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Geometrical embeddings for distributions into algebras of generalized functions

We use spectral theory to produce embeddings of distributions in the algebras of generalized functions on a closed Riemannian manifold. These embeddings are invariant under isometries and preserve the singularity structure of the distributions.


Introduction
Algebras of generalized functions as introduced by Colombeau [5,6,4,19,12] provide a consistent framework for carrying out nonlinear analysis on spaces of distributions, especially in nonlinear partial differential equations and regularity theory [19,11,14,15,8]. There are various choices of embedding distributions into these algebras, representing fundamental aspects of nonlinear modeling depending on the choice of regularizing process.
The special algebra of generalized functions [5,12] over an open set in R n or on a manifold, is characterized in terms of a regularizing net of smooth functions and asymptotic estimates in terms of the regularizing parameter ε. More precisely, let I = (0, 1] then the special algebra over R n is defined by the following quotient construction. Firstly the moderate nets are defined as E M (R n ) = {(U ε ) ε∈I ∈ C ∞ (R n ) I |∀ semi-norms ρ on C ∞ (R n ) ∃N ∈ Z such that ρ(U ε ) ∼ O(ε N )}.
The nets of negligible growth are given by form an ideal in E M (R n ) and the special Colombeau algebra is defined as the quotient G(R n ) := E M (R n )/N (R n ).
Supported by FWF grant Y237-N13 of the Austrian Science Fund.
Given a moderate net of smooth functions (u ε ) ε one usually denotes the corresponding element in the quotient by [(u ε ) ε ] or by u ε . We shall often identify the net with the element it represents if no confusion can arise. Also, for the sake of simplicity we shall omit the index M in the notation of moderate nets E M henceforth and simply write E. The algebra C ∞ (R n ) of smooth functions can of course be embedded into G(R n ) as a subalgebra by the constant embedding σ : To embed nonsmooth distributions one first picks a mollifier ρ ∈ S (R n ) such that the net ρ ε := ε −n ρ(x/ε) suitably approximates the delta distribution. The compactly supported distributions are embedded by ω → [(ω * ρ ε ) ε ]. This embedding can be extended to all distributions in D ′ (R n ) by suitable partitions of unity and cut off functions using the sheaf structure. The resulting embedding ι ρ : D ′ (R n ) → G(R n ) commutes with partial derivations and matches up with the constant embedding σ of smooth functions mentioned above. This implies that the inclusion of smooth functions via ι ρ is in fact an algebra homomorphism. Taking a closer look at the above process one realizes that: a) Convolution by the smooth mollifier is in fact a smoothing operator, that is it maps tempered distributions to smooth function. b) Since the net ρ ε → δ, the convolution by this net gives a net of smoothing operators that approximate the identity operator (in a sense to be made precise below).
In view of the above observations we consider embedding distributions into algebras of generalized functions using a net of smoothing operators approximating the identity operator. In case of a compact manifold one could rely on operators defined either by smooth kernels or by functional calculus to generate such nets of smoothing operators. In this paper we shall work with a compact Riemannian manifold M with associated Laplace operator ∆. We recall that by functional calculus any Schwartz class function F ∈ S (R) defines a smoothing operator F (∆). Let us set F ε (x) = F (εx). The main result of this paper is then as follows: Theorem 1.1. If a Schwartz function F is 1 near the origin then the net of smoothing operators F ε (∆) provides an embedding F ε : D ′ (M) → G(M). Such an embedding preserves the multiplication on C ∞ (M) (coincides with the constant embedding for smooth functions) and is invariant under isometries.
We emphasize the following interesting features of our approach. The embeddings F ε are of course globally constructed. Thus we do not need to work with local coordinates and use the sheaf structure of the algebra G. In addition the global construction makes it compatible with the geometry of the Riemannian manifold as these embeddings commute with the Laplace operator and hence are preserved under isometries. The embeddings introduced so far in the literature ( [1,7,16]) are 'nongeometric' in that they depend on choices of partitions of unity, cut-offs, etc. (cf. [12], Sec. 3.2.2 for a discussion).
