Table Of ContentMicrolocal analysis in the dual of a Colombeau
algebra: generalized wave front sets and
noncharacteristic regularity
Claudia Garetto
Institut fu¨r Technische Mathematik,
6
Geometrie und Bauniformatik,
0
0 Universit¨at Innsbruck, Austria
2 [email protected]
n
a
J
Abstract
9
Weintroducedifferent notionsof wave front set for thefunctionals in thedualof theColombeau
P] algebraGc(Ω)providingawaytomeasuretheGandtheG∞-regularityinL(Gc(Ω),C). Forthesmaller
family of functionals having a “basic structure” we obtain a Fourier transform-characterization for
A this typeof generalized wave front sets and results of noncharacteristic G and G∞e-regularity.
.
h
t
a
0 Introduction
m
[
The past decade has seen the emergence of a differential-algebraic theory of generalized functions of
2
v Colombeau type [1, 2, 3, 17, 35, 38, 45] that answered a wealth of questions on solutions to linear
7 and nonlinear partial differential equations involving non-smoothcoefficients and strongly singular data.
9 InterestingresultswereobtainedinLiegroupinvarianceofgeneralizedfunctions[5,25,40,42],nonlinear
2
hyperbolic equations with generalized function data [4, 33, 34, 36, 39, 41, 43, 44], distributional metrics
1
in general relativity [26, 27, 28, 29, 30, 31], propagation of strong singularities in linear hyperbolic
1
5 equationswithdiscontinuouscoefficients[19,20,32,37],microlocalanalysis,pseudodifferentialoperators
0 and Fourier integral operators with non-smooth symbols [10, 14, 15, 16, 21, 22, 23, 24].
/
h Some “key technologies” for the regularity theory of partial differential equations in the Colombeau
t
a context have been developed in [10, 13, 14, 15]. They consist in a complete theory of generalized pseu-
m dodifferential operators (including a parametrix-construction for operators with generalized hypoelliptic
: symbol) [10, 14] and the application of those pseudodifferential techniques to the microlocal analysis of
v
generalizedfunctions [15]. Particularattention hasbeen givento the dualof a Colombeaualgebrawhich
i
X playsamainroleinthekerneltheoryforgeneralizedpseudodifferentialoperators[10,14]. Itisnownatu-
r raltoextendthepseudodifferentialoperator’sactiontothedualandtoshiftthemicrolocalinvestigations
a from the level of generalized functions to the level of C-linear functionals. This will require notions of
local and microlocal regularity in the dual of a Colombeau algebra and suitable ways of measuring such
e
kinds of regularity. Microlocal analysis is essential for a full understanding of the generalized pseudodif-
ferential 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 investigate the dual of a
Colombeaualgebra. Basedonthe dualitytheorydevelopedwithinthe Colombeauframeworksin[11,12]
it extends and adapts the microlocal results for Colombeau generalized functions stated in [15]. In the
usual Colombeau context a generalized function u ∈ G(Ω) is said to be regular if it belongs to the
subalgebra G∞(Ω). This allows to set up a regularity theory for G(Ω) which is coherent with the usual
concept of regularity for distributions since G∞(Ω)∩D′(Ω) = C∞(Ω). In [6, 35] a notion of generalized
1
wave front set is defined for u ∈ G(Ω) as a G∞-wave front set. It means that the conic regions of
“microlocalregularity”we dealwithinthe cotangentspaceareregionsofG∞-regularity. Comingnowto
the dual L(G (Ω),C), i.e. the space of all continuous and C-linear functionals on G (Ω), where C is the
c c
ring of complex generalized numbers, by continuous embedding it contains both G∞(Ω) and G(Ω). As a
e e e
consequence, two levels of regularity concern a functional in L(G (Ω),C): the regularity with respect to
c
G(Ω) and the regularity with respect to G∞(Ω). In this paper in order to measure such different kinds
e
of regularity of T ∈ L(G (Ω),C) we introduce the notions of G-wave front set (WF (T)) and G∞-wave
c G
front set (WFG∞(T)). e
Inspired by [15] and making use of the theory of pseudodifferential operators with generalized symbols
elaboratedin[14,15],WF (T)andWF∞(T)aredefinedasintersectionofsuitableregionsofgeneralized
G G
non-ellipticityofthosepseudodifferentialoperatorswhichmapT inG(Ω)andG∞(Ω)respectively. Coreof
thepaperisaFouriertransform-characterizationofWFG(T)andWFG∞(T)asin[9,Theorem8.56]which
consistsinthedirectinvestigationofthepropertiesoftheFouriertransformofT aftermultiplicationbya
suitable cut-off function. For this purpose special spacesof generalizedfunctions with rapidly decreasing
behaviorona conic subset ofRn are introduced andamong allthe functionals of L(G (Ω),C) we restrict
c
to consider those elements which have a “basic structure”. More precisely we assume T ∈ L(G (Ω),C)
e c
being defined by a net of distributions (T ) which fulfills a continuity assumption uniform with respect
ε ε e
to ε (Definition 1.3) and the equality Tu = [(T u ) ] ∈ C for all u = [(u ) ] ∈ G (Ω). Even though the
ε ε ε ε ε c
G-wavefrontsetandtheG∞-wavefrontsetcanbedefinedonanyfunctionalofthe dualL(G (Ω),C), the
e c
main theorems and propositions presented here are proven to be valid for basic functionals. In addition,
e
all the results of microlocal regularity have a double version: the G-version and the G∞-version.
