On regularization of vector distributions on manifolds
aa r X i v : . [ m a t h . F A ] A p r On regularization of vector distributionson manifolds
E. A. Nigsch ∗ July 4, 2018
Abstract.
One can represent Schwartz distributions with values in a vectorbundle E by smooth sections of E with distributional coefficients. More-over, any linear continuous operator which maps E -valued distributions tosmooth sections of another vector bundle F can be represented by sections ofthe external tensor product E ∗ ⊠ F with coefficients in the space L ( D ′ , C ∞ )of operators from scalar distributions to scalar smooth functions. We estab-lish these isomorphisms topologically, i.e., in the category of locally convexmodules, using category theoretic formalism in conjunction with L. Schwartz’notion of ε -product. Keywords.
Vector valued distributions, Distributions on manifolds, Topo-logical tensor product, Regularization.
Primary 46T30; secondary46A32. ∗ Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. E-Mail:[email protected], Phone: +43 4277 50760
Our aim is to show, given any vector bundles E → M and F → N , theisomorphisms D ′ ( M, E ) ∼ = Γ( M, E ) ⊗ C ∞ ( M ) D ′ ( M ) ∼ = L C ∞ ( M ) (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) (1) L (cid:0) D ′ ( M, E ) , Γ( N, F ) (cid:1) ∼ = Γ( M × N, E ∗ ⊠ F ) ⊗ C ∞ ( M × N ) L (cid:0) D ′ ( M ) , C ∞ ( N ) (cid:1) ∼ = L C ∞ ( M × N ) (cid:0) Γ( M × N, E ⊠ F ∗ ) , L ( D ′ ( M ) , C ∞ ( N )) (cid:1) (2)in the category of locally convex modules (see Section 2 for notation).(1) is fundamental for the the extension of L. Schwartz’ theory of distribu-tions to the case of distributions on manifolds with values in vector bundles.In fact, it enables one to view distributional sections as smooth sectionswith distributional coefficients, and hence allows their description by localcoordinates. Naturally, it is desirable to establish that this is a topologicalisomorphism, for instance in order to obtain convergence of a sequence in D ′ ( M, E ) from convergence of its coordinates.The motivation to consider (2) comes from the field of nonlinear generalizedfunctions (or Colombeau algebras). Such algebras, containing distributionsas a vector subspace and smooth functions as a faithful subalgebra (whilsthaving optimal properties in light of the Schwartz impossibility result aboutmultiplication of distributions [23]), are commonly constructed by represent-ing Schwartz distributions by families of smooth functions, which amounts toregularizing them in a particular way (cf. [21]). Although this is straightfor-ward in the scalar case, the construction of a (diffeomorphism invariant) alge-bra of generalized tensor fields is considerably more complicated (cf. [9, 10]).The construction in [10] involves the ingredients Γ( M × M, E ∗ ⊠ E ) and L ( D ′ ( M ) , C ∞ ( M )) (the latter albeit only in disguise) in order to regularizedistributions in D ′ ( M, E ) in a coordinate-independent manner, but it wasnot exploited there that any linear continuous mapping D ′ ( M, E ) → Γ( M, E )necessarily is of the form exhibited by (2). The spaces of so-called smoothingoperators L ( D ′ ( M ) , C ∞ ( M )) (scalar case) and L ( D ′ ( M, E ) , Γ( M, E )) (vectorvalued case) were found to be the optimal starting points for a general, geo-metric construction of Colombeau algebras ([21]; cf. also [7, 9, 19]). Hence,it is desirable to obtain isomorphism (2) in a topological setting for two rea-sons: first, it allows to relate the construction of [10] to the new, more naturalapproach to Colombeau algebras given in [21]; and second, it allows to splitthe regularization of vector valued distributions into a smooth vectorial part2egularization of vector distributions July 4, 2018Γ( M × N, E ∗ ⊠ F ) and a regularizing part L ( D ′ ( M ) , C ∞ ( N )). This splittingis expected to be of essential practical importance in the further developmentof spaces of nonlinear generalized sections applicable to problems of nonlineardistributional geometry ([15, 16, 27]).It is evident that the above isomorphisms are straightforward to obtain onthe algebraic level by reduction to the trivial line bundles (see Section 4). Onthe topological level, however, they require the proper handling of topolo-gies on modules and their tensor products as well as related spaces of linearmappings. Here we draw on concepts of A. Grothendieck and L. Schwartzconcerning topological tensor products and the theory of vector valued dis-tributions ([11, 24, 25, 26]): first, we use the idea of endowing the tensorproduct H ⊗ K with a topology such that linear continuous mappings on itcorrespond to bilinear mappings on
H × K which are hypocontinuous withrespect to certain families of bounded sets; second, we employ the notionof ε -product; and third, a key element of our proof (Lemma 5 below) maybe regarded as an application of L. Schwartz’ Th´eor`eme de croisement , acornerstone of his theory of vector valued distributions.We remark that a version of (1) in the bornological setting (i.e., using thebornological tensor product) was obtained in [20].
