aa r X i v : . [ m a t h . OA ] S e p A Dixmier-Malliavin theorem for Lie groupoids
Michael FrancisDepartment of MathematicsPennsylvania State UniversityUniversity Park, PA 16802, USA [email protected]
September 27, 2020
Abstract
A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supportedfunction on a Lie group can be expressed as a finite sum in which each term is theconvolution, with respect to Haar measure, of two such functions. We establish thatthe same holds for a Lie groupoid. Most of the heavy lifting is done by a lemma in theoriginal work of Dixmier-Malliavin. We also need the technology of Lie algebroids andthe corresponding notion of exponential map. As an application, we obtain a resulton the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arisingfrom functions vanishing to given order on an invariant submanifold of the unit space.
In a 1960 paper, Ehrenpreis [10] posed a number of related questions including whether everysmooth, compactly-supported function ϕ ∈ C ∞ c ( R n ) can be “deconvolved” as ϕ = f ∗ g where f, g ∈ C ∞ c ( R n ). The latter question became known as the Ehrenpreis factorization problem .In 1978, Rubel-Squires-Taylor [17] showed that the answer is “no” if n ≥
3. In the sameyear, Dixmier-Malliavin [9] showed that the answer is still “no” if n = 2. The remainingcase n = 1 was eventually settled in 1999 by Yulmukhametov [20] who showed the answeris “yes” for the real line. Also in the positive direction, [9] gives the answer to a weakerform of the factorization question to be “yes” for any Lie group whatsoever. This is the Dixmier-Malliavin theorem . Theorem 1 (3.1 Th´eor`eme, [9]) . Let G be a Lie group and form the smooth convolutionalgebra C ∞ c ( G ) . Then, every ϕ ∈ C ∞ c ( G ) can be expressed as f ∗ ψ + . . . + f N ∗ ψ N More accurately, one of several closely-related Dixmier-Malliavin theorems. here f i , ψ i ∈ C ∞ c ( G ) . Moreover, we can choose this factorization such that, for every i , supp( ψ i ) ⊆ supp( ϕ ) and supp( f i ) ⊆ W , where W ⊆ G is a neighbourhood of the identityfixed in advance. The main goal of this paper is to establish the following analogous result for Lie groupoids.
Theorem 2.
Let G be a Lie groupoid and form the smooth convolution algebra C ∞ c ( G ) .Then, every ϕ ∈ C ∞ c ( G ) can be decomposed as f ∗ ψ + . . . + f N ∗ ψ N , where f i , ψ i ∈ C ∞ c ( G ) .Moreover, we can choose this factorization such that, for every i , supp( ψ i ) ⊆ supp( ϕ ) and supp( f i ) ⊆ W , where W ⊆ G is a neighbourhood of the unit space of G fixed in advance. Note that defining a convolution product on C ∞ c ( G ) requires a choice of Haar system but, asdifferent Haar systems lead to canonically isomorphic algebras, issues relating to factorizationdo not depend on this choice. One could also avoid making a choice entirely by workingwith appropriate densities in place of functions. Additionally, the open set W does notactually need to contain the whole unit space of G , only a suitable compact subset. Thesetechnicalities, and others, shall be entered into in more detail later in the document. A morecomprehensive statement is given in Theorem 13.We furthermore apply our Dixmier-Malliavin theorem to obtain results on the arithmetic ofcertain ideals in the smooth convolution algebra of a Lie groupoid. Given a closed, invariantsubmanifold X of the unit space of a Lie groupoid G , one obtains ideals J p ⊆ C ∞ c ( G )consisting of the functions which vanish to order p on the restricted groupoid G X ⊆ G . Ourmain findings here are: J ∞ ∗ J ∞ = J ∞ ( J ) ∗ p = J p . In practical terms, this means that a function vanishing to infinite order on G X can bewritten as a finite sum in which each term is a convolution of two functions vanishing toinfinite order on G X , and that a function vanishing to p th order on G X can be written as afinite sum in which each term is a p -fold convolution of functions which vanish on G X . Notethese results on ideals are only interesting after one has generalized to the groupoid setting.In the group case, the unit space consists of a single point and these ideals do not arise atall.Elsewhere ([11]), we apply the results of this article to investigate the structure of the smoothconvolution algebra C ∞ c ( G ) when G is the holonomy groupoid of a singular foliation ([14],[15], [8], [2]). The holonomy groupoid of the singular foliation of R given by vector fields thatvanish to k th order at the origin ([2], Example 1.3 (3)) is isomorphic to the transformationgroupoid R ⋊ k R of an appropriate flow fixing the origin to k th order. We show, by wayof an analysis of the ideal structure, that the smooth algebras C ∞ c ( R ⋊ k R ) are pairwisenonisomorphic. The C*-completions, on the other hand, fall into two isomorphism classesaccording to the parity of k . This demonstrates that, even in very simple cases, the smoothalgebra of a singular foliation can contain information that is washed away after one passesto the foliation C*-algebra. 2e now explain the organization of this article. In Section 2, we use a key lemma ofDixmier-Malliavin to obtain, as a corollary, the preliminary factorization result Theorem 3.This allows us, given a smooth R -action on a manifold M , to express functions on M astwo-term sums in which both terms are the convolution of a function on R with a functionon M . Section 3 largely serves a notational purpose. In it, we lay out our conventionsfor the Lie algebroid of a Lie groupoid, for Haar measures, and for Lie groupoid actions.Section 4 is quite technical and culminates with Lemma 11, giving a framework in whichfunctions on R can be convolved to yield a function on a Lie groupoid with appropriateproperties. In Section 5, we derive the main result, Theorem 13. Section 6 prepares theground for the final section by analyzing the product structure of ideals in the (commutative)algebra of smooth functions on a manifold under pointwise multiplication. This is againsomewhat technical, though we do make use of an interesting inversion principle whichconnects Schwartz functions to bump functions (Lemma 19). Finally, in Section 7, weobtain our results on the product structure of ideals in the smooth convolution algebra of aLie groupoid. The main finding here is Theorem 25. R actions Let X be a complete vector field on a smooth manifold M . We denote by t e tX thecorresponding 1-parameter group of diffeomorphisms (in other words the flow) that is relatedto X by ddt (cid:12)(cid:12)(cid:12) t =0 f ( e tX m ) = ( Xf )( m ) . This action of R on M can alternatively be encoded by a representation π of the smoothconvolution algebra C ∞ c ( R ) on C ∞ c ( M ) which we call the integrated form of the action. Thisrepresentation π is defined by the following formula:( π ( f ) ψ )( m ) = Z R f ( − t ) ψ ( e tX m ) dt. (1)The essential ingredient in our Lie groupoid generalization of Dixmier-Malliavin is the fol-lowing preliminary factorization result: Theorem 3.
Let R act smoothly on a manifold M via a complete vector field X and let π be the representation of C ∞ c ( R ) on C ∞ c ( M ) defined by (1) . Then, for any ϕ ∈ C ∞ c ( M ) , thereexists f , f ∈ C ∞ c ( R ) and ψ , ψ ∈ C ∞ c ( M ) such that ϕ = π ( f ) ψ + π ( f ) ψ . (2) Moreover, this factorization can be taken such that, for i = 0 , , supp( ψ i ) ⊆ supp( ϕ ) and supp( f i ) ⊆ ( − ǫ, ǫ ) , where ǫ > is fixed in advance. f i . It is far easier to achieve the factorization if the f i are only required to be differentiableof a large, but finite, order (see [4], pp. 23). The means by which this simpler result isachieved suggest a strategy for the more difficult result, so it seems worthwhile to provide ashort outline here. Lemma 4.
