New York Journal of Mathematics
New York J. Math. 12(2006)275–318.
Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and
noncharacteristic regularity
Claudia Garetto
Abstract. We introduce different notions of wave front set for the functionals in the dual of the Colombeau algebraGc(Ω) providing a way to measure theG and theG∞- regularity inL(Gc(Ω),C). For the smaller family of functionals having a “basic structure” we obtain a Fourier transform-characterization for this type of generalized wave front sets and results of noncharacteristicGand G∞-regularity.
Contents
0. Introduction 276
0.1. Topology and duality theory 278
1. Duality theory in the Colombeau context: basic maps and functionals 282 1.1. Action of basic functionals on generalized functions in two
variables 284
1.2. Composition of a basic functional with an integral operator 287 1.3. Convolution of Colombeau generalized functions and functionals 288 1.4. Fourier transform in the dualL(GS(Rn),C) 294 2. Generalized pseudodifferential operators onL(Gc(Ω),C) andL(G(Ω),C) 295 3. G-wave front set andG∞-wave front set of a functional inL(Gc(Ω),C) 303 3.1. The generalized wave front sets WFG(T) and WFG∞(T) 304 3.2. Fourier transform-characterization of WFG(T) and WFG∞(T)
whenT is a basic functional 306
4. NoncharacteristicG andG∞-regularity 314
References 315
Received February 20, 2006.
Mathematics Subject Classification. 46F30, 35A18, 46A20, 35D10.
Key words and phrases. Algebras of generalized functions, wave front sets, duality theory.
Supported by FWF (Austria), grant P16820-N04 and TWF (Tyrol), grant UNI-0404/305.
ISSN 1076-9803/06
275
0. Introduction
The past decade has seen the emergence of a differential-algebraic theory of generalized functions of Colombeau type [2, 4, 5, 22, 41, 44, 51] that answered a wealth of questions on solutions to linear and nonlinear partial differential equations involving nonsmooth coefficients and strongly singular data [3, 6, 38, 43]. Interesting results were obtained in Lie group invariance of generalized functions [8, 31, 46, 48], nonlinear hyperbolic equations with generalized function data [7, 39, 40, 42, 45, 47, 49, 50], distributional metrics in general relativity [32, 33, 34, 35, 36, 37], propagation of strong singularities in linear hyperbolic equations with discontinuous coefficients [27, 28, 38, 43], microlocal analysis, pseudodifferential operators and Fourier integral operators with nonsmooth symbols [13, 18, 19, 20, 25, 26, 29, 30].
The elements of Colombeau algebras are given by classes of regularizations, i.e., sequences of smooth functions, subject to asymptotic conditions with respect to the regularization parameterε. Distributions are embedded via convolution with a mollifierϕε(x) =ε−nϕ(ε−1x) and smooth functions are embedded as a subalgebra, where the multiplication agrees with the classical ones. It follows that Colombeau algebras can be considered as a unifying and well-structured framework for deal- ing with equations of the type Aε(x, D)uε(x) =fε(x), ε > 0, which arise in the study of singularly perturbed partial differential equations, in semiclassical analy- sis, or when regularizing partial differential operators with nonsmooth coefficients or pseudodifferential operators with irregular symbols [18, 27, 28, 25]. By adopt- ing the point of view of asymptotic analysis, the regularity of the right-hand side and of the solution as well as the mapping properties of the operator are described by means of asymptotic estimates in terms of the parameter ε → 0. This leads to introducing different scales of growth in the parameter ε in order to measure different kinds of regularity (e.g., moderate nets, negligible nets, logarithmic and slow scale nets [18, 29]). A strong motivation for the use of Colombeau techniques with refined scales comes from important models in acoustic wave propagation, as illustrated in [1, 5, 21, 43].
Some “key technologies” for the regularity theory of partial differential equa- tions in the Colombeau context have been developed in [13, 17, 18, 19]. They consist in a complete theory of generalized pseudodifferential operators (includ- ing a parametrix-construction for operators with generalized hypoelliptic symbol) [13, 18] and the application of those pseudodifferential techniques to the microlo- cal analysis of generalized functions [19]. Particular attention has been given to the dual of a Colombeau algebra which plays a main role in the kernel theory for generalized pseudodifferential operators [13, 18]. It is now natural to extend the pseudodifferential operator’s action to the dual and to shift the microlocal investiga- tions from the level of generalized functions to the level ofC-linear functionals. This will require notions of local and microlocal regularity in the dual of a Colombeau algebra and suitable ways of measuring such kinds of regularity. Microlocal analysis is essential for a full understanding of the generalized pseudodifferential operator’s action and propagation of singularities and has to be developed in the dual context since the kernels of such operators are not always Colombeau generalized functions but functionals.