We now describe in detail the contents of the sections.
Section 1 provides the needed theoreticalbackgroundof basic functionals and refer for topologicalissues
to[11,12]. Afterthefirstdefinitionsandbasicproperties,theactionofabasicfunctionalonaColombeau
generalizedfunction in twovariablesis investigatedinSubsection 1.1. Togetherwith someresults onthe
composition 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 functions and functionals in Subsection 1.3.
ThealgebrasofgeneralizedfunctionshereinvolvedareG (Ω),G(Ω) andG (Rn)while thefunctionalsare
c S
elements of the duals L(G (Ω),C), L(G(Ω),C) and L(G (Rn),C). A regularizationof basic functionals is
c S
obtained via convolution with a generalized mollifier. Finally in Subsection 1.4 we extends the natural
e e e
notion of Fourier transform on G (Rn) to the dual L(G (Rn),C) and we study the Fourier transform of
S S
a basic functional in L(G(Ω),C).
e
In the recent Colombeau litereature a pseudodifferential operator with generalized symbol is a C-linear
continuous operator which maps G (Ω) into G(Ω). Section 2 extends the action of such generalized
c e
pseudodifferentialoperatortothedualsL(G (Ω),C)andL(G(Ω),C). Theextensionprocedureisobtained
c
via transpositionandgives interesting mapping properties concerning the subspacesof basic functionals.
e e
A variety of symbols (and amplitudes) is considered: generalized symbols 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 [14, 15]). 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 that R is G-regularizing on the basic functionals of L(G(Ω),C), in the sense
that RT ∈ G(Ω) if T ∈ L(G(Ω),C) is basic. Analogously R is an integral operator with kernel in
e
G∞(Ω×Ω) if and only if it is a pseudodifferential operator with regular amplitude of order −∞ and
e
it is G∞-regularizing on the basic functionals of L(G(Ω),C). A G-pseudolocality property is obtained
for properly supported pseudodifferential operators with generalized symbols while a G∞-pseudolocality
e
property is valid when the symbols are regular. Section 2 ends by adapting the result of G∞-regularity
in [14] for pseudodifferentialoperatorswith generalizedhypoelliptic symbols to the dual context of basic
functionals.
2
AG-microlocalanalysisanda G∞-microlocalanalysisfor the dualL(G (Ω),C) aresettled anddeveloped
c
inSection3. The additionalassumptionofbasic structureonthe functionalT is employedinSubsection
e
3.1 in provingthat the projections onΩ ofWFG(T) and WFG∞(T)coincide with the G-singularsupport
and the G∞-singular support of T respectively. The Fourier transform-characterizationsof WF (T) and
G
WFG∞(T) are the result of the G and the G∞-microlocal investigations of pseudodifferential operators
elaborated throughout Subsection 3.2 in the dual L(G (Ω),C). Concerning the notion of slow scale
c
micro-ellipticity here employed this has been already introduced in [15] while the concept of generalized
e
microsupport of a generalized symbol in [15, Definition 3.1] is transformed into G-microsupport and
G∞-microsupport (Definition 3.6).
Section 4 concludes the paper with a theorem on noncharacteristic G and G∞-regularity for pseudodif-
ferentialoperatorswith slow scale symbols when they acton basic functionals ofL(G (Ω),C). This is an
c
extension and adaptation to the dual L(G (Ω),C) of Theorem 4.1 in [15].
c e
For the advantage of the reader we recall in theesequel some topological issues discussed in [11, 12] 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 [11, 12, 13, 46, 47, 48] setting the foundations of duality theory in the recent work on topological
and locally convex topological C-modules [11, 12, 13]. Without presenting the technical details of this
theoretical construction, we recall that a suitable adaptation of the classical notion of seminorm, called
e
ultra-pseudo-seminorm[11, Definition 1.8], allowsto characterizea locally convexC-linear topology as a
topologydeterminedbyafamilyofultra-pseudo-seminorms. ThemostcommonColombeaualgebrascan
e
be introduced as C-modules of generalized functions based on a locally convex topological vector space
E. Such a C-module G is the quotient of the set
e E
(0.1) e M :={(u ) ∈E(0,1] : ∀i∈I ∃N ∈N p (u )=O(ε−N)asε→0}
E ε ε i ε
of E-moderate nets with respect to the set
(0.2) N :={(u ) ∈E(0,1] : ∀i∈I ∀q ∈N p (u )=O(εq)asε→0},
E ε ε i ε
of E-negligible nets, and it is naturally endowed with a locally convex C-linear topology usually called
sharp topology in [35, 46, 47, 48]. Given a family of seminorms {p } on E, the sharp topology on G
i i∈I e E
is determined by the ultra-pseudo-seminorms Pi(u):=e−vpi(u), where vpi is the valuation
v ([(u ) ]):=v ((u ) ):=sup{b∈R: p (u )=O(εb) as ε→0}
pi ε ε pi ε ε i ε
(see [11, Subsection 3.1] for further explanations). Note that valuations and ultra-pseudo-seminormsare
defined on M and extended to the factor space G in a second time. It is clear that the ring C of
E E
complexgeneralizednumbersisanexampleofG -spaceobtainedbychoosingE =C. Thevaluationand
E e
ultra-pseudo-norm on C obtained as above by means of the absolute value on C are denoted by v and
C
|·| respectively.
