New York Journal of Mathematics
New York J. Math. 15 (2009) 37–72.
Generalized Gevrey ultradistributions
Khaled Benmeriem and Chikh Bouzar
Abstract. We first introduce new algebras of generalized functions
containing Gevrey ultradistributions and then develop a Gevrey microlocal analysis suitable for these algebras. Finally, we give an application
through an extension of the well-known Hörmander’s theorem on the
wave front of the product of two distributions.
Contents
1. Introduction
2. Generalized Gevrey ultradistributions
3. Generalized point values
4. Embedding of Gevrey ultradistributions with compact support
5. Sheaf properties of G σ
6. Equalities in G σ (Ω)
7. Regular generalized Gevrey ultradistributions
8. Generalized Gevrey wave front
9. Generalized Hörmander’s theorem
References
37
39
43
45
50
54
55
60
69
71
1. Introduction
The theory of generalized functions initiated by J. F. Colombeau (see
[4] and [5]) in connection with the problem of multiplication of Schwartz
distributions [22], has been developed and applied in nonlinear and linear
problems, [5], [19] and [18]. The recent book [9] gives further developments
Received December 4, 2007; revised September 10, 2008, and January 26, 2009.
Mathematics Subject Classification. 46F30, 46F10, 35A18.
Key words and phrases. Generalized functions, Gevrey ultradistributions, Colombeau generalized functions, Gevrey wave front, Microlocal analysis, Product of ultradistributions.
ISSN 1076-9803/09
37
38
Khaled Benmeriem and Chikh Bouzar
and applications of such generalized functions. Some methods of constructing algebras of generalized functions of Colombeau type are given in [1], [9]
and [17].
Ultradistributions, important in theoretical as well applied fields (see [15],
[16] and [21]), are natural generalizations of Schwartz distributions, and
the problem of multiplication of ultradistributions is still posed. So, it is
natural to search for algebras of generalized functions containing spaces of
ultradistributions, to study and to apply them. This is the purpose of this
paper.
First, we introduce new differential algebras of generalized Gevrey ultradistributions G σ (Ω) defined on an open set Ω of Rm as the quotient algebra
G σ (Ω) =
σ (Ω)
Em
,
N σ (Ω)
σ (Ω) is the space of (f ) ∈ C ∞ (Ω)]0,1] satisfying for every compact
where Em
ε ε
,
∃k
>
0, ∃c > 0, ∃ε0 ∈ ]0, 1] , ∀ε ≤ ε0 ,
subset K of Ω, ∀α ∈ Zm
+
1
sup |∂ α fε (x)| ≤ c exp kε− 2σ−1 ,
x∈K
and N σ (Ω) is the space of (fε )ε ∈ C ∞ (Ω)]0,1] satisfying for every compact
subset K of Ω, ∀α ∈ Zm
+ , ∀k > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , ∀ε ≤ ε0 ,
1
sup |∂ α fε (x)| ≤ c exp −kε− 2σ−1 .
x∈K
The functor Ω → G σ (Ω) being a sheaf of differential algebras on Rm , we
show that G σ (Ω) contains the space of Gevrey ultradistributions of order
(3σ − 1), and the following diagram of embeddings is commutative:
/ G σ (Ω)
LLL
O
LLL
LLL
L%
E σ (Ω)
(Ω).
D3σ−1
We then develop a Gevrey microlocal analysis adapted to these algebras
in the spirit of [12], [21] and [18]. The starting point of the Gevrey microlocal
analysis in the framework of the algebra G σ (Ω) consists first in introducing
the algebra of regular generalized Gevrey ultradistributions G σ,∞ (Ω) and
then proving the following fundamental result:
(Ω) = E σ (Ω).
G σ,∞ (Ω) ∩ D3σ−1
The functor Ω → G σ,∞ (Ω) is a subsheaf of G σ . This permits us to define
the generalized Gevrey singular support and then, with the help of the
Fourier transform, the generalized Gevrey wave front of f ∈ G σ (Ω), denoted
W Fgσ (f ), and further to give its main properties, as W Fgσ (T ) = W F σ (T ),
Generalized Gevrey ultradistributions
39
(Ω) ∩ G σ (Ω), and
if T ∈ D3σ−1
W Fgσ (P (x, D) f ) ⊂ W Fgσ (f ) ∀f ∈ G σ (Ω),
aα (x) Dα is a partial differential operator with G σ,∞ (Ω)
if P (x, D) =
|α|≤m
coefficients.
