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Regularity, Local and Microlocal Analysis in Theories of Generalized Functions

We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a"frequential microlocal analysis"and into a"microlocal asymptotic analysis". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations.


Introduction
The notion of regularity in algebras or spaces of generalized functions can be formulated in a general way with the help of sheaf theory. In section 2, when A is a presheaf of algebras or vector spaces on a topological space X, and B a subpresheaf of A, for each open set Ω in X, we consider B (Ω) as the space or algebra of some regular elements of A (Ω). This leads to the notion of B-singular support which refines the notion of support of a section u ∈ A (Ω) provided the localization principle (F 1 ) holds: if u and v are global sections of A which agree on each open set of a family (Ω i ) i∈I of open set in X, they agree on the union ∪ i∈I Ω i .
We can give many examples of this situation in the framework of theories of generalized functions: distributions [38] or Colombeau-type algebras [1,7,8,9,10]. To illustrate this, let us consider the following sequence of sheaf embeddings, defined for each Ω open set in R n by C ∞ (Ω) → L (C ∞ c (Ω), C) = D ′ (Ω) → G (Ω) → L G c (Ω), C G (Ω) being the Colombeau algebra, C the ring of Colombeau's generalized numbers, G c (Ω) the set of elements in G (Ω) with compact support and L G c (Ω), C defined in [17] the space of all continuous and C linear functionals on G c (Ω). Each term of the sequence can be considered as the B (Ω) regular space or algebra of the following algebra or space A (Ω). It is the basis for a local analysis of the elements in A (Ω). Some results on propagation of singularities under Bcompatible operators permit to explain and summarize the classical results involving differential or pseudo-differential ones. But if we want to define a more precise "microlocal" analysis which gives some informations not only on the locus, but on the causis of the singularities described as a fibered space above that locus, we have first to give a precise local characterization of the singularities under consideration. A review on the ideas, technics and results on microlocalization is given in section 3. The first step was to follow the Hörmander ideas about the wave front set W F (u) of a distribution u, whose construction is deduced from the classical Fourier characterization of smoothness of distributions with compact support. For a general v ∈ E ′ (R n ) Hörmander introduces the cone Σ (v) of all η ∈ R n \0 having no conic neighbourhood V such that the Fourier transform v is rapidly decreasing in V . Lemma 8.1.1. in [22] proves that if Φ ∈ D (R n ) and v ∈ E ′ (R n ) then Σ (Φv) ⊂ Σ (v). It follows that if Ω is an open set in R n and u ∈ D ′ (Ω), setting: Σ x (u) = ∩ Φ Σ (Φu) ; Φ ∈ D (Ω) , Φ (x) = 0, one can define the wave front set of u as W F (u) = {(x, ξ) ∈ Ω × (R n \0) ; ξ ∈ Σ x (u)}.
This way was led in the sheaf A = G of Colombeau simplified algebras by Nedeljkov, Pilipovic and Scarpalezos [31] by taking B = G ∞ as regular subsheaf of G. This subsheaf, introduced by Oberguggenberger [32], generalizes in a natural way in G the regular properties of C ∞ in D ′ . We can find in the literature a description of the main properties of Fourier transform of compacted supported elements in G ∞ (Ω), which leads to a frequential microlocal analysis similar to the Hörmander's one (see [37] for instance). The crucial point was the conservation of the power of the lemma 8.1.1. leading to the definition of the generalized wave front set of u ∈ G(Ω) denoted W F g (u).
Recently, A. Delcroix has extended in [12] the G ∞ regularity to a so called G R regularity, which still preserves the statements of the above quoted lemma, and gives a G R frequential microanalysis. We can chose R such that G R countains an embedding of D ′ into G, which is not the case for G ∞ (G ∞ ∩ D ′ =C ∞ is a result of [32]). Then it becomes possible to investigate the frequential D ′ -singularities of u ∈ G(Ω).
Inspired by the classical theories, many results on propagation of singularities and pseudodifferential techniques have been obtained during the last years by De Hoop, Garetto, Hörmann, Gramchev, Grosser, Kunzinger, Steinbauer and others (see [15,18,19,25,26,27]). For example, when u ∈ G(Ω), Hörmann and Garetto [18] obtain characterisations of W F g (u) in terms of intersections of some domains corresponding to pseudodifferential operators similarly to Hörmander's characterizations of W F (u) for u ∈ D ′ (Ω) [23]. Following the ideas and technics of [18] and making use of the theory of pseudodifferential operators with generalized symbols ( [15,18]), Garetto [17] has recently extended the definition of W F g (u) when u ∈ G(Ω) to the definitions of W F G (T ) and W F G ∞ (T ) when T ∈ L G c (Ω), C . She can also give a Fourier-transform characterization of these wave front sets when T is a basic functional. Nevertheless, these very interesting and deep results are still mainly limited to linear cases, at least in the framework developed above. More precisely, even when A is a sheaf of factor algebras, we don't know any study on the microlocal behaviour of singularities under nonlinear operations by means of frequential methods based on the Fourier transform.
However, such studies exist in a classical framework involving some spaces of Sobolev type. In section 8 of [24], Hörmander uses the results on microlocal H loc (s) -regularity of nonlinear operations for tempered distributions in S ′ (R n ) to discuss semi-linear equations, following Rauch [35]. By means of paradifferential techniques some general results for quasilinear equations are given in Bony [4]. Then, from a general result on propagation of singularities for pseudo-differential operators and a Bony's linearization theorem, Hörmander ([24], section 11) can discuss fully nonlinear equations and obtain precise propagation results for hyperbolic second order semilinear equations. Extensions of the previous results can be found in works of Beals [2,3] and Bony [5].
The Fourier transform is still the main tool involved in other generalized cases, where the G ∞ -regularity is subordinated to an additional condition (such as an estimate on the growth of derivatives) characterizing a special property such as to belong to an analytic, Gevrey or C L class in Hörmander sense ( [22], section 8.4). It is the case of "analytic" algebra: G A studied by Pilipovic, Scarpalezos and Valmorin [34], of "C L class"algebra: G L introduced by Marti [30], which are subalgebras of G, of "regular Gevrey ultradistributions" algebra: G σ,∞ of Bouzar and Benmeriem [6] which is a subalgebra of G σ , the "generalized Gevrey ultradistributions". In these examples, the aim is always to perform the B-frequential microlocalization of generalized functions from the starting algebra A (Ω), when B is G A , G L or G s,∞ and A is G or G σ . All these cases are special cases of the more general one obtained when taking A = G r which extends the Colombeau algebra G and B = G r,R,L which generalizes all the previous regularity cases. The Fourier transform is still used to characterize the B-regularity with the corresponding constraints. But this is not so easy or natural. For instance, in [30] one starts by giving a characterization of local G L -regularity by means of some sequence u k of generalized functions with compact support whose the Fourier transform u k verifies an estimate involving a special sequence (L k ) k∈N . Indeed u k is constructed as product of u ∈ G(Ω) and a suitable cutoff sequence X k whose derivatives are controled up to the order k. This leads to define the G L -wave front set of a generalized function: ⊂ Ω × (R n \ 0) and prove, by refining the cutoff sequence X k , that its projection on Ω is the G L -singular support of u. Then, W F L g (u) gives a spectral decomposition of sing supp L u. A generalization of these results to the case of local G r,R,L -regularity of elements in G r (Ω) is given in [13]. They lead to define the G r,R,L -wave front set of G r -generalized functions.
These sophisticated constructions give a synthetic description of the frequential microanalysis but the proofs seem to tell that the Fourier transform is not really the good tool to perform this description in the above cases. Perhaps the Fourier-Bros-Iagolnitzer transform would permit to give a better approach of the problem in the future. For the analytic generalized wave front set, we also can think of refering to boundary values techniques which embed distributions into hyperfunctions. But we don't expect results about nonlinear cases in these ways.
We recall here that generalized functions in the initial definition (sections of the sheaf G) are classes of families (u ε ) ε of classical functions. But in the definitions of generalized (frequential) wave front set considered above (when B is G ∞ , G R , G A or G L ), the parameter ε does not play a specific role. It has only to ensure the correct use of some notions as regularity, rapid decrease, analyticity, and so on in the definition of the different algebras under consideration. For example the generalized wave front set of any Dirac-delta function ∆ in the generalized framework is exactly the same as the classical wave front set of the distribution δ. It is still the same as the generalized wave front set of any power ∆ m of ∆ without possibility to compare them. The main reason lies in the structure of Fourier transform. A paradigmatic alternative can be found in the concept of asymptotic analysis.
The idea of an "asymptotic" analysis [14,28,29] of u = [u ε ] ∈ A (Ω) = G(Ω) is the following. Let F be a subsheaf of vector spaces (or algebras) of G. One defines first the sheaf B such that, for any open set V in R n , B(V ) is the space of elements u = [u ε ] ∈ A(V ) such that u ε has a limit in F(V ) when ε tends to 0. Then O F G (u) is the set of all x ∈ Ω such that u agrees with a section of B above some neighbourhood of x. The F-singular (or B-singular) support of u is Ω\O F G (u). For fixed x and u, N x (u) is the set of all r ∈ R + such that ε r u ε tends to a section of F above some neighbourhood of x. The F-singular spectrum of u is the set of all (x, r) ∈ Ω × R + such that r ∈ R + \N x (u). It gives a spectral decomposition of the F-singular support of u. As example, take δ ε (x) = 1 ε ϕ x ε where ϕ ∈ D(R), ϕ ≥ 0 and ϕ (x) dx = 1. Then, for m ≥ 1, is a generalized function in G(R). Except for m = 1, for which ∆ is associated with δ ∈ D ′ (R), ∆ m is not locally associated with an element of D ′ (V ) in any neighbourhood V of 0.
But, for r ≥ m − 1, [ε r ] ∆ m is locally associated with such an element. It follows that for all m, the D ′ -singular support of ∆ m is {0} but its D ′ -singular spectrum is the set {(0, ∅)} if m = 1 or {(0, [0, m − 1))} if m > 1. It gives a more precise description of the singularities of ∆ m that its D ′ -singular support and even that its frequential generalized wave front set {(0, R\0)} which doesn't depend upon m.
This asymptotic analysis is extended to (C, E, P) algebras. This gives the general asymptotic framework, in which the net (ε r ) ε is replaced by any net a satisfying some technical conditions, leading to the concept of the (a, F)-singular parametric spectrum. The main advantage is that this asymptotic analysis is compatible with the algebraic structure of the presheaf F asymptotically associated to (C, E, P) algebras. Thus the (a, F)-singular asymptotic spectrum inherits good properties with respect to nonlinear operations when F is a presheaf of topological algebras. Moreover, even when F is a presheaf (or sheaf) of vector spaces (for instance F = D ′ ), some results on microlocal analysis are still obtained for nonlinear operations (see paragraph 4.3.1) on (a, D ′ )-singular asymptotic spectrum of powers of δ functions. In [14], various examples of propagation of singularities through nonlinear differential operators are given, connected to some results of Oberguggenberger, Rauch, Reeds and Travers ([33, 36, 39]).
The paper is organized as follows. In section 2 we introduce the local analysis of generalized functions. Subsections of 2 give the basic ingredients, some examples in algebraic or duality theories, and define G r,R,L -local analysis. G-local analysis of functional sections of L G c , C and F-local analysis for sections of some (C, E, P) algebra are recalled. Section 3 is devoted to the frequential microlocal analysis, with characterization of G r,R and G r,R,L -local regularities and corresponding wave front sets. We also give the result proved in [17] on the Fourier transform characterisation of W F G (T ) when T is a basic functional. The asymptotic microlocal analysis studied in [14] is detailed in section 4, with examples and applications to nonlinear partial differential equations.