e e e
As proved in [11, Corollary 1.17] for an arbitrary locally convex topological C-module (G,{Q } ), a
j j∈J
C-linear map T :G →G is continuous if and only if for all j ∈J there exists a finite subset I ⊆I and
E e 0
a constant C >0 such that for all u∈G
e E
Q (Tu)≤CmaxP (u).
j i
i∈I0
The Colombeau algebras G(Ω), G (Ω), G (Rn)
c S
The Colombeau algebra G(Ω) is the C-module of G -type given by E = E(Ω). Equipped with the
E
family of seminorms p (f) = sup |∂αf(x)| where K ⋐ Ω, the space E(Ω) induces on G(Ω)
K,i x∈Ke,|α|≤i
3
a metrizable and complete locally convex C-linear topology which is determined by the ultra-pseudo-
seminorms PK,i(u) = e−vpK,i(u). For coheerence with some well-established notations in Colombeau
theory we write M =E (Ω) and N =N(Ω).
E(Ω) M E(Ω)
The Colombeau algebra G (Ω) of generalized functions with compact support is topologized by means
c
of a strict inductive limit procedure. More precisely, setting G (Ω) := {u ∈ G (Ω) : suppu ⊆ K} for
K c
K ⋐Ω, Gc(Ω) is the strictinductive limit ofthe sequence (GKn(Ω))n∈N, where (Kn)n∈N is anexhausting
sequenceofcompactsubsetsofΩsuchthatK ⊆K . WerecallthatthespaceG (Ω)isendowedwith
n n+1 K
the topology induced by G where K′ is a compact subset containing K in its interior. In detail
DK′(Ω)
we consider on GK(Ω) the ultra-pseudo-seminorms PGK(Ω),n(u) = e−vK,n(u). Note that the valuation
v (u) := v (u) is independent of the choice of K′ when acts on G (Ω). As observed in [13,
K,n pK′,n K
Subsection 1.2.2] the Colombeau algebra G (Ω) is isomorphic to the factor space E (Ω)/N (Ω) where
c c,M c
Ec,M(Ω) and Nc(Ω) are obtained by intersecting EM(Ω) and N(Ω) with ∪K⋐ΩDK(Ω)(0,1] respectively.
The ColombeaualgebraG (Rn) of generalizedfunctions basedonS(Rn) is obtainedas a G -module by
S E
choosingE =S(Rn). Itis a Fr´echetC-module accordingto the topologyofthe ultra-pseudo-seminorms
Ph(u) = e−vph(u), where ph(f) = supex∈Rn,|α|≤h(1+|x|)h|∂αf(x)|, f ∈ S(Rn), h ∈ N. In the course of
the paper we will use the notations ES(Rn) and NS(Rn) for the spaces of nets MS(Rn) and NS(Rn)
respectively.
The regular Colombeau algebras G∞(Ω), G∞(Ω), G∞(Rn)
c S
Given a locally convex topological space (E,{p } ) the C-module G∞ of regular generalized functions
i i∈I E
based on E is defined as the quotient M∞/N , being
E E e
M∞ :={(u ) ∈E(0,1] : ∃N ∈N ∀i∈I p (u )=O(ε−N)as ε→0}
E ε ε i ε
the set of E-regular nets. The moderateness properties of M∞ allows to define the valuation
E
v∞((u ) ):=sup{b∈R: ∀i∈I p (u )=O(εb) as ε→0}
E ε ε i ε
which extends to GE∞ and leads to the ultra-pseudo-norm PE∞(u) := e−vE∞(u). This topological model
is employed in endowing the Colombeau algebras G∞(Ω), G∞(Ω), G∞(Rn) and G∞(Rn) with a locally
c S τ
convex C-linear topology.
We begien by recalling that G∞(Ω) is the subalgebra of all elements u of G(Ω) having a representative
(u ) belonging to the set
ε ε
E∞(Ω):={(u ) ∈E[Ω]: ∀K ⋐Ω∃N ∈N∀α∈Nn sup|∂αu (x)|=O(ε−N) as ε→0}.
M ε ε ε
x∈K
G∞(Ω) can be seen as the intersection∩K⋐ΩG∞(K), where G∞(K) is the space of all u∈G(Ω) having a
representative (u ) satisfying the condition: ∃N ∈N ∀α∈Nn, sup |∂αu (x)|=O(ε−N). The ultra-
ε ε x∈K ε
pseudo-seminormsPG∞(K)(u):=e−vG∞(K), where vG∞(K) :=sup{b∈R: ∀α∈Nn supx∈K|∂αuε(x)|=
O(εb)}, equips G∞(Ω) with the topological structure of a Fr´echet C-module.