Let us note that in [3], the authors introduced a general well adapted local
and microlocal ultraregular analysis within the Colombeau algebra G(Ω).
Finally, we give an application of the introduced generalized Gevrey microlocal analysis. The product of two generalized Gevrey ultradistributions
always exists, but there is no final description of the generalized wave front
of this product. Such a problem is also still posed in the Colombeau algebra.
In [13], the well-known result of Hörmander on the wave front of the product
of two distributions has been extended to the case of two Colombeau generalized functions. We show this result in the case of two generalized Gevrey
ultradistributions, namely we obtain the following result: let f, g ∈ G σ (Ω),
satisfying ∀x ∈ Ω, (x, 0) ∈
/ W Fgσ (f ) + W Fgσ (g), then
W Fgσ (f g) ⊆ W Fgσ (f ) + W Fgσ (g) ∪ W Fgσ (f ) ∪ W Fgσ (g) .
2. Generalized Gevrey ultradistributions
To define the algebra of generalized Gevrey ultradistributions, we first
introduce the algebra of moderate elements and its ideal of null elements
depending on the Gevrey order σ ≥ 1. The set Ω is a nonvoid subset of Rm .
σ (Ω), is the space
Definition 1. The space of moderate elements, denoted Em
∞
]0,1]
satisfying for every compact subset K of Ω, ∀α ∈
of (fε )ε ∈ C (Ω)
,
∃k
>
0,
∃c
>
0,
∃ε
Zm
0 ∈ ]0, 1] , ∀ε ≤ ε0 ,
+
1
(2.1)
sup |∂ α fε (x)| ≤ c exp kε− 2σ−1 .
x∈K
The space of null elements, denoted N σ (Ω), is the space of
(fε )ε ∈ C ∞ (Ω)]0,1]
satisfying for every compact subset K of Ω, ∀α ∈ Zm
+ , ∀k > 0, ∃c > 0, ∃ε0 ∈
]0, 1], ∀ε ≤ ε0 ,
1
(2.2)
sup |∂ α fε (x)| ≤ c exp −kε− 2σ−1 .
x∈K
σ (Ω) and N σ (Ω) are given in the
The main properties of the spaces Em
following proposition.
σ (Ω) is an algebra
Proposition 1.
(1) The space of moderate elements Em
stable under derivation.
σ (Ω).
(2) The space N σ (Ω) is an ideal of Em
40
Khaled Benmeriem and Chikh Bouzar
σ (Ω) and K be a compact subset of Ω. Then
Proof. (1) Let (fε )ε , (gε )ε ∈ Em
m
∀β ∈ Z+ , ∃k1 = k1 (β) > 0, ∃c1 = c1 (β) > 0, ∃ε1β ∈ ]0, 1], ∀ε ≤ ε1β ,
1
β
(2.3)
sup ∂ fε (x) ≤ c1 exp k1 ε− 2σ−1 ,
x∈K
∀β ∈
Zm
+,
(2.4)
Let α ∈
∃k2 = k2 (β) > 0, ∃c2 = c2 (β) > 0, ∃ε2β ∈ ]0, 1] , ∀ε ≤ ε2β ,
1
sup ∂ β gε (x) ≤ c2 exp k2 ε− 2σ−1 .
x∈K
Zm
+.
Then
α α α−β
fε (x) ∂ β gε (x) .
|∂ (fε gε ) (x)| ≤
∂
β
α
β=0
For
k = max {k1 (β) : β ≤ α} + max {k2 (β) : β ≤ α} ,
ε ≤ min {ε1β , ε2β ; |β| ≤ |α|}
and x ∈ K, we have
α 1
1
α
− 2σ−1
α
|∂ (fε gε ) (x)| ≤
exp −k1 ε− 2σ−1 ∂ α−β fε (x)
exp −kε
β
β=0
1
× exp −k2 ε− 2σ−1 ∂ β gε (x)
α α
c1 (α − β) c2 (β) = c (α) ,
≤
β
β=0
σ (Ω). It is clear, from
Em
Zm
+ , ∃k1 = k1 (β + 1) >
(2.3) that for every compact subset
i.e., (fε gε )ε ∈
0, ∃c1 = c1 (β + 1) > 0, ∃ε1β ∈ ]0, 1]
K of Ω, ∀β ∈
such that ∀x ∈ K, ∀ε ≤ ε1β ,
β
− 1
∂ (∂fε ) (x) ≤ c1 exp k1 ε 2σ−1 ,
σ (Ω).
