### An intrinsic definition of the Colombeau generalized functions

Jiˇr´ı Jel´ınek

Abstract. A slight modification of the definition of the Colombeau generalized functions
allows to have a canonical embedding of the space of the distributions into the space of
the generalized functions on aC^{∞}manifold. The previous attempt in [5] is corrected,
several equivalent definitions are presented.

Keywords: Colombeau generalized function, distribution, canonical embedding, mani- fold

Classification: 46F, 46F05

Introduction

The aim of Colombeau’s paper [5] was to avoid the drawback that the embed-
ding of the space D^{′} of the Schwartz distributions into the algebra (and sheaf)
of Colombeau generalized functions is not intrinsic: This canonical embedding
(even of the space C of continuous functions) defined by [4] is not kept under
coordinate diffeomorphisms. More precisely: If Ω,Ω are open sets in Euclideane
spaceR^{d},T a distribution on Ω, then by [4],T is identified with the generalized
functionhRihaving the function R(ϕ, x) =hT, ϕ(•−x)ias a representative (pro-
vided suppϕ⊂Ω−x). For a diffeomorphismµ:Ωe →Ω, the inverse image of the
distributionT, denoted byµ^{∗}T orT◦µorT(µ(x)), is defined in the usual way as
a distribution onΩ ([16]), while by [4], the inverse imagee µ^{∗}hRi is a generalized
functionhRihaving as a representative the function

ϕ,ex7→R(ϕ, µ(ex)) (xe∈Ω ).e

The distribution µ^{∗}T turns out to be associated with the generalized function
µ^{∗}hRi, but in general not identified in the above sense. For this reason, we cannot
define an algebraG(M) of generalized functions on aC^{∞} manifold M in such a
way that the spaceD^{′}(M) is canonically embedded inG(M). This inconvenience
can be removed by a slight change of the definition of the Colombeau generalized
functions and of their inverse image, which is attempted in [5].

Note that there are also simplified definitions of generalized functions of hyper-
function type where a representative is a sequence or a net ofC^{∞}functions (see

Supported by Research Grant GA ˇCR 201/97/1161.

[15], [9]). With these definitions,C^{∞} sheaf morphisms can be easily extended to
generalized functions and the generalized functions can be easily defined on aC^{∞}
manifold. There are embeddings ofD^{′} into such a space of generalized functions,
however no embedding is canonical. One cannot agree completely with a remark
in [15] referring to [1] that there is no need for a canonical embedding, since
in applications it matters to find a suitable embedding adapted to the problem
considered. The existence of an embedding suitable for all applications would
simplify the task. For instance in [15] it is proved in a rather complicated way
that there is a sheaf morphism (in the category of linear spaces) σ : D^{′} → G
identical onC^{∞} and such that the image of a distribution is associated with it.

Certainly, the constructive proof in [15] gives more, but the only formulation does not ensure even that the product of a continuous function with a Dirac measure is preserved (up to the association).

In [15] it is said: for a sheaf morphismσone cannot expect that it is compatible
with theC^{∞}module structure nor that it commutes with the differentiation in all
coordinates. As for the latter, we will see that in our case the canonical embed-
ding is a sheaf morphism commuting with the differentiation and, of course, with
coordinate diffeomorphisms. Moreover, thanks to the existence of the canonical
embedding, it is possible to define for instance the Colombeau product of distri-
butions on a manifold as it is done onR^{d}in [11].

Colombeau’s definitions

In the following, Ω will always be an open set in the Euclidean spaceR^{d}.
Notation 1 (by [4]).

A_{q}(R^{d}) :={ϕ∈ D(R^{d});R

ϕ= 1, R

ϕ(x)x^{β}dx= 0 for β∈N_{0}^{d},1≤ |β| ≤q},
A_{q}(M) :=A_{q}∩ D(M) for M ⊂R^{d}.

If there is no danger of misunderstanding, we writeA_{q} instead ofA_{q}(M). We de-
note by A := A_{0} − A_{0} and we do not introduce any special symbol for
A_{q}− A_{q} (q6= 0).

Originally, the notationϕε was used for the function

(1) ϕε(x) = 1

ε^{d}ϕ x
ε

(ε∈]0,1] ).

In [5], this notion is replaced with C^{∞} bounded paths of functions (ϕ^{ε})_{ε∈}_{]0,1]},
(ϕ^{ε} ∈ A_{0}), and with the unbounded paths (ϕε)_{ε∈}_{]0,1]}, developed from it by
ϕ_{ε}(x) = ^{1}

ε^{d}ϕ^{ε x}_{ε}

. We will accept this notation. There is another change in [5]:

A_{q} are no more sets of functions as above but sets of bounded paths satisfying
Z

x^{α}ϕ^{ε}(x) dx=O(ε^{q}) if α∈N^{d}

0,1<|α| ≤q, εց0.

Since we need both meanings of A_{q}, we keep Notation 1 above, used in [4], and
unlike in [5] we introduce semi-normsaqas follows.

Notation 2. Forϕ∈ A_{0}, we define
aq(ϕ) = sup

Z

x^{α}ϕ(x) dx

; α∈N_{0}^{d},1<|α| ≤q

. So we haveAq=

ϕ∈ A_{0};aq(ϕ) = 0 .

A similar change is done in [5] with the definition ofE[Ω], too: the set E[Ω]

containing the set of representativesE_{M}[Ω] is no more a set of functionsR(ϕ, x)
but the set of allC^{∞}mapsR(Φ, x) intoC^{]0,1]}where Φ = (ϕ^{ε})_{ε∈}_{]0,1]}is a bounded
path,x∈Ω and

R(Φ, x) = R(ϕε, x)

0<ε≤1.

If it is the case (and if there is not a misunderstanding), then the formula define a one-to-one mappingR ↔R, and there is no reason for accepting this change here:

E[Ω] will stand for the space of functionsR(ϕ, x) like in [4] and paths will only be
used to define the moderate growth and other similar notions. However, unlike
in [4] and as in [5], R(ϕ, x) are C^{∞} complex valued functions in both variables
ϕ∈ A_{0}, x∈ Ω simultaneously. Other notions defined in [5], like the set of the
moderate functionsE_{M}[Ω]⊂ E[Ω], will be introduced or recalled later.

3. Now, ifµ:Ωe →Ω is a diffeomorphism, a representativeRe of the composition
hRi◦µ(i.e. of the inverse imageµ^{∗}hRi) is defined in [5] by the formula

R(ϕe _{ε},ex) =R ϕf_{ε}, µ(x)e

(ex∈Ω ),e

whereϕfεis defined by a rather complicated formula in order to obtain a compo- sition for the generalized functions equal to the classical one for the distributions.

