Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

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Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

Antoine Delcroix

To cite this version:

Antoine Delcroix. Regular rapidly decreasing nonlinear generalized functions. Application to microlo- cal regularity. Journal of Mathematical Analysis and Applications, Elsevier, 2007, 327, pp.564-584.

�10.1016/j.jmaa.2006.04.045�. �hal-00020280�

ccsd-00020280, version 1 - 8 Mar 2006

Regular rapidly decreasing nonlinear generalized functions.

Application to microlocal regularity

Antoine Delcroix

Equipe Analyse Alg´ebrique Non Lin´eaire Laboratoire Analyse, Optimisation, Contrˆole

Facult´e des sciences - Universit´e des Antilles et de la Guyane 97159 Pointe-`a-Pitre Cedex Guadeloupe

March 8, 2006

Abstract

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the Colombeau simplified model. This generalizes the notion ofG^∞-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.

Mathematics Subject Classification (2000): 35A18, 35A27, 42B10, 46E10, 46F30.

Keywords: Colombeau generalized functions, rapidly decreasing generalized functions, Fourier transform, microlocal regularity.

1 Introduction

The various theories of nonlinear generalized functions are suitable frameworks to set and solve differential or integral problems with irregular operators or data. Even for linear problems, these theories are efficient to overcome some limitations of the distributional framework. We follow in this paper the theory introduced by J.-F. Colombeau [2], [3], [10], [19]. To be short, a special Colombeau type algebra is a factor space G = X/N of moderate modulo negligible nets. The moderateness (respectively the negligibility) of nets is defined by their asymptotic behavior when a real parameter εtends to 0.

A local and microlocal analysis of singularities of nonlinear generalized functions has been developed during the last decade, based on the notion ofG^∞-regularity [20]. A generalized func- tion isG^∞-regular if it has uniform growth bounds, with respect to the regularization parameter ε, for all derivatives. In fact, this notion appears to be the exact generalization of the C^∞- regularity for distributions, in the sense given by the result of [20] which asserts thatG^∞∩ D^′ is equal to C^∞.

In this paper we include G^∞ and G in a new framework of spaces of R-regular nonlinear generalized functions, in which the growth bounds are defined with the help of spaces R of sequences satisfying natural conditions of stability. One main property of those spaces is that the elements with compact support can be characterized by a “R-property” of their Fourier transform. (Those Fourier Transforms belong to some regular subspaces of spaces of rapidly

decreasing generalized functions [8], [22].) Thus, the parallel is complete with the C^∞-regularity of compactly supported distributions. Moreover, from this characterization, we deduce that the microlocal behavior of a generalized function with respect to a given R-regularity is completely similar to the one of a distribution with respect to the C^∞-regularity. In particular, we can handle theR-wavefront of an element of G as the C^∞one of a distribution. Finally, the G^∞-regularity for an element of G appears as a remarkable particular case.

With this new notion of R-regularity, we enlarge the possibility for the study of the propa- gation of singularities through differential and pseudo differential operators, since we give a less restrictive framework than theG^∞-regularity. This will be studied in forthcoming papers.

Let us also quote that spaces ofR-regular generalized functions have been used in a problem of Schwartz kernel type theorem. More precisely, we showed in [4] that some nets of linear maps (parametrized by ε∈ (0,1]), satisfying some growth conditions similar to those introduced for R-regular spaces, give rise to linear maps between spaces of generalized functions. Moreover, those maps can be represented by generalized integral kernel on some specialR-regular subspaces ofG(Ω) in which the growth bounds are at most sublinear with respect tol. This reinforces the interest of this framework.

The paper is organized as follows. In section 2, we introduce the spaces ofR-regular general- ized functions and we precise some classical results about the embedding ofD^′ into these spaces.

Section 3 is devoted to the study of the space GS of rapidly decreasing generalized functions. In particular, we show that O_C^′ , the space of rapidly decreasing distributions, is embedded in GS. Thus, G_S plays for O^′_C the role that G plays for D^′. Section 4 contains the material related to Fourier transform of elements of GS and especially an exchange theorem which is, in the context ofR-regularity, an analogon and a generalization of the classical exchange theorem betweenO^′_C and O_M. Section 5 gives the above mentioned characterization by Fourier transform of com- pactly supported R-regular generalized functions whereas, in section 6, we present the R-local and R-microlocal analysis of generalized functions.

2 The sheaf of Colombeau simplified algebras and related sub- sheaves

2.1 Sheaves of regular generalized functions Notation 1 For two sequences (N1, N2)∈ R^N

+

2

, we say thatN1 is smaller or equal toN2 and note N₁4N₂ iff ∀n∈N N₁(n)≤N₂(n).

Definition 1 We say that a subspaceR of R^N₊ is regularif Ris non empty and (i) R is “overstable” by translation and by maximum

∀N ∈ R, ∀ k, k^′

∈N², ∃N^′ ∈ R, ∀n∈N N(n+k) +k^′ ≤N^′(n), (1)

∀N1∈ R, ∀N2∈ R, ∃N ∈ R, ∀n∈N max (N₁(n), N₂(n))≤N(n). (2) (ii) For all N₁ and N₂ in R, there existsN ∈ R such that

∀(l1, l2)∈N² N1(l1) +N2(l2)≤N(l1+l2). (3) Example 1

(i) The set B of bounded sequences and the set A of affine sequences are regular subsets of R^N

+, which is itself regular.

(ii) The set L_og =

N ∈R^N

+ | ∃b∈R₊,N :n7→lnn+b is not regular ((3) is not satisfied),

whereasL¹_og =

N ∈R^N

+

∃(a, b)∈R²

+,N :n7→alnn+b is regular. ((3) comes, for example, from lnx+ lny≤2 ln(x+y), forx >0 andy >0.)