For every nonnegative integer k , there exists f ∈ C kc ( R ) , g ∈ C ∞ c ( R ) such that δ = f ( k +2) + g where δ denotes the delta distribution at .Proof. Antidifferentiate the delta distribution k + 2 times, always picking the antiderivativethat vanishes on the negative half line. The result is the C k function F vanishing on thenegative-half line and satisfying F ( t ) = k +1)! t k +1 for t ≥
0. Of course, F ( k +2) = δ , but F isnot compactly-supported. Let G be a C ∞ function that agrees with F outside of a boundedinterval. Then, δ = F ( k +2) = ( F − G ) ( k +2) + G ( k +2) = f ( k +2) + g , where f = F − G ∈ C kc ( R )and g = G ( k +2) ∈ C ∞ c ( R ).This elementary lemma has a weak version of Theorem 3 as a corollary. Note the represen-tation π defined by (1) still makes sense for functions that aren’t smooth, but only, say, con-tinuous. Moreover, π ( f ) ψ still belongs to C ∞ c ( M ) when f ∈ C c ( R ), provided ψ ∈ C ∞ c ( M ).We can even extend the representation π to compactly-supported distributions on R , forinstance π ( δ ) = id C ∞ c ( M ) . This is mainly a notational point—it is entirely possible to elim-inate distributions from the discussion. However, because the corollary below is only beingincluded for motivational reasons, it hardly seems worth it to do so. Corollary 5.
Let X be a complete vector field on a manifold M and let π be the representa-tion of C c ( R ) on C ∞ c ( M ) defined by (1) . Then, for any ϕ ∈ C ∞ c ( M ) and any integer k ≥ ,one can write ϕ = π ( f ) ψ + π ( f ) ψ where f ∈ C kc ( R ) , f ∈ C ∞ c ( R ) and ψ , ψ ∈ C ∞ c ( M ) .Proof sketch. Applying the above lemma, we can write δ = f ( k +2) + g where f ∈ C k ( R ), g ∈ C ∞ c ( R ). Noting the relation π ( h ′ ) ψ = π ( h ) Xψ (an instance of the integration by partsformula), we get ϕ = π ( δ ) ϕ = π ( f ( k +2) ) ϕ + π ( g ) ϕ = π ( f ) X k +2 ϕ + π ( g ) ϕ so, setting f = f , f = g , ψ = X k +2 ϕ , ψ = ϕ , we are finished.A major innovation of [9] is to achieve approximate representations of δ , analogous to thatof Lemma 4, but in terms of C ∞ functions. Precisely, they prove the following.4 heorem 6 (2.5 Lemme, [9]) . Given any positive sequence c k > , there exist functions f, g ∈ C ∞ c ( R ) and scalars a k with | a k | ≤ c k such that δ = g + ∞ X k =1 a k f ( k ) , where the convergence is in the sense of compactly-supported distributions. Note that an obvious rescaling argument implies that the functions f, g of the above theoremcan be taken to have support contained in ( − ǫ, ǫ ), for any ǫ > λ = { < λ < λ < . . . } that will guarantee that thefunction χ λ defined as the restriction to R of the reciprocal of the entire function representedby the infinite product Q (1 + z λ i ) defines a function in the Schwartz space S ( R ). Functionsof this type are used to prove a Schwartz function analog of Theorem 6 which, in turn, isused to prove Theorem 6.Theorem 6 is exactly the tool needed to give the Proof of Theorem 3.
Fix ϕ ∈ C ∞ c ( M ). Note that Xϕ, X ϕ, . . . have supports contained inthe support of ϕ . We can choose a sequence of positive constants c k that decays rapidlyenough that, for any sequence of scalars a k with | a k | < c k , the sum P k a k X k ϕ is uniformlyconvergent to a C ∞ function (this requires a little diagonal selection trick, but is elementary).Obviously, the support of the sum is contained in that of ϕ . From Theorem 6, we canchoose scalars a k , | a k | < c k and f, g ∈ C ∞ c ( R ) with supports contained in ( − ǫ, ǫ ) such that g + P k a k f ( k ) → δ , in the distributional sense. Thus, defining h n = g + P n a k f ( k ) , we havethat π ( h n ) ϕ → ϕ pointwise over M . On the other hand, π ( h n ) ϕ = π ( g ) ϕ + n X a k π ( f ( k ) ) ϕ = π ( g ) ϕ + π ( f ) n X a k X k ϕ. By the choice of constants c k , the series P n a k X k ϕ converges uniformly to a function ψ ∈ C ∞ c ( R ) which furthermore satisfies supp( ψ ) ⊆ supp( ϕ ). Thus, we arrive at the weakfactorization ϕ = π ( f ) ψ + π ( g ) ϕ , which has all the features advertised in the statement. To a large extent, the purpose of the present section is to sufficiently fix our notations andconventions that precise discussion of Lie groupoids is possible later on. Throughout, G ⇒ B Explicitly, h n ∈ C ∞ c ( R ) converge to δ in the distributional sense if their supports are uniformly boundedand R h n ( x ) f ( x ) dx → f (0) for every f ∈ C ∞ c ( R ). s and target map t . The inversion map is denoted ı : G → G or, frequently, just γ γ − . For peace of mind, it is assumed that s and t aresubmersions with k -dimensional fibres and that the unit space B is a closed submanifoldof G . Multiplication is performed from right to left so that γ γ is defined if and only if s ( γ ) = t ( γ ). We use the (standard) notations G x = s − ( x ) and G x = t − ( x ) for the sourceand target fibres. Examples. Transformation groupoid:
Let H be a Lie group acting smoothly on the left of themanifold B . Set G = H × B and define s, t : G → B by s ( h, b ) = b and t ( h, b ) = hb .Define the product by ( h , h b )( h , b ) = ( h h , b ).2. Pair groupoid:
Define G = B × B and take s = pr , t = pr , the two coordinateprojections. Define the product by ( b , b )( b , b ) = ( b , b ).With some superficial differences, our conventions accord with those found in [13] and [16].We follow [13] in viewing sections of the Lie algebroid of a Lie groupoid as right-invariantvector fields, but depart from [16] by working with right Haar measures in place of left Haarmeasures. This is done so that our measures and our vector fields will both live along thesame fibers (namely the source fibres), but is of no real importance; left and right and canalways be exchanged using the inversion map. The Lie algebroid of a Lie groupoid
A vector field on G is said to be right-invariant if it is tangent to the source fibres (i.e. is asection of the distribution ker( ds ) ⊆ T G ) and satisfies( R γ ) ∗ X = X γ ∈ G, where R γ denotes right-multiplication by γ . The equation above is a bit imprecise because R γ is not defined on all of G , but instead is a diffeomorphism G t ( γ ) → G s ( γ ) . A more accurateformulation would be( R γ ) ∗ (cid:16) X (cid:12)(cid:12) G y (cid:17) = X (cid:12)(cid:12) G x γ ∈ G x ∩ G y , where the restricted vector fields make sense because X was assumed tangent to the sourcefibres. Because we can write any γ ∈ G as R γ ( t ( γ )), it follows that a right-invariant vectorfield is completely determined by its restriction to B . Definition 7 (Definition 3.1, [13]) . As a vector bundle over the base manifold B , the Liealgebroid AG of G is the restriction of the source fibre tangent bundle ker( ds ) ⊆ T G to B .Every section of AG extends uniquely to a right-invariant vector field on G . Thus, right-invariant vector fields on G are in 1-1 correspondence with sections of the Lie algebroid AG .This is the Lie groupoid counterpart of the analogous pair of descriptions for the Lie algebra6f a Lie group. We shall tend to abuse notation, denoting a right-invariant vector and itsrestriction to B (a section of AG ) by the same symbol. Example. If G is the pair groupoid B × B , then the source fibres are just the slices B × { b } , b ∈ B . A right-invariant vector field on G amounts to a single vector field X on B copiedon each slice. Meanwhile, the Lie algebroid obviously identifies with T B , so sections of theLie algebroid also correspond in an obvious way to vector fields on B .There is a bundle map AG → T B called the anchor map defined simply by restrictingthe differential dt : T G → T B of the target submersion to AG . It is common practiceto denote the anchor map by AG → T B , but we shall avoid this notation becauseit overlaps with common notation for fundamental vector fields in the context of a Liegroup actions, and we will be considering groupoids acting on manifolds. Instead, given X ∈ C ∞ ( B, AG ) = { right-invariant vector fields on G } , we write X B for the correspondingvector field on B . Thus, X B ( b ) = ddt t ( e tX b ) (cid:12)(cid:12)(cid:12) t =0 . (3)The vector fields X and X B are t -related; if X is a complete, right-invariant vector fieldon G , then X B is a complete vector field on B and the target submersion t : G → B isequivariant for the R -actions on G and B .An irritation that does not arise in the Lie group context, but does for Lie groupoids, is thepotential for right-invariant vector fields to have incomplete flows. For example, in the caseof the pair groupoid G = B × B , the flow of a right-invariant vector field is just the flow of anarbitrary vector field on B , copied on each slice B × { b } . The following proposition, however,at least shows that complete, right-invariant vector fields are in plentiful supply. Proposition 8.