Let the field K be fixed as R or C throughout. All locally convex spaces willbe assumed to be Hausdorff and over K . For two locally convex spaces H and K we denote by L ( H , K ) the space of continuous linear mappings from H to K . Endowed with the topology of simple (or pointwise) convergence it will bedenoted by L σ ( H , K ) and if it carries the topology of bounded convergenceby L β ( H , K ) ([22, Chapter III, §
3, p. 81]).By
H ⊗ λ K for λ ∈ { β, ι } we denote the algebraic tensor product H ⊗ K endowed with the finest locally convex topology such that the canonical map-ping ⊗ : H × K → H ⊗ K is λ -continuous, which means separately continuousin case λ = ι and hypocontinuous in case λ = β (cf. [26, p. 10]); when we sayhypocontinuous, if not specified otherwise we always mean this with respectto the families of bounded subsets of the respective spaces. There would bemore possible choices for λ but we will not need these here. Note that inboth cases H ⊗ λ K is Hausdorff. H ⊗ λ K is the unique locally convex space (up to isomorphism) with the3egularization of vector distributions July 4, 2018following universal property: for each λ -continuous bilinear mapping from H × K into any locally convex space M there exists a unique continuousmapping ˜ f : H ⊗ λ K → M such that f = ˜ f ◦ ⊗ . This correspondence definesa linear isomorphism between the vector spaces of all λ -continuous bilinearmappings H × K → M and L ( H ⊗ λ K , M ) ([26, p. 10]).Let H i , K i , M i be locally convex spaces and f i ∈ L ( H i , K i ) for i = 1 , f ⊗ f ∈ L ( H ⊗ λ H , K ⊗ λ K ) for λ ∈ { β, ι } ([26, p. 14]) and for g i ∈ L ( K i , M i ) ( i = 1 ,
2) we have ( g ⊗ g ) ◦ ( f ⊗ f ) = ( g ◦ f ) ⊗ ( g ◦ f ),which turns ⊗ λ into a functor LCS × LCS → LCS (see below).All manifolds will be assumed to be smooth, Hausdorff, second countableand finite dimensional. Given a manifold M and a vector bundle E → M we denote by C ∞ ( M ), C ∞ c ( M ), Γ( M, E ) and Γ c ( M, E ) the spaces ofsmooth functions M → K , compactly supported smooth functions M → K , smooth section of E and compactly supported smooth sections of E ,respectively. C ∞ ( M ) and Γ( M, E ) carry their usual Fr´echet topology ([5,Chapter XVII, Section 2, p. 236]) and Γ c ( M, E ) the corresponding (LF)-topology. Writing Vol( M ) for the volume bundle of M ([17, Chapter 16,p. 429]) and E ∗ for the dual bundle of E , the spaces of scalar and E -valueddistributions on M are defined as the dual spaces D ′ ( M ) := Γ c (cid:0) M, Vol( M ) (cid:1) ′ and D ′ ( M, E ) := Γ c (cid:0) M, E ∗ ⊗ Vol( M ) (cid:1) ′ , respectively, both endowed with thestrong dual topology ([8, Definition 3.1.4, p. 231]). Finally, E ⊠ F denotesthe external tensor product of two vector bundles E and F ([6, Chapter II,Problem 4, p. 84]).We will employ some notions from category theory, using [2, 3] for generalbackground reference. Given a category C and any two of its objects, A and B , the set of morphisms from A to B will be denoted by C ( A, B ). Wewill employ the following categories: VB M , the category of smooth vectorbundles over a fixed manifold M with morphisms given by smooth vectorbundle homomorphisms covering the identity mapping of M ; LCS , the cat-egory of locally convex spaces with morphisms given by continuous linearmappings; A − Mod , the category of A -modules with morphisms given by A -linear mappings; and A − LCMod , the category of locally convex A -moduleswith morphisms given by A -linear continuous mappings, as defined in Section3.We will need certain functors to commute with coproducts. This will beobtained very easily in our setting because the categories and functors weare dealing with are additive . We recall the relevant definitions from [3]: a preadditive category is a category C together with an abelian group struc-4egularization of vector distributions July 4, 2018ture on each set C ( A, B ) of morphisms such that the composition mappings C ( A, B ) × C ( B, C ) → C ( A, C ), ( f, g ) g ◦ f are group homomorphisms ineach variable. A convenient feature of preadditive categories is that finitecoproducts and finite products are the same objects ([3, Proposition 1.2.4,p. 4]) and hence are called biproducts . An additive category then is a pread-ditive category with a zero object and such that all finite biproducts exist.A functor F : A → B between two preadditive categories is called additive iffor all objects A, A ′ in A , the mapping F : A ( A, A ′ ) → B (cid:0) F ( A ) , F ( A ′ ) (cid:1) , f F ( f )is a group homomorphism. Most importantly, a functor is additive if andonly if it preserves biproducts ([3, Proposition 1.3.4, p. 9]).Concerning our setting it is easy to see that the categories VB M and LCS areadditive; moreover, the functors ∗ : VB M → VB M (dual bundle), ⊗ : VB M × VB M → VB M (tensor product of vector bundles) and ⊠ : VB M × VB N → VB M × N (external tensor product) as well as ⊗ λ : LCS × LCS → LCS areadditive. We omit the detailed proofs here because they amount to routineverification of well-known properties.
In this sections we are going to recall some needed definitions and propertiesof locally convex algebras as well as locally convex modules and their tensorproducts (see [28, 12, 18] for additional information).A locally convex algebra A is a locally convex space together with a sepa-rately continuous multiplication A × A → A turning it into an associativecommutative unitary algebra over K . Given a locally convex algebra A , a lo-cally convex A -module H is a locally convex space which is an A -module suchthat module multiplication A × H → H is separately continuous. For fixed A , the locally convex A -modules whose multiplication is λ -continuous (with λ ∈ { ι, β } as before) are the objects of an additive category A − LCMod − λ whose morphisms are continuous A -linear mappings and whose biproductsare formed in LCS .We simply write A − LCMod instead of A − LCMod − ι .Given a locally convex algebra A and two locally convex A -modules H and K we denote by L A ( H , K ) the space of all continuous A -linear mappings from H to K . Endowed with the topology of simple or bounded convergence (i.e.,the trace topology with respect to L σ ( H , K ) or L β ( H , K ), respectively) wedenote it by by L A,σ ( H , K ) or L A,β ( H , K ), respectively.5egularization of vector distributions July 4, 2018 L A,σ ( H , K ) with its canonical A -module structure is a locally convex A -module.Moreover, if K has β -continuous multiplication then L A,β ( H , K ) has β -continuous multiplication as well. In fact, fix a 0-neighbourhood U B,V = { ℓ : ℓ ( B ) ⊆ V } in L A,β ( H , K ) where B ⊆ H is bounded and V ⊆ K is a0-neighborhood. Given a bounded subset A ′ ⊆ A , choose a 0-neighborhood V ′ ⊆ K such that A ′ · V ′ ⊆ V ; then U B,V ′ is a 0-neighborhood in L A,β ( H , K )such that A ′ · U B,V ′ ⊆ U B,V . On the other hand, given a bounded subset L ⊆L A,β ( H , K ) we can find a 0-neighborhood W ⊆ A such that W · L ( B ) ⊆ V (note that L ( B ) is bounded); this means that W · L ⊆ U B,V , which provesthe claim.Moreover, L A,σ ( − , − ) is an additive functor A − LCMod × A − LCMod → A − LCMod which is contravariant in the first argument and covariant inthe second argument. Similarly, L A,β is an additive functor A − LCMod × A − LCMod − β → A − LCMod − β . Lemma 1.