The aim of this paper is to provide tools of microlocal analysis suited to inves- tigate the dual of a Colombeau algebra. Based on the duality theory developed
within the Colombeau framework in [14, 15] it extends and adapts the microlocal results for Colombeau generalized functions stated in [19]. In the usual Colombeau context a generalized functionu∈ G(Ω) is said to be regular if it belongs to the sub- algebraG∞(Ω). This allows to set up a regularity theory forG(Ω) which is coherent with the usual concept of regularity for distributions sinceG∞(Ω)∩D(Ω) =C∞(Ω).
In [9, 41] a notion of generalized wave front set is defined foru∈ G(Ω) as a G∞- wave front set. It means that the conic regions of “microlocal regularity” we deal with in the cotangent space are regions ofG∞-regularity. Coming now to the dual L(Gc(Ω),C), i.e., the space of all continuous and C-linear functionals on Gc(Ω), where C is the ring of complex generalized numbers, by continuous embedding it contains bothG∞(Ω) andG(Ω). As a consequence, two levels of regularity concern a functional inL(Gc(Ω),C): the regularity with respect toG(Ω) and the regularity with respect toG∞(Ω). In this paper in order to measure such different kinds of reg- ularity ofT ∈ L(Gc(Ω),C) we introduce the notions ofG-wave front set (WFG(T)) andG∞-wave front set (WFG∞(T)).
Inspired by [19] and making use of the theory of pseudodifferential operators with generalized symbols elaborated in [18, 19], WFG(T) and WFG∞(T) are defined as intersection of suitable regions of generalized nonellipticity of those pseudodiffer- ential operators which mapT in G(Ω) andG∞(Ω) respectively. Core of the paper is a Fourier transform-characterization of WFG(T) and WFG∞(T) as in [12, Theo- rem 8.56] which consists in the direct investigation of the properties of the Fourier transform ofT after multiplication by a suitable cut-off function. For this purpose special spaces of generalized functions with rapidly decreasing behavior on a conic subset of Rn are introduced and among all the functionals of L(Gc(Ω),C) we re- strict to consider those elements which have a “basic structure”. More precisely we assumeT ∈ L(Gc(Ω),C) is defined by a net of distributions (Tε)ε which fulfills a continuity assumption uniform with respect to ε(Definition 1.3) and the equality T u= [(Tεuε)ε]∈C for all u= [(uε)ε]∈ Gc(Ω). Even though theG-wave front set and theG∞-wave front set can be defined on any functional of the dualL(Gc(Ω),C), the main theorems and propositions presented here are proven to be valid for ba- sic functionals. In addition, all the results of microlocal regularity have a double version: theG-version and theG∞-version.
We now describe in detail the contents of the sections.
Section 1 provides the needed theoretical background of basic functionals and refers for topological issues to [14, 15]. After the first definitions and basic prop- erties, the action of a basic functional on a Colombeau generalized function in two variables is investigated in Subsection 1.1. Together with some results on the com- position of a basic functional with an integral operator in Subsection 1.2, it gives the essential tools for dealing with the convolution of Colombeau generalized func- tions and functionals in Subsection 1.3. The algebras of generalized functions here involved areGc(Ω),G(Ω) andGS(Rn) while the functionals are elements of the duals L(Gc(Ω),C),L(G(Ω),C) andL(GS(Rn),C). A regularization of basic functionals is obtained via convolution with a generalized mollifier. Finally in Subsection 1.4 we extends the natural notion of Fourier transform onGS(Rn) to the dualL(GS(Rn),C) and we study the Fourier transform of a basic functional inL(G(Ω),C).
In the recent Colombeau literature a pseudodifferential operator with general- ized symbol is a C-linear continuous operator which mapsGc(Ω) into G(Ω). Sec- tion 2 extends the action of such generalized pseudodifferential operator to the dualsL(Gc(Ω),C) andL(G(Ω),C). The extension procedure is obtained via trans- position and gives interesting mapping properties concerning the subspaces of basic functionals. A variety of symbols (and amplitudes) is considered: generalized sym- bols of order m and type (ρ, δ), regular symbols, slow scale symbols, generalized symbols of order −∞, regular symbols of order −∞ and generalized symbols of refined order (see [18, 19]). A connection is shown to exist between G-regularity, generalized symbols of order −∞ and basic functionals as well as between G∞- regularity, regular symbols of order−∞and basic functionals. More precisely we prove that R is an integral operator with kernel in G(Ω×Ω) if and only if it is a pseudodifferential operator with generalized amplitude of order−∞and thatR is G-regularizing on the basic functionals ofL(G(Ω),C), in the sense thatRT ∈ G(Ω) if T ∈ L(G(Ω),C) is basic. Analogously R is an integral operator with kernel in G∞(Ω×Ω) if and only if it is a pseudodifferential operator with regular amplitude of order −∞and it isG∞-regularizing on the basic functionals of L(G(Ω),C). A G-pseudolocality property is obtained for properly supported pseudodifferential op- erators with generalized symbols while aG∞-pseudolocality property is valid when the symbols are regular. Section 2 ends by adapting the result of G∞-regularity in [18] for pseudodifferential operators with generalized hypoelliptic symbols to the dual context of basic functionals.