We also note that this functional calculus technique can be naturally applied in case of embedding distributional sections of a (complex) vector bundle into the corresponding space of generalized sections.
The proof of Theorem 1.1 is based on the functoriality of the construction of generalizedC-modules over a locally convex space. The negligibility and moderateness estimate are hence obtained as direct consequences of Weyl's estimates for eigenvalues of positive elliptic operators. For instance the Weyl estimates being asymptotic provides the well known continuity of the map and hence establish the moderateness estimates.

Locally convex spaces
Let X be a locally convex (Hausdorff) topological vector space then one can associate a generalized locally convex space G X (see [10,9], as well as [22]) as follows. Let I ⊂ R be the interval (0, 1]. Define the moderate nets in X I to be, Similarly we can define the negligible nets to be N (X) :={(x ε ) : ∀ continuous semi-norm ρ on X and ∀m The generalized locally convex space of X is then defined to be the quotient, G X := E(X)/N (X). One notes that in defining E(X), N (X) it suffices to restrict to a family of semi-norms that generate the locally convex topology for X. When X = C ∞ (M) is the space of smooth functions on a manifold M then we also write G(M) := G C ∞ (M ) . Also for X = C we call G C the space of generalized numbers and denote it byC.C is the ring of constants in G C . Every G X is naturally a module over the ringC, and hence is often referred to as theC-module associated with X. The sharp topology oñ C is the topology generated by sets of the form U x,p where x ∈C, p is an integer and Any continuous semi-norm ρ on a locally convex spaces X by definition provides a mapρ : E(X) → E(C) by applying ρ to each component. In factρ descends to a map G X →C. The sharp topology on anỹ C-module G X shall be defined as the weakest topology that makes all the aboveρ's continuous.
We recall the functoriality of the above construction.
Lemma 2.1. If φ : X → Y is a continuous linear map between locally convex spaces X and Y then there is a natural induced map φ * : G X → G Y defined on the representatives as φ * ((x ε )) = (φ(x ε )). Further φ * is continuous with respect to the sharp topology.
Proof. If τ is a continuous semi-norm on Y then τ • φ is a continuous semi-norm on X. Thus if x ε satisfies an estimate with τ • φ in E(X) or N (X) then φ(x ε ) satisfies the exact same estimates with respect to τ in E(Y ) or N (Y ). Thus φ * is well-defined. Since basic open sets U in G Y are pull-backs of open sets inC by some semi-norm ρ, thus φ −1 * (U) is a pullback of an open set with respect to ρ • φ.
For example any smooth map between two manifold f : M → N gives rise to a pull back map f * : G(N) → G(M).

Embedding of distributions
Let M be a compact manifold without boundary. A continuous linear operator T : D ′ (M) → C ∞ (M) is called a smoothing operator. We shall denote the space of all smoothing operators by Ψ −∞ (M). Each smoothing operator is a pseudodifferential operator and extends a map T : C ∞ (M) → C ∞ (M). The following is a well know characterization of smoothing operators: Let dy be a Riemannian density on M. An operator T is smoothing if and only if there is a smooth function k(x, y) ∈ C ∞ (M × M) such that for any u ∈ C ∞ (M), It also follows that any such identification gives an isomorphism of Ψ −∞ (M) with C ∞ (M × M) as locally convex spaces. In the sequel we shall always assume a Riemannian density is provided to us. This in particular shall imply that we are given an embedding of C ∞ (M) into D ′ (M).
Definition 3.1. A parametrized family T ε of smoothing operator is called a special embedding if (1) For any semi-norm ρ on C ∞ (M) and any distribution u there exists an integer N such that, is a smooth function on M then for all semi-norms ρ on C ∞ (M) and given any integer N, The above definition is tailored to obtain an embedding: If a collection of smoothing operators T ε is a special embedding then it provides an embedding of the space of distributions which coincides with the constant embedding of smooth functions and hence preserves the multiplication of smooth functions.