Given a locally convex algebra A and a locally convex A -module H , H ∼ = L A,σ ( A, H ) in A − LCMod . If H has hypocontinuous multiplicationthen H ∼ = L A,β ( A, H ) in A − LCMod − β .Proof. Algebraically, the isomorphism ϕ : H → L A ( A, H ) is given by ϕ ( x )( a ) := a · x with inverse ϕ − ( l ) := l (1). Continuity of ϕ and ϕ − is clear in bothcases.Finally, we note that Γ , Γ c : VB M → C ∞ ( M ) − LCMod − β (sections andcompactly supported sections) are additive functors.Following [4] we will now give the construction of the tensor product of locallyconvex modules. Let A be a locally convex algebra and H , K locally convex A -modules. Define J as the sub- Z -module of H ⊗ K (the tensor product over K ) generated by all elements of the form ma ⊗ n − m ⊗ an with a ∈ A , m ∈ H and n ∈ K . The vector spaces H ⊗ A K and ( H ⊗ K ) /J are isomorphic ([4,Theorem I.5.1, p. 9]). Noting that the closure J again is a sub- Z -module of H ⊗ λ K , we have a locally convex space H ⊗
A,λ K := ( H ⊗ λ K ) /J . Denotingby q : H ⊗ λ K → ( H ⊗ λ K ) /J the quotient mapping we obtain a bilinearmapping ⊗ A,λ := q ◦ ⊗ : H × K → H ⊗
A,λ K .We call H ⊗
A,λ K the λ -tensor product of H and K over A . It is a locallyconvex A -module, but in general its multiplication m : A × H ⊗ A,λ
K →H ⊗
A,λ K is only separately continuous. In fact, for given a ∈ A we define themapping m a : H⊗ A,λ
K → H⊗
A,λ K as the tensor product of the multiplicationmapping x ax on H and the identity on K , which both are continuous.6egularization of vector distributions July 4, 2018We then set m ( a, z ) := m a ( z ), which is continuous in z . For continuity in a ,given z = q ( P x i ⊗ y i ) ∈ H ⊗ A,λ K with x i ∈ H and y i ∈ K , the value m ( a, z )is given by q ( P ax i ⊗ y i ) which obviously is continuous in a .The λ -tensor product of locally convex modules has the following universalproperty. Proposition 2.
Let A be a locally convex algebra and H , K , M locally convex A -modules. Then given any λ -continuous A -bilinear mapping f : H×K → M there exists a unique continuous A -linear mapping g : H ⊗
A,λ
K → M suchthat f = g ◦ ⊗ A,λ . Conversely, given any g ∈ L A ( H ⊗
A,λ K , M ) the mapping g ◦ ⊗ A,λ is λ -continuous and A -bilinear from H × K to M .This correspondence gives a vector space isomorphism between the space ofall λ -continuous A -bilinear mappings H × K → M and the space L A ( H ⊗
A,λ K , M ) .Proof. To given f as in the statement there corresponds a continuous map-ping ˜ f : H ⊗ λ K → M ; moreover, J ⊆ ker ˜ f and by continuity of ˜ f , also J ⊆ ker ˜ f . Hence, there exists a continuous mapping g : H ⊗
A,λ
K → M such that f = g ◦ q ◦ ⊗ = g ◦ ⊗ A,λ . The converse is obvious.Let locally convex A -modules H i , K i , M i and A -linear continuous mappings f i ∈ L A ( H i , K i ) be given for i = 1 ,
2. With p, q, r the respective quotientmappings, the continuous mapping q ◦ ( f ⊗ f ) ∈ L ( H ⊗ λ H , K ⊗ A,λ K )vanishes on the kernel of p , hence induces the continuous A -linear mapping f ⊗ A,λ f ∈ L A ( H ⊗ A,λ H , K ⊗ A,λ K ). For g i ∈ L a ( K i , M i ) ( i = 1 ,
2) wehave ( g ◦ g ) ⊗ A,λ ( f ◦ f ) = ( g ⊗ A,λ g ) ◦ ( f ⊗ A,λ f ), as is easily seen fromthe following diagram: H ⊗ λ H g ◦ f ) ⊗ ( g ◦ f ) & & f ⊗ f (cid:15) (cid:15) p / / H ⊗ A,λ H f ⊗ A,λ f (cid:15) (cid:15) ( g ◦ g ) ⊗ A,λ ( f ◦ f ) x x K ⊗ λ K g ⊗ g (cid:15) (cid:15) q / / K ⊗ A,λ K g ⊗ A,λ g (cid:15) (cid:15) M ⊗ λ M r / / M ⊗ A,λ M In fact we only need to use that p is a quotient mapping and the wholediagram except for the part in question is commutes.This makes ⊗ A,λ a functor A − LCMod × A − LCMod → A − LCMod whichobviously is additive. 7egularization of vector distributions July 4, 2018Given a locally convex algebra A and a locally convex A -module H with λ -continuous multiplication, we have a canonical continuous A -linear mapping A ⊗ A,λ
H → H whose inverse is given by m ⊗ A,λ m ; this defines anisomorphism A ⊗ A,λ
H ∼ = H in A − LCMod . (3)The following Lemma will be needed in Section 6. Lemma 3.