Let G ⇒ B be a Lie groupoid, X a compactly-supported section of the Liealgebroid AG . Then, X , considered as a right-invariant vector field on G , is complete.Proof. Let φ : W → G be the (maximal) flow of X . So, W is an open subset of R × G containing { } × G . To see φ is complete, it suffices to show ( − ǫ, ǫ ) × G ⊆ W for some ǫ >
0. Let K ⊆ B be a compact set containing { b ∈ B : X ( b ) = 0 } . By an easy compactnessargument, there exists an ǫ > − ǫ, ǫ ) × K ⊆ W . In fact, since X vanishes on B − K , we actually have ( − ǫ, ǫ ) × B ⊆ W . But then, for any γ ∈ G , we have, using theright-invariance, the integral curve t φ t ( t ( γ )) γ : ( − ǫ, ǫ ) → G passing through γ at t = 0,and so ( − ǫ, ǫ ) × G ⊆ W , as was to be proven.Note that right-invariance of a complete vector field X on G can also be formulated as aproperty of its flow; each diffeomorphism e tX should be right-invariant in the sense that itpreserves the s -fibers and satisfies e tX ( γ γ ) = ( e tX γ ) γ whenever γ γ is defined. Possibly no longer compactly-supported; consider what happens if G is a noncompact Lie group. aar systems and convolution Defining a convolution product on C ∞ c ( G ) for a Lie groupoid G ⇒ B requires a choice of(smooth) Haar system. This choice is ultimately unimportant; the algebras associated todifferent Haar measures are canonically isomorphic. Indeed, by replacing functions on G withsections of an appropriate density bundle, it is possible to obtain the convolution algebrain a fully intrinsic way which does not require a choice of Haar system. See the discussionfollowing Definition 2 in [7], Section 2.5. In this article, we will stick with functions, however,in large part to better resemble the classical Dixmier-Malliavan theorem.For a given a submersion p : W → B , say with k -dimensional fibres, a (smooth) fibrewisemeasure is a collection λ = ( λ b ) of smooth measures on the fibres p − ( b ), b ∈ B such that,in any open subset of W small enough to be identified with R k × U for U an open set in B in such a way that p is identified with the factor projection R k × U → U , the measurestake the form λ b = ρ ( · , b ) dt · · · dt k , b ∈ U , where ρ is a smooth, positive-valued functionon R k × U and dt · · · dt k is the standard volume measure copied on each fibre { b } × R k .Equivalently, one can think of λ as a globally positive section of the density bundle of the p -vertical subbundle ker( dp ) ⊆ T W . We list a few basic properties:1. Fibrewise measures are unique up to rescaling (corresponding to the fact that thedensity bundle is trivial and oriented); if λ and ν are fibrewise measures for p : W → B ,then there is a unique positive-valued, smooth function ρ on W such that ν = ρλ .2. A fibrewise measure λ for p : W → B determines a corresponding “integration alongfibres” map p ! : C ∞ c ( W ) → C ∞ c ( B ).3. Fibrewise measures can be pulled back. Suppose, p : W → B is a submersion and µ : M → B is a smooth map so that pr : W × p,µ M → M is a submersion. W × p,µ M WM B pr pr pµ Then, a fibrewise measure λ for p determines a fibrewise measure λ M for pr by iden-tifying pr − ( m ) with p − ( µ ( m )) in the obvious way. If preferable, one may obtain thispullback construction in two stages, expressing it in terms of appropriate product andrestriction constructions.A (smooth, right) Haar system λ for a Lie groupoid G ⇒ B is a fibrewise measure λ = ( λ b ) b ∈ B for the source submersion s : G → B that is right-invariant in the sense that, for any γ ∈ G ,the right-multiplication R γ is a measure isomorphism from ( G t ( γ ) , λ t ( γ ) ) to ( G s ( γ ) , λ s ( γ ) ). The fibre product W × p,µ M = { ( γ, m ) : p ( γ ) = µ ( m ) } is a closed submanifold of W × M . This followsfrom writing it as the preimage of the diagonal ∆ ⊆ M × M under the map p × µ : G × M → M × M andnoting that, because p is a submersion, the latter map is transverse to ∆. G ⇒ B proceeds along analogous lines: for any γ ∈ G , the right-translation R γ : G t ( γ ) → G s ( γ ) sends t ( γ ) γ . Thus, any globally-positive section of thedensity bundle of AG → B can be canonically extended to a globally-positive section ofthe density bundle of ker( ds ) ⊆ T G , the subbundle of tangent spaces to the source fibres.By construction, this extension is right-invariant in an obvious sense. Along the same lines,one sees that the (smooth) Haar measure of a Lie groupoid is unique up to multiplication(appropriately defined) by a smooth, positive-valued function on the base manifold.Once a Haar measure λ has been fixed for G , we obtain a convolution operation ∗ withrespect to which C ∞ c ( G ) becomes a (generally noncommutative) algebra.( f ∗ g )( γ ) = Z G t ( γ f ( γ − ) g ( γγ ) dλ t ( γ ) (4) Lie groupoid actions
Let G ⇒ B be a Lie groupoid. Let M be a manifold with a given smooth map µ : M → B called the momentum map . By definition, a left action of G on M is a smooth product G × s,µ M ∋ ( γ, m ) γ · m ∈ M , such that µ ( γ · m ) = t ( γ ) for all ( γ, m ) ∈ G × s,µ M , µ ( m ) · m = m for all m ∈ M and ( γ γ ) · m = γ · ( γ · m ) for all γ , γ ∈ G , m ∈ M with s ( γ ) = t ( γ ), s ( γ ) = µ ( m ). Examples.