Let Ω be an open subset of R^d (d∈N) and consider the algebra C^∞(Ω) of complex valued smooth functions, endowed with its usual topology. This topology can be described by the family of seminorms (p_K,l)_{K⋐Ω,l∈N}defined by

p_K,l(f) = sup

x∈K,|α|≤l

|∂^αf(x)|. For any regular subsetRof R^N₊, we set

X^R(Ω) =n

(f_ε)_ε ∈C^∞(Ω)^(0,1]

∀K⋐Ω, ∃N ∈ R, ∀l∈N p_K,l(f_ε) = O

ε^−N(l)

asε→0o N^R(Ω) =n

(f_ε)_ε ∈C^∞(Ω)^(0,1]

∀K⋐Ω, ∀m∈ R, ∀l∈N p_K,l(f_ε) = O ε^m(l)

asε→0o . Proposition 1

(i) For any regular subspace R of R^N₊, the functor Ω → X^R(Ω) defines a sheaf of differential algebras over the ring

X(C) =n

(r_ε)_ε∈C^(0,1]

∃q∈N |rε|= O ε^−q

asε→0o . (ii) The functorN^R: Ω→ N^R(Ω)defines a sheaf of ideals of the sheaf X^R(·).

(iii) For any regular subspacesR1 andR2 of R^N₊, withR1 ⊂ R2, the sheafX^R1(Ω)is a subsheaf of the sheaf X^R2(Ω).

Proof. We split the proof in two parts.

(a)Algebraical properties.- Let us first show that for any open set Ω⊂R^d,X^R(Ω) is a subalgebra of C^∞(Ω)^(0,1]. Take (f_ε)_ε and (g_ε)_ε inX^R(Ω) and K ⋐ Ω. There exist N_f ∈ R and N_g ∈ R such that

∀l∈N p_K,l(h_ε) = O

ε^−Nh(l)

asε→0, for h_ε=f_ε, g_ε. We get immediately that p_K,l(f_ε+g_ε) = O

ε^−max(^Nf(l),Ng(l))

asε→ 0, with max (N_f, N_g)4 N for someN ∈ R according to (2). Then, (f_ε+g_ε)_ε belongs to X^R(Ω).

For (cε)_ε ∈ X(C), there exists qc such that |cε| = O (ε^−qc) as ε → 0. Then p_K,l(cεfε) = O ε^{−Nf(l)−qc}

. From (1), there exists N ∈ R such that N_f +q_c 4N. Thus, (c_εf_ε)_ε ∈ X^R(Ω).

It follows that X^R(Ω) is a submodule of C^∞(Ω)^(0,1] over X(C).

Consider now l∈N andα ∈N^d with|α|=l. By Leibniz’ formula, we have, for allε∈(0,1]

and x∈K,

|∂^α(f_εg_ε) (x)|=X

γ≤α

C_α^γ∂^γf_ε (x)∂^α−γg_ε(x)≤X

γ≤α

C_α^γp_K,|γ|(f_ε)p_K,|α−γ|(g_ε), whereC_α^γ is the generalized binomial coefficient. We have, for allγ ≤α,

p_K,|γ|(f_ε)p_K,|α−γ|(g_ε) = O

ε^−Nf(|γ|)−N g(|α−γ|)

asε→0.

Asγ ≤α, we get|γ|+|α−γ|=|α|=l. According to (3), there existsN ∈ R such that, for all kand k^′ ≤k inN,N_f(k^′) +N_g(k−k^′)≤N(k). Then

sup

x∈K

|∂^α(f_εg_ε) (x)|= O

ε^−N(l)

asε→0.

Thusp_K,l(f_εg_ε) = O ε^−N(l)

asε→0, and (f_εg_ε)_ε∈ X^R(Ω).

For the properties related to N^R(Ω), we have the following fundamental lemma:

Lemma 2 The set N^R(Ω)is equal to Colombeau’s ideal N (Ω) =n

(f_ε)_ε ∈C^∞(Ω)^(0,1] | ∀K⋐Ω, ∀l∈N, ∀m∈N p_K,l(f_ε) = O (ε^m) asε→0o .

Indeed, take (f_ε)_ε∈ N^R(Ω). For anyK ⋐Ω,l∈Nandm∈N, chooseN ∈ R. According to (1) there exists N^′ ∈ Rsuch thatN +m4N^′. Thus,p_K,l(fε) = O

ε^N′(l)

= O (ε^m) asε→ 0 and (fε)_ε ∈ N(Ω). Conversely, given (fε)_ε ∈ N(Ω) and N ∈ R, we havep_K,l(fε) = O ε^N(l) asε→0, since this estimates holds for all m∈N.

(b)Sheaf properties.- The proof follows the same lines as the one of Colombeau simplified algebras.

(See for example [10], theorem 1.2.4.) First, the definition of restriction (by the mean of the restriction of representatives) is straightforward as in Colombeau’s case. For the sheaf properties, we have to replace Colombeau’s usual estimates by X^R-estimates. But, at each place this happens, we have only to consider a finite number of terms, by compactness properties. Thus, the stability by maximum ofR (property (2)) induces the result. Furthermore, lemma 2 shows that nothing changes for estimates dealing with the ideal. Finally, point (iii) of the proposition follows directly from the obvious inclusion X^R1(Ω)⊂ X^R2(Ω).

Definition 2 For any regular subsetR of R^N

+, the sheaf of algebras G^R(·) =X^R(·)/N^R(·)

is called the sheaf of R-regular algebras of (nonlinear) generalized functions.

According to lemma 2, we have N^R = N. From now on, we shall write all algebras with Colombeau’s ideal N. The sheaf G^R(·) turns out to be a sheaf of differential algebras and a sheaf of modules over the factor ringC=X(C)/N (C) with

N (C) =n

(rε)_ε∈C^(0,1] | ∀p∈N |rε|= O (ε^p) asε→0o .

