Sheaves of nonlinear generalized function spaces
aa r X i v : . [ m a t h . F A ] J u l SHEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES
ANDREAS DEBROUWERE AND EDUARD A. NIGSCH
Abstract.
We provide a framework for the construction of diffeomorphism invariantsheaves of nonlinear generalized functions spaces. As an application, global algebrasof generalized functions for distributions on manifolds and diffeomorphism invariantalgebras of generalized functions for ultradistributions are constructed. Introduction
The theory of generalized functions developed by L. Schwartz [21] suffers from the factthat in general one cannot define nonlinear operations (like multiplication) on distri-butions, so the use of this theory for nonlinear problems is limited. In the 1980s, differ-ential algebras of nonlinear generalized functions were developed by J. F. Colombeau[2, 3] in order to study nonlinear PDEs with singular data or coefficients. TheseColombeau algebras have found numerous applications for instance in connection withPDEs involving singular data and/or coefficients, singular differential geometry andgeneral relativity. In particular, a diffeomorphism invariant formulation of the theorywas developed in [9, 11] and recently extended to the vector-valued setting in [18].Spaces of nonlinear generalized ultradistributions were studied in [1, 5, 6, 8, 20]. Themost recent variant, developed in [4], is optimal in the sense that the embedding ofultradistributions of class M p there preserves the product of all ultradifferentiable func-tions of class M p ; this is an improvement over the previous variants where only theproduct of ultradifferentiable functions of a strictly more regular class had been pre-served. The setting of [4] is that of special Colombeau algebras, which allows for asimpler development of the theory but makes it impossible to obtain diffeomorphisminvariance (cf. [10, Chapter 2]).
Full
Colombeau algebras, on the other hand, are techni-cally more involved but allow for an embedding of distributions that is diffeomorphisminvariant and in addition commutes with arbitrary derivatives (cf. [9, 11, 17]). Hence,for applications in a geometric context it is essential that a formulation of the theoryin the full setting is obtained.
Date : July 2016.2010
Mathematics Subject Classification. primary: 46F30, secondary: 46T30.
Key words and phrases.
Colombeau algebras, multiplication of ultradistributions, nonlinear gener-alized functions, full algebra.A. Debrouwere gratefully acknowledges support by Ghent University, through a BOF Ph.D.-grant.E. A. Nigsch was supported by grant P26859 of the Austrian Science Fund (FWF).
As a continuation of the development of [17, 16] and [4] we work out the abstractformulation of the construction of sheaves of nonlinear generalized function spaces. Inparticular, it turns out that very little structure is needed on the underlying spaces ofgeneralized functions as the respective arguments mainly concern the sheaf structure.Our construction applies at the same time to distributions and to ultradistributions,both of Beurling and Roumieu type, which leads to the following results.
Theorem 6.1.
Let M be a paracompact Hausdorff manifold. There is an associa-tive commutative algebra G loc ( M ) with unit containing D ′ ( M ) injectively as a linearsubspace and C ∞ ( M ) as a subalgebra. G loc ( M ) is a differential algebra, where thederivations b L X extend the usual Lie derivatives from D ′ ( M ) to G loc ( M ) , and G loc is afine sheaf of algebras over M . As customary, we write ∗ instead of ( M p ) or { M p } to treat the Beurling and Roumieucase simultaneously. For the following theorem, let M p be a weight sequence satisfying ( M. , ( M. , and ( M. ′ . Theorem 7.6.
For each open set Ω ⊆ R n there is an associative commutative algebrawith unit G ∗ loc (Ω) containing D ∗′ (Ω) injectively as a linear subspace and E ∗ (Ω) as a sub-algebra. G ∗ loc (Ω) is a differential algebra, where the partial derivatives b ∂ i , i = 1 , . . . , n ,extend the usual partial derivatives from D ∗ (Ω) to G ∗ loc (Ω) , and G ∗ loc is a fine sheaf ofalgebras over Ω . Moreover, the construction is invariant under real-analytic coordi-nate changes, i.e., if µ : Ω ′ → Ω is a real-analytic diffeomorphism then there is a map b µ : G ∗ loc (Ω ′ ) → G ∗ loc (Ω) compatible with the canonical embeddings ι and σ . The structure of this article is as follows. • We collect some preliminary notions in Section 2. • The basic spaces containing the representatives of nonlinear generalized func-tions are introduced in Section 3. • The quotient construction, which ensures that the product of smooth or ofultradifferentiable functions is preserved, is detailed in Section 4. • Sheaf properties of the quotient space are established in Section 5. • The construction of diffeomorphism invariant differential algebras of distribu-tions and ultradistributions is given in Section 6 and Section 7, respectively.2.
Preliminaries
Our general references are [21] for distribution theory, [12, 13, 14] for ultradistributionsand [2, 3, 19, 10] for Colombeau algebras.We set I = (0 , , R + = [0 , ∞ ) and N = { , , , . . . } . Given a set M , id M (or simply id if the set is clear from the context) denotes the identity mapping on M . For anelement λ ∈ ( R + ) I we write λ ( ε ) = λ ε . Furthermore, Landau’s O -notations are alwaysmeant for ε → + . Given two locally convex spaces E and F , L b ( E, F ) denotes HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 3 the space of continuous linear mappings from E into F endowed with the topologyof bounded convergence. L σ ( E, F ) is this space endowed with the weak topologyinstead. We denote by csn( E ) the set of continuous seminorms on E . An algebraalways means an associative commutative algebra over C , and a locally convex algebrais an algebra endowed with a locally convex topology such that its multiplication isjointly continuous. C ∞ ( E, F ) is the space of smooth functions E → F in the sense ofconvenient calculus [15], with C ∞ ( E ) := C ∞ ( E, C ) ; in this context, d k f denotes the k th differential of a mapping f ∈ C ∞ ( E, F ) .Colombeau algebras are usually defined by means of a quotient construction employingcertain asymptotic scales. Most frequently a polynomial scale is used for this purpose,but we will employ more general scales based on [7] instead, which increases the flexi-bility regarding applications. Definition 2.1.
A set
A ⊆ ( R + ) I is said to be an asymptotic growth scale if(i) ∀ λ, µ ∈ A ∃ ν ∈ A : λ ε + µ ε = O ( ν ε ) ,(ii) ∀ λ, µ ∈ A ∃ ν ∈ A : λ ε µ ε = O ( ν ε ) ,(iii) ∃ λ ∈ A : lim inf ε → + λ ε > .A set I ⊆ ( R + ) I is said to be an asymptotic decay scale if(iv) ∀ λ ∈ I ∃ µ, ν ∈ I : µ ε + ν ε = O ( λ ε ) ,(v) ∀ λ ∈ I ∃ µ, ν ∈ I : µ ε ν ε = O ( λ ε ) ,(vi) ∃ λ ∈ I : lim ε → + λ ε = 0 .We call a pair ( A , I ) an admissible pair of scales if A is an asymptotic growth scale, I is an asymptotic decay scale, and the following two properties are satisfied:(vii) ∀ λ ∈ I ∀ µ ∈ A ∃ ν ∈ I : µ ε ν ε = O ( λ ε ) ,(viii) ∃ λ ∈ A ∃ µ ∈ I : µ ε = O ( λ ε ) .The prototypical scale to keep in mind is given by the polynomial scale(2.1) A = I = { ε ε k | k ∈ Z } , which is easily verified to give an admissible pair. For a detailed study of asymptoticscales we refer to [6, 7]. 3. The basic space
A main principle behind Colombeau algebras is to represent singular functions by reg-ular ones and thus define classical operations like multiplication on the former throughthe latter. Usually the roles of singular and regular functions are played by D ′ and C ∞ ,respectively, but for our considerations we will replace these spaces by a more generalpair of locally convex spaces E and F . Such a pair ( E, F ) is called a test pair if F ⊆ E A. DEBROUWERE AND E.A. NIGSCH and the topology on F is finer than the one induced by E . Throughout this section wefix a test pair ( E, F ) . Definition 3.1.
We define the basic space as E ( E, F ) := C ∞ ( L b ( E, F ) , F ) and the canonical linear embeddings of E and F into E ( E, F ) via ι : E → E ( E, F ) , ι ( u )(Φ) := Φ( u ) ,σ : F → E ( E, F ) , σ ( ϕ )(Φ) := ϕ. There are three common ways of transferring classical operations T on E and F toelements R of the basic space E ( E, F ) . These are, in brief, given as follows: ( e T R )(Φ) := T ( R (Φ)) , ( T ∗ R )(Φ) := T ( R ( T − ◦ Φ ◦ T )) , ( b T R )(Φ) := − d R (Φ)( T ◦ Φ − Φ ◦ T ) + T ( R (Φ)) . We will now specify in which situation they are well-defined on the basic space, andwhen each variant is employed.The first one amounts to applying an operation on F after inserting the parameter Φ ∈ L ( E, F ) . This defines the vector space structure of E ( E, F ) and its algebra struc-ture if F is a locally convex algebra. Moreover, this is used for extending directionalderivatives and especially the covariant derivative in geometry (see [18]). For multilin-ear mappings it is formulated as follows: Lemma 3.2.
Let T : F × · · · × F → F be a jointly continuous multilinear mapping.Then, the mapping e T : E ( E, F ) × · · · × E ( E, F ) → E ( E, F ) given by (3.1) e T ( R , . . . , R n )(Φ) := T ( R (Φ) , . . . , R n (Φ)) commutes with the embedding σ in the sense that e T ( σ ( ϕ ) , . . . , σ ( ϕ n )) = σ ( T ( ϕ , . . . , ϕ n )) . Corollary 3.3.
Suppose that F is a locally convex algebra. Then, E ( E, F ) is an algebrawith multiplication given by (3.2) ( R · R )(Φ) := R (Φ) · R (Φ) and σ is an algebra homomorphism. The second variant of extending operations to the basic space applies to isomorphismson E which restrict to isomorphisms on F . This will be used for isomorphisms ondistribution spaces coming from diffeomorphisms of the respective domains. HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 5
Lemma 3.4.
Let ( E , F ) and ( E , F ) be two test pairs. Suppose that f : E → E is alinear topological isomorphism such that also the restriction f | F is a linear topologicalisomorphism F → F . Then, the mapping f ∗ : E ( E , F ) → E ( E , F ) given by (3.3) ( f ∗ R )(Φ) := f ( R ( f − ◦ Φ ◦ f )) is a vector space isomorphism that makes the following diagrams commutative: E f / / ι (cid:15) (cid:15) E ι (cid:15) (cid:15) E ( E , F ) f / / E ( E , F ) F f / / σ (cid:15) (cid:15) F σ (cid:15) (cid:15) E ( E , F ) f / / E ( E , F ) Finally, the third variant of extending operations to the basic space applies to theextension of derivatives to E ( E, F ) : Lemma 3.5.
Let T ∈ L ( E, E ) with T | F ∈ L ( F, F ) . Then, the mapping T RO : L ( E, F ) → L ( E, F ) , Φ T ◦ Φ − Φ ◦ T is linear and continuous, and the mapping b T : E ( E, F ) → E ( E, F ) given by (3.4) ( b T R )(Φ) := T ( R (Φ)) − d R (Φ)( T RO Φ) is a well defined linear mapping that makes the following diagrams commutative: E T / / ι (cid:15) (cid:15) E ι (cid:15) (cid:15) E ( E, F ) b T / / E ( E, F ) F T / / σ (cid:15) (cid:15) F σ (cid:15) (cid:15) E ( E, F ) b T / / E ( E, F ) The quotient construction
Colombeau algebras are defined as the quotient of moderate by negligible functions,which permits the product of regular functions to be preserved. While originally theseproperties were determined by inserting translated and scaled test functions into therepresentatives of generalized functions, the functional analytic formulation of the the-ory makes it possible to give a very elegant formulation of this testing procedure inmore general terms. Our next goal is to give a proper definition of moderateness andnegligibility of elements of the basic space in our setting. We start by introducing testobjects for a test pair ( E, F ) . Definition 4.1.