Definition 3.3. Let k > 0 be an integer. We say that a parametrized family T ε is an order-k embedding if T ε satisfies (1) and (2) above and the condition that for any smooth function f ∈ C ∞ (M) we have Example 3.4. Let M = R n and let ρ ∈ S (R n ) be in the Schwartz class such thatρ(0) = 1 and all moments of ρ vanish. Let ρ ε (x) = 1 ε n ρ(x/ε) be the approximate unit converging to the delta function. The operators T ε (f ) := ρ ε * f form a one parameter family of operators mapping the tempered distributions to smooth functions which satisfies all the above properties in the class of tempered distributions and smooth tempered functions. Of course it does not include all distributions on R n because of non-compactness. Now we fix a Riemannian metric on M and let ∆ be the corresponding Laplace operator. Let f ∈ S (R) then by functional calculus the operator f (∆) is a smoothing operator. We shall use smoothing operators obtained from the geometric Laplace operator to obtain geometrical embeddings.

Geometric Laplace operators and Functional calculus
Let M be a closed Riemannian manifold. Let ∆ be the associated scalar Laplace operator. Let φ k in L 2 (M) be eigenfunctions of ∆ with eigenvalues λ k . The classical theorem of Hermann Weyl implies that there exist a, b > 0 such that λn an b → 1 as n → ∞. This together with elliptic regularity of ∆ gives that the "Fourier series" expansion provides an isomorphism In the following⊗ as usual denotes the projective tensor product.
Proof. We give an outline of the proof here. See [13] for original and more detailed discussion. Fix a metric on both M and N. Let ∆ M and ∆ N be Laplace operators on M and N respectively. Then the Laplace on M × N with the product metric is given by ∆ M ×N := ∆ M ⊗ 1 + 1 ⊗ ∆ N . Then one can check that the following diagram commutes, Note that the map on the top row is an isomorphism since the one on the lower row is.
We view the smooth functions in C ∞ (M ×M) as kernels of smoothing operators. Thus in particular the above proposition Ψ −∞ (M) gives the following .
and hence defines a smoothing operator.
For any Schwartz class function f ∈ S (R) we denote by f (∆) the operator k f (λ k )φ k ⊗ φ k . Since f (∆) is diagonal with the same eigenspaces as ∆ it automatically commutes with ∆.
We thus obtain a map of locally convex spaces S (R) → Ψ −∞ (M) namely f → f (∆) which, abusing notation, we also denote by ∆. Proof. Let λ n , φ n be a spectral decomposition for ∆ as above. Then by the kernel theorem (1) and by Proposition 4.1 one can identify Ψ ∞ (M) with S (N × N) isomorphically. The map ∆ now becomes The ideal I sp(∆) := {g ∈ S (R)|g(λ k ) = 0 for all k} is a closed ideal. Now the map ∆ is a composition of two maps q : S (R) → S (N), the quotient map with respect to the ideal I sp(∆) , and the diagonal embedding S (N) → S (N×N). Note that the quotient S (R)/I sp(∆) ≃ S (N) since by Weyl's theorem the λ n grow moderately.
As a consequence of Lemma 2.1 we get an induced map The map ∆ * will provide us with estimates on G Ψ −∞ (M ) in terms of those in G S (R) .

Geometrical Embeddings
Let F ∈ S (R) and F ≡ 1 near the origin in an interval (−t, t). Set F ε (x) := F (εx) for 0 < ε < 1, then the F ε form a net of approximate units in S (R). We fix one such net for the current section. Let M be a closed Riemannian manifold and ∆ the associated scalar Laplace operator. Further let (λ k , φ k ) be eigenvalues and eigenvectors of ∆, where 0 ≤ λ 1 ≤ λ 2 . . . are counted with multiplicity. We would like to analyze the net F ε (∆) of smoothing operators. To begin with we set Proof. This is a direct consequence of Weyl's estimates which provide an asymptotic estimate for the spectral counting function  Hence for α ≥ m 2 we have N ε ≥ Cε −α for ε small enough.