There is however an apparent inconsistency: Re seems to depend on ε. In our
new notation the formulas will be simpler and will not containε. Unfortunately
there is a true inconsistency, too: ϕfε depends onx and the definition ofE_{M}[Ω]

does not deal with test functions depending on x (i.e. on the second variable of R). As a consequence, it may happen that Re is not moderate even if R is.

For instance, if hRi is a constant generalized function on R with a representa- tiveR(ϕ, x) = exp iexpR

|ϕ(x)|^{2}dx

, thenR∈ E_{M}[Ω], and one can check using
formulas in [5] (see also (42) later) that, for arbitrary non-linear coordinate dif-
feomorphism µ, the first derivative of Re does not have a moderate growth. In
order to correct it, we have to modify the definition ofE_{M}[Ω] and, as consequence,
to restrict the set of generalized functions only accepting those one which have
moderate growth in all coordinate systems.

Change 4in notation. The representative which is denoted byR(ϕ, x) in [4] will be denoted by R(ϕ(•−x), x) here. In other words, our notationR(ϕ, x) means what was denoted byR(ϕ(x+•), x) in [4].

According to the definition of the null idealN in [4], only the valuesR(ϕ, x) matter for determining the generalized functionhRi, where suppϕis in an arbi- trarily chosen neighborhood of 0. In our notation, only the valuesR(ϕ, x) matter where suppϕis in a neighborhood of the pointx. So the values for suppϕ⊂Ω suffice and we can formulate the definition ofE[Ω] as follows.

Definition 5. Ω being an open set inR^{d}, we defineE[Ω] to be the set of allC^{∞}
maps

R:A_{0}(Ω)×Ω→C
ϕ, x 7→R(ϕ, x).

Thus the test functions have their supports in Ω. This is more natural and
will simplify the definition of generalized functions on a C^{∞} manifolds: in this
caseϕ will be defined on this manifold. With this change the embedding ofD^{′}
into G becomes simpler: iff is a distribution, then the function ϕ7→ hf, ϕiis a
representative off as a generalized function. However, some other notions become
more complicated, the formula (1) forϕ_{ε}is even useless in this simple form. Also,
the notion of a constant generalized function becomes less natural (anyway, on a
manifold this notion has no sense) and the definition of the derivative becomes
more complicated. For this reason, we are introducing the notation (R)εreplacing
the notation (1).

Notation 6. IfR∈ E[Ω], we denote by (R)_{ε} or simplerR_{ε}, if there is no danger
of misunderstanding, the function defined on a part ofA_{0}(R^{d})×Ω by

Rε(ϕ, x) =R(ϕx,ε, x) with ϕx,ε(ξ) =ε^{−d}ϕ(^{ξ−x}_{ε} )

(provided suppϕ_{x,ε}⊂Ω ). Equivalently, R(ψ, x) =R_{ε} ε^{d}ψ(x+•ε), x
.

By Change 4, for ε = 1 we get the original notion of representative introduced
in [4]. Only the values R_{ε}(ϕ, x) with suppϕ in a neighborhood of zero matter
for determining the generalized function hRi. Note that suppϕx,ε −→ {x} for
ε ց 0 (uniformly when ϕ runs over a set of functions with uniformly bounded
supports).

7. As we have already noticed, the definition of moderate growth of the represen-
tativesR(ϕ, x) of the generalized functions must be modified, taking into account
the dependence ofϕonx. Thus the definition becomes more complicated. On the
other hand, we simplify this definition, requiring the moderate growth ofR(ϕ, x)
for all paths (ϕ^{ε})_{ε}, unlike Definition 3 in [5], where this was required only for
(ϕ^{ε})_{ε}∈ A_{N} (using the notation in [5]). We can see later, using Theorem 21, that
this restriction does not restrict the set of generalized functions.

It does not matter that the paths (ϕ^{ε})_{ε} in [5] areC^{∞}in the variableε. So we
replace them simply with bounded sets of test functions.

Notation. IfF is a locally convex space, denote byE(Ω→F) the locally convex
space of allC^{∞} maps (vector valued functions)

Φ = (ϕ_{x})_{x∈Ω}: Ω→ F
x7→ϕx

with the usual topology of locally uniform convergence of every derivative with
respect tox. ByE(Ω→A_{q}) we mean the topological (affine) subspace ofE(Ω→D)
consisting of theAq-valued functions.

It is useful to consider the convergence

εց0limΦ^{ε}= Φ ( Φ∈ E(Ω→F) ),

even in the case when the maps Φ^{ε} ∈ E(Ω_{ε}→F) are not defined on the same
set. We only need that every compact K ⋐Ω is contained in Ω_{ε} for allε > 0
sufficiently small.

Definition 8. E_{M}[Ω] is the set of all R ∈ E[Ω] such that ∀K ⋐ Ω (compact),
α ∈ N^{d}

0 ∃N ∈ N such that _{∂x}^{∂} _{α}

Rε(ϕx, x) = O(ε^{−N}) (ε ց 0) uniformly
whenx∈Kand (ϕ_{x})_{x∈Ω} runs over any bounded subset ofE Ω→A_{0}(R^{d})

.
If the variable (ϕx)_{x∈Ω} runs over a bounded subset of E Ω→A_{0}(R^{d})

, then
the values ϕ_{x}, for x ∈ K, remain bounded in A_{0}(R^{d}). Hence their supports
are uniformly bounded in R^{d}. It is easy to check from the definition ofR_{ε} that
R_{ε}(ϕ_{x}, x) is always defined (andC^{∞}with respect tox) for allεsufficiently small
independently on theseϕx andx∈K.

Remark. Evidently, the moderate growth condition in this definition can be
equivalently formulated as follows. ∀K⋐Ω (compact), α∈N^{d}

0 ∃N ∈Nsuch that, for every bounded path

(ϕ^{ε}_{x})_{x∈Ω};ε∈]0,1] ⊂ E Ω→A_{0}(R^{d})

, we have

∂

∂x

α

R_{ε}(ϕ^{ε}_{x}, x) = O(ε^{−N}) (ε ց 0) uniformly with respect to x ∈ K. Here

“bounded path” means simply a bounded set of elements depending onε∈]0,1].

The smoothness with respect toεis not required. However, if the smoothness is required, it can be easily shown that the above formulation remains equivalent.

We will do a similar thing in details in the proof of Equivalent definitions 18.

Differential calculus

9. We recall some theorems from differential calculus ([2], [17]) which we will need
later. Theorems are usually formulated for vector valued functions defined on an
open subset of a locally convex space; however they can be evidently generalized
for functions defined on an open subset of an affine space, for instanceA_{0}provided
the derivatives are taken with respect to vectors belonging toA=A_{0}−A_{0}. While
applying differential calculus, we consider a complex linear structure to be a real
one, the differential means the Fr´echet differential.