Example 2 Taking R=R^N₊, we recover the sheaf of Colombeau simplifiedor special algebras.

Notation 2 In the sequel, we shall writeG(Ω)(resp. XM(Ω)) instead ofG^RN+(Ω)(resp. X^RN+(Ω)).

For(f_ε)_εin X_M(Ω)or X^R(Ω),[(f_ε)_ε]will be its class inG(Ω)or inG^R(Ω), since these classes are obtained modulo the same ideal. (We consider G^R(Ω) as a subspace of G(Ω).)

Example 3 Taking R = B, introduced in example 1, we obtain the sheaf of G^∞-generalized functions [20].

Example 4 Take a in [0,+∞]and set R₀ =

N ∈R^N₊ lim

l→+∞(N(l)/l) = 0

; For a >0 :R_a=

N ∈R^N₊

limsup

l→+∞

(N(l)/l)< a

. For any a in [0,+∞], Ra is a regular subset of R^N₊. The corresponding sheaves G^Ra(·) are the sheaves of algebras of generalized functions with slow growth introduced in [4]. Note that, for a in (0,+∞], a sequence N is in Ra iff there exists (a^′, b) ∈ (R⁺)² with a^′ < a such that N(l)≤a^′l+b. The growth of the sequenceN is at most linear.

Remark 1 We have the obvious sheaves inclusions X^∞(·)⊂ X^Ra(·)⊂ X(·).

For a given regular subspaceRofR^N

+, the notion of support of a sectionf ∈ G^R(Ω) (Ω open subset of R^d) makes sense since G^R(·) is a sheaf. The following definition will be sufficient for this paper.

Definition 3 The support of a generalized function f ∈ G^R(Ω) is the complement in Ω of the largest open subset of Ω where f is null.

Notation 3 We denote byG_C^R(Ω) the subset of G^R(Ω)of elements with compact support.

Lemma 3 Every f ∈ G^R_C has a representative (f_ε)_ε, such that each f_ε has the same compact support.

We shall not prove this lemma here, since lemma 12 below gives the main ideas of the proof.

2.2 Some embeddings For any regular subspaceRofR^N

+and any Ω open subset ofR^d, C^∞(Ω) is embedded intoG^R(Ω) by the canonical embedding

σ: C^∞(Ω)→ G^R(Ω) f →[(fε)_ε] withfε=f for all ε∈(0,1].

We refine here the well known result about the embedding of D^′(Ω) into G(Ω). Following the ideas of [19], we consider ρ∈ S R^d

such that Rρ(x) dx= 1, R

x^mρ(x) dx= 0 for allm ∈N^d\ {0}. (Such a map can be chosen as the Fourier transform of a function of D R^d

equal to 1 on a neighborhood of 0.) We now chooseχ∈ D R^d

such that 0≤χ≤1,χ≡1 onB(0,1) andχ≡0 on R^d\B(0,2). We define

∀ε∈(0,1], ∀x∈R^d θ_ε(x) = 1 ε^dρx

ε

χ(|lnε|x) . Finally, consider (κ_ε)_ε ∈ D R^d(0,1]

such that

∀ε∈(0,1), 0≤κ_ε≤1 κ_ε≡1 on n

x∈Ω

d(x,R^d\Ω)≥εand d(x,0) ≤1/εo . With these ingredients, the map

ι:D^′(Ω)→ G(Ω) T 7→(κ_εT ∗θ_ε)_ε+N(Ω) (4) is an embedding of D^′(Ω) intoG(Ω) such that ι_|C^∞_(Ω) =σ.

The proof is mainly based on the following property of (θ_ε)_ε: Rθ_ε(x) dx= 1 + O

ε^k

asε→0, ∀m∈N^d\ {0} R

x^mθ_ε(x) dx= O ε^k

asε→0. (5) Set

R1 =n

N ∈R^N

+ | ∃b∈R₊,∀l∈R N(l)≤l+bo

. (6)

One can verify that the setR is regular and we setG⁽¹⁾(·) = G^R1(·).

Proposition 4 The image of D^′(Ω)by the embedding, defined by (4), is included in G⁽¹⁾(Ω).

Proof. The proof is a refinement of the classical proof (see [5]), and we shall focus on the estimation of the growth of (κεT∗θε)_ε which is the main novelty. We shall do it for the case Ω =R^d, for which the additional cutoff by (κ_ε)_ε is not needed. Then, for a given T ∈ D^′ R^d

, we have ι(T) = (T ∗θ_ε)_ε+N R^d

, with (T ∗θ_ε)_ε=hT, θε(y− ·)i.

Let us fix a compact setKand considerW an open subset ofR^dsuch thatK ⊂W ⊂W ⋐R^d. With the above definitions, the function x 7→ θ_ε(y−x) belongs to D(W) for all y ∈K and ε small enough, since the support of θ_ε shrinks to {0} when ε tends to 0. Therefore, for β ∈ N^d and εsmall enough, we have

∀y∈K ∂^β(T ∗θ_ε) (y) =D

T, ∂^β{x7→θ_ε(y−x)}E

=D

T_|W, ∂^β{x7→θ_ε(y−x)}E

= (−1)^|β|D T_|W,n

x7→

∂^βθ_ε

(y−x)oE .

By using the local structure of distributions [24], we can writeT_|W =∂_x^αf wheref is a compactly supported continuous function having its support included in W. It follows

∀y∈K ∂^β(T∗θ_ε) (y) = (−1)^|β|D

∂_x^αf,

∂^βθ_ε

(y− ·)E

= (−1)^|α|+|β|D f,

∂^α+βθ_ε

(y− ·)E

= (−1)^|α|+|β|

Z

W

f(x)∂^α+βθ_ε(y−x) dx.