Lemma 5.2. Let f ∈ C ∞ (M) then for any natural number N, Proof. By Weyl's estimates as before the Fourier expansion in eigenfunctions . Let α be as in Lemma 5.1. Given any integer N, we can find a k such that for all l > k Since F ε (λ n ) are all uniformly bounded, Let D := (1 + ∆)  Proof. Let k(x, y) be the smooth kernel of T . Let u(y) be a smooth function, then using Cauchy-Schwartz on the integral operator, (2) For any u ∈ D ′ (M) we shall show that lim ε→0 F ε (∆)u = u therefore the map D ′ (M) → G(M) is injective. First when u ∈ L 2 (M) then we write a Fourier expansion u = a n φ n . And for ε < t λ k one observes that u−F ε (∆)(u) ≤ n>k a n (F ε (λ n )−1)φ n , where the right hand side tends to zero in norm as k → ∞. For u in any other Sobolev space H s (M) we notice that the above argument can be applied to (1+∆) − s 2 u and that these operators commute with F ε (∆).
(3) By Gårding's inequality , hence by applying Lemma 5.2 one obtains ∀N, Repeating this process inductively yields all Sobolev estimates.
Remark 5.6. The proof of Proposition 5.5 shows that any function F ∈ S (R) with F (0) = 1 gives an embedding from D ′ (M) to G(M), but such an embedding will in general fail to preserve multiplication on smooth functions. An example is provided by the solution operator to the heat equation e −ε∆ .

A larger class of embeddings
Here we generate new embeddings from old ones. Lemma 6.1. If T ε is a special embedding and N ε a negligible net in E(Ψ −∞ (M)) then T ε + N ε is a special embedding.
Proof. The negligibility of N ε in particular implies that for all k, the Sobolev operator norm N ε k ∼ O(ε N ) where N is any integer. Thus in all Sobolev estimates one obtains no contributions from N ε . Proposition 6.2. Let T n,ε be a special embedding for all positive integers n, such that a) For any distribution u, T n,ε u → u in D ′ (M) as ε → 0. b) T n,ε is a moderate net in E(Ψ −∞ (M)) and class T n,ε converges in sharp topology in G Ψ −∞ (M ) to T = T ε .
Then any representative of T defines a special embedding.
Remark 6.3. The condition 6.2 a) above holds when T ε → Id in weak- * operator topology in every B(H k (M)).
Proof. Since T ε is a moderate net in Ψ −∞ (M), for any u ∈ H s (M) ∃N such that Let u ∈ H s (M). Due to the convergence of T n → T in sharp topology there is an n such the operator norm T ε − T n,ε H s (M )→C(M ) < Cε 2 for small enough ε. Therefore lim ε→0 (T ε − T n,ε )u = 0 in C(M).

Thus by embedding of continuous functions into
The invariance of multiplication of smooth functions follows analogously.
Now we proceed to enlarge our class of Schwartz functions. Let F n be a sequence that converges to F in S (R) fast enough that is for every semi-norm ρ there exists a constant C ρ > 0 such that, Given an integer l we set Then we define F  Proof. Since F n converge to F in S (R) it is a bounded set of Schwartz functions and hence the moderateness of F [l] is immediate.
Let N be a given integer and given α, β we observe that for all l > N + |α| + |β| one has, where C α,β are given by (4) ε → F ε in sharp topology. Remark 6.5. Given a net of equi-continuous injection maps i ε : X → X so that x → i ε (x) defines an embedding of X into its generalized space G X one could by the above recipe construct an sequence Y Now let F n → F be a sequence in S (R) such that F n ≡ 1 in an interval (−t n , t n ). We define analogous to (3) for our nets F [l] ε ; Lemma 6.6. Let the sequence of intervals t n be such that for any α > 0 there exists N α such that t n > n −α for all n > N α . Then there exist C > 0, τ > 0 with N l (ε) > Cε −τ for all l and small enough ε.