Notation. LetX, Y be locally convex spaces,U an open subset ofX,R:U →Y
a mapping. We denote the value of the k-th Fr´echet differential ofRat the point
u∈Uwith respect to the vectorsx_{1}, . . . , x_{k} ∈ Xby d^{k}R(u)[x_{1}, . . . , x_{k}]. Different
brackets, used for clarity, are not obligatory. Another notation d^{k}_{x}_{1}_{,...,x}_{k}R(u) is
used mainly for the first differential.

Theorem 10 ([2, 1.2.5], [17, 1.8.2]). The k-th differential d^{k}R(u), if it exists,
belongs to the space L_{s}(^{k}X→Y) of all hypo-continuous symmetric poly-linear
(here: k-linear)maps of X^{k} into Y, endowed with the topology of the uniform
convergence on the cartesian products of kbounded subsets of X.

Note that ifXis a Fr´echet spaces (and this is always here), any hypo-continuous poly-linear map is continuous.

If the mapu7→d^{k}R(u) is continuous, thenR is said to beC^{k} (or of the class
C^{k}). If it is so for allk∈N_{0}, thenRis said to be of the classC^{∞}(d^{0}RmeansR).

Theorem 11(Mean value theorem [17, 1.3.3.4^{◦}]). If Ris C^{1}on an open neigh-
borhoodU of a segment[u, u+x]⊂X then

R(u+x)−R(u)∈conv{dR(u+tx)[x];t∈[0,1]}

(a closed convex hull).

12. For the theorem on the differentiation of a composition ([17, 1.5.3]), we in- troduce the following notations. For a finite set I ⊂ N, we denote by #I its cardinality and by I =

i_{1}, . . . , i_{#I} its elements in the increasing order. If we
have elementsx1, x2, . . ., then we denote the finite sequencex_{i}_{1}, . . . , x_{i}_{#I} byx_{I}.
By a decomposition ofI we mean a subset I ={I_{1}, . . . , I_{k}} of expIr{∅}such
that the setsI_{1}, . . . , I_{k}are non-empty, pairwise disjoint andS

I_{j} =I.

Theorem. LetX, Y, Z be locally convex spaces,U, V open sets in X, Y respec-
tively,R :U →Y, S :V →Z maps of the class C^{n},(n∈N), R(U)⊂V. Then
S◦R is a map of the classC^{n}and, foru∈U andx_{1}, . . . , x_{n} ∈ X, we have

d^{n}(T◦S)(u)[x_{1}, . . . , x_{n}]

= Xn k=1

X

{I1,...,Ik} pairwise disjoint,

S

Ij={1,2,...,n}

d^{k}T(S(u))h

d^{#I}^{1}S(u)[x_{I}_{1}], . . . ,d^{#I}^{k}S(u)[x_{I}_{k}]i
,

where the summation is extended over all decompositionsI={I_{1}, . . . , I_{k}} of the
multi-indexI={1, . . . , n}.

As a special case, we have for the first differential d(T◦S)(u)[x] = dT(S(u)) [dS(u)[x]].

13. The following theorems concern mappings of two variables. According to our
needs we will formulate them for a mapping of an open subset of A ×R^{d} (or
A_{q}×R^{d}) with values in a locally convex space Z. In order to avoid the use of
indexes in the notation of partial differentials, we will denote the total differential
by the letterd, the partial differential with respect to the variableϕ∈ A resp.

x∈R^{d}by d resp. ∂. For the latter we also use the symbol∂^{α} (α∈N^{d}

0 ), which
denotes the α-th derivative. Thus∂^{α}R(ϕ, x) = _{∂x}^{∂} α

R(ϕ, x), provided ϕdoes not depend onx.

Theorem 14 ([17, 1.11.2]). If the first differential dR exists in a point
(ϕ, x)∈ A_{0}(Ω)×R^{d}, then dRand∂R exist in(ϕ, x)and

dR(ϕ, x)[ψ, h] = d_{ψ}R(ϕ, x) +∂_{h}R(ϕ, x) (ψ∈ A, h∈R^{d}).

It follows for the differentials of higher degree

d^{n}R(ϕ, x)[(ψ1, h1), . . . ,(ψn, hn)] = (d_{ψ}_{1} +∂_{h}_{1}). . .(d_{ψ}_{n}+∂_{h}_{n})R(ϕ, x)

= X

I⊂(1,2,...,n)

d^{#I}_{ψ}

I∂_{h}^{n−#I}

(1,...,n)rIR(ϕ, x)

(using the notation for Theorem 12).

Theorem 15([17, 1.11.3]). A mapRis of the classC^{1}iff the partial differentials
dR and∂Rexist and are continuous.

Theorem 16 (Schwartz, [17, 1.11.5.2^{◦}]). If dR and∂R exist and if d_{ψ}∂_{h}R or

∂_{h}d_{ψ}R is continuous on a neighborhood of a point (ϕ, x), then d_{ψ}∂_{h}R(ϕ, x) =

∂_{h}d_{ψ}R(ϕ, x).

Remark. Ifd^{2}R(ϕ, x) exists, then d_{ψ}∂_{h}R(ϕ, x) =∂_{h}d_{ψ}R(ϕ, x).

Indeed, by Theorem 14,

dR(ϕ, x)[(ψ,0)] = d_{ψ}R(ϕ, x),
d^{2}R(ϕ, x)[(ψ,0),(0, h)] =∂_{h}d_{ψ}R(ϕ, x)

and the bilinear mappingd^{2}R(ϕ, x) on the left hand side is symmetric by Theo-
rem 10.

Note that we deal only withC^{∞}maps in this paper, hence the order of taking
derivatives does not matter.

Example (The differential of the product). If
F :R^{2} →R

x, y7→xy,

then

dF(u, v)[(x, y)] =uy+vx
d^{2}F(u, v)[(x1, y1),(x2, y2)] =x1y2+x2y1

d^{n}F = 0 for n≥3.

Results

Theorem 17 (Equivalent definition of representatives). ForR∈ E[Ω], we have
R ∈ E_{M}[Ω]iff the partial differentials d^{k}R_{ε} have a moderate growth in the fol-
lowing sense: ∀K⋐Ω, α∈N^{d}

0, k∈N_{0} ∃N∈Nsuch that
(2) ∂^{α}d^{k}R_{ε}(ϕ, x)[ψ_{1}, . . . , ψ_{k}] =O(ε^{−N}) (εց0 )

uniformly whenx∈K, ϕis in a bounded subset ofA_{0}(R^{d})andψ1, . . . , ψ_{k}are in
a bounded subset of A(R^{d}).

This means: if we include partial differentials in the definition of the moderate growth, we do not need to considerϕdepending onx(unlike in Definition 8).