Let A be a barrelled locally convex algebra. Then for locally con-vex A -modules H , K at least one of which has hypocontinuous multiplication, M ⊗ A,β N is a locally convex A -module with hypocontinuous multiplication.In other words, we have an additive functor ⊗ A,β : A − LCMod − β × A − LCMod → A − LCMod − β. Proof.
Suppose that H has hypocontinuous multiplication f : A × H → K (the case in which K does is similar). H ⊗
A,β K has separately continuousmultiplication m : A × H ⊗ A,β
K → H ⊗
A,β K given by m ( a, x ) := ( f a ⊗ A,β id)( x ), where f a := f ( a, . ) ∈ L ( H , H ). As A is barrelled, m is hypocontinuouswith respect to bounded subsets of H ⊗
A,β K ([14, § A ′ ⊆ A the set { f a ⊗ A,β id | a ∈ A ′ } ⊆ L ( H ⊗
A,β K , H ⊗
A,β K )is equicontinuous. For this it suffices ([26, p. 11]) to know that { f a | a ∈ A ′ } is equicontinuous, which holds by assumption. We will now describe the general (classical) principle at work behind theproofs of isomorphisms (1) and (2). Given a manifold M , consider two covari-ant functors T, T ′ from VB M into any additive subcategory of C ∞ ( M ) − Mod .Suppose we have a natural transformation ν : T → T ′ , i.e., for each vectorbundle E there is a morphism ν E : T ( E ) → T ′ ( E ) such that for each vectorbundle homomorphism µ : E → E ′ covering the identity of M the followingdiagram commutes: T ( E ) ν E / / T ( µ ) (cid:15) (cid:15) T ′ ( E ) T ′ ( µ ) (cid:15) (cid:15) T ( E ′ ) ν E ′ / / T ′ ( E ′ ) . (4)8egularization of vector distributions July 4, 2018Suppose we want to show that ν E is a monomorphism, epimorphism or iso-morphism for all E . We will show that this can easily be reduced to the caseof the trivial line bundle E = M × K if T and T ′ are additive functors.For this it is essential that for every vector bundle E there exists a vectorbundle F such that E ⊕ F is trivial ([6, Section 2.23, Theorem I, p. 76]).Denoting by ι E : E → E ⊕ F and π E : E ⊕ F → E the canonical injectionand projection, respectively, we obtain the diagram T ( E ) ν E / / T ( ι E ) (cid:15) (cid:15) T ′ ( E ) T ′ ( ι E ) (cid:15) (cid:15) T ( E ⊕ F ) ν E ⊕ F / / T ( π E ) (cid:15) (cid:15) T ′ ( E ⊕ F ) T ′ ( π E ) (cid:15) (cid:15) T ( E ) ν E / / T ′ ( E )which commutes because ν is natural. We see that if ν E ⊕ F is a monomor-phism, epimorphism or isomorphism, ν E has the same property; in the lastcase the inverse of ν E is given by T ( π E ) ◦ ν − E ⊕ F ◦ T ′ ( ι E ).This way, the problem is reduced to the case where E is a trivial vectorbundle. But then it is of the form ( M × K ) ( n ) , i.e., the direct sum of n copiesof the trivial line bundle, and by the same reasoning as before the discussionis reduced to the case where E = M × K .Hence, the main work lies in showing that T , T ′ and ν have the desiredproperties, which is easy as soon as the respective categories are identifiedand a candidate for ν is found.As an example, we have the isomorphism in C ∞ ( M ) − LCMod ψ E,F : Γ(
M, E ) ⊗ C ∞ ( M ) ,ι Γ c ( M, F ) → Γ c ( M, E ⊗ F ) (5)given by taking the fiberwise tensor product, i.e., ψ ( s ⊗ t )( p ) := s ( p ) ⊗ t ( p ). T = ⊗ C ∞ ( M ) ,ι ◦ (Γ × Γ c ) and T ′ = Γ c ◦ ⊗ are additive because Γ, Γ c , ⊗ C ∞ ( M ) ,ι and ⊗ are, and ψ is easily seen to be a natural transformation. By theabove procedure the claim that ψ E,F is an isomorphism can be reduced tothe case E = F = M × K , which amounts to establishing C ∞ ( M ) ⊗ C ∞ ( M ) ,ι C ∞ c ( M ) ∼ = C ∞ c ( M ). This follows using (3) from the fact that the modulemultiplication C ∞ ( M ) × C ∞ c ( M ) → C ∞ c ( M ) is separately continuous and ψ is an isomorphism. Note that the ι -tensor product and the β -tensor productcoincide here because Γ( M, E ) and Γ c ( M, F ) are barrelled.9egularization of vector distributions July 4, 2018
Let H be a locally convex C ∞ ( M )-module with multiplication m : C ∞ ( M ) × H → H and (via Proposition 2) associated linear continuous mapping e m : C ∞ ( M ) ⊗ C ∞ ( M ) ,ι H → H . In order to obtain the second isomorphism of (1) (for which we set H = D ′ ( M )) we endow Γ( M, E ) ⊗ C ∞ ( M ) H with the ι -tensor product topologyand L C ∞ ( M ) (cid:0) Γ( M, E ∗ ) , H (cid:1) with the simple topology.We first show the following: Theorem 4.