Proof. First we note that from (5) K(l, ε) > ε −l . Therefore from the hypothesis on the t n 's the corresponding intervals t K(l,ε) > ε lα for any α > 0. In particular And the result follows again just as before by Weyl's asymptotic formula for λ k 's. Definition 6.7. We say a Schwartz function F ∈ S (R) is admissible (or is almost 1 near 0) if there exists a sequence of Schwartz functions F n with the following properties: a) F n converges to F fast enough, that is for every continuous seminorm ρ on S (R) there exists a constant C ρ > 0 such that, b) There exists t n such that F n ≡ 1 on an interval (−t n , t n ) and such that for any α > 0 there exists N α such that t n > n −α for all n > N α .
It is immediate that if a Schwartz function F is admissible then F (0) = 1 and ∂ α F (0) = 0 for all α.
Before we state our main result we need the following lemma.
Lemma 6.8. Let F n → F be a sequence in S (R) with F n ≡ 1 in an interval (−t n , t n ) such that for any α > 0 there exists N α such that t n < n −α for all n > N α . Then the nets Proof. This is a consequence of Lemma 6.6. Firstly since each F ε (∆)u is a moderate net in G(X) for any distribution u.
Just like in Proposition 5.5 for any u ∈ D ′ (M) one checks that First let us assume that u be in L 2 (M) so we may write a Fourier expansion u = a n φ n . Since by Lemma 6.6 N l (ε) → ∞ for any integer k, there exist ε 0 such that ε < ε (λ n ) are uniformally bounded. For u in any other Sobolev space H s (M) we notice that the above argument can be applied to (1 + ∆) − s 2 u and that these operators commute with F [l] ε (∆). The multiplicativity on smooth functions follows by a similar argument as before. We prove below the L 2 (M) estimate and all other estimates follow from it using Gårding's inequality.
Let the Fourier expansion of a smooth function f ∈ C ∞ (M) be, f = a k φ k then (a k ) ∈ S (N). Given any integer N, we can find a k such that for all l > k n>l |a n | 2 < 1 where τ > 0 is chosen so that N l (ε) > (ε −τ ). Now choose ε 0 such that λ k < N l (ε) for all 0 < ε < ε 0 . Then for any ε < ε 0 , ε (λ n ) are all uniformly bounded, Proposition 6.9. Let F be an admissible Schwartz function. Then F ε (∆) is a special embedding.
Proof. Since F is admissible there is a sequence F n of Schwartz function approximating it in S (R) satisfying Definition 6.7. Then by Lemma 6.4 the sequence F ε defines a moderate net in G S (R) converging in sharp topology to F ε . Thus by applying ∆ * , we get that F Proof. Since φ is an isometry φ * (∆) = ∆. Therefore φ preserves the eigenspaces for all operators of the form F (∆) and hence φ * (F (∆) = F (∆). Therefore the result follows.
Of course all pseudodifferential operators which commute with the Laplace ∆ also are preserved under embeddings obtained from ∆. 7.2. Support and singular support. We recall that G ∞ (M) is a sub-algebra of G(M) so that u ∈ G ∞ (M) iff there exists an N such that for any continuous semi-norm ρ on C ∞ (M), ρ(u) ∼ O(ε N ). That is, the moderateness estimates in G ∞ (M) are independent of the seminorms. As introduced in [19] on R n the convolution embedding has the property that ι ρ (D ′ ) ∩ G ∞ = C ∞ . We shall need a generalization of this result this result to our geometrical embeddings: Proposition 7.3. Let F be a Schwartz function such that F ≡ 1 on (−t, t). Then the embedding of distributions under T ε = F ε (∆) satisfies Proof. It is clear that any smooth function embeds into G ∞ (M). We shall prove that any non-smooth distribution is not in G ∞ (M). To start with we make the following claim: Proof. Let φ k be the L 2 (M) spectral basis for Laplace with nondecreasing eigenvalues. Since (1 + ∆) Let a k = (1 + λ) − s 2 b k . By Weyl's theorem λ k ∼ Ck 2 m . Therefore with our assumptions on u and on s we have: Hence by using (6), given a constant C > 0 we get a sequence k n such that |a kn | > Ck − s m − 1 2 −δ . (This is seen by comparing the divergent series |a k | 2 k 2s m +δ with the convergent series Ck −1−δ .) Now pick ε n such that (a) if −1 − 2s m − δ > 0 then Since k n → ∞ therefore ε n → 0 and Since G( ) is in fact a sheaf of algebras (cf. [7,12]), there is a welldefined notion of support of its elements.