Proof: I. If the condition is fulfilled, we calculate _{∂x}^{∂} α

Rε(ϕx, x) using Theo- rem 12 on differentiation of a composition.

II. Suppose R ∈ E[Ω] (Definition 8). We have to prove (2) for a suitable N
(depending onαandk), uniformly forx∈K,ϕ∈ A_{0}∩ Bandψ1, . . . , ψ_{k}∈ A ∩ B
(B is a bounded subset ofD(R^{d})). Fora= (a_{1}, . . . , a_{d})∈K,x= (x_{1}, . . . , x_{d})∈
Ω,ϕ, ψ1, . . . , ψ_{k}running over bounded sets as above andt1, . . . , t_{k}attaining values
0,1, . . . , k, let us define

(3)

ϕx=ϕx,t1,...,tk :=ϕ+ Xk j=1

t_{j}ψ_{j}

(|α|+k^{2}+j)!(x_{d}−a_{d})^{|α|+k}^{2}^{+j},
p:=

Xk j=1

|α|+k^{2}+j

=k|α|+k^{3}+
k+ 1

2

.

By Definition 8, there is a numberN ∈N_{0} (depending onK, αandp) such that

(4) _{∂x}^{∂}_{α} _{∂}

∂xd

_{p}

Rε(ϕx, x) =O(ε^{−N}) (εց0)

uniformly ifx∈K and if (ϕx)_{x∈Ω} runs over a bounded subset ofE Ω→A_{0}(R^{d})
.
We will only use it forx=a∈K. The derivative at the left hand side of (4)
is the value of the differential with respect to the vectors

(5) h_{1}, h_{2}, . . . , h_{|α|+p}

such that exactly α_{j} of them are equal to the coordinate unit vector e_{j} =
(0, . . . ,0,1,0, . . . ,0) (j = 1, . . . , d−1) and α_{d}+p of them are equal to e_{d}. We
apply Theorem 12 on the differentiation of a composition to the composition of
R withx7→(ϕ_{x}, x) atx=a. The inner mappingx7→(ϕ_{x}, x) has the following
value and derivatives atx=a:

(ϕx, x) = (ϕ, a)

∂

∂x_{j}(ϕx, x) = (0, e_{j})
(6)

∂

∂x_{d}

|α|+k^{2}+j

(ϕx, x) = (t_{j}ψ_{j},0) (j= 1,2, . . . , k).

The other derivatives with respect to coordinate unit vectors are = 0 atx=a.

So, only those decompositions I of the multi-index I = (1,2, . . . ,|α|+p) can
give non-zero terms in the sum in Theorem 12, that every element of I either
is a singleton (i.e. has the cardinality 1) or has the cardinality |α|+k^{2} +j for
some j = 1, . . . , k. Moreover, every h_{i} 6= e_{d} must belong to a singleton of I.

The number ek of elements of I that are not singleton (even if they have the
less possible cardinality|α|+k^{2}+ 1) cannot be greater thenk: if ek=k+ 1 we
would have (k+ 1)(|α|+k^{2}+ 1)≤ |α|+p, which contradicts (3). It follows from
Theorem 12 that _{∂x}^{∂} α ∂

∂xd

p

Rε(ϕx,t1,...,tk) at x=a equals to a sum of terms of the form

(7)

d^{e}^{k+|}^{α|}^{e}R_{ε}(ϕ, a)

(t_{j}_{1}ψ_{j}_{1},0), . . . ,(t_{j}

ekψ_{j}

ek,0),(0, h_{n}_{1}), . . . ,(0, h_{n}_{|}

α|e)

=t_{j}_{1}. . . t_{j}

ekd^{e}^{k}∂^{α}^{e}R_{ε}(ϕ, a)

ψ_{j}_{1}, . . . , ψ_{j}

ek

(ek≤k)

(the numbersek,α, je 1, . . . , jek depend on I and can be the same for different de-
compositionsI). By (3) we see that there is at least one decomposition I of I
giving the termt_{1}·. . .·t_{k}∂^{α}d^{k}Rε(ϕ, a)

ψ_{1}, . . . , ψ_{k}

. Choose coefficientsc_{0}, . . . , c_{k}
fulfilling the equations

(8)

Xk j=0

c_{j}·j = 1
Xk

j=0

c_{j}·j^{n}= 0 for n= 0 or 2,3, . . . , k.

It follows from (4) that (9)

Xk t1=0

· · · Xk tk=0

c_{t}_{1}. . . c_{t}_{k} _{∂x}^{∂} _{α} _{∂}

∂xd

_{p}

R_{ε}(ϕ_{x,t}_{1}_{,...,t}_{k}, x) =O(ε^{−N})

(uniformly under the requirements as above). By (7) and (8), the left hand side is
the sum only of terms of the form∂^{α}^{e}d^{k}Rε(ϕ, a)

ψ1, . . . , ψ_{k}

for some multi-index e

αand there is at least once the term∂^{α}d^{k}Rε(ϕ, a)

ψ_{1}, . . . , ψ_{k}

. As everyh_{i}6=e_{d}
belongs to a singleton, we haveαe_{j} =α_{j} forj < d. Considering the cardinalities
of the elements ofI, we see that|α|e =|α|, soαe=α. Thus the left hand side of
(9) is a natural multiple of∂^{α}d^{k}Rε(ϕ, a)

ψ_{1}, . . . , ψ_{k}

and this is what we wanted

to prove.

18. While we had to modify the definition ofE_{M}[Ω] in [5], there is no need to do
the same with the definition of the idealN, serving as representatives of the null
generalized function, thanks to the following equivalences.

Equivalent definitions. The ideal N[Ω] ⊂ E_{M}[Ω] is defined to be the set of
all representatives fulfilling one of the following equivalent conditions (A_{q} means
Aq(R^{d})).

1^{◦} (the definition in [4], where only the uniformity with respect to ϕis not re-
quired). ∀K⋐Ω, α∈N^{d}

0, n∈N ∃q∈Nsuch that ∀ boundedB ⊂ D(R^{d}), we
have

∂^{α}Rε(ϕ, x) =O(ε^{n})
uniformly forϕ∈ A_{q}∩ B, x∈K.

2^{◦} (the same for the differentials). ∀K ⋐ Ω, α∈ N^{d}

0, k ∈N_{0}, n ∈N ∃q ∈N
such that∀ boundedB ⊂ D(R^{d}), we have

(10) ∂^{α}d^{k}Rε(ϕ, x)[ψ1, . . . , ψ_{k}] =O(ε^{n})
uniformly for

(11) ϕ∈ A_{q}∩ B, ψ_{1}, . . . , ψ_{k}∈(A_{q}− A_{q})∩ B, x∈K.