1. A Lie group H acting smoothly on the left of a manifold M can be considered as aLie groupoid action by taking G = H , B = { G } and s, t, µ to be the collapsing mapsonto the one-point space B .2. Every Lie groupoid G ⇒ B acts on its own arrow space G by taking the momentummap to be the target submersion t : G → B and the action map to be the groupoidmultiplication.3. Every Lie groupoid G ⇒ B acts on its own unit space B by taking the momentummap to be the identity map on B . The fibre product G × s, id B canonically identifieswith G via, γ ( γ, s ( γ )) and, under this identification, the action map is just thetarget submersion t : G → B .Recall that, when a Lie group H acts on a manifold M , each Lie algebra element X ∈ h determines a corresponding fundamental vector field X on M . Similarly, when a Liegroupoid G acts on a manifold M with momentum map µ , each Lie algebroid section X ∈ ∞ c ( B, AG ) determines a vector field X M on a M by the formula: X M ( m ) = ddt h e tX µ ( m ) · m i t =0 . (5)The vector field X M is complete if X is and satisfies the following right-invariance condition: e tX M ( γ · m ) = ( e tX γ ) · m whenever γ · m is defined.If, additionally, a Haar system λ has been specified for G , then an action of G on M determines a representation π of the convolution algebra C ∞ c ( G ) = C ∞ c ( G, λ ) on C ∞ c ( M )called the integrated form of the action according to the following formula:( π ( f ) ψ )( m ) = Z G µ ( m ) f ( γ − ) ψ ( γ · m ) dλ µ ( m ) . (6)Take note that, in the special case when G is acting on itself from the left, we recover (4),the convolution product on C ∞ c ( G ).The action of G on M can be packaged as a Lie groupoid G ⋉ M ⇒ M called the transfor-mation groupoid of the action. This is done by taking G ⋉ M = G × s,µ M with structuremaps defined as follows:source σ : ( γ, m ) m target τ : ( γ, m ) γ · m inversion : ( γ, m ) ( γ − , γ · m )multiplication : ( γ , γ · m )( γ , m ) = ( γ γ , m )Note that the relation τ = σ ◦ shows that the action map is in fact a submersion. The Haarmeasure λ on G determines a corresponding Haar system λ M on G ⋉ M , using the obviousidentification of each ( G ⋉ M ) m = G µ ( m ) × { m } with G µ ( m ) . R actions to groupoid actions This section is devoted to proving the somewhat technical Lemma 11 below. Accordingly,most of the notations set down below can safely be forgotten once this end has been achieved.As always, G ⇒ B is a Lie groupoid with source s , target t and inversion map ı and Haarsystem λ . Let M be a G -space with momentum map µ : M → B . Let π be the correspondingrepresentation of C ∞ c ( G ) on C ∞ c ( M ) defined by (6).( π ( f ) ψ )( m ) = Z G µ ( m ) f ( γ − ) ψ ( γ · m ) dλ µ ( m ) Let X , . . . , X k ∈ C ∞ c ( B, AG ), thought of as complete, right-invariant vector fields on G .Correspondingly, we have complete vector fields X B , . . . , X Bk on B and and X M , . . . , X Mk on10 defined by (3) and (5), respectively. Let π , . . . , π k be the representations of C ∞ c ( R ) on C ∞ c ( M ) associated to the complete vector fields X M , . . . , X Mk in accordance with (1).( π i ( f ) ψ )( m ) = Z R f ( − t ) ψ ( e tX Mi m ) dt Our basic goal is to work out the relationship between π and π , . . . , π k in neighbourhoodsof G that are parametrized by the map u : R k × B → G defined by u ( t , . . . , t k , b ) = e t X · · · e t k X k b. We find it helpful to introduce the following operation, as an intermediary between π andthe π i . Given f ∈ C ∞ c ( R k × B ), we define a convolution operation e π ( f ) on C ∞ c ( M ) by( e π ( f ) ψ )( m ) = Z · · · Z R k f ( − t k , · · · , − t , µ ( e t X M · · · e t k X Mk m )) ψ ( e t X M · · · e t k X Mk m ) dt · · · dt k . The reason for using precisely the above expression will hopefully be made clear shortly.For now, let us note that the following relationship between π , . . . , π k and e π is a trivialconsequence of the definitions. Lemma 9.
Suppose f , . . . , f k ∈ C ∞ c ( R ) , χ ∈ C ∞ c ( B ) and define f ∈ C ∞ c ( R k × B ) by f = f k ⊗ · · · ⊗ f ⊗ χ . Then, e π ( f ) ψ = π k ( f k ) · · · π ( f ) ψ holds whenever ψ ∈ C ∞ c ( M ) has χ ≡ on µ (supp( ψ )) . Next, we relate e π to π . Lemma 10.
Suppose that W is an open subset of R k × B that is mapped diffeomorphicallyby u onto an open subset of G . Then, there exists a linear bijection θ W from C ∞ c ( W ) ⊆ C ∞ c ( R k × B ) to C ∞ c ( u ( W )) ⊆ C ∞ c ( G ) such that e π ( f ) ψ = π ( θ W ( f )) ψ holds for all f ∈ C ∞ c ( W ) and ψ ∈ C ∞ c ( M ) . We remark that the bijection θ W above is independent of the manifold M and the givenaction of G ; the same θ W works for all G -sets. The basic idea is to define θ W ( f ) as thepushforward of f by u , followed by multiplication by a suitable Jacobian factor.Before proceeding to the proof of Lemma 10 we find it useful to re-express the representations π and e π in a somewhat more abstract form. Form the transformation groupoid G ⋉ M ⇒ M with source σ , target τ , inversion map and induced Haar system λ M . Given f ∈ C ∞ c ( G )and ψ ∈ C ∞ c ( M ), we define f × ψ ∈ C ∞ c ( G ⋉ M ) by restricting f ⊗ ψ to G ⋉ M ⊆ G × M ( f × ψ )( γ, m ) = f ( γ ) ψ ( m ) .
11e can express π in terms of the above operations as follows: π ( f ) ψ = σ ! (( f × ψ ) ◦ ) f ∈ C ∞ c ( G ) , ψ ∈ C ∞ c ( M ) , (7)where σ ! : C ∞ c ( G ⋉ M ) → C ∞ c ( M ) is the integration along fibres map associated to λ M .Next, we find an analogous expression for e π . First, define the following maps: u : R k × B → G ( t , . . . , t k , b ) e t X · · · e t k X k b e s : R k × B → B ( t , . . . , t k , b ) b e ı : R k × B → R k × B ( t , . . . , t k , b ) ( − t k , . . . , − t , e t X B · · · e t k X Bk b ) v : R k × M → G ⋉ M ( t , . . . , t k , m ) ( e t X · · · e t k X k µ ( m ) , m ) e σ : R k × M → M ( t , . . . , t k , m ) m e : R k × M → R k × M ( t , . . . , t k , m ) ( − t k , . . . , − t , e t X M · · · e t k X Mk m ) . We give e s and e σ the obvious fibrewise measures, copying the standard volume measure of R k on each fibre, and denote the associated integration along fibre maps by e s ! and e σ ! . Noticethat e ı and e are order-2 diffeomorphisms and that the following intertwining relations aresatisfied. u ◦ e ı = ı ◦ u e s = s ◦ u v ◦ e = ◦ v e σ = σ ◦ v Given f ∈ C ∞ c ( R k × B ) and ψ ∈ C ∞ c ( M ), we define f × ψ ∈ C ∞ c ( R k × M ) by( f × ψ )( t , . . . , t k , m ) = f ( t , . . . , t k , µ ( m )) ψ ( m ) . By analogy with (7), we give the following expression for e π in terms of the above opera-tions. e π ( f ) ψ = e σ ! (( f × ψ ) ◦ e ) f ∈ C ∞ c ( R k × B ) , ψ ∈ C ∞ c ( M ) (8)With these preparations and notations, we proceed to the Proof of Lemma 10.
