be of class C k for k ∈ N ⊔ {∞} if it is of class C k when considered Thus, for each neighbourhood U ⊆ G of e , there exists some n U ∈ N with exp( − τ · ˙ µ (0)) · µ ( τ /n ) n ∈ U for each n ≥ n U and τ ∈ [0 , ℓ ]. We explicitly refer to Definition 3.1 and Definition 3.3 in [2]. A review of the corresponding differential calculus –including the standard differentiation rules used in this paper – can be found, e.g., in Appendix A.1 that essentiallyequals Sect. 3.3.1 in [7].
5s a map between the manifolds I and M . If D is an arbitrary interval, then γ is said to be of class C k for k ∈ N ⊔ {∞} if γ = γ ′ | D holds for a C k -curve γ ′ : I → M that is defined on an open interval I containing D – we write γ ∈ C k ( D, M ) in this case. If γ : D → M is of class C , then we denotethe corresponding tangent vector at γ ( t ) ∈ M by ˙ γ ( t ) ∈ T γ ( t ) M . The above conventions also holdif M ≡ F is a Hausdorff locally convex vector space with system of continuous seminorms Q . Inthis case, we let F denote the completion of F ; as well as q : F → R ≥ the continuous extension of q to F , for each q ∈ Q . We furthermore defineB q ,ε := { X ∈ F | q ( X ) < ε } as well as B q ,ε := { X ∈ F | q ( X ) ≤ ε } for all q ∈ Q and ε >
0. If
X, Y are sets, then Map(
X, Y ) ≡ Y X denotes the set of all mappings X → Y . Let F be a Hausdorff locally convex vector space with system of continuous seminorms Q . • By C lip ([ r, r ′ ] , F ) we denote the set of all Lipschitz curves on [ r, r ′ ] ∈ K ; i.e., all curves γ : [ r, r ′ ] → F , such that Lip( q , γ ) := sup n q ( γ ( t ) − γ ( t ′ )) | t − t ′ | (cid:12)(cid:12)(cid:12) t, t ′ ∈ [ r, r ′ ] , t = t ′ o ∈ R ≥ exists for each q ∈ Q – i.e., we have q ( γ ( t ) − γ ( t ′ )) ≤ Lip( q , γ ) · | t − t ′ | ∀ t, t ′ ∈ [ r, r ′ ] , q ∈ Q . • By C c ([ r, r ′ ] , F ) we denote the set of all constant curves on [ r, r ′ ] ∈ K ; i.e., all curves of the form γ X : [ r, r ′ ] → F, t X for some X ∈ F .We define c + 1 := ∞ , ∞ + 1 := ∞ , lip + 1 := 1; as well as q lip ∞ ( γ ) := max( q ∞ ( γ ) , Lip (cid:0) q , γ ) (cid:1) ∀ γ ∈ C lip ([ r, r ′ ] , F ) q s ∞ ( γ ) := sup (cid:8) q (cid:0) γ ( m ) ( t ) (cid:1) (cid:12)(cid:12) ≤ m ≤ s , t ∈ [ r, r ′ ] (cid:9) ∀ γ ∈ C k ([ r, r ′ ] , F ) q ∞ ( γ ) := q ∞ ( γ ) ∀ γ ∈ C ([ r, r ′ ] , F )for each q ∈ Q , k ∈ N ⊔ {∞ , c } , s (cid:22) k , and [ r, r ′ ] ∈ K – Here, s (cid:22) k means • s = lip for k = lip, • N ∋ s ≤ k for k ∈ N , • s ∈ N for k = ∞ , • s = 0 for k = c.The C k -topology on C k ([ r, r ′ ] , F ) for k ∈ N ⊔ { lip , ∞ , c } is the Hausdorff locally convex topologythat is generated by the seminorms q s ∞ for all q ∈ Q and s (cid:22) k . Remark 1.
In the Lipschitz case, the above conventions deviate from the conventions used, e.g.,in [7, 9] as there the p ∞ -seminorms, i.e., the C -topology is considered on C lip ([ r, r ′ ] , F ) . ‡ Finally, we let CP ([ r, r ′ ] , F ) denote the set of piecewise C -curves on [ r, r ′ ] ∈ K ; i.e., all γ : [ r, r ′ ] → F , such that there exist r = t < . . . < t n = r ′ and γ [ p ] ∈ C ([ t p , t p +1 ] , F ) with γ | ( t p ,t p +1 ) = γ [ p ] | ( t p ,t p +1 ) for p = 0 , . . . , n − . (2)6 .1.2 Lie Groups In this paper, G will always denote an infinite dimensional Lie group in the sense of [2] (cf. Definition3.1 and Definition 3.3 in [2]) that is modeled over the Hausdorff locally convex vector space E , withcorresponding system of continuous seminorms P . We denote the Lie algebra of G by ( g , [ · , · ] ), fixa chart Ξ : G ⊇ U → V ⊆ E, with V convex, e ∈ U and Ξ( e ) = 0; and define (cid:5) p := p ◦ d e Ξ ∀ p ∈ P . We let m : G × G → G denote the Lie group multiplication, R g := m( · , g ) the right translation by g ∈ G , inv : G ∋ g g − ∈ G the inversion, and Ad : G × g → g the adjoint action – i.e., we haveAd( g, X ) ≡ Ad g ( X ) := d e Conj g ( X ) with Conj g : G ∋ h g · h · g − ∈ G for each g ∈ G and X ∈ g . We furthermore recall the product ruled ( g,h ) m( v, w ) = d g R h ( v ) + d h L g ( w ) ∀ g, h ∈ G, v ∈ T g G, w ∈ T h G. (3) Let µ ∈ Map([ r, r ′ ] , G ), { µ n } n ∈ N ⊆ Map([ r, r ′ ] , G ), and { µ h } h ∈ D δ ⊆ Map([ r, r ′ ] , G ) for δ > • lim ∞ n µ n = µ if for each open neighbourhood U ⊆ G of e , there exists some n U ∈ N with µ − · µ n ∈ U for each n ≥ n U . • lim ∞ h → µ h = µ if for each open neighbourhood U ⊆ G of e , there exists some 0 < δ U < δ with µ − · µ h ∈ U for each h ∈ D δ U .Evidently, then we have Lemma 1.
Suppose δ > and { µ h } h ∈ D δ ⊆ C ([ r, r ′ ] , G ) are given. If lim ∞ n µ h n = e holds for eachsequence D δ ⊇ { h n } n ∈ N → , then we have lim ∞ h → µ h = e .Proof. If the claim is wrong, then there exists a neighbourhood U ⊆ G of e , such that the followingholds: For each n ∈ N , there exists some h n = 0 with | h n | ≤ n as well as some τ n ∈ [ r, r ′ ], such that µ − ( τ n ) · µ h n ( τ n ) / ∈ U holds. Since we have { h n } n ∈ N →
0, this contradicts the presumptions.The same conventions (and Lemma 1) also hold if ( G, · ) ≡ ( F, +) is a Hausdorff locally convexvector space (or its completion) – In this case, we use the following convention:Let δ > α : D δ × [ r, r ′ ] → F be given, with α ( h, · ) ∈ Map([ r, r ′ ] , F ) for each h ∈ D δ . Then,for β ∈ Map([ r, r ′ ] , F ), we write dd h (cid:12)(cid:12) ∞ h =0 α = β def. ⇐⇒ lim ∞ h → (cid:2) /h · α ( h, · ) (cid:3) = β. Remark 2.
In this paper, the above convention will mainly be used in the following form: F = E will be the completion of a Hausdorff locally convex vector space E ; and we will have α : D δ × [ r, r ′ ] → E ⊆ E as well as β ∈ Map([ r, r ′ ] , E ) . ‡ In this subsection, we provide the relevant facts and definitions concerning the right logarithmicderivative and the evolution map. 7 .2.1 Basic Definitions
We define C k ∗ ([ r, r ′ ] , G ) := { µ ∈ C k ([ r, r ′ ] , G ) | µ ( r ) = e } ∀ [ r, r ′ ] ∈ K , k ∈ N ⊔ {∞} . The right logarithmic derivative is given by δ r : C ([ r, r ′ ] , G ) → C ([ r, r ′ ] , g ) , µ d µ R µ − ( ˙ µ )for each [ r, r ′ ] ∈ K ; and we define D [ r,r ′ ] := δ r ( C ([ r, r ′ ] , G )) for each [ r, r ′ ] ∈ K , as well as D k [ r,r ′ ] := D [ r,r ′ ] ∩ C k ([ r, r ′ ] , g ) ∀ [ r, r ′ ] ∈ K , k ∈ N ⊔ { lip , ∞ , c } . Then, δ r restricted to C ∗ ([ r, r ′ ] , G ) is injective for each [ r, r ′ ] ∈ K ; so thatEvol : F [ r,r ′ ] ∈ K D [ r,r ′ ] → F [ r,r ′ ] ∈ K C ∗ ([ r, r ′ ] , G )is well defined by Evol : D [ r,r ′ ] → C ∗ ([ r, r ′ ] , G ) , δ r ( µ ) µ · µ ( r ) − for each [ r, r ′ ] ∈ K . Here, Evol | D k [ r,r ′ ] : D k [ r,r ′ ] → C k +1 ([ r, r ′ ] , G )holds for each [ r, r ′ ] ∈ K , and each k ∈ N ⊔ { lip , ∞ , c } . The product integral is given by R ts φ := Evol (cid:0) φ | [ s,t ] (cid:1) ( t ) ∈ G ∀ [ s, t ] ⊆ dom[ φ ] , φ ∈ F [ r,r ′ ] ∈ K D [ r,r ′ ] ;and we let R φ ≡ R r ′ r φ as well as R cc φ := e for φ ∈ D [ r,r ′ ] and c ∈ [ r, r ′ ]. Moreover, we setevol k [ r,r ′ ] ≡ R (cid:12)(cid:12) D k [ r,r ′ ] ∀ k ∈ N ⊔ { lip , ∞ , c } , [ r, r ′ ] ∈ K ;and let evol k ≡ evol k [0 , as well as D k ≡ D k [0 , for each k ∈ N ⊔ { lip , ∞ , c } . We furthermore letevol ≡ evol : D ≡ D → G. Then, we have the following elementary identities, cf., [3, 11] or Sect. 3.5.2 in [7] a) For each φ, ψ ∈ D [ r,r ′ ] , we have φ + Ad R • r φ ( ψ ) ∈ D [ r,r ′ ] , with R tr φ · R tr ψ = R tr φ + Ad R • r φ ( ψ ) . b) For each φ, ψ ∈ D [ r,r ′ ] , we have Ad [ R • r φ ] − ( ψ − φ ) ∈ D [ r,r ′ ] , with (cid:2) R tr φ (cid:3) − (cid:2) R tr ψ (cid:3) = R tr Ad [ R • r φ ] − ( ψ − φ ) . c) For r = t < . . . < t n = r ′ and φ ∈ D [ r,r ′ ] , we have R tr φ = R tt p φ · R t p t p − φ · . . . · R t r φ ∀ t ∈ ( t p , t p +1 ] , p = 0 , . . . , n − . d) For ̺ : [ ℓ, ℓ ′ ] → [ r, r ′ ] of class C and φ ∈ D [ r,r ′ ] , we have ˙ ̺ · ( φ ◦ ̺ ) ∈ D [ ℓ,ℓ ′ ] , with R ̺r φ = (cid:2) R • ℓ ˙ ̺ · ( φ ◦ ̺ ) (cid:3) · (cid:2) R ̺ ( ℓ ) r φ (cid:3) . e) For each homomorphism Ψ : G → H between Lie groups G and H that is of class C , we haveΨ ◦ R • r φ = R • r d e Ψ ◦ φ ∀ φ ∈ D [ r,r ′ ] . We say that G is C k -semiregular for k ∈ N ⊔ { lip , ∞} if D k = C k ([0 , , g ) holds; which, by d) , isequivalent to that D k [ r,r ′ ] = C k ([ r, r ′ ] , g ) holds for each [ r, r ′ ] ∈ K , cf. e.g., Lemma 12 in [7].8 .2.3 The Exponential Map The exponential map is defined byexp : dom[exp] ≡ i − (D c ) → G, X R φ X | [0 , = (evol c ◦ i )( X )with i : g ∋ X → φ X | [0 , ∈ C c ([0 , , g ).Then, instead of saying that G is C c -semiregular, in the following we will rather say that G admitsan exponential map. We furthermore remark that d) implies R · dom[exp] ⊆ dom[exp]; and that t exp( t · X ) is a 1-parameter group for each X ∈ dom[exp], withexp( t · X ) = R t · φ X | [0 , d) = R t φ X | [0 , ∀ t ≥ , (4)cf., e.g., Remark 2.1) in [7]. Finally, if G is abelian, then X + Y ∈ dom[exp] holds for all X, Y ∈ dom[exp], because we haveexp( X ) · exp( Y ) a) = R φ X | [0 , · R φ Y | [0 , = R φ X + Y | [0 , . We say that G is C k -continuous for k ∈ N ⊔{ lip , ∞ , c } if evol k is continuous w.r.t. the C k -topology.We explicitly remark that under the identification i : g → { φ X | [0 , | X ∈ g } , the C c -topology justequals the subspace topology on dom[exp] that is inherited by the locally convex topology on g . So,instead of saying that G is C c -continuous if evol c is continuous w.r.t. this topology, we will rathersay that the exponential map is continuous. Let F be a Hausdorff locally convex vector space with system of continuous seminorms Q , andcompletion F . For each q ∈ P , we let q : F → R ≥ denote the continuous extension of q to F . TheRiemann integral of γ ∈ C ([ r, r ′ ] , F ) (for [ r, r ′ ] ∈ K ) is denoted by R γ ( s ) d s ∈ F ; and we define R ba γ ( s ) d s := R γ | [ a,b ] ( s ) d s, R ab γ ( s ) d s := − R ba γ ( s ) d s, R cc γ ( s ) d s := 0 (5)for r ≤ a < b ≤ r ′ and c ∈ [ r, r ′ ]. Clearly, the Riemann integral is linear, with R ca γ ( s ) d s = R ba γ ( s ) d s + R cb γ ( s ) d s ∀ r ≤ a < b < c ≤ r ′ (6) γ − γ ( r ) = R • r ˙ γ ( s ) d s ∀ γ ∈ C ([ r, r ′ ] , F ) , (7) q ( γ − γ ( r )) ≤ R • r q ( ˙ γ ( s )) d s ∀ q ∈ Q , γ ∈ C ([ r, r ′ ] , F ) , (8)as well as q (cid:0) R • r γ ( s ) d s (cid:1) ≤ R • r q ( γ ( s )) d s ∀ q ∈ Q , γ ∈ C ([ r, r ′ ] , F ) . (9)We furthermore have the substitution formula R ̺ ( t ) r γ ( s ) d s = R tℓ ˙ ̺ ( s ) · ( γ ◦ ̺ )( s ) d s (10)for each γ ∈ C ([ r, r ′ ] , F ), and each ̺ : K ∋ [ ℓ, ℓ ′ ] → [ r, r ′ ] of class C with ̺ ( ℓ ) = r and ̺ ( ℓ ′ ) = r ′ .Moreover, if E is a Hausdorff locally convex vector space, and L : F → E is a continuous linearmap, then we have R γ ( s ) d s ∈ F for γ ∈ C ([ r, r ′ ] , F ) = ⇒ L ( R γ ( s ) d s ) = R L ( γ ( s )) d s. (11)9inally, for γ ∈ CP ([ r, r ′ ] , F ) with γ [0] , . . . , γ [ n −
1] as in (2), we define R γ ( s ) d s := P n − p =0 R γ [ p ]( s ) d s. (12)A standard refinement argument in combination with (6) then shows that this is well defined; i.e.,independent of any choices we have made. We define R ba γ ( s ) d s , R ab γ ( s ) d s and R cc γ ( s ) d s as in (5);and observe that (12) is linear and fulfills (6). Let F , . . . , F n , E be Hausdorff locally convex vector spaces with corresponding system of continuousseminorms Q , . . . , Q n , P . We recall that Lemma 2.