i.e., (∂fε )ε ∈ Em
(2) If (gε )ε ∈ N σ (Ω), for every compact subset K of Ω, ∀β ∈ Zm
+ , ∀k2 >
0, ∃c2 = c2 (β, k2 ) > 0, ∃ε2β ∈ ]0, 1],
1
− 2σ−1
α
, ∀x ∈ K, ∀ε ≤ ε2β .
|∂ gε (x)| ≤ c2 exp −k2 ε
Let α ∈ Zm
+ and k > 0. Then
1
exp kε− 2σ−1 |∂ α (fε gε ) (x)|
α 1
α α−β
− 2σ−1
fε (x) ∂ β gε (x) .
≤ exp kε
∂
β
β=0
Generalized Gevrey ultradistributions
41
Let k2 = max {k1 (β) ; β ≤ α} + k and ε ≤ min {ε1β , ε2β ; β ≤ α}, then ∀x ∈
K,
α 1
1
α
− 2σ−1
α
|∂ (fε gε ) (x)| ≤
exp −k1 ε− 2σ−1 ∂ α−β fε (x)
exp kε
β
β=0
1
× exp k2 ε− 2σ−1 ∂ β gε (x)
α α
c1 (α − β) c2 (β, k2 ) = c (α, k) ,
≤
β
β=0
which shows that (fε gε )ε ∈
N σ (Ω).
σ (Ω) is not necessary staRemark 1. The algebra of moderate elements Em
ble under σ-ultradifferentiable operators, because the constant c in (2.1)
depends on α.
According to the topological construction of Colombeau type algebras of
generalized functions, we introduce the desired algebras.
Definition 2. The algebra of generalized Gevrey ultradistributions of order
σ ≥ 1, denoted G σ (Ω), is the quotient algebra
G σ (Ω) =
σ (Ω)
Em
.
N σ (Ω)
A comparison of the structure of our algebras G σ (Ω) and the Colombeau
algebra G(Ω) is given in the following remark.
Remark 2. The Colombeau algebra G(Ω) :=
Em (Ω)
N (Ω) ,
where Em (Ω) is the
space of (fε )ε ∈ C ∞ (Ω)]0,1] satisfying for every compact subset K of Ω,
∀α ∈ Zm
+ , ∃k > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , ∀ε ≤ ε0 ,
sup |∂ α fε (x)| ≤ cε−k ,
x∈K
and N (Ω) is the space of (fε )ε ∈ C ∞ (Ω)]0,1] satisfying for every compact
subset K of Ω, ∀α ∈ Zm
+ , ∀k > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , ∀ε ≤ ε0 ,
sup |∂ α fε (x)| ≤ cεk .
x∈K
Due to the inequality
1
exp −ε− 2σ−1 ≤ ε, ∀ε ∈ ]0, 1] ,
τ (Ω) ⊂
we have the strict inclusions N σ (Ω) ⊂ N τ (Ω) ⊂ N (Ω) ⊂ Em (Ω) ⊂ Em
σ
Em (Ω), with σ < τ .
We have the null characterization of the ideal N σ (Ω).
42
Khaled Benmeriem and Chikh Bouzar
σ (Ω), then (u ) ∈ N σ (Ω) if and only if for
Proposition 2. Let (u ) ∈ Em
every compact subset K of Ω, ∀k > 0, ∃c > 0, ∃ε0 ∈ ]0, 1], ∀ε ≤ ε0 ,
1
(2.5)
sup |fε (x)| ≤ c exp −kε− 2σ−1 .
x∈K
σ (Ω) satisfy (2.5). We will show that (∂ u ) also
Proof. Let (u )ε ∈ Em
i satisfy (2.5) when i = 1, . . . , m, and then it will follow by induction that
(u ) ∈ N σ (Ω).
Suppose that u has real values. In the complex case we do the calculus
separately for the real and imaginary part of u . Let K be a compact subset
of Ω. For δ = min (1, dist (K, ∂Ω)), set L = K + B 0, 2δ . Then
K ⊂⊂ L ⊂⊂ Ω.
