Colombeau algebras without asymptotics

We present a construction of algebras of generalized functions of Colombeau-type which, instead of asymptotic estimates with respect to a regularization parameter, employs only topological estimates on certain spaces of kernels for its definition.


Introduction
Colombeau algebras, as introduced by Colombeau [1,2], today represent the most widely studied approach to embedding the space of Schwartz distributions into an algebra of generalized functions such that the product of smooth functions as well as partial derivatives of distributions are preserved. These algebras have found numerous applications in situations involving singular objects, differentiation and nonlinear operations (see, e.g., [9,12,15]).
All constructions of Colombeau algebras so far incorporate certain asymptotic estimates for the definition of the spaces of moderate and negligible functions, the quotient of which constitutes the algebra. There is a certain degree of freedom in the asymp-Dedicated to the memory of Prof. Todor Gramchev.
In this article we will present an algebra of generalized functions which instead of asymptotic estimates employs only topological estimates on certain spaces of kernels for its definition. This is a direct generalization of the usual seminorm estimates valid for distributions.
We will first develop the most general setting in the local scalar case, namely that of diffeomorphism invariant full Colombeau algebras. We will then derive a simpler variant, similar to Colombeau's elementary algebra. Finally, we give canonical mappings into the most important Colombeau algebras, which points to a certain universality of the construction offered here.

Preliminaries
N and N 0 denote the sets of positive and non-negative integers, respectively, and R + the set of nonnegative real numbers. Concerning distribution theory we use the notation and terminology of L. Schwartz [18].
Given any subsets K , L ⊆ R n (with n ∈ N) the relation K L means that K is compact and contained in the interior L • of L.
Note that · K ,m , · K ,m;L ,l and · K ,m;B are continuous seminorms on the respective spaces. We define δ ∈ C ∞ ( , E ( )) by δ(x) := δ x for x ∈ , where δ x is the delta distribution at x. D L ( ) is the space of test functions on with support in L. For two locally convex spaces E and F, L(E, F) denotes the space of linear continuous mappings from E to F, endowed with the topology of bounded convergence. By U x ( ) we denote the filter base of open neighborhoods of a point x in , and by U K ( ) the filter base of open neighborhoods of K . By csn(E) we denote the set of continuous seminorms of a locally convex space E. B r (x) := {y ∈ R n : y−x < r } is the open Euclidean ball of radius r > 0 at x ∈ R n , and for any subset K ⊆ R n we define B r (K ) := x∈K B r (x).
Our notion of smooth functions between arbitrary locally convex spaces is that of convenient calculus [11]. In particular, d k f denotes the k-th differential of a smooth mapping f .

Construction of the algebra
Throughout this section let ⊆ R n be a fixed open set. Let C be the category of locally convex spaces with smooth mappings in the sense of convenient calculus as morphisms.
Definition 1 Consider C ∞ (−, D( )) and C ∞ (−) as sheaves with values in C. We define the basic space of nonlinear generalized functions on to be the set of sheaf homomorphisms Hence, an element of B( ) is given by a family (R U ) U of mappings We will casually write R in place of R U .

Remark 2
The basic space B( ) can be identified with the set of all mappings R ∈ C ∞ (C ∞ ( , D( )), C ∞ ( )) such that for any open subset U ⊆ and ϕ, ψ ∈ C ∞ ( , D( )) the equality ϕ| U = ψ| U implies R( ϕ)| U = R( ψ)| U (cf. [10]). [18, Chap. IV, §1, Th. II, p. 105] or [20,Theorem 40.2,p. 416]). More abstractly, this can be seen using the theory of topological tensor products [16,17,20] as follows: where C ∞ (U ) ⊗D( ) denotes the completed projective tensor product of C ∞ (U ) and D( ). The assignment ϕ → u, ϕ is smooth, being linear and continuous [11, 1.3, p. 9]. Hence, we have the following embeddings of distributions and smooth functions into B( ): Note that both D X and D X satisfy the Leibniz rule. We have While D X is C ∞ ( )-linear in X , D X is only C-linear in X . We refer to [13,14] for a discussion of the role of these derivatives in differential geometry.