Let X be a topological space; and let Φ : X × F × . . . × F n → E be continuous with Φ( x, · ) n -multilinear for each x ∈ X . Then, for each compact K ⊆ X and each p ∈ P , there existseminorms q ∈ Q , . . . , q n ∈ Q n as well as O ⊆ X open with K ⊆ O , such that ( p ◦ Φ)( y, X , . . . , X n ) ≤ q ( X ) · . . . · q n ( X n ) ∀ y ∈ O holds for all X ∈ F , . . . , X n ∈ F n .Proof. Confer, e.g., Corollary 1 in [7].Next, given Hausdorff locally convex vector spaces F , F , and a continuous linear map Φ : F → F ,we denote its unique continuous linear extension by Φ : F → F (cf., 2. Theorem in Sect. 3.4in [10]). We recall that Lemma 3.
Let F , F be Hausdorff locally convex vector spaces; and let f : F ⊇ U → F be of class C . Suppose that γ : D → F ⊆ F is continuous at t ∈ D , such that lim h → /h · ( γ ( t + h ) − γ ( t )) =: X ∈ F exists. Then, we have lim h → /h · ( f ( γ ( t + h )) − f ( γ ( t ))) = d γ ( t ) f ( X ) . Proof.
Confer, e.g., Lemma 7 in [7].
Remark 3.
Let F be a Hausdorff locally convex vector space, let U ⊆ F be open, and let G be aLie group. A map f : U → G is said to be • differentiable at x ∈ U if there exists a chart (Ξ ′ , U ′ ) of G with f ( x ) ∈ U ′ , such that ( D Ξ ′ v f )( x ) := lim h → /h · ((Ξ ′ ◦ f )( x + h · v ) − (Ξ ′ ◦ f )( x )) ∈ E ∀ v ∈ F (13) exists. Then, Lemma 3 applied to coordinate changes shows that (13) holds for one chart around f ( x ) if and only if it holds for each chart around f ( x ) – and that d x f ( v ) := (cid:0) d Ξ ′ ( f ( x )) Ξ ′− ◦ ( D Ξ ′ v f ) (cid:1) ( x ) ∈ T f ( x ) G ∀ v ∈ F is independent of the explicit choice of (Ξ ′ , U ′ ) . • differentiable if f is differentiable at each x ∈ U . ‡ In particular, Lemma 2 provides us with the following statements (cf. also Sect. 3.4.1 in [7]):I) Since Ad : G × g ∋ ( g, X ) Ad g ( X ) ∈ g is smooth as well as linear in the second argument(by Lemma 2), to each compact C ⊆ G and each v ∈ P , there exists some v ≤ w ∈ P , suchthat (cid:5) v ◦ Ad g ≤ (cid:5) w holds for each g ∈ C. 10I) By Lemma 2 applied to Φ ≡ Ad and K ≡ { e } , to each m ∈ P , there exists some m ≤ q ∈ P ,as well as O ⊆ G symmetric open with e ∈ O , such that (cid:5) m ◦ Ad g ≤ (cid:5) q holds for each g ∈ O .III) Suppose that im[ µ ] ⊆ U holds for µ ∈ C ([ r, r ′ ] , G ). Then, we have δ r ( µ ) = ω (Ξ ◦ µ, ∂ t (Ξ ◦ µ )) , (14)for the smooth map ω : V × E ∋ ( x, X ) d Ξ − ( x ) R [Ξ − ( x )] − (d x Ξ − ( X )) ∈ g . Since ω is linearin the second argument, (by Lemma 2) for each q ∈ P , there exists some q ≤ m ∈ P with( (cid:5) q ◦ ω )( x, X ) ≤ m ( X ) ∀ x ∈ B m , , X ∈ E. (15)IV) Suppose that im[ µ ] ⊆ U holds for µ ∈ C ([ r, r ′ ] , G ). Then, we have ∂ t (Ξ ◦ µ ) = υ (Ξ ◦ µ, δ r ( µ )) , (16)for the smooth map υ : V × g ∋ ( x, X ) (cid:0) d Ξ − ( x ) Ξ ◦ d e R Ξ − ( x ) (cid:1) ( X ) ∈ E . Since υ is linear inthe second argument, (by Lemma 2) for each q ∈ P , there exists some u ≤ m ∈ P with( u ◦ υ )( x, X ) ≤ (cid:5) m ( X ) ∀ x ∈ B m , , X ∈ g . For each µ ∈ C ([ r, r ′ ] , G ) with im[Ξ ◦ µ ] ⊆ B m , , we thus obtain from (16), (7), and (8) that u (Ξ ◦ µ ) = u (cid:0) R • r υ ((Ξ ◦ µ )( s ) , δ r ( µ )( s )) d s (cid:1) ≤ R • r (cid:5) m ( δ r ( µ )( s )) d s. (17)For instance, we immediately obtain from (17) that Lemma 4.
For each u ∈ P , there exist u ≤ m ∈ P , and U ⊆ G open with e ∈ U , such that ( u ◦ Ξ)( R • r χ ) ≤ R • r m ( χ ( s )) d s holds, for each χ ∈ D [ r,r ′ ] with R • r χ ∈ U ; for all [ r, r ′ ] ∈ K . Moreover,
Lemma 5.
We have Ad µ ( φ ) ∈ C k ([ r, r ′ ] , g ) for each µ ∈ C k +1 ([ r, r ′ ] , G ) , φ ∈ C k ([ r, r ′ ] , g ) , and k ∈ N ⊔ { lip , ∞} .Proof. Confer, e.g., Lemma 13 in [7].
Lemma 6.
Let [ r, r ′ ] ∈ K , k ∈ N ⊔ {∞} , and φ ∈ D k [ r,r ′ ] be fixed. Then, for each p ∈ P and s (cid:22) k ,there exists some p ≤ q ∈ P with (cid:5) p p ∞ (cid:0) Ad [ R • r φ ] − ( ψ ) (cid:1) ≤ (cid:5) q p ∞ ( ψ ) ∀ ψ ∈ C k ([ r, r ′ ] , g ) , ≤ p ≤ s . Proof.
Confer, e.g., Lemma 14 in [7].Then, modifying the argumentation used in the proof of the Lipschitz case in Lemma 13 in [7] toour deviating convention concerning the topology on the set of Lipschitz curves, we also obtain
Lemma 7.
Let [ r, r ′ ] ∈ K , and φ ∈ D [ r,r ′ ] be fixed. Then, for each p ∈ P , there exists some p ≤ q ∈ P with (cid:5) p lip ∞ (cid:0) Ad [ R • r φ ] − ( ψ ) (cid:1) ≤ (cid:5) q lip ∞ ( ψ ) ∀ ψ ∈ C lip ([ r, r ′ ] , g ) . Proof.
Confer Appendix A.2. 11 .5 Continuity Statements
For h ∈ G , we define Ξ h ( g ) := Ξ( h − · g ) for each g ∈ h · U ; and recall that, cf. Lemma 8 in [7] Lemma 8.
Let C ⊆ U be compact. Then, for each p ∈ P , there exists some p ≤ u ∈ P , and asymmetric open neighbourhood V ⊆ U of e with C · V ⊆ U and B u , ⊆ Ξ( V ) , such that p (Ξ( q ) − Ξ( q ′ )) ≤ u (Ξ g · h ( q ) − Ξ g · h ( q ′ )) ∀ q, q ′ ∈ g · V, h ∈ V holds for each g ∈ C . Now, combining Lemma 4 with Lemma 8, we obtain the following variation of Proposition 1 in [7]:
Lemma 9.
For each p ∈ P , there exist p ≤ q ∈ P and V ⊆ G open with e ∈ V , such that p (cid:0) Ξ (cid:0) R • r φ (cid:1) − Ξ (cid:0) R • r ψ (cid:1)(cid:1) ≤ R • r (cid:5) q ( φ ( s ) − ψ ( s )) d s holds for all φ, ψ ∈ D [ r,r ′ ] with R • r φ, R • r ψ ∈ V ; for each [ r, r ′ ] ∈ K .Proof. We let p ≤ u ∈ P and V be as in Lemma 8 for C ≡ { e } there (i.e., V is symmetric withB u , ⊆ Ξ( V )). We choose U ⊆ G and u ≤ m ∈ P as in Lemma 4. We furthermore let m ≤ q ∈ P and O ⊆ G be as in II). Then, shrinking V if necessary, we can assume that V − · V ⊆ U as well as V ⊆ O holds. Then, for φ, ψ as in the presumptions, Lemma 8 applied to q ≡ R • r φ, q ′ ≡ R • r ψ, h ≡ R • r φ ∈ V ,and g ≡ e gives p (cid:0) Ξ (cid:0) R • r φ (cid:1) − Ξ (cid:0) R • r ψ (cid:1)(cid:1) ≤ u (cid:0) Ξ R • r φ (cid:0) R • r φ (cid:1) − Ξ R • r φ (cid:0) R • r ψ (cid:1)(cid:1) = ( u ◦ Ξ) (cid:0) [ R • r φ ] − [ R • r ψ ] (cid:1) . By assumption, for each t ∈ [ r, r ′ ], we have U ⊇ V − · V ∋ [ R tr φ ] − [ R tr ψ ] b) = R tr Ad [ R • r φ ] − ( ψ − φ ) with [ R • r φ ] − ∈ V − = V ⊆ O. We obtain from Lemma 4 and II) that( u ◦ Ξ) (cid:0) [ R tr φ ] − [ R tr ψ ] (cid:1) ≤ R tr (cid:5) m (cid:0) Ad [ R sr φ ] − ( ψ ( s ) − φ ( s )) (cid:1) d s ≤ R tr (cid:5) q ( ψ ( s ) − φ ( s )) d s holds for each t ∈ [ r, r ′ ]; which proves the claim.We furthermore observe that Lemma 10.
Suppose that exp : dom[exp] → G is continuous; and let X ∈ dom[exp] be fixed. Then,for each open neighbourhood V ⊆ G of e , there exists some m ∈ P , such that (cid:5) m ( Y − X ) ≤ for Y ∈ dom[exp] = ⇒ R • φ Y | [0 , ∈ R • φ X | [0 , · V. Proof.
By assumption, α : [0 , × dom[exp] ∋ ( t, Y ) exp( t · X ) − · exp( t · Y ) is continuous; andwe have α ( · , X ) = e . For τ ∈ [0 ,
1] fixed, there thus exists an open interval I τ ⊆ R containing τ , aswell as an open neighbourhood O τ ⊆ g of X , such that we haveexp( t · X ) − · exp( t · Y ) ∈ V ∀ t ∈ I τ ∩ [0 , , Y ∈ O τ ∩ dom[exp] . (18)We choose τ , . . . , τ n ∈ [0 ,
1] with [0 , ⊆ I τ ∪ . . . ∪ I τ n ; and define O := O τ ∩ . . . ∩ O τ n . Then,(18) holds for each t ∈ [0 ,
1] and Y ∈ O ∩ dom[exp]; so that the claim holds for each fixed m ∈ P with B m , ⊆ O . In this section, we introduce the continuity notions that we will need to formulate our main results.We furthermore provide some elementary continuity statements that we will need in the main text.12 .1 Sets of Curves
Let [ r, r ′ ] ∈ K be fixed. We will tacitly use in the following that C k ([ r, r ′ ] , g ) is a real vector spacefor each k ∈ N ⊔ { lip , ∞ , c } . We will furthermore use that:A) For each k ∈ N ⊔{ lip , ∞} , φ ∈ D k [ r,r ′ ] , and ψ ∈ C k ([ r, r ′ ] , g ), we have Ad [ R • r φ ] − ( ψ ) ∈ C k ([ r, r ′ ] , g )by Lemma 5. Evidently, the same statement also holds for k = c if G is abelian.B) For each k ∈ N ⊔ { lip , ∞ , c } , φ ∈ D k [ r,r ′ ] , [ ℓ, ℓ ′ ] ∈ K , and ̺ : [ ℓ, ℓ ′ ] → [ r, r ′ ] , t r + ( t − ℓ ) · ( r ′ − r ) / ( ℓ ′ − ℓ ) , we have ˙ ̺ · ( φ ◦ ̺ ) = ( r ′ − r ) / ( ℓ ′ − ℓ ) · ( φ ◦ ̺ ) ∈ D k [ ℓ,ℓ ′ ] by d) ; with (cid:5) p s ∞ ( ˙ ̺ · ( φ ◦ ̺ )) = h ( r ′ − r )( ℓ ′ − ℓ ) i s+1 · (cid:5) p s ∞ ( φ ) with s (cid:22) k for k ∈ N ⊔ {∞ , c } , Lip( (cid:5) p , ˙ ̺ · ( φ ◦ ̺ )) = h ( r ′ − r )( ℓ ′ − ℓ ) i · Lip( (cid:5) p , φ ) for k = lip . We say that g is k-complete for k ∈ N ⊔ { lip , ∞ , c } if R Ad [ R sr φ ] − ( χ ( s )) d s ∈ g (19)holds for all φ, χ ∈ D k [ r,r ′ ] , for each [ r, r ′ ] ∈ K . Then, Remark 4. • g is c -complete if G is abelian. • g is k -complete for k ∈ N ⊔ { lip , ∞ , c } if and only if (19) holds for [ r, r ′ ] ≡ [0 , .For this, let φ, χ ∈ D k [ r,r ′ ] be given. Then, for ̺ : [0 , → [ r, r ′ ] as in B) with [ ℓ, ℓ ′ ] ≡ [0 , there,we have ˙ ̺ · ( φ ◦ ̺ ) , ˙ ̺ · ( χ ◦ ̺ ) ∈ D k [0 , with R Ad [ R sr φ ] − ( χ ( s )) d s = R ̺ (1) r Ad [ R sr φ ] − ( χ ( s )) d s (10)= R ˙ ̺ ( s ) · Ad [ R ̺ ( s ) r φ ] − ( χ ( ̺ ( s ))) d s = R Ad [ R ̺ ( s ) r φ ] − (( ˙ ̺ · ( χ ◦ ̺ ))( s )) d s d) = R Ad [ R s ˙ ̺ · ( φ ◦ ̺ )] − (( ˙ ̺ · ( χ ◦ ̺ ))( s )) d s. In particular, Point A) then shows: • If G is C -semiregular, then g is -complete if and only if g is integral complete – i.e., if andonly if R φ ( s ) d s ∈ g holds for each φ ∈ C ([0 , , g ) . • If G is C k -semiregular for k ∈ N ≥ ⊔ { lip , ∞} , then g is k -complete if and only if g is Mackey-complete. ‡ Recall that g is Mackey complete if and only if R φ ( s )d s ∈ g holds for each φ ∈ C k ([0 , , g ), for any k ∈ N ≥ ⊔{ lip , ∞} ,cf., 2.14 Theorem in [12]. .2 Weak Continuity A pair ( φ, ψ ) ∈ C ([ r, r ′ ] , g ) × C ([ r, r ′ ] , g ) is said to be • admissible if φ + ( − δ, δ ) · ψ ⊆ D [ r,r ′ ] holds for some δ > • regular if it is admissible withlim ∞ h → R • r φ + h · ψ = R • r φ. Then,
Remark 5.