The relation u e ı = ıu implies that W ′ := e ı ( W ) is also mapped diffeo-morphically by u onto an open subset of G . Thus, there exists a smooth, positive-valuedfunction ρ W ′ on W ′ such that the pullback of the Haar measure along u to W ′ ⊆ R k × B equals ρ W ′ dt · · · dt k . Thus, for all f ∈ C ∞ c ( u ( W ′ )), we have e s ! (( f ◦ u | W ′ ) ρ W ′ ) = s ! ( f ) . Let Ω = (id × µ ) − ( W ), an open subset of R k × M , and note that v maps Ω diffeomorphicallyonto an open subset of G ⋉ M . The relation v e = v implies that Ω ′ := e (Ω) is also mappeddiffeomorphically by v onto an open subset of G ⋉ M . The pullback of the Haar measure λ M G ⋊ M along v | Ω ′ is ρ Ω ′ dt . . . dt k , where ρ Ω ′ = ρ W ′ (id × µ ). Thus, for all f ∈ C ∞ c ( v (Ω ′ )),we have e σ ! (( f ◦ v | Ω ′ ) ρ Ω ′ ) = σ ! ( f ) . (9)Take θ W to be the bijection C ∞ c ( W ) → C c ( u ( W )) determined by( θ W ( f ) ◦ u | W )( ρ W ′ ◦ e ı ) = f f ∈ C ∞ c ( W ) . Then, for any f ∈ C ∞ c ( W ) and ψ ∈ C ∞ c ( M ), a simple calculation shows that f × ψ = (( θ W ( f ) ◦ u | W )( ρ W ′ ◦ e ı )) × ψ = (( θ W ( f ) × ψ ) ◦ v | Ω )( ρ Ω ′ ◦ e )and so ( f × ψ ) ◦ e = (( θ W ( f ) × ψ ) ◦ ◦ v | Ω ′ ) ρ Ω ′ . Thus, applying (7), (8) and (9), we find that e π ( f ) ψ = e σ ! (( f × ψ ) ◦ e ) = σ ! (( θ W ( f ) × ψ ) ◦ ) = π ( f ) ψ, as was to be proven.Combining Lemma 10 and Lemma 9, we obtain the following technical lemma below, whichhas been our goal throughout the section. Lemma 11.
Let G ⇒ B be a Lie groupoid with a given Haar system. Let K be a compact sub-set of B and let W be an open subset of G containing K . Suppose X , . . . , X k ∈ C ∞ c ( B, AG ) constitute a base for AG over each point of K . Then, there exists an ǫ > such that, for any f , . . . , f k ∈ C ∞ c ( R ) having supp( f i ) ⊆ ( − ǫ, ǫ ) , there exists an f ∈ C ∞ c ( G ) with supp( f ) ⊆ W with the property that, for any G -space M with momentum map µ and any ψ ∈ C ∞ c ( M ) with µ (supp( ψ )) ⊆ K , one has π ( f ) ψ = π ( f ) · · · π k ( f k ) ψ where π , . . . , π k are the representations of C ∞ c ( R ) on C ∞ c ( M ) associated by (1) to the cor-responding complete vector fields X M , · · · , X Mk on M and π denotes the representation of C ∞ c ( G ) on C ∞ c ( M ) given by (6) .Proof. Let u : R k × M → G be defined by u ( t , . . . , t k , b ) = e t X . . . e t k X k b . By an inversefunction theorem/compactness argument, there exists an open subset U of B containing K and an ǫ > u maps ( − ǫ, ǫ ) k × U diffeomorphically onto an open subset of G .By possibly shrinking ǫ and U , we may assume that u (( − ǫ, ǫ ) k × U ) ⊆ W . Let χ ∈ C ∞ c ( B )satisfy supp( χ ) ⊆ U and χ ≡ K . Let f , . . . , f k ∈ C ∞ c ( R ) have supp( f i ) ⊆ ( − ǫ, ǫ )so that f = f k ⊗ · · · ⊗ f ⊗ χ ∈ C ∞ c ( R k × B ) has supp( f ) ⊆ ( − ǫ, ǫ ) k × U . Then, for any ψ ∈ C ∞ c ( M ) with supp( ψ ) ⊆ K , we have π k ( f k ) · · · π ( f ) ψ = e π ( f ) ψ = π ( θ W ( f )) ψ where the first equality comes from Lemma 9 and the second from Lemma 10. Since θ W ( f )is supported in u (( − ǫ, ǫ ) k × U ) ⊆ W , we are finished.13 Proof of main theorem
We have made the necessary preparations to prove Theorem 2, stated in the introduction.In fact, we can prove the following more general result.
Theorem 12.
Let G ⇒ B be a Lie groupoid with a given Haar system, let M be a G -spacewith momentum map µ and let π be the corresponding representation of C ∞ c ( G ) = C ∞ c ( G, λ ) on C ∞ c ( M ) , i.e. the integrated form of the action defined by (6) . Then, for every ϕ ∈ C ∞ c ( M ) , there exist f , . . . , f N ∈ C ∞ c ( G ) and ψ , . . . , ψ N ∈ C ∞ c ( M ) such that ϕ = π ( f ) ψ + . . . + π ( f N ) ψ N . Moreover, this factorization can be taken such that, for all i , supp( ψ i ) ⊆ supp( ϕ ) and supp( f i ) ⊆ W , where W is a prescribed open subset of G containing µ (supp( ϕ )) .Proof. Let K = µ (supp( ϕ )) and fix an open set W in G containing K . It is enough toprove the theorem under the additional hypothesis that the Lie algebroid AG is trivial (asa bundle) over a neighbourhood of K . Indeed, we can use a partition of unity on B towrite any ϕ as a finite sum of functions that satisfy this extra hypothesis and have supportcontained in that of the original ϕ . If each summand can be decomposed in the desired way,then so can the sum, simply by adding up the decompositions.Under this extra assumption, there exist sections X , . . . , X k ∈ C ∞ c ( B, AG ) that constitutea frame of AG over each point in K . Let X M , . . . , X Mk denote the corresponding completevector fields on M and let π , . . . , π k denote the corresponding representations of C ∞ c ( R ).Let ǫ > X = X Mk and ψ = ϕ , we can write ϕ = π k ( f ) ψ + π k ( f ) ψ where f , f ∈ C ∞ c ( R ) with supports contained in ( − ǫ, ǫ ) and ψ , ψ ∈ C ∞ c ( M ) with supportscontained in supp( ϕ ). Applying Theorem 3 with X = X Mk − for ψ = ψ and ψ = ψ thengives ϕ = π ( f ) π ( f ) ψ + π ( f ) π ( f ) ψ + π ( f ) π ( f ) ψ + π ( f ) π ( f ) ψ where the f s are in C ∞ c ( R ) with supports contained in ( − ǫ, ǫ ) and ψ s are in C ∞ c ( M ) withsupports contained in supp( ϕ ). Continuing in this manner, we eventually get ϕ as the sumof 2 k terms of the form π ( f ) · · · π k ( f k ) ψ where f , . . . , f k ∈ C ∞ c ( R ) have supports contained in ( − ǫ, ǫ ) and ψ ∈ C ∞ c ( M ) has supportcontained in supp( ϕ ). If ǫ is sufficiently small, then Lemma 11 guarantees that each of theseterms can be written as π ( f ) ψ where f ∈ C ∞ c ( G ) has supp( f ) ⊆ W .14pecializing to the case where G is acting on itself, we obtain the desired generalized Dixmier-Malliavin theorem as a corollary. Theorem 13.
Let G ⇒ B be a Lie groupoid with a given Haar system. Then, for any ϕ ∈ C ∞ c ( G ) , there exist f , . . . , f N , ψ , . . . , ψ N ∈ C ∞ c ( G ) such that ϕ = f ∗ ψ + . . . + f N ∗ ψ N . Moreover, this factorization can be taken such that, for all i , supp( ψ i ) ⊆ supp( ϕ ) and supp( f i ) ⊆ W , where W is a prescribed open subset of G containing t (supp( ϕ )) . In this section, we study ideals of smooth functions vanishing to given order along a subman-ifold when the operation is pointwise multiplication. This will lay the groundwork for thesubsequent section in which the operation is convolution. The result we wish to generalizeto the convolution setting is the following.
Theorem 14.