Using the definition of (θ_ε)_ε, we get

∀ξ ∈R ∂^α+βθ_ε(ξ) = X

γ≤α+β

C^γ_α+β∂^γρ_ε(ξ) ∂^α+β−γ(χ(ξ|lnε|)) (with ρ_ε(·) = 1 ε^dρ·

ε

)

= X

γ≤α+β

C^γ_α+βε^−d−|γ||lnε|^|α+β−γ|(∂^γρ) ξ

ε ∂^α+β−γχ

(ξ|lnε|), with|α+β−γ|=|α|+|β| − |γ|sinceγ ≤α+β. For all γ ≤α+β, we have

ε^−d−|γ||lnε|^|α+β−γ|= O

ε−d−1−|α|−|β|

asε→0.

As ρ and χ are bounded, as well as their derivatives, there exists C₁ >0, depending on (α, β), such that

∀ξ∈R

∂^α+β(θ_ε(ξ))

≤C₁ε−d−1−|α|−|β|. We get

∀y∈K

∂^β(T ∗θ_ε) (y)

≤C₁ sup

ξ∈W

|f(ξ)|vol W

ε−d−1−|α|−|β|.

From this last inequality, we deduce that for any l ∈ N, there exists some constant C₂ > 0, depending on (l, K), such that

p_K,l((T∗θ_ε))≤C₂ε^{−d−1−|α|−l}=C₂ε^−N(l) withN(l) =l+d+ 1 +|α|, which shows that (T∗θ_ε)_ε∈ X^R1 R^d

.

We can summarize these results in the following commutative diagram:

C^∞(Ω) −→ D^′(Ω)

↓σ ↓ι

G^∞(Ω) −→ G⁽¹⁾(Ω) −→ G(Ω).

(7)

3 Rapidly decreasing generalized functions

3.1 Definition and first properties

Spaces of rapidly decreasing generalized functions have been introduced in the literature ([8], [22], [23]), notably in view of the definition of the Fourier transform in convenient spaces of nonlinear generalized functions. We give here a more complete description of this type of space in the framework of R-regular spaces.

Definition 4 We say that a subspaceR^′ of the space R^N2

+ of maps fromN² to R₊ is regular if (i) R^′ is “overstable” by translation and by maximum

∀N ∈ R^′, ∀ k, k^′, k^′′

∈N³, ∃N^′ ∈ R^′, ∀(q, l)∈N² N q+k, l+k^′

+k^′′ ≤N^′(q, l), (8)

∀N1∈ R^′, ∀N2 ∈ R^′, ∃N ∈ R^′, ∀(q, l)∈N² max (N1(q, l), N2(q, l))≤N(q, l). (9) (ii) For any N1 and N2 in R^′, there existsN ∈ R^′ such that

∀(q₁, q₂, l₁, l₂)∈N⁴ N₁(q₁, l₁) +N₂(q₂, l₂)≤N(q₁+q₂, l₁+l₂). (10) Example 5

(i) The set B^′ of bounded maps from N² to R₊ is a regular subset of R^N₊². (ii) The set R^N2

+ of all maps from N² to R₊ is a regular set.

We consider Ω an open subset of R^d and the space S(Ω) of rapidly decreasing functions defined on Ω, endowed with the family of seminorms Q(Ω) = (µ_q,l)_(q,l)∈N2 defined by

µ_q,l(f) = sup

x∈Ω,|α|≤l

(1 +|x|)^q|∂^αf(x)|. LetR^′ be a regular subset ofR^N₊² and set

X_S^R′(Ω) =n

(f_ε)_ε ∈ S(Ω)^(0,1]

∃N ∈ R^′, ∀(q, l)∈N² µ_q,l(f_ε) = O

ε^−N(q,l)

asε→0o , N_S^R′(Ω) =n

(f_ε)_ε ∈ S(Ω)^(0,1]

∀m∈ R^′, ∀(q, l)∈N² µ_q,l(f_ε) = O

ε^m(q,l)

asε→0o .

As for lemma 2, we have N_S^R′(Ω) =NS(Ω), with N_S(Ω) =n

(f_ε)_ε ∈ S(Ω)^(0,1]

∀(q, l)∈N², ∀m∈N µ_q,l(f_ε) = O (ε^m) asε→0o .

Proposition 5

(i) For any regular subspace R^′ of R^N2

+, the functor Ω→ X_S^R′(Ω) defines a presheaf (it allows restrictions) of differential algebras over the ring X(C).

(ii) The functorNS : Ω→ NS(Ω)defines a presheaf of ideals of the presheaf X_S^R′(·).

(iii) For any regular subspaces R^′₁ and R^′₂ of R^N₊², with R^′₁ ⊂ R^′₂, the presheaf X^R

′ 1

S (Ω) is a subpresheaf of the presheaf X_S^R′2(Ω).

Proof.

(a) Algebraical properties.- Let us first prove that X_S^R′(Ω) is a subalgebra of S(Ω)^(0,1]. The proof thatX_S^R′(Ω) is a sublinear space of C^∞(Ω)^(0,1]goes along the same lines as in proposition

1. For the product, take (f_ε)_ε and (g_ε)_ε in X_S^R′(Ω). According to the definitions, there exist N_f ∈ R^′ and Ng ∈ R^′ such that

∀(q, l)∈N² µ_q,l(h_ε) = O

ε^−Nh(q,l)

asε→0, forh_ε=f_ε, g_ε. Consider (q, l)∈N² and α∈N^d with|α|=l. By Leibniz’ formula, we have

∀ε∈(0,1] ∂^α(f_εg_ε) =X

γ≤α

C_α^γ∂^γf_ε∂^α−γg_ε.