1) It follows from c) that ( φ, χ ) is admissible/regular if and only if ( φ | [ ℓ,ℓ ′ ] , χ | [ ℓ,ℓ ′ ] ) is admissi-ble/regular for each r ≤ ℓ < ℓ ′ ≤ r ′ .2) Each (0 , i ( X )) with X ∈ dom[exp] is regular; because we have R t h · φ X | [0 ,
1] (4) = R th · φ X | [0 ,
1] (4) = R th φ X | [0 , for each t ∈ [0 , , and each h ∈ R . ‡ We say that G is weakly k-continuous for k ∈ N ⊔ { lip , ∞ , c } if each admissible ( φ, ψ ) ∈ C k ([0 , , g ) × C k ([0 , , g ) is regular. Lemma 11. If G is weakly k -continuous for k ∈ N ⊔ { lip , ∞ , c } , then each admissible ( φ, ψ ) ∈ C k ([ r, r ′ ] , g ) × C k ([ r, r ′ ] , g ) (for each [ r, r ′ ] ∈ K ) is regular.Proof. We define ̺ : [0 , ∋ t r + t · ( r ′ − r ) ∈ [ r, r ′ ]; and observe that R ̺r φ d) = R • ˙ ̺ · ( φ ◦ ̺ ) , R ̺r [ φ + h · ψ ] d) = R • [ ˙ ̺ · ( φ ◦ ̺ ) + h · ˙ ̺ · ( ψ ◦ ̺ )]holds for h > ̺ · ( φ ◦ ̺ ) , ˙ ̺ · ( ψ ◦ ̺ ) ∈ C k ([0 , , g ) by Point B), theclaim is clear from the presumptions. Lemma 12. G is weakly k -continuous for k ∈ N ⊔ { lip , ∞} if and only if lim ∞ h → R • h · χ = e (20) holds, for each χ ∈ D k with ( − δ, δ ) · χ ⊆ D k for some δ > . The same statement also holds for k = c if G is abelian.Proof. The one implication is evident. For the other implication, we suppose that ( φ, ψ ) ∈ C k ([0 , , g ) × C k ([0 , , g ) is admissible. Since φ ∈ D k [0 , holds, we have[ R t φ ] − [ R t φ + h · ψ ] b) = R t h · Ad [ R • φ ] − ( ψ ) ∀ t ∈ [0 , χ := Ad [ R • φ ] − ( ψ ) ∈ C k ([0 , , g ) by Point A). The claim is thus clear from (20). Corollary 1. If G is abelian, then G is weakly c -continuous.Proof. This is clear from Lemma 12 and Remark 5.2).14 .3 Mackey Continuity
We write { φ n } n ∈ N ⇀ m . k φ for k ∈ N ⊔ { lip , ∞ , c } , { φ n } n ∈ N ⊆ C k ([ r, r ′ ] , g ), and φ ∈ C k ([ r, r ′ ] , g ) if (cid:5) p s ∞ ( φ − φ n ) ≤ c s p · λ n ∀ n ≥ l s p , p ∈ P , s (cid:22) k (21)holds for certain { c s p } s (cid:22) k, p ∈ P ⊆ R ≥ , { l s p } s (cid:22) k, p ∈ P ⊆ N , and R ≥ ⊇ { λ n } n ∈ N → Remark 6.
Suppose that D k [ r,r ′ ] ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k [ r,r ′ ] holds for k ∈ N ⊔ { lip , ∞ , c } . Then, { φ ι ( n ) } n ∈ N ⇀ m . k φ holds for each strictly increasing ι : N → N . ‡ We say that G is Mackey k-continuous ifD k ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k = ⇒ lim ∞ n R • r φ n = R • r φ. (22)In analogy to Lemma 11, we obtain Lemma 13. G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞ , c } if and only if D k [ r,r ′ ] ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k [ r,r ′ ] = ⇒ lim ∞ n R • r φ n = R • r φ, for each [ r, r ′ ] ∈ K .Proof. The one implication is evident. For the other implication, we suppose that (22) holds. Then,for [ r, r ′ ] ∈ K fixed, we let ̺ : [0 , ∋ t r + t · ( r ′ − r ) ∈ [ r, r ′ ]; and obtain D k [ r,r ′ ] ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k [ r,r ′ ] B) = ⇒ D k ⊇ { ˙ ̺ · ( φ n ◦ ̺ ) } n ∈ N ⇀ m . k ˙ ̺ · ( φ ◦ ̺ ) ∈ D k = ⇒ lim ∞ n R • r ˙ ̺ · ( φ n ◦ ̺ ) = R • r ˙ ̺ · ( φ ◦ ̺ ) d) = ⇒ lim ∞ n R • r φ n = R • r φ, whereby the second step is due to the presumptions.In analogy to Lemma 12, we obtain Lemma 14. G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞} if and only if D k ⊇ { φ n } n ∈ N ⇀ m . k ⇒ lim ∞ n R • φ n = e. (23) The statement also holds for k = c if G is abelian.Proof. The one implication is evident. For the other implication, we suppose that D k ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k holds; and observe that[ R t φ ] − [ R t φ n ] b) = R t Ad [ R • φ ] − ( φ n − φ ) | {z } ψ n ∈ D k ∀ n ∈ N , t ∈ [0 , k ⊇ { ψ n } n ∈ N ⇀ m . k
0; from whichthe claim is clear.We furthermore observe that
Lemma 15. If G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞ , c } , then G is weakly k -continuous. roof. If G is not weakly k-continuous, then there exists an admissible ( φ, ψ ) ∈ C k ([ r, r ′ ] , g ) × C k ([ r, r ′ ] , g ), an open neighbourhood U ⊆ G of e , as well as sequences { τ n } n ∈ N ⊆ [ r, r ′ ] and R =0 ⊇ { h n } →
0, such that [ R τ n r φ ] − [ R τ n r φ + h n · ψ ] / ∈ U ∀ n ∈ N holds. Then, G cannot be Mackey k-continuous, because we have { φ + h n · ψ } n ∈ N ⇀ m . k φ .Remarkably, the uniform convergence on the right side of (23) in Lemma 14 can be replaced by aweaker convergence property; namely, Lemma 16. G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞} if and only if D k ⊇ { φ n } n ∈ N ⇀ m . k ⇒ lim n R φ n = e. (24) The statement also holds for k = c if G is abelian.Proof. The one implication is evident. For the other implication, we suppose that (24) holds; andthat G is not Mackey k-continuous. By Lemma 14, there exist D k ⊇ { φ n } n ∈ N ⇀ m . k U ⊆ G openwith e ∈ G , a sequence { τ n } n ∈ N ⊆ [0 , ι : N → N strictly increasing, such that R τ n φ ι n / ∈ U ∀ n ∈ N (25)holds. For each n ∈ N , we defineD k ∋ χ n := ˙ ̺ n · ( φ ι n ◦ ̺ n ) with ̺ n : [0 , ∋ t t · τ n ∈ [0 , τ n ];and conclude from Remark 6 and Point B) that { χ n } n ∈ N ⇀ m . k n R τ n φ ι n d) = lim n R χ n = e, which contradicts (25). We write { φ n } n ∈ N ⇀ s . k φ for k ∈ N ⊔ { lip , ∞ , c } , { φ n } n ∈ N ⊆ C k ([ r, r ′ ] , g ), and φ ∈ C k ([ r, r ′ ] , g ) iflim n (cid:5) p s ∞ ( φ − φ n ) = 0 ∀ p ∈ P , s (cid:22) k holds. We say that G is sequentially k-continuous ifD k ⊇ { φ n } n ∈ N ⇀ s . k φ ∈ D k = ⇒ lim ∞ n R • r φ n = R • r φ. Remark 7.
1) Suppose that G is sequentially k -continuous for k ∈ N ⊔ { lip , ∞ , c } . Evidently, then G is Mackey k -continuous; thus, weakly k -continuous by Lemma 15.2) G is sequentially k -continuous for k ∈ N ⊔ { lip , ∞ , c } if and only if D k [ r,r ′ ] ⊇ { φ n } n ∈ N ⇀ s . k φ ∈ D k [ r,r ′ ] = ⇒ lim ∞ n R • r φ n = R • r φ holds for each [ r, r ′ ] ∈ K . This just follows as in Lemma 13.3) Let k ∈ N ⊔ { lip , ∞ , c } , with G abelian for k = c . Then, the same arguments as in Lemma 14show that G is sequentially k -continuous if and only if D k ⊇ { φ n } n ∈ N ⇀ s . k ⇒ lim n R • φ n = e. ) Let k ∈ N ⊔ { lip , ∞ , c } , with G abelian for k = c . Then, the same arguments as in Lemma 16show that G is sequentially k -continuous if and only if D k ⊇ { φ n } n ∈ N ⇀ s . k ⇒ lim n R φ n = e.
5) If G is C k -continuous for k ∈ N ⊔ { lip , ∞ , c } , then G is sequentially k -continuous – This is clear • for k ∈ N ⊔ { lip , ∞} from Point 4), • for k = c from Lemma 10.6) Let k ∈ N ⊔ { lip , ∞ , c } be given; and suppose that the C k -topology on C k ([0 , , g ) is first count-able, and that G is sequentially k -continuous. Then, G is C k -continuous.In fact, if evol k is not C k -continuous, then there exists U ⊆ G open, such that W := evol − ( U ) ⊆ D k is not open, i.e., not a neighbourhood of some φ ∈ W . Since C k ([0 , , g ) (thus, D k ) is firstcountable, there exists a sequence { φ n } n ∈ N ⊆ D k \ W with φ n ⇀ s . k φ . We obtain evol k ( φ n ) ∈ A := G \ U for each n ∈ N ; thus, evol k ( φ ) = lim n evol k ( φ n ) ∈ A , as evol k is sequentiallycontinuous, and since A is closed. This contradicts that evol k ( φ ) ∈ U = G \ A holds. ‡ We now finally discuss piecewise integrable curves. Specifically, we provide the basic facts anddefinitions ; and furthermore show that sequentially 0-continuity and Mackey 0-continuity carryover to the piecewise integrable category. This will be used in Sect. 5 to generalize Theorem 1in [8]. For k ∈ N ⊔ { lip , ∞ , c } and [ r, r ′ ] ∈ K , we let D P k ([ r, r ′ ] , g ) denote the set of all ψ : [ r, r ′ ] → g , suchthat there exist r = t < . . . < t n = r ′ and ψ [ p ] ∈ D k [ t p ,t p +1 ] with ψ | ( t p ,t p +1 ) = ψ [ p ] | ( t p ,t p +1 ) for p = 0 , . . . , n − . (26)In this situation, we define R rr ψ := e , as well as R tr ψ := R tt p ψ [ p ] · R t p t p − ψ [ p − · . . . · R t t ψ [0] ∀ t ∈ ( t p , t p +1 ] . (27)A standard refinement argument in combination with c) then shows that this is well defined; i.e.,independent of any choices we have made. It is furthermore not hard to see that for φ, ψ ∈ D P k ([ r, r ′ ] , g ), we have Ad [ R • r φ ] − ( ψ − φ ) ∈ D P k ([ r, r ′ ] , g ) with (cid:2) R tr φ (cid:3) − (cid:2) R tr ψ (cid:3) = R tr Ad [ R • r φ ] − ( ψ − φ ) ∀ t ∈ [ r, r ′ ] . (28)We write • { φ n } n ∈ N ⇀ s φ for { φ n } n ∈ N ⊆ D P ([ r, r ′ ] , g ) and φ ∈ D P ([ r, r ′ ] , g ) iflim n (cid:5) p ∞ ( φ − φ n ) = 0 ∀ p ∈ P holds. • { φ n } n ∈ N ⇀ m φ for { φ n } n ∈ N ⊆ D P ([ r, r ′ ] , g ) and φ ∈ D P ([ r, r ′ ] , g ) if (cid:5) p ∞ ( φ − φ n ) ≤ c p · λ n ∀ n ≥ l p , p ∈ P holds for certain { c p } p ∈ P ⊆ R ≥ , { l p } p ∈ P ⊆ N , and R ≥ ⊇ { λ n } n ∈ N → Confer Sect. 4.3 in [7] for the statements mentioned but not proven here. .5.2 A Continuity Statement We recall the construction made in Sect. 4.3 in [7].i) We fix (a bump function) ρ : [0 , → [0 ,
2] smooth with ρ | (0 , > , R ρ ( s ) d s = 1 as well as ρ ( k ) (0) = 0 = ρ ( k ) (1) ∀ k ∈ N . (29)Then, given [ r, r ′ ] ∈ K and r = t < . . . < t n = r ′ , we let ρ p : [ t p , t p +1 ] → [0 , , t ρ (( t − t p ) / ( t p +1 − t p )) ∀ p = 0 , . . . , n − ρ : [ r, r ′ ] → [0 ,
2] by ρ | [ t p ,t p +1 ] := ρ p ∀ p = 0 , . . . , n − . Then, ρ is smooth, with ρ ( k ) ( t p ) = 0 for each k ∈ N , p = 0 , . . . , n ; and (10) shows that ̺ : [ r, r ′ ] → [ r, r ′ ] , t r + R tr ρ ( s ) d s holds, with ̺ ( t p ) = t p for p = 0 , . . . , n − ψ ∈ D P ([ r, r ′ ] , g ) with r = t < . . . < t n = r ′ as well as ψ [0] , . . . , ψ [ n −
1] as in (26), welet ̺ : [ r, r ′ ] → [ r, r ′ ] and ρ ≡ ˙ ̺ : [ r, r ′ ] → [0 ,
2] be as in i). Then, it is straightforward from thedefinitions that χ := ρ · ( ψ ◦ ̺ ) ∈ D r,r ′ ] holds, with R ̺r ψ = R • r χ and (cid:5) p ∞ ( χ ) ≤ · (cid:5) p ∞ ( ψ )for each p ∈ P . We obtain
Lemma 17.
1) If G is sequentially -continuous, then D P ([ r, r ′ ] , g ) ⊇ { φ n } n ∈ N ⇀ s φ ∈ D P ([ r, r ′ ] , g ) = ⇒ lim ∞ n R • r φ n = R • r φ.