A G-microlocal analysis and a G∞-microlocal analysis for the dualL(Gc(Ω),C) are settled and developed in Section 3. The additional assumption of basic structure on the functionalT is employed in Subsection 3.1 in proving that the projections on Ω of WFG(T) and WFG∞(T) coincide with the G-singular support and the G∞-singular support of T respectively. The Fourier transform-characterizations of WFG(T) and WFG∞(T) are the result of the G and the G∞-microlocal investiga- tions of pseudodifferential operators elaborated throughout Subsection 3.2 in the dual L(Gc(Ω),C). Concerning the notion of slow scale micro-ellipticity here em- ployed this has been already introduced in [19] while the concept of generalized microsupport of a generalized symbol in [19, Definition 3.1] is transformed into G-microsupport andG∞-microsupport (Definition 3.6).
Section 4 concludes the paper with a theorem on noncharacteristicG and G∞- regularity for pseudodifferential operators with slow scale symbols when they act on basic functionals of L(Gc(Ω),C). This is an extension and adaptation to the dualL(Gc(Ω),C) of Theorem 4.1 in [19].
For the advantage of the reader we recall in the sequel some topological issues discussed in [14, 15] and we fix some notations.
0.1. Notions of topology and duality theory for spaces of Colombeau type. A topological investigation into spaces of generalized functions of Colombeau type has been initiated in [14, 15, 17, 52, 53, 54] setting the foundations of duality theory in the recent work on topological and locally convex topologicalC-modules [14, 15, 17]. Without presenting the technical details of this theoretical construc- tion, we recall that a suitable adaptation of the classical notion of seminorm, called ultra-pseudo-seminorm [14, Definition 1.8], allows us to characterize a locally convex
C-linear topology as a topology determined by a family of ultra-pseudo-seminorms.
The most common Colombeau algebras can be introduced as C-modules of gen- eralized functions based on a locally convex topological vector space E. Such a C-moduleGE is the quotient of the set
ME:=
(uε)ε∈E(0,1] : ∀i∈I ∃N ∈N pi(uε) =O(ε−N) asε→0 (0.1)
ofE-moderate nets with respect to the set NE:=
(uε)ε∈E(0,1] : ∀i∈I ∀q∈N pi(uε) =O(εq) asε→0 , (0.2)
of E-negligible nets, and it is naturally endowed with a locally convex C-linear topology usually called sharp topology in [41, 52, 53, 54]. Given a family of semi- norms{pi}i∈I onE, the sharp topology on GE is determined by the ultra-pseudo- seminormsPi(u) := e−vpi(u), where vpi is thevaluation
vpi([(uε)ε]) := vpi((uε)ε) := sup{b∈R: pi(uε) =O(εb) as ε→0}
(see [14, Subsection 3.1] for further explanations). Note that valuations and ultra- pseudo-seminorms are defined on ME and extended to the factor space GE in a second time. It is clear that the ring C of complex generalized numbers is an example ofGE-space obtained by choosingE=C. The valuation and ultra-pseudo- norm onC obtained as above by means of the absolute value onCare denoted by vCand| · |e respectively.
As proved in [14, Corollary 1.17] for an arbitrary locally convex topologicalC- module (G,{Qj}j∈J), a C-linear map T :GE → G is continuous if and only if for allj∈J there exists a finite subsetI0⊆I and a constant C >0 such that for all u∈ GE
Qj(T u)≤Cmax
i∈I0Pi(u).
The Colombeau algebras G(Ω), Gc(Ω), GS(Rn). The Colombeau algebraG(Ω) is the C-module of GE-type given by E = E(Ω). Equipped with the family of seminorms pK,i(f) = supx∈K,|α|≤i|∂αf(x)| where K Ω, the space E(Ω) induces on G(Ω) a metrizable and complete locally convex C-linear topology which is de- termined by the ultra-pseudo-seminormsPK,i(u) = e−vpK,i(u). For coherence with some well-established notations in Colombeau theory we write ME(Ω) = EM(Ω) andNE(Ω)=N(Ω).