3^{◦} ∀K ⋐ Ω, α ∈ N^{d}

0, n ∈ N ∃q ∈ N such that, for every bounded path
(ϕ^{ε}_{x})_{x∈Ω}; ε∈]0,1] ⊂ E Ω→Aq(R^{d})

, which isC^{∞}with respect toε, we have

(12) _{∂x}^{∂} _{α}

R_{ε}(ϕ^{ε}_{x}, x) =O(ε^{n})
uniformly forx∈K.

4^{◦} (the definition in [5]). ∀K⋐Ω, α∈N^{d}

0, n∈N ∃q∈N such that, for every bounded path

(ϕ^{ε}_{x})_{x∈Ω};ε∈]0,1] ⊂ E Ω→A_{0}(R^{d})

, which isC^{∞} with respect
toεand fulfills

aq(ϕ^{ε}_{x}) =O(ε^{q}) (εց0 )

uniformly forx∈K(foraq see Notation 2), we have (12) uniformly forx∈K.

Proof of 1^{◦} ⇔ 2^{◦}: ⇐ being evident, we are going to deduce 2^{◦} from 1^{◦} by
induction. Denote byS(k) (k∈N_{0}) the statement

S(k) : ∀K⋐Ω, α∈N^{d}

0, n∈N ∃q∈Nsuch that∀ boundedB ⊂ D(R^{d})
(10) holds uniformly under the requirements (11).

S(0) is the definition 1^{◦}. SupposingS(k−1), we will proveS(k) by contradiction
(k ∈ N). If S(k) does not hold, choose K ⋐ Ω, α ∈ N^{d}

0, n ∈ N such that

∀q∈N ∃ Bfor which (10) does not hold uniformly under the requirements (11).

ChooseN by Theorem 17 (an equivalent definition of representatives) and then chooseqby the induction hypothesisS(k−1) such that

d^{k+1}∂^{α}R_{ε}(ϕ, x)[ψ_{1}, . . . , ψ_{k−1}, ψ_{k}, ψ_{k}] =O(ε^{−N}),
(13)

d^{k−1}∂^{α}Rε(ϕ, x)[ψ_{1}, . . . , ψ_{k−1}] =O(ε^{2n+N}^{+2}),
(14)

uniformly under the requirements (11) for any boundedB ⊂ D(R^{d}). Since, for
thisq, (10) does not hold uniformly, there are bounded sequences of test functions

ϕ_{j} ∈ A_{q}, ψ_{1,j}, . . . , ψ_{k,j}∈ A_{q}− A_{q} (j∈N)
andxj ∈K, εj ց0, εj ∈]0,1] such that

(15) d^{k}∂^{α}R_{ε}_{j}(ϕ_{j}, x_{j})[ψ_{1,j}, . . . , ψ_{k,j}]≥2ε_{j}^{n}.
By (13) we get

d^{k+1}∂^{α}R_{ε}_{j}(ϕ_{j}+tψ_{k,j}, x_{j})[ψ_{1,j}, . . . , ψ_{k−1,j}, ψ_{k,j}, ψ_{k,j}]≤ε_{j}^{−N}^{−1}
for alljsufficiently great independently ont∈[0,1]. Therefore, for

0< t≤ε_{j}^{n+N+1}
we obtain from the Mean Value Theorem (Theorem 11)

d^{k}∂^{α}R_{ε}_{j}(ϕ_{j}+tψ_{k,j}, x_{j})[ψ_{1,j}, . . . , ψ_{k,j}]−d^{k}∂^{α}R_{ε}_{j}(ϕ_{j}, x_{j})[ψ_{1,j}, . . . , ψ_{k,j}]≤

sup

t^{′}∈[0,t]

d^{k+1}∂^{α}Rεj(ϕ_{j}+t^{′}ψ_{k,j}, x_{j})[ψ_{1,j}, . . . , ψ_{k,j}, tψ_{k,j}]≤ε_{j}^{−N}^{−1}ε_{j}^{n+N}^{+1}=ε_{j}^{n}.
This means

(16) d^{k}∂^{α}Rεj(ϕ_{j}+tψ_{k,j}, x_{j})[ψ_{1,j}, . . . , ψ_{k,j}] ∈ B(d_{j}, ε_{j}^{n})
(the closed ball inR), where we have denoted by

(17) d_{j} :=d^{k}∂^{α}Rεj(ϕ_{j}, x_{j})[ψ_{1,j}, . . . , ψ_{k,j}].

Again from the Mean Value Theorem and (16), we get

d^{k−1}∂^{α}Rεj(ϕ_{j}+ε_{j}^{n+N+1}ψ_{k,j}, x_{j})[ψ_{1,j}, . . . , ψ_{k−1,j}]

−d^{k−1}∂^{α}Rεj(ϕ_{j}, x_{j})[ψ_{1,j}, . . . , ψ_{k−1,j}] ∈
convn

d^{k}∂^{α}Rεj(ϕ_{j}+tψ_{k,j}, x_{j})[ψ_{1,j}, . . . , ψ_{k−1,j}, ε^{n+N+1}_{j} ψ_{k,j}];t∈(0, ε^{n+N+1}_{j} )o

⊂ B(ε_{j}^{n+N+1}d_{j}, ε_{j}^{2n+N}^{+1}).

Thanks to (15) and (17), it follows

(18)

d^{k−1}∂^{α}R_{ε}_{j}(ϕ_{j}+ε_{j}^{n+N+1}ψ_{k,j}, x_{j})[ψ_{1,j}, . . . , ψ_{k−1,j}]

−d^{k−1}∂^{α}Rεj(ϕ_{j}, x_{j})[ψ_{1,j}, . . . , ψ_{k−1,j}] ≥ ε_{j}^{2n+N}^{+1}.

The functionsϕ_{j}+ε_{j}^{n+N+1}ψ_{k,j} form a bounded set, hence by (14) the left hand
side of (18) should be =O(ε_{j}^{2n+N+2}). This contradicts (18).

Proof of1^{◦} or2^{◦}⇔3^{◦}: 2^{◦}⇒3^{◦} can be calculated using Theorem 12 (on the
differentiation of a composition) and 14.

If 1^{◦}does not hold, there are K⋐Ω, α∈N^{d}

0, n∈Nsuch that for everyq∈N
we can find sequencesε_{j} ց0 withε_{1}> ε_{2} > . . ., x_{j} ∈K and

ϕ_{j} bounded in
A_{q} such that

(19) ∂^{α}R_{ε}_{j}(ϕ_{j}, x_{j})6=O(ε^{n}).