2) If G is Mackey -continuous, then D P ([ r, r ′ ] , g ) ⊇ { φ n } n ∈ N ⇀ m φ ∈ D P ([ r, r ′ ] , g ) = ⇒ lim ∞ n R • r φ n = R • r φ. Specifically, in both situations, for each p ∈ P , there exist some p ≤ q ∈ P and n p ∈ N with ( p ◦ Ξ)([ R • r φ ] − [ R • r φ n ]) ≤ R (cid:5) q ( φ n ( s ) − φ ( s )) d s ∀ n ≥ n p . Proof.
Let φ ∈ D P ([ r, r ′ ] , g ) and { φ n } n ∈ N ⊆ D P ([ r, r ′ ] , g ) be given. For p ≡ u ∈ P fixed, wechoose U ⊆ G and u ≤ m ∈ P as in Lemma 4. We furthermore let m ≤ q ≡ w ∈ P be as in I), forC ≡ im[[ R • r φ ] − ] and v ≡ m there. Then, for each n ∈ N , we let ̺ n ≡ ̺ , ρ n ≡ ρ , and χ n ≡ χ be asin ii), for ψ ≡ ψ n := Ad [ R • r φ ] − ( φ n − φ ) ∈ D P ([ r, r ′ ] , g )there. Then, • we have[ R ̺ n ( t ) r φ ] − [ R ̺ n ( t ) r φ n ] (28) = R ̺ n ( t ) r Ad [ R • r φ ] − ( φ n − φ ) ii) = R tr χ n ∀ n ∈ N , t ∈ [ r, r ′ ] . (30) In the proof of Lemma 24 in [7], this statement was more generally verified for the case that k ∈ N ⊔ { lip , ∞} holds. we have (cid:5) m ∞ ( χ n ) ≤ · (cid:5) m ∞ ( ψ n ) ≤ · (cid:5) q ∞ ( φ n − φ ) for each n ∈ N by Lemma 6, which shows that – D r,r ′ ] ⊇ { χ n } n ∈ N ⇀ s . , – D r,r ′ ] ⊇ { χ n } n ∈ N ⇀ m . .In both situations, there thus exists some n p ∈ N with R • r χ n ∈ U for each n ≥ n p .We obtain from Lemma 4 (second step), and I) (last step) that ( p ◦ Ξ)([ R ̺ n ( t ) r φ ] − [ R ̺ n ( t ) r φ n ]) (30) = ( p ◦ Ξ)( R tr χ n ) ≤ R tr ( (cid:5) m ◦ χ n )( s ) d s (10) = R ̺ n ( t ) r ( (cid:5) m ◦ ψ n )( s ) d s ≤ R ̺ n ( t ) r (cid:5) q ( φ n ( s ) − φ ( s )) d s holds for all n ≥ n p and t ∈ [ r, r ′ ]; which proves the claim. In this section, we show that
Theorem 1. If G is C k -semiregular for k ∈ N ⊔ { lip , ∞} , then G is Mackey k -continuous. The proof of Theorem 1 is based on a bump function argument similar to that one used in the proofof Theorem 2 in [7]. It furthermore makes use of the fact that [0 , ∋ t R t φ ∈ G is continuousfor each φ ∈ D k . However, before we can provide the proof, we need some technical preparationfirst. Suppose we are given ̺ : [ r, r ′ ] → [ r, r ′ ]; and let ρ ≡ ˙ ̺ as well as C [ ρ, s] := max (cid:0) , max ≤ m,n ≤ s (sup {| ρ ( m ) ( t ) | n +1 | t ∈ [ r, r ′ ] } ) (cid:1) ∀ s ∈ N . We observe the following: • Let ψ ∈ C k ([ r, r ′ ] , g ) for k ∈ N ⊔ {∞} and s (cid:22) k be given. By c) , d) in Appendix A.1, we have( ρ · ( ψ ◦ ̺ )) (s) = P s q,m,n =0 h s ( q, m, n ) · (cid:0) ρ ( m ) (cid:1) n +1 · ( ψ ( q ) ◦ ̺ ) , for a map h s : (0 , . . . , s) → { , } that is independent of ̺, ρ, φ . We obtain (cid:5) p (cid:0) ( ρ · ( ψ ◦ ̺ )) (s) (cid:1) ≤ (s + 1) · C [ ρ, s] · (cid:5) p s ∞ ( ψ ) (31)for each p ∈ P , 0 ≤ s (cid:22) k , and ψ ∈ C k ([ r, r ′ ] , g ). • Let ψ ∈ C lip ([ r, r ′ ] , g ) be given. Then we have (cid:5) p (( ρ · ( ψ ◦ ̺ ))( t ) − ( ρ · ( ψ ◦ ̺ ))( t ′ )) ≤ | ρ ( t ) − ρ ( t ′ ) | · (cid:5) p ( ψ ( ̺ ( t ))) + | ρ ( t ′ ) | · (cid:5) p ( ψ ( ̺ ( t )) − ψ ( ̺ ( t ′ ))) ≤ | t − t ′ | · C [ ρ, · (cid:5) p ∞ ( ψ ) + C [ ρ, · Lip( (cid:5) p , ψ ) · | ̺ ( t ) − ̺ ( t ′ ) | | {z } ≤ C [ ρ, · | t − t ′ | ≤ · C [ ρ, · (cid:5) p lip ∞ ( ψ ) · | t − t ′ | For the third step observe that ρ n ≥ n ∈ N . More concretely, h s ( q, m, n ) are the coefficients appearing in the Leibniz rule for iterated derivatives of compositions. t, t ′ ∈ [ r, r ′ ]; thus, Lip( (cid:5) p , ρ · ( ψ ◦ ̺ )) ≤ · C [ ρ, · p lip ∞ ( ψ ) . (32)Let now ρ : [0 , → [0 ,
2] be a fixed bump function as in (29); as well as { t n } n ∈ N ⊆ [0 ,
1] strictlydecreasing with t = 1. For each n ∈ N , we define 0 < δ n := t n − t n +1 <
1, as well as ρ n := δ − n · ( ρ ◦ κ n ) for κ n : [ t n +1 , t n ] ∋ t δ − n · ( t − t n +1 ) ∈ [0 , . We obtain from (10) that ̺ n : [ t n +1 , t n ] ∋ t R tt n +1 ρ n ( s ) d s ∈ [0 , ∀ n ∈ N holds; and furthermore observe that C [ ρ n , s] ≤ δ − (s+1) n · C [ ρ , s] ∀ n ∈ N . (33) Suppose we are given { φ n } n ∈ N ⊆ D k [0 , with k ∈ N ⊔ { lip , ∞} ; and let ρ , ρ n , ̺ n , { τ n } n ∈ N , { δ n } n ∈ N be as in Sect. 4.1. Then, • We obtain from (31) and (33) that (cid:5) p (cid:0) ( ρ n · ( φ n ◦ ̺ n )) (s) (cid:1) (31) ≤ (s + 1) · C [ ρ n , s] · (cid:5) p s ∞ ( φ n ) (33) ≤ (s + 1) · δ − (s+1) n · C [ ρ , s] · (cid:5) p s ∞ ( φ n ) (34)holds, for each p ∈ P , s (cid:22) k , and n ∈ N . • We obtain from (32) and (33) thatLip( (cid:5) p , ρ n · ( φ n ◦ ̺ n )) (32) ≤ · C [ ρ n , · (cid:5) p lip ∞ ( φ n ) (33) ≤ · δ − n · C [ ρ , · (cid:5) p lip ∞ ( φ n ) (35)holds, for each p ∈ P and n ∈ N .We define φ : [0 , → g , by φ (0) := 0 and φ | [ t n +1 ,t n ] := ρ n · ( φ n ◦ ̺ n ) ∀ n ∈ N . (36)Then, it is straightforward to see that φ | [ t n +1 , ∈ D k [ t n +1 , holds for each n ∈ N , with R ̺ n φ n d) = R • t n +1 φ | [ t n +1 ,t n ] ∀ n ∈ N . (37)Moreover, for k = lip, we obtain from (35) thatLip( (cid:5) p , φ | [ t n +1 , ) ≤ · C [ ρ , · max (cid:0) δ − · (cid:5) p lip ∞ ( φ ) , . . . , δ − n · (cid:5) p lip ∞ ( φ n ) (cid:1) (38)holds, cf. Appendix A.3. The technical details can be found, e.g., in the proof of Lemma 24 in [7]. .3 Proof of Theorem 1 We are ready for the
Proof of Theorem 1.
Suppose that the claim is wrong, i.e., that G is C k -semiregular for k ∈ N ⊔ { lip , ∞} but not Mackey k-continuous. Then, by Lemma 14, there exists a sequence D k ⊇{ φ n } n ∈ N ⇀ m . k { c s p } s ≺ k, p ∈ P ⊆ R ≥ , { l s p } s (cid:22) k, p ∈ P ⊆ N , and R ≥ ⊇ { λ n } n ∈ N → φ ≡ U ⊆ G open with e ∈ U , such that im[ R • φ n ] U holds for infinitely many n ∈ N . Passing to a subsequence if necessary, we thus can assume thatim[ R • φ n ] U and λ n ≤ − ( n +1) ∀ n ∈ N (39)holds. We let t := 1, and t n := 1 − P nk =1 − k for each n ∈ N ≥ ; so that δ n = t n − t n +1 = 2 − ( n +1) holds for each n ∈ N . We construct φ : [0 , → g as in (36) in Sect. 4.2; and fix V ⊆ U open with e ∈ V and V · V − ⊆ U .Suppose now that we have shown that φ is of class C k ; i.e., that φ ∈ D k holds as G is C k -semiregular.Since [0 , ∋ t R t φ ∈ G is continuous, there exists some ℓ ≥ R t φ ∈ V for each t ∈ [0 , t ℓ ];thus, R ̺ n ( t )0 φ n (37) = R tt n +1 φ | [ t n +1 ,t n ] c) = [ R t φ ] · [ R t n +1 φ ] − ∈ V · V − ⊆ U for each t ∈ [ t n +1 , t n ] with n ≥ ℓ , which contradicts (39).To prove the claim, it thus suffices to show that φ is of class C k : • Suppose first that k ∈ N ⊔ {∞} holds. Then, it suffices to show thatlim (0 , ∋ h → /h · φ (s) ( h ) = 0 ∀ s (cid:22) k holds, because φ is of class C k on (0 , h ∈ [ t n +1 , t n ] for n ∈ N be given; and observe that h ≥ t n = t n − t n +1 + t n +1 = 2 − ( n +1) + 1 − P n +1 k =1 − k ≥ − ( n +1) holds. Then, for p ∈ P fixed and n ≥ l s p , we obtain from (34) (with δ n = 2 − ( n +1) ) as well as (39)that 1 /h · (cid:5) p (cid:0) φ (s) ( h ) (cid:1) ≤ n +1 · (cid:5) p (cid:0) ( ρ n · ( φ n ◦ ̺ n )) (s) (cid:1) (34) ≤ (s + 1) · ((s+1) +1) · ( n +1) · C [ ρ , s] · (cid:5) p s ∞ ( φ n ) (39) ≤ (s + 1) · C [ ρ , s] · c s p · ((s+1) +1) · ( n +1) − ( n +1) = (s + 1) · C [ ρ , s] · c s p · (((s+1) +1) − ( n +1)) · ( n +1) holds; which clearly tends to zero for n → ∞ . • Suppose now that k = lip holds. The previous point then shows φ ∈ C ([0 , , g ). For p ∈ P fixed, we thus have (cid:5) p ∞ ( φ ) < ∞ . We let l p ≡ l lip p for each p ∈ P , define D p := max (cid:0) · δ − · (cid:5) p lip ∞ ( φ ) , . . . , · δ − l p · (cid:5) p lip ∞ ( φ l p ) (cid:1) , n ≥ l p thatLip( (cid:5) p , φ | [ t n +1 , ) (38) ≤ C [ ρ , · max (cid:0) · δ − · (cid:5) p lip ∞ ( φ ) , . . . , · δ − n · (cid:5) p lip ∞ ( φ n ) (cid:1) (21) ≤ C [ ρ , · max (cid:0) D p , c lip p · max (cid:0) l p +2) · λ l p +1 , . . . , n +1) · λ n (cid:1)(cid:1) (39) ≤ C [ ρ , · max (cid:0) D p , c lip p · max (cid:8) ℓ +1) · − ( ℓ +1) (cid:12)(cid:12) l p + 1 ≤ ℓ ≤ n (cid:9)(cid:1) = C [ ρ , · max (cid:0) D p , c lip p · max (cid:8) − ℓ +6 ℓ +8 (cid:12)(cid:12) l p + 1 ≤ ℓ ≤ n (cid:9)(cid:1) = C [ ρ , · max (cid:0) D p , c lip p · max (cid:8) − ( ℓ − (cid:12)(cid:12) l p + 1 ≤ ℓ ≤ n (cid:9)(cid:1) ≤ C [ ρ , · max (cid:0) D p , c lip p · (cid:1) =: L holds; thus, (cid:5) p ( φ ( t ) − φ ( t ′ )) ≤ L · | t − t ′ | ∀ t, t ′ ∈ (0 , . Moreover, since φ is continuous with φ (0) = 0, for each t ∈ [0 ,
1] we have (cid:5) p ( φ (0) − φ ( t )) = lim ℓ →∞ (cid:5) p ( φ (0) − φ (1 /ℓ ) + φ (1 /ℓ ) − φ ( t )) ≤ lim ℓ →∞ (cid:5) p ( φ (1 /ℓ )) + lim ℓ →∞ (cid:5) p ( φ (1 /ℓ ) − φ ( t ))= lim ℓ →∞ (cid:5) p ( φ (1 /ℓ ) − φ ( t )) ≤ lim ℓ →∞ L · | /ℓ − t | = L · | − t | . This shows Lip( (cid:5) p , φ ) ≤ L , i.e., φ ∈ C lip ([0 , , g ). In this Section, we want to give a brief application of the notions introduced so far. For this, werecall that a Lie group G is said to have the strong Trotter property if (1) holds; and now will show Proposition 1.
1) If G is sequentially -continuous, then G has the strong Trotter property.2) If G is Mackey -continuous, then (1) holds for each µ ∈ C ∗ ([0 , , G ) with ˙ µ (0) ∈ dom[exp] and δ r ( µ ) ∈ C lip ([0 , , g ) . Here, • By Remark 7.5), Proposition 1.1) generalizes Theorem 1 in [8], stating that G admits the strongTrotter property if G is locally µ -convex (recall that, by Theorem 1 in [7], local µ -convexity isequivalent to that G is C -continuous). • By Theorem 1, the presumptions made in Proposition 1.2) are always fulfilled, e.g., if G is C -semiregular, and µ is of class C with ˙ µ (0) ∈ dom[exp].We will need the following observations: Let ℓ > µ ∈ C ∗ ([0 , , U ) be given; and define φ := δ r ( µ ), X := ˙ µ (0) = φ (0), as well as µ τ : [0 , /ℓ ] ∋ t µ ( τ · t ) ∈ G ∀ τ ∈ [0 , ℓ ] . Then, for each t ∈ [0 , /ℓ ], we have δ r ( µ τ )( t ) − τ · X (14) = ω ((Ξ ◦ µ τ )( t ) , ∂ t (Ξ ◦ µ τ )( t )) − τ · X = τ · ω ((Ξ ◦ µ )( τ · t ) , ∂ t (Ξ ◦ µ )( τ · t )) − τ · X = τ · δ r ( µ )( τ · t ) − τ · X = τ · ( φ ( τ · t ) − X ) . p ∈ P , τ ∈ [0 , ℓ ], and s ≤ /ℓ , we thus obtain (cid:5) p ∞ ( δ r ( µ τ ) | [0 ,s ] − τ · X ) = ℓ · sup { (cid:5) p ( φ ( τ · t ) − φ (0)) | t ∈ [0 , s ] } ; (40)whereby, for the case that φ ∈ C lip ([0 , , g ) holds, we additionally havesup { (cid:5) p ( φ ( τ · t ) − φ (0)) | t ∈ [0 , s ] } ≤ s · ℓ · Lip( (cid:5) p , φ ) . (41)We are ready for the Proof of Proposition 1.