The local analysis of generalized functions
The purpose of this section is to localize the singularities of some generalized functions. We refer the reader to [20] for more details on the sheaf theory involved in the sequel.

The basis ingredients
The basis ingredients of such an analysis are very simple and general ; even in this subsection no algebraic condition is required.
• A is a given sheaf of sets (or presheaf with localization principle (F 1 ) in addition) over a topological space X.
• B is a given subsheaf (or subpresheaf) of A.

Definition 1 : B-global regularity
For any open set Ω in X, the elements in B (Ω) are considered as regular, and called B-regular elements of A (Ω).

Definition 3 : B-regular open set
We denote by O B A (u) the set of all x ∈ Ω such that u is B-regular at x. We also can write This very simple framework suffices to state the following For any section u ∈ A(Ω) , Ω any open set in X, the B-singular support of u is Remark 1 (i) The gluing principle (F 2 ) is not needed to get the notion of B-singular support of a section u ∈ A(Ω). More precisely, when {b} is the constant presheaf defined by a gobal section of B, the localization principle (F 1 ) is sufficient to prove the following: the set We make this precise in the following The family (f i ) i∈I is coherent by assumption: and Ω B A (u) is clearly the largest open subset of Ω having this property.
A (u) and the assertion (ii) holds.

Remark 2 When
A is a sheaf and B a subpresheaf of A, we can associate to B a subsheaf of A as follows. When Ω is an open set of X, we note u ∈ B (x) if u ∈ A (Ω) is B-regular at x according to Definition 2. Set Let B * be the functor Ω → B * (Ω). We intend to prove that B * is a subsheaf of A. The presheaf structure of A induces immediately the same one for B * and principle (F 1 ) is also fullfiled. To prove that the gluing principle (F 2 ) holds, we consider a collection (Ω i ) i∈I of open subset Ω i of Ω such that Ω = ∪ i∈I Ω i and a coherent family (u i ) i∈I of elements u i ∈ B * (Ω i ). First, we can glue the u i into u ∈ A (Ω). Now, we have to prove that u ∈ B * (Ω). For any x ∈ Ω choose i such that x ∈ Ω i . Then, we have which proves that u ∈ B (x) for each x ∈ Ω. Then B * is a sheaf. In fact it is the sheaf associated to B in the sense of [20]. Its construction is simplified by using the sheaf structure of A. Roughly speaking, B * is constructed thanks to a local procedure which adds many sections to the B ones. Then, one wishes to compare the corresponding singular supports of the same u ∈ A (Ω). The answer is given by the following proposition which shows that the singularities of sections of A don't decrease when replacing B by B * .
Proposition 2 Suppose that A is a sheaf and B a subpresheaf of A. Let B * the subsheaf of A associated to B. Then, for any section u of A over the open set Ω of X we have Proof. For u ∈ A (Ω), the presheaf mapping: B → B * leads immediately to the set inclusion: .
which gives the converse one for the respective singular supports and leads to the required equality.