Thus

sup

x∈Ω

(1 +|x|)^q|∂^α(f_εg_ε) (x)| ≤X

γ≤α

C_α^γµ_q,|γ|(f_ε) µ_0,|α−γ|(f_ε), with, for allγ ≤α,µ_q,|γ|(fε)µ_0,|α−γ|(fε) = O ε^{−Nf(q,|γ|)−Ng(0,|α−γ|)}

asε→0. Since γ≤α, we get|γ|+|α−γ|=|α|=l. According to (10), there existsN ∈ R^′ such that, for allkand k^′≤k inN,N₁(q, k^′) +N₂(0, k−k^′)≤N(q, k). Then

sup

x∈Ω

|(1 +|x|)^q∂^α(f_εg_ε) (x)|= O

ε^−N(q,l)

asε→0.

Thusµ_q,l(fεgε) = O ε^−N(q,l)

asε→0 and (fεgε)_ε∈ X_S^R′(Ω).

The same kind of estimates shows also that NS(Ω) (or N_S^R′(Ω)) is an ideal ofX_S^R′(Ω).

(b)Presheaf properties.- Take Ω₁and Ω₂ two open subsets ofR^d, with Ω₁ ⊂Ω₂, (u_ε)_ε∈ X_S^R′(Ω₂) (resp. NS(Ω₂)). As u_ε ∈C^∞(Ω₂) for allε∈(0,1], it admits a restrictionu_ε|Ω1, and obviously

u_ε|Ω1

ε∈ X_S^R′(Ω₁) (resp. NS(Ω₂)). So, restrictions are well defined in X_S^R′(·) and NS(·).

Definition 5 The presheaf G_S^R′(·) =X_S^R′(·)/N_S(·) is called the presheaf of R^′-regular rapidly decreasing generalized functions.

As for the case ofG^R(·), the presheafG_S^R′(·) is a presheaf of differential algebras and a sheaf of modules over the factor ring C=X(C)/N(C).

Example 6 Taking R^′ =R^N₊², we obtain the presheaf of algebras of rapidly decreasing general- ized functions ([8], [22], [23]).

Notation 4 In the sequel, we shall note GS(Ω) (resp. XS(Ω)) instead of G^R

N2 +

S (Ω) (resp.

X^R

N2 +

S (Ω)). For all regular subset R^′ and (f_ε)_ε ∈ X_S^R′(Ω), [(f_ε)_ε]_S denotes its class in G_S^R′(Ω).

Example 7 Taking R^′ =B^′, we obtain the presheaf of G_S^∞ generalized functions or of regular rapidly decreasing generalized functions.

Set

NS∗(Ω) =n

(fε)_ε ∈C^∞(Ω)^(0,1] | ∀m∈N, ∀q∈N µq,0(fε) = O (ε^m) asε→0o

. (11) We have the exact analogue of theorems 1.2.25 and 1.2.27 of [10].

Lemma 6 If the open set Ωis a box, i.e. the product of dopen intervals of R(bounded or not) then N_S(Ω)is equal to N_S∗(Ω)∩ X_S(Ω).

Proof. We use similar technics as in the proof of Theorem 1.2.27 in [10]. It suffices to show the result for XS(Ω) (which is the biggest subspace of S(Ω)^(0,1] we may have to consider) and for real valued functions. We prove here that a derivative ∂_j =∂ /∂x_j (1 ≤j ≤d) satisfies the 0-order estimate of the definition ofNS∗ (11). The proof for higher order derivatives, which goes by induction, is left to the reader. Let (f_ε)_εbe in NS∗(Ω)∩ XS(Ω),q inNand minN. As (f_ε)_ε is in X_S(Ω), there exists N such that

∀q^′∈ {0, ..., q} sup

x∈Ω

(1 +|x|)^q′

∂_j²fε(x)

= O ε^−N

. (12)

As (f_ε)_ε is inNS∗(Ω), we get

∀q^′ ∈N sup

x∈Ω

(1 +|x|)^q′|fε(x)|= O ε^N+2m

. (13)

Since the open set Ω is a box, for εsufficiently small (but independent ofx), either the segment x, x+ε^N+me_j

or

x, x−ε^N+me_j

is included in Ω. Suppose it is

x, x+ε^N+me_j

. Taylor’s theorem gives the existence ofθ∈(0,1) such that

∂_jf_ε(x) = f_ε x+ε^N+me_j

−f_ε(x)

ε^−N−m−(1/2)∂_j²f_ε(x_θ)ε^N+m, x_θ =x+θε^N+me_j. This gives

(1 +|x|)^q|∂jf_ε(x)| ≤(1 +|x|)^q

f_ε x+ε^N+me_jε^−N−m

| {z }

∗

+ (1 +|x|)^q|fε(x)|ε^−N−m

| {z }

∗∗

+ ε^N+m(1 +|x|)^q∂_j²f_ε(x_θ)

| {z }

∗∗∗

.

From (13), we get immediately that (∗∗) is of order O (ε^m). For (∗), we have (1 +|x|)^q≤ 1 +

x+ε^N+me_j

+ε^N+mq

≤ Xq k=0

C^k_q 1 +

x+ε^N+me_j

^q−kε^(N+m)k.

Then

(1 +|x|)^q

f_ε x+ε^N+me_jε^−N−m

≤ Xq k=0

C^k_q 1 +

x+ε^N+mej

^q−k

fε x+ε^N+mejε^(N+m)(k−1), and (13) implies that (∗) is of order O (ε^m). Finally, the same method shows that (∗ ∗ ∗) is also of order O (ε^m).

3.2 Embeddings

3.2.1 The natural embeddings of S R^d

and O^′_C R^d

into GS R^d The embedding of S R^d

intoGS R^d

is done by the canonical injective map σ_S:S

R^d

→ G_S R^d

f 7→[(f_ε)_ε]_S withf_ε=f for all ε∈(0,1]. In fact, the image ofσ_S is included inG_S^R′ R^d

for any regular subset of R^′ ⊂R^N2

+ .