Let φ := δ r ( µ ) and X := ˙ µ (0) = φ (0), for1) µ ∈ C ∗ ([0 , , G ) with ˙ µ (0) ∈ dom[exp] if G is sequentially 0-continuous.2) µ ∈ C ∗ ([0 , , G ) with ˙ µ (0) ∈ dom[exp] and φ ∈ C lip ([0 , , g ) if G is Mackey 0-continuous.We suppose that (1) is wrong; i.e., that there exists some ℓ >
0, an open neighbourhood U ⊆ G of e , a sequence { τ n } n ∈ N ⊆ [0 , ℓ ], and a strictly increasing sequence { ι n } n ∈ N ⊆ N ≥ ∩ [ ℓ, ∞ ) withexp( − τ n · X ) · µ ( τ n /ι n ) ι n / ∈ U ∀ n ∈ N . (42)Passing to a subsequence if necessary, we can additionally assume that lim n τ n = τ ∈ [0 , ℓ ] exists.We choose V ⊆ G open with e ∈ V and V · V ⊆ U , and fix some n V ∈ N withexp(( τ − τ n ) · X ) ∈ V ∀ n ≥ n V . (43)Moreover, for each n ∈ N : • We define χ n := δ r ( µ τ n ) | [0 , /ι n ] ∈ D k [0 , /ι n ] with µ τ n : [0 , /ℓ ] ∋ t µ ( t · τ n ) . • We define t n,m := m/ι n for m = 0 , . . . , ι n ; as well as φ n [ m ] : [ t n,m , t n,m +1 ] ∋ t χ n ( · − t n,m ) ∈ g for m = 0 , . . . , ι n − . Then, we have R φ n [ m ] d) = R χ n = µ τ n (1 /ι n ) = µ ( τ n /ι n ) ∀ m = 0 , . . . , ι n − . (44) • We define φ n ∈ D P ([0 , , g ) by φ n | [ t n,m ,t n,m +1 ) := φ n [ m ] | [ t n,m ,t n,m +1 ) ∀ m = 0 , . . . , ι ( n ) − ,φ n | [ t n,ιn − ,t n,ιn ] := φ n [ ι n − R φ n (27) = R t n,ιn t n,ιn − φ n [ ι n − · . . . · R t n, t n, φ n [0] (44) = µ ( τ n /ι n ) ι n ∀ n ∈ N . (45)Then, for each n ∈ N and p ∈ P , we have (cid:5) p ∞ ( φ n − τ · φ X ) ≤ (cid:5) p ∞ ( φ n − τ n · X ) + (cid:5) p ( τ n · X − τ · X )= (cid:5) p ∞ (cid:0) δ r ( µ τ n ) | [0 , /ι n ] − τ n · X (cid:1) + | τ − τ n | · (cid:5) p ( X ) (40) ≤ ℓ · sup { (cid:5) p ( φ ( τ n · t ) − φ (0)) | t ∈ [0 , /ι n ] } + | τ − τ n | · (cid:5) p ( X ) . (46) In the first step below, d) is applied with ̺ : [ t n,m , t n,m +1 ] t − t n,m ∈ [0 , /ι n ]. φ ∈ C lip ([0 , , g ) holds, we furthermore obtain (cid:5) p ∞ ( φ n − τ · φ X ) (46) , (41) ≤ ℓ /ι n · Lip( (cid:5) p , φ ) + | τ − τ n | · (cid:5) p ( X ) ≤ (Lip( (cid:5) p , φ ) + (cid:5) p ( X )) | {z } c p · ( ℓ /ι n + | τ − τ n | ) | {z } λ n for each n ∈ N and p ∈ P ; whereby lim n λ n = 0 holds. We thus have D P ([0 , , g ) ⊇{ φ n } n ∈ N ⇀ s τ · φ X as well as D P ([0 , , g ) ⊇{ φ n } n ∈ N ⇀ m τ · φ X if φ ∈ C lip ([0 , , g ) holds . In both cases, by Lemma 17, there exists some N ∋ n ′ V ≥ n V with[ R τ · φ X | [0 , ] − · [ R φ n ] ∈ V ∀ n ≥ n ′ V ; (47)and we obtain for n ≥ n ′ V thatexp( − τ n · X ) · µ ( τ n /ι n ) ι n = (43) ∈ V z }| { exp (cid:0) ( τ − τ n ) · X (cid:1) · (cid:0) = [ R τ · φ X | [0 , ] − z }| { exp( − τ · X ) · (45) = R φ n z }| { µ ( τ n /ι n ) ι n (cid:1)| {z } (47) ∈ V ∈ V · V ⊆ U holds, which contradicts (42). In this section, we discuss several differentiability properties of the evolution map. The wholediscussion is based on the following generalization of Proposition 7 in [7].
Proposition 2.
Let { ε n } n ∈ N ⊆ C ([ r, r ′ ] , g ) , χ ∈ C ([ r, r ′ ] , g ) , and R =0 ⊇ { h n } n ∈ N → be givenwith { h n · χ + h n · ε n } n ∈ N ⊆ D [ r,r ′ ] , such thati) lim n ε n ( t ) = 0 holds for each t ∈ [ r, r ′ ] ,ii) sup { p ∞ ( ε n ) | n ∈ N ) } < ∞ holds for each p ∈ P .Then, the following two conditions are equivalent:a) lim ∞ n Ξ( R • r h n · χ + h n · ε n ) = 0 .b) lim ∞ n /h n · Ξ (cid:0) R • r h n · χ + h n · ε n (cid:1) = R • r (d e Ξ ◦ χ )( s ) d s ∈ E . The proof of Proposition 2 will be established in Sect. 6.4. We now first use this proposition, todiscuss the differential of the evolution maps as well as the differentiation of parameter-dependentintegrals.
We will need the following variation of Proposition 2:
Corollary 2.
Let δ > , { ε h } h ∈ D δ ⊆ C ([ r, r ′ ] , g ) , and χ ∈ C ([ r, r ′ ] , g ) be given with { h · χ + h · ε h } h ∈ D δ ⊆ D [ r,r ′ ] , such that i) lim h → ε h ( t ) = 0 holds for each t ∈ [ r, r ′ ] , ii) sup { p ∞ ( ε h ) | h ∈ D δ p } < ∞ holds for some < δ p ≤ δ , for each p ∈ P . hen, the following conditions are equivalent: a) lim ∞ h → Ξ( R • r h · χ + h · ε h ) = 0 . b) lim ∞ n Ξ( R • r h n · χ + h n · ε h n ) = 0 for each sequence D δ ⊇ { h n } n ∈ N → . c) lim ∞ n /h n · Ξ (cid:0) R • r h n · χ + h n · ε h n (cid:1) = R • r (d e Ξ ◦ χ )( s )d s ∈ E for each sequence D δ ⊇ { h n } n ∈ N → . d) dd h (cid:12)(cid:12) ∞ h =0 Ξ (cid:0) R • r h · χ + h · ε h (cid:1) = R • r (d e Ξ ◦ χ )( s ) d s ∈ E .Proof. By Lemma 1 (applied to ( G, +) ≡ ( E, + ) there), a) is equivalent to b). Moreover, byProposition 2, b) is equivalent to c), because − Condition i) implies Condition i) in Proposition 2, for ε n ≡ ε h n there, − Condition ii) implies Condition ii) in Proposition 2, for ε n ≡ ε h n there.Finally, by Lemma 1 (applied to ( G, +) ≡ ( E, + ) there), c) is equivalent to d).Given a net { ψ α } α ∈ I ⊆ C ([ r, r ′ ] , g ), and some ψ ∈ C ([ r, r ′ ] , g ), we write { ψ α } α ∈ I ⇀ n . ψ iflim α (cid:5) p ( ψ − ψ α ) = 0 ∀ p ∈ P . Lemma 18.
Suppose that G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞ , c } . Suppose furthermorethat we are given D k [ r,r ′ ] ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k [ r,r ′ ] as well as D k [ r,r ′ ] ⊇ { ψ α } α ∈ I ⇀ n . ψ ∈ D k [ r,r ′ ] for [ r, r ′ ] ∈ K , such that the expressions ξ ( φ, ψ ) := d e L R φ (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) ξ ( φ n , ψ α ) := d e L R φ n (cid:0) R Ad [ R sr φ n ] − ( ψ α ( s )) d s (cid:1) ∀ n ∈ N are well defined; i.e., such that the occurring Riemann integrals exist in g . Then, we have lim ( n,α ) ξ ( φ n , ψ α ) = ξ ( φ, ψ ) . Proof.
This follows by the same arguments as in Corollary 13 and Lemma 41 in [7]. For complete-ness, the adapted argumentation is provided in Appendix A.4.
We now discuss the differential of the evolution map – for which we recall the conventions fixed inRemark 3. Then, Corollary 2 (with ε h ≡ Proposition 3.
Suppose that ( φ, ψ ) is admissible, with dom[ φ ] , dom[ ψ ] = [ r, r ′ ] .1) The pair ( φ, ψ ) is regular if and only if we have dd h (cid:12)(cid:12) ∞ h =0 Ξ (cid:0) [ R • r φ ] − [ R • r φ + h · ψ ] (cid:1) = R • r (d e Ξ ◦ Ad [ R sr φ ] − )( ψ ( s )) d s ∈ E.
2) If ( φ, ψ ) is regular, then ( − δ, δ ) ∋ h R φ + h · ψ ∈ G is differentiable at h = 0 (for δ > suitably small) if and only if R Ad [ R sr φ ] − ( ψ ( s )) d s ∈ g holds. In this case, we have dd h (cid:12)(cid:12) h =0 R φ + h · ψ = d e L R φ (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) . (48) Recall Remark 2 for the notation used in d). roof.
1) For | h | < δ , with δ > (cid:0) [ R tr φ ] − [ R tr φ + h · ψ ] (cid:1) b) = Ξ (cid:0) R tr h · =: χ z }| { Ad [ R • r φ ] − ( ψ ) (cid:1) ∀ t ∈ [ r, r ′ ] . (49)We obtain from the Equivalence of a) and d) in Corollary 2 for ε h ≡ ∞ h → R • r φ + h · ψ = R • r φ ⇐⇒ lim ∞ h → Ξ (cid:0) [ R • r φ ] − [ R • r φ + h · ψ ] (cid:1) = 0 (49) ⇐⇒ lim ∞ h → Ξ( R • r h · χ ) = 0 ⇐⇒ dd h (cid:12)(cid:12) ∞ h =0 Ξ( R • r h · χ ) = R • r (d e Ξ ◦ χ )( s ) d s ∈ E (49) ⇐⇒ dd h (cid:12)(cid:12) ∞ h =0 Ξ (cid:0) [ R • r φ ] − [ R • r φ + h · ψ ] (cid:1) = R • r (d e Ξ ◦ Ad [ R sr φ ] − )( ψ ( s )) d s ∈ E.
2) Let ( φ, ψ ) be regular; and µ : ( − δ, δ ) ∋ h R φ + h · ψ ∈ G . • Suppose that R Ad [ R sr φ ] − ( ψ ( s )) d s ∈ g holds; and let (shrink δ > γ : ( − δ, δ ) → V , h Ξ([ R φ ] − [ R φ + h · ψ ]) . Then, we have˙ γ (0) Part R (d e Ξ ◦ Ad [ R sr φ ] − )( ψ ( s )) d s (11) = d e Ξ (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) . (50)Since µ = Ξ ′− ◦ γ (thus, γ = Ξ ′ ◦ µ ) holds for the chartΞ ′ : R φ · U =: U ′ → V , g Ξ([ R φ ] − · g ) , (51)(50) shows that µ is differentiable at 0 – Specifically, we have, cf. Remark 3˙ µ (0) = d Ξ ′− (cid:0) dd h (cid:12)(cid:12) h =0 (Ξ ′ ◦ µ )( h ) (cid:1) (50) = (d Ξ ′− ◦ d e Ξ) (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) = (cid:0) d e L R φ ◦ d Ξ − ◦ d e Ξ (cid:1)(cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) = d e L R φ (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) ;which shows (48). • Suppose that µ is differentiable at h = 0. Then, for Ξ ′ as in (51) we have, cf. Remark 3 E ∋ dd h (cid:12)(cid:12) h =0 (Ξ ′ ◦ µ )( h ) = dd h (cid:12)(cid:12) h =0 Ξ (cid:0) [ R φ ] − [ R φ + h · ψ ] (cid:1) Part R (d e Ξ ◦ Ad [ R sr φ ] − )( ψ ( s )) d s. We obtain g ∋ d Ξ − (cid:0) R (d e Ξ ◦ Ad [ R sr φ ] − )( ψ ( s )) d s (cid:1) (11) = R Ad [ R sr φ ] − ( ψ ( s )) d s. In particular, (48) holds by the previous point.26 .2.1 The Generic Case
Combining Proposition 3 with Theorem 1 and Lemma 15, we obtain
Theorem 2.
Suppose that G is C k -semiregular for k ∈ N ⊔ { lip , ∞} . Then, evol k is differentiableif and only if g is k -complete. In this case, evol k [ r,r ′ ] is differentiable for each [ r, r ′ ] ∈ K , with d φ evol k [ r,r ′ ] ( ψ ) = d e L R φ (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) ∀ φ, ψ ∈ C k ([ r, r ′ ] , g ) . In particular,a) d φ evol k [ r,r ′ ] : C k ([ r, r ′ ] , g ) → T R φ G is linear and C -continuous for each φ ∈ C k ([ r, r ′ ] , g ) ,b) for each sequence { φ n } n ∈ N ⇀ m . k φ , and each net { ψ α } α ∈ I ⇀ n . ψ , we have lim ( n,α ) d φ n evol k [ r,r ′ ] ( ψ α ) = d φ evol k [ r,r ′ ] ( ψ ) . Proof.
The first part is clear from Theorem 1, Lemma 15, Remark 4, and Proposition 3.2). Then, b) is clear from Lemma . Moreover, (by the first part) d φ evol k [ r,r ′ ] is linear; with (cf. (3))d φ evol k [ r,r ′ ] ( ψ ) = d ( R φ,e ) m(0 , Γ φ ( ψ )) for Γ φ : C k ([0 , , g ) ∋ ψ → R Ad [ R sr φ ] − ( ψ ( s )) d s ∈ g . Then, since Γ φ is C -continuous by (9) and I), a) is clear from smoothness of the Lie groupmultiplication. Corollary 3.