Elementary algebraic properties
Proposition 3 We suppose that B and A are presheaves of K-vector spaces, (resp. algebras). Let (u j ) 1≤j≤p be any finite family of elements in A(Ω) and (λ j ) 1≤j≤p any finite family of elements in K. We have: In the resp. case, we have in addition: In particular, if The result follows by taking the complementary sets in Ω.

B-compatible operators and propagation of singularities
. Thus we can simplify the notations and write A instead A V when acting on sections over V .  The following weakened form of locally B-compatibility may be more practical for applications

Definition 5 Let A be a presheaf operator in A (Ω). We say that
It is easy to see that a B-compatible operator is locally B-compatible but with the above definition we can get some useful results. The simplest one concerns the composition product, with an obvious proof.

Proposition 5 If a presheaf operator A in
Adding some algebraic hypothesis leads to the following definitions and results as corollaries of propositions 3 and 4. When B is a presheaf of algebras and A a presheaf of vector spaces and a B-module we recall that the external sheaf product B × A → A extending the usual algebra product B × B → B is defined for any Ω ∈ O X and (b,

Definition 7
We suppose that B is a presheaf of algebras and A a presheaf of vector space and a B-module. Let b be a given element in B (Ω). We define the B-operator of multiplication in A (Ω) by the map A (Ω) → A (Ω) such that B (u) = bu.
When v is in B (V ), the external product b | V v agrees with the standard product in the algebra B (V ) and lies in it. Then, for each pair

Corollary 7
We keep the same assumption as above and consider a family (b α ) α∈A of elements in B (Ω) and another family of When v belongs to B (V ), A α (v) belongs to B (V ) from the hypothesis on A α . Then, the external product b α | V A α (v) agrees with the standard product in the algebra B (V ) and lies in it. Then, for each pair . We suppose that B and A are presheaves of algebras. Then P is B-compatible.
Proof. It suffices to remark that for each j ∈ N the map A j : u → u j is a B-compatible presheaf operator of A (Ω). Putting α = j in the above corollary gives the result.
Collecting all these informations we can summarize the previous results in the following: We suppose that B is a presheaf of algebras and A a presheaf of vector spaces and a B-module. A being a set of indices, let (b α ) α∈A be a family of elements in B (Ω), (A α ) α∈A a family of B-compatible operators in A (Ω) and (p α ) α∈A a family of positive integers. Then,

Examples
where sing supp u is, in the Hörmander sense, the closet subset aff all x ∈ Ω having no neighbourhood in which the distribution u is smooth.
Let A = G and B = G ∞ the "regular" subsheaf of G, the sheaf of Colombeau's generalized functions. G ∞ (Ω) is defined as the sections of G (Ω) having a representative verifying In [12] the G ∞ -regularity is extended into a G R one. Starting from a set R of sequences of positive numbers, G R is defined by the sections u ∈ G (Ω) having a representative verifying Under certain stability conditions on the set R exposed in [12], G R is a subsheaf of differential algebras of G and when R consists of the set of all bounded sequences, then

Example 4 : Local analysis of G L -type
In [30] one constructs B = G L as a special regular sub(pre)sheaf of A = G, extending in a generalized sense the C L classes of Hörmander [22] containing analytic and Gevrey classes and constructed from an increasing sequence L k of positive numbers such that L 0 =1 and k ≤ L k , L k+1 ≤ CL k for some constant C. When taking L k = k + 1, we obtain the analytic case G A studied in [34] involving special properties of holomorphic generalized functions which give to G A a sheaf property.
In [6] Bouzar and Benmeriem introduce a sheaf of algebra G σ = G of Gevrey ultradistributions with another asymptotic scale than the Colombeau one by replacing the estimate elements When taking L k = (k + 1) σ they can construct a subpresheaf G σ,∞ of G σ and give a study of G s,∞ -singularity.
By choosing R as a regular subset of R N + , Delcroix has extended the G ∞ -regularity into the G R one, and in a work in progress Bouzar replaces the classical regularity by the R-regularity to extend the regular generalized Gevrey ultradistributions. We follow this way in view of constructing a general model containing all the previous examples but we have to add two other parameters: an asymptotic scale r = (r λ ) λ ∈ R * + Λ and a sequence L k of positive numbers such that L 0 =1 and k ≤ L k , L k+1 ≤ CL k for some constant C.

2.4
G r,R and G r,R,L -local analysis in G r (Ω) 2.4.1 The G r sheaf of algebras

Let us consider
• E = C ∞ as starting sheaf of algebras, for each open set Ω in R n , C ∞ (Ω) is endowed by the usual family p K,α of seminorms • Λ a set of indices left-filtering for the given (partial) order relation ≺.
• an asymptotic scale r = (r λ ) λ ∈ R * + Λ such that lim Λ r λ = 0, (or r λ → 0), that is to say: Define the functors X r (resp. N r ): Ω → X r (Ω) (resp. N r (Ω)) by it is not difficult to prove with the same techniques as Colombeau ones [7] that X r and N r are respectively a sheaf of differential algebras and a subsheaf of ideals of X r over the ring Then X r /N r = G r is a priori a factor presheaf of Colombeau type. It is well known that G is a sheaf and even a fine sheaf. The first assumption (a result from Aragona and Biagioni [1]) is based on the existence of a C ∞ partition of unity associated to any open covering of Ω (due to the fact that R d is a locally compact Hausdorff space). On the other hand, we can notice that C ∞ is a fine sheaf because multiplication by a smooth function defines a sheaf homomorphism in a natural way. Hence the usual topology and C ∞ partition of unity define the required sheaf partition of unity according to the definition in sheaf theory. This leads very easily to the second assumption, from the well known result that any sheaf of modules on a fine sheaf is itself a fine sheaf: it is precisely the case of G which is a sheaf of C ∞ modules. And it is the same for G r which is a fine sheaf of C ∞ modules and also a sheaf of differential algebras over the ring The G r,R regularity of G r generalizes the G ∞ regularity of G. We begin by defining a regular subspace R of R n + in the Delcroix sense [12]: (ii) For all N 1 and N 2 in R, there exists N ∈ R such that Then, for any regular subset R of R N + we can set

Proposition 10
(i) For any regular subspace R of R N + , the functor Ω → X r,R (Ω) defines a sheaf of differential algebras over the ring X r (C).
(ii) The set N r,R (Ω) is equal to N r (Ω). Thus, the functor N R : Ω → N R (Ω) defines a sheaf of ideals of the sheaf X R (·).
(iii) For any regular subspaces R 1 and Proof. The proof follows the same lines as in the case of G R algebras (see [12], Proposition 1.). We have to verify that our asymptotic scale (r λ ) λ involving a more general parametrization doesn't modify the results. We deduce assertion (i) from the assertion (i) in the definition of R.
since this estimate holds for all m ∈ N. For the sheaf properties we have to replace Colombeau's estimates by X r,R estimates and consider only a finite number of terms by compactness. Thus, from (ii) in definition 8 we have the results.
According to same arguments as those used for G r the presheaf G r,R = X r,R /N r,R = X r,R /N r turns to be a sheaf of differentiable algebras on the ring X r, Moreover, from (ii) in the above proposition, G r,R is a subsheaf of G r .

Definition 9
For any regular subset R of R N + , the sheaf of algebras (subsheaf of G r ) is called the sheaf of (r, R)-regular algebras of (nonlinear) generalized functions.
2σ−1 and R = R N + , we obtain the sheaf of so called G sgeneralized functions in [6].

2.4.3
The G r,R,L subpresheaf of G r and (r,R,L)-analysis Let L k be an increasing sequence of positive numbers such that L 0 =1 and k ≤ L k , L k+1 ≤ CL k for some constant C. According to Hörmander definition given in subsection 8.4 of [22], we shall denote by C L the sheaf of K-algebras on R n (K = R or C) such that, for any open set Ω ⊂ R n When L k = k + 1, C L is the sheaf A of analytical functions. If L k = (k + 1) a , a > 1, C L is the sheaf G a of the Gevrey class of order a.
It is possible to enlarge the above definition into a generalized one involving three parameters r, R, L corresponding to a choice of some asymptotic scale r = (r λ ) λ ∈ R * + Λ , a regular subset Definition 10 Let us define the functors X r,R,L (resp. N r,R,L ): Proof. For each Ω, X r,R,L (Ω) is a subalgebra of X r,R (Ω), and the restriction and localization processes are obvious. Let us try to glue together the bits, giving some family ( From the assumption there exits some λ 0 ∈ Λ such that λ 0 ≺ λ 0,i and N ∈ R such that N ≥ N i for 1 ≤ i ≤ p. Set c = max 1≤i≤L c i . Then for each α ∈ N n and λ ≺ λ 0 we have Thus U belongs to X L (Ω). It is easy to see that for each Ω, N L * (Ω) is an ideal of X L (Ω), and the same proof as above leads to the sheaf structure of N L * . Then we can define a new factor presheaf of C L -type algebras which is a subpresheaf of G r,R and G r , according to the definition of the ideal N r,R,L (Ω) = X r,R,L (Ω) ∩ N r (Ω).

Definition 11
For any asymptotic scale r = (r λ ) λ ∈ R * + Λ , any regular subset R of R N + , and any sequence L = (L k ) k , the presheaf of algebras (subpresheaf of G r ) G r,R,L = X r,R,L /N r,R,L is called the sheaf of (r, R, L)-regular algebras of (nonlinear) generalized functions.

Remark 3
We proved that G r,R,L is a presheaf ; the localization principle (F 1 ) is not difficult to prove. However, we lack some suitable partition of unity or other argument which preserve the X r,R,L estimates and permit to glue together the bits and get (F 1 ). But instead of introducing the sheaf associated to G r,R,L , one can keep its presheaf structure in the following localization processes.
Taking A = G r and B = G r,R,L , the G r,R,L -singularities of u ∈ G r (Ω) are localized in where sing supp . Let H r,R,L (Ω) be the set of all u ∈ G r (Ω) which are locally G r,R,L at x ∈ Ω. It is not difficult to prove that the sheaf associated to G r,R,L is the functor Ω → H r,R,L (Ω). It is a subsheaf of algebras of G r . But in the general case, we cannot prove that u ∈ H r,R,L (Ω) has a global representative in X r,R,L (Ω). However this is fulfilled when taking r ε = ε, R = Bo, L k = k + 1, corresponding to the analytic case studied in [34] involving special properties of holomorphic generalized functions. And then H r,R,L = G A , the subsheaf of generalized analytic functions of the sheaf G. But when A = G and B = G A , the G A -singularities of u ∈ G(Ω) are always localized in where sing supp A u is the set of all x ∈ Ω having no neighbourhood V such that u | V ∈ G A (V ).
Here, the sheaf structure of G A provides the following precision: from lemma 2 involving the (F 2 ) principe, we can prove that Ω\sing supp A u is also the largest open set O such that u ∈ G A (O).

Canonical embeddings
Lemma 12 We have the following commutative diagram in which the arrows are canonical embeddings Proof. The canonical (pre)sheaf embedding of C L into G r,R,L (resp. of C ∞ into G r,R ) is defined for each open set Ω ⊂ R n in by the canonical map C L (Ω) → G r,R,L (Ω) (resp. C ∞ → G r,R ) : u → [u λ ] , with u λ = u for λ ∈ Λ which is an injective homomorphism of algebras, [u λ ] being the class of u ∈ C L (Ω) (resp. C ∞ (Ω)) in the factor algebra G r,R,L (Ω) (resp.G r,R (Ω)). In order to construct the sheaf embedding : G r,R,L → G r,R , we recall that X r,R,L (Ω) ⊂ X r,R (Ω) ; N r,R,L (Ω) = X r,R,L (Ω) ∩ N r (Ω) is a necessary and sufficient condition to embed G r,R,L (Ω) into G r,R (Ω), from which we deduce the required sheaf embedding.
Remark 5 When λ = ε and r ε = ε, we can suppress the symbol r in the previous formulation. When taking R = Bo, the symbol R becomes ∞. For example, when we do that simultaneously, we have G r = G ; G r,R = G ∞ ; G r,R,L = G L .
2.5 G-local analysis in L G c (Ω), C

Duality in the Colombeau context
Starting from the usual family of semi norms (p i ) i∈I defining the topology of C ∞ (Ω) by the so-called sharp topology of G(Ω) [31]is defined by the family of ultra-pseudo-seminorms The valuation on the ring C of generalized numbers given for each r leads to the ultra-pseudo-norm on C : |r| e = e −v(r) . It is proved in [16] that a C-linear map T : G c (Ω) → C is continuous for the above topologies if and only if there exists a finite subset I 0 ⊂ I and a constant C > 0 such that, for all u ∈ G c (Ω) In [17], the topological dual L G c (Ω), C is endowed with the topology of uniform convergence on bounded subsets which is defined by the ultra-pseudo-seminorms with B varying in the family of all bounded subsets of G c (Ω), i.e., for each i ∈ I, sup u∈B P i (u) < ∞.

Localization of G-singularities
The sheaf embedding G → L G c , C is defined, for each open set Ω of R n , by the continuous (as recalled in [17] where K is an arbitrary compact set containing suppv in its interior. From Definition 2.9 in [17], the G-singular support of T ∈ L G c (Ω), C denoted by (singsupp G (T )) is the complement of the set of all points x ∈ Ω such that the restriction of T to some neighborhood V of x belongs to G (V ). Then we still have with our standard notations (A = L G c , C , 2.6 F -local analysis in (C, E, P)-algebras A (Ω) The algebraic structure of a (C, E, P) algebra We summarize the construction of the so-called (C, E, P) algebras [14,29] which generalize many cases met in the literature. K is the real or complex field and Λ a set of indices. C is the factor ring A/I where I is an ideal of A, a given subring of K Λ . (E, P) is a sheaf of topological Kalgebras on a topological space X. A presheaf of (C, E, P) algebras on X is a presheaf A = H/J of factor algebras where J is an ideal of H, a subsheaf of E Λ . The sections of H (resp. J ) of X have to verify some estimates given by means of P and A (resp. I).
The above construction needs some technical conditions given in [14] on the structure of C and we suppose that for any open set Ω in X, the algebra E(Ω) is endowed with a family P(Ω) = (p i ) i∈I(Ω) of semi-norms. Then, we set The factor H (A,E,P) /J (I A ,E,P) is a presheaf verifying the localization principle (F 1 ) but generally not the gluing principle (F 2 ). The element in A(Ω) defined by (u λ ) λ∈Λ ∈ H (A,E,P) (Ω) is denoted by [u λ ]. For u ∈ A (Ω), the notation (u λ ) λ∈Λ ∈ u means that (u λ ) λ∈Λ is a representative of u.