Definition 5
For k ∈ N 0 we set P k := R + [y 0 , . . . , y k ], More explicitly, P k is the commutative semiring of polynomials in the k + 1 commuting variables y 0 , . . . , y k with coefficients in R + . Similarly, I k is the commutative semiring in the 2(k + 1) commuting variables y 0 , . . . , y k , z 0 , . . . , z k with coefficients in R + and such that, if λ ∈ I k is given by the finite sum λ = α,β∈N k+1 0 λ αβ y α z β , then λ α0 = 0 for all α. Note that P k is a subsemiring of P k+1 and I k a subsemiring of I k+1 . Furthermore, I k is an ideal in P k if P k is considered as a subsemiring of R + [y 0 , . . . , y k , z 0 , . . . , z k ]. Given λ ∈ P k and y i ≤ y i for i = 0 . . . k we have λ(y) ≤ λ(y ). For λ, μ ∈ P k we write λ ≤ μ if λ(y) ≤ μ(y) for all y ∈ (R + ) k+1 , and similarly for λ, μ ∈ I k .
We can now formulate the following definitions of moderateness and negligibility, not involving any asymptotic estimates: The subset of all moderate elements of B( ) is denoted by M( ).

The subset of all negligible elements of B( ) is denoted by N ( ).
It is worthwile to discuss possible simplifications of these definitions, which at this stage should be considered more as a proof of concept than as the definite form they should have. First, we note that we cannot replace (∀x ∈ ) (∃U ∈ U x ( )) (∀K , L U ) by (∀K , L ). In fact, in the second case K and L can be distant from each other, while in the first case it suffices to control the situation where K and L are close to each other. However, the following result shows that we can always assume K L and that the ϕ 0 , . . . , ϕ k are given merely on an arbitrary open neighborhood of K , i.e., as elements of the direct limit C ∞ (K , D L ( )) := lim − →V ∈U K ( ) C ∞ (V, D L ( )):

Similarly, R is negligible if and only if
Proof Obviously each of these conditions is weaker than the corresponding one of Definition 6 or Definition 7.
Suppose we are given R ∈ B( ) such that the condition stated for moderateness holds. Given x ∈ there hence exists some U ∈ U x ( ). Now given arbitrary K , L U we choose a set L U such that K ∪ L L . Fixing m, k ∈ N 0 for the moderateness test, for (K , L ) we hence obtain c, l ∈ N 0 and λ ∈ P k . Now fix some ϕ 0 , . . . , ϕ k ∈ C ∞ (U, D L (U )); each of those represents an element of C ∞ (K , D L (U )), whence we have the estimate where the last equality follows because the ϕ 0 , . . . , ϕ k take values in D L (U ). This shows that R is moderate.
For the case of negligibility we proceed similarly until we obtain c, l which proves negligibility of R.
If the test of Definition 6, Definition 7 or Definition 8 holds on some U then clearly it also holds on any open subset of U . The following characterization of moderateness and negligiblity is obtained by applying polarization identities to the differentials of R: Proof We assume k ≥ 1, as for k = 0 the statements are identical. (i) "⇒": One obtains λ ∈ P k such that with λ ∈ P 1 given by λ (y 0 , y 1 ) = λ(y 0 , y 1 , . . . , y 1 ). "⇐": One obtains λ ∈ P 1 . We then use the polarization identity [19, eq. (7), p. 471] where S J := i∈J ϕ i and we have set with λ ∈ P k given by with λ ∈ I k given by "⇐": We obtain λ ∈ I 1 such that, as above, with λ ∈ I k given by Note that the polarization identities could be applied also in the formulation of Proposition 8.