By the moderateness of (u )ε , we have ∃k1 > 0, ∃c1 > 0, ∃ε1 ∈ ]0, 1] , ∀ε ≤ ε1
1
(2.6)
sup ∂i2 u (x) ≤ c1 exp k1 ε− 2σ−1 .
x∈L
By the assumption (2.5), ∀k > 0, ∃c2 > 0, ∃ε2 ∈ ]0, 1] , ∀ε ≤ ε2
1
(2.7)
sup |u (x)| ≤ c2 exp − (2k + k1 ) ε− 2σ−1 .
x∈L
1
Let x ∈ K, ε sufficiently small and r = exp − (k + k1 ) ε− 2σ−1 < δ2 . By
Taylor’s formula, we have
(u (x + rei ) − u (x)) 1 2
− ∂i u (x + θrei ) r,
∂i u (x) =
r
2
th
where ei is i vector of the canonical base of Rm , hence (x + θrei ) ∈ L, and
then
1
|∂i u (x)| ≤ |u (x + rei ) − u (x)| r −1 + ∂i2 u (x + θrei ) r.
2
1
−1
From (2.6) and (2.7): |u (x + rei ) − u (x)| r ≤ c2 exp −kε− 2σ−1 and
2
1
∂ u (x + θrei ) r ≤ c1 exp −kε− 2σ−1 , so
i
1
|∂i u (x)| ≤ c exp −kε− 2σ−1 ,
which gives the proof.
Proposition 3. If P is a polynomial function and f = cl (fε )ε ∈ G σ (Ω),
then P (f ) = (P (fε ))ε + N σ (Ω) is a well-defined element of G σ (Ω).
σ (Ω), P (ξ) =
aα ξ α and K be a compact subset
Proof. Let (fε )ε ∈ Em
|α|≤m
of Ω. Then we have ∀α ∈ Zm
+ , ∃k = k (α) > 0, ∃c = c (α) > 0, ∃ε0 = ε (α) ∈
]0, 1] , ∀ε ≤ ε0 ,
1
(2.8)
sup |∂ α fε (x)| ≤ c exp kε− 2σ−1 .
x∈K
Generalized Gevrey ultradistributions
Let β ∈ Zm
+ , so
43
β
|aα | ∂ β fεα (x) .
∂ P (fε ) (x) ≤
|α|≤m
By the Leibniz formula and (2.8), we obtain
nα,γ
1
β
cα,γ exp kα,γ ε− 2σ−1
,
∂ P (fε ) (x) ≤
|α|≤m
γ≤β
where cα,γ > 0 and nα,γ ∈ Z+ . Hence
β
− 1
∂ P (fε ) (x) ≤ c exp kε 2σ−1 .
One can easily cheek that if (fε )ε ∈ N σ (Ω), then (P (fε ))ε ∈ N σ (Ω).
The space of functions slowly increasing, denoted OM (Km ), is the space
of C ∞ -functions all derivatives growing at most like some power of |x|, as
|x| → +∞, where Km Rm or R2m .
Corollary 4. If v ∈ OM (Km ) and f = (f1 , f2 , . . . , fm ) ∈ G σ (Ω)m , then
v ◦ f := (v ◦ fε )ε + N σ (Ω) is a well-defined element of G σ (Ω).
3. Generalized point values
The ring of Gevrey generalized complex numbers, denoted C σ , is defined
by the quotient
Eσ
C σ = 0σ ,
N0
where
E0σ = (aε )ε ∈ C]0,1] ; ∃k > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , such that
1
∀ε ≤ ε0 , |aε | ≤ c exp kε− 2σ−1
and
N0σ = (aε )ε ∈ C]0,1] ; ∀k > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , such that
1
∀ε ≤ ε0 , |aε | ≤ c exp −kε− 2σ−1 .
It is not difficult to see that E0σ is an algebra and N0σ is an ideal of E0σ . The
ring C σ motivates the following, easy to prove, result.
Proposition 5. If u ∈ G σ (Ω) and x ∈ Ω, then the element u(x) represented
by (uε (x))ε is an element of C σ independent of the representative (uε )ε of u.
44
Khaled Benmeriem and Chikh Bouzar
A generalized Gevrey ultradistribution is not defined by its point values.
We give here an example of a generalized Gevrey ultradistribution f =
/ N σ (R), but [(fε (x))ε ] ∈ N0σ for every x ∈ R. Let ϕ ∈ D (R) such
[(fε )ε ] ∈
that ϕ (0) = 0. For ε ∈ ]0, 1], define
x
1
, x ∈ R.
fε (x) = x exp −ε− 2σ−1 ϕ
ε
σ (R). Let K be a compact neighborhood of 0, then
It is clear that (fε )ε ∈ Em
1
sup f (x) ≥ fε (0) = exp −ε− 2σ−1 |ϕ (0)| ,
K
which
/ N σ (R). For any x0 ∈ R, there exists ε0 such that
x shows that (fε )ε ∈
0
ϕ ε = 0, ∀ε ≤ ε0 , i.e., f (x0 ) ∈ N0σ .