Let X be a closed submanifold of a smooth manifold M . Let I p ⊆ C ∞ c ( M ) denote the ideal of functions vanishing to p th order on X . Then,1. ( I ∞ ) = I ∞ ( I ) p = I p for every positive integer p . The relation ( I ∞ ) = I ∞ actually remains true even when X is any closed subset of M , andnot necessarily a submanifold. This stronger result is due to Tougeron. See [19], Propo-sition V.2.3 as well as [18], Section 4. Note that, although the results in these referencesare stated in terms of germs of functions, it is a simple matter to use partitions of unityto convert them into statements about compactly-supported functions. The second relation( I ) p = I p is much more elementary than the first and can be established by applying Taylor’stheorem locally.Theorem 14 is not quite sufficient for our purposes, however. We need to consider a submer-sion π : N → M (later taken to be the source or target projection of Lie groupoid) and theresulting C ∞ c ( N )-module structure on C ∞ c ( M ). It will be convenient for us to combine thecases of infinite and finite vanishing order into a single statement, but we hasten to point outthat the case of infinite vanishing order is by far the more substantive one. Note also thatthe MathOverflow question [3] (still not fully resolved at time of writing) centers aroundquite similar issues. Theorem 15.
Let π : N → M be a submersion. View C ∞ c ( N ) as a C ∞ c ( M ) -module withproduct f · g = ( f ◦ π ) g , where f ∈ C ∞ c ( M ) , g ∈ C ∞ c ( N ) . Let X be a closed submanifold of and set Y := π − ( Y ) . For p ∈ N ∪ {∞} , write I p ⊆ C ∞ c ( M ) and J p ⊆ C ∞ c ( M ) for theideals of functions that vanish to p th order on X and Y respectively. Then, the relation J p + q = I p · J q is satisfied for all p, q ∈ N ∪ {∞} , where I p · J q means the set of all sums of products g · h ,where g ∈ I p , h ∈ J q . The problem is obviously local in nature; one can use a partition of unity to chop up a function f on N into smaller functions all of which vanish to the same order as f on Y . In fact, it isenough to consider the case where N = R k × R ℓ , M = R k , X = { } and π is the standardprojection, so that Y = { } × R ℓ . Throughout this section, n = k + ℓ and R n = R k × R ℓ hascoordinates ( x, y ) = ( x , . . . , x k , y , . . . , y ℓ ). We use the usual multi-index notation for partialderivatives: given γ = ( α, β ) ∈ N n = N k × N ℓ we write ∂ γ := ∂ α ∂x α · · · ∂ αk ∂x αkk ∂ β ∂y β · · · ∂ βℓ ∂y βℓℓ .We treat the p < ∞ and p = q = ∞ cases of the local problem separately in the followingtwo lemmas. The bulk of our effort will go towards establishing the second of these. Lemma 16. If f ∈ C ∞ ( R n ) vanishes to order p + q on { } × R ℓ , where p ∈ N , q ∈ N ∪ {∞} ,then one can write f ( x, y ) = X | α | = p x α f α ( x, y ) , where each f α belongs to C ∞ ( R n ) and vanishes to order q on { } × R ℓ . Lemma 17. If f ∈ C ∞ c ( R n ) vanishes to order ∞ on { } × R ℓ , then one can write f ( x, y ) = ρ ( x ) h ( x, y ) , where ρ ∈ C ∞ ( R k ) has a zero of order ∞ at and is strictly positive on R k \ { } , and h ∈ C ∞ ( R n ) vanishes to order ∞ on { } × R ℓ . It is a simple matter to derive Theorem 15 from these lemmas.
Proof of Theorem 15.
We just need to show J p + q ⊆ I p · J q , the reverse containment beingobvious. Suppose, therefore, that f ∈ J p + q . Using a partition of unity argument andstandard facts about the local structure of submersions and submanifolds, we may assumeone of the following two alternatives holds:(i) The support of f is disjoint from Y .(ii) The support of f is contained in an open set U such that:(a) U is diffeomorphic to R k × R ℓ × R ℓ and π ( U ) is diffeomorphic to R k × R ℓ ,(b) under these diffeomorphisms, π : U → π ( U ) identifies with the standard projection R k × R ℓ × R ℓ → R k × R ℓ , and 16c) under these diffeomorphisms, X ∩ π ( U ) identifies { } × R ℓ , so that Y ∩ U identifieswith { } × R ℓ × R ℓ .If (i) is satisfied, take any g ∈ C ∞ c ( M ) that is equal to 1 on π (supp( f )) and equal to 0outside of some open set not intersecting X . Then, f = ( g ◦ π ) f , where g vanishes to order ∞ ≥ p on X and f vanishes to order p + q ≥ q on Y .If (ii) is satisfied and p < ∞ , then, applying Lemma 16 in the given chart with ℓ = ℓ + ℓ ,we may assume that f = x α h , where | α | = p and h ∈ C ∞ c ( R k × R ℓ × R ℓ ) vanishes to order q on { } × R ℓ × R ℓ . Then, f = ( g ◦ π ) h , where g ∈ C ∞ c ( R k × R ℓ ) is given by g = x α ϕ for an appropriate cutoff function ϕ ∈ C ∞ c ( R k × R ℓ ). Obviously, g vanishes to order p on { } × R ℓ . The expression f = ( g ◦ π ) h can be made global simply by extending g and h tobe identically 0 outside of π ( U ) and U , respectively.If (ii) is satisfied and q = ∞ , we proceed in the same way using Lemma 17. First, in thegiven chart, write f ( x, y ) = ρ ( x ) h ( x, y ) where ρ ∈ C ∞ c ( R k ) has a zero of infinite order at 0and is strictly positive on R k \ { } , and h ∈ C ∞ c ( R k × R ℓ × R ℓ ) vanishes to order ∞ on { } × R ℓ × R ℓ . Then, f = ( g ◦ π ) h , where g ∈ C ∞ c ( R k × R ℓ ) is given by g = ρ · ϕ for anappropriate cutoff function ϕ ∈ C ∞ c ( R k × R ℓ ). Obviously g vanishes to order ∞ ≥ p on { } × R ℓ .It remains to prove the Lemmas 16 and 17. Proof of Lemma 16. If p = 0, there is nothing to prove, so assume p ≥
1. It clearly sufficesto prove that we can write f ( x, y ) = x f ( x, y ) + . . . x k f k ( x, y )where f i ∈ C ∞ ( R n ) vanish to order p + q − f , . . . , f k defined by f ( x, y ) = Z ∂f∂x ( tx , x , . . . , x k , y , y , . . . , y ℓ ) dtf ( x, y ) = Z ∂f∂x (0 , tx , x , . . . , x k , y , y , . . . , y ℓ ) dt ... f k ( x, y ) = Z ∂f∂x k (0 , . . . , , tx k , y , y , . . . , y ℓ ) dt serve this purpose.The proof of Lemma 17 will rely on several further lemmas, specifically Lemmas 21, 23and 24. Recall that a function f on [1 , ∞ ) is rapidly decaying if lim t →∞ t m f ( t ) = 0 forevery nonnegative integer m . We say f is a Schwartz function if it is C ∞ and it and all its17erivatives are rapidly decaying. Since there are many more functions of rapid decay thanthere are Schwartz functions, it seems plausible that there could exist a function of rapiddecay that vanishes more slowly than any Schwartz function. The following lemma showsthis does not occur by providing a “Schwartz envelope” for any rapidly decaying function.The original reference for this fact may be [6], Lemma 3.6, pp. 127. One can also find it inthe expository note [12]. Lemma 18.