Let S = ( A , I ) be an admissible pair of scales. We define TO(
E, F, S ) as the set consisting of all (Φ ε ) ε ∈ L ( E, F ) I that satisfy (TO) ∀ p ∈ csn( L σ ( E, F )) ∃ λ ∈ A : p (Φ ε ) = O ( λ ε ) , (TO) ∀ p ∈ csn( L σ ( F, F )) ∀ λ ∈ I : p (Φ ε | F − id F ) = O ( λ ε ) , (TO) Φ ε → id E in L σ ( E, E ) . A. DEBROUWERE AND E.A. NIGSCH
Elements of
TO(
E, F, S ) are called test objects (with respect to S ) . If S is clear fromthe context, we shall simply write TO(
E, F, S ) = TO( E, F ) .Similarly, we define TO ( E, F ) = TO ( E, F, S ) as the set consisting of all (Ψ ε ) ε ∈L ( E, F ) I that satisfy (TO) ∀ p ∈ csn( L σ ( E, F )) ∃ λ ∈ A : p (Ψ ε ) = O ( λ ε ) , (TO) ∀ p ∈ csn( L σ ( F, F )) ∀ λ ∈ I : p (Ψ ε | F ) = O ( λ ε ) , (TO) Ψ ε → in L σ ( E, E ) .Elements of TO ( E, F, S ) are called -test objects (with respect to S ) . Again, we write TO ( E, F, S ) = TO ( E, F ) if S is clear from the context.We shall need the following result later on. Lemma 4.2. (i) Let T i ∈ L ( E, E ) , i = 0 , . . . , N ∈ N , be given such that T i | F ∈L ( F, F ) and P Ni =0 T i = id . Then, (cid:16)P Ni =0 T i ◦ Φ i,ε (cid:17) ε ∈ TO(
E, F ) for all (Φ i,ε ) ε ∈ TO(
E, F ) , i = 0 , . . . , N .(ii) Let T ∈ L ( E, E ) be such that T | F ∈ L ( F, F ) . Then, ( T ◦ Φ ε ) ε ∈ TO ( E, F ) forall (Φ ε ) ε ∈ TO ( E, F ) .(iii) Let T ∈ L ( E, E ) with T | F ∈ L ( F, F ) . Then, ( T ◦ Φ ε − Φ ε ◦ T ) ε ∈ TO ( E, F ) for all (Φ ε ) ε ∈ TO(
E, F ) ∪ TO ( E, F ) . Having test objects at our disposal, we are now able to define moderateness and neg-ligibility.
Definition 4.3.
Let S = ( A , I ) be an admissible pair of scales and let Λ ⊆ TO(
E, F, S ) , Λ ⊆ TO ( E, F, S ) be nonempty. An element R ∈ E ( E, F ) is called moderate (withrespect to Λ , Λ , and S ) if ∀ p ∈ csn( F ) ∀ l ∈ N ∀ (Φ ε ) ε ∈ Λ ∀ (Ψ ,ε ) ε , . . . , (Ψ l,ε ) ε ∈ Λ ∃ λ ∈ A : p (d l R (Φ ε )(Ψ ,ε , . . . , Ψ l,ε )) = O ( λ ε ) , and negligible (with respect to Λ , Λ , and S ) if ∀ p ∈ csn( F ) ∀ l ∈ N ∀ (Φ ε ) ε ∈ Λ ∀ (Ψ ,ε ) ε , . . . , (Ψ l,ε ) ε ∈ Λ ∀ λ ∈ I : p (d l R (Φ ε )(Ψ ,ε , . . . , Ψ l,ε )) = O ( λ ε ) . The set of all moderate (negligible, respectively) elements is denoted by E M ( E, F ) = E M ( E, F, Λ , Λ , S ) ( E N ( E, F ) = E N ( E, F, Λ , Λ , S ) , respectively).The following important properties follow immediately from our definitions. In fact,we chose our definitions in such a way precisely for these properties to hold. Proposition 4.4. (i) E M ( E, F ) is a vector space and E N ( E, F ) is a subspace of E M ( E, F ) ,(ii) ι ( E ) ⊆ E M ( E, F ) , σ ( F ) ⊆ E M ( E, F ) , HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 7 (iii) ι ( E ) ∩ E N ( E, F ) = { } , σ ( F ) ∩ E N ( E, F ) = { } ,(iv) ( ι − σ )( F ) ⊆ E N ( E, F ) . We now construct the quotient.
Definition 4.5.
Let S = ( A , I ) be an admissible pair of scales and let Λ ⊆ TO(
E, F, S ) , Λ ⊆ TO ( E, F, S ) be nonempty. The nonlinear extension of the test pair ( E, F ) (withrespect to Λ , Λ , and S ) is defined as G ( E, F ) = G ( E, F, Λ , Λ , S ) := E M ( E, F, Λ , Λ , S ) / E N ( E, F, Λ , Λ , S ) . The equivalence class of R ∈ E M ( E, F ) is denoted by [ R ] .Proposition 4.4 implies that ι : E → G ( E, F ) , ι ( u ) := [ ι ( u )] ,σ : E → G ( E, F ) , σ ( ϕ ) := [ σ ( ϕ )] are linear embeddings such that ι | F = σ . The name "nonlinear extension" is justifiedby the following lemma. Lemma 4.6.
Let T : F × · · · × F → F be a jointly continuous multilinear mappingand consider the multilinear mapping e T : E ( E, F ) × · · · × E ( E, F ) → E ( E, F ) given by (3.1) . Then, e T preserves moderateness, i.e., e T ( E M ( E, F ) , . . . , E M ( E, F )) ⊆ E M ( E, F ) ,and e T ( R , . . . , R n ) is negligible if at least one of the R i is negligible. Consequently, e T : G ( E, F ) × . . . × G ( E, F ) → G ( E, F ) e T ([ R ] , . . . , [ R n ]) := [ T ( R , . . . , R n )] is a well-defined multilinear mapping such that e T ( σ ( ϕ ) , . . . , σ ( ϕ n )) = σ ( T ( ϕ , . . . , ϕ n )) . Proof.
This follows from Lemma 3.2 and the continuity of T . (cid:3) Corollary 4.7.
Suppose that F is a locally convex algebra. Then, E M ( E, F ) is analgebra with multiplication given by (3.2) and E N ( E, F ) is an ideal of E M ( E, F ) . Con-sequently, G ( E, F ) is an algebra with multiplication given by [ R ] · [ R ] := [ R · R ] and σ is an algebra homomorphism. Lemma 4.8.
Let ( E , F ) and ( E , F ) be two test pairs. Suppose that f : E → E is alinear topological isomorphism such that also the restriction f | F is a linear topologicalisomorphism F → F . Let S = ( A , I ) be an admissible pair of scales and let Λ i ⊆ TO( E i , F i , S ) , Λ i ⊆ TO ( E i , F i , S ) be nonempty for i = 1 , such that ( f − ◦ Φ ε ◦ f ) ε ∈ Λ ∀ (Φ ε ) ε ∈ Λ , ( f − ◦ Ψ ε ◦ f ) ε ∈ Λ ∀ (Ψ ε ) ε ∈ Λ A. DEBROUWERE AND E.A. NIGSCH and ( f ◦ Φ ε ◦ f − ) ε ∈ Λ ∀ (Φ ε ) ε ∈ Λ , ( f ◦ Ψ ε ◦ f − ) ε ∈ Λ ∀ (Ψ ε ) ε ∈ Λ . Consider the mapping f ∗ : E ( E , F ) → E ( E , F ) given by (3.3) . Set E M ( E i , F i ) = E M ( E i , F i , Λ i , Λ i , S ) , E N ( E i , F i ) = E N ( E i , F i , Λ i , Λ i , S ) , for i = 1 , . Then, f ∗ preserves moderateness and neglibility. Consequently, the map-ping f ∗ : G ( E , F ) → G ( E , F ) given by f ∗ ([ R ]) := [ f ∗ ( R )] is an isomorphism that makes the following diagram commutative. E f / / ι (cid:15) (cid:15) E ι (cid:15) (cid:15) G ( E , E ) f / / G ( E , F ) Proof.
This follows from Lemma 3.4 and the continuity of f . (cid:3) Lemma 4.9.
Let T ∈ L ( E, E ) with T | F ∈ L ( F, F ) . Consider the mapping b T : E ( E, F ) →E ( E, F ) given by (3.4) . Then, b T preserves moderateness and negligibility. Conse-quently, the mapping b T : G ( E, F ) → G ( E, F ) given by b T ([ R ]) := [ b T ( R )] is a well-defined linear mapping that makes the following diagram commutative: E T / / ι (cid:15) (cid:15) E ι (cid:15) (cid:15) G ( E, F ) b T / / G ( E, F ) Proof.
This follows from Lemma 3.5 and continuity of T . (cid:3) Sheaf properties
In this section we study the sheaf theoretic properties of our generalized functionspaces. After introducing the necessary terminology, we first look in detail at testobjects. Satisfying a certain localizability condition, the spaces of test objects and0-test objects themselves form sheaves. This is used for showing the existence of globaltest objects by gluing together local ones, and for extending and restricting test objectsin the proof of the sheaf property of the Colombeau quotient.
HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 9
Locally convex sheaves.
Let X be a Hausdorff locally compact paracompacttopological space. For open subsets V, U ⊆ X we write V ⋐ U to indicate that V ⊂ U and V is relatively compact in X . We shall only use this notation for open sets.A presheaf (of vector spaces) E assigns to each open set U ⊆ X a vector space E ( U ) andgives, for every inclusion of open sets V ⊆ U , a linear mapping ρ V,U : E ( U ) → E ( V ) such that for all W ⊆ V ⊆ U the identities ρ W,U = ρ W,V ◦ ρ V,U and ρ U,U = id hold. Theelements of E ( U ) are called sections of E over U and the mappings ρ V,U restrictionmappings.A presheaf E is a sheaf if for all open subsets U ⊆ X and all open coverings ( U i ) i of U the following properties are satisfied:(S1) If u ∈ E ( U ) satisfies ρ U i ,U ( u ) = 0 for all i then u = 0 .(S2) If u i ∈ E ( U i ) are given such that ρ U i ∩ U j ,U i ( u i ) = ρ U i ∩ U j ,U j ( u j ) for all i, j thenthere exists u ∈ E ( U ) such that ρ U i ,U ( u ) = u i for all i .A section u ∈ E ( U ) is said to vanish on an open set V ⊆ U if ρ V,U ( u ) = 0 . The supportof u , denoted by supp u , is defined as the complement in U of the union of all open setson which u vanishes. The restriction of the sheaf E to an open set U ⊆ X is denotedby E | U .A locally convex sheaf E is a sheaf E such that E ( U ) is a locally convex space for eachopen set U ⊆ X , the restriction mappings are continuous, and for all open sets U ⊆ X and all open coverings ( U i ) i of U the following property is satisfied:(S3) the topology on E ( U ) coincides with the projective topology on E ( U ) withrespect to the mappings ρ U i ,U .Property (S3) and the fact that X is locally compact imply the canonical isomorphismof locally convex spaces(5.1) E ( U ) ∼ = lim ←− W ⋐ U E ( W ) , where the projective limit is taken with respect to the restriction mappings. Noticethat the algebraic isomorphism in (5.1) holds because of (S1) and (S2).Let E and E be (locally convex) sheaves. A sheaf morphism µ : E → E consistsof (continuous) linear mappings µ U : E ( U ) → E ( U ) for each open set U ⊆ X suchthat, for every inclusion of open sets V ⊆ U , the identity ρ V,U ◦ µ U = µ V ◦ ρ V,U holds.The set of all sheaf morphisms from E into E is denoted by Hom( E , E ) . Theassignment U → Hom( E | U , E | U ) together with the canonical restriction mappings isa sheaf. By abuse of notation we shall also denote this sheaf by Hom( E , E ) . Moregenerally, let E , . . . , E n , E be (locally convex) sheaves on X . A multilinear sheafmorphism T : E × · · · × E n → E consists of (jointly continuous) multilinear mappings T U : E ( U ) × · · · × E n ( U ) → E ( U ) for each open set U ⊆ X such that, for every inclusion of open sets V ⊆ U , we have ρ V,U ( T U ( u , . . . , u n )) = T V ( ρ V,U ( u ) , . . . , ρ V,U ( u n )) if u i ∈ E i ( U ) for i = 1 , . . . , n .A (locally convex) sheaf E is called a (locally convex) sheaf of algebras if for each openset U ⊆ X the space E ( U ) is a (locally convex) algebra and the multiplication is abilinear sheaf morphism.A (locally convex) sheaf E is called fine if for all closed subsets A, B of X with A ∩ B = ∅ there is µ ∈ Hom(
E, E ) and open neighbourhoods U and V of A and B , respectively,such that µ U = id and µ V = 0 . Or, equivalently, if for every open covering ( U i ) i of X there is a family ( η i ) i ⊂ Hom(
E, E ) such that the family of supports of the η i is locallyfinite, supp η i ⊆ U i for all i , and P i η i = id . The family ( η i ) i is called a partitionof unity subordinate to the covering ( U i ) i . We shall often use the following extensionprinciple for (locally convex) fine sheaves E : Let U, V, W be open subsets of X suchthat W ⊂ V ⊆ U . Then, there is a (continuous) linear mapping τ : E ( V ) → E ( U ) such that ρ W,V = ρ W,U ◦ τ .5.2. Localizing regularization operators.