Choose a decomposition of unityP

λ_{j} = 1 on the interval ]0,1] with test functions
λ_{j} ∈ D(]ε_{j+1}, ε_{j−1[}) (j= 2,3, . . .), λ_{1}∈ D(]ε_{2},∞[), λ_{j}(ε_{j}) = 1. Then the path
of constantAq-valued functions

P_{∞}

j=1λ_{j}(ε)ϕ_{j}

x∈Ω ∈ E Ω→A_{q}(R^{d})

(ε∈]0,1] )

has, for ε = ε_{j} and x = x_{j}, the values ϕ_{j}, therefore due to (19) it does not

satisfy 3^{◦}.

Proof of 3^{◦} ⇔ 4^{◦}: ⇐ being evident, we are proving ⇒. For the given K
and α take first a number N by Theorem 17 (an equivalent definition of the
representatives) such that

(20) ∂^{α}dR_{ε}(ϕ, x)[ψ] =O(ε^{−N})

uniformly ifx∈K and if ϕ, ψ run over bounded sets in A_{0}(R^{d}), A(R^{d}) respec-
tively. Then, having chosenn, letq satisfy 3^{◦} and at the same time

(21) q≥n+N.

LetB⊂R^{d}be a bounded set containing the supports of allϕ^{ε}_{x}withx∈Kand let
ε_{0} >0 be such thatRε(ϕ, x) is always defined whenever 0< ε≤ε_{0},ϕ∈ A_{0}(B)
andx∈K.

Recall a known lemma of functional analysis ([14, II.3, Lemma 5): If linear
formsf_{0}, f_{1}, . . . , f_{k} on a linear spaceE are linearly independent, then there is a
pointx∈E such thatf_{0}(x) = 1,f_{1}(x) =· · ·=f_{k}(x) = 0.

Since the functionsx7→x^{β} considered as distributions∈ D^{′}(B) with β∈N^{d}

0, 0 ≤ |β| ≤ q, are linearly independent, there are test functions ψα ∈ D(B), 0≤ |α| ≤q, fulfilling

Z

ψα(ξ)·ξ^{α}dξ= 1
(22)

Z

ψα(ξ)·ξ^{β}dξ= 0 for β6=α,0≤ |β| ≤q.

(23)

Henceψ_{α}∈ A(B), except forψ_{0}, which we will not need. If we denote

(24) cα,x,ε:=

Z

ϕ^{ε}_{x}(ξ)ξ^{α}dξ,
we have

(25) κ^{ε}_{x}:=ϕ^{ε}_{x}− X

α∈N^{d}
1≤|α|≤q0

c_{α,x,ε}ψ_{α} ∈ A_{q}.

By the hypothesis of 4^{◦}, the definition ofaq (in Notation 2) and (24), we have

(26) c_{α,x,ε}=O(ε^{q}).

Let us order the summation indexesαin (25) into a sequenceα1, . . . , αm. Using the Mean Value Theorem 11, we have

∂^{α}Rε ϕ^{ε}_{x}, x

−∂^{α}Rε κ^{ε}_{x}, x

= Xm

j=1

∂^{α}Rε κ^{ε}_{x}+Pj

i=1cαi,x,εψαi, x

−∂^{α}Rε κ^{ε}_{x}+Pj−1

i=1cαi,x,εψαi, x

∈ Xm

j=1

convn

∂^{α}dRε κ^{ε}_{x}+Pj−1

i=1cαi,x,εψαi+tcαj,x,εψαj

cαj,x,εψαj

; t∈]0,1]o .

Due to (26) and (20), it follows

∂^{α}Rε ϕ^{ε}_{x}, x

−∂^{α}Rε κ^{ε}_{x}, x

=O(ε^{q−N})

uniformly for x ∈ K. It is also = O(ε^{n}) due to (21). Now, by (25) and 3^{◦},

∂^{α}R_{ε} κ^{ε}_{x}, x

=O(ε^{n}), hence also∂^{α}R_{ε} ϕ^{ε}_{x}, x

=O(ε^{n}).

19. We can easily see like in [4] that N[Ω] is an ideal in the algebra E_{M}[Ω], so
we can defineG[Ω] as follows.

Definition. The space of generalized functions on Ω is the quotient algebraG =

EM[Ω]

N[Ω] .

Notation. The generalized function with the representative R, i.e. the class of the representatives defining the same generalized function asR, is denoted byhRi.

Proposition 1^{◦} (Moderate growth as a local property). A function R ∈ E[Ω]

belongs to E_{M}[Ω] iff ∀x∈ Ωthere is an open neighborhood U of xin Ωsuch
thatR∈ E_{M}[U] (after the restriction ofRonA_{0}(U)×U).

2^{◦}. (N as a local property). A representativeN belongs toN[Ω]iff ∀x∈Ωthere
is an open neighborhoodU of xin Ω such thatR∈ N[U] (after the restriction
ofRonA_{0}(U)×U).

3^{◦}. G is a sheaf.

Proof: The statements 1^{◦} and 2^{◦} are an easy consequence of the following
observation (see Notation 6): If ε ց 0, then suppϕx,ε tends to {x} uniformly
with respect toϕrunning over a bounded subset ofA_{0}(R^{d}).

3^{◦} is similar as in [4] ( 1.3, Local properties. . . ).

Notation 20(the values of a representative which matter). 1^{◦}. Letx7→q_{x}∈N_{0}
be an upper semi-continuous function on Ω and (U_{x})_{x∈Ω}be a family of open neigh-
borhoods of pointsx, contained in Ω, which are locally uniform in the following
sense: for everyx∈Ω there is a neighborhoodV ofxsuch that T

y∈V

(U_{y}−y) is a
neighborhood of 0. Under these hypotheses we denote

U=U (Ux, qx)_{x∈Ω}
:=

(ϕ, x);x∈Ω,suppϕ⊂Ux, ϕ(•−x)∈ Aqx .
If R ∈ E[Ω] is a representative, then we can check from Definition 18.1^{◦} of N
that only the values R(ϕ, x) matter for which (ϕ, x) ∈ U. This means that if
two representatives are equal for these pairs (ϕ, x), they determine the same
generalized function.

2^{◦}. Let (V_{i})_{i∈I} be an open covering of Ω withV_{i} ⊂Ω for alli∈I, whereI is an
arbitrary set of indexes, and let{q_{i}}_{i∈I} be a family of numbersq_{i}∈N_{0}. Denote
by

V=V (Vi, qi)_{i∈I}
:=

(ϕ, x);∃i∈Isuch thatx∈V_{i},suppϕ⊂V_{i}, ϕ(•−x)∈ A_{q}_{i} .

IfR∈ E[Ω] is a representative, then only the valuesR(ϕ, x) matter for determining hRi, for which (ϕ, x)∈V(see the following proposition).