Proposition 10 N ( ) ⊆ M( ).
Proof Let R ∈ N ( ) and fix x ∈ for the moderateness test. By negligibility of R there exists U ∈ U x ( ) as in Definition 7. Let K , L U and m, k ∈ N 0 be arbitrary. Then there exist c, l, λ and B such that the estimate of Definition 7 holds. We know that λ ∈ I k is given by a finite sum It suffices to show that there are λ 1 , λ 2 ∈ P 0 such that for any ϕ ∈ C ∞ (U, D L (U )) we have the estimates In fact, these inequalities imply with λ ∈ P k given by Inequality (1) is seen as follows:

Proposition 11 M( ) is a subalgebra of B( ) and N ( ) is an ideal in M( ).
Proof This is evident from the definitions.

Theorem 12
Let u ∈ D ( ) and f ∈ C ∞ ( ). Then Proof (i): Fix x for the moderateness test and let U ∈ U x ( ) be arbitrary. Fix any K , L U and m ∈ N 0 . Then there are constants C = C(L) ∈ R + and l = l(L) ∈ N 0 such that | u, ϕ | ≤ C ϕ L ,l for all ϕ ∈ D L ( ). Hence, we see that with λ(y 0 ) = Cy 0 . Moreover, we have with λ(y 0 , y 1 ) = Cy 1 . Higher differentials of ιu vanish and the moderateness test is satisfied with λ = 0 for k ≥ 2.
(ii): Fix x and let U ∈ U x ( ) be arbitrary. For any K , L U and m ∈ N 0 we have : Fix x and let U ∈ U x ( ) be arbitrary. For any K , L U and m, k ∈ N 0 we have Hence, with c = m, l = 0 and B = { f } the negligibility test is satisfied with λ(y 0 , z 0 ) = z 0 for k = 0, λ(y 0 , y 1 , z 0 , z 1 ) = z 1 for k = 1 and λ = 0 for k ≥ 2.

Because of the estimates
which may be verified by a direct calculation, we have by the choice of q, which means that (ιu)( ϕ ε )| V → 0 in C(V ) and hence also in D (V ). On the other hand, we have , as is easily verified. This completes the proof.
Proof The claims for D X are clear because for some constant C depending on X . As to D X , we have to deal with terms of the form d k+1 R( ϕ)(D SK X ϕ, ψ, . . . , ψ) and d k R( ϕ)(D SK X ψ, ψ, . . . , ψ) for which we use the estimate for some constant C depending on X .
We now come to the quotient algebra.