The Colombeau algebraGc(Ω) of generalized functions with compact support is topologized by means of a strict inductive limit procedure. More precisely, setting GK(Ω) :={u∈ Gc(Ω) : suppu⊆K} forKΩ,Gc(Ω) is the strict inductive limit of the sequence (GKn(Ω))n∈N, where (Kn)n∈Nis an exhausting sequence of compact subsets of Ω such thatKn⊆Kn+1. We recall that the spaceGK(Ω) is endowed with the topology induced byGDK(Ω)whereK is a compact subset containingKin its interior. In detail we consider onGK(Ω) the ultra-pseudo-seminormsPGK(Ω),n(u) = e−vK,n(u). Note that the valuation vK,n(u) := vp
K,n(u) is independent of the choice of K when acts on GK(Ω). As observed in [17, Subsection 1.2.2] the Colombeau algebra Gc(Ω) is isomorphic to the factor space Ec,M(Ω)/Nc(Ω) where Ec,M(Ω) and Nc(Ω) are obtained by intersecting EM(Ω) and N(Ω) with ∪KΩDK(Ω)(0,1]
respectively.
The Colombeau algebra GS(Rn) of generalized functions based on S(Rn) is obtained as a GE-module by choosing E = S(Rn). It is a Fr´echet C-module according to the topology of the ultra-pseudo-seminormsPh(u) = e−vph(u), where ph(f) = supx∈Rn,|α|≤h(1 +|x|)h|∂αf(x)|, f ∈S(Rn), h∈N. In the course of the paper we will use the notationsES(Rn) andNS(Rn) for the spaces of netsMS(Rn)
andNS(Rn)respectively.
The regular Colombeau algebras G∞(Ω), G∞c (Ω), GS∞(Rn). Given a locally convex topological space (E,{pi}i∈I) theC-moduleGE∞of regular generalized func- tions based onE is defined as the quotientM∞E/NE, where
M∞E :=
(uε)ε∈E(0,1]: ∃N ∈N∀i∈I pi(uε) =O(ε−N) asε→0 , the set ofE-regular nets. The moderateness properties ofM∞E allows to define the valuation
vE∞((uε)ε) := sup{b∈R: ∀i∈I pi(uε) =O(εb) asε→0}
which extends toGE∞ and leads to the ultra-pseudo-normPE∞(u) := e−v∞E(u). This topological model is employed in endowing the Colombeau algebrasG∞(Ω),Gc∞(Ω), GS∞(Rn) andGτ∞(Rn) with a locally convexC-linear topology.
We begin by recalling that G∞(Ω) is the subalgebra of all elements u of G(Ω) having a representative (uε)ε belonging to the set
EM∞(Ω) :=
(uε)ε∈ E[Ω] : ∀KΩ∃N ∈N∀α∈Nn sup
x∈K|∂αuε(x)|=O(ε−N) asε→0
. G∞(Ω) can be seen as the intersection ∩KΩG∞(K), where G∞(K) is the space of all u ∈ G(Ω) having a representative (uε)ε satisfying the condition: ∃N ∈ N
∀α∈Nn, supx∈K|∂αuε(x)|=O(ε−N). The ultra-pseudo-seminorms PG∞(K)(u) := e−vG∞(K), where
vG∞(K):= sup
b∈R: ∀α∈Nn sup
x∈K|∂αuε(x)|=O(εb)
, equipG∞(Ω) with the topological structure of a Fr´echetC-module.
The algebraGc∞(Ω) is the intersection of G∞(Ω) with Gc(Ω). On G∞K(Ω) :={u∈ G∞(Ω) : suppu⊆KΩ}
we define the ultra-pseudo-normPGK∞(Ω)(u) = e−v∞K(u) where v∞K(u) := v∞D
K(Ω)(u) and K is any compact set containing K in its interior. At this point, given an exhausting sequence (Kn)n of compact subsets of Ω, the strict inductive limit pro- cedure determines a complete and separated locally convex C-linear topology on Gc∞(Ω) =∪nGK∞n(Ω). ClearlyGc∞(Ω) is isomorphic to Ec∞,M(Ω)/Nc(Ω), where
Ec∞,M(Ω) :=EM∞(Ω)∩
∪KΩDK(Ω)(0,1]
.
FinallyGS∞(Rn) is theC-module of regular generalized functions based onE= S(Rn). In coherence with the notations already in use we setM∞S(Ω) = ES∞(Ω) for any open subset Ω ofRn.