Suppose that G is C k -semiregular for k ∈ N ⊔ { lip , ∞} , and that g is k -complete.Then, µ : R ∋ h R φ + h · ψ is of class C for each φ, ψ ∈ C k ([ r, r ′ ] , g ) and [ r, r ′ ] ∈ K .Proof. Theorem 1 and Lemma 15 show that µ is continuous. Moreover, for each t ∈ R , and eachsequence { h n } n ∈ N →
0, we have { φ + ( t + h n ) · ψ } n ∈ N ⇀ m . k ( φ + t · ψ ); thus,lim n ˙ µ ( t + h n ) = lim n d φ +( t + h n ) · ψ evol k [ r,r ′ ] ( ψ ) = d φ + t · ψ evol k [ r,r ′ ] ( ψ ) = ˙ µ ( t )by Theorem 2. b) . This shows that ˙ µ is continuous, i.e., that µ is of class C . Remark 8.
1) It is straightforward from Corollary 3, the differentiation rules d) and c) , as well as (3) , (12) ,c), and e) (for Ψ ≡ Conj g there) that (48) also holds for all φ, ψ ∈ D P k ([ r, r ′ ] , g ) , for each [ r, r ′ ] ∈ K .2) Expectably, µ as defined in Corollary 3 is even of class C ∞ . A detailed proof of this fact,however, would require further technical preparation – which we do not want to carry out at thispoint.3) Expectably, the equivalence lim ∞ h → Ξ( R • r h · χ ) = 0 ⇐⇒ dd h (cid:12)(cid:12) ∞ h =0 Ξ (cid:0) R • r h · χ (cid:1) = R • r (d e Ξ ◦ χ )( s ) d s ∈ E also holds for χ ∈ D P ([ r, r ′ ] , g ) – implying that Proposition 3.1) carries over to the piecewisecategory. This might be shown by the same arguments (Taylor expansion) as used in the proofof Lemma 7 in [7] (cf. Lemma 3) additionally using (27) as well as that for n ∈ N fixed, f : G n → G, ( g , . . . , g n ) g · . . . · g n is smooth, with d ( e,...,e ) f ( X , . . . , X n ) = X + · · · + X n for all X , . . . , X n ∈ g . The details,however, appear to be quite technical, so that we leave this issue to another paper. ‡ .2.2 The Exponential Map We recall the conventions fixed in Sect. 2.2.3, specifically that exp = evol c[0 , ◦ i | dom[exp] holds. Then,Proposition 3.2), for k ≡ c and [ r, r ′ ] ≡ [0 ,
1] there, reads as follows.
Corollary 4.
Suppose that ( i ( X ) , i ( Y )) is regular for X, Y ∈ g . Then, ( − δ, δ ) ∋ h exp( X + h · Y ) is differentiable at h = 0 (for δ > suitably small) if and only if R Ad exp( − s · X ) ( Y ) d s ∈ g holds. Inthis case, we have dd h (cid:12)(cid:12) h =0 exp( X + h · Y ) = d e L exp( X ) (cid:0) R Ad exp( − s · X ) ( Y ) d s (cid:1) . Remark 9.
1) Suppose that G admits an exponential map; and that G is weakly c -continuous. Then, Corollary4 shows that we have dd h (cid:12)(cid:12) h =0 exp( X + h · Y ) = d e L exp( X ) (cid:0) R Ad exp( − s · X ) ( Y ) d s (cid:1) ∀ X, Y ∈ g (52) if and only if g is c -complete. For instance, G is weakly c -continuous, and g is c -complete if • exp : g → G is of class C , by Remark 7.1), Remark 7.5), and Corollary 4. • G is abelian, by Corollary 1 and Remark 4.2) Suppose that g is c -complete; and that G admits a continuous exponential map. Then, G is C c -semiregular; and G is weakly c -continuous by Remark 7.1) and Remark 7.5). More formally, (52) then reads d φ evol c ( ψ ) = d e L R φ (cid:0) R Ad [ R sr ψ ] − ( ψ ( s )) d s (cid:1) ∀ φ, ψ ∈ C c ([0 , , g ) . (53) The same arguments as in [7] then show that evol c (thus exp ) is of class C . More specifically,one has to replace Lemma 23 by Lemma 10 in the proof of Lemma 41 in [7]. Then, substitutingEquation (95) in [7] by (53) , the proof of Corollary 13 in [7] just carries over to the case where k = c holds (a similar adaption has been done in the proof of Lemma 18).As in the Lipschitz case, cf. Remark 7 in [7], it is to be expected that a (quite elaborate andtechnical) induction shows that exp is even smooth if g is Mackey complete (or, more generally,if all the occurring iterated Riemann integrals exist in g ). ‡ Given an open interval J ⊆ R as well as x ∈ J , in the following, we denote J [ x ] := { h ∈ R =0 | x + h ∈ J } . The next theorem generalizes Theorem 5 in [7] (with significantly simplified proof).
Theorem 3.
Suppose that G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞ , c } – additionally abelianif k = c holds. Let Φ : I × [ r, r ′ ] → g ( I ⊆ R open) be given with Φ( z, · ) ∈ D k [ r,r ′ ] for each z ∈ I .Then, dd h (cid:12)(cid:12) ∞ h =0 Ξ (cid:0) [ R • r Φ( x, · )] − [ R • r Φ( x + h, · )] (cid:1) = R • r (d e Ξ ◦ Ad [ R sr Φ( x, · )] − )( ∂ z Φ( x, s )) d s ∈ E holds for x ∈ I , provided thata) We have ( ∂ z Φ)( x, · ) ∈ C k ([ r, r ′ ] , g ) . More specifically, this means that for each t ∈ [ r, r ′ ], the map I ∋ z Φ( z, t ) is differentiable at z = x withderivative ( ∂ z Φ)( x, t ), such that ( ∂ z Φ)( x, · ) ∈ C k ([ r, r ′ ] , g ) holds. The latter condition in particular ensures that (cid:5) p s ∞ (( ∂ z Φ)( x, · )) < ∞ holds for each p ∈ P and s (cid:22) k , cf. ii). ) For each p ∈ P and s (cid:22) k , there exists L p , s ≥ , as well as I p , s ⊆ I open with x ∈ I p , s , such that / | h | · (cid:5) p s ∞ (Φ( x + h, · ) − Φ( x, · )) ≤ L p , s ∀ h ∈ I p , s [ x ] . In particular, we have dd h (cid:12)(cid:12) h =0 R Φ( x + h, · ) = d e L R Φ( x, · ) (cid:0) R Ad [ R sr Φ( x, · )] − ( ∂ z Φ( x, s )) d s (cid:1) if and only if the Riemann integral on the right side exists in g .Proof. The last statement follows from the first statement and Lemma 3 – just as in the proof ofProposition 3.2). Now, for x + h ∈ I , we haveΦ( x + h, t ) = Φ( x, t ) + h · ∂ z Φ( x, t ) + h · ε ( x + h, t ) ∀ t ∈ [ r, r ′ ] , with ε : I × [ r, r ′ ] → g such thati) lim h → ε ( x + h, t ) = ε ( x, t ) = 0 ∀ t ∈ [ r, r ′ ],ii) (cid:5) p s ∞ ( ε ( x + h, · )) ≤ L p , s + (cid:5) p s ∞ (( ∂ z Φ)( x, · )) =: C p , s < ∞ ∀ h ∈ I p , s [ x ] for all p ∈ P , s (cid:22) k .We let α := R • r Φ( x, · ); and obtain[ R • r Φ( x, · )] − [ R • r Φ( x + h, · )] = R • r ψ h z }| { h · Ad α − ( ∂ z Φ( x, · )) | {z } χ + h · Ad α − ( ε ( x + h, · )) | {z } ε h (54)with ψ h ∈ D k [ r,r ′ ] , because our presumptions ensure that χ, ε h ∈ C k ([ r, r ′ ] , g ) holds. By Lemma 6and Lemma 7, for each p ∈ P and s (cid:22) k , there exists some p ≤ q ∈ P with (cid:5) p s ∞ ( ψ h ) ≤ | h | · (cid:5) q s ∞ ( ∂ z Φ( x, · ) + ε ( x + h, · )) b ) ≤ | h | · L q , s ∀ h ∈ I q , s [ x ] . For each fixed sequence I [ x ] ⊇ { h n } n ∈ N →
0, we thus have ψ h n ⇀ m . k
0. Since G is Mackeyk-continuous, this implieslim ∞ n Ξ( R • r h n · χ + h n · ε h n ) = lim ∞ n Ξ( R • r ψ h n ) = 0 . (55)Now, for δ > D δ ⊆ I [ x ] holds, by i) and ii), { ε h } h ∈ D δ fulfills the presumptions inCorollary 2. We thus havelim ∞ h → Ξ (cid:0) [ R • r Φ( x, · )] − [ R • r Φ( x + h, · )] (cid:1) = R • r (d e Ξ ◦ Ad [ R sr Φ( x, · )] − )( ∂ z Φ( x, s )) d s ∈ E by (54), (55), as well as the equivalence of b) and d) in Corollary 2.We immediately obtain Corollary 5.
Suppose that G is C k -semiregular for k ∈ N ⊔ { lip , ∞} ; and that g is k -complete. Let Φ : I × [ r, r ′ ] → g ( I ⊆ R open) be given with Φ( z, · ) ∈ D k [ r,r ′ ] for each z ∈ I . Then, dd h (cid:12)(cid:12) h =0 R Φ( x + h, · ) = d e L R Φ( x, · ) (cid:0) R Ad [ R sr Φ( x, · )] − ( ∂ z Φ( x, s )) d s (cid:1) holds for x ∈ I , provided that the conditions a) and b) in Theorem 3 are fulfilled.Proof. This is clear from Theorem 1 and Theorem 3.We furthermore obtain the following generalization of Corollary 11 in [7]. If k = c holds, we can just choose s = 0 and q = p , because G is presumed to be abelian in this case. orollary 6. Suppose that G is Mackey k -continuous for k ∈ {∞ , c } – additionally abelian if k = c holds. Suppose furthermore that X : I → dom[exp] ⊆ g is of class C ; and define α := exp ◦ X .Then, for x ∈ I , we have dd h (cid:12)(cid:12) h =0 α ( x + h ) = d e L exp( X ( x )) (cid:0) R Ad exp( − s · X ( x )) ( ˙ X ( x )) d s (cid:1) , provided that the Riemann integral on the right side exists in g . If this is the case for each x ∈ I ,then α is of class C .Proof. We let Φ : I × [0 , ∋ ( z, t ) X ( z ); and observe that α ( z ) = R Φ( z, · ) holds for each z ∈ I .Then, the first statement is clear from Theorem 3. For the second statement, we suppose that˙ α ( x ) = d e L exp( X ( x )) (cid:0) R Ad exp( − s · X ( x )) ( ˙ X ( x )) d s (cid:1) ∀ x ∈ I is well defined; i.e., that the Riemann integral on the right side exists for each x ∈ I . We fix x ∈ I and δ > x − δ, x + δ ] ⊆ I ; and observe that (cid:5) p ( X ( x + h ) − X ( x )) ≤ | h | · sup { (cid:5) p ( ˙ X ( z )) | z ∈ [ x − δ, x + δ ] } ∀ p ∈ P , | h | ≤ δ holds by (8). For each sequence I ⊇ { h n } n ∈ N →
0, we thus haveD c ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D c for φ := i ( X ( x )) and φ n := i ( X ( x + h n )) ∀ n ∈ N . Moreover, since X is of class C , we haveD c ⊇ { ψ n } n ∈ N ⇀ s . ψ ∈ D c for ψ := i ( ˙ X ( x )) and ψ n := i ( ˙ X ( x + h n )) ∀ n ∈ N ;so that Lemma 18 showslim n →∞ ˙ α ( x + h n ) = lim ( n,n ) → ( ∞ , ∞ ) ξ ( φ n , ψ n ) = ξ ( φ, ψ ) = ˙ α ( x ) . This shows that ˙ α is continuous at x . Since x ∈ I was arbitrary, it follows that α is of class C .For instance, we obtain the following generalization of Remark 2.3) in [7]. Example 1.
Suppose that G is Mackey c -continuous and abelian. Then, for each φ ∈ C ([ r, r ′ ] , g ) with [ r, r ′ ] ∋ t R tr φ ( s ) d s ∈ dom[exp] , we have, cf. Appendix A.5 R φ = exp (cid:0) R φ ( s ) d s (cid:1) . (56) In particular, if dom[exp] = g holds ( G admits an exponential map), then G is C k -semiregular for k ∈ N ⊔ { lip , ∞} if g is k -complete. ‡ In this subsection, we prove Proposition 2. We start with some general remarks:Let [ r, r ′ ] ∈ K and χ ∈ C ([ r, r ′ ] , g ) be given. For m ≥ t k := r + k/m · ( r ′ − r ) for k = 0 , . . . , m ; as well as X k := χ ( t k ) for k = 0 , . . . , m −
1. We furthermore define χ m ∈ C ([ r, r ′ ] , g )by χ m ( r ) := X and χ m ( t ) = X k + ( t − t k ) / ( t k +1 − t k ) · ( X k +1 − X k ) ∀ t ∈ ( t k , t k +1 ] , k = 0 , . . . , m − . Then, { χ m } m ≥ ⊆ C ([ r, r ′ ] , g ) constructed in this way, admits the following properties: a) We have lim m (cid:5) p ∞ ( χ − χ m ) = 0 for each p ∈ P . b) We have γ h,m := h · d e Ξ( R • r χ m ( s ) d s ) ∈ E for each h ∈ R and m ≥ ) Since im[ χ ] ⊆ g is bounded, also { im[ χ m ] } m ≥ ⊆ g , { im[d e Ξ( R • r χ m ( s ) d s )] } m ≥ ⊆ E arebounded. Thus, • For each p ∈ P , there exists some C p > p ∞ ( γ h,m ) ≤ | h | · C p ∀ h ∈ R , m ≥ . (57) • For δ > µ h,m := Ξ − ◦ γ h,m ∈ C ([ r, r ′ ] , G )is well defined for each | h | ≤ δ , m ≥
1; and we define χ h,m := δ r ( µ h,m ) (14) = h · ω ( γ h,m , d e Ξ( χ m )) ∀ | h | ≤ δ, m ≥ . (58)Moreover, for each fixed open neighbourhood V ⊆ G of e , there exists some 0 < δ V ≤ δ with V ∋ µ h,m = R • r χ h,m ∀ | h | ≤ δ V , m ≥ . (59)Modifying the proof of Proposition 7 in [7], we obtain the Proof of Proposition 2.
Suppose first that b) holds; and let A := Ξ( R • r h n · χ + h n · ε n ) B := R • r (d e Ξ ◦ χ )( s ) d s. Then, a) is clear from p ∞ ( A ) ≤ | h n | · p ∞ (1 /h n · A − B ) + | h n | · p ∞ ( B ).Suppose now that a) holds – i.e. that we have lim ∞ n Ξ( R • r ψ n ) = 0 with ψ n := h n · χ + h n · ε n ∀ n ∈ N . We now have to show that for p ∈ P fixed, the expression∆ n := 1 / | h n | · p ∞ (cid:0) Ξ( R • r ψ n ) − h n · R • r d e Ξ( χ ( s )) d s (cid:1) tends to zero for n → ∞ . For this, we choose p ≤ q ∈ P and e ∈ V ⊆ G as in Lemma 9; and let { χ m } m ≥ , { χ h,m } m ≥ , { γ h,m } m ≥ , { µ h,m } m ≥ , δ V > δ V additionally such small that R • r ψ n ∈ V holds for each n ∈ N with | h n | < δ V .We choose ℓ ∈ N such large that { h n } n ≥ ℓ ⊆ ( − δ V , δ V ) holds. Then, for each n ≥ ℓ and m ≥
1, weobtain from (9) (second step), (59) and Lemma 9 (fifth step), as well as (58) (last step) that∆ n ≤ / | h n | · p ∞ (cid:0) Ξ( R • r ψ n ) − h n · d e Ξ( R • r χ m ( s ) d s ) (cid:1) + p ∞ (cid:0) R • r d e Ξ( χ ( s )) d s − R • r d e Ξ( χ m ( s )) d s (cid:1) ≤ / | h n | · p ∞ (cid:0) Ξ( R • r ψ n ) − γ h n ,m (cid:1) + R ( p ◦ d e Ξ)( χ ( s ) − χ m ( s )) d s = 1 / | h n | · p ∞ (cid:0) Ξ( R • r ψ n ) − Ξ( µ h n ,m ) (cid:1) + R (cid:5) p ( χ ( s ) − χ m ( s )) d s = 1 / | h n | · p ∞ (cid:0) Ξ( R • r ψ n ) − Ξ( R • r χ h n ,m ) (cid:1) + R (cid:5) p ( χ ( s ) − χ m ( s )) d s ≤ / | h n | · R (cid:5) q (cid:0) ψ n ( s ) − χ h n ,m ( s ) (cid:1) d s + R (cid:5) p ( χ ( s ) − χ m ( s )) d s ≤ R (cid:5) q ( ε n ( s )) d s + R (cid:5) q (cid:0) χ ( s ) − ω ( γ h n ,m ( s ) , d e Ξ( χ m ( s ))) (cid:1) d s + ( r ′ − r ) · (cid:5) p ∞ ( χ − χ m )holds. By Lebesgue’s dominated convergence theorem and i) , ii) , the first term tends to zero for n → ∞ ; and, by a) , the third term tends to zero for m → ∞ . Thus, ε > ℓ ε ≥ ℓ , such that both the first-, and the third term is bounded by ε/ m, n ≥ ℓ ε . Moreover,since χ = ω (0 , d e Ξ( χ )) holds (second step), we can estimate the second term by R (cid:5) q (cid:0) χ ( s ) − ω ( γ h n ,m ( s ) , d e Ξ( χ m ( s ))) (cid:1) d s ≤ ( r ′ − r ) · (cid:5) q ∞ (cid:0) χ − ω ( γ h n ,m , d e Ξ( χ m )) (cid:1) = ( r ′ − r ) · (cid:5) q ∞ (cid:0) ω (0 , d e Ξ( χ )) − ω ( γ h n ,m , d e Ξ( χ m )) (cid:1) ≤ ( r ′ − r ) · (cid:5) q ∞ (cid:0) ω (0 , d e Ξ( χ )) − ω ( γ h n ,m , d e Ξ( χ )) (cid:1) + ( r ′ − r ) · (cid:5) q ∞ (cid:0) ω ( γ h n ,m , d e Ξ( χ − χ m )) (cid:1) . (60)31 Since im[ χ ] is compact, increasing ℓ ε if necessary, by (57), we can achieve that the fourth line in(60) is bounded by ε/ n, m ≥ ℓ ε . • To estimate the last line in (60), we choose q ≤ m ∈ P as in (15); and increase ℓ ε in such a way(use (57)) that m ∞ ( γ h n ,m ) ≤ n, m ≥ ℓ ε ; thus, (cid:5) q ∞ (cid:0) ω ( γ h n ,m , d e Ξ( χ − χ m )) (cid:1) (15) ≤ (cid:5) m ∞ ( χ − χ m ) . Then, it is clear from a) that for ℓ ′ ε ≥ ℓ ε suitably large, the last line in (60) is bounded by ε/ m, n ≥ ℓ ′ ε .We thus have ∆ n ≤ ε for each n ≥ ℓ ′ ε ∈ N ; which shows lim n ∆ n = 0. We recall that a Hausdorff locally convex vector space is said to be metrizable if it admits a metricthat generates the topology thereon. We furthermore recall that G is said to be C k -regular if G is C k -semiregular such that evol k is smooth w.r.t. the C k -topology.After this paper had been put on the arXiv, the author’s attention was drawn by Gl¨ockner andSchmeding to the fact that in metrizable locally convex vector spaces, convergence of a sequenceimplies its Mackey convergence (and vice versa). Specifically, it was argued that the following tworesults will hold: Lemma 19.
Suppose that g is metrizable; and let k ∈ N ⊔{ lip , ∞ , c } . Then, the following conditionsare equivalent:i) G is C k -continuous.ii) G is sequentially k -continuous.iii) G is Mackey k -continuous. Corollary 7.
Suppose that g is a Fr´echet space; and let k ∈ N ⊔ {∞} . Then, G is C k -regular ifand only if G is C k -semiregular.Proof. The one implication is evident. Suppose thus that G is C k -semiregular. Then, G is Mackeyk-continuous by Theorem 1; so that evol k is C k -continuous by Lemma 19. Since g is complete(thus, integral complete and Mackey complete), Theorem 4 in [7] shows that evol k is smooth, i.e.,that G is C k -regular.The rest of this section is dedicated to a selfcontained proof of Lemma 19. Some Standard Facts:
Let F be a Hausdorff locally convex vector space, with system of continuous seminorms Q . Asubsystem H ⊆ Q is said to be a fundamental system if { B h , ε (0) } h ∈ H ,ε> is a local base of zero in F . We recall that Lemma 20.
Let H ⊆ Q be a fundamental system, and S ⊆ Q a subsystem. Then, the followingstatements are equivalent: S is a fundamental system. To each h ∈ H , there exist c > and s ∈ S with h ≤ c · s .Proof. If S is a fundamental system, then follows from Proposition 22.6 in [14] when applied tothe identity id F . Suppose thus that holds; and let V ⊆ F be open with 0 ∈ V . We choose h ∈ H with B h ,ε (0) ⊆ V , fix c > s ∈ S with h ≤ c · s ; and observe that B s , εc (0) ⊆ B h ,ε (0) ⊆ V holds.Since B s , εc (0) ⊆ F is open, follows. 32 emma 21. The following statements are equivalent: F is metrizable. There exists a countable fundamental system { q [ m ] | m ∈ N } ⊆ Q . There exists { q [ m ] | m ∈ N } ⊆ Q as in with q [ m ] ≤ q [ m + 1] for each m ∈ N .Proof. The equivalence of and is covered by Proposition 25.1 in [14]. It is furthermore clearthat implies . Let thus { q [ m ] | m ∈ N } ⊆ Q be as in ; and define S := { o [ m ] ≡ q [0] + . . . + q [ m ] | m ∈ N } ⊆ Q . Since q [ m ] ≤ o [ m ] holds for each m ∈ N , Lemma 20 shows that S is a fundamental system; whichestablishes .Let H ⊆ Q be a fundamental system. We write { X n } n ∈ N ⇀ m X for { X n } n ∈ N ⊆ F and X ∈ F if h ( X − X n ) ≤ c h · λ n ∀ n ≥ l h , h ∈ H (61)holds for certain { c h } h ∈ H ⊆ R ≥ , { l h } h ∈ H ⊆ N , and R ≥ ⊇ { λ n } n ∈ N → Remark 10.
It is immediate from Lemma 20 that the definition made in (61) does not depend onthe explicit choice of the fundamental system H . ‡ We obtain
Lemma 22.
Suppose that F is metrizable; and let { X n } n ∈ N ⊆ F be a sequence with { X n } n ∈ N → X ∈ F . Then, we have { X n } n ∈ N ⇀ m X .Proof. Although this statement is well known from the literature (cf., e.g., 4. Proposition in Sect.10.1 in [10]), for completeness reasons, we provide an elementary proof that is adapted to ourparticular formulation of Mackey convergence, cf. Appendix A.6.We recall that the C k -topology on F k := C k ([0 , , F ) for k ∈ N ⊔ { lip , ∞ , c } is the Hausdorff locallyconvex topology that is generated by the seminorms H k := { q s ∞ | q ∈ Q , s (cid:22) k } (cf. Sect. 2.1.1).Since H k is a fundamental system, the definition made in (61) coincides with the definition madein (21). We furthermore recall that Lemma 23. If F is metrizable, then C k ([0 , , F ) is metrizable for each k ∈ N ⊔ { lip , ∞ , c } .Proof. Confer, e.g., Appendix A.7.
The Proof of Lemma 19:
We obtain from Lemma 21 and Lemma 23:
Corollary 8.
Suppose that g is metrizable; and let k ∈ N ⊔ { lip , ∞ , c } . Then, G is sequentially k -continuous if and only if G is Mackey k -continuous.Proof. Let { φ n } n ∈ N ⊆ D k , and φ ∈ D k be given. • Evidently, { φ n } n ∈ N ⇀ m . k φ implies { φ n } n ∈ N ⇀ s . k φ ; so that G is Mackey k-continuous if G issequentially k-continuous. • By Lemma 23, C k ([0 , , g ) is metrizable. Lemma 22 thus shows that { φ n } n ∈ N ⇀ s . k φ implies { φ n } n ∈ N ⇀ m . k φ . Consequently, G is sequentially k-continuous if G is Mackey k-continuous.We are ready for the Proof of Lemma 19.
The equivalence of ii) and iii) is covered by Corollary 8. Moreover, sinceLemma 23 shows that C k ([0 , , g ) is metrizable (thus, first countable), the equivalence of i) and ii)is clear from Remark 7.5) as well as Remark 7.6).33 PPENDIXA Appendix
A.1 Bastiani’s Differential Calculus
In this Appendix, we recall the differential calculus from [2, 6, 15, 18], cf. also Sect. 3.3.1 in [7].Let E and F be Hausdorff locally convex vector spaces. A map f : U → E , with U ⊆ F open, issaid to be differentiable at x ∈ U if( D v f )( x ) := lim h → /h · ( f ( x + h · v ) − f ( x )) ∈ E exists for each v ∈ F . Then, f is said to be differentiable if it is differentiable at each x ∈ U . Moregenerally, f is said to be k -times differentiable for k ≥ D v k ,...,v f ≡ D v k ( D v k − ( . . . ( D v ( f )) . . . )) : U → E is well defined for each v , . . . , v k ∈ F – implicitly meaning that f is p -times differentiable for each1 ≤ p ≤ k . In this case, we defined px f ( v , . . . , v p ) ≡ d p f ( x, v , . . . , v p ) := D v p ,...,v f ( x ) ∀ x ∈ U, v , . . . , v p ∈ F for p = 1 , . . . , k ; and let d f ≡ d f , as well as d x f ≡ d x f for each x ∈ U . Then, f is said to be • of class C if it is continuous – In this case, we let d f ≡ f . • of class C k for k ≥ k -times differentiable, such thatd p f : U × F p → E, ( x, v , . . . , v p ) D v p ,...,v f ( x )is continuous for each p = 0 , . . . , k . In this case, d px f is symmetric and p -multilinear for each x ∈ U and p = 1 , . . . , k , cf. [2]. • of class C ∞ if it is of class C k for each k ∈ N .We have the following differentiation rules [2]: a) A map f : F ⊇ U → E is of class C k for k ≥ f is of class C k − when considered as a map F ′ ⊇ U ′ → E for F ′ ≡ F × F and U ′ ≡ U × F . b) If f : U → F is linear and continuous, then f is smooth; with d x f = f for each x ∈ E , as wellas d k f = 0 for each k ≥ c) Suppose that f : F ⊇ U → U ′ ⊆ F ′ and f ′ : F ′ ⊇ U ′ → F ′′ are of class C k for k ≥
1, forHausdorff locally convex vector spaces
F, F ′ , F ′′ . Then, f ′ ◦ f : U → F ′′ is of class C k withd x ( f ′ ◦ f ) = d f ( x ) f ′ ◦ d x f ∀ x ∈ U. d) Let F , . . . , F m , E be Hausdorff locally convex vector spaces, and let f : F × . . . × F m ⊇ U → E be of class C . Then, f is of class C if and only if for p = 1 , . . . , m , the “partial derivative” ∂ p f : U × F p ∋ (( x , . . . , x m ) , v p ) lim h → /h · ( f ( x , . . . , x p + h · v p , . . . , x m ) − f ( x , . . . , x m ))exists in E , and is continuous. In this case, we haved ( x ,...,x m ) f ( v , . . . , v m ) = P mp =1 ∂ p f (( x , . . . , x m ) , v p )= P mp =1 d f (( x , . . . , x m ) , (0 , . . . , , v p , , . . . , x , . . . , x m ) ∈ U , and v p ∈ F p for p = 1 , . . . , m .34 .2 Proof of Lemma 7 In this appendix, we prove
Lemma 7.
Let [ r, r ′ ] ∈ K , and φ ∈ D [ r,r ′ ] be fixed. Then, for each p ∈ P , there exists some p ≤ q ∈ P with (cid:5) p lip ∞ (cid:0) Ad [ R • r φ ] − ( ψ ) (cid:1) ≤ (cid:5) q lip ∞ ( ψ ) ∀ ψ ∈ C lip ([ r, r ′ ] , g ) . Proof.