Proposition 15 Let R ∈ B( ) and ⊆ be open. If R is moderate then R| is moderate; if R is negligible then R| is negligible.
Proof Suppose that R ∈ M( ). Fix x ∈ , which gives U ∈ U x ( ). Set U := U ∩ ∈ U x ( ) and let K , L U and m, k ∈ N 0 be arbitrary. Then there are c, l, λ as in Definition 6. Let now ϕ 0 , . . . , ϕ k ∈ C ∞ (U , D L (U )) be given. Choose Hence, the moderateness test is satisfied for R| . Now suppose that R ∈ N ( ). For the negligibility test fix x ∈ , which gives U ∈ U x ( ). Set U := U ∩ and let K , L U and m, k ∈ N 0 be arbitrary. Then ∃c, l, B, λ as in Definition 7. Let now ϕ 0 , . . . , ϕ k ∈ C ∞ (U , D L (U )) be given. Choose ρ ∈ D(U ) such that ρ ≡ 1 on a neighborhood of K . Then ρ · ϕ i ∈ C ∞ (U, D L (U )) (i = 0 . . . k) and We first remark that if R ∈ B(X ) satisfies R| X i ∈ N (X i ) for all i then R ∈ N (X ), as is evident from the definition of negligibility.
Suppose now that we are given be a partition of unity subordinate to (X i ) i , i.e., a family of mappings χ i ∈ C ∞ (X ) such that 0 ≤ χ i ≤ 1, (supp χ i ) i is locally finite, i χ i (x) = 1 for all x ∈ X and supp χ i ⊆ X i . Choose functions ρ i ∈ C ∞ (X i , D(X i )) which are equal to 1 on an open neighborhood of the diagonal in X i × X i for each i.
For showing smoothness of R V consider a curve c ∈ C ∞ (R, C ∞ (V, D(X ))). We have to show that t → R V (c(t)) is an element of C ∞ (R, C ∞ (V )). By [11, 3.8, p. 28] it suffices to show that for each open subset W ⊆ V which is relatively compact in V the curve t → R V (c(t))| W = R W (c(t)| W ) is smooth, but this holds because the sum in (3) then is finite. Hence, Fix x ∈ X for the moderateness test. There is a finite index set F and an open neighborhood U as well as m, k ∈ N 0 be arbitrary. For each i ∈ F there are c i , l i , λ i such that for any and hence, for ϕ 0 , . . . , ϕ k ∈ C ∞ (U, D L (U )), We see that with c = max j∈F c j , l = max j∈F l j , some constant C(m) coming from the Leibniz rule, and λ ∈ P k given by This shows that R is moderate. Finally, we claim that R| X j − R j ∈ N (X j ) for all j. For this we first note that for ϕ ∈ C ∞ (X j , D(X j )). Again, for x ∈ X j there is a finite index set F and an open neighborhood W ∈ U x (X ) such that W ∩ supp χ i = ∅ implies i ∈ F, and we can assume that x ∈ i∈F X i . Let Y be a neighborhood of x such that ρ i ≡ 1 on Y × Y for all i ∈ F and let U i ∈ U x (X i ∩ X j ) be given by the negligibility test of As above, we then have with c = max i∈F c i , l = max i∈F l i , B = i∈F B i , and λ ∈ I k given by This completes the proof.

An elementary version
We will now give a variant of the construction of Sect. 3 similar in spirit to Colombeau's elementary algebra [2]: if we only consider derivatives along the coordinate lines of R n we can replace the smoothing kernels ϕ ∈ C ∞ (U, D L ( )) by convolutions. This way, one can use a simpler basic space which does not involve calculus on infinite dimensional locally convex spaces anymore: and define B c ( ) to be the set of all mappings R : U ( ) → C such that R(ϕ, ·) is smooth for fixed ϕ.
Note that this is almost the basic space used originally by Colombeau (see [2, 1.2.1, p. 18] or [9, Definition 1.4.3, p. 59]) but with D(R n ) in place of the space of test functions whose integral equals one. We now introduce a notation for the convolution kernel determined by a test function.
In fact, with this definition we have u, ϕ = u * φ, where as usually we setφ(y) := ϕ(−y). Furthermore, for c ∈ N 0 we write The direct adaptation of Definition 6,7 then looks as follows:

The subset of all moderate elements of B c ( ) is denoted by M c ( ).
Similarly, R is called negligible if The subset of all negligible elements of B c ( ) is denoted by N c ( ).
It is convenient to work with the following simplification of these definitions.

Similarly, R ∈ B c ( ) is negligible if and only if
Proof Suppose R is moderate and fix K . We can cover K by finitely many open sets U i obtained from Definition 19 and write K = i K i with K i U i . Choose r > 0 such that L i := B r (K i ) U i for all i. Fixing m, by moderateness there exist c i and λ i for each i. Set c = max i c i and choose λ with λ ≥ λ i for all i. Now given ϕ ∈ D(R n ) with supp ϕ ⊆ B r (0) we also have K i + supp ϕ ⊆ L i and we can estimate Conversely, suppose the condition holds and fix x ∈ for the moderateness test. Choose a > 0 such that B a (x) . By assumption there is r > 0 with B r +a (x) . Set U := B r/2 (x). Then, fix K L U and m for the moderateness test. There are c and λ by assumption. Now given ϕ with K + supp ϕ ⊆ L, we see that for y ∈ supp ϕ and an arbitrary point z ∈ K we have |y| ≤ |y + z − x| + |z − x| < r , which means that supp ϕ ⊆ B r (0). But then R(ϕ, .) K ,m ≤ λ( ϕ c ) as desired.
If R is negligible we proceed similarly until the choice of K i L i U i and m gives c i , λ i and B i . Choose χ i ∈ D(U i ) with χ i ≡ 1 on a neighborhood of L i , and define B : The converse is seen as for moderateness by restricting the elements of B ⊆ C ∞ ( ) to U .
The embeddings now take the following form.