Let X be a Hausdorff locally compactparacompact topological space and E and F locally convex sheaves. We call ( E, F ) a test pair of sheaves if the following three properties are satisfied:(i) F is a subsheaf of E .(ii) ( E ( U ) , F ( U )) is a test pair for each open set U ⊆ X .Given a sheaf morphism µ ∈ Hom(
E, E ) we write µ | F for its restriction to F . Hence µ | F ∈ Hom(
F, F ) means that µ U | F ( U ) is a continuous linear operator from F ( U ) intoitself for each open set U ⊆ X . The third property can then be formulated as follows:(iii) For all open sets U ⊆ X and all closed subsets A, B of U with A ∩ B = ∅ there is µ ∈ Hom( E | U , E | U ) with µ | F ∈ Hom( F | U , F | U ) such that µ V = id and µ W = 0 for some open neighbourhoods V and W (in U ) of A and B , respectively. Or,equivalently, to the fact that for any open set U of X and any open covering ( U i ) i of U there is a partition of unity ( η i ) i ⊂ Hom( E | U , E | U ) subordinate to ( U i ) i such that η i | F | U ∈ Hom( F | U , F | U ) for all i .In particular, property (iii) implies that E | U and F | U are fine sheaves for all open sets U ⊆ X . Moreover, it implies that for all open subsets U, V, W of X with W ⊂ V ⊆ U there is τ ∈ L ( E ( V ) , E ( U )) such that ρ W,V = ρ W,U ◦ τ and τ | F ( V ) ∈ L ( F ( V ) , F ( U )) .Since F is a subsheaf of E , there is no need to make a distinction between the restric-tion mappings on E and F , respectively. These mappings will be denoted by ρ U,V .Furthermore, we introduce the shorthand notation
RO( U ) = L ( E ( U ) , F ( U )) , where RO stands for “regularization operator”. HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 11
Definition 5.1.
Let U ⊆ X be open. An element (Φ ε ) ε ∈ RO( U ) I is called localizing if ( ∀ V, V ⊆ X : V ⋐ V ⋐ U ) ( ∃ ε ∈ I ) ( ∀ ε < ε ) ( ∀ u ∈ E ( U ))( ρ V ,U ( u ) = 0 ⇒ ρ V,U (Φ ε ( u )) = 0) . We write RO loc ( U ) for the set of all localizing elements in RO( U ) I . Furthermore, wedefine TO loc ( U ) = TO loc ( U, S ) := TO( E ( U ) , F ( U ) , S ) ∩ RO loc ( U )TO ( U ) = TO ( U, S ) := TO ( E ( U ) , F ( U ) , S ) ∩ RO loc ( U ) , where S is an admissible pair of scales. Remark . Throughout this subsection we shall always assume that the space TO loc ( U ) is nonempty. Definition 5.3.
Let U ⊆ X be open. We define NO( U ) as the vector space consistingof all (Φ ε ) ε ∈ RO( U ) I such that for all V ⋐ U we have ρ V,U ◦ Φ ε = 0 for ε small enough.Define g RO loc ( U ) := RO loc ( U ) / NO( U ) , g TO ( U ) := TO ( U ) / NO( U ) . For (Φ ε ) ε , (Φ ′ ε ) ε ∈ RO( U ) I we write (Φ ε ) ε ∼ (Φ ′ ε ) ε if (Φ ε − Φ ′ ε ) ε ∈ NO( U ) . Set g TO loc ( U ) := TO loc ( U ) / ∼ . The main goal of this section is to show that one can define a natural sheaf structureon U → g RO loc ( U ) . We start with defining the restriction mappings. Lemma 5.4.
Let
U, V be open subsets of X with V ⊆ U . There is a linear mapping ρ RO V,U : RO( U ) → RO( V ) which is continuous for the strong topologies on RO( U ) and RO( V ) and such that for all (Φ ε ) ε ∈ RO loc ( U ) the following properties hold:(i) We have that ( ∀ W, W ⊆ X : W ⋐ W ⋐ V ) ( ∃ ε ∈ I ) ( ∀ ε < ε ) ( ∀ u ∈ E ( U )) ( ∀ v ∈ E ( V ))( ρ W ,U ( u ) = ρ W ,V ( v ) ⇒ ρ W,V ( ρ RO V,U (Φ ε )( v )) = ρ W,U (Φ ε ( u ))) . (ii) For all W ⋐ V and all τ ∈ L ( E ( V ) , E ( U )) with ρ W ,U ◦ τ = ρ W ,V for some W ⋐ W ⋐ V we have that ρ W,U ◦ Φ ε ◦ τ = ρ W,V ◦ ρ RO V,U (Φ ε ) for ε small enough.(iii) For all W ⋐ V we have that ρ W,V ◦ ρ RO V,U (Φ ε ) ◦ ρ V,U = ρ W,U ◦ Φ ε for ε small enough. (iv) For all W ⋐ V and Φ , Φ ∈ RO loc ( U ) that satisfy ρ W,U ◦ Φ = ρ W,U ◦ Φ we have that ρ W,V ◦ ρ RO V,U (Φ ) = ρ W,V ◦ ρ RO V,U (Φ ) . Proof.
Let ( V i ) i be an open covering of V such that V i ⋐ V for all i . Let ( η i ) i ⊂ Hom( F | V , F | V ) be a partition of unity subordinate to ( V i ) i and choose τ i ∈ L ( E ( V ) , E ( U )) such that ρ V i ,V = ρ V i ,U ◦ τ i for all i . We define ρ RO V,U (Φ) := X i η iV ◦ ρ V,U ◦ Φ ◦ τ i . For all W ⋐ V it holds that supp η i ∩ W = ∅ except for i belonging to some finiteindex set J . Hence(5.2) ρ W,V ◦ ρ RO V,U (Φ) = X i ∈ J η iW ◦ ρ W,U ◦ Φ ◦ τ i , By (5.1) we then have that ρ RO V,U (Φ) ∈ RO( V ) . The linearity and continuity of ρ RO V,U and also (iv) are clear from this expression. We now show (i). Let W ⋐ V and W ⋐ W ⋐ V be arbitrary. Suppose that the representation (5.2) holds for some finiteindex set J . Choose V ′ i ⋐ V i such that supp η i ⊂ V ′ i . Since (Φ ε ) ε is localizing, there is ε ∈ I such that for all i ∈ J , ε < ε , and u ∈ E ( U ) it holds that(5.3) ρ W ∩ V i ,U ( u ) = 0 ⇒ ρ W ∩ V ′ i ,U (Φ ε ( u )) = 0 . Assume that u ∈ E ( U ) and v ∈ E ( V ) are given such that ρ W ,U ( u ) = ρ W ,V ( v ) . Since ρ W,U ◦ Φ ε = X i ∈ J η iW ◦ ρ W,U ◦ Φ ε and supp η i ⊂ V ′ i it suffices to show that ρ W ∩ V ′ i ,U (Φ ε ( u − τ i ( v ))) = 0 for all i ∈ J . This follows from (5.3) and our choice of τ i . Properties (ii) and (iii) arespecial cases of (i). (cid:3) Lemma 5.5.
Let
U, V be open subsets of X with V ⊆ U . Then, for all (Φ ε ) ε ∈ RO loc ( U ) it holds that(i) ( ρ RO V,U (Φ ε )) ε ∈ RO loc ( V ) ,(ii) if (Φ ε ) ε ∼ , then ( ρ RO V,U (Φ ε )) ε ∼ ,(iii) for W ⊆ V ⊆ U it holds that (( ρ RO W,V ◦ ρ RO V,U )(Φ ε )) ε ∼ ( ρ RO W,U (Φ ε )) ε .Proof. (i) Let W ⋐ V and W ⋐ W ⋐ V be arbitrary. Since (Φ ε ) ε is localizing there is ε ∈ I such that such that for all ε < ε and all u ∈ E ( U ) it holds that(5.4) ρ W ,U ( u ) = 0 ⇒ ρ W,U (Φ ε ( u )) = 0 . HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 13
Choose τ ∈ L ( E ( V ) , E ( U )) such that ρ W ,U ◦ τ = ρ W ,V . By Lemma 5.4 (ii) there is ε ∈ I such that ρ W,V ◦ ρ RO V,U (Φ ε ) = ρ W,U ◦ Φ ε ◦ τ for all ε < ε . Set ε = min( ε , ε ) . Let v ∈ E ( V ) be such that ρ W ,V ( v ) = 0 . Hence ρ W,V ( ρ RO V,U (Φ ε )( v )) = ρ W,U (Φ ε ( τ ( v ))) = 0 for all ε < ε .(ii) Let W ⋐ V be arbitrary. Choose τ ∈ L ( E ( V ) , E ( U )) such that ρ W ,U ◦ τ = ρ W ,V .By Lemma 5.4 (ii) we have that ρ W,V ◦ ρ RO V,U (Φ ε ) = ρ W,U ◦ Φ ε ◦ τ = 0 for ε small enough because (Φ ε ) ε ∼ .(iii) Let W ⋐ W be arbitrary. Fix an open set W ′ such that W ⋐ W ′ ⋐ W . Choose τ ∈ L ( E ( V ) , E ( U )) such that ρ W ′ ,U ◦ τ = ρ W ′ ,V and τ ′ ∈ L ( E ( W ) , E ( V )) such that ρ W ′ ,V ◦ τ = ρ W ′ ,W . Hence τ ◦ τ ′ ∈ L ( E ( W ) , E ( U )) and ρ W ′ ,U ◦ τ ◦ τ ′ = ρ W ′ ,W ByLemma 5.4 (ii) we have that ρ W ,W ◦ ρ RO W,V ( ρ RO V,U (Φ ε )) = ρ W ,V ◦ ρ RO V,U (Φ ε ) ◦ τ = ρ W ,U ◦ Φ ε ◦ τ ′ ◦ τ = ρ W ,W ◦ ρ RO W,U (Φ ε ) for ε small enough. (cid:3) Lemma 5.5 implies that the mappings ρ RO V,U ([(Φ ε ) ε ]) := [( ρ RO V,U (Φ ε )) ε ] define a presheaf structure on U → g RO loc ( U ) . We now show that it is in fact a sheaf. Proposition 5.6. g RO loc is a sheaf of vector spaces.Proof. Let U ⊆ X be open and let ( U i ) i be an open covering of U .(S1) Suppose that [(Φ ε ) ε ] ∈ g RO loc ( U ) such that ρ RO U i ,U ([(Φ ε ) ε ]) = 0 for all i . We need to show that (Φ ε ) ε ∼ . Let W ⋐ U be arbitrary. We mayassume without loss of generality that W ⋐ U i for some i . By Lemma 5.4 (iii) and ourassumption we have that ρ W,U ◦ Φ ε = ρ W,U i ◦ ρ RO U i ,U (Φ ε ) ◦ ρ U i ,U = 0 for ε small enough.(S2) Since X is locally compact we may assume without loss of generality that U i ⋐ U for all i . Suppose that [(Φ i,ε ) ε ] ∈ g RO loc ( U i ) are given such that ρ RO U i ∩ U j ,U i ([(Φ i,ε ) ε ]) = ρ RO U i ∩ U j ,U j ([(Φ j,ε ) ε ]) for all i, j . Let ( η i ) i ⊂ Hom( F | U , F | U ) be a partition of unity subordinate to thecovering ( U i ) i . Choose τ i ∈ L ( F ( U i ) , F ( U )) such that ρ V i ,U ◦ τ i = ρ V i ,U i for some V i ⋐ U i with supp η i ⊂ V i . We define Φ ε = X i η iU ◦ τ i ◦ Φ i,ε ◦ ρ U i ,U for all ε ∈ I . Notice that Φ ε ∈ RO( U ) because of (5.1) and the fact that the family ofsupports of the η i is locally finite. We now show that (Φ ε ) ε is localizing. Let W ⋐ U and W ⋐ W ⋐ U be arbitrary and suppose that supp η i ∩ W = ∅ except for i belongingto some finite index set J . Choose V ′ i ⋐ U i such that V i ⋐ V ′ i . Since the (Φ i,ε ) ε arelocalizing there is ε ∈ I such that for all i ∈ J , ε < ε , and u ∈ E ( U i ) it holds that(5.5) ρ W ∩ V ′ i ,U i ( u ) = 0 ⇒ ρ W ∩ V i ,U i (Φ i,ε ( u )) = 0 . Now suppose that u ∈ E ( U ) satisfies ρ W ,U ( u ) = 0 . Since ρ W,U (Φ ε ( u )) = X i ∈ J η iW ( ρ W,U ( τ i (Φ i,ε ( ρ U i ,U ( u ))))) and supp η i ⊂ V i it suffices to show that ρ W ∩ V i ,U ( τ i (Φ i,ε ( ρ U i ,U ( u )))) = ρ W ∩ V i ,U i (Φ i,ε ( ρ U i ,U ( u ))) = 0 for all i ∈ J . This follows from (5.5). Finally, we show that ρ RO U i ,U ([(Φ ε ) ε ]) = [(Φ i,ε ) ε ] for all i . Let W ⋐ U i be arbitrary and suppose that supp η j ∩ W = ∅ except for j belonging to some finite index set J . Let τ ∈ L ( E ( U i ) , E ( U )) be such that ρ W ,U ◦ τ = ρ W ,U i where W is some open set such that W ⋐ W ⋐ U i . Lemma 5.4 (ii) yields that ρ W,U i ◦ ρ RO U i ,U (Φ ε ) − ρ W,U i ◦ Φ i,ε = ρ W,U ◦ Φ ε ◦ τ − ρ W,U i ◦ Φ i,ε = X j ∈ J η jW ◦ ( ρ W,U ◦ τ j ◦ Φ j,ε ◦ ρ U j ,U ◦ τ − ρ W,U i ◦ Φ i,ε ) . Since supp η j ⊂ V j it suffices to show that ρ W ∩ V j ,U ◦ τ j ◦ Φ j,ε ◦ ρ U j ,U ◦ τ − ρ W ∩ V j ,U i ◦ Φ i,ε = 0 for all j ∈ J and ε small enough. Our choice of τ j and Lemma 5.4 (ii) and (iii) implythat ρ W ∩ V j ,U ◦ τ j ◦ Φ j,ε ◦ ρ U j ,U ◦ τ − ρ W ∩ V j ,U i ◦ Φ i,ε = ρ W ∩ V j ,U j ◦ Φ j,ε ◦ ρ U j ,U ◦ τ − ρ W ∩ V j ,U i ◦ Φ i,ε = ρ W ∩ V j ,U i ∩ U j ◦ ρ RO U i ∩ U j ,U j (Φ j,ε ) ◦ ρ U i ∩ U j ,U ◦ τ − ρ W ∩ V j ,U i ◦ Φ i,ε = ρ W ∩ V j ,U i ∩ U j ◦ ρ RO U i ∩ U j ,U i (Φ i,ε ) ◦ ρ U i ∩ U j ,U ◦ τ − ρ W ∩ V j ,U i ◦ Φ i,ε = ρ W ∩ V j ,U i ◦ Φ i,ε ◦ ρ U i ,U ◦ τ − ρ W ∩ V j ,U i ◦ Φ i,ε which equals zero for ε small enough because (Φ i,ε ) ε is localizing. (cid:3) HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 15
Lemma 5.7.