1. If f is a bounded, rapidly decaying function on [1 , ∞ ) , then there exists a positive-valued, monotone decreasing Schwartz function g on [1 , ∞ ) such that | f | ≤ g .2. If ( f k ) is a sequence of rapidly decaying functions on [1 , ∞ ) , then there exists a positive-valued Schwartz function g on [1 , ∞ ) such that lim t →∞ f k ( t ) g ( t ) = 0 for all k .Proof. For the first part, assume without loss of generality that f is monotone decreasing,or else replace it by t sup s ≥ t | f ( s ) | . Let ϕ ∈ C ∞ ( R ) be a nonnegative-valued functionwith support contained in [0 ,
1] satisfying R R ϕ ( t ) dt = 1. Define g to be the convolution ϕ ∗ f , that is, g ( t ) = R ϕ ( s ) f ( t − s ) ds . We remark that there is a small issue with thisdefinition of g near t = 1, but this is easily fixed by enlarging the domain of f , say by defining f ( t ) = sup s ≥ f ( s ) for t ≤
1. It is easy to see that g is monotone decreasing and that f ≤ g .One can check that the convolution of two rapidly decaying functions is rapidly decaying(imposing sufficient regularity properties so that convolution makes sense), and it followsthat the convolution of a rapidly decaying function with a Schwartz function is Schwartz(since the derivatives can be put on the Schwartz function).For the second part, assume without loss of generality that each f k is bounded and usethe first part to produce, for each k , a positive-valued Schwartz function g k such thatlim t →∞ f k ( t ) g k ( t ) = 0 (if f k ≤ g k holds, but f k g k does not vanish at infinity, replace g k with t tg k ( t )). An easy diagonal selection argument guarantees the existence of constants c k > g = P c k g k is a Schwartz function. Since g > g k , it is clear that f k g vanishesat infinity for every k .The next step is to convert Lemma 18 into a statement about smooth functions with aninfinite order zero at 0 by performing an inversion in the variable. Much more sophisticatedaccounts of the connection between Schwartz functions and functions that remain smoothafter being extended by zero can be found in the literature, see [1], Theorem 5.4.1. Forpresent purposes, the simple-minded lemma below is enough. Lemma 19.
The inversion map t /t : (0 , → [1 , ∞ ) puts functions f on (0 , with lim t → + f ( t ) t − m = 0 for all positive integers m into bijection with the rapidly decaying func-tions on [1 , ∞ ) , and also puts the smooth functions f on (0 , for which putting f ( t ) = 0 for t ≤ yields a smooth extension into bijection with the Schwartz functions on [1 , ∞ ) . To prove Lemma 19, we need the following simple fact.18 emma 20.
Suppose f is a smooth function on (0 , ∞ ) with lim t → + f ( m ) ( t ) = 0 for everynonnegative integer m . Then, in setting f ( t ) = 0 for t ≤ , one obtains a C ∞ extension of f to all of R .Proof. An application of the mean value theorem shows the extension is differentiable withderivative 0 at the origin. The statement follows by induction.
Proof of Lemma 19.
The first correspondence is obvious. Towards the second, suppose f isa Schwartz function on [1 , ∞ ) and define g on (0 ,
1] by g ( t ) = f (1 /t ). Then, g ′ ( t ) = f (1 /t ),where f ( t ) = − t f ′ ( t ) is yet another Schwartz function. By induction, each derivative of g has the form f k (1 /t ) for some Schwartz function f k on [1 , ∞ ). In particular, lim t → + g ( k ) ( t ) =0 for all k so that, by Lemma 20, setting g ( t ) = 0 for t ≤ g . The converse direction, that f ( t ) = g (1 /t ) is a Schwartz function on [1 , ∞ ) when g is asmooth function with g ( t ) = 0 for t ≤
0, proceeds similarly.Applying the correspondence of Lemma 19, the second part of Lemma 18 translates to thefollowing.
Lemma 21.
Let f k be a sequence of functions on [0 , ∞ ) that vanish to infinite order at ,i.e. f k (0) = 0 and lim t → + f k ( t ) t − m = 0 for all m . Then, there exists a C ∞ function g on R with g ( t ) = 0 for t ≤ , and g ( t ) > for t > such that lim t → + f k ( t ) g ( t ) = 0 for all k .Remark. A direct proof of Lemma 21 was given by George Lowther at [3] (Lemma 2).Nonetheless, as the Schwartz function formulation of this result appears to be better known,it seemed worthwhile to draw out this connection here.Lemmas 23 and 24, stated and proved below, will rely on the following mild generalizationof Lemma 20 whose proof we omit. Recall that n = k + ℓ and R n = R k × R ℓ has coordinates( x, y ) = ( x , . . . , x k , y , . . . , y ℓ ). Lemma 22.
Let f be a smooth function on R n \ ( { } × R ℓ ) such that, for every γ ∈ N n , thepartial derivative ∂ γ f has limit zero at every point of { } × R ℓ . Then, f extends to a C ∞ function on all of R n vanishing to infinite order on { } × R ℓ . In particular, when ℓ = 0, the above says that a smooth function on R k \ { } with all higherpartials vanishing at the origin extends smoothly to all of R k . This is helpful in checking thefollowing. Lemma 23.
Let ϕ be a smooth function on R with a zero of infinite order at . Then, thefunction f on R k defined by f ( x ) = ϕ ( | x | ) , where | x | = p x + . . . + x k , is a C ∞ functionon R k with a zero of infinite order at .Proof. Obviously f is smooth on R k \ { } . On the latter domain, ∂f∂x i ( x ) = ψ ( | x | ) where ψ ( t ) = ( − ϕ ( t ) t t = 00 t = 0 is another C ∞ function on R with a zero of infinite order at 0.19y induction, f satisfies the conditions of Lemma 22 (with ℓ = 0), whence is smooth asclaimed.The next lemma gives sufficient conditions under which the quotient of two smooth functionson R n that vanish to infinite order on { } × R ℓ is another such function. Lemma 24.
Let f and g be C ∞ functions on R n that vanish to infinite order on { } × R ℓ and assume g > on R n \ ( { } × R ℓ ) . If, for every γ ∈ N n and m ∈ N , the function ∂ γ fg m haslimit at each point of { } × R ℓ , then fg extends to a C ∞ function on all of R n vanishing toinfinite order on { } × R ℓ .Proof. Let F denote the collection of all smooth functions on R n \ ( { } × R ℓ ) obtainedas C ∞ ( R n )-linear combinations of the functions ∂ γ fg m . By assumption, the functions in F all have limit 0 at each point { } × R ℓ . Observe that F is closed under taking partialderivatives. Indeed, if γ is a multi-index, m is a positive integer, h ∈ C ∞ ( R n ) and ∂ is oneof the first-order partials ∂∂x i or ∂∂y j , then ∂ ( h ∂ γ fg m ) = ( ∂h ) ∂ γ fg m + h ∂ ◦ ∂ γ fg m − m ( ∂g ) h ∂ γ fg m +1 . Thus,thinking of fg as a smooth function on R n \ ( { } × R ℓ ), we have by induction that all of itshigher order partial derivatives have limit 0 at every point of { } × R ℓ and so, by Lemma 22, fg extends to a smooth function on R n that vanishes to infinite order on { } × R ℓ .We are now in a position to give the Proof of Lemma 17.