The algebra G∞(Ω) is the intersection of G∞(Ω) with G (Ω). OneG∞(Ω) := {u ∈ G∞(Ω) : suppu ⊆
c c K
K ⋐ Ω} we define the ultra-pseudo-norm PGK∞(Ω)(u) = e−vK∞(u) where vK∞(u) := vD∞K′(Ω)(u) and K′
is any compact set containing K in its interior. At this point, given an exhausting sequence (K )
n n
of compact subsets of Ω, the strict inductive limit procedure determines on a complete and separated
locallyconvexC-lineartopologyonG∞(Ω)=∪ G∞ (Ω). ClearlyG∞(Ω)isisomorphictothefactorspace
c n Kn c
Ec∞,M(Ω)/Nc(Ω)ewhere Ec∞,M(Ω):=EM∞(Ω)∩(∪K⋐ΩDK(Ω)(0,1]).
Finally G∞(Rn) is the C-module of regular generalized functions based on E = S(Rn). In coherence
S
with the notations already in use we set M∞ =E∞(Ω) for any open subset Ω of Rn.
e S(Ω) S
4
The Colombeau algebras of tempered generalized functions G (Rn) and G∞(Rn)
τ τ
The Colombeau algebra of tempered generalized functions G (Rn) is defined as E (Rn)/N (Rn), where
τ τ τ
E (Rn) is the space
τ
{(u ) ∈O (Rn)(0,1] : ∀α∈Nn∃N ∈N sup(1+|x|)−N|∂αu (x)|=O(ε−N) as ε→0}
ε ε M ε
x∈Rn
of τ-moderate nets and N (Rn) is the space
τ
{(u ) ∈O (Rn)(0,1] : ∀α∈Nn∃N ∈N∀q ∈N sup(1+|x|)−N|∂αu (x)|=O(εq) as ε→0}
ε ε M ε
x∈Rn
ofτ-negligiblenets. ThesubalgebraG∞(Rn)ofregularandtemperedgeneralizedfunctionsisthequotient
τ
E∞(Rn)/N (Rn), where E∞(Rn) is the set of all (u ) ∈O (Rn)(0,1] satisfying the following condition:
τ τ τ ε ε M
∃N ∈N∀α∈Nn∃M ∈N sup(1+|x|)−M|∂αu (x)|=O(ε−N).
ε
x∈Rn
The topological duals L(G (Ω),C), L(G (Ω),C), L(G (Rn),C)
c c S
e e e
Throughout the paper the topological duals L(G (Ω),C), L(G (Ω),C), L(G (Rn),C) are endowed with
c c S
the corresponding topologies of uniform convergence on bounded subsets. These topologies, denoted by
e e e
β (L(G (Ω),C),G (Ω)), β (L(G(Ω),C),G(Ω)) and β (L(G (Rn),C),G (Rn)) respectively, are determined
b c c b b S S
by the ultra-pseudo-seminorms P (T) := sup |Tu| with B varying in the family of all bounded
e Be u∈B e e
subsets of G (Ω), G(Ω) and G (Rn) respectively. As in the classical functional analysis a subset B of a
c S
locallyconvexC-module(G,{P } )isboundedifandonlyifeveryultra-pseudo-seminormP isbounded
i i∈I i
onB,i.e. sup P (u)<∞. Withrespecttothetopologiescollectedinthissubsectionandthetopology
ue∈B i
on G (Rn) introduced in [11, Example 3.9] we have that the following chain of inclusions
τ
G∞(Ω) ⊆ G(Ω) ⊆ L(G (Ω),C),
c
G∞(Ω) ⊆ G (Ω) ⊆ L(G(Ω),Ce),
c c
G∞(Rn) ⊆ G (Rn) ⊆ G (Rn) ⊆ L(Ge (Rn),C)
S S τ S
are continuous [12, Theorems 3.1, 3.8]. Moreover Ω → L(G (Ω),C) is a esheaf and the dual L(G(Ω),C)
c
can be identified with the set of functionals in L(G (Ω),C) having compact support [12, Theorem 1.2].
c e e
e
1 Duality theory in the Colombeau context: basic maps and
functionals
This section is devoted to maps and functionals defined on C-modules of Colombeau type. Before
considering topics more related to the duals of the Colombeau algebras G (Ω), G(Ω) and G (Rn) in
e c S
Subsections1.1,1.2,1.3and1.4wefocusourattentiononthesetL(G ,G )ofallC-linearandcontinuous
E F
maps from G to G . Among all the elements of L(G ,G ) we study those elements whose action has a
E F E F e
“basic structure” at the level of representatives.
Definition 1.1. Let (E,{p } ) and (F,{q } ) be locally convex topological vector spaces. We say
i i∈I j j∈J
that T ∈ L(G ,G ) is basic if there exists a net (T ) of continuous linear maps from E to F fulfilling
E F ε ε
the continuity-property
(1.3) ∀j ∈J∃I ⊆Ifinite ∃N ∈N∃η ∈(0,1]∀u∈E∀ε∈(0,η] q (T u)≤ε−Nmaxp (u),
0 j ε i
i∈I0
such that Tu=[(T (u )) ] for all u∈G .
ε ε ε E
5
Note that the equality Tu = [(T (u )) ] holds for all the representatives of (u ) since (1.3) entails
ε ε ε ε ε
(T v ) ∈M if (v ) ∈M and (T v ) ∈N if (v ) ∈N .