In order to give a solution to this situation, set
(3.1) ΩσM =
− 1
]0,1]
kε 2σ−1
: ∃k > 0, ∃c > 0, ∃ε0 > 0, ∀ε ≤ ε0 , |xε | ≤ ce
.
(xε )ε ∈ Ω
Define in ΩσM the equivalence relation ∼ by
(3.2) xε ∼ yε
⇐⇒ ∀k > 0, ∃c > 0, ∃ε0 > 0, ∀ε ≤ ε0 ,
|xε − yε | ≤ ce−kε
1
− 2σ−1
.
σ = Ωσ / ∼ is called the set of generalized Gevrey
Definition 3. The set Ω
M
points. The set of compactly supported Gevrey points is defined by
σc =
(3.3) Ω
σ : ∃K a compact set of Ω, ∃ε0 > 0, ∀ε ≤ ε0 , xε ∈ K .
x
= [(xε )ε ] ∈ Ω
σc -property does not depend on the
Remark 3. It is easy to see that the Ω
choice of the representative.
σ , then the generalized
= [(xε )ε ] ∈ Ω
Proposition 6. Let f ∈ G σ (Ω) and x
c
Gevrey point value of f at x
, i.e.,
f (
x) = [(fε (xε ))ε ]
is a well-defined element of the algebra of generalized Gevrey complex numbers C σ .
σc , there exists a compact subset
= [(xε )ε ] ∈ Ω
Proof. Let f ∈ G σ (Ω) and x
K of Ω such that xε ∈ K for ε small, then ∃k > 0, ∃c > 0, ∃ε0 > 0, ∀ε ≤ ε0 ,
1
|fε (xε )| ≤ sup |fε (x)| ≤ c exp kε− 2σ−1 .
x∈K
E0σ ,
and it is clear that if f ∈ N σ (Ω), then (fε (xε ))ε ∈
Therefore (fε (xε ))ε ∈
σ
x) does not depend on the choice of the representative (fε )ε .
N0 , i.e., f (
Generalized Gevrey ultradistributions
45
Let now x
= [(xε )ε ] ∼ y = [(yε )ε ], then ∀k > 0, ∃c > 0, ∃ε0 > 0, ∀ε ≤ ε0 ,
1
|xε − yε | ≤ c exp −kε− 2σ−1 .
Since (fε )ε ∈ E σ (Ω), so for every compact subset K of Ω, ∀j ∈ {1, m} , ∃kj >
0, ∃cj > 0, ∃εj > 0, ∀ε ≤ εj ,
∂
1
− 2σ−1
fε (x) ≤ cj exp kj ε
.
sup x∈K ∂xj
We have
|fε (xε ) − fε (yε )| ≤ |xε − yε |
m j=1
0
∂
∂xj fε (xε + t (yε − xε )) dt,
1 and xε + t (yε − xε ) remains within some compact subset K of Ω for ε ≤ ε .
Let k > 0, then for k + k = sup kj and ε ≤ min (ε , ε0 , εj : i = 1, m), we
j
have
1
|fε (xε ) − fε (yε )| ≤ c exp −k ε− 2σ−1 ,
which gives (fε (xε ) − fε (yε ))ε ∈ N0σ .
The characterization of nullity of f ∈ G σ (Ω) is given by the following
theorem.
Theorem 7. Let f ∈ G σ (Ω). Then
σc .
x) = 0 in C σ for all x
∈Ω
f = 0 in G σ (Ω) ⇐⇒ f (
σc .
x) ∈ N0σ , ∀
x ∈ Ω
Proof. It is easy to see that if f ∈ N σ (Ω), then f (
σ
σ
Suppose that f = 0 in G (Ω). Then by the characterization of N (Ω) we
have, there exists a compact subset K of Ω, ∃k > 0, ∀c > 0, ∀ε0 > 0, ∃ε ≤ ε0 ,
1
sup |fε (x)| > c exp −kε− 2σ−1 .
K
So there exists a sequence εm 0 and xm ∈ K such that ∀m ∈ Z+ ,
1
− 2σ−1
.
(3.4)
|fεm (xm )| > exp −kεm
For ε > 0 we set xε = xm when εm+1 < ε ≤ εm . We have (xε )ε ∈ ΩσM with
σ
σ and (3.4) means that (fε (xε )) ∈
values in K, so x
= [(xε )ε ] ∈ Ω
c
ε / N0 , i.e.,
σ
f (
x) = 0 in C .
4. Embedding of Gevrey ultradistributions with
compact support
We recall some definitions and results on Gevrey ultradistributions (see
[15], [16] or [21]).
46
Khaled Benmeriem and Chikh Bouzar
Definition 4. A function f ∈ E σ (Ω), if f ∈ C ∞ (Ω) and for every compact
subset K of Ω, ∃c > 0, ∀α ∈ Zm
+,
sup |∂ α f (x)| ≤ c|α|+1 (α!)σ .
x∈K
Obviously we have E t (Ω) ⊂ E σ (Ω) if 1 ≤ t ≤ σ. It is well-known that
= A(Ω) is the space of all real analytic functions in Ω. Denote by
σ
D (Ω) the space E σ (Ω) ∩ C0∞ (Ω). Then Dσ (Ω) is nontrivial if and only if
σ > 1. The topological dual of D σ (Ω), denoted Dσ (Ω), is called the space
of Gevrey ultradistributions of order σ. The space Eσ (Ω) is the topological
dual of E σ (Ω) and is identified with the space of Gevrey ultradistributions
with compact support.
E 1 (Ω)
Definition 5. A differential operator of infinite order
aγ Dγ
P (D) =
γ∈Zm
+
is called a σ-ultradifferential operator, if for every h > 0 there exist c > 0
such that ∀γ ∈ Zm
+,
(4.1)
|aγ | ≤ c
h|γ|
.
(γ!)σ
The importance of σ-ultradifferential operators lies in the following result.
Proposition 8. Let T ∈ Eσ (Ω), σ > 1 and supp T ⊂ K. Then there exist
a σ-ultradifferential operator
aγ Dγ ,
P (D) =
γ∈Zm
+
M > 0 and continuous functions fγ ∈ C0 (K) such that
sup
γ∈Zm
+ ,x∈K
T =
|fγ (x)| ≤ M,
aγ Dγ fγ .
γ∈Zm
+
The space S (σ) (Rm ), σ > 1 (see [10]) is the space of functions ϕ ∈
such that ∀b > 0, we have
|x||β|
|∂ α ϕ (x)| dx < ∞.
(4.2)
ϕb,σ = sup
|α+β| α!σ β!σ
m
b
α,β∈Z+
C ∞ (Rm )
Lemma 9. There exists φ ∈ S (σ) (Rm ) satisfying
φ (x) dx = 1 and
xα φ (x) dx = 0, ∀α ∈ Zm
+ \{0}.
Generalized Gevrey ultradistributions
47
Proof. For an example of function φ ∈ S (σ) satisfying these conditions,
take the Fourier transform of a function of the class D (σ) (Rm ) equal 1
in a neighborhood of the origin. Here D (σ) (Rm ) denotes the projective
Gevrey space of order σ, i.e., D (σ) (Rm ) = E (σ) (Rm ) ∩ C0∞ (Rm ), where
f ∈ E (σ) (Rm ), if f ∈ C ∞ (Rm ) and for every compact subset K of Rm , ∀b >
0, ∃c > 0, ∀α ∈ Zm
+,
sup |∂ α f (x)| ≤ cb|α| (α!)σ .
x∈K
Definition 6. The net φε = ε−m φ (./ε) , ε ∈ ]0, 1], where φ satisfies the
conditions of Lemma 9, is called a net of mollifiers.
The space E σ (Ω) is embedded into G σ (Ω) by the standard canonical injection
I : E σ (Ω) −→ G σ (Ω)
(4.3)
f −→ [f ] = cl (fε ) ,
where fε = f , ∀ε ∈ ]0, 1].
The following proposition gives the natural embedding of Gevrey ultradistributions into G σ (Ω).
Theorem 10. The map
(Ω) −→ G σ (Ω)
J0 : E3σ−1
(4.4)
T −→ [T ] = cl (T ∗ φε )/Ω
ε
is an embedding.
(Ω) with supp
T ⊂ K. Then there exists an (3σ − 1)Proof. Let T ∈ E3σ−1
aγ Dγ and continuous functions fγ
ultradifferential operator P (D) =
with supp fγ ⊂ K, ∀γ ∈
Zm
+,
γ∈Zm
+
and
sup
γ∈Zm
+ ,x∈K
T =
|fγ (x)| ≤ M , such that
aγ Dγ fγ .
γ∈Zm
+
We have
(−1)|γ|
aγ |γ|
T ∗ φε (x) =
ε
γ∈Zm
fγ (x + εy) Dγ φ (y) dy.
+
Let α ∈
Zm
+.
Then
|∂ α (T ∗ φε (x))| ≤
γ∈Zm
+
aγ
1
ε|γ+α|
|fγ (x + εy)| Dγ+α φ (y) dy.
From (4.1) and the inequality
(4.5)
(β + α)!t ≤ 2t|β+α| α!t β!t , ∀t ≥ 1,
48
Khaled Benmeriem and Chikh Bouzar
we have, ∀h > 0, ∃c > 0, such that
h|γ|
1
α
c 3σ−1 |γ+α|
|fγ (x + εy)| Dγ+α φ (y) dy
|∂ (T ∗ φε (x))| ≤
γ!
ε
m
γ∈Z+
≤
cα!3σ−1
γ∈Zm
+
2(3σ−1)|γ+α| h|γ| 1
b|γ+α|
(γ + α)!2σ−1 ε|γ+α|
×
|fγ (x + εy)|
|Dγ+α φ (y)|
dy.
b|γ+α| (γ + α)!σ
Then for h > 12 ,
1
α!3σ−1
i.e.,
(4.6)
3σ bh |γ+α|
2
1
|∂ α (T ∗ φε (x))| ≤ φb,σ M c
2−|γ|
2σ−1
|γ+α|
(γ + α)!
ε
γ∈Zm
+
1
≤ c exp k1 ε− 2σ−1 ,
1
|∂ α (T ∗ φε (x))| ≤ c (α) exp k1 ε− 2σ−1 ,
1
where k1 = (2σ − 1) 23σ bh 2σ−1 .
Suppose that (T ∗ φε )ε ∈ N σ (Ω). Then for every compact subset L of Ω,
∃c > 0, ∀k > 0, ∃ε0 ∈ ]0, 1],
1
(4.7)
|T ∗ φε (x)| ≤ c exp −kε− 2σ−1 , ∀x ∈ L, ∀ε ≤ ε0 .
Let χ ∈ D 3σ−1 (Ω) and χ = 1 in a neighborhood of K. Then ∀ψ ∈ E 3σ−1 (Ω),
T, ψ = T, χψ = lim (T ∗ φε ) (x) χ (x) ψ (x) dx.
ε→0
Consequently, from (4.7), we obtain
1
(T ∗ φε ) (x) χ (x) ψ (x) dx ≤ c exp −kε− 2σ−1 ,
which gives T, ψ = 0.
∀ε ≤ ε0 ,
Remark 4. We have c (α) = α!3σ−1 φb,σ M c in (4.6).
In order to show the commutativity of the following diagram of embeddings
/ G σ (Ω)
D σ (Ω)
KKK
KKK
KKK
K%
O
(Ω),
E3σ−1
we have to prove the following fundamental result.
Generalized Gevrey ultradistributions
49
Proposition 11. Let f ∈ D σ (Ω) and (φε )ε be a net of mollifiers. Then
f − (f ∗ φε )/Ω ∈ N σ (Ω) .
ε
Proof. Let f ∈
D σ (Ω).
Then there exists a constant c > 0, such that
α
|∂ f (x)| ≤ c|α|+1 α!σ ,
∀α ∈ Zm
+ , ∀x ∈ Ω.
Let α ∈ Zm
+ . Taylor’s formula and the properties of φε give
(εy)β
∂ α+β f (ξ) φ (y) dy,
∂ α (f ∗ φε − f ) (x) =
β!
|β|=N
where x ≤ ξ ≤ x + εy. Consequently, for b > 0, we have
|y|N |∂ α (f ∗ φε − f ) (x)| ≤ εN
∂ α+β f (ξ) |φ (y)| dy
β!
|β|=N
α+β
f (ξ)
∂
σ N
2σ−1 σ|α+β| |β|
β!