Lemma 7.5. The embeddings F ε (∆) preserve the support of distributions, that is supp(u) = supp(F ε (∆)u) for any distribution u with compact support.
Proof. This is a consequence of finite propagation speed for operators with kernels supported near the diagonal. Given δ > 0 we pick a cutoff function φ δ (x) = 1 |x] < δ 2 0 |x| > δ Then (1 − φ δ (x))F ε (x) is a negligible net in N S (R) and therefor the classes F ε (∆)u is the same as φ δ (∆))F ε (∆)u. Since the operators φ δ (∆))F ε (∆) are supported in a δ neighborhood of the diagonal the support of φ δ (x))F ε (∆)u is in 2δ of supp u. Since δ was arbitrary we have the desired result.
The singular support for generalized functions can be defined in terms of G ∞ (M) as This is analogous to the definition of singular support for a distribution. Writing a distribution u as we note that by Lemma 7.3 Moreover, by Lemma 7.5 if φ is supported near x then F ε (∆)(1 − φ)u is supported away from x. Therefore, Corollary 7.6. The embeddings F ε (∆) preserve the singular support of distributions, that is singsupp(u) = singsupp(F ε (∆)u) for any distribution u.
Let P be a (classical) pseudodifferential operator of order 0 with σ(P ) its principal symbol. The characteristic set of P is char(P ) = σ(P ) −1 (0) ⊆ T * M \ 0. We follow the results in [11] to define the generalized wavefront set of u ∈ G(M) to be The preservation of wavefront set under our embeddings shall be considered elsewhere.

Sections of vector bundles
We begin by reviewing the construction of generalized sections of a vector bundle ( [16,17,12]) and then construct embeddings of distributional sections into generalized sections preserving various properties of interest. Let E → M be a (complex) vector bundle over M. We fix a fiberwise inner product on E. Let Γ ∞ (E) = Γ ∞ (M, E) be the space of smooth sections of E and L 2 (E) its closure with respect to the inner product We shall denote the norm on this Hilbert space by L 2 (E) . Let D : Γ ∞ (E) → Γ ∞ (E) be an order one positive elliptic operator on E. Then the Sobolev spaces H t (E) as the completion of Γ ∞ (E) with respect to the norm: Γ ∞ (E) is a locally convex space with respect to the family of norms t . TheC-module associated with the Fréchet space Γ ∞ (E) will be denoted by G E (M) or G E when no confusion can arise. We note the following obvious properties.   Proof. Part (1) and (2) are obvious. For (3) we note that x ε ⊗ y ε → x ε ⊗y ε is the required injection, as Γ ∞ (E)⊗ C ∞ (M ) Γ ∞ (F ) ≃ Γ ∞ (E⊗F ) are isomorphic. (4) is a special case of Lemma 2.1.
Let D be a positive elliptic operator on sections of a vector bundle E such that the order of D is ≥ 1. Then Weyl's estimate holds for the spectrum of D, namely there is a spectral decomposition of L 2 (E) in terms of eigenvectors of E and the corresponding eigenvalues satisfy asymptotic estimates of rational order. This in particular implies that we can use the Schwartz functional calculus. For any F ∈ S (R) the operator F (D) is a smoothing operator on E. Such an operator has kernel in the big homomorphism bundle where Ω M is the 1-density bundle on M and E ′ is the dual bundle. We could again fix a Riemannian density for once and normalize the kernel. The following can be proven in exactly the same way as Proposition 6.9. In particular the following diagram commutes with respect to the two module actions, We can now for example define a generalized connection on a vector bundle E as aC-linear map such that ∇(s ⊗ α) = ∇(s) ⊗ α + (−1) k s ⊗d(α), s ∈ G E , α ∈ G Ω k (M ) .
Of course any usual connection on E gives rise to a generalized connections. Generalized connections are studied in [18]