The Colombeau algebras of tempered generalized functions Gτ(Rn) and Gτ∞(Rn). The Colombeau algebra of tempered generalized functionsGτ(Rn) is de- fined asEτ(Rn)/Nτ(Rn), whereEτ(Rn) is the space
(uε)ε∈ OM(Rn)(0,1]: ∀α∈Nn∃N ∈N sup
x∈Rn(1 +|x|)−N|∂αuε(x)|=O(ε−N) asε→0 ofτ-moderate nets andNτ(Rn) is the space
(uε)ε∈ OM(Rn)(0,1]: ∀α∈Nn∃N ∈N∀q∈N sup
x∈Rn(1 +|x|)−N|∂αuε(x)|=O(εq) asε→0
ofτ-negligible nets. The subalgebraG∞τ (Rn) of regular and tempered generalized functions is the quotient Eτ∞(Rn)/Nτ(Rn), whereEτ∞(Rn) is the set of all (uε)ε∈ OM(Rn)(0,1] satisfying the following condition:
∃N∈N∀α∈Nn∃M ∈N sup
x∈Rn(1 +|x|)−M|∂αuε(x)|=O(ε−N). The topological duals L(Gc(Ω),C), L(Gc(Ω),C), L(GS(Rn),C). Throughout the paper the topological duals L(Gc(Ω),C), L(Gc(Ω),C), L(GS(Rn),C) are en- dowed with the corresponding topologies of uniform convergence on bounded sub- sets. These topologies, denoted by βb(L(Gc(Ω),C),Gc(Ω)), βb(L(G(Ω),C),G(Ω)) andβb(L(GS(Rn),C),GS(Rn)), are determined by the ultra-pseudo-seminorms
PB(T) := sup
u∈B|T u|e
with B varying in the family of all bounded subsets of Gc(Ω), G(Ω) and GS(Rn) respectively. As in the classical functional analysis a subsetB of a locally convex topological C-module (G,{Pi}i∈I) is bounded if and only if every ultra-pseudo- seminormPiis bounded onB, i.e., supu∈BPi(u)<∞. With respect to the topolo- gies collected in this subsection and the topology on Gτ(Rn) introduced in [14, Example 3.9] we have that the following chains of inclusions
G∞(Ω) ⊆ G(Ω) ⊆ L(Gc(Ω),C), Gc∞(Ω) ⊆ Gc(Ω) ⊆ L(G(Ω),C),
GS∞(Rn) ⊆ GS(Rn) ⊆ Gτ(Rn) ⊆ L(GS(Rn),C)
are continuous [15, Theorems 3.1, 3.8]. Moreover Ω→ L(Gc(Ω),C) is a sheaf and the dual L(G(Ω),C) can be identified with the set of functionals in L(Gc(Ω),C) having compact support [15, Theorem 1.2].
1. Duality theory in the Colombeau context: basic maps and functionals
This section is devoted to maps and functionals defined onC-modules of Colom- beau type. Before considering topics more related to the duals of the Colombeau algebrasGc(Ω),G(Ω) andGS(Rn) in Subsections1.1, 1.2, 1.3 and 1.4 we focus our attention on the setL(GE,GF) of allC-linear and continuous maps fromGEtoGF. Among all the elements ofL(GE,GF) we study those elements whose action has a
“basic structure” at the level of representatives.
Definition 1.1. Let (E,{pi}i∈I) and (F,{qj}j∈J) be locally convex topological vector spaces. We say that T ∈ L(GE,GF) is basic if there exists a net (Tε)ε of continuous linear maps fromE toF fulfilling the continuity property
(1.1) ∀j∈J∃I0⊆Ifinite∃N∈N∃η∈(0,1]∀u∈E∀ε∈(0, η]
qj(Tεu)≤ε−Nmax
i∈I0 pi(u), such thatT u= [(Tε(uε))ε] for allu∈ GE.
Note that the equalityT u= [(Tε(uε))ε] holds for all the representatives of (uε)ε since (1.1) entails (Tεvε)ε∈ MF if (vε)ε∈ ME and (Tεvε)ε∈ NF if (vε)ε∈ NE. Remark 1.2.
(i) If the net (Tε)ε satisfies the condition
(1.2) ∀j∈J∃I0⊆Ifinite∀q∈N∃η∈(0,1]∀u∈E∀ε∈(0, η]
qj(Tεu)≤εqmax
i∈I0 pi(u), then (Tε+Tε)ε defines the mapT, in the sense that for allu∈ GE,
T u= [(Tεuε)ε] = [((Tε+Tε)(uε))ε] inGF.
Inspired by the established language of moderateness and negligibility in Co- lombeau theory we define the space
M(E, F) :=
(Tε)ε∈ L(E, F)(0,1]: (Tε)εsatisfies (1.1) ofmoderate nets and the space
N(E, F) :=
(Tε)ε∈ L(E, F)(0,1]: (Tε)εsatisfies (1.2)
ofnegligible nets. By the previous considerations it follows that the classes of M(E, F)/N(E, F) generate maps inL(GE,GF) which are basic. One easily proves that ifEis a normed space with dimE <∞then the space of all basic maps in L(GE,GF) can be identified with the quotient M(E, F)/N(E, F).
Moreover by Proposition 3.22 in [14] it follows that for any normed spaceE the ultra-pseudo-normed C-moduleGE is isomorphic to the set of all basic functionals inL(GE,C).