2
b
≤ α! ε
(α + β)!σ
|β|=N
×
|y||β|
|φ (y)| dy.
b|β| β!σ
Let k > 0 and T > 0. Then
N 2σ−1 −N
T
k
|∂ α (f ∗ φε − f ) (x)| ≤ α!σ εN 2σ−1
α+β
∂
f
(ξ)
|β|
×
2σ|α+β| k2σ−1 bT
(α + β)!σ
|β|=N
|y||β|
× |β| σ |φ (y)| dy
b β!
N 2σ−1 −N
σ
T
k
≤ α! εN 2σ−1
σ 2σ−1 |β| |β|
× c φb,σ (2σ c)|α|
bT
c ,
2 k
|β|=N
2σ k2σ−1 bT c
≤
1
2a ,
with a > 1, we obtain
N 2σ−1 −N
(4.8) |∂ α (f ∗ φε − f ) (x)| ≤ α!σ εN 2σ−1
T
k
hence, taking
× c φb,σ (2σ c)|α| a−N
1 |β|
2
|β|=N
N 2σ−1 −N −N
≤ φb,σ c|α|+1 α!σ εN 2σ−1
T
a .
k
1
Let ε0 ∈ ]0, 1] such that ε02σ−1 lnka < 1 and take T > 22σ−1 . Then
1
1
ln a 2σ−1
ε
, ∀ε ≤ ε0 .
T 2σ−1 − 1 > 1 >
k
50
Khaled Benmeriem and Chikh Bouzar
In particular, we have
−1
−1
1
1
1
ln a 2σ−1
ln a 2σ−1
ε
ε
T 2σ−1 −
> 1.
k
k
Then, there exists N = N (ε) ∈ Z+ , such that
−1
−1
1
1
1
ln a 2σ−1
ln a 2σ−1
ε
ε
<N <
T 2σ−1 ,
k
k
i.e.,
1≤
(4.9)
1
1
ln a 2σ−1
ε
N ≤ T 2σ−1 ,
k
which gives
−N
a
1
≤ exp −kε− 2σ−1
and
εN 2σ−1
≤
k2σ−1 T
1
ln a
2σ−1
< 1,
if we choose ln a > 1. Finally, from (4.8), we have
1
(4.10)
|∂ α (f ∗ φε − f ) (x)| ≤ c exp −kε− 2σ−1 ,
i.e., f ∗ φε − f ∈ N σ (Ω).
From the proof (see (4.8)) we obtained in fact the following result.
Corollary 12. Let f ∈ D σ (Ω). Then for every compact subset K of Ω, ∀k >
0, ∃c > 0, ∃ε0 ∈ ]0, 1] , ∀α ∈ Zm
+ , ∀ε ≤ ε0 ,
1
(4.11)
sup |∂ α (f ∗ φε − f ) (x)| ≤ c|α|+1 α!σ exp −kε− 2σ−1 .
x∈K
5. Sheaf properties of G σ
Let Ω be an open subset of Ω and let f = (fε )ε + N σ (Ω) ∈ G σ (Ω), the
restriction of f to Ω , denoted f/Ω , is defined as
fε/Ω ε + N σ Ω ∈ G σ Ω .
Theorem 13. The functor Ω → G σ (Ω) is a sheaf of differential algebras on
Rm .
Proof. Let Ω be a nonvoid open of Rm and (Ωλ )λ∈Λ be an open covering
of Ω. We have to show the properties
(S1) If f, g ∈ G σ (Ω) such that f/Ωλ = g/Ωλ , ∀λ ∈ Λ, then f = g.
(S2) If for each λ ∈ Λ, we have fλ ∈ G σ (Ωλ ), such that
fλ/Ωλ ∩Ωμ = fμ/Ωλ ∩Ωμ
for all λ, μ ∈ Λ with Ωλ ∩ Ωμ = φ,
then there exists a unique f ∈ G σ (Ω) with f/Ωλ = fλ , ∀λ ∈ Λ.
Generalized Gevrey ultradistributions
51
To show (S1), take K a compact subset of Ω. Then there exist compact
sets K1 , K2 , . . . , Km and indices λ1 , λ2 , . . . , λm ∈ Λ such that
K⊂
m
and Ki ⊂ Ωλi ,
Ki
i=1
where (fε − gε )ε satisfies the N σ -estimate on each Ki . Then it satisfies the
N σ -estimate on K which means (fε − gε )ε ∈ N σ (Ω).
∞
To show (S2), let (χj )∞
j=1 be a C -partition of unity subordinate to the
covering (Ωλ )λ∈Λ . Set
f := (fε )ε + N σ (Ω),
∞