Suppose that f ∈ C ∞ c ( R n ) vanishes to order ∞ on { } × R ℓ . Given γ ∈ N n and m, r ∈ N , define a continuous function f γ,m,r on [0 , ∞ ) by f γ,m,r ( t ) = sup x ∈ R k | x |≤ t sup y ∈ R ℓ | y |≤ r | ∂ γ f ( x, y ) | /m . The assumption that f vanishes to infinite order on { }× R ℓ implies that each f γ,m,r vanishesto infinite order at t = 0, i.e. lim t → + f γ,m,r ( t ) t − s = 0 for any positive integer s . It thereforefollows from Lemma 21 that there exists a C ∞ function ϕ on R vanishing to infinite orderat t = 0 with ϕ ( t ) > t > t → + f γ,m,r ( t ) ϕ ( t ) = 0 for all γ, m, r . By Lemma 23,the function ρ on R k defined by ρ ( x ) = ϕ ( | x | ) is a C ∞ function, positive on R k \ { } andvanishing to infinite order at 0. By design, for any γ ∈ N n and m, r ∈ N , one has the bound (cid:12)(cid:12)(cid:12)(cid:12) ∂ γ f ( x, y ) ρ ( x ) m (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) f γ,m,r ( | x | ) ϕ ( | x | ) (cid:19) m for x = 0 and | y | < r , which shows that the left hand side vanishes as x →
0. Thus, applyingLemma 24, one has that h ( x, y ) = f ( x,y ) ρ ( x ) extends smoothly to a function on all of R n thatvanishes to infinite order on { } × R ℓ , completing the proof.20 Product structure of ideals in the smooth convolu-tion algebra of a Lie groupoid
In this final section, we apply our generalization of the Dixmier-Malliavin theorem to obtainthe Lie groupoid analog of Theorem 14 by reducing it to its commutative counterpart.Throughout, G denotes a Lie groupoid over the manifold M with fixed Haar system λ and weassume that X ⊆ M is an invariant closed submanifold in the sense that s − ( X ) = t − ( X ).The restriction G X := s − ( X ) = t − ( X ) of G to X is, in its own right, a Lie groupoid G X ⇒ X . The Haar system λ on G can be restricted to a Haar system λ X on G X anddoing so makes the restriction map C ∞ c ( G ) → C ∞ c ( G X ) into a homomorphism of the smoothconvolution algebras. The kernel of this homomorphism is the ideal J ⊆ C ∞ c ( G ) of functionsthat vanish on G X . More generally, one can consider J p ⊆ C ∞ c ( G ), the functions whichvanish to p th order on G X . It is simple to confirm that each J p is an ideal with respect tothe convolution product (either by arguing directly, or by applying Proposition 27 below).The quotients C ∞ c ( G ) /J p for p > C ∞ c ( G X ), fitting as they do into exact sequences of the form0 → J /J p → C ∞ c ( G ) /J p → C ∞ c ( G X ) → . Roughly speaking, the kernel J /J p contains Taylor series information up to order p − G X .The Lie groupoid algebra analog of Theorem 14 is the following: Theorem 25.
Let G ⇒ M be a Lie groupoid with given Haar system. Let X be an invariant,closed submanifold of M and let G X := s − ( X ) = t − ( X ) . Let J p ⊆ C ∞ c ( G ) denote the ideal,with respect to convolution, of functions that vanish to order p on G X . Then,1. J ∞ ∗ J ∞ = J ∞ ( J ) ∗ p = J p for every positive integer p . As in the preceding section, for the sake of efficiency, we shall in fact prove a more generalresult which treats the cases of finite and infinite vanishing order on equal footing.
Theorem 26.
Let G ⇒ M be a Lie groupoid with a given Haar system. Let X be aninvariant, closed submanifold of M and let G X := s − ( X ) = t − ( X ) . For p ∈ N ∪ {∞} , let J p ⊆ C ∞ c ( G ) denote the ideal of functions that vanish to order p on G X . Then, J p + q = J p ∗ J q holds for all p, q ∈ N ∪ {∞} . It is easy to see that J p ∗ J q ⊆ J p + q is satisfied (again, either by arguing directly or by applyingProposition 27 below). The goal is therefore to sharpen these containments to equalities.Note that, whereas in the commutative setting the p = q = 0 is trivial, in Theorem 2621bove the p = q = 0 case is exactly Theorem 13, our extension of the Dixmier-Malliavintheorem . Conversely, Theorem 13, in tandem with Proposition 27 below, reduces the proofof Theorem 26 to a formal manipulation.Recall that C ∞ c ( G ) is a C ∞ c ( M )-bimodule with respect to the products defined by f · ϕ = ( f ◦ t ) ϕ ϕ · f = ϕ ( f ◦ s )and, moreover, that these products satisfy the expected associativity identities f · ( ϕ ∗ ψ ) = ( f · ϕ ) ∗ ψ ( ϕ ∗ ψ ) · f = ϕ ∗ ( ψ · f ) , where f ∈ C ∞ c ( M ) and ϕ, ψ ∈ C ∞ c ( G ).The following proposition shows that the ideals I p ⊆ C ∞ c ( M ) of functions vanishing to p thorder on X determine the ideals J p ⊆ C ∞ c ( G ) of functions vanishing to p th order on G X byway of this module structure; one may write J p = I p · C ∞ c ( G ) = C ∞ c ( G ) · I p . It is a quickcorollary of the results in the preceding section. Proposition 27.
Let G ⇒ M be Lie groupoid with a given Haar system. Let X be aninvariant closed submanifold of M . For each p ∈ N ∪{∞} , let I p ⊆ C ∞ c ( M ) and J p ⊆ C ∞ c ( G ) denote the collection of functions vanishing to p th order on X and G X respectively. Then, J p + q = I p · J q = J q · I p holds for all p, q ∈ N ∪ {∞} .Proof. Apply Theorem 15 with N = G and π = s , respectively π = t .Theorem 26 is now a trivial consequence of Theorem 13 and Proposition 27. Proof of Theorem 26.
We have J p ∗ J q = I p · C ∞ c ( G ) ∗ C ∞ c ( G ) · I q = I p · C ∞ c ( G ) · I q = J p · I q = J p + q , where the second equality holds by Theorem 13 and the rest hold by Proposition 27. References [1] A. Aizenbud and D. Gourevitch. Schwartz functions on Nash manifolds.
Int. Math.Res. Not. IMRN , (5):Art. ID rnm 155, 37, 2008.[2] I. Androulidakis and G. Skandalis. The holonomy groupoid of a singular foliation.
J.Reine Angew. Math. , 626:1–37, 2009. 223] M. B¨achtold. Are submersions of differentiable manifolds flat morphisms? http://mathoverflow.net/q/22350 . Accessed: 2019-08-31.[4] P. Cartier. Vecteurs diff´erentiables dans les repr´esentations unitaires des groupes deLie. pages 20–34. Lecture Notes in Math., Vol. 514, 1976.[5] B. Casselman. The theorem of Dixmier-Malliavin. . Ac-cessed: 2019-01-14.[6] C. Chevalley.
Theory of distributions . Lectures given at Columbia University, 1950-1951.[7] A. Connes.
Noncommutative geometry . Academic Press, Inc., San Diego, CA, 1994.[8] C. Debord. Holonomy groupoids of singular foliations.
J. Differential Geom. , 58(3):467–500, 2001.[9] J. Dixmier and P. Malliavin. Factorisations de fonctions et de vecteurs ind´efinimentdiff´erentiables.
Bull. Sci. Math. (2) , 102(4):307–330, 1978.[10] L. Ehrenpreis. Solution of some problems of division. IV. Invertible and elliptic opera-tors.
Amer. J. Math. , 82:522–588, 1960.[11] M. Francis. The smooth algebra of a one-dimensional singular foliation. In preparation.[12] P. Garrett. Weil-Schwartz envelopes for rapidly decreasing functions. .Accessed: 2019-08-31.[13] K. Mackenzie.
Lie groupoids and Lie algebroids in differential geometry , volume 124 of
London Mathematical Society Lecture Note Series . Cambridge University Press, Cam-bridge, 1987.[14] J. Pradines. How to define the differentiable graph of a singular foliation.
CahiersTopologie G´eom. Diff´erentielle Cat´eg. , 26(4):339–380, 1985.[15] J. Pradines and B. Bigonnet. Graphe d’un feuilletage singulier.
C. R. Acad. Sci. ParisS´er. I Math. , 300(13):439–442, 1985.[16] J. Renault.
A groupoid approach to C ∗ -algebras , volume 793 of Lecture Notes in Math-ematics . Springer, Berlin, 1980.[17] L. A. Rubel, W. A. Squires, and B. A. Taylor. Irreducibility of certain entire functionswith applications to harmonic analysis.
Ann. of Math. (2) , 108(3):553–567, 1978.[18] V. Thilliez. Division by flat ultradifferentiable functions and sectorial extensions.
ResultsMath. , 44(1-2):169–188, 2003.[19] J.-C. Tougeron.
Id´eaux de fonctions diff´erentiables . Springer-Verlag, Berlin-New York,1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71.2320] R. S. Yulmukhametov. Solution of the L. Ehrenpreis problem on factorization.