(ii) Any continuous linear mapt:E→F produces a natural example of a basic element ofL(GE,GF). Indeed, as observed in [14, Remark 3.14], it is sufficient to take the constant net (t)ε and the corresponding map
T :GE → GF :u→[(tuε)ε].
(iii) A certain regularity of the basic operator T ∈ L(GE,GF) can be already viewed at the level of the net (Tε)ε. Indeed, if we assume that (Tε)εbelongs to the subsetM∞(E, F) ofM(E, F) obtained by replacing the string
∀j∈J ∃I0⊆Ifinite ∃N ∈N with
∃N∈N∀j∈J ∃I0⊆Ifinite in (1.1), we have thatT mapsGE∞into GF∞.
Definition 1.3. Let E = span(∪γ∈Γιγ(Eγ)),ιγ :Eγ → E be the inductive limit of the locally convex topological vector spaces (Eγ,{pi,γ}i∈Iγ)γ∈Γ and F be a lo- cally convex topological vector space. Let G = C-span(∪γ∈Γιγ(GEγ)) ⊆ GE be the inductive limit of the locally convex topologicalC-modules (GEγ)γ∈Γ. We say that T ∈ L(G,GF) is basic if there exists a net (Tε)ε ∈ L(E, F)(0,1] fulfilling the continuity property
(1.3) ∀γ∈Γ∀j∈J∃I0,γ ⊆Iγfinite∃N∈N∃η∈(0,1]∀u∈Eγ∀ε∈(0, η] qj(Tειγ(u))≤ε−N max
i∈I0,γpi,γ(u), such thatT u= [(Tε(uε))ε] for allu∈ G.
It is clear that (Tε)ε∈ L(E, F)(0,1] defines a basic mapT ∈ L(G,GF) if and only if (Tε◦ιγ)εdefines a basic mapTγ∈ L(GEγ,GF) such thatT◦ιγ =Tγ for allγ∈Γ.
We recall that nets (Tε)εwhich define basic maps as in Definitions1.1 and 1.3 were already considered in [10, 11] with slightly more general notions of moderateness and different choices of notations and language.
Particular choices ofE andF in the lines above yield the following statements:
(i) A functionalT ∈ L(G(Ω),C) is basic if it is of the formT u= [(Tεuε)ε], where (Tε)εis a net of distributions inE(Ω) satisfying the following condition:
∃KΩ∃j∈N∃N ∈N∃η∈(0,1]∀u∈ C∞(Ω)∀ε∈(0, η]
|Tε(u)| ≤ε−N sup
x∈K,|α|≤j|∂αu(x)|.
(ii) A functionalT ∈ L(Gc(Ω),C) is basic if it is of the formT u= [(Tεuε)ε], where (Tε)εis a net of distributions inD(Ω) satisfying the following condition:
∀KΩ∃j∈N∃N ∈N∃η∈(0,1]∀u∈ DK(Ω)∀ε∈(0, η]
|Tε(u)| ≤ε−N sup
x∈K,|α|≤j|∂αu(x)|.
Note that in analogy with distribution theory there exists a natural multiplication between functionals inL(Gc(Ω),C) and generalized functions inG(Ω) given by
uT(v) =T(uv), v∈ Gc(Ω).
It provides aC-linear operator fromL(Gc(Ω),C) to L(Gc(Ω),C) which maps basic functionals into basic functionals. Moreover, ifu∈ Gc(Ω) then uT ∈ L(G(Ω),C)⊆ L(G(Rn),C).
1.1. Action of basic functionals on generalized functions in two variables.
In this subsection we study the action of a basic functionalT belonging to the duals L(Gc(Ω),C), L(G(Ω),C) orL(GS(Rn),C) on a generalized function u(x, y) in two variables. Throughout the paper π1 : Ω×Ω → Ω and π2 : Ω×Ω → Ω are the projections of Ω×Ω on Ω and Ω respectively. We recall that V is a proper subset of Ω×Ω if for allK Ω andK Ω we haveπ2(V ∩π1−1(K))Ω and π1(V ∩π2−1(K))Ω.
Proposition 1.4. Let Ω be an open subset of Rn. Let T be a basic functional of L(Gc(Ω),C).
(i) If u∈ Gc(Ω×Ω)thenT(u(x,·)) := [(Tε(uε(x,·)))ε]is a well-defined element of Gc(Ω).
(ii) If u∈ Gc∞(Ω×Ω)thenT(u(x,·))∈ Gc∞(Ω).
(iii) Ifu∈ G(Ω×Ω)andsuppuis a proper subset ofΩ×ΩthenT(u(x,·))defines a generalized function inG(Ω).
(iv) G can be replaced byG∞ in(iii).
Let T be a basic functional of L(G(Ω),C).