Association process
We assume further that A is unitary and Λ is left-filtering for the given (partial) order relation ≺. Let us denote by: • F a given sheaf of topological K-vector spaces (resp. K-algebras) over X containing E as a subsheaf, • a a map from R + to A + such that a(0) = 1 (for r ∈ R + , we denote a (r) by (a λ (r)) λ ).
For (u λ ) λ ∈ H (A,E,P) (Ω), we denote by lim Λ F (Ω) u λ the limit of (u λ ) λ for the F-topology when it exists. We recall that lim We suppose also that we have, for each open subset V ⊂ Ω, Now, consider u = [u λ ] ∈ A(Ω), r ∈ R + , V an open subset of Ω and f ∈ F(V ). We say that u is a (r)-associated to f in V : In particular, if r = 0, u and f are said associated in V . To ensure the independence of the definition with respect to the representative of u, we must have, for any (η λ ) λ ∈ J (I A ,E,P) (Ω), lim Thus, our claim follows from the above assumption.

Using the previous notations and according to the previous assumption we have, for any open set Ω in
is well defined because if (η λ ) λ belongs to J (I A ,E,P) (Ω), we have lim Λ F (Ω) η λ = 0. Moreover, F A is a subpresheaf of vector spaces (resp. algebras) of A. Roughly speaking, it is the presheaf whose sections above some open set Ω are the generalized functions in A (Ω) associated to an element of F (Ω).

Localization of F-singularities
We refer to definition and results given in section 2 and take here B = F A . When u belongs to A (Ω), we can consider the set O F A (u) (= O B A (u)) of all x ∈ Ω having a neighborhood V on which u is associated to f ∈ F (V ), that is: V x being the set of all the neighborhoods of x. This leads to the following definition: The F-singular support of u ∈ A(Ω) is denoted S F A (u) and defined as Since the support of u ∈ A(Ω) is defined by it is clear that S F A (u) is a closed subset containing supp(u).

Some results
We can directly deduce the algebraic properties of S F A (u) (see [14]) from subsection 2.2. For the differential ones we suppose that F is a sheaf of topological differential vector spaces, with continuous differentiation, admitting E as a subsheaf of topological differential algebras. Then the presheaf A is also a presheaf of differential algebras with, for any α ∈ N n and u ∈ A (Ω), The independence of ∂ α u on the choice of representative follows directly from the definition of J (I A ,E,P) .) The behaviour of S F A (u) under differential operations is linked to the following Proposition 13 Under the above hypothesis ∂ α is a F A -compatible presheaf operator of A (Ω).
Proof. Let V be any open set of Ω. We have This result permits to obtain in the following subsection all the expected results on the propagation of local F A -singularities under differential operations.

Example 8
Taking E = C ∞ ; F = D ′ ; A = G leads to the D ′ -singular support of an element of the Colombeau algebra. this notion is complementary to the usual concept of local association in the Colombeau sense. We refer the reader to [28,29] for more details.

Example 9
Those following examples are considered for X = R d , E = C ∞ and A = G.

It suffices to verify that
sure that the previous definition don't depends upon the representative (u ε ) ε of u ∈ G σ (Ω).
The subspace of G s (Ω), the Gevrey generalized ultradistributions associated to ultradistributions

B-compatibility of differential or pseudo-differential operators
Here is a list of particular cases of subpresheaf B of presheaf A of interest to us: is always a presheaf (or a sheaf) of differential algebras and A is a differential B-module with a differentiation ∂ α (α ∈ A = N n ) extending the B-one. Then in each case and each open set V in Ω (open set of X = R n ) it is easy to prove that ∂ α v maps B (V ) into itself. Thus ∂ α is a presheaf operator B-compatible in A (Ω) according to Definition 6. If we give now a family (b α ) α∈N n of elements in B (Ω), then, Moreover at least in some cases, when A is D ′ , (resp.G, L G c (Ω), C ), a pseudo-differential maps D (Ω) into C ∞ (Ω). Via the theory of oscillatory integrals, the above formula can be extended into Au (x) = R n e i (x−y),ξ a(x, y, ξ)u(y)dydξ which defines a pseudo-differential operator A [23] when a = a(x, y, ξ) in S m ρ,δ (Ω × Ω × R n ) of Hörmander symbols of order m and type (ρ, δ). A extends continuously to a map E ′ (Ω) → D ′ (Ω).
We can find (for example in [11]) a definition of generalized pseudo-differential operators with generalized symbols extending the classical one. The set S m ρ,δ (Ω × R n ) of generalized symbols can be described [17] as the algebra G S m ρ,δ (Ω × R n ) based on S m ρ,δ (Ω × R n ) and obtained as a G Emodule by choosing E = S m ρ,δ (Ω × R n ). Then, the pseudo differential operator with generalized symbols b ∈ S m ρ,δ (Ω × R n ) is the map G c (Ω) → G (Ω) given by One can define more generally pseudo differential operators by means of symbols in S m ρ,δ (Ω × Ω × R n ) and generalized oscillatory integrals (see [15]). Such an operator A is given by Au := Ω×R n e i (x−y),ξ a(x, y, ξ)u(y)dydξ := Ω×R n e i (x−y),ξ a ε (x, y, ξ)u ε (y)dydξ which defines a generalized function in G (Ω) when u is in G c (Ω). We can find in ( [17], def. 2.5) an extension of the action of A to the dual L G c (Ω), C namely where t A (the transpose of A) is the pseudo-differential operator defined by t Au := Ω×R n e i (x−y),ξ a(x, y, −ξ)u(y)dydξ.
When A is D ′ , (resp.G, L G c (Ω), C ), it is proved in each case that if A is a properly supported pseudo-differential operator, it maps A (Ω) into itself. Moreover, when B is C ∞ , (resp.G ∞ , G), for each open set V in Ω, A maps B (V ) into itself. In other words A is a B-compatible operator in A (Ω).
Therefore proposition 4 allows to deduce the classical inclusions from the presheaf property of an operator B-compatible. Through Proposition 9, we can even obtain some non linear results, when A is a presheaf of algebras as where p is any positive integer and (p α ) α∈N n any given family of positive integers.

Remark 6 In Definition 1.3 of [17], a functional T ∈ L G c (Ω) , C is said to be basic if it
is of the form T, u = [ T ε , u ε ] where (T ε ) ε is a net of distributions in D ′ (Ω) satisfying the following condition When T is a basic functional in L G c (Ω), C , Theorem 2.10 proves that the inclusion is valid for any pseudo differential operator A with amplitude in S m ρ,δ (Ω × Ω × R n ) and a fortiori when A is properly supported. In this case our above remark on the presheaf property of an operator B-compatible shows that this result remains valid for any functional in L G c (Ω) , C .

The frequential microlocal analysis
After a short overview of classical results in distribution theory and propagation of singularities under linear and nonlinear operators, we will give a characterisation of the local regularity in the two more general cases of generalized functions which summarize the other ones. This leads to the definition of corresponding wave front sets which gives a general frequential microanalysis of generalized singularities.

Wave front set and microlocal regularity of product
As it is recalled in Introduction it follows from Lemma 8.1.1. in [22] that if Ω is an open set in R n and u ∈ D ′ (Ω), one can set: Σ x (u) = ∩ Φ Σ (Φu) ; Φ ∈ D (Ω) , Φ (x) = 0 and define the wave

Pseudo-differential operators
We recall that the space S m ρ,δ (R n × R n ) of symbols of order m and type (̺, δ) consist in all a ∈ C ∞ (R n × R n ) such that We consider here the simplified case S m = S m 1,0 which defines a pseudodifferential operator (belonging to OpS m ) If Ω is open then a continuous linear map A : D (Ω) → C ∞ (Ω) is said to be a pseudodifferential operator of order m in Ω (an element of Ψ m (Ω)) if for any ϕ,ψ ∈ D (Ω) the operator S (R n ) ∋ u → ϕA (ψu) is in OpS m . If u ∈ E ′ (Ω) and A ∈ Ψ m (Ω), then Au ∈ D ′ (Ω) is well defined and leads to the inclusion sing suppAu ⊂ sing suppu which is the projection of the microlocal property when A is properly supported and u belongs to D ′ (Ω).
The propagation of non characteristic regularity for semi-linear equations studied in [35] is given by where Ω is an open set in R n and s > n/2, be a solution of the semi-linear equation where J k u = (∂ α u) |α≤k| , f and the coefficients of P are C ∞ and k is smaller than the order m of P (x, D). If P is noncharacteristic at (x, ξ) ∈ Ω × (R n \0), it follows that u ∈ H loc (2s+m−n/2) at (x, ξ).

Application of paradifferential calculus
The paradifferentiel calculus of Bony [4] is based on some regularization of non smooth symbols. We don't intend to develop this theory here but look at it as a powerfull tool to prove good results for nonlinear problems such as the following Theorem (10.3.6 in [24]): Let u ∈ H loc (s+m−1/2) (Ω), s > max((n − 1) /2 , n/4), and assume that u verifies the quasilinear differential equation where a α and c are C ∞ . Then it follows that u ∈ H loc (2s+m−n/2) at every non characteristic point (x, ξ).

Propagation of singularities
Roughly speaking we know that under some conditions for linear or pseudo-differential equations, the singularities of solutions propagate along bicharacteristics This remains valid for nonlinear equations in the sense of Theorem (11.4.1 in [24]: Let u ∈ H loc (s+m) (Ω) , s > n/2 + 1,be a real valued solution of the differential equation

is contained in the characteristic set and it is invariant under the Hamilton flow defined by the principal symbol of the linearized equation.
Beals [2,3] has studied the case of second order hyperbolic equations (extended by [5] to arbitrary order) for which we can give a special version as Theorem (11.5.10 in [24]: Let u ∈ H loc (s) (Ω) , s > n/2 be a solution of of the hyperbolic second order semi-linear equation where f ∈ C ∞ . If u ∈ H loc (s) at a characteristic point (x, ξ) and if s ≤ σ < 3s − n + 1,it follows that u ∈ H loc (s) at the bicharacteristic γ through (x, ξ).

The frequential microlocal analysis in G r
We refer the reader to [12,13] for more details.

Characterization of G r,R -local regularity
We consider an open subset Ωof R d and the Schwartz space S (Ω) of rapidly decreasing functions defined on Ω, endowed with the family of seminorms Q (Ω) = (µ q,α ) (q,α)∈N×N d defined by The space of "rough"rapidly decreasing functions can be defined as In order to make easier the comparison between the distributional case and the generalized case, we begin by recalling the classical theorem and complete it by some equivalent statements given in the following result (Theorem 16 in [12]): for u in E ′ (R n ), the following equivalences hold: where F is the classical Fourier transform defined as topological automorphism of S R d . The result is based on the following inclusions The Fourier transform has been extended to some spaces of rapidly decreasing generalized functions (like G s (R n ) = X S (R n ) /N S (R n )) and more completely described in [12] in the framework of R-regular spaces. We can point out that in any framework, the elements with compact support have always a Fourier transform.
Definition 12 Let R be a regular subset of R N + and Ω an open subset or R n . Set One can show that X r,R S * (Ω) is a subalgebra of S * (Ω) (0,1] and that N r S * (Ω) is an ideal of X r,R S * (Ω). (The proof is similar to that of Proposition 1 in [12]).

Definition 13
The algebra G r,R s * (Ω) = X r,R S * (Ω) /N r S * (Ω) is called the algebra of (r, R)-regular rough rapidly decreasing generalized functions.
Theorem 14 Let x 0 ∈ Ω ⊂ R n and u ∈ G r (Ω). Then, u is G r,L at x 0 (in the sense of definition 2) iff there exist some neighbourhood W of x 0 , some ϕ ∈ D (W ), ϕ (x 0 ) = 0, such that ϕu ∈ G r,R s * (R n ) Proof. (Sketch). Let u be an element in G r (Ω) G r,R at x 0 . There exists a neighbourhood W of x 0 such that u | W ∈ G r,R (W ). We can extend any given ϕ ∈ D (W ), ϕ (x 0 ) = 0, into ϕ ∈ D R d such that ϕu ∈ G r,R c R d = G c (R n )∩G r,R (R n ). We follow the arguments of Theorem 22 in [12], replacing the ε-estimates by the r λ -ones. This leads to prove that ϕu ∈ G r,R s * (R n ). Conversely, this last assertion with the above hypothesis permits to prove that ϕu ∈ G r,R c (R n ), and then there exits a neighbourhood V of x 0 such that u | V ∈ G r,R (V ). In this last part one needs to define an inverse Fourier transform F −1 in G r,R s * (R n ) for which one introduces the space G r,R B (R n ) of (r, R)-regular bounded generalized functions such that F −1 G r,R s * (R n ) ⊂ G r,R B (R n ). The result follows from the equality G r,R

Definition 15
The G r,R -generalized wave front set of u ∈ G r (Ω), denoted by W F (r,R) (u) is the complement in Ω × (R n \0) of the set of all pairs (x 0 , ξ 0 ) such that u is microlocally (r, R)-regular at (x 0 , ξ 0 ).

Theorem 15
The projection of W F (r,R) (u) in Ω is equal to sing supp (r,R) (u) .
The proof follows from the arguments involved in [22] using lemma 8.1.1.

Characterization of G r,R,L -local regularity
When starting from previous cases (like G, G R or G L ) the problem is to change simultaneously the asymptotic scale into a new one, and the G ∞ -regularity subordinated to L-conditions into G R -regularity subordinated to L-conditions. To do that we have to mix carefully the techniques used in [12] and [30]. This study is done in [13]. In this subsection, we only give the definitions and results without proofs.
Theorem Let x 0 ∈ Ω ⊂ R n and u ∈ G r (Ω). Then, u is G r,R,L at x 0 (in the sense of Definition 2) iff there exist some neighbourhood W of x 0 , a compact K such W ⊂ K ⋐ Ω, a sequence of functions χ k, each in D K (Ω) and valued in [0, 1] with χ k u = u on W , a representative (u λ ) λ of u, a regular sequence N ∈ R, a positive constant c, and λ 0 ∈ Λ such that for all ξ ∈ R

3.2.4
The G r,R,L -generalized wave front set Definition 16 An element u ∈ G r (Ω) is said to be microlocally (r, R, L)-regular at (x 0 , ξ 0 ) ∈ Ω×(R n \0)(we set: u ∈ G r,R,L (x 0 , ξ 0 )) if there exist a neighborhood W of x 0 , a conic neighborhood Γ of ξ 0 , a sequence (u k = χ k u) k∈N of generalized functions where each χ k is valued in [0, 1] and is in D K (Ω), with W ⊂ K ⋐ Ω, u k being equal to u in W , a sequence N ∈ R, a positive constant c, and λ 0 ∈ Λ such that (*) holds when ξ ∈ Γ.

Definition 17
The G r,R,L -generalized wave front set of u∈ G r (Ω), denoted by W F (r,R,L) (u) is the complement in Ω × (R n \0) of the set of all pairs (x 0 , ξ 0 ) such that u is microlocally (r, R, L)-regular at (x 0 , ξ 0 ).
W F r,R,L (u) is a closed subset of Ω × (R n \0), and its projection in Ω is given by the following result: Theorem The projection of W F (r,R,L) (u) in Ω is equal to sing supp (r,R,L) (u) .
3.2.5 Propagation of singularities under differential (or pseudo-differential) operators a) We can summarize the first investigations in the following results proved in [13] Proposition If the coefficients a α lie in G r,R (Ω) (resp. in G r,R,L (Ω)), then we have b) In the special case of G ∞ singularities of G, we can quote the results based on pseudodifferential operators and pseudodifferential techniques. In [18] analogues of Hörmander definition of the distributional wave front set given in [23] are obtained by characterizations of generalized wave front set in terms of intersection over some non-ellipticity domains. This intersection is taken over all slow scale pseudo-differential operators a ∈ S m sc (Ω × R n ) (def. 1.1) verifying some other regularity conditions. More precisely, if Ell sc (a) denote the set of all (x, ξ) ∈ Ω × T * (Ω) \0 where a is slow scale micro-elliptic (def. 1.2), Theorem 2.1 proves that for all u ∈ G (Ω) where pr Ψ 0 (Ω) denote the set of all properly supported slow scale operators of order 0. Another pseudo-differential characterisation of W F g (u) is given by Theorem 2.1.1 which proves that for all u ∈ G (Ω) where the intersection is taken over all classical properly supported classical pseudo-differential operators A such that Au belongs to G ∞ (Ω). Following these characterizations some refined results on propagation of singularities can be obtained. For example, Theorem 3.1 proves that if A = a (x, D) is a properly supported pseudo-differential operator with slow scale symbol and u ∈ G (Ω)

3.3
The frequential microlocal analysis in L G c (Ω), C 3.3.1 The generalized wave front set W F G (T ) Inspired by the results and definitions of [18] recalled in the previous subsection, Garetto (Def. 3.3,[17]) defines the G-wave front set of a functional T ∈ L G c (Ω), C as Ell sc (a) c .
And even the G ∞ -wave front set of T is defined in the same way by replacing G (Ω) by G ∞ (Ω). Proposition 3.5 shows that the projection on Ω of W F G (T ) is exactly singsupp G T .
When A is a properly supported pseudo-differential operator with symbol a ∈ S m ρ,δ (Ω × R n ), the inclusion can be refined by introducting the concept of G-microsupport of a, denoted µsupp G (a). It is the complement of all (x, ξ) ∈ Ω × T * (Ω) \0 where a is G-smoothing (Def.3.6). Then we have (Corollary 3.9) This result is reformulated in terms of G-microsupport of the operator A (µsupp G (A)) in the form where the G-microsupport of A is defined (Def.3.11) by

Fourier transform characterisation of T and propagation of singularities
When ϕ ∈ D (Ω) and T ∈ D ′ (Ω), we recall that the regularity of ϕT can be measured by the rapid decay of its Fourier transform in some conic region Γ ⊂ R n \0. Following this idea, Garetto introduces the subset G S,0 (Γ) of G τ (R n ) (algebra of tempered generalized functions) such that which is similar to G r,R s * (Γ) introduced in Definition 13. This leads to the Fourier transform characterization of T given in Then an extension of Theorem 4.1 in [15] follows: Theorem 4.1 in [17]: If A = a (x, D)is a properly supported pseudo-differential operator with symbol a ∈ S m sc (Ω × R n ) and T a basic functional in L G c (Ω) , C , then

The asymptotic microlocal analysis
Let Ω be an open set in X. Fix u = [u λ ] ∈ A(Ω) and x ∈ Ω. The idea of the (a, F)-microlocal analysis is the following: (u λ ) λ may not tend to a section of F above a neighborhood of x, that is, there may not exist V ∈ V x and f ∈ F (V ) such that lim Λ F (V ) u λ = f . Nevertheless, in this case, there may exist V ∈ V x , r ≥ 0 and f ∈ F (V ) such that lim Λ F (V ) a λ (r)u λ = f , that is [a λ (r)u λ | V ] is in the subspace (resp. subalgebra) F A (V ) of A(V ) introduced in Subsection 2.5. These preliminary remarks lead to the following concept and results which we summarize from the results given in [28,29,14].

The (a, F )-singular parametric spectrum
We recall that a is a map from R + to A + such that a(0) = 1 and F is a presheaf of topological vector spaces (or topological algebras). For any open subset Ω of X, u = [u λ ] ∈ A(Ω) and x ∈ Ω, set It is easy to check that N (a,F ),x (u) does not depend on the representative of u. If no confusion may arise, we shall simply write N (a,F ),x (u) = N x (u).

Linear and differential properties
It is easy to prove that for any u, v ∈ A(Ω), we have As a corollary: for any u, u 0 , u 1 in A(Ω) with Assume that F is a sheaf of topological differential vector spaces, with continuous differentiation, admitting E as a subsheaf of topological differential algebras. Then the sheaf A is also a sheaf of differential algebras with, for any α ∈ N d and u ∈ A (Ω), The independence of ∂ α u on the choice of representative follows directly from the definition of J (I A ,E,P) .) It follows that if u is in A(Ω) and g in E(Ω), for all ∂ α , α ∈ N d , we have This leads to the more general statement: Let P (∂) = |α|≤m C α ∂ α be a differential polynomial with coefficients in E(Ω). For any u ∈ A(Ω), we have

Nonlinear properties
When F is a presheaf of algebras, the (a, F)-singular spectrum inherits new properties with respect to nonlinear operations. It is the purpose of following results.
Theorem (15 in [14]) We suppose that F is a presheaf or algebras. For u and v ∈ A(Ω), let D i ( i = 1, 2, 3) be the following disjoint sets: Then the ( a,F )-singular asymptotic spectrum of uv verifies Given now a test function ψ ∈ D(R d ), we have

The singular spectrum of solutions to semilinear hyperbolic equations
The singular spectrum of solutions of a semilinear transport problem where λ and F are smooth functions of their arguments. may decrease or increase with respect to the one of the data, depending on the function F . We observe that by a change of coordinates we may assume without loss of generality that λ ≡ 0. a) For F (u ε (x, t)) = −u 3 ε (x, t) (the dissipative case: Example 8 in [14]), the problem (P 0 ) has the solution u ε (x, t) = u 0ε (x) 2tu 2 0ε (x) + 1 .
When the initial data are given by a power of the delta function, u 0ε (x) = ϕ m ε (x), the solution formula shows that u ε (x, t) is a bounded function (uniformly in ε) supported on the line {x = 0}. Thus u ε (x, t) converges to zero in D ′ (R×]0, ∞[), and so x ∈ R, t > 0 ( [14], Example 9), the problem (P 0 ) has the solution u ε (x, t) = u 0ε (x) cosh t + 1 + u 2 0ε (x) sinh t. b 1 ) with a delta function as initial value, that is, u 0ε (x) = ϕ ε (x) we obtain for ψ ∈ D(R 2 ). Thus in this case b 2 ) with the derivative of a delta function as initial value, u 0ε (x) = ϕ ′ ε (x), a similar calculation shows that and so S Example 10 in [14] shows that it is quite possible for the singular spectrum to increase with time. c) when taking F (u ε (x, t)) = u ε (x, t) + 1) log u ε (x, t) + 1 , x ∈ R, t > 0, the problem (P 0 ) has the solution u ε (x, t) = u 0ε (x) + 1 provided u 0ε > −1 in which case the function on the right hand side of the differential equation is smooth in the relevant region. To demonstrate the effect, we take a power of the delta function as initial value, that is u 0ε (x) = ϕ m ε (x). Then

Blow-up in finite time
In situations where blow-up in finite time occurs, microlocal asymptotic methods allow to extract information beyond the point of blow-up. This can be done by regularizing the initial data and truncating the nonlinear term. This is shown in Example 11 of [14] for a simple situation. The problem to be treated is formally the initial value problem where H denotes the Heaviside function. Clearly, the local solution u(x, t) = H(x)/(1 − t) blows up at time t = 1 when x > 0. Choose χ ε ∈ C ∞ (R) with 0 ≤ χ ε (z) ≤ 1 ; χ ε (z) = 1 if |z| ≤ ε −s , χ ε (z) = 0 if |z| ≥ 1 + ε −s , s > 0.
When x < 0 and ε is sufficiently small, u ε (x, t) = 0 for all t ≥ 0. For x > 0, u ε (x, t) = 1/(1 − t) as long as t ≤ 1 − ε s . The cut-off function is chosen in such a way that |χ ε (z)z 2 | ≤ (1 + ε −s ) 2 for all z ∈ R. Therefore, ∂ t u ε ≤ (1 + ε −s ) 2 always and ∂ t u ε = 0 when |u ε | ≥ 1 + ε −s . Some computations and estimates permit to obtain the following C 0 -singular support and a, C 0 -singular spectrum (for a ε (r) = ε r ) of u = [u ε ]: These results give a microlocal precision on the the blow-up: The C 0 -singularities (resp. a, C 0 -singularities) of u are described by means of two sets: S 1 (u) and S 2 (u) (resp. S 1 (u)×{0} and S 2 (u) × [0, s]). The set S 1 (u) (resp. S 1 (u) × {0}) is related to the data C 0 (resp. a, C 0 )singularity. The set S 2 (u) (resp. S 2 (u)×[0, s]) is related to the singularity due to the nonlinearity of the equation giving the blow-up at t = 1. The blow-up locus is the edge {x ≥ 0, t = 1} of S 2 (u) and the strength of the blow-up is measured by the length s of the fiber [0, s] above each point of the blow-up locus. This length is closely related to the diameter of the support of the regularizing function χ ε and depends essentially on the nature of the blow-up: Changing simultaneously the scales of the regularization and of the cut-off (i.e. replacing ε by some function h(ε) → 0 in the definition of ϕ ε and χ ε ) does not change the fiber and characterizes a sort of moderateness of the strength of the blow-up.

The strength of a singularity and the sum law
We point out the following remark ( [14], subsection 4.3): when studying the propagation and interaction of singularities in semilinear hyperbolic systems, Rauch and Reed [36] defined the strength of a singularity of a piecewise smooth function. This notion is recalled in the onedimensional case. Assume that the function f : R → R is smooth on ] − ∞, x 0 ] and on [x 0 , ∞[ for some point x 0 ∈ R. The strength of the singularity of f at x 0 is the order of the highest derivative which is still continuous across x 0 . For example, if f is continuous with a jump in the first derivative at x 0 , the order is 0; if f has a jump at x 0 , the order is −1. Travers [39] later generalized this notion to include delta functions. Slightly deviating from her definition, but in line with the one of [36], it is possible to define the strength of singularity of the k-th derivative of a delta function at x 0 , ∂ k x δ(x − x 0 ), by −k − 2. The significance of these definitions is perceived in the description of what Rauch and Reed termed anomalous singularities in semilinear hyperbolic systems. This effect is demonstrated in a paradigmatic example, also due to [36], the (3 × 3)-system w(x, 0) = 0 Assume that u 0 has a singularity of strength n 1 ≥ −1 at x 1 = −1 and v 0 has a singularity of strength n 2 ≥ −1 at x 2 = +1. The characteristic curves emanating from x 1 and x 2 are straight lines intersecting at the point x = 0, t = 1. Rauch and Reed showed that, in general, the third component w will have a singularity of strength n 3 = n 1 + n 2 + 2 along the half-ray {(0, t) : t ≥ 1}. This half-ray does not connect backwards to a singularity in the initial data for w, hence the term anomalous singularity. The formula n 3 = n 1 + n 2 + 2 is called the sum law. Travers extended this result to the case where u 0 and v 0 were given as derivatives of delta functions at x 1 and x 2 . We are going to further generalize this result to powers of delta functions, after establishing the relation between the strength of a singularity of a function f at x 0 and the singular spectrum of f * ϕ ε .
We consider a function f : R → R which is smooth on (−∞, x 0 ] and on [x 0 , ∞) for some point x 0 ∈ R; actually only the local behavior near x 0 is relevant. A mollifier ϕ ε (x) = 1 ε ϕ( x ε ) is fixed as in 4.3.1 and the corresponding embedding of D ′ (R) into the (C, E, P)-algebra A(R) is denoted by ι. In particular, ι(f ) = [f * ϕ ε ].
If f is continuous at x 0 , then lim ε→0 f * ϕ ε = f in C 0 . If f has a jump x 0 , this limit does not exist in C 0 , but lim ε→0 ε r f * ϕ ε = 0 in C 0 for every r > 0. The following result is Proposition (16 in [14]) Let x 0 ∈ R. If f : R → R is a smooth function on (−∞, Here n ∈ N and a ε (r) = ε r . When returning to the model equation we find that the sum law remains valid when the initial data are powers of delta functions. Suitable (C, E, P)-algebras A(R) and A(R 2 ) are exhibited in which the initial value problem can be uniquely solved. When the scale is taken as a ε (r) = ε r , the following result is obtained: Proposition (17 in [14]) Let u 0 (x) = δ m (x + 1), v 0 (x) = δ n (x − 1) for some m, n ∈ N * . Let w ∈ A(R 2 ) be the third component of the solution to problem (**). Then w(x, t) vanishes at all points (x, t) with x = 0 as well as (0, t) with t < 1, and Σ (a,C 1 ),(0,t) w ⊂ [0, m + n] for t ≥ 1.

Microlocal characterisation of some regular subalgebras
We recall that the subsheaf G ∞ of regular Colombeau functions of the sheaf G is defined as follows [32]: Given an open subset Ω of R d , the algebra G ∞ (Ω) comprises those elements u of G(Ω) whose representatives (u ε ) ε satisfy the condition ∀K ⋐ Ω, ∃m ∈ N, ∀l ∈ N : p K,l (u ε ) = o(ε −m ) as ε → 0.
In relation with regularity theory of solutions to nonlinear partial differential equations, a further subalgebra of G(Ω) has been introduced in [33] -the algebra of Colombeau functions of total slow scale type. It consists of those elements u of G(Ω) whose representatives (u ε ) ε satisfy the condition ∀K ⋐ Ω, ∀r > 0, ∀l ∈ N : p K,l (u ε ) = o(ε −r ) as ε → 0.