Definition 21
We define ι c : D ( ) → B c ( ) and σ c : Partial derivatives on B c ( ) then can be defined via differentiation in the second variable: R(ϕ, x)).
Proof This is evident from the definitions.

Proposition 24
We have x). The second claim is clear.
Proof The result follows from for suitable λ 1 and c 1 , which is seen as in the proof of Proposition 10.
Similarly to Proposition 11 we have: Theorem 27 Let u ∈ D ( ) and f ∈ C ∞ ( ). Then The proof is almost identical to that of Theorem 12 and hence omitted. As before, one may show that G c is a sheaf.

Canonical mappings
In this section we show that the algebra G constructed above is near to being universal in the sense that there exist canonical mappings from it into most of the classical Colombeau algebras which are compatible with the embeddings.
We begin by constructing a mapping G( ) → G c ( ).

Definition 29 Given
where ϕ ∈ C ∞ ( , D( )) is chosen such that ϕ = ϕ in a neighborhood of x.
This definition is meaningful: given (ϕ, x) in U ( ) we have supp ϕ(. − x ) ⊆ for x in a neighborhood V of x. Choosing ρ ∈ D( ) with supp ρ ⊆ V and ρ ≡ 1 in a neighborhood of x, we can take ϕ(x) := ρ ϕ. Obviously, R(ϕ, x) does not depend on the choice of ϕ(x) and R(ϕ, .) is smooth, so indeed we have R ∈ B c ( ).
Proposition 30 Let R ∈ B( ). Then the following holds: (ii) is clear. (iii): Suppose that R ∈ M( ). Fixing x ∈ , we obtain U as in Proposition 8. Let K L U and m be given, set k = 0, and choose L such that L L U . Then Proposition 8 gives c, l, λ such that for ϕ ∈ C ∞ (K , D L (U )), which proves that R ∈ M c ( ). (iv): Similarly, if R ∈ N ( ) then for x ∈ we have U as in Proposition 8. For K L U , m given, k = 0, and L such that L L U , we obtain c, l, λ, B as in Proposition 8 such that and hence which gives negligibility of R.

Moreover, with
we choose functions κ ε ∈ D( ) such that 0 ≤ κ ε ≤ 1 and κ ε ≡ 1 on K ε . Then the special algebra G s ( ) is given by Proof (i) and (ii) are clear.
For (iii) it suffices to show the needed estimate locally. Fix x ∈ , which gives U ∈ U x ( ) as in Proposition 8. Choose any K , L such that x ∈ K L U , fix m, and set k = 0. Then there are c, l, λ as in Proposition 8. Because supp ψ ε (x) ⊆ B 2|ln ε| −1 (x) we have ψ ε ∈ C ∞ (K , D L (U )) for ε small enough, which gives R s ε K ,m ≤ λ( ψ ε K ,c;L ,l ).
For negligibility we proceed similarly; the claim then follows by using that for a bounded subset B ⊆ C ∞ (U ) we have ψ ε − δ K ,c;B = O(ε N ) for all N ∈ N, which is seen as in [4,Prop. 12,p. 38] and actually merely a restatement of the fact that ι s f − σ s f = O(ε N ) for all N uniformly for f ∈ B.

Definition 34 For
Proof which hold by definition of the spaces S( ) and S 0 ( ).

The elementary algebra
For Colombeau's elementary algebra we employ the formulation of [9, Section 1.4], Sect. 1.4. For k ∈ N 0 we let A k (R n ) be the set of all ϕ ∈ D(R n ) with integral one such that, if k ≥ 1, all moments of ϕ order up to k vanish.