By definition, there exists some µ ∈ C ( I, G ), for I an open interval containing [ r, r ′ ], with δ r ( µ ) | [ r,r ′ ] = φ and µ ( r ) = e . We now have to show that C lip ([ r, r ′ ] , g ) ∋ χ : [ r, r ′ ] ∋ t Ad µ − ( t ) ( ψ ( t ))holds, for each fixed ψ ∈ C lip ([ r, r ′ ] , g ). For this, we let p ∈ P be fixed; and obtain (cid:5) p ( χ ( t ) − χ ( t ′ )) ≤ (cid:5) p (cid:0) Ad µ − ( t ) ( ψ ( t ) − ψ ( t ′ )) (cid:1) + (cid:5) p (cid:0)(cid:0) Ad µ − ( t ) − Ad µ − ( t ′ ) (cid:1) ( ψ ( t ′ )) (cid:1) . (62) • We let C := im[ µ − ], choose p ≤ w ∈ P as in I) for v ≡ p there; and obtain (cid:5) p ( χ ( t )) ≤ (cid:5) w ( ψ ( t )) ∀ t ∈ [ r, r ′ ] , (63) (cid:5) p (cid:0) Ad µ − ( t ) ( ψ ( t ) − ψ ( t ′ )) (cid:1) ≤ (cid:5) w ( ψ ( t ) − ψ ( t ′ )) ≤ Lip( (cid:5) w , ψ ) · | t − t ′ | ∀ t, t ′ ∈ [ r, r ′ ] . (64) • The map α : I × g ∋ ( s, X ) → ∂ s Ad µ − ( s ) ( X ) is well defined, continuous, and linear in the secondargument. By Lemma 2 applied to K ≡ C, there thus exists some p ≤ m ∈ P with( (cid:5) p ◦ α )( s, X ) ≤ (cid:5) m ( X ) ∀ s ∈ [ r, r ′ ] , X ∈ g . Then, we obtain from (8) that (cid:5) p (cid:0)(cid:0) Ad µ − ( t ) − Ad µ − ( t ′ ) (cid:1) ( ψ ( t ′ )) (cid:1) ≤ R tt ′ (cid:5) p (cid:0) ∂ s Ad µ − ( s ) ( ψ ( t ′ )) (cid:1) d s = R tt ′ ( (cid:5) p ◦ α )( s, ψ ( t ′ )) d s ≤ (cid:5) m ∞ ( ψ ) · | t − t ′ | (65)holds, for each t, t ′ ∈ [ r, r ′ ] with t ′ ≤ t .We choose q ∈ P with q ≥ · max( m , w ) (i.e., p , m , w ≤ q ); and obtain (cid:5) p ∞ ( χ ) (63) ≤ (cid:5) w ∞ ( ψ ) ≤ (cid:5) q ∞ ( ψ ) . (66)We furthermore obtain from (62), (64), (65) that (cid:5) p ( χ ( t ) − χ ( t ′ )) ≤ Lip( (cid:5) w , ψ ) · | t − t ′ | + (cid:5) m ∞ ( ψ ) · | t − t ′ | ≤ (cid:5) q lip ∞ ( ψ ) · | t − t ′ | holds for each t, t ′ ∈ [ r, r ′ ]; thus,Lip( (cid:5) p , χ ) ≤ (cid:5) q lip ∞ ( ψ ) (66) = ⇒ (cid:5) p lip ∞ ( χ ) ≤ (cid:5) q lip ∞ ( ψ ) , which proves the claim. 35 .3 Proof of Equation (38) In this appendix, we showLip( (cid:5) p , φ | [ t n +1 , ) ≤ · C [ ρ , · max (cid:0) δ − · (cid:5) p lip ∞ ( φ ) , . . . , δ − n · (cid:5) p lip ∞ ( φ n ) (cid:1) . (38) Proof of Equation (38) . We let ϕ n := ρ n · ( φ n ◦ ̺ n ) for each n ∈ N ; so thatLip( (cid:5) p , ϕ n ) ≤ · δ − n · C [ ρ , · (cid:5) p lip ∞ ( φ n )holds by (35). Then, for t, t ′ ∈ [ t ℓ +1 , t ℓ ] with ℓ ∈ N , we have (cid:5) p ( φ ( t ) − φ ( t ′ )) = (cid:5) p ( ϕ ℓ ( t ) − ϕ ℓ ( t ′ )) ≤ Lip( (cid:5) p , ϕ ℓ ) · | t − t ′ |≤ · C [ ρ , · δ − ℓ · (cid:5) p lip ∞ ( φ ℓ ) · | t − t ′ | . (67)Moreover, for t ∈ [ t ( ℓ +1)+ m , t ℓ + m ] and t ′ ∈ [ t ℓ +1 , t ℓ ], with m ≥ ℓ ∈ N , we have (cid:5) p ( φ ( t ) − φ ( t ′ )) ≤ (cid:5) p ( φ ( t ) − φ ( t ℓ + m ))+ P k = m − (cid:5) p ( φ ( t ( ℓ +1)+ k ) − φ ( t ℓ + k ))+ (cid:5) p ( φ ( t ℓ +1 ) − φ ( t ′ )) ≤ (cid:5) p ( ϕ ℓ + m ( t ) − ϕ ℓ + m ( t ℓ + m ))+ P k = m − (cid:5) p ( ϕ ℓ + k ( t ( ℓ +1)+ k ) − ϕ ℓ + k ( t ℓ + k ))+ (cid:5) p ( ϕ ℓ ( t ℓ +1 ) − ϕ ℓ ( t ′ )) ≤ Lip( (cid:5) p , ϕ ℓ + m ) · | t − t ℓ + m | + P k = m − Lip( (cid:5) p , ϕ ℓ + k ) · | t ( ℓ +1)+ k − t ℓ + k | + Lip( (cid:5) p , ϕ ℓ ) · | t ℓ +1 − t ′ |≤ · δ − ℓ + m · C [ ρ , · (cid:5) p lip ∞ ( φ ℓ + m ) · | t − t ℓ + m | + P k = m − · δ − ℓ + k · C [ ρ , · (cid:5) p lip ∞ ( φ ℓ + k ) · | t ( ℓ +1)+ k − t ℓ + k | + 2 · δ − ℓ · C [ ρ , · (cid:5) p lip ∞ ( φ ℓ ) · | t ℓ +1 − t ′ |≤ · C [ ρ , · max ≤ k ≤ m (cid:0) δ − ℓ + k · (cid:5) p lip ∞ ( φ ℓ + k ) (cid:1) · | t − t ′ | . (68)Combining (67) with (68), we obtain (38). A.4 Proof of Lemma 18
In this appendix, we prove
Lemma 18.
Suppose that G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞ , c } . Suppose furthermorethat we are given D k [ r,r ′ ] ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k [ r,r ′ ] as well as D k [ r,r ′ ] ⊇ { ψ α } α ∈ I ⇀ n . ψ ∈ D k [ r,r ′ ] for [ r, r ′ ] ∈ K , such that the expressions ξ ( φ, ψ ) := d e L R φ (cid:0) R Ad [ R sr φ ] − ( ψ ( s )) d s (cid:1) ξ ( φ n , ψ α ) := d e L R φ n (cid:0) R Ad [ R sr φ n ] − ( ψ α ( s )) d s (cid:1) ∀ n ∈ N are well defined; i.e., such that the occurring Riemann integrals exist in g . Then, we have lim ( n,α ) ξ ( φ n , ψ α ) = ξ ( φ, ψ ) . (69)36or this, we first show the following analogue to Lemma 41 in [7]. Lemma 24.
Suppose that G is Mackey k -continuous for k ∈ N ⊔ { lip , ∞ , c } ; and let [ r, r ′ ] ∈ K befixed. Let Γ : G × g → g be continuous; and define b Γ : D k [ r,r ′ ] × C k ([ r, r ′ ] , g ) → g , ( φ, ψ ) R Γ (cid:0) R sr φ, ψ ( s ) (cid:1) d s. Then, for each sequence D k [ r,r ′ ] ⊇ { φ n } n ∈ N ⇀ m . k φ ∈ D k [ r,r ′ ] and each net C k ([ r, r ′ ] , g ) ⊇ { ψ α } α ∈ I ⇀ n . ψ ∈ C k ([ r, r ′ ] , g ) , we have lim ( n,α ) b Γ( φ n , ψ α ) = b Γ( φ, ψ ) . Proof.
By (9), it suffices to show that for e Γ : D k [ r,r ′ ] × C k ([ r, r ′ ] , g ) → C ([ r, r ′ ] , g ) , ( φ, ψ ) (cid:2) t Γ (cid:0) R tr φ, ψ ( t ) (cid:1)(cid:3) , we have lim ( n,α ) e Γ( φ n , ψ α ) = e Γ( φ, ψ ) w.r.t. the C -topology; i.e., that for p ∈ P and ε > N ε ∈ N and α ε ∈ I with (cid:5) p ∞ (cid:0)e Γ( φ n , ψ α ) − e Γ( φ, ψ ) (cid:1) < ε ∀ n ≥ N ε , α ≥ α ε . (70)For this, we let µ := R • r φ , and consider the continuous map α : G × g × G × g → g , (( g, X ) , ( g ′ , X ′ )) (cid:5) p (Γ( g, X ) − Γ( g ′ , X ′ )) . Then, for t ∈ [ r, r ′ ] fixed, there exists an open neighbourhood W [ t ] ⊆ G of e , as well as U [ t ] ⊆ g open with 0 ∈ U [ t ], such that α (( g, X ) , ( g ′ , X ′ )) < ε ∀ ( g, X ) , ( g ′ , X ′ ) ∈ (cid:2) µ ( t ) · W [ t ] (cid:3) × (cid:2) ψ ( t ) + U [ t ] (cid:3) (71)holds. We choose a ) V [ t ] ⊆ G open with e ∈ V [ t ] and V [ t ] · V [ t ] ⊆ W [ t ]. b ) O [ t ] ⊆ g open with 0 ∈ O [ t ] and O [ t ] + O [ t ] ⊆ U [ t ]. c ) J [ t ] ⊆ R open with t ∈ J , such that for D [ t ] := J [ t ] ∩ [ r, r ′ ], we have µ ( D [ t ]) ⊆ µ ( t ) · V [ t ] ⊆ µ ( t ) · ⊆ W [ t ] and ψ ( D [ t ]) ⊆ ψ ( t ) + O [ t ] ⊆ ψ ( t ) + U ( t ) . (72)Since [ r, r ′ ] is compact, there exist t , . . . , t n ∈ [ r, r ′ ], such that [ r, r ′ ] ⊆ D ∪ . . . ∪ D n holds. • We define V := V [ t ] ∩ . . . ∩ V [ t n ].Since G is Mackey k-continuous, there exists some N ε ∈ N with R • r φ n ∈ R • r φ · V ∀ n ≥ N ε . (73) • We define O := O [ t ] ∩ . . . ∩ O [ t n ].Since { ψ α } α ∈ I ⇀ n . ψ holds, there exists α ε ∈ I with( ψ α ( t ) − ψ ( t )) ∈ O ∀ t ∈ [ r, r ′ ] , α ≥ α ε . (74)Then, for τ ∈ D p with 0 ≤ p ≤ n , as well as n ≥ N ε and α ≥ α ε , we obtain from (73), (74), as wellas (72) for t ≡ t p there that • µ ( t p ) − · R τr φ n = (cid:0) µ ( t p ) − · µ ( τ ) (cid:1) · (cid:0) [ R τr φ ] − [ R τr φ n ] (cid:1) ∈ V · V ⊆ W [ t p ], • ψ α ( τ ) − ψ ( t p ) = ( ψ α ( τ ) − ψ ( τ )) + ( ψ ( τ ) − ψ ( t p )) ∈ O + O ⊆ U [ t p ].The claim is thus clear from (71) and (72). Proof of Lemma 18.
For each χ, χ ′ ∈ D k [ r,r ′ ] , we have, cf. (3) ξ ( χ, χ ′ ) = d ( R χ,e ) m (cid:0) , b Γ( χ, χ ′ ) (cid:1) for Γ ≡ Ad(inv( · ) , · );so that (69) holds by Lemma 24, because the Lie group multiplication is smooth.37 .5 Proof of Equation (56) In this appendix, we show R φ = exp (cid:0) R φ ( s ) d s (cid:1) . (56) Proof of Equation (56) . We fix I ≡ ( ι, ι ′ ) with ι < r < r ′ < ι ′ , define ψ ∈ C ([ ι, ι ′ ] , g ) by ψ | [ ι,r ) := φ ( r ) ψ | [ r,r ′ ] := φ ψ | ( r ′ ,ι ′ ] := φ ( r ′ );and observe that X : I → dom[exp] , x
7→ − ( r − ι ) · φ ( r ) + R xι ψ ( s ) d s fulfills the presumptions in Corollary 6, withexp (cid:0) R • r φ ( s ) d s (cid:1) = α | [ r,r ′ ] for α := exp ◦ X . By Corollary 6, we thus have α ∈ C ( I, G ), with δ r ( α )( x ) = dd h (cid:12)(cid:12) h =0 α ( x + h ) · α ( x ) − = dd h (cid:12)(cid:12) h =0 α ( x ) − · α ( x + h )= d exp( X ( x )) L exp( − X ( x )) (cid:0) dd h (cid:12)(cid:12) h =0 α ( x + h ) (cid:1) = (cid:0) d exp( X ( x )) L exp( − X ( x )) ◦ d e L exp( X ( x )) (cid:1)(cid:0) R Ad exp( − s · X ( x )) ( ˙ X ( x )) d s (cid:1) = R ˙ X ( x ) d s = ˙ X ( x ) = ψ ( x ) = φ ( x )for each x ∈ [ r, r ′ ]. Here, we have used in the second-, and the fifth step that G is abelian. A.6 Proof of Lemma 22
In this appendix, we prove
Lemma 22.
Suppose that F is metrizable; and let { X n } n ∈ N ⊆ F be a sequence with { X n } n ∈ N → X ∈ F . Then, we have { X n } n ∈ N ⇀ m X .Proof. We choose H := { q [ m ] | m ∈ N } ⊆ Q as in Lemma 21. ; and let l : N → N be strictlyincreasing with q [ m ]( X − X n ) ≤ m ∀ n ≥ l q [ m ] := l ( m ) , m ∈ N . (75) • We define λ n := m for each n ∈ N with l ( m ) ≤ n < l ( m + 1); and observe that lim n λ n = 0 holds. • For m, d, n ∈ N with l ( m + d ) ≤ n < l ( m + d + 1), we obtain q [ m ]( X − X n ) ≤ q [ m + d ]( X − X n ) (75) ≤ m + d = λ n . This shows that q [ m ]( X − X n ) ≤ λ n holds for each n ≥ l q [ m ] = l ( m ); thus, { X n } n ∈ N ⇀ m X . Recall the last statement in Sect. 2.2.3 for the fact that im[ X ] ⊆ dom[exp] holds. .7 Proof of Lemma 23 In this appendix, we prove
Lemma 23. If F is metrizable, then C k ([0 , , F ) is metrizable for each k ∈ N ⊔ { lip , ∞ , c } .Proof. Let { q [ m ] | m ∈ N } ⊆ Q be as in Lemma 21. . For each m ∈ N , we defines[lip , m ] := lip s[c , m ] := 0 s[ ∞ , m ] := m as well as s[ k, m ] := k ∀ k ∈ N . Moreover, for each k ∈ N ⊔ { lip , ∞ , c } , we let s [ k, m ] := q [ m ] s[ k,m ] ∞ ∀ m ∈ N as well as S k := { s [ k, m ] | m ∈ N } . Let now k ∈ N ⊔ { lip , ∞ , c } , q ∈ Q , s (cid:22) k be fixed. By Lemma 20, there exist c > ℓ ∈ N with q ≤ c · q [ ℓ ]. We define m := ( ℓ for k ∈ N ⊔ { lip , c } , max(s , ℓ ) for k = ∞ , observe that q ≤ c · q [ m ] as well as s ≤ s[ k, m ] holds; and obtain q s ∞ ≤ c · q [ m ] s[ k,m ] ∞ = c · s [ k, m ] . Then, Lemma 20 shows that S k is a fundamental system; and, since S k is countable, the claimfollows from Lemma 21. References [1] J. J. Duistermaat and J. A. C. Kolk,
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