ε ε ε F ε ε E ε ε ε F ε ε E
Remark 1.2.
(i) If the net (T′) satisfies the condition
ε ε
(1.4) ∀j ∈J∃I ⊆Ifinite ∀q ∈N∃η ∈(0,1]∀u∈E∀ε∈(0,η] q (T′u)≤εqmaxp (u),
0 j ε i
i∈I0
then (T +T′) defines the map T, in the sense that for all u∈G ,
ε ε ε E
Tu=[(T u ) ]=[((T +T′)(u )) ]in G .
ε ε ε ε ε ε ε F
Inspired by the established language of moderateness and negligibility in Colombeau theory we define
the space
M(E,F):={(T ) ∈L(E,F)(0,1] : (T ) satisfies (1.3)}
ε ε ε ε
of moderate nets and the space
N(E,F):={(T ) ∈L(E,F)(0,1] : (T ) satisfies (1.4)}
ε ε ε ε
ofnegligible nets. BythepreviousconsiderationsitfollowsthattheclassesofM(E,F)/N(E,F)generate
maps in L(G ,G ) which are basic. One easily proves that if E is a normed space with dimE <∞ then
E F
thespaceofallbasicmapsinL(G ,G )canbeidentifiedwiththequotientM(E,F)/N(E,F). Moreover
E F
by Proposition 3.22 in [11] it follows that for any normed space E the ultra-pseudo-normed C-module
GE′ is isomorphic to the set of all basic functionals in L(GE,C). e
e
(ii) Any continuous linear map t : E → F produces a natural example of basic element of L(G ,G ).
E F
Indeed,asobservedin[11,Remark3.14],itissufficienttotaketheconstantnet(t) andthecorresponding
ε
map T :G →G :u→[(tu ) ].
E F ε ε
(iii) A certain regularity of the basic operator T ∈ L(G ,G ) can be already viewed at the level of the
E F
net (T ) . Indeed, if we assume that (T ) belongs to the subset M∞(E,F) of M(E,F) obtained by
ε ε ε ε
substituting the string
∀j ∈J ∃I ⊆Ifinite ∃N ∈N
0
with
∃N ∈N ∀j ∈J ∃I ⊆Ifinite
0
in (1.3), we have that T maps G∞ into G∞.
E F
Definition 1.3. Let E = span(∪ ι (E )), ι : E → E be the inductive limit of the locally convex
γ∈Γ γ γ γ γ
topological vector spaces (E ,{p } ) and F be a locally convex topological vector space. Let G =
γ i,γ i∈Iγ γ∈Γ
C−span(∪ ι (G ))⊆G be the inductive limit of the locally convex topological C-modules (G ) .
γ∈Γ γ Eγ E Eγ γ∈Γ
We say that T ∈ L(G,G ) is basic if there exists a net (T ) ∈ L(E,F)(0,1] fulfilling the continuity-
e F ε ε e
property
(1.5) ∀γ ∈Γ∀j ∈J∃I ⊆I finite ∃N ∈N∃η ∈(0,1]∀u∈E ∀ε∈(0,η]
0,γ γ γ
q (T ι (u))≤ε−N max p (u),
j ε γ i,γ
i∈I0,γ
such that Tu=[(T (u )) ] for all u∈G.
ε ε ε
It is clear that (T ) ∈ L(E,F)(0,1] defines a basic map T ∈ L(G,G ) if and only if (T ◦ι ) defines a
ε ε F ε γ ε
basic map T ∈ L(G ,G ) such that T ◦ι = T for all γ ∈ Γ. We recall that nets (T ) which define
γ Eγ F γ γ ε ε
basic maps as in Definitions 1.1 and 1.3 where already considered in [7, 8] with slightly more general
notions of moderateness and different choices of notations and language.
Particular choices of E and F in the lines above yield the following statements:
6
(i) a functional T ∈ L(G(Ω),C) is basic if it is of the form Tu = [(T u ) ], where (T ) is a net of
ε ε ε ε ε
distributions in E′(Ω) satisfying the following condition:
e
∃K ⋐Ω∃j ∈N∃N ∈N∃η ∈(0,1]∀u∈C∞(Ω)∀ε∈(0,η] |T (u)|≤ε−N sup |∂αu(x)|.
ε
x∈K,|α|≤j
(ii) A functional T ∈ L(G (Ω),C) is basic if it is of the form Tu = [(T u ) ], where (T ) is a net of
c ε ε ε ε ε
distributions in D′(Ω) satisfying the following condition:
e
∀K ⋐Ω∃j ∈N∃N ∈N∃η ∈(0,1]∀u∈D (Ω)∀ε∈(0,η] |T (u)|≤ε−N sup |∂αu(x)|.