(v) If u∈ G(Ω×Ω)then T(u(x,·))∈ G(Ω).
(vi) G can be replaced byG∞ in(v).
(vii) If u∈ G(Ω×Ω)andsuppuis a proper subset ofΩ×Ωthen T(u(x,·))∈ Gc(Ω).
(viii) G can be replaced byG∞ in(vii).
Finally, letT be a basic functional of L(GS(Rn),C).
(ix) If u∈ Gτ(R2n)has a representative (uε)ε satisfying the condition (1.4) ∀α∈Nn∀s∈N∃N ∈N
sup
x∈Rn(1 +|x|)−N sup
y∈Rn,|β|≤s
(1 +|y|)s|∂xα∂βyuε(x, y)|=O(ε−N) asε→0, thenT(u(x,·))is a well-defined element ofGτ(Rn).
(x) If (uε)ε fulfills the property (1.5) ∃M ∈N∀α∈Nn∀s∈N∃N ∈N
sup
x∈Rn(1 +|x|)−N sup
y∈Rn,|β|≤s(1 +|y|)s|∂xα∂yβuε(x, y)|=O(ε−M) asε→0, thenT(u(x,·))∈ Gτ∞(Rn).
Proof. For the sake of brevity we give a detailed proof of the first and the fourth assertions only. We recall that the final statements (ix) and (x) are easily obtained by employing the moderateness conditions which characterize the objects involved there and the point value theory for tempered generalized functions (cf. [23, Section 1.2.3]).
(i)–(ii) Letu∈ Gc(Ω×Ω) and (uε)εbe a representative ofusuch that suppuε⊆ K1×K2,K1Ω,K2Ω for allε∈(0,1]. By definition of basic functional there exists a net (Tε)ε∈ D(Ω)(0,1],N ∈N,j∈Nandη∈(0,1] such that
|Tε(uε(x,·))| ≤ε−N sup
y∈K2,|β|≤j|∂yβuε(x, y)| (1.6)
for all x ∈ Ω and for all ε ∈ (0, η]. From (1.6) it follows immediately that (uε)ε∈ Ec,M(Ω×Ω) implies (Tε(uε(x,·)))ε∈ Ec,M(Ω), (uε)ε∈ Nc(Ω×Ω) implies (Tε(uε(x,·)))ε∈ Nc(Ω) and (uε)ε∈ Ec∞,M(Ω×Ω) implies (Tε(uε(x,·)))ε∈ Ec∞,M(Ω).
To complete the proof thatT(u(x,·)) is a well-defined generalized function we still have to prove that it does not depend on the choice of the net (Tε)ε which deter- minesT. Let (Tε)ε∈ D(Ω)(0,1]be another net definingTandxa generalized point ofΩc. Sinceu(x, ·) := [(uε(xε,·))ε] belongs toGc(Ω) we have that
((Tε−Tε)(uε(xε,·)))ε∈ N,
i.e., the generalized functions [(Tε(uε(x,·)))ε]∈ Gc(Ω) and [(Tε(uε(x,·)))ε]∈ Gc(Ω) have the same point values. By point value theory this means ((Tε−Tε)(uε(x,·)))ε∈ Nc(Ω).
(iii)–(iv) Let us now assume that u ∈ G(Ω ×Ω) and that suppu is a proper subset of Ω×Ω. Let χ(x, y) be a proper smooth function on Ω×Ω identically 1 in a neighborhood of suppu. Clearly we can write χu = u in G(Ω×Ω). By the previous reasoning we have that for anyψ ∈ C∞c (Ω) the generalized function ψ(x)T(u(x,·)) = [(ψ(x)Tε(χ(x,·)uε(x,·)))ε] belongs to Gc(Ω) if u ∈ G(Ω ×Ω) and to Gc∞(Ω) if u ∈ G∞(Ω×Ω). Finally, let (Ωλ)λ∈Λ be a locally finite open covering of Ω with Ωλ Ω and (ψλ)λ∈Λ be a family of cut-off functions such that ψλ = 1 in a neighborhood of π2(suppu ∩ π1−1(Ωλ)). One can easily see thatψλ(x)T(u(x,·))|Ωλ ∈ G(Ωλ) determines a coherent family of generalized func- tions for λ varying in Λ and therefore, by the sheaf properties of G(Ω) it de- fines a generalized functionT(u(x,·)) inG(Ω) when u∈ G(Ω×Ω). Analogously T(u(x,·))∈ G∞(Ω) if u∈ G∞(Ω×Ω). We use the notationT(u(x,·)) since the definition of this generalized function does not depend on (Ωλ)λ∈Λand (ψλ)λ∈Λ. Remark 1.5. By means of the continuous map
ν:GS(Rn)→ G(Rn) : (uε)ε+NS(Rn)→(uε)ε+N(Rn) the dualL(G(Rn),C) can be embedded intoL(GS(Rn),C) as follows:
L(G(Rn),C)→ L(GS(Rn),C) (1.7)
T →(u→T(ν(u))).