Every sheaf morphism µ ∈ Hom(
F, F ) induces a sheaf morphism µ ∈ Hom( g RO loc , g RO loc ) via (5.6) µ U ([(Φ ε ) ε ]) := [( µ U ◦ Φ ε ) ε ] , with U an open subset of X .Proof. Clearly, µ U : g RO loc ( U ) → g RO loc ( U ) is a well-defined linear mapping for each U ⊆ X open. We now show that µ is a sheaf morphism. Let V, U be open subsets of X such that V ⊆ U . It suffices to show that for all W ⋐ V and all (Φ ε ) ε ∈ RO loc ( U ) it holds that ρ W,V ◦ ρ RO V,U ( µ U ◦ Φ ε ) = ρ W,V ◦ µ V ◦ ρ RO V,U (Φ ε ) for ε small enough. Let τ ∈ L ( E ( V ) , E ( U )) be such that ρ W ,U ◦ τ = ρ W ,V for someopen set W such that W ⋐ W ⋐ V . By Lemma 5.4 (ii) we have that ρ W,V ◦ ρ RO V,U ( µ U ◦ Φ ε ) = ρ W,U ◦ µ U ◦ Φ ε ◦ τ = µ W ◦ ρ W,U ◦ Φ ε ◦ τ = µ W ◦ ρ W,V ◦ ρ RO V,U (Φ ε )= ρ W,V ◦ µ V ◦ ρ RO V,U (Φ ε ) for ε small enough. (cid:3) We now turn our attention to spaces of test objects.
Lemma 5.8.
Let U ⊆ X be open and let ( U i ) i be an open covering of U . Let (Φ ε ) ε ∈ RO loc ( U ) . Then, (Φ ε ) ε ∈ TO loc ( U ) ( (Φ ε ) ε ∈ TO ( U ) , respectively) if and only if ( ρ RO U i ,U (Φ ε )) ε ∈ TO loc ( U i ) ( ( ρ RO U i ,U (Φ ε )) ε ∈ TO ( U i ) , respectively) for all i .Proof. We only show the statement for TO loc , the proof for TO is similar. Let (Φ ε ) ε ∈ RO loc ( U ) . We first assume that (Φ ε ) ε satisfies (TO) j with j = 1 , or , andprove that ( ρ RO U i ,U (Φ ε )) ε does so as well. j = 1 : It suffices to show that for all u ∈ E ( U i ) and all p ∈ csn( F ( W )) , with W ⋐ U i arbitrary, there is λ ∈ A such that p ( ρ W,U i ( ρ RO U i ,U (Φ ε )( u )) = O ( λ ε ) . Let τ ∈ L ( E ( U i ) , E ( U )) such that ρ W ,U ◦ τ = ρ W ,U i where W is an open set suchthat W ⋐ W ⋐ U i . By Lemma 5.4 (ii) we have that ρ W,U i ( ρ RO U i ,U (Φ ε )( u )) = ρ W,U (Φ ε ( τ ( u ))) for ε small enough. The result now follows from our assumption and the fact that ρ W,U ∈ L ( F ( U ) , F ( V )) . j = 2 : It suffices to show that for all ϕ ∈ F ( U i ) , all p ∈ csn( F ( W )) , with W ⋐ U i arbitrary, and all λ ∈ I it holds that p ( ρ W,U i ( ρ RO U i ,U (Φ ε )( ϕ ) − ϕ )) = O ( λ ε ) . Let τ ∈ L ( F ( U i ) , F ( U )) such that ρ W ,U ◦ τ = ρ W ,U i where W is an open set such that W ⋐ W ⋐ U i . By Lemma 5.4 (ii) we have that ρ W,U i ( ρ RO U i ,U (Φ ε )( ϕ ) − ϕ ) = ρ W,U (Φ ε ( τ ( ϕ )) − τ ( ϕ )) for ε small enough. The result now follows from our assumption and the fact that ρ W,U ∈ L ( F ( U ) , F ( W )) . j = 3 : It suffices to show that for all u ∈ E ( U i ) and all p ∈ csn( E ( W )) , with W ⋐ U i arbitrary, it holds that p ( ρ W,U i ( ρ RO U i ,U (Φ ε )( u ) − u )) → . Let τ ∈ L ( E ( U i ) , E ( U )) such that ρ W ,U ◦ τ = ρ W ,U i where W is an open set suchthat W ⋐ W ⋐ U i . By Lemma 5.4 (ii) we have that ρ W,U i ( ρ RO U i ,U (Φ ε )( u ) − u ) = ρ W,U (Φ ε ( τ ( u )) − τ ( u )) for ε small enough. The result now follows from our assumption and the fact that ρ W,U ∈ L ( E ( U ) , E ( V )) .Conversely, assume that ( ρ RO U i ,U (Φ ε )) ε satisfies TO j with j = 1 , or for each i . Wewill prove that (Φ ε ) ε does so as well. j = 1 : It suffices to show that for all u ∈ E ( U ) and all p ∈ csn( F ( W )) , with W ⋐ U i (for some i ) arbitrary, there is λ ∈ A such that p ( ρ W,U ((Φ ε ( u )))) = O ( λ ε ) . By Lemma 5.4 (iii) we have that ρ W,U (Φ ε )( u ) = ρ W,U i ( ρ RO U i ,U (Φ ε )( ρ U i ,U ( u ))) for ε small enough and the result follows from our assumption and the fact that ρ W,U i ∈L ( F ( U i ) , F ( W )) . j = 2 : It suffices to show that for all ϕ ∈ F ( U ) and all p ∈ csn( F ( W )) , with W ⋐ U i (for some i ) arbitrary, and all λ ∈ I it holds that p ( ρ W,U ((Φ ε )( ϕ ) − ϕ )) = O ( λ ε ) . By Lemma 5.4 (iii) we have that ρ W,U ((Φ ε )( ϕ ) − ϕ ) = ρ W,U i ( ρ RO U i ,U (Φ ε )( ρ U i ,U ( ϕ )) − ρ U i ,U ( ϕ )) for ε small enough and the result follows from our assumption and the fact that ρ W,U i ∈L ( F ( U i ) , F ( W )) . j = 3 : It suffices to show that for all u ∈ E ( U ) and all p ∈ csn( E ( W )) , with W ⋐ U arbitrary, it holds that p ((Φ ε )( u ) − u ) → . By Lemma 5.4 (iii) we have that ρ W,U ((Φ ε )( u ) − u ) = ρ W,U i ( ρ RO U i ,U (Φ ε )( ρ U i ,U ( u )) − ρ U i ,U ( u )) for ε small enough and the result follows from our assumption and the fact that ρ W,U i ∈L ( E ( U i ) , E ( W )) . (cid:3) HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 17
The following result is an immediate consequence of Lemma 4.2 and Lemma 5.7.
Lemma 5.9. (i) Let µ i ∈ Hom(
E, E ) , i = 0 , . . . , N , N ∈ N , be such that µ i | F ∈ Hom(
F, F ) and P Ni =0 µ i = id . Then, (cid:16)P Ni =0 µ iU ◦ Φ i,ε (cid:17) ε ∈ TO loc ( U ) for all (Φ i,ε ) ε ∈ TO loc ( U ) , i = 0 , . . . , N .(ii) Let µ ∈ Hom(
E, E ) , be such that µ | F ∈ Hom(
F, F ) . Then, ( µ U ◦ Φ ε ) ε ∈ TO ( U ) for all (Φ ε ) ε ∈ TO ( U ) .(iii) Let µ ∈ Hom(
E, E ) be such that µ | F ∈ Hom(
F, F ) . Then, ( µ U ◦ Φ ε − Φ ε ◦ µ U ) ε ∈ TO ( U ) for all (Φ ε ) ε ∈ TO loc ( U ) ∪ TO ( U ) . We conclude this subsection with a lemma that will be very useful later on.
Lemma 5.10.
Let
W, V, U be open sets in X such that W ⋐ V ⊆ U . For every (Φ ε ) ε ∈ RO loc ( V ) ( (Φ ε ) ε ∈ TO loc ( V ) , (Φ ε ) ε ∈ TO ( V ) respectively) there is (Φ ′ ε ) ε ∈ RO loc ( U ) ( (Φ ′ ε ) ε ∈ TO loc ( U ) , (Φ ′ ε ) ε ∈ TO ( U ) respectively) such that ρ W,V ◦ Φ ε ◦ ρ V,U = ρ W,U ◦ Φ ′ ε for ε small enough.Proof. We only show the statement for (Φ ε ) ε ∈ TO loc ( V ) , the other cases can betreated similarly. Choose open sets W , W such that W ⋐ W ⋐ W ⋐ V and let µ ∈ Hom(
E, E ) be such that µ | F ∈ Hom(
F, F ) , µ W = id , and µ U \ W = 0 . Furthermore,pick an arbitrary element (Φ ′′ ε ) ε ∈ TO loc ( U ) . By Lemma 5.8 we have that ρ RO U \ W ,U ([(Φ ′′ ε ) ε ]) ∈ g TO loc ( U \ W ) , and by Lemma 5.9 it holds that µ V ([(Φ ε ) ε ]) + (id − µ ) V ( ρ RO V,U ([(Φ ′′ ε ) ε ])) ∈ g TO loc ( V ) . Since ρ RO U \ W ∩ V,U \ W ( ρ RO U \ W ,U ([(Φ ′′ ε ) ε ])) = ρ RO U \ W ∩ V,V ( µ V ([(Φ ε ) ε ]) + (id − µ ) V ( ρ RO V,U ([(Φ ′′ ε ) ε ]))) , Proposition 5.6 and Lemma 5.8 imply that there is an element (Φ ′ ε ) ε ∈ TO loc ( U ) suchthat ρ RO W ,U ([(Φ ′ ε ) ε ]) = ρ RO W ,V ( µ V ([(Φ ε ) ε ]) + (id − µ ) V ( ρ RO V,U ([(Φ ′′ ε ) ε ]))) = ρ RO W ,V ([(Φ ε ) ε ]) . The result now follows from Lemma 5.4 (iii). (cid:3)
Sheaves of nonlinear extensions.
Let ( E, F ) be a test pair of sheaves. Wewrite E ( U ) = E ( E ( U ) , F ( U )) . Definition 5.11. R ∈ E ( U ) is called local if for all V ⊆ U and all Φ , Φ ∈ RO( U ) theimplication ( ρ V,U ◦ Φ = ρ V,U ◦ Φ ) = ⇒ ( ρ V,U ( R (Φ )) = ρ V,U ( R (Φ ))) holds. The set of all local elements of E ( U ) is denoted by E loc ( U ) . Remark . If R ∈ E ( U ) is local then the identities ρ V,U ◦ Φ = ρ V,U ◦ Φ , ρ V,U ◦ Ψ i, = ρ V,U ◦ Ψ i, , with Φ , Φ , Ψ ,i , Ψ ,i ∈ RO( U ) for i = 1 , . . . , l imply that ρ V,U (d l R (Φ )(Ψ , , . . . , Ψ l, )) = ρ V,U ((d l R )(Φ )(Ψ , , . . . , Ψ l, )) . Next, we define a restriction mapping on E loc . Lemma 5.13.
Let
U, V be open subsets of X with V ⊆ U . There is a unique linearmapping ρ E V,U : E loc ( U ) → E loc ( V ) such that(i) for all W ⋐ V , Φ ∈ RO( V ) , and Φ ′ ∈ RO( U ) which satisfy ρ W,V ◦ Φ ◦ ρ V,U = ρ W,U ◦ Φ ′ , we have ρ W,V ( ρ E V,U ( R )(Φ)) = ρ W,U ( R (Φ ′ )) . Moreover, the following properties are satisfied:(ii) For all l ∈ N and all W ⋐ V it holds that if Φ ∈ RO( V ) , Φ ′ ∈ RO( U ) and Ψ i ∈ RO( V ) , Ψ ′ i ∈ RO( U ) , i = 0 , . . . , l , satisfy ρ W,V ◦ Φ ◦ ρ V,U = ρ W,U ◦ Φ ′ , ρ W,V ◦ Ψ i ◦ ρ V,U = ρ W,U ◦ Ψ ′ i for all i = 1 , . . . , l , then ρ W,V ((d l ( ρ E V,U ( R )))(Φ)(Ψ , . . . , Ψ l )) = ρ W,U ((d l R )(Φ ′ )(Ψ ′ , . . . , Ψ ′ l )) . (iii) For W ⊆ V ⊆ U it holds that ρ E W,V ◦ ρ E V,U = ρ E W,U .Proof.
Let ( V i ) i be an open covering of V such that V i ⋐ V for all i and let ( η i ) i ⊂ Hom( F | V , F | V ) be a partition of unity subordinate to ( V i ) i . Choose τ i ∈ L ( F ( V ) , F ( U )) such that ρ V i ,U ◦ τ i = ρ V i ,V . For each i we define the mapping f i ∈ L (RO( V ) , RO( U )) via f i (Φ) := τ i ◦ Φ ◦ ρ V,U . Note that(5.7) ρ V i ,U ◦ f i (Φ) = ρ V i ,V ◦ Φ ◦ ρ V,U . We set ρ E V,U ( R ) := X i η iV ◦ ρ V,U ◦ R ◦ f i . We start by showing that ρ E V,U ( R ) is smooth. By [15, Lemma 3.8] it suffices to showthat ρ W,V ◦ ρ E V,U ( R ) : RO( V ) → F ( W ) is smooth for all W ⋐ V . Since(5.8) ρ W,V ◦ ρ E V,U ( R ) = X i ∈ J η iW ◦ ρ W,U ◦ R ◦ f i HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 19 for some finite index set J , this follows from the fact that η iW , ρ W,U , and f i are contin-uous linear mappings. Next, we show that ρ E V,U ( R ) is local. It suffices to show that forall W ⋐ V , ρ W,V ◦ Φ = ρ W,V ◦ Φ , Φ , Φ ∈ RO( V ) , implies ρ W,V ( ρ E V,U ( R )(Φ )) = ρ W,V ( ρ E V,U ( R )(Φ )) . The mapping ρ W,V ◦ ρ E V,U ( R ) can be represented as (5.8) for some finite index set J .Since supp η i ⊂ V i it suffices to show that ρ W ∩ V i ,U ( R ( f i (Φ ))) = ρ W ∩ V i ,U ( R ( f i (Φ ))) for all i ∈ J . By locality of R this follows from (5.7) and our assumption. Thelinearity of the mapping ρ E V,U is clear. We now show (i). Given W ⋐ V , the mapping ρ W,V ◦ ρ E V,U ( R ) can be represented as (5.8) for some finite index set J . Since ρ W,U ( R (Φ ′ )) = X i ∈ J η iW ( ρ W,U ( R (Φ ′ ))) and supp η i ⊂ V i it is enough to show that ρ W ∩ V i ,U ( R ( f i (Φ))) = ρ W ∩ V i ,U ( R (Φ ′ )) for all i ∈ J . Again, by locality of R this follows from (5.7) and our assumption. Themapping ρ E V,U is unique because for any W ⋐ V and any Φ ∈ RO( V ) one can find Φ ′ ∈ RO( U ) such that ρ W,V ◦ Φ ◦ ρ V,U = ρ W,U ◦ Φ ′ ; this follows from the fact that F isfine. We continue with showing (ii). We use induction on l . The case l = 0 has beentreated in (i). Now suppose that the statement holds for l − and let us show it for l . ρ W,V ((d l ( ρ E V,U ( R )))(Φ)(Ψ , . . . , Ψ l ))= ρ W,V (cid:18) dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 (d l − ( ρ E V,U ( R )))(Φ + t Ψ )(Ψ , . . . , Ψ l ) (cid:19) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ρ W,V ((d l − ( ρ E V,U ( R )))(Φ + t Ψ )(Ψ , . . . , Ψ l ))= dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ρ W,V ((d l − ( ρ E V,U ( R )))(Φ ′ + t Ψ ′ )(Ψ ′ , . . . , Ψ ′ l ))= ρ W,V (cid:18) dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 (d l − ( ρ E V,U ( R )))(Φ ′ + t Ψ ′ )(Ψ ′ , . . . , Ψ ′ l ) (cid:19) = ρ W,V ((d l ( ρ E V,U ( R )))(Φ ′ )(Ψ ′ , . . . , Ψ ′ l )) . Finally, we prove (iii). Let R ∈ E loc ( U ) be arbitrary. It suffices to show that for all Φ ∈ RO( W ) and all W ⋐ W it hold that ρ W ,W ( ρ E W,V ( ρ E V,U ( R ))(Φ)) = ρ W ,W ( ρ E W,U ( R )(Φ)) . Choose Φ ′ ∈ RO( V ) such that ρ W ,W ◦ Φ ◦ ρ W,V = ρ W ,V ◦ Φ ′ and Φ ′′ ∈ RO( U ) such that ρ W ,V ◦ Φ ′ ◦ ρ V,U = ρ W ,U ◦ Φ ′′ . Hence also ρ W ,W ◦ Φ ◦ ρ W,U = ρ W ,U ◦ Φ ′′ . Therefore (i) implies that ρ W ,W ( ρ E W,V ( ρ E V,U ( R ))(Φ)) = ρ W ,V ( ρ E V,U ( R )(Φ ′ ))= ρ W ,U ( R (Φ ′′ ))= ρ W ,W ( ρ E W,U ( R )(Φ)) . (cid:3) We now discuss the extension of sheaf morphisms to E . Lemma 5.14.
Let T : F ×· · ·× F → F be a multilinear sheaf morphism. For each opensubset U ⊆ X consider the mapping e T U : E ( U ) × · · · × E ( U ) → E ( U ) given by e T U := f T U as in (3.1) . Then, e T preserves locality, i.e., e T U ( E loc ( U ) , . . . , E loc ( U )) ⊆ E loc ( U ) and ρ E V,U ( e T U ( R , . . . , R n )) = e T V ( ρ E V,U ( R ) , . . . , ρ E V,U ( R n )) for all open subsets U, V of X with V ⊆ U .Proof. The mappings T U are well-defined by Lemma 3.2. Moreover, the fact that the T U preserve locality is clear from their definition. In order to show the last property itsuffices to show that ρ W,V ( ρ E V,U ( T U ( R , . . . , R n ))(Φ)) = ρ W,V ( T V ( ρ E V,U ( R ) , . . . , ρ E V,U ( R n ))(Φ)) for all Φ ∈ RO( V ) and all W ⋐ V . Choose Φ ′ ∈ RO( U ) such that ρ W,V ◦ Φ ◦ ρ V,U = ρ W,U ◦ Φ ′ . Lemma 5.13 (i) implies that ρ W,V ( ρ E V,U ( T U ( R , . . . , R n ))(Φ)) = ρ W,U ( T U ( R , . . . , R n )(Φ ′ ))= ρ W,U ( T U ( R (Φ ′ ) , . . . , R n (Φ ′ )))= T W ( ρ W,U ( R (Φ ′ )) , . . . , ρ W,U ( R n (Φ ′ )))= T W ( ρ W,V ( ρ E V,U ( R )(Φ)) , . . . , ρ W,V ( ρ E V,U ( R n )(Φ)))= ρ W,V ( T V ( ρ E V,U ( R )(Φ) , . . . , ρ E V,U ( R n )(Φ)))= ρ W,V ( T V ( ρ E V,U ( R ) , . . . , ρ E V,U ( R n ))(Φ)) . (cid:3) Lemma 5.15.
Let ( E , F ) , ( E , F ) be test pairs of sheaves. Suppose we are given asheaf isomorphism µ : E → E such that its restriction to F is a sheaf isomorphism µ : F → F . For each open subset U ⊆ X consider the mapping ( µ ∗ ) U : E ( E ( U ) , F ( U )) →E ( E ( U ) , F ( U )) given by ( µ ∗ ) U := ( µ U ) ∗ as in (3.3) . Then, µ ∗ preserves locality, i.e., ( µ ∗ ) U ( E loc ( U )) ⊆ E loc ( U ) , and ρ E V,U (( µ ∗ ) U R ) = ( µ ∗ ) V ( ρ E V,U ( R )) . HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 21
Proof.
Suppose we are given open subsets
V, U ⊆ X with V ⊆ U , R ∈ E loc ( E ( U ) , F ( U )) and Φ , Φ ∈ RO( U ) with ρ V,U ◦ Φ = ρ V,U ◦ Φ . We first need to show that ρ V,U ((( µ ∗ ) U R )(Φ )) = ρ V,U ((( µ ∗ ) U R )(Φ )) . For this we notice that ρ V,U ( µ U ( R ( µ − U ◦ Φ ◦ µ U ))) = µ V ( ρ V,U ( R ( µ − U ◦ Φ ◦ µ U )))= µ V ( ρ V,U ( R ( µ − U ◦ Φ ◦ µ U ))) = ρ V,U ( µ U ( R ( µ − U ◦ Φ ◦ µ U ))) because R is local and ρ V,U ◦ µ − U ◦ Φ ◦ µ U = µ − V ◦ ρ V,U ◦ Φ ◦ µ U = µ − V ◦ ρ V,U ◦ Φ ◦ µ U = ρ V,U ◦ µ − U ◦ Φ ◦ µ U . For the second statement it suffices to show that ρ W,V ( ρ E V,U (( µ ∗ ) U R )(Φ)) = ρ W,V (( µ ∗ ) V ( ρ E V,U ( R ))(Φ)) for all W ⋐ V and all Φ ∈ RO( V ) . Choose Φ ′ ∈ RO( U ) such that ρ W,V ◦ Φ ◦ ρ V,U = ρ W,U ◦ Φ ′ . Then, ρ W,V ( ρ E V,U (( µ ∗ ) U R )(Φ)) = ρ W,U ((( µ ∗ ) U R )(Φ ′ ))= ρ W,U ( µ U ( R ( µ − U ◦ Φ ′ ◦ µ U )))= µ W ( ρ W,U ( R ( µ − U ◦ Φ ′ ◦ µ U )))= µ W ( ρ W,V (( ρ E V,U R )( µ − V ◦ Φ ◦ µ V )))= ρ W,V (( µ ∗ ) V ( ρ E V,U ( R ))(Φ)) where we used that ρ W,V ◦ ( µ − V ◦ Φ ◦ µ V ) ◦ ρ V,U = ρ W,U ◦ ( µ − U ◦ Φ ′ ◦ µ U ) . (cid:3) Lemma 5.16.
Let T : E → E be a sheaf morphism such that T | F : F → F is a sheafmorphism. For any open subset U ⊆ X consider the mapping b T U : E ( U ) → E ( U ) given by b T U := c T U as in (3.4) . Then, b T preserves locality, i.e., b T U ( E loc ( U )) ⊆ E loc ( U ) and ρ E V,U ( b T U ( R )) = b T V ( ρ E V,U ( R )) .Proof. Suppose we are given open sets
U, V with V ⊆ U , R ∈ E loc ( U ) and Φ , Φ ∈ RO( U ) with ρ V,U ◦ Φ = ρ V,U ◦ Φ . We see that ρ V,U (( b T U R )(Φ )) = ρ V,U ( T U ( R (Φ )) − d R (Φ )( T U ◦ Φ − Φ ◦ T U ))= T V ( ρ V,U ( R (Φ ))) − ρ V,U (d R (Φ )( T U ◦ Φ − Φ ◦ T U ))= T V ( ρ V,U ( R (Φ ))) − ρ V,U (d R (Φ )( T U ◦ Φ − Φ ◦ T U ))= ρ V,U (( b T U R )(Φ )) because ρ V,U ◦ ( T U ◦ Φ − Φ ◦ T U ) = T V ◦ ρ V,U ◦ Φ − ρ V,U ◦ Φ ◦ T U = T V ◦ ρ V,U ◦ Φ − ρ V,U ◦ Φ ◦ T U = ρ V,U ◦ ( T U ◦ Φ − Φ ◦ T U ) . For the second statement, let W ⋐ V and Φ ∈ RO( V ) . Choose Φ ′ ∈ RO( U ) such that ρ W,V ◦ Φ ◦ ρ V,U = ρ W,U ◦ Φ ′ . Then, ρ W,V ( ρ E V,U ( b T U ( R ))(Φ)) = ρ W,U ( b T U ( R )(Φ ′ ))= ρ W,U ( T U ( R (Φ ′ )) − d R (Φ ′ )( T U ◦ Φ ′ − Φ ′ ◦ T U ))= T W ( ρ W,U ( R (Φ ′ ))) − ρ W,V (d( ρ E V,U R )(Φ)( T V ◦ Φ − Φ ◦ T V ))= ρ W,V ( T V (( ρ E V,U R )(Φ)) − d( ρ E V,U R )(Φ)( T V ◦ Φ − Φ ◦ T V ))= ρ W,V ( b T V ( ρ E V,U R )(Φ)) . (cid:3) We now make the quotient construction (see Definition 4.5).
Definition 5.17.
Let S be an admissible pair of scales. For any open subset U ⊆ X we define the space of moderate elements of E loc ( U ) (with respect to S ) as E M , loc ( U ) = E M , loc ( U, S ) := E M ( E ( U ) , F ( U ) , TO loc ( U ) , TO ( U ) , S ) ∩ E loc ( U ) , and the space of negligible elements (with respect to S ) as E N , loc ( U ) = E N , loc ( U, S ) := E N ( E ( U ) , F ( U ) , TO loc ( U ) , TO ( U ) , S ) ∩ E loc ( U ) . We set G loc ( U ) = G loc ( U, S ) := E M , loc ( U ) / E N , loc ( U ) . Lemma 5.18.
Let U ⊆ X be open and let ( U i ) i be an open covering. Let R ∈ E loc ( U ) .Then, R is moderate (negligible, respectively) if and only if ρ E U i ,U ( R ) is moderate (neg-ligible, respectively) for all i .Proof. Let R ∈ E loc ( U ) be moderate or negligible. The moderateness or negligibility of ρ E U i ,U ( R ) is determined by p ( ρ W,U i ((d l ( ρ E U i ,U ( R )))(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))) for ε small enough, where l ∈ N , Φ ε ∈ TO loc ( U i ) , Ψ j,ε ∈ TO ( U i ) for j = 1 , . . . , l , W ⋐ U i , and p ∈ csn( F ( W )) are arbitrary. By Lemma 5.10 there are (Φ ′ ε ) ∈ TO loc ( U ) and Ψ ′ j,ε ∈ TO ( U ) such that ρ W,U i ◦ Φ ε ◦ ρ U i ,U = ρ W,U ◦ Φ ′ ε , ρ W,U i ◦ Ψ j,ε ◦ ρ U i ,U = ρ W,U ◦ Ψ ′ j,ε for all j = 1 , . . . , l and ε small enough. Hence Lemma 5.13 (ii) implies that ρ W,U i ((d l ( ρ E U i ,U ( R )))(Φ ε )(Ψ ,ε , . . . , Ψ l,ε )) = ρ W,U ((d l ( R ))(Φ ′ ε )(Ψ ′ ,ε , . . . , Ψ ′ l,ε )) for ε small enough. The moderateness or negligibility of ρ E U i ,U ( R ) therefore follows fromthe corresponding property of R and the continuity of ρ W,U . Conversely, suppose that ρ E U i ,U ( R ) is moderate or negligible for all i . The moderateness of R is determined by p ( ρ W,U ((d l R )(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))) HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 23 for ε small enough, where l ∈ N , Φ ε ∈ TO loc ( U ) , Ψ j,ε ∈ TO ( U ) for j = 1 , . . . , l , W ⋐ U i (for some i ), and p ∈ csn( F ( W )) are arbitrary. Lemma 5.4 (iii) and Lemma 5.13(ii) imply that ρ W,U ((d l R )(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))= ρ W,U i ((d l ( ρ E U i ,U ( R )))( ρ RO U i ,U (Φ ε ))( ρ RO U i ,U (Ψ ,ε ) , . . . , ρ RO U i ,U (Ψ l,ε ))) for ε small enough. The moderateness or negligibility of R therefore follows from thecorresponding property of ρ E U i ,U ( R ) and the continuity of ρ W,U i . (cid:3) Lemma 5.13 and Lemma 5.18 imply that the mappings ρ G V,U ([ R ]) := [ ρ E V,U ( R )] define a presheaf structure on U → G loc ( U ) . We now show that it is in fact a sheaf. Proposition 5.19. G loc is a sheaf of vector spaces.Proof. (S1) Immediate consequence of Lemma 5.18.(S2) Let U ⊆ X be open and let ( U i ) i be an open covering of U . Since X is locallycompact we may assume without loss of generality that U i ⋐ U for all i . Supposethat [ R i ] ∈ G loc ( U i ) are given such that ρ G U i ∩ U j ,U i ([ R i ]) = ρ G U i ∩ U j ,U j ([ R j ]) for all i, j .Let ( η i ) i ⊂ Hom( F | U , F | U ) be a partition of unity subordinate to ( U i ) i . Choose τ i ∈L ( F ( U i ) , F ( U )) such that ρ V i ,U ◦ τ i = ρ V i ,U i for some V i ⋐ U i with supp η i ⊂ V i . Wedefine R := X i η iU ◦ τ i ◦ R i ◦ ρ RO U i ,U . We start with showing that R ∈ C ∞ (RO( U ) , F ( U )) . By [15, Lemma 3.8] it suffices toshow that ρ W,U ◦ R : RO( U ) → F ( W ) is smooth for all W ⋐ U . Since(5.9) ρ W,U ◦ R = X i ∈ J η iW ◦ ρ W,U ◦ τ i ◦ R i ◦ ρ RO U i ,U for some finite index set J , this follows from the fact that the linear mappings η iW , ρ W,U , τ i , and ρ RO U i ,U are continuous (see Lemma 5.4). Next, we show that R is local. Weneed to show that for all W ⋐ U the equality ρ W,U ◦ Φ = ρ W,U ◦ Φ , Φ , Φ ∈ RO( U ) , implies ρ W,U ( R (Φ )) = ρ W,U ( R (Φ )) . The mapping ρ W,U ◦ R can be represented as (5.9) for some finite index set J . Since supp η i ⊂ V i and ρ V i ,U ◦ τ i = ρ V i ,U i , it suffices to show that ρ W ∩ V i ,U i ( R i ( ρ RO U i ,U (Φ ))) = ρ W ∩ V i ,U i ( R i ( ρ RO U i ,U (Φ ))) for all i ∈ J . By locality of R i this follows from Lemma 5.4 (iv) and our assumption.We continue with showing that R is moderate. The moderateness of R is determinedby the values of the sequence p ( ρ W,U ((d l R )(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))) for ε small enough, where l ∈ N , Φ ε ∈ TO loc ( U ) , Ψ j,ε ∈ TO ( U ) for j = 1 , . . . , l , W ⋐ U , and p ∈ csn( F ( W )) are arbitrary. Since ρ W,U ((d l R )(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))= (d l ( ρ W,U ◦ R ))(Φ ε )(Ψ ,ε , . . . , Ψ l,ε )= d l X i ∈ J η iW ◦ ρ W,U ◦ τ i ◦ R i ◦ ρ RO U i ,U !! (Φ ε )(Ψ ,ε , . . . , Ψ l,ε )= X i ∈ J ( η iW ◦ ρ W,U ◦ τ i )((d l R i )( ρ RO U i ,U (Φ ε ))( ρ RO U i ,U (Ψ ,ε ) , . . . , ρ RO U i ,U (Ψ l,ε ))) for some finite index set J , the moderateness of R follows from the continuity ofthe mapping η iW ◦ ρ W,U ◦ τ i and the moderateness of the R i . Finally, we show that ρ G U i ,U ([ R ]) = [ R i ] for all i . We need to show that ρ E U i ,U ( R ) − R i is negligible. Thenegligibility is determined by p ( ρ W,U i ((d l ( ρ E U i ,U ( R ) − R i ))(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))) for ε small enough, where l ∈ N , Φ ε ∈ TO loc ( U i ) , Ψ j,ε ∈ TO ( U i ) for j = 1 , . . . , l , W ⋐ U i , and p ∈ csn( F ( W )) are arbitrary. By Lemma 5.10 there are Φ ′ ε ∈ TO loc ( U ) , Ψ ′ j,ε ∈ TO ( U ) for j = 1 , . . . , l such that ρ W,U i ◦ Φ ε ◦ ρ U i ,U = ρ W,U ◦ Φ ′ ε , ρ W,U i ◦ Ψ j,ε ◦ ρ U i ,U = ρ W,U ◦ Ψ ′ j,ε for all j = 1 , . . . , l and ε small enough. Hence Lemma 5.4 (ii) yields that ρ W,U i ((d l ( ρ E U i ,U ( R )))(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))= ρ W,U ((d l R )(Φ ′ ε )(Ψ ′ ,ε , . . . , Ψ ′ l,ε ))= (d l ( ρ W,U ◦ R ))(Φ ′ ε )(Ψ ′ ,ε , . . . , Ψ ′ l,ε )= d l X j ∈ J η jW ◦ ρ W,U ◦ τ j ◦ R j ◦ ρ RO U j ,U !! (Φ ′ ε )(Ψ ′ ,ε , . . . , Ψ ′ l,ε )= X j ∈ J η jW ( ρ W,U ( τ j ((d l R j )( ρ RO U j ,U (Φ ′ ε ))( ρ RO U j ,U (Ψ ′ ,ε ) , . . . , ρ RO U j ,U (Ψ ′ l,ε ))))) for ε small enough. On the other hand, Lemma 5.4 (ii) and the fact that (Φ ε ) ε islocalizing imply that ρ W,U i ◦ Φ ε = ρ W,U i ◦ ρ RO U i ,U (Φ ′ ε ) , ρ W,U i ◦ Ψ j,ε = ρ W,U i ◦ ρ RO U i ,U (Ψ ′ j,ε ) HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 25 for all j = 1 , . . . , l and ε small enough. By Remark 5.12 we obtain that ρ W,U i ((d l R i )(Φ ε )(Ψ ,ε , . . . , Ψ l,ε ))= ρ W,U i ((d l R i )( ρ RO U i ,U (Φ ′ ε ))( ρ RO U i ,U (Ψ ′ ,ε ) , . . . , ρ RO U i ,U (Ψ ′ l,ε )))= X j ∈ J η jW ( ρ W,U i ((d l R i )( ρ RO U i ,U (Φ ′ ε ))( ρ RO U i ,U (Ψ ′ ,ε ) , . . . , ρ RO U i ,U (Ψ ′ l,ε )))) for ε small enough. Since supp η j ⊂ V j and ρ V j ,U ◦ τ j = ρ V j ,U j , it suffices to estimate ρ W ∩ V j ,U j ((d l R j )( ρ RO U j ,U (Φ ′ ε ))( ρ RO U j ,U (Ψ ′ ,ε ) , . . . , ρ RO U j ,U (Ψ ′ l,ε ))) − ρ W ∩ V j ,U i ((d l R i )( ρ RO U i ,U (Φ ′ ε ))( ρ RO U i ,U (Ψ ′ ,ε ) , . . . , ρ RO U i ,U (Ψ ′ l,ε ))) . for all j ∈ J . By Lemma 5.13 (ii) we have that ρ W ∩ V j ,U j ((d l R j )( ρ RO U j ,U (Φ ′ ε ))( ρ RO U j ,U (Ψ ′ ,ε ) , . . . , ρ RO U j ,U (Ψ ′ l,ε )))= ρ W ∩ V j ,U i ∩ U j ((d l ( ρ E U i ∩ U j ,U j ( R j )))( ρ RO U i ∩ U j ,U (Φ ′ ε ))( ρ RO U i ∩ U j ,U (Ψ ′ ,ε ) , . . . , ρ RO U i ∩ U j ,U (Ψ ′ l,ε ))) and ρ W ∩ V j ,U i ((d l R i )( ρ RO U i ,U (Φ ′ ε ))( ρ RO U i ,U (Ψ ′ ,ε ) , . . . , ρ RO U i ,U (Ψ ′ l,ε )))= ρ W ∩ V j ,U i ∩ U j ((d l ( ρ E U i ∩ U j ,U i ( R i )))( ρ RO U i ∩ U j ,U (Φ ′ ε ))( ρ RO U i ∩ U j ,U (Ψ ′ ,ε ) , . . . , ρ RO U i ∩ U j ,U (Ψ ′ l,ε ))) . The negligibility now follows from the assumption. (cid:3)
Next, we discuss the embedding of E into G loc . For U ⊆ X open consider the canonicalembeddings (see Definition 3.1) ι U : E ( U ) → E ( U ) , σ U : F ( U ) → E ( U ) . Clearly, ι U ( E ( U )) ⊆ E loc ( U ) and σ U ( F ( U )) ⊆ E loc ( U ) . Hence Proposition 4.4 impliesthat the mappings ι U : E ( U ) → G loc ( U ) , ι ( u ) := [ ι ( u )] σ : F ( U ) → G loc ( U ) , σ ( ϕ ) := [ σ ( ϕ )] are linear embeddings such that ι U | F ( U ) = σ U . Proposition 5.20.
The embeddings ι : E → G loc and σ : F → G loc are sheaf morphisms,and ι | F = σ .Proof. We already noticed that ι U | F ( U ) = σ U for all U ⊆ X open. Since F is a subsheafof E it therefore suffices to show that ι is a sheaf morphism. Let U, V be open subsetsof X such that V ⊆ U . We need to show that for all u ∈ E ( U ) it holds that ρ E V,U ( ι U ( u )) − ι V ( ρ V,U ( u )) is negligible. It suffices to show that for all W ⋐ V and all (Φ ε ) ε ∈ RO loc ( V ) it holdsthat ρ W,V ( ρ E V,U ( ι U ( u ))(Φ ε )) = ρ W,V ( ι V ( ρ V,U ( u ))(Φ ε )) for ε small enough. By Lemma 5.10 there is (Φ ′ ε ) ε ∈ RO loc ( U ) such that ρ W,V ◦ Φ ε ◦ ρ V,U = ρ W,U ◦ Φ ′ ε . Hence Lemma 5.13 (i) yields that ρ W,V ( ρ E V,U ( ι U ( u ))(Φ ε )) = ρ W,U ( ι U ( u )(Φ ′ ε ))= ρ W,U (Φ ′ ε ( u )) = ρ W,V (Φ ε ( ρ V,U ( u )))= ρ W,V ( ι V ( ρ V,U ( u ))(Φ ε )) for ε small enough. (cid:3) We end this section by showing how one can extend sheaf morphisms to G loc . Lemma 5.21.
Let T : F ×· · ·× F → F be a multilinear sheaf morphism. The mappings b T U : G loc ( U ) × · · · × G loc ( U ) → G loc ( U ) given by b T U ([ R ] , . . . [ R n ]) := [ b T U ( R , . . . , R n )] are well-defined multilinear mappings such that b T U ( σ U ( ϕ ) , . . . , σ U ( ϕ n )) = σ U ( T U ( ϕ , . . . , ϕ n )) . Moreover, T is a multilinear sheaf morphism. Lemma 5.22.
Let ( E , F ) , ( E , F ) be test pairs of sheaves. Suppose we are given asheaf isomorphism µ : E → E such that its restriction to F is a sheaf isomorphism µ : F → F . The mappings ( µ ∗ ) U : G loc ( E ( U ) , F ( U )) → G loc ( E ( U ) , F ( U )) given by ( µ ∗ ) U [ R ] := [( µ ∗ ) U R ] are well-defined multilinear mappings such that ( µ ∗ ) U ◦ ι U = ι U ◦ µ U and ( µ ∗ ) U ◦ σ U = σ U ◦ µ U . Lemma 5.23.
Let T : E → E be a sheaf morphism such that T | F : F → F is a sheafmorphism. Then, the mappings b T U : G loc ( U ) → G loc ( U ) given by b T U [ R ] := [ b T U R ] arewell-defined such that b T U ◦ ι U = ι U ◦ b T U and b T U ◦ σ U = σ U ◦ b T U . We obtain the following two important corollaries.
Corollary 5.24.
For every open set U in X the sheaf G loc | U is fine.Proof. Let A and B be closed sets in U such that A ∩ B = ∅ . Let τ ∈ Hom( F | U , F | U ) be such that τ V = id and τ W = 0 for some open neighbourhoods V and W (in U ) of A and B , respectively. Consider the associated sheaf morphism τ ∈ Hom( G loc | U , G loc | U ) .Then, τ V ([ R ]) = [ τ V ( R )] = [ τ V ◦ R ] = [ R ] for all [ R ] ∈ G loc ( V ) . Similarly, one can show that τ W = 0 . (cid:3) Corollary 5.25.
Suppose that F is a locally convex sheaf of algebras. Then, G loc is asheaf of algebras and the σ -embedding is a sheaf homomorphism of algebras. HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 27 Diffeomorphism invariant algebras of distributions
The space of distributions on a paracompact Hausdorff manifold M is defined as D ′ ( M ) := (Γ c ( M, Vol( M ))) ′ where Γ c ( M, Vol( M )) denotes the space of compactly supported sections of the volumebundle Vol( M ) , endowed with its natural (LF)-topology (see [10, Section 3.1]). It iswell known that D ′ and C ∞ are locally convex sheaves on M , so ( D ′ , C ∞ ) is a test pairof sheaves.Given any open subset U ⊆ M , the space TO loc ( U ) is nonempty – we refer to [18]for the concrete construction of localizing test objects for ( D ′ , C ∞ ) , which is done byconvolution with smooth mollifiers in local charts.Fix the asymptotic scale to be the polynomial scale (2.1). By the previous section weobtain a fine sheaf G loc of algebras such that σ : C ∞ → G loc is a sheaf homomorphism ofalgebras and ι : D ′ → G loc is an injective sheaf homomorphism of vector spaces. Givenany vector field X on M , the Lie derivative L X of distributions and smooth functionssatisfies the assumptions of Lemma 3.5, so it defines a mapping b L X : G loc ( M ) → G loc ( M ) which commutes with ι .Moreover, given any diffeomorphism µ : M → N , we apply Lemma 5.22 to the functors E = D ′ , E = D ′ ◦ µ , F = C ∞ , F = C ∞ ◦ µ , which gives an induced action µ ∗ : G loc ( M ) → G loc ( N ) which commutes with ι .Hence, we have easily obtained the following result of [11]: Theorem 6.1.
Let M be a paracompact Hausdorff manifold. There is an associa-tive commutative algebra G loc ( M ) with unit containing D ′ ( M ) injectively as a linearsubspace and C ∞ ( M ) as a subalgebra. G loc ( M ) is a differential algebra, where thederivations b L X extend the usual Lie derivatives from D ′ ( M ) to G loc ( M ) , and G loc is afine sheaf of algebras over M . Diffeomorphism invariant algebras of ultradistributions
Spaces of ultradifferentiable functions and their duals.
Let ( M p ) p ∈ N be asequence of positive reals (with M = 1 ). We will make use of the following conditions: ( M. M p ≤ M p − M p +1 , p ∈ Z + , ( M. M p + q ≤ AH p + q M p M q , p, q ∈ N , for some A, H ≥ , ( M. ′ ∞ X p =1 M p − M p < ∞ . We refer to [12] for the meaning of these conditions. For α ∈ N d we write M α = M | α | .As usual, the relation M p ⊂ N p between two weight sequences means that there are C, h > such that M p ≤ Ch p N p , p ∈ N . The stronger relation M p ≺ N p means thatthe latter inequality remains valid for every h > and a suitable C = C h > . Theassociated function of M p is defined as M ( t ) := sup p ∈ N log t p M p , t > , and M (0) := 0 . We define M on R d as the radial function M ( x ) = M ( | x | ) , x ∈ R d .Under ( M. , the assumption ( M. holds [12, Prop 3.6] if and only if M ( t ) ≤ M ( Ht ) + log A, t > . Unless otherwise explicitly stated, M p will always stand for a weight sequence satisfying ( M. , ( M. , ( M. ′ .For a regular compact set K in R d and h > we write E M p ,h ( K ) for the Banach spaceconsisting of all ϕ ∈ C ∞ ( K ) such that(7.1) k ϕ k K,h := sup α ∈ N d sup x ∈ K | ϕ ( α ) ( x ) | h | α | M | α | < ∞ . The space D M p ,hK consists of all ϕ ∈ C ∞ ( R d ) with support in K that satisfy (7.1). Let Ω ⊆ R d be open. We define E ( M p ) (Ω) = lim ←− K ⋐ Ω lim ←− h → + E M p ,h ( K ) , E { M p } (Ω) = lim ←− K ⋐ Ω lim −→ h →∞ E M p ,h ( K ) , and D ( M p ) (Ω) = lim −→ K ⋐ Ω lim ←− h → + D M p ,hK , D { M p } (Ω) = lim −→ K ⋐ Ω lim −→ h →∞ D M p ,hK . Elements of E ( M p ) (Ω) and E { M p } (Ω) are called ultradifferentiable functions of class ( M p ) of Beurling type on Ω and ultradifferentiable functions of class { M p } of Roumieu typeon Ω , respectively. These spaces are complete Montel locally convex algebras (underpointwise multiplication) [12, Th. 2.6, Th. 5.12, Th. 2.8]. Elements of the dual spaces D ′ ( M p ) (Ω) and D ′{ M p } (Ω) are called ultradistributions of class ( M p ) of Beurling type on Ω and ultradistributions of class { M p } of Roumieu type on Ω , respectively. We endowthese spaces with the strong topology. D ′ ( M p ) (Ω) and D ′{ M p } (Ω) are complete Montellocally convex spaces [12, Th. 2.6] and Ω ′ → D ′ ( M p ) (Ω ′ ) , Ω ′ → D ′{ M p } (Ω ′ ) are locallyconvex sheaves on Ω [12, Th. 5.6].We write R for the family of positive real sequences ( r j ) j ∈ N (with r = 1 ) which increaseto infinity. This set is partially ordered and directed by the relation r j (cid:22) s j , whichmeans that there is an j ∈ N such that r j ≤ s j for all j ≥ j . By [14, Prop. 3.5] afunction ϕ ∈ C ∞ (Ω) belongs to E { M p } (Ω) if and only if k ϕ k K,r j := sup α ∈ N d sup x ∈ K | ϕ ( α ) ( x ) | M α Q | α | j =0 r j < ∞ , HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 29 for all K ⋐ Ω and r j ∈ R . Moreover, the topology of E { M p } (Ω) is generated by thesystem of seminorms {k k K,r j : K ⋐ Ω , r j ∈ R} .In the sequel we shall write ∗ instead of ( M p ) or { M p } if we want to treat both casessimultaneously. In addition, we shall often first state assertions for the Beurling casefollowed in parenthesis by the corresponding statements for the Roumieu case.7.2. Nonlinear extensions of spaces of ultradistributions.
We apply the generaltheory developed in Section 3 and Section 5 to construct algebras containing spacesof ultradistributions which are invariant under real-analytic diffeomorphisms. Let usremark that this construction is a novelty, as the previous construction in [4] was givenin the context of special Colombeau algebras and therefore cannot be diffeomorphisminvariant.In order to not having to develop the theory of ultradistributions on manifolds here,we restrict the considerations to the local case, i.e., to open subsets of R n , wherediffeomorphism invariance can be stated easily.By the remarks in Section 7.1 and the existence of partitions of unity of ultradiffer-entiable functions of class ∗ [12, Prop. 5.2] it is clear that the pair ( D ′∗ , E ∗ ) is a testpair of sheaves on R d , giving rise to the corresponding presheaf E loc of basic spaces(Definition 5.11).We now choose appropriate asymptotic scales. Given r j ∈ R we write M r j for theassociated function of the weight sequence M p Q pj =0 r j . Definition 7.1.