K ε
x∈K,|α|≤j
Note that in analogy with distribution theory there exists a natural multiplication between functionals
in L(G (Ω),C) and generalized functions in G(Ω) given by
c
e uT(v)=T(uv), v ∈G (Ω).
c
It provides a C-linear operator from L(G (Ω),C) to L(G (Ω),C) which maps basic functionals into basic
c c
functionals. Moreover,if u∈G (Ω) then uT ∈L(G(Ω),C)⊆L(G(Rn),C).
e c e e
e e
1.1 Action of basic functionals on generalized functions in two variables
In this subsection we study the action of a basic functional T belonging to the duals L(G (Ω),C),
c
L(G(Ω),C) or L(G (Rn),C) on a generalized function u(x,y) in two variables. Throughout the pa-
S e
per π : Ω′×Ω → Ω′ and π : Ω′×Ω → Ω are the projections of Ω′×Ω on Ω′ and Ω respectively. We
1 e e2
recall that V is a proper subset of Ω′×Ω if for all K′ ⋐ Ω′ and K ⋐ Ω we have π (V ∩π−1(K′)) ⋐ Ω
2 1
and π (V ∩π−1(K))⋐Ω′.
1 2
Proposition 1.4. Let Ω′ be an open subset of Rn′ and T be a basic functional of L(G (Ω),C).
c
e
(i) If u∈G (Ω′×Ω) then T(u(x,·)):=[(T (u (x,·))) ] is a well-defined element of G (Ω′);
c ε ε ε c
(ii) if u∈G∞(Ω′×Ω) then T(u(x,·))∈G∞(Ω′);
c c
(iii) ifu∈G(Ω′×Ω)andsuppuisapropersubsetofΩ′×ΩthenT(u(x,·))definesageneralizedfunction
in G(Ω′);
(iv) G can be replaced by G∞ in (iii).
Let T be a basic functional of L(G(Ω),C).
e
(v) If u∈G(Ω′×Ω) then T(u(x,·))∈G(Ω′);
(vi) G can be replaced by G∞ in (v);
(vii) if u∈G(Ω′×Ω) and suppu is a proper subset of Ω′×Ω then T(u(x,·))∈G (Ω′);
c
(viii) G can be replaced by G∞ in (vii).
Finally, let T be a basic functional of L(G (Rn),C).
S
e
(ix) If u∈G (R2n) has a representative (u ) satisfying the condition
τ ε ε
(1.6) ∀α∈Nn∀s∈N∃N ∈N
sup(1+|x|)−N sup (1+|y|)s|∂α∂βu (x,y)|=O(ε−N) as ε→0
x y ε
x∈Rn y∈Rn,|β|≤s
then T(u(x,·)) is a well-defined element of G (Rn);
τ
7
(x) if (u ) fulfills the property
ε ε
(1.7) ∃M ∈N∀α∈Nn∀s∈N∃N ∈N
sup(1+|x|)−N sup (1+|y|)s|∂α∂βu (x,y)|=O(ε−M) as ε→0
x y ε
x∈Rn y∈Rn,|β|≤s
then T(u(x,·))∈G∞(Rn).
τ
Proof. (i)−(ii) Let u ∈ G (Ω′ ×Ω) and (u ) be a representative of u such that suppu ⊆ K ×K ,
c ε ε ε 1 2
K ⋐Ω′, K ⋐Ω for all ε∈(0,1]. By definition of basic functional there exists a net (T ) ∈D′(Ω)(0,1],
1 2 ε ε
N ∈N, j ∈N and η ∈(0,1] such that
(1.8) |T (u (x,·))|≤ε−N sup |∂βu (x,y)|
ε ε y ε
y∈K2,|β|≤j
for all x∈Ω′ and for all ε ∈(0,η]. From (1.8) it follows immediately that (u ) ∈E (Ω′×Ω) implies
ε ε c,M
(T (u (x,·))) ∈E (Ω′), (u ) ∈N (Ω′×Ω) implies (T (u (x,·))) ∈N (Ω′) and (u ) ∈E∞ (Ω′×Ω)
ε ε ε c,M ε ε c ε ε ε c ε ε c,M
implies (T (u (x,·))) ∈ E∞ (Ω′). To complete the proof that T(u(x,·)) is a well-defined generalized
ε ε ε c,M
function we still have to prove that it does not depend on the choice of the net (T ) which determines
ε ε
T. Let (T′) ∈ D′(Ω)(0,1] be another net defining T and x a generalized point of Ω′ . Since u(x,·) :=
ε ε c
[(u (x ,·)) ] belongs to G (Ω) we have that
ε ε ε c e f e
((T −T′)(u (x ,·))) ∈N
ε ε ε ε ε
i.e., the generalizedfunctions [(T (u (x,·))) ]∈G (Ω′) and[(T′(u (x,·))) ]∈G (Ω′) havethe same point
ε ε ε c ε ε ε c
values. By point value theory this means that ((T −T′)(u (x,·))) ∈N (Ω′).