Proposition. For each set U according to 1^{◦} there is a set V ⊂ U according
to2^{◦}. For each setV according to2^{◦} there is a setU⊂V according to1^{◦}.
Proof: I. Using the uniformity condition in 1^{◦}, forx∈ Ω choose its neighbor-
hood V_{x} such that everyU_{y} fory ∈ V_{x} contains an open ball B(y, r) (r > 0 is
independent ony). Then changeV_{x} for a smaller one so that its diameter

(27) diamVx≤r

and that the function y 7→ qy is bounded on Vx by a number eqx ∈ N_{0}. Thus
we obtainV =V (V_{x},qe_{x})_{x∈Ω}

⊂U. Indeed, if (ϕ, y)∈V, we have for somex:

y∈V_{x}, suppϕ⊂V_{x}⊂B(y, r) by (27). HenceV_{x}⊂U_{y} andϕ(•−y)∈ Aeqx ⊂ A_{q}_{y}.
This means (ϕ, y)∈U.

II. Taking a refining, we can suppose without a loss of generality that (V_{i})_{i∈I}is
locally finite andV_{i} are relatively compact in Ω. Let (W_{i})_{i∈I}be an open covering
of Ω with W_{i} ⊂V_{i} for every i∈ I (this is possible for instance according to [7,
Chapter 5, p. 207, Lemma 1] in a normal space (even with a point finite open
covering)). We defineUas follows:

Ux:= \

x∈Wi

V_{i}, qx:= max

q_{i};x∈W_{i} ,

where the intersection and the maximum are extended over those i for which
x ∈ Wi. Fix a point x ∈ Ω. As (Wi)_{i∈I} is locally finite, there is an open
neighborhoodV ofxsuch thatV is compact in Ω and does not meet anyW_{i}with
x /∈W_{i}. Thus, for y∈V, it isU_{y} ⊃U_{x} andq_{y} ≤q_{x}. Hence the functionx7→q_{x}
is upper semi-continuous. SinceU_{x} is an open neighborhood of the compact set
V, the neighborhoodsU_{x}−yof pointsy∈V are uniform. Therefore,U_{y}−y are

uniform as well.

21. The following useful theorem shows that a representative need not be defined on the whole setA(Ω)×Ω. It is sufficient for determining hRionly to define R on a setUorV defined in Notation 20.

Theorem. 1^{◦}. Let R^{◦} be a C^{∞} function defined on a setV (V_{i}, q_{i})_{i∈I}
. Then
there is a C^{∞} function R on A_{0}(R^{d}) ×Ω coinciding with R^{◦} on some set
U (Ux, qx)_{x∈Ω}

.

2^{◦}. Suppose in addition thatR^{◦} satisfies the moderate growth condition in Defi-
nition 8, whenever(ϕx)_{x∈Ω} runs over such a bounded set ofE Ω→A_{0}(R^{d})

that

∂x∂

_{α}

(R^{◦})ε(ϕx, x)is defined forx∈K andεsufficiently small independently on
x∈K. ThenR from the part1^{◦} can be chosen in addition∈ E_{M}[Ω].

Proof of 1^{◦}: Taking a refining, we can suppose that (V_{i})_{i∈I} is in addition
locally finite and that V_{i} ⋐ Ω. Choose a locally finite open covering (W_{i})_{i∈I}
of Ω, with W_{i} ⊂ V_{i}, and a smooth partition of unity (τ_{i})_{i∈I} subordinated to
(W_{i})_{i∈I} :τ_{i}∈ D(W_{i}),P

τ_{i}= 1 on Ω. The idea of the proof is to define
R(ϕ, x) :=X

τ_{i}(x)R_{i}(ϕ, x)
(28)

with R_{i}(ϕ, x) :=R^{◦} π_{i}(ϕ), x

(x in a neighborhood of suppτ_{i}),
whereπ_{i} is an appropriate mapping (depending onx) of A_{0}(R^{d}) intoA_{0}(V_{i}). If
x /∈suppτ_{i}, then the termτ_{i}(x)R_{i}(ϕ, x) is considered to be = 0 even ifR_{i}(ϕ, x)
is not defined. For the sake of simplicity of the notation, we do not indicate the
dependence ofπ_{i} onx. Here are all required properties ofπ_{i}:

(i) the mapϕ, x 7→ π_{i}(ϕ) is defined andC^{∞}forxin a neighborhood of suppτ_{i}
and for allϕ∈ A_{0}(R^{d});

(ii) suppπ_{i}(ϕ)⊂V_{i};
(iii) π_{i}(ϕ)(x+•) ∈ A_{q}_{i};

(iv) ifϕ(x+•) ∈ Aqi and suppϕ⊂W_{i}, thenπ_{i}(ϕ) =ϕ.

Under these requirements (which we will prove), we haveR(ϕ, x) =R^{◦}(ϕ, x)
whenever

ϕ(x+•) ∈ A_{max}_{q}_{i} and suppϕ⊂ \

x∈suppτi

W_{i},

where max andT

are taken over those i for whichx∈suppτ_{i}. Thus, we have
got U_{x} := T

x∈suppτi

W_{i} and q_{x} := max

x∈suppτi

q_{i}. For proving the first part of
the theorem, we only have to construct the mapπ_{i} with the required properties.

Denote byB_{1}=B(0,1) the open unit ball inR^{d} and byB=B(0, ρ) the ball
inR^{d}with Lebesgue measure Λ(B) = 1. Thus

(29) Λ(B_{1}) =ρ^{−d}.

Fixi∈I and choose a numberr_{i}>0 such that

(30) suppτ_{i}+^{r}_{2}^{1}B_{1}⊂W_{i} and W_{i}+^{r}_{2}^{1}B_{1}⊂V_{i}.

Choose 0≤ϑ_{i}∈ D(V_{i}) withϑ_{i} = 1 onW_{i}, and 0≤ϑ∈ D([−1,1]) withϑ= 1 on
[−^{1}_{2},^{1}_{2}]. We will defineπ_{i} andR_{i} for

(31) x∈

x;x+^{r}_{2}^{i}B_{1} ⊂W_{i} .

By (30), this is a neighborhood of suppτ_{i}, contained in W_{i}. Forϕ∈ A_{0}(R^{d}) put
ϕ^{◦} :=ϑ_{i}·ϕ ( so ϕ^{◦} ∈ D(V_{i}) ),

(32)

k:=

kϕ^{◦}k^{2}

ρ^{d} + ϑ (^{r}_{ρ}^{i})^{d}· ^{kϕ}_{2}^{◦}^{k}^{2}
r_{i}^{d}

1 d

(k kis theL^{2}-norm).