We define A ( M p ) := { e M ( λ/ε ) : λ > } , I ( M p ) := { e − M ( λ/ε ) : λ > } , A { M p } := { e M rj (1 /ε ) : r j ∈ R} , I { M p } := { e − M rj (1 /ε ) : r j ∈ R} . Condition ( M. ensures that sc ∗ := ( A ∗ , I ∗ ) are admissible pair of scales . For Ω ⊆ R d open we set TO ∗ loc (Ω) := TO loc (Ω , D ′∗ , E ′∗ , sc ∗ ) , TO , ∗ loc (Ω) := TO (Ω , D ′∗ , E ′∗ , sc ∗ ) . Remark . It follows from [4, Prop. 4.4] that (Φ ε ) ε ∈ L ( D ′{ M p } (Ω) , E { M p } (Ω)) I satisfies (TO) and (TO) (with respect to the scale sc { M p } ) if and only if(i) ∀ u ∈ D ′{ M p } (Ω) ∀ K ⋐ Ω ∀ λ > ∃ h > k Φ ε ( u ) k K,h = O ( e M ( λ/ε ) ) ,(ii) ∀ ϕ ∈ E { M p } (Ω) ∀ K ⋐ Ω ∃ λ > ∃ h > k Φ ε ( ϕ ) − ϕ k K,h = O ( e − M ( λ/ε ) ) .In order to be able to apply the results from Section 5 we must show that TO ∗ loc (Ω) isnonempty for every open set Ω ⊆ R d . For this we shall need the following lemma. We do not use the notation S ∗ for the pair of scales ( A ∗ , I ∗ ) since this is the standard notation forGelfand-Shilov type spaces. Lemma 7.3.
Let M p and N p be two weight sequences satisfying ( M. such that N p ≺ M p and let M and N be the associated functions of M p and N p , respectively. Then,there is an increasing net ( r ε ) ε of positive reals with lim ε → + r ε = 0 such that for every λ > there is ε > such that for all ε < ε it holds that M ( t ) ≤ N ( r ε t ) + M ( λ/ε ) , t > . Proof.
By [12, Lemma 3.10] there is a continuous increasing function ρ : (0 , ∞ ) → (0 , ∞ ) with lim t →∞ ρ ( t ) t = 0 such that M ( t ) = N ( ρ ( t )) for all t > . One can readily verify that r ε := sup t ≥ / √ ε ρ ( t ) t satisfies all requirements. (cid:3) By [12, Lemma 4.3] there is a weight sequence N p satisfying ( M. and ( M. ′ suchthat N p ≺ M p . Pick ψ ∈ D ( M p ) ( R d ) even with ≤ ψ ≤ , supp ψ ⊂ B (0 , , and ψ ≡ on B (0 , , and χ ∈ D ( N p ) ( R d ) even with supp χ ⊂ B (0 , and χ ≡ on B (0 , .Choose ( r ε ) ε according to Lemma 7.3. We define θ ε ( x ) := 1 ε d F − ( ψ )( x/ε ) χ ( x/r ε ) , x ∈ R d , where we fix the constants in the Fourier transform as follows F ( ϕ )( ξ ) = b ϕ ( ξ ) := Z R d ϕ ( x ) e − ixξ d x. Next, let ( K n ) n ∈ N be an exhaustion by compacts of Ω and choose κ n ∈ D ( M p ) (Ω) suchthat κ n ≡ on K n . For ε ∈ I we set κ ε = κ n if n ≤ ε − < n + 1 . Finally, we define Φ ε ( u ) := ( κ ε u ) ∗ θ ε = h u ( x ) , κ ε ( x ) θ ε ( · − x ) i , u ∈ D ′∗ (Ω) . Lemma 7.4. (Φ ε ) ε ∈ TO ∗ loc (Ω) . The proof of Lemma 7.4 is based on the following growth estimates of the Fouriertransforms of the θ ε . Lemma 7.5. (i) For all ε ∈ I it holds that sup ξ ∈ R d | b θ ε ( ξ ) | ≤ π ) d k b χ k L ( R d ) , (ii) for all h, λ > there is ε > such that sup ε<ε sup | ξ |≥ /ε | b θ ε ( ξ ) | e M ( ξ/h ) − M ( λ/ε ) < ∞ . HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 31 (iii) for all λ > there is ε > such that sup ε<ε sup | ξ |≤ /ε | − b θ ε ( ξ ) | e M ( λ/ε ) < ∞ . Proof.
Property (i) is clear. We now show (ii). Let ε ∈ I be arbitrary. We have that | b θ ε ( ξ ) | = r dε (2 π ) d (cid:12)(cid:12)(cid:12)(cid:12)Z R d ψ ( εη ) b χ ( r ε ( ξ − η ))d η (cid:12)(cid:12)(cid:12)(cid:12) ≤ r dε (2 π ) d Z | η |≤ ε | b χ ( r ε ( ξ − η )) | d η = 1(2 π ) d Z | ξ − trε | ≤ ε | b χ ( t ) | d t. By [12, Lemma 3.3] there is
C > such that | b χ ( t ) | ≤ Ce − N (2 Ht/h ) ≤ ACe − N (2 t/h ) , t ∈ R d . Furthermore, notice that for ξ, t ∈ R d it holds that | ξ | ≥ ε and (cid:12)(cid:12)(cid:12)(cid:12) ξ − tr ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε → | t | ≥ r ε | ξ | . Hence we obtain that | b θ ε ( ξ ) | ≤ C ′ e − N ( r ε | ξ | /h ) , | ξ | ≥ ε , where C ′ = AC (2 π ) d Z R d e − N (2 t/h ) d t < ∞ . The result now follows from Lemma 7.3. Finally, we show (iii). Let ε ∈ I be arbitrary.We have that | − b θ ε ( ξ ) | = (cid:12)(cid:12)(cid:12)(cid:12) − r dε (2 π ) d Z R d ψ ( εη ) b χ ( r ε ( ξ − η ))d η (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) r dε (2 π ) d Z R d (1 − ψ ( εη )) b χ ( r ε ( ξ − η ))d η (cid:12)(cid:12)(cid:12)(cid:12) ≤ r dε (2 π ) d Z | η |≥ ε | b χ ( r ε ( ξ − η )) | d η = 1(2 π ) d Z | ξ − trε | ≥ ε | b χ ( t ) | d t. By [12, Lemma 3.3] there is
C > such that | b χ ( t ) | ≤ Ce − N (2 H λt ) ≤ ACe − N (2 Hλt ) , t ∈ R d . Furthermore, notice that for ξ, t ∈ R d it holds that | ξ | ≤ ε and (cid:12)(cid:12)(cid:12)(cid:12) ξ − tr ε (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε → | t | ≥ r ε ε . Hence we obtain that | − b θ ε ( ξ ) | ≤ C ′ e − N ( Hλr ε /ε ) , | ξ | ≤ ε , where C ′ = AC (2 π ) d Z R d e − N (2 Hλt ) d t < ∞ . By Lemma 7.3 there is ε > such that M ( t ) ≤ N ( r ε t ) + M ( λ/ε ) , t > , for all ε < ε . By setting t = Hλ/ε we obtain that N ( Hλr ε /ε ) ≥ M ( Hλ/ε ) − M ( λ/ε ) ≥ M ( λ/ε ) − log A, for all ε < ε and the result follows. (cid:3) Proof of Lemma 7.4.
For ε ∈ I fixed we have that Φ ε ∈ L ( D ′∗ (Ω) , E ∗ (Ω)) by [12, Prop.6.10]. The fact that (Φ ε ) ε is localizing follows easily from lim ε → + r ε = 0 . We now showthat (Φ ε ) ε satisfies (TO) j , j = 1 , , . In the Roumieu case we use Remark 7.2. (TO) :We need to show that ∀ u ∈ D ′ ( M p ) (Ω) ∀ K ⋐ Ω ∀ h > ∃ λ > k Φ ε ( u ) k K,h = O ( e M ( λ/ε ) ) , ( ∀ u ∈ D ′{ M p } (Ω) ∀ K ⋐ Ω ∀ λ > ∃ h > k Φ ε ( u ) k K,h = O ( e M ( λ/ε ) )) . There is N ∈ N such that supp θ ε ( x − · ) ⊆ K N , x ∈ K, for ε small enough. Hence Φ ε ( u )( x ) = ( κu ∗ θ ε )( x ) , x ∈ K, where κ = κ N , for ε small enough. By [12, Lemma 3.3] it suffices to show that for all h > there is λ > (for all λ > there is h > ) such that Z R d | c κu ( ξ ) || b θ ε ( ξ ) | e M ( ξ/h ) d ξ = O ( e M ( λ/ε ) ) . There are µ > and C > (for every µ > there is C > ) such that | c κu ( ξ ) | ≤ Ce M ( µξ ) , ξ ∈ R d . Lemma 7.5 (ii) implies that for all h, λ > (both in the Beurling and Roumieu case) Z | ξ |≥ ε | c κu ( ξ ) || b θ ε ( ξ ) | e M ( ξ/h ) d ξ = O ( e M ( λ/ε ) ) . On the other hand, by Lemma 7.5 (i), we have that Z | ξ |≤ ε | c κu ( ξ ) || b θ ε ( ξ ) | e M ( ξ/h ) d ξ ≤ AC k b χ k L ( R d ) (2 π ) d Z | ξ |≤ ε e M ( µξ )+ M ( Hξ/h ) − M ( ξ/h ) d ξ ≤ C ′ e M ( λ /ε ) HEAVES OF NONLINEAR GENERALIZED FUNCTION SPACES 33 where λ = 4 H max( µ, H/h ) and C ′ = AC k b χ k L ( R d ) (2 π ) d Z R d e − M ( ξ/h ) d ξ < ∞ . The Beurling case follows at once while the Roumieu case follows by noticing that λ can be made as small as desired by choosing µ small enough and h big enough. (TO) : By [4, Prop. 4.2] it suffices to show that ∀ ϕ ∈ E ( M p ) (Ω) ∀ K ⋐ Ω ∀ λ > x ∈ K | Φ ε ( ϕ )( x ) − ϕ ( x ) | = O ( e − M ( λ/ε ) ) , ( ∀ ϕ ∈ E { M p } (Ω) ∀ K ⋐ Ω ∃ λ > x ∈ K | Φ ε ( ϕ )( x ) − ϕ ( x ) | = O ( e − M ( λ/ε ) )) . There is N ∈ N such that supp θ ε ( x − · ) ⊆ K N , x ∈ K, for ε small enough. Hence Φ ε ( ϕ )( x ) − ϕ ( x ) = ( κϕ ∗ θ ε )( x ) − κ ( x ) ϕ ( x ) , x ∈ K, where κ = κ N , and, thus, sup x ∈ K | Φ ε ( ϕ )( x ) − ϕ ( x ) | = sup x ∈ K | ( κϕ ∗ θ ε )( x ) − κ ( x ) ϕ ( x ) |≤ π ) d Z R d | c κϕ ( ξ ) || − b θ ε ( ξ ) | d ξ for ε small enough. Therefore it suffices to show that for all λ > (for some λ > ) itholds that Z R d | c κϕ ( ξ ) || − b θ ε ( ξ ) | d ξ = O ( e − M ( λ/ε ) ) . For every µ > there is C > (there are µ, C > ) such that | c κϕ ( ξ ) | ≤ Ce − M ( Hµξ ) ≤ ACe − M ( µξ ) , ξ ∈ R d . Lemma 7.5 (iii) implies that for all λ > (both in the Beurling and Roumieu case) Z | ξ |≤ ε | c κϕ ( ξ ) || − b θ ε ( ξ ) | d ξ = O ( e − M ( λ/ε ) ) . On the other hand, by Lemma 7.5 (i), we have that Z | ξ |≥ ε | c κϕ ( ξ ) || − b θ ε ( ξ ) | d ξ ≤ C ′ e − M ( µ/ (2 ε )) where C ′ = (cid:18) k b χ k L ( R d ) (2 π ) d (cid:19) AC Z R d e − M ( µξ ) d ξ < ∞ . (TO) : Since the space D ∗ (Ω) is Montel it suffices to show that for all u ∈ D ′∗ (Ω) itholds that lim ε → + Z R d Φ ε ( u )( x ) ϕ ( x )d x = h u, ϕ i , ϕ ∈ D ∗ (Ω) . There is N ∈ N such that supp θ ε ( x − · ) ⊆ K N , x ∈ supp ϕ, for ε small enough. Hence, for κ = κ N , we have that Z R d Φ ε ( u )( x ) ϕ ( x )d x = Z R d h u ( y ) , κ ( y ) θ ε ( x − y ) i ϕ ( x )d x = h u ( y ) , κ ( y ) Z R d θ ε ( x − y ) ϕ ( x )d x i = h u ( y ) , κ ( y )Φ ε ( ϕ )( y ) i for ε small enough. The result now follows from (TO) and the continuity of u . (cid:3) As in Section 6 we now obtain a fine sheaf G ∗ loc of algebras such that σ : E ∗ → G ∗ loc is asheaf homomorphism of algebras and ι : D ∗′ is a sheaf homomorphism of vector spaces.The partial derivative ∂ i , i = 1 , . . . , n , satisfies the assumptions of Lemma 3.5, so itdefines a mapping b ∂ i : G ∗ loc (Ω) → G ∗ loc (Ω) which commutes with ι .Moreover, as seen in [13, p. 626] real analytic coordinate transformations induce con-tinuous mappings on the spaces D ∗′ and E ∗ , so by Lemma 4.8 we obtain correspondingactions on the quotient spaces G ∗ loc . Theorem 7.6.
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