ε ε ε ε c
(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 neighborhoodof suppu. Clearly we can
write χu=u in G(Ω′×Ω). By the previous reasoning we have that for any ψ ∈C∞(Ω′) the generalized
c
function ψ(x)T(u(x,·))=[(ψ(x)T (χ(x,·)u (x,·))) ] belongs to G (Ω′) if u∈G(Ω′×Ω) and to G∞(Ω′) if
ε ε ε c c
u∈G∞(Ω′×Ω). Finally, let (Ω′) be a locally finite open covering of Ω′ with Ω′ ⋐ Ω′ and (ψ )
λ λ∈Λ λ λ λ∈Λ
be a family of cut-off functions such that ψ = 1 in a neighborhood of π (suppu ∩ π−1(Ω′)). One can
λ 2 1 λ
veaasriylyingseientΛhaatnψdλt(hxe)rTef(our(ex,,b·y))|tΩh′λe s∈heGa(fΩp′λr)opdeertteiremsoinfeGs(aΩ′c)oihtedreenfitnefsamaiglyenoefragleizneedrafluiznecdtiofunnTct(iuo(nxs,·f)o)riλn
G(Ω′) when u ∈ G(Ω′ ×Ω). Analogously T(u(x,·)) ∈ G∞(Ω′) if u ∈ G∞(Ω′×Ω). We use the notation
T(u(x,·)) since the definition of this generalized function does not depend on (Ω′) and (ψ ) .
λ λ∈Λ λ λ∈Λ
The proof of (v) and (vi) is clear arguing at the level of representatives.
(vii)−(viii)When the supportofuis properwecanchooseasmoothproperfunctionχ(x,y) identically
1 in a neighborhood of suppu and a cut-off function ψ(y) identically 1 in a neighborhood of suppT.
Hence, T(u(x,·))=T(ψ(·)u(x,·))=T(ψ(·)u(x,·)χ(x,·)), where ψ(y)u(x,y)χ(x,y)∈G (Ω′×Ω). By the
c
first and the second assertion of this proposition we have that T(u(x,·)) belongs to G (Ω′) or G∞(Ω′) if
c c
u is an element of G(Ω′×Ω) or G∞(Ω′×Ω) respectively.
(ix)−(x) Let us consider T ∈ L(G (Rn),C) defined by (T ) ∈ S′(Rn)(0,1]. Recall that (T ) has the
S ε ε ε ε
following continuity-property:
e
∃j ∈N∃N ∈N∃η ∈(0,1]∀u∈S(Rn)∀ε∈(0,η] |T (u)|≤ε−N sup (1+|y|)j|∂βu(y)|.
ε
y∈Rn,|β|≤j
Hence, if (u ) satisfies (1.6) one gets that for all x∈Rn and for all ε small enough
ε ε
(1.9) |∂αT (u (x,·))|=|T (∂αu (x,·))|≤ε−N′ sup (1+|y|)j|∂α∂βu (x,y)|≤ε−N′−N(1+|x|)N,
x ε ε ε x ε x y ε
y∈Rn,|β|≤j
where N depends on α and j. This means that (T (u (x,·))) ∈ E (Rn). As already observed in the
ε ε ε τ
proof of [13, Proposition1.2.25]if (u ) and (u′) are two representativesof u both satisfying (1.6) then
ε ε ε ε
8
the difference v :=u −u′ fulfills the property
ε ε ε
∀α∈Nn∀s∈N∃N ∈N∀q ∈N
sup(1+|x|)−N sup (1+|y|)s|∂α∂βv (x,y)|=O(εq) as ε→0.
x y ε
x∈Rn y∈Rn,|β|≤s
As a consequence (T (u (x,·)−u′(x,·))) ∈ N (Rn). In order to claim that T(u(x,·)) is a generalized
ε ε ε ε τ
function in G (Rn) it remains to show that different nets (T ) and (T′) defining T lead to ((T −
τ ε ε ε ε ε
T′)(u (x,·))) ∈ N (Rn). This is due to the fact that for every x ∈ Rn, u(x,·) := [(u (x ,·)) ] is a
ε ε ε τ ε ε ε
generalizedfunction in G (Rn) andthen by definition of T the net ((T −T′)(u (x ,·))) is negligible. It
follows that the tempereSd generalizedfunctions [(T (u (x,·))) ] ande[(Tε′(fuε(x,·ε)e))ε] havεe the same point
ε ε ε ε ε ε
values. Thus, ((T −T′)(u (x,·))) ∈ N (Rn). Finally, arguing as in (1.9) it is clear that T(u(x,·))
ε ε ε ε τ
belongs to G∞(Rn) when u has a representative satisfying (1.7).
τ
Remark 1.5. By means of the continuous map
ν :GS(Rn)→G(Rn):(uε)ε+NS(Rn)→(uε)ε+N(Rn)
the dual L(G(Rn),C) can be embedded into L(G (Rn),C) as follows:
S
e e
(1.10) L(G(Rn),C)→L(G (Rn),C):T →(u→T(ν(u))).
S
e e
Indeed, by composition of continuous maps, u→T(ν(u)) belongs to the dual L(G (Rn),C) and, taking
S
a cut-off function χ ∈ C∞(Rn) identically 1 in a neighborhood of suppT, if u → T(ν(u)) is the null
c e
functional in L(G (Rn),C) we get that
S
e
T(u)=T(χu)=T(ν(χu))=0in C
forallu∈G(Rn). Thisshowsthatthemapin(1.10)isinjective. Obvieouslyallthepreviousconsiderations
hold for basic functionals.
Before stating the next proposition we recall that every tempered generalized function can be viewed as
an element of G(Rn) via the map
ν :G (Rn)→G(Rn):(u ) +N (Rn)→(u ) +N(Rn).