For the embedding of O_C^′ R^d

, we consider ρ∈ S R^d

which satisfies Rρ(x) dx= 1, R

x^mρ(x) dx= 0 for allm∈N^d\ {0} (14) Set

∀ε∈(0,1], ∀x∈R^d ρ_ε(x) = 1 ε^dρx

ε

. (15)

Note that, contrary to the case of the embedding ofD^′(Ω) inG(Ω), we don’t need an additional cutoff.

Theorem 7 The map

ι_S :O_C^′ R^d

→ GS

R^d

u7→[(u∗ρ_ε)_ε]_S (16) is a linear embedding which commutes with partial derivatives.

Proof. Takeu∈ O^′_C R^d

. First, asu∈ O_C^′ R^d

andρ_ε ∈ S R^d

,u_ε∗ρ_εis inS R^d for all ε∈(0,1]. (This result is classical: Our proof, with a slight adaptation, shows also this result.)

Consider q∈N. Using the structure of elements ofO_C^′ R^d

[24], we can find a finite family (fj)_1≤j≤l(q) of continuous functions such that (1 +|x|)^qfj is bounded (for 1 ≤ j ≤ l(q)), and (α_j)_1≤j≤l(q)∈ N^dl(q)

such thatu=Pl(q)

j=1∂^αjf_j. In order to simplify notations, we shall suppose that this family is reduced to one elementf, that is u=∂^αf. Take nowβ∈N^d. We have

∀x∈R^d ∂^β(u∗ρ_ε) (x) =∂^β(∂^αf∗ρ_ε) (x) =

f∗∂^α+β(ρ_ε) (x),

= Z

f(x−y)∂^α+β(ρε) (y) dy.

Asρ_ε(y) =ε^−dρ(y/ε), we have∂^α+β(ρ_ε) (y) =ε^{−d−|α|−|β|} ∂^α+βρ

(y/ε) and

∀x∈R^d ∂^β(u∗ρ_ε) (x) =ε^{−|α|−|β|}

Z

f(x−εv)∂^α+βρ(v) dv.

On one hand, there exists a constant C₁ >0 such that

∀(x, v)∈R^2d |f(x−εv)| ≤C₁(1 +|x−εv|)^−q. On the other hand, as ρ is rapidly decreasing, there existsC2>0 such that

∂^α+βρ(v)≤C₂(1 +|v|)^−q−d−1.

These estimates imply the existence of a constant C₃ such that, for allx∈R^d,

∂^β(u∗ρ_ε) (x)

≤C₃ε^{−|α|−|β|}

Z

((1 +|x−εv|) (1 +|v|))^−q(1 +|v|)^−d−1 dv.

We have (1 +|x−εv|) ≥ (1 +||x| −ε|v||) and a short study of the family of functions φ_|x|,ε : t7→(1 +||x| −εt|) (1 +t) for positivetshows that φ_|x|,ε(t)≥1 +|x|. Consequently

∀x∈R^d

∂^β(u∗ρε) (x)

≤C3ε^{−|α|−|β|}(1 +|x|)^−q Z

(1 +|v|)^−d−1 dv

≤C₄ε^{−|α|−|β|}(1 +|x|)^−q (C₄ positive constant).

It follows thatµ_q,l(u∗ρ_ε) = O ε^−N(q,l)

asε→0 with N(q, l) =|α|+l. (α may depends onq) This shows that (u∗ρ_ε)_ε belongs toXS R^d

. Finally, it is clear that (u∗ρ_ε)_ε ∈ NS R^d

implies that u_ε∗ρ_ε → 0 in S^′, as ε → 0. As u_ε∗ρ_ε→u asε→0,u is therefore null.

Theorem 8 We have: ι_S|^S(^Rd) =σ_S.

Proof. We shall prove this assertion in the cased= 1, the general case only differs by more complicate algebraic expressions.

Let f be inS(R) and set ∆ =ι_S(f)−σ(f).One representative of ∆ is given by

∆ε:R→ F(C^∞(R)) y 7→(f∗ρε) (y)−f(y) = Z

f(y−x)ρε(x) dx−f(y).

Using the fact that R

ρ_ε(x) dx= 1, we get

∀y∈R ∆_ε(y) = Z

(f(y−x)−f(y))ρ_ε(x) dx.

Letm be an integer. Taylor’s formula gives

∀(x, y)∈R² f(y−x)−f(y) = Xm j=1

(−x)^j

j! f⁽ⁱ⁾(y) + (−x)^m+1 m!

Z ₁

0

f^(m+1)(y−ux) (1−u)^m du, and, for ally∈R,

∆_ε(y) = Xm j=1

(−1)^j

j! f⁽ⁱ⁾(y) Z

xⁱρ_ε(x) dx+

Z (−x)^m+1 m!

Z 1

0

f^(m+1)(y−ux) (1−u)^mdu

ρ_ε(x) dx

| {z }

Rε(m,y)

.

According to the choice of mollifiers, we have R

xⁱρε(x) dx= 0 for ε∈(0,1] andj ∈ {0, . . . , m}

and consequently ∆_ε(y) =R_ε(m, y).

Setting v=x/ε, we get

∀y∈R ∆ε(y) =ε^m+1

Z (−v)^m+1 m!

Z 1

0

f^(m+1)(y−εuv) (1−u)^m du

ρ(v) dv.

Consider q ∈N. As ρ (resp. f) is inS(R), we get a constantC₁ >0 (resp. C₂ >0) such that

|ρ(t)| ≤C₁(1 +|t|)^−m−3−q (respectively

f^(m+1)(t)

≤C₂(1 +|t|)^−q) for all t∈R. Thus, there exists a constant C₃>0 such that

∀y∈R |∆ε(y)| ≤C₃ε^m+1 m!