Indeed, by composition of continuous maps, u → T(ν(u)) belongs to the dual L(GS(Rn),C) and, taking a cut-off functionχ ∈ Cc∞(Rn) identically 1 in a neigh- borhood of suppT, if u→ T(ν(u)) is the null functional in L(GS(Rn),C) we get that T(u) =T(χu) =T(ν(χu)) = 0 in C for all u∈ G(Rn). This shows that the map in (1.7) is injective. Obviously all the previous considerations hold for basic functionals.
Before stating the next proposition we recall that every tempered generalized function can be viewed as an element ofG(Rn) via the map
ντ :Gτ(Rn)→ G(Rn)
(uε)ε+Nτ(Rn)→(uε)ε+N(Rn). Proposition 1.6. Let T be a basic functional of L(G(Rn),C).
(i) If u∈ Gτ(R2n)thenT((ντu)(ξ,·))∈ Gτ(Rn).
(ii) If u∈ Gτ∞(R2n)thenT((ντu)(ξ,·))∈ Gτ∞(Rn).
(iii) If u∈ Gτ(R2n)has a representative (uε)ε fulfilling the condition
∀α, β, γ∈Nn∃N ∈N sup
y∈Rn,ξ∈Rn(1 +|y|)−N|ξβ∂ξα∂yγuε(ξ, y)|=O(ε−N), (1.8)
thenT((ντu)(ξ,·))∈ GS(Rn).
(iv) If u∈ Gτ(R2n)has a representative (uε)ε fulfilling the condition (1.9) ∃M ∈N∀α, β, γ∈Nn∃N∈N
sup
y∈Rn,ξ∈Rn(1 +|y|)−N|ξβ∂ξα∂γyuε(ξ, y)|=O(ε−M), thenT((ντu)(ξ,·))∈ GS∞(Rn).
Proof. We begin by observing that the generalized function T((ντu)(ξ,·)) is de- fined by the net (Sε(ξ))ε:= (Tε(uε(ξ,·)))ε, where (uε)ε∈ Eτ(R2n) and (Tε)εsatisfies the following condition:
(1.10) ∃KRn∃j∈N∃N∈N∃η∈(0,1]∀u∈ C∞(Rn)∀ε∈(0, η]
|Tε(u)| ≤ε−N sup
y∈K,|β|≤j|∂βu(y)|.
Consequently if (uε)ε∈ Eτ(R2n) then for allα∈Nn there existsN ∈Nsuch that for allεsmall enough the estimate
|∂αSε(ξ)|=|Tε(∂ξαuε(ξ, y))| ≤ε−N sup
y∈K,|β|≤j|∂ξα∂yβuε(ξ, y)| ≤cε−N−N(1 +|ξ|)N holds. This proves that (Sε)ε∈ Eτ(Rn). In an analogous way we obtain that (Sε)ε∈ Nτ(Rn) when (uε)ε∈ Nτ(R2n) and that (Sε)ε∈ Eτ∞(Rn) when (uε)ε ∈ Eτ∞(R2n).
Note that for all ξ ∈Rn, u(ξ,·) := (uε(ξε,·))ε+N(Rn)∈ G(Rn). Therefore, for (Tε)εand (Tε)εdifferent nets definingT and (uε)ε∈ Eτ(R2n) one has that
(Sε(ξε)−Sε(ξε)) := (Tε(uε(ξε,·))−Tε(uε(ξε,·)))ε
is negligible. Sinceξis arbitrary this implies that (Sε−Sε)ε ∈ Nτ(Rn) and com- pletes the proof of (i) and (ii).
Let us assume thatu∈ Gτ(R2n) has a representative fulfilling (1.8). Then the corresponding net (Sε)ε, which is already known to belong toEτ(Rn), satisfies the following estimate:
sup
ξ∈Rn|ξβ∂αSε(ξ)|= sup
ξ∈Rn|ξβTε(∂ξαuε(ξ, y))|
≤ε−N sup
ξ∈Rn sup
y∈K,|γ|≤j|ξβ∂ξα∂γyuε(ξ, y)|
≤ε−N−Nsup
y∈K(1 +|y|)N,
uniformly for small values ofε. This means thatT((ντu)(ξ,·))∈ GS(Rn).
Finally, when (uε)ε satisfies (1.9) then sup
ξ∈Rn|ξβ∂αSε(ξ)| ≤ε−N sup
ξ∈Rn sup
y∈K,|γ|≤j|ξβ∂ξα∂yγuε(ξ, y)| ≤ε−N−M sup
y∈K(1 +|y|)N,