For any locally convex C ∞ ( M ) -module H , the following iso-morphism of locally convex C ∞ ( M ) -modules holds: Γ( M, E ) ⊗ C ∞ ( M ) ,ι H ∼ = L C ∞ ( M ) ,σ (cid:0) Γ( M, E ∗ ) , H (cid:1) . (6)In order to apply the procedure of Section 4 we introduce some notation:let E, E ′ be vector bundles over M and µ ∈ VB M ( E, E ′ ); then Γ( µ ) = µ ∗ ∈ L C ∞ ( M ) (cid:0) Γ( M, E ) , Γ( M, E ′ ) (cid:1) denotes pushforward of sections and (Γ ◦ ∗ )( µ ) = µ ∗ ∈ L C ∞ ( M ) (cid:0) Γ( M, ( E ′ ) ∗ ) , Γ( M, E ∗ ) (cid:1) , which is defined via con-traction by ( µ ∗ s ) · t := s · ( µ ∗ t ) for s ∈ Γ( M, ( E ′ ) ∗ ) and t ∈ Γ( M, E ), isthe pullback of sections of the dual bundle. The functors
T, T ′ : VB M → C ∞ ( M ) − LCMod and the natural transformation ν : T → T ′ are defined asfollows: • T ( E ) := Γ( M, E ) ⊗ C ∞ ( M ) ,ι H , T ( µ ) := µ ∗ ⊗ C ∞ ( M ) ,ι id H . In other words, T = ( ⊗ C ∞ ( M ) ,ι H ) ◦ Γ. • T ′ ( E ) := L C ∞ ( M ) ,σ (cid:0) Γ( M, E ∗ ) , H (cid:1) , T ′ ( µ ) := [ ℓ ℓ ◦ µ ∗ ]. In other words, T ′ = L C ∞ ( M ) ,σ ( , H ) ◦ Γ ◦ ∗ . • ν E ∈ L C ∞ ( M ) (cid:0) T ( E ) , T ′ ( E ) (cid:1) is given by ν E ( x )( s ) := e m (cid:0) ( c s ⊗ C ∞ ( M ) ,ι id H )( x ) (cid:1) for x ∈ T ( E ) and s ∈ Γ( M, E ∗ ), where c s ∈ L C ∞ ( M ) (cid:0) Γ( M, E ) , C ∞ ( M ) (cid:1) denotes contraction with s . Given x ∈ T ( E ), which can always bewritten in the form x = q ( P t i ⊗ h i ) with t i ∈ Γ( M, E ), h i ∈ H and q the quotient mapping, one has ν E ( x )( s ) = P i m ( t i · s, h i ), from whichcontinuity in s and also in x is obvious.10egularization of vector distributions July 4, 2018Because the functors T and T ′ are given by the composition of additive func-tors they also are additive. Next, we show that ν is a natural transformation:because for s ∈ Γ( M, ( E ′ ) ∗ ) and t ∈ Γ( M, E ), c µ ∗ s ( t ) = µ ∗ s · t = s · µ ∗ t =( c s ◦ µ ∗ )( t ) we have (cid:0) T ′ ( µ ) ◦ ν E (cid:1) ( x )( s ) = ν E ( x )( µ ∗ s ) = e m (cid:0) ( c µ ∗ s ⊗ C ∞ ( M ) ,ι id H )( x ) (cid:1) = e m (cid:0) (( c s ◦ µ ∗ ) ⊗ C ∞ ( M ) ,ι ⊗ id H )( x ) (cid:1) = ν E ′ (cid:0) ( µ ∗ ⊗ C ∞ ( M ) ,ι id H )( x ) (cid:1) ( s ) = (cid:0) ν E ′ ◦ T ( µ ) (cid:1) ( x )( s ).Hence, (6) follows because by Lemma 1 and (3) ν M × K : C ∞ ( M ) ⊗ C ∞ ( M ) ,ι H → L C ∞ ( M ) ,σ (cid:0) C ∞ ( M ) , H (cid:1) is an isomorphism with inverse l ⊗ C ∞ ( M ) ,ι l (1) in C ∞ ( M ) − LCMod .In order to prove (1) we will show the outer spaces to be isomorphic, whichmeans that D ′ ( M, E ) ∼ = L C ∞ ( M ) ,σ (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) in C ∞ ( M ) − LCMod . Once more, we reduce everything to the case of trivialbundles. Let ψ E ∗ : Γ( M, E ∗ ) ⊗ C ∞ ( M ) ,ι Γ c (cid:0) M, Vol( M ) (cid:1) → Γ c (cid:0) M, E ∗ ⊗ Vol( M ) (cid:1) denote isomorphism (5). With µ as before we define our functors T and T ′ : VB M → C ∞ ( M ) − LCMod and the natural transformation ν : T → T ′ as follows: • T ( E ) := D ′ ( M, E ), T ( µ )( u ) := u ◦ ψ E ∗ ◦ ( µ ∗ ⊗ C ∞ ( M ) ,ι id) ◦ ψ − E ′ ) ∗ ∈D ′ ( M, E ′ ) for u ∈ D ′ ( M, E ). • T ′ ( E ) := L C ∞ ( M ) ,σ (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) and T ′ ( µ )( ℓ ) := ℓ ◦ µ ∗ for each ℓ ∈ L C ∞ ( M ) (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) . • ν E ( u )( s )( ω ) := h u, ψ E ∗ ( s ⊗ C ∞ ( M ) ,ι ω ) i for u ∈ D ′ ( M, E ), s ∈ Γ( M, E ∗ )and ω ∈ Γ c (cid:0) M, Vol( M ) (cid:1) . That ν E ( u )( s ) is in D ′ ( M ) and ν E ( u ) is C ∞ ( M )-linear is clear. Finally, ν E ( u ) is continuous: in fact, let s n → M, E ∗ ). Then because Γ c (cid:0) M, Vol( M ) (cid:1) is barrelled, we have [ ω u, ψ E ∗ ( s n ⊗ C ∞ ( M ) ,ι ω ) i ] → D ′ ( M ) if it converges pointwise, whichobviously is the case.As before, T and T ′ are additive because they are given by the compositionof additive functors; one easily verifies that ν is a natural transformation.Hence, the claim is reduced to the trivial line bundle M × K , for which itreads D ′ ( M ) ∼ = L C ∞ ( M ) ,σ (cid:0) C ∞ ( M ) , D ′ ( M ) (cid:1) . (7)11egularization of vector distributions July 4, 2018By Lemma 1 this is an isomorphism in C ∞ ( M ) − LCMod with inverse ℓ ℓ (1). This completes the proof of (1).In order to round off this first result we show that we can in fact use strongertopologies: Γ( M, E ) ⊗ C ∞ ( M ) ,ι D ′ ( M ) = Γ( M, E ) ⊗ C ∞ ( M ) ,β D ′ ( M ) , L C ∞ ( M ) ,σ (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) = L C ∞ ( M ) ,β (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) . (8)For the first, we simply note that Γ( M, E ) and D ′ ( M ) are barrelled, henceevery seperately continuous bilinear mapping from Γ( M, E ) × D ′ ( M ) intoany locally convex space is hypocontinuous ([14, §
40 2.(5) a), p. 159]).For the second, we need to show that every 0-neighborhood of the β -topologyon L C ∞ ( M ) (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) , which can be taken to be of the form U B,V := { ℓ : ℓ ( B ) ⊆ V } for B ⊆ Γ( M, E ∗ ) bounded and V ⊆ D ′ ( M ) a 0-neighborhood,contains a 0-neighborhood of the σ -topology, i.e., one of the form U B ′ ,V ′ where B ′ ⊆ Γ( M, E ∗ ) is finite and V ′ again is a 0-neighborhood in D ′ ( M ).For this we need to use the fact that Γ( M, E ∗ ) is a finitely generated C ∞ ( M )-module as follows: let F be such that E ∗ ⊕ F is trivial and choose a basis( b i ) i =1 ...n of the C ∞ ( M )-module Γ( E ∗ ⊕ F ) with corresponding dual basis( β i ) i =1 ...m . Denoting by ι and π the canonical injection of E ∗ into E ∗ ⊕ F andthe corresponding projection, respectively, we can write any s ∈ Γ( M, E ∗ ) as s = π ∗ ( ι ∗ s ) = π ∗ (cid:0) n X i =1 β i ( ι ∗ s ) · b i ) = n X i =1 β i ( ι ∗ s ) · π ∗ b i . Hence, B ′ := ( π ∗ b i ) i =1 ...n is a generating set of Γ( M, E ∗ ). Because β i ◦ ι ∗ isa continuous linear mapping Γ( M, E ∗ ) → C ∞ ( M ) the set D := { s i | s ∈ B, i = 1 . . . n } ⊆ C ∞ ( M ) with with s i := β i ( ι ∗ s ) for s ∈ B is bounded.Choose 0-neighborhoods V ′ , V ′′ in D ′ ( M ) such that V ′′ + . . . + V ′′ ( n sum-mands) is contained in V and D · V ′ ⊆ V ′′ . Then U B ′ ,V ′ is a 0-neighborhoodfor the topology of simple convergence and for s ∈ B and ℓ ∈ U B ′ ,V ′ we have ℓ ( s ) = ℓ ( X i s i π ∗ b i ) = X i s i ℓ ( π ∗ b i ) ∈ X i D · V ′ ⊆ X i V ′′ ⊆ V which means that U B ′ ,V ′ ⊆ U B,V . Hence, the topologies of simple andbounded convergence coincide. 12egularization of vector distributions July 4, 2018
In this section we are going to show isomorphism (2), for which we requiresome preliminaries. First of all, we recall L. Schwartz’ notion of ε -productfrom [25]. For any locally convex space H , let H ′ c denote the dual space of H endowed with the topology of uniform convergence on absolutely convexcompact subsets of H . The ε -product H ε K of two locally convex spaces H and K is defined as the vector space of all bilinear mappings H ′ c × K ′ c → K which are hypocontinuous with respect to equicontinuous subsets of H ′ and K ′ . It is endowed with the topology of uniform convergence on products ofequicontinuous subsets of H ′ and K ′ ([25, §
1, Definition, p. 18]). There is acanonical isomorphism H ε K ∼ = L ε ( H ′ c , K ), where the latter space is endowedwith the topology of uniform convergence on equicontinuous subsets of H ′ ([25, §
1, Corollaire 2 to Proposition 4, p. 34]). By [25, §
1, Proposition 3,p. 29] H ε K is complete if H and K are complete. Noting that H ⊗ K iscanonically contained in H ε K ([25, p. 19]), given locally convex spaces H i , K i for i = 1 , f : H → H and g : K → K there is a canonical continuous linear map f ε g : H ε K → H ε K extendingthe map f ⊗ g : H ⊗ K → H ⊗ K ([25, § ε -product in two ways. First, fix two vector bundles E → M and F → N . Viewing Γ( M, E ) ⊗ Γ( N, F ) (endowed with theprojective tensor topology) as a dense subspace of Γ( M × N, E ⊠ F ), itscompletion Γ( M, E ) b ⊗ Γ( N, F ) is isomorphic as a locally convex space toΓ(
M, E ) ε Γ( N, F ) ([25, §
1, Corollaire 1 to Proposition 11, p. 47]) becauseΓ(
M, E ) and Γ(
N, F ) are complete and have the approximation property.Second, we note that on D ′ ( M, E ) the topology of bounded convergence co-incides with the topology of absolutely convex compact convergence. BecauseΓ c (cid:0) M, E ∗ ⊗ Vol( M ) (cid:1) is barrelled, on L ( D ′ ( M, E ) , Γ( N, F )) the topologies ofbounded and equicontinuous convergence coincide as a consequence of theBanach-Steinhaus theorem. Summarizing, we haveΓ(
M, E ) b ⊗ Γ( N, F ) = Γ( M × N, E ⊠ F ) ∼ = Γ( M, E ) ε Γ( N, F ) , L β (cid:0) D ′ ( M, E ) , Γ( N, F ) (cid:1) = Γ c (cid:0) M, E ∗ ⊗ Vol( M ) (cid:1) ε Γ( N, F ) , L β ( D ′ ( M ) , C ∞ ( N )) = Γ c ( M, Vol( M )) ε C ∞ ( N ) . In order to define a locally convex C ∞ ( M × N )-module structure on thesespaces and establish the desired isomorphism (2) we will employ the followingLemma. While its proof can be based on L. Schwartz’ Th´eor`eme de croise-ment ([26, §
2, Proposition 2, p. 18]; cf. also [1, Proposition 2]) we prefer togive a direct proof. 13egularization of vector distributions July 4, 2018
Lemma 5.
Let A and B be nuclear Fr´echet spaces and M , M , N , N lo-cally convex spaces with M and N complete. Suppose we are given twohypocontinuous bilinear mappings f : A × M → M and g : B × N → N .Then there exists a unique separately continuous bilinear mapping σ : A b ⊗ B × M ε N → M ε N satisfying σ ( a ⊗ b, u ) := (cid:0) f ( a, · ) ε g ( b, · ) (cid:1) ( u ) ∀ a ∈ A, b ∈ B, u ∈ M ε M . (9) Moreover, σ even is hypocontinuous.Proof. For a ∈ A and b ∈ B we set f a := f ( a, · ) ∈ L ( M , M ) and g b := g ( b, · ) ∈ L ( N , N ). We define a trilinear map ˜ σ : A × B × M ε N → M ε N by ˜ σ ( a, b, u ) := ( f a ε g b )( u ). In order to show that it is hypocontinuous fixbounded subsets A ′ ⊆ A , B ′ ⊆ B and D ⊆ M ε N and a 0-neighborhood U in M ε N of the form U = ( X × Y ) ◦ , where X ⊆ M ′ and Y ⊆ N ′ arearbitrary equicontinuous subsets.(i) As { f a | a ∈ A ′ } and { g b | b ∈ B ′ } are equicontinuous by assumption, X ′ := { m ′ ◦ f a | m ′ ∈ X, a ∈ A ′ } ⊆ M ′ and Y ′ := { n ′ ◦ g b | n ′ ∈ Y, b ∈ B ′ } ⊆ N ′ are equicontinuous as well and V := ( X ′ × Y ′ ) ◦ is a 0-neighborhood in M ε N such that ˜ σ ( A ′ , B ′ , V ) ⊆ U .(ii) We need to find a 0-neighborhood V in A such that ˜ σ ( V, B ′ , D ) ⊆ U .By [25, §
1, Proposition 2 bis, p. 28] D is an ε -equihypocontinuous subset of L (cid:0) ( M ) ′ β × ( N ) ′ β , K (cid:1) . This means that there exists a 0-neighborhood of theform C ◦ in ( M ′ ) b , with C ⊆ M bounded, such that u ( C ◦ , Y ′ ) ∈ D for all u ∈ D , where D is the closed unit disk of K . Hence, we only need to choose V such that m ′ ◦ f ( a, c ) ∈ D for all m ′ ∈ X , a ∈ V and c ∈ C , which ispossible by the assumption on f . This means that m ′ ◦ f ( a, · ) ∈ C ◦ andhence ˜ σ ( V, B ′ , D )( X, Y ) ⊆ D ( X ◦ f ( V, · ) , Y ′ ) ∈ D .(iii) Similarly, one can find a 0-neighborhood V ⊆ B such that ˜ σ ( A ′ , V, D ) ⊆ U .This shows that ˜ σ is hypocontinuous as claimed. Because M εN is complete([25, §
1, Proposition 3, p. 29]), for each u the (by [14, §
40 2.(1) a), p. 158])continuous map ˜ σ u := ˜ σ ( ., ., u ) has a unique extension to a linear continuousmap σ u : A b ⊗ B → M ε N . We now define σ : A b ⊗ B × M ε N → M ε N by σ ( x, u ) := σ u ( x ), which by definition is linear and continuous in x . Forlinearity in u , it suffices by continuity in x to verify σ ( x, u + λu ) = σ ( x, u )+ λσ ( x, u ) for u , u ∈ M ε M and λ ∈ K for x in the dense subspace A ⊗ B ,14egularization of vector distributions July 4, 2018where it is evident. For hypocontinuity let Z ⊆ A b ⊗ B be bounded and D, U as above. Then by [13, Theorem 21.5.8, p. 495]) there exist bounded subsets A ′ ⊆ A and B ′ ⊆ B such that Z ⊆ acx( A ′ ⊗ B ′ ), where acx denotes theabsolutely convex closed hull of a set. Choosing V as in (i) above we thenhave σ ( Z, V ) ⊆ acx( σ ( A ′ ⊗ B ′ , V )) = acx(˜ σ ( A ′ , B ′ , V )) ⊆ acx U = U because σ is continuous in the first variable and U is closed and absolutely convex. Onthe other hand, { σ ( ., u ) | u ∈ D } is equicontinuous because { ˜ σ ( ., ., u ) | u ∈ D } is separately equicontinuous ([14, §
40 2.(2), p. 158]). Because A ⊗ B is densein A b ⊗ B , (9) uniquely defines σ . Corollary 6.
Let A and B be nuclear Fr´echet locally convex algebras.(i) The map A ⊗ B × A ⊗ B → A ⊗ B , ( a ⊗ b , a ⊗ b ) a a ⊗ b b extends uniquely to a continuous bilinear map A b ⊗ B × A b ⊗ B → A b ⊗ B ,turning A b ⊗ B into a locally convex algebra.(ii) Given a complete locally convex A -module M and a complete locallyconvex B -module N such that the multiplications f of M and g of N arehypocontinuous, the map A ⊗ B × M εN → M εN , ( a ⊗ b, u ) ( f a εg b )( u ) extends uniquely to a hypocontinuous bilinear mapping A b ⊗ B × M εN → M ε N turning
M ε N into a locally convex A b ⊗ B -module withhypocontinuous multiplication.(iii) In particular, ε is an additive functor A − LCMod − β × B − LCMod − β → A b ⊗ B − LCMod − β. Proof.
Lemma 5 gives the necessary mappings. The axioms for the algebraand module structure are easily verified by restricting to the dense subspace A ⊗ B of A b ⊗ B .In order to apply the procedure of Section 4 we define functors T, T ′ : VB M × VB N → C ∞ ( M × N ) − LCMod − β by T := ε ◦ (Γ c × Γ) ◦ (cid:0) ( ⊗ Vol( M )) × id (cid:1) ◦ ( ∗ × id) ,T ′ := L C ∞ ( M × N ) ,β (cid:0) , Γ c ( M, Vol( M )) ε C ∞ ( N ) (cid:1) ◦ b ⊗ ◦ (Γ × Γ) ◦ (id × ∗ )Because they are compositions of additive functors, T and T ′ are additive.In order to define ν E,F : T ( E, F ) → T ′ ( E, F ) we apply Lemma 5 to A =Γ( M, E ), B = Γ( N, F ∗ ), M = Γ c ( M, E ∗ ⊗ Vol( M )), N = Γ( N, F ), M =15egularization of vector distributions July 4, 2018Γ c ( M, Vol( M )), N = C ∞ ( N ), g : Γ( N, F ∗ ) × Γ( N, F ) → C ∞ ( N ) given bycontraction and f defined viaΓ( M, E ) × Γ c (cid:0) M, E ∗ ⊗ Vol( M ) (cid:1) f / / id × ψ − (cid:15) (cid:15) Γ c (cid:0) M, Vol( M ) (cid:1) Γ( M, E ) × Γ( M, E ∗ ) b ⊗ Γ c (cid:0) M, Vol( M ) (cid:1) ( t,s ) ( c t ⊗ id)( s ) / / C ∞ ( M ) b ⊗ Γ c (cid:0) M, Vol( M ) (cid:1) M O O where c t : Γ( M, E ∗ ) → C ∞ ( M ) is contraction with t ∈ Γ( M, E ), which is a C ∞ ( M )-linear continuous map, ψ is the isomorphism (5) and M : C ∞ ( M ) b ⊗ Γ c (cid:0) M, Vol( M ) (cid:1) → Γ c (cid:0) M, Vol( M ) (cid:1) denotes the linear continous mapping canonically associated to the modulemultiplication of Γ c (cid:0) M, Vol( M ) (cid:1) . Explicitly, f is given by the map f ( t, s ) := M (cid:0) ( c t ⊗ id)( ψ − ( s )) (cid:1) and obviously is hypocontinuous. This way, Lemma 5defines a mapping σ : Γ( M × N, E ⊠ F ∗ ) × Γ c (cid:0) M, E ∗ ⊗ Vol( M ) (cid:1) ε Γ( N, F ) → Γ c (cid:0) M, Vol( M ) (cid:1) εC ∞ ( N )which is hypocontinuous with respect to bounded subsets of Γ( M × N, E ⊠ F ∗ ), thus by [14, § ν E,F : Γ c (cid:0) M, E ∗ ⊗ Vol( M ) (cid:1) ε Γ( N, F ) → L β (cid:0) Γ( M × N, E ⊠ F ∗ ) , Γ c ( M, Vol( M )) ε C ∞ ( N ) (cid:1) by setting ( νu )( s ) := σ ( s, u ). It is easily verified that for a ⊗ b ∈ C ∞ ( M ) ⊗ C ∞ ( N ) and s ⊗ t ∈ Γ( M, E ) ⊗ Γ( N, F ∗ ), ( νu )(( a ⊗ b ) · ( s ⊗ t )) = ( a ⊗ b ) · (( νu )( s ⊗ t )) which implies that νu is C ∞ ( M × N )-linear. The equality ν ( h · u )( v ) = h · ν ( u )( v ) holds for all h = a ⊗ b and v = s ⊗ t as above,and hence also for h ∈ C ∞ ( M × N ) and v ∈ Γ( M × N, E ⊠ F ∗ ) becauseboth sides are continuous in h and v separately; this means that ν itself is C ∞ ( M × N )-linear.Summarizing, for each pair E, F we have defined a C ∞ ( M × N )-linear con-tinuous map ν E,F : T ( E, F ) → T ′ ( E, F ). Moreover, one easily verifies that ν is a natural transformation from T to T ′ . It follows that in order for ν tobe a natural isomorphism, it suffices to verify this for the case of trivial linebundles E = M × K and F = N × K ; but in this case one can immediatelywrite down the inverse of ν , namely ν − ℓ := ℓ (1) with 1 ∈ C ∞ ( M × N ).Together with Theorem 4 and (8) this establishes (2).16egularization of vector distributions July 4, 2018 The topological variant of isomorphisms (1) and (2) hence reads as follows: D ′ ( M, E ) ∼ = Γ( M, E ) ⊗ C ∞ ( M ) ,β D ′ ( M ) ∼ = L C ∞ ( M ) ,β (cid:0) Γ( M, E ∗ ) , D ′ ( M ) (cid:1) L β (cid:0) D ′ ( M, E ) , Γ( N, F ) (cid:1) ∼ = Γ( M × N, E ∗ ⊠ F ) ⊗ C ∞ ( M × N ) ,β L β (cid:0) D ′ ( M ) , C ∞ ( N ) (cid:1) ∼ = L C ∞ ( M × N ) ,β (cid:0) Γ( M × N, E ⊠ F ∗ ) , L β ( D ′ ( M ) , C ∞ ( N )) (cid:1) . Acknowledgements.
This work was supported by the Austrian ScienceFund (FWF) grant P26859-N25.
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