Z

|v|^m+1(1 +|v|)^−m−3 Z 1

0

(1 +|y−εuv|)^−q(1 +|v|)^−qdu

dv.

Using the same technic as in the proof of theorem 7, we obtain that the underlined term is less than (1 +|y|)^−q, for all ε∈(0,1], u∈[0,1] andv∈R. This implies the existence of a constant C₄>0 such that

∀y∈R |∆ε(y)| ≤C₄ε^m+1(1 +|y|)^−q. As (∆_ε)_ε ∈ XS(R) and sup_y∈R(1 +|y|)^q|∆ε(y)| = O ε^m+1

for all m > 0, we can conclude directly (without estimating the derivatives) that (∆_ε)_ε∈ NS(R), by using lemma 6.

Consider

R^′₁ =n

N^′ ∈R^N2

+

∃N ∈ R1 N^′ = 1⊗No

, (17)

where R₁ is defined by (6). (This amounts to: N ∈ R^′₁ iff there exists b ∈ R₊ such that N(q, l)≤l+b.) The set R^′₁ is clearly regular and we note

G_S⁽¹⁾ R^d

=X_S^R′1 R^d

/NS

R^d

. (18)

Proposition 9 The image of O^′_M R^d

byιS is included in G_S⁽¹⁾ R^d . Proof. Letube inO_M^′ R^d

. According to the characterization of elements ofO_M^′ R^d [11], there exists a finite family (f_j)_1≤j≤l of rapidly decreasing continuous functions and (α_j)_1≤j≤l ∈

N^dl

such thatu=P_l

j=1∂^αjf_j. In order to simplify, we shall suppose that this family is reduced to one elementf, that is u=∂^αf. For β∈N^d, the same estimates as in proof of theorem 7 lead to the following property

∀q∈N, ∃Cq>0, ∀x∈R^d (1 +|x|)^q

∂^β(u∗ρ_ε) (x)

≤C_qε^{−|α|−|β|},

since, in the present case, f is rapidly decreasing. (The only difference is here that f and α do not depend on the chosen integerq.) Then

µ_q,l(u∗ρ_ε)≤C_qε^−l−|α|.

Our claim follows, with N^′(q, l) =l+|α|, where|α|only depends on u.

We can summarize theorems 7 and 8 and proposition 9 in the following commutative diagram in which all arrows are embeddings (compare with diagram (7)):

S R^d

−→ O^′_M R^d

−→ O_C^′ R^d ցσ_S ↓ι_S ↓ι_S

G⁽¹⁾_S R^d

−→ GS R^d .

(19)

3.2.2 Embedding of S^′ R^d

into G_S R^d In order to embedS^′ R^d

into an algebra playing the role ofG R^d

forD^′ R^d

, a spaceG_τ R^d of tempered generalized functions is often introduced (see [2], [10]). This spaceGτ R^d

does not fit in the general scheme of construction of Colombeau type algebras, since the growth estimates for Gτ R^d

are not based on the natural topology of the spaceOM R^d

, which replaces C^∞ R^d in this case. Although it is possible to construct a spaceGτ R^d

based on the topology ofOM R^d , we don’t need it in the sequel and we only show thatS^′ R^d

can be embedded intoGS R^d by means of a cutoff of the embeddingιS :O^′_C R^d

→ GS R^d . Proposition 10 With the notations (14) and (15), the map

ι_S^′ :S^′ R^d

→ GS

R^d

, u7→[((u∗ρε)ρbε)_ε]_S is a linear embedding.

Proof. Foru∈ S^′ R^d

, the net ((u∗ρε))_ε belongs to the space Xτ

R^d

=n

(f_ε)_ε∈C^∞(Ω)^(0,1]

∀α∈N^d, ∃q∈N, ∃N ∈N µ_−q,α(f_ε) = O ε^−N

asε→0o . (See, for example, theorem 1.2.27 in [10]: The proof is similar to the one of theorem 7.) A straightforward calculus shows thatXS is an ideal ofXτ. It follows that ((u∗ρ_ε)ρb_ε)_ε∈ XS R^d since (ρbε)_ε∈ XS R^d

. Then, the map ι_S^′ is well defined.

Note that, for u ∈ S^′ R^d

, we have u∗ρ_ε ^S

′

−→ u as ε→ 0 and, therefore, u∗ρ_ε ^D

′

−→ u as ε→0. Asρb_ε= 1 on a compact setK_ε such that K_ε →R^d asε→0, we get that

(u∗ρ_ε)ρb_ε−→^D′ u asε→0. (20) Finally, takeu∈ S^′ R^d

withι_S^′(u) = 0, that is ((u∗ρ_ε)ρb_ε)_ε∈ NS R^d

. We have (u∗ρ_ε)ρb_ε−→^S′ 0 and, consequently, (u∗ρ_ε)ρb_ε ^D

′

−→0. Thus,u= 0 according to (20).

4 Fourier transform and exchange theorem

4.1 Fourier transform in GS R^d

The Fourier transformF is a continuous linear map fromS R^d

toS R^d

. According to propo- sition 3.2. of [7], F has a canonical extension FS from GS to GS defined by

GS

R^d

→ GS

R^d

u7→ub=

x7→

Z

e^−ixξu_ε(ξ) dξ

ε

S

, (21)

where (u_ε)_ε∈ XS R^d

is any representative ofu.

The proof of this result uses mainly the continuity ofF. More precisely, any linear continuous map is continuously moderate in the sense of [7] and, therefore, admits such a canonical extension.

(The proof of lemma 15 below, with a slight adaptation, shows directly this result for the case of F.)

Definition 6 The map FS defined by (21) is called the Fourier transform inGS.

In the same way, we can define F_S⁻¹ by GS

R^d

→ GS

R^d

u7→

x7→(2π)^−d Z

e^ixξu_ε(ξ) dξ

ε

S

, (22)

where (u_ε)_ε∈ XS R^d

is any representative ofu.