(33)

We have k ≥ _{r}^{1}

i since either ^{kϕ}_{ρ}^{◦}d^{k}^{2} ≥ ^{1}

r_{i}^{d} or ϑ(. . .) = 1. Let ψα ∈ D(B_{1}) be
functions fulfilling (22), (23) for 0≤ |α| ≤q_{i}. Put

(34) π_{i}(ϕ) :=ϕ^{◦} − X

α∈N^{d}
0≤|α|≤q0 i

cαψα (•−x)k

with such coefficients c_{α} that π_{i}(ϕ)(x+•)∈ A_{q}_{i}. By (22) and (23), c_{α} are well
defined for 0≤ |α| ≤q_{i} and we have

(35)

k^{d}(R

ϕ^{◦}−1) =c0

k^{|α|+d}
Z

ϕ^{◦}(x+ξ)ξ^{α}dξ=cα (1≤ |α| ≤q_{i}).

The properties (i) and (iii) are evident. As k ≥ _{r}^{1}

i and suppψ_{α} ⊂ B_{1}, (ii)
easily follows from (30) and (32), ifx fulfills (31). Now, let ϕ(x+•) ∈ A_{q}_{i} and
suppϕ⊂W_{i}. Then by (32) and the definition ofϑ_{i}, we have ϕ=ϕ^{◦}. Evidently
π_{i}(ϕ) =ϕ^{◦} =ϕand so the requirements (i)–(iv) are proved.

Proof of 2^{◦}: Since the sum in (28) is locally finite, it suffices to prove the
moderate growth ofR_{i}. If the vector valued function (ϕ_{x})_{x∈Ω} ∈ E Ω→A_{0}(R^{d})
runs over a bounded set, its valuesϕ_{x} for x∈V_{i} ⋐Ω remain in a bounded set
ofA_{0}, so their supports are contained in a common ballB(0, A)⊂R^{d} (A >0).

Hence, ifε_{0}:= _{2A}^{r}^{i}, then ∀ε∈]0, ε_{0}],x∈V_{i} the support of the function

(36) ϕ_{x,ε}=ε^{−d}ϕ_{x} ^{•}−x

ε

is contained inx+^{r}_{2}^{i}B_{1} ⊂W_{i} by (31) (see Notation 6 definingR_{ε}). By (32) we
haveϕ^{◦}_{x,ε}=ϕ_{x,ε}. The H¨older inequality gives

kϕ_{x,ε}k · kχ_{x+}ri
2B1k ≥

Z

ϕ_{x,ε}= 1 (χis the characteristic function).

Due to (29), this means

(37) kϕ_{x,ε}k^{2}·r_{i}

2ρ d

≥1.

By (28) and Notation 6, we have

(38) (R_{i})_{ε}(ϕx, x) =R_{i}(ϕx,ε, x) =R^{◦} π_{i}(ϕx,ε), x

= (R^{◦})_{ε} ε^{d}π_{i}(ϕ_{x,ε})(x+•ε), x

whereπ_{i}is defined by (34) and (35). Denote the numberkin (33) for the function
ϕx,ε=ϕ^{◦}_{x,ε}bykε, taking into account that it depends onx, too. It follows from
(37), due to the definition ofϑ, thatk_{ε} is given by a simpler formula then (33):

(39) k_{ε} = kϕ_{x,ε}k^{2/d}

ρ .

Considering thatϕ^{◦}_{x,ε}=ϕx,ε, we get from (38), (36) and (34)

(40) R_{i}

ε(ϕx, x) = R^{◦}

ε ϕx − ε^{d}P

cαψα(•εkε), x .

From (39) and (36) we calculateεk_{ε}=ε_{0}k_{ε}_{0}. From (35) we calculatec_{0}= 0 and,
for|α| ≥1,

ε^{d}c_{α}=ε^{d}k_{ε}^{|α|+d}
Z

ϕ_{x,ε}(x+ξ)ξ^{α}dξ=ε^{|α|+d}k^{|α|+d}_{ε}
Z

ϕ_{x} ^{ξ}_{ε}

· ^{ξ}_{ε}_{α}
ε^{−d}dξ,
which does not depend on ε due to the preceding result. So the test function
on the right hand side of (40) does not depend on ε and remains bounded in
E Ω→A_{0}(R^{d})

if (ϕx)_{x} runs over a bounded set. Moreover, the right hand side
of (40) is defined, being equal toR^{◦} π_{i}(ϕ_{x,ε}), x

with π_{i}(ϕ_{x,ε}), x

∈V, thanks to the points (ii) and (iii) of the first part of the proof. Hence, by hypothesis, it

has a moderate growth.

Remark 22(Definition of the derivative). We define the derivative∂_{e}_{j}hRiof a
generalized functionhRi(with respect to the j-th coordinate unit vectore_{j}) in
the same way as it is defined in [4]: If R1 is a representative of hRi according
to the definitions in [4], a representative of ∂ejhRiis defined there to be ϕ, x 7→

∂

∂xjR_{1}(ϕ, x). As a consequence of Change 4 in notation, we have R_{1}(ϕ, x) =
R(ϕ(•−x), x). It follows

∂

∂x_{j}R_{1}(ϕ, x) = dR(ϕ(•−x), x)[−∂_{e}_{j}ϕ(•−x)] + ∂ejR(ϕ(•−x), x).

Hence (in our notation)∂_{e}_{j}hRi=hR^{′}iwith

R^{′}(ϕ, x) =−dR(ϕ, x)[∂_{e}_{j}ϕ] +∂ejR(ϕ, x).

Recall our definition of the canonical embedding of D^{′} into G: the canonical
image of a distributionf in Ghas the functionϕ, x 7→ hf, ϕi(independent onx)
as a representative. Thus, with the usual definition of the differentiation of the
distributions (by [16]) and with the definition above, the canonical embedding
evidently commutes with the differentiation.

Action of aC^{∞} diffeomorphism

Change 23(which we will not always keep). In the expressionR(ϕ, x), we con- sider the test function ϕas a test density ([8]). While we are not dealing with coordinate diffeomorphisms, this change has no influence, as there is a one-to-one correspondence between a test function ϕ and the corresponding test densityϕ given by the formula

(41) ϕ(x) =ϕ(x)dx,

where dxstands for the Lebesgue measure onR^{d}. According to [10], we denote
all odd differential forms (including densities) by underline letters. In the same
way we denote also the spaces of odd differential forms. For instance, D(^{d} R^{d}) is
the space of all test densities onR^{d}. When the first variable of a representative
is a test density, we will denote the representative and the spaces of representa-
tives by underline letters as well, for instance R(ϕ, x). We have to use this type
of representatives, when we deal with generalized functions on a C^{∞} manifold
(different from Ω ⊂ R^{d}), but this is not necessary for generalized functions on
Ω⊂R^{d}. Recall that similarly, for defining the distributions on aC^{∞}manifold of
the dimensiond, the space of the test functionsDis replaced withD. Thanks to^{d}
the notion of density, we can define the image by a coordinate diffeomorphism in
an easy and natural way.