τ τ ε ε τ ε ε
Proposition 1.6. Let T be a basic functional of L(G(Rn),C).
e
(i) If u∈G (R2n) then T((ν u)(ξ,·))∈G (Rn);
τ τ τ
(ii) if u∈G∞(R2n) then T((ν u)(ξ,·))∈G∞(Rn);
τ τ τ
(iii) if u∈G (R2n) has a representative (u ) fulfilling the condition
τ ε ε
(1.11) ∀α,β,γ ∈Nn∃N ∈N sup (1+|y|)−N|ξβ∂α∂γu (ξ,y)|=O(ε−N)
ξ y ε
y∈Rn,ξ∈Rn
then T((ν u)(ξ,·))∈G (Rn);
τ S
(iv) if u∈G (R2n) has a representative (u ) fulfilling the condition
τ ε ε
(1.12) ∃M ∈N∀α,β,γ ∈Nn∃N ∈N sup (1+|y|)−N|ξβ∂α∂γu (ξ,y)|=O(ε−M)
ξ y ε
y∈Rn,ξ∈Rn
then T((ν u)(ξ,·))∈G∞(Rn).
τ S
9
Proof. WebeginbyobservingthatthegeneralizedfunctionT((ν u)(ξ,·))isdefinedbythenet(S (ξ)) :=
τ ε ε
(T (u (ξ,·))) , where (u ) ∈E (R2n) and (T ) satisfies the following condition:
ε ε ε ε ε τ ε ε
(1.13)
∃K ⋐Rn∃j ∈N∃N ∈N∃η ∈(0,1]∀u∈C∞(Rn)∀ε∈(0,η] |T (u)|≤ε−N sup |∂βu(y)|.
ε
y∈K,|β|≤j
Consequently if (u ) ∈E (R2n) then for all α∈Nn there exists N′ ∈N such that for all ε small enough
ε ε τ
the estimate
|∂αS (ξ)|=|T (∂αu (ξ,y))|≤ε−N sup |∂α∂βu (ξ,y)|≤cε−N−N′(1+|ξ|)N′
ε ε ξ ε ξ y ε
y∈K,|β|≤j
holds. This proves that (S ) ∈ E (Rn). In an analogous way we obtain that (S ) ∈ N (Rn) when
ε ε τ ε ε τ
(u ) ∈ N (R2n) and that (S ) ∈ E∞(Rn) when (u ) ∈ E∞(R2n). Note that for all ξ ∈ Rn, u(ξ,·) :=
ε ε τ ε ε τ ε ε τ
(u (ξ ,·)) +N(Rn)∈G(Rn). Therefore,for(T ) and(T′) differentnetsdefiningT and(u ) ∈E (R2n)
ε ε ε ε ε ε ε εfε τe
one has that
(S (ξ )−S′(ξ )):=(T (u (ξ ,·))−T′(u (ξ ,·)))
ε ε ε ε ε ε ε ε ε ε ε
is negligible. Since ξ is arbitrary this implies that (S −S′) ∈ N (Rn) and completes the proof of (i)
ε ε ε τ
and (ii). Let us assume that u ∈G (R2n) has a representative fulfilling (1.11). Then the corresponding
e τ
net (S ) , which is already known to belong to E (Rn), satisfies the following estimate:
ε ε τ
sup |ξβ∂αS (ξ)|= sup |ξβT (∂αu (ξ,y))|
ε ε ξ ε
ξ∈Rn ξ∈Rn
≤ε−N sup sup |ξβ∂α∂γu (ξ,y)|≤ε−N−N′ sup(1+|y|)N′,
ξ y ε
ξ∈Rny∈K,|γ|≤j y∈K
uniformly for small values of ε. This means that T((ν u)(ξ,·)) ∈ G (Rn). Finally, when (u ) satisfies
τ S ε ε
(1.12) then
sup |ξβ∂αS (ξ)|≤ε−N sup sup |ξβ∂α∂γu (ξ,y)|≤ε−N−M sup(1+|y|)N′,
ε ξ y ε
ξ∈Rn ξ∈Rny∈K,|γ|≤j y∈K
for some N,N′,M ∈ N. Since N and M do not depend on α,β we have that (S ) ∈ E∞(Rn) and
ε ε S
therefore T((ν u)(ξ,·))∈G∞(Rn).
τ S
Remark 1.7. As a straightforward application of the previous proposition we consider the action of a
basic functional T ∈ L(G(Rn),C) on e−iyξ ∈ G∞(R2n). Omitting the notation ν for simplicity, we are
τ τ
allowed to claim that
e
T(e−i·ξ):=(T (e−i·ξ)) +N (Rn)
ε ε τ
is a generalized function in G∞(Rn).
τ
1.2 Composition of a basic functional with an integral operator
Inthe sequelfor the advantageofthe readerwe recallthe results onintegraloperatorselaboratedin [14,
Proposition 2.14]. They are needed in stating and proving Proposition1.9.
Proposition 1.8. Let us consider the expression
(1.14) k(x,y)u(x)dx.
ZΩ′
(i) If k ∈G(Ω′×Ω) then (1.14) defines a C-linear continuous map
e
u→ k(x,y)u(x)dx
ZΩ′
from G (Ω′) into G(Ω);
c
10