Theorem 11 FS :GS R^d

→ GS R^d

is a one to one linear map, whose inverse is F_S⁻¹. Proof. Letu be in G_S R^d

and (u_ε)_ε ∈ X_S R^d

be one of its representative. As represen- tative ofF F⁻¹(u)

, we can choose (eu_ε)_ε defined by

∀ε∈(0,1], ∀x∈R^d ue_ε(x) = (2π)^−d Z

e^ixξbu_ε(ξ) dξ.

Since the Fourier transform is an isomorphism in S R^d

, we get ue_ε=u_ε, for all ε∈(0,1], and F_S F_S⁻¹(u)

= [(u_ε(x))_ε]_S =u.

4.2 Regular sub presheaves of GS(·)

We introduce here some regular sub presheaves ofGS(·) needed for our further microlocal anal- ysis.

Let Rbe a regular subset ofR^N₊ and set Ru =n

N^′ ∈R^N₊²

∃N ∈ R N^′ = 1⊗No

; R∂ =n

N^′ ∈R^N₊² | ∃N ∈ R N =N ⊗1o . In other words, N^′ ∈ Ru (resp. R∂) iff there exists N ∈ R such that N^′(q, l) = N(l) (resp.

N^′(q, l) =N(q)) or, equivalently, iff N only depends (in aR-regular way) ofl (resp. q).

Notation 5 We shall write, with a slight abuse,Ru ={1} ⊗ R, R_∂ =R ⊗ {1}. Obviously, R_u (resp. R_∂) is a regular subset ofR^N₊².

Example 8 Take R=R^N

+. We set: G_S^u(·) =G_S^Ru(·) (resp. G_S^∂(·) =G^R_S^∂(·)). In this case, we have R_u={1} ⊗R^N

+ (resp. R_∂ =R^N

+⊗ {1}).

The elements of G_S^u(Ω) (Ω open subset of R^d) have uniform growth bounds with respect to the regularization parameter ε for all factors (1 +|x|)^q. For G_S^∂(Ω), those bounds are uniform for all derivatives. ForG_S^∞(Ω), introduced in example 7, the uniformity is global, in some sense stronger than the G^∞-regularity considered for the algebra G. (In this last case, the uniformity is not required with respect to the compact sets.)

We have the obvious inclusions (for Rregular subset of R^N₊):

G_S^R∂(Ω) −→ G_S^∂(Ω) G_S^∞(Ω) ր

ց

ց

ր GS(Ω) G_S^Ru(Ω) −→ G_S^u(Ω)

. (23)

Example 9 The algebra G_S⁽¹⁾ R^d

=X_S^R′1 R^d

/NS R^d

introduced in (18) for the embedding of O_M^′ R^d

into GS R^d

(proposition 9) can be written as G_S^(R1)u, with

R₁ =n

N ∈R^N

+ | ∃b∈R₊, ∀l∈R N(l)≤l+bo .

As a first illustration of the properties of these spaces, we can show the existence of a canonical embedding of algebras of compactly supported generalized functions into particular spaces of rapidly decreasing generalized functions.

Lemma 12 Let R be a regular subset of R^N

+ and u be in G_C^R(Ω) (Ω open subset of R^d), with (u_ε)_ε a representative of u. Let κ be inD(Ω), with 0≤κ ≤1 and κ ≡1 on a neighborhood of suppu.Then(κu_ε)_ε belongs to X_S^Ru R^d

and[(κu_ε)_ε]_S only depends onu, but not on(u_ε)_ε and κ.

Proof. We first show that (κu_ε)_εis inX_S^Ru R^d

and then the independence with respect to the representation.

(a) There exists a compact setK⊂Ω such that, for allε∈(0,1], suppκu_ε⊂K. It follows that κuε is compactly supported and therefore rapidly decreasing. Moreover

∀(q, l)∈N², ∀ε∈(0,1] µ_q,l(κu_ε)≤sup_x∈K(1 +|x|)^qp_K,l(κu_ε)≤C_K,qp_K,l(κu_ε), C_K,q >0.

Thus, (κu_ε)_ε belongs to X_S^Ru R^d

. Indeed, by using Leibniz’ formula for estimatingp_K,l(κu_ε), we can find a constant C_K,q,κ>0 such that

∀(q, l)∈N², ∀ε∈(0,1] µ_q,l(κu_ε)≤C_K,q,κp_K,l(u_ε). (24) (b) Let (ue_ε)_ε be another representative of u and eκ be in D(Ω), with 0 ≤eκ ≤ 1 andκe = 1 on a neighborhood of suppeu. Let L be a compact set such that suppκu_ε ∪suppκeue_ε ⊂ L ⊂ Ω.

According to the previous estimate, we have

∀(q, l)∈N², ∀ε∈(0,1] µ_q,l(κu_ε−eκeu_ε)≤µ_q,l((κ−eκ)u_ε) +µ_q,l(eκ(u_ε−ue_ε))

≤CL,qp_L,l((κ−eκ)uε) +CL,qp_L,l(eκ(uε−ueε)). As κ = eκ on a closed neighborhood V of suppu, it follows that p_V,l((κ−eκ)u_ε) = 0. More- over, for all m ∈ N, p_L\V,l((κ−eκ)u_ε) = O (ε^m) as ε → 0, since (L\V)∩suppu = ∅. Then p_L,l((κ−eκ)u_ε) = O (ε^m) asε→ 0. As [(u_ε)_ε] = [(ue_ε)_ε], we have p_L,l(eκ(u_ε−ue_ε)) = O (ε^m) as ε→0. Thenµ_q,l(κu_ε−eκue_ε) = O (ε^m) and [(κu_ε)_ε]_S = [(eκeu)_ε]_S.

From lemma 12, we deduce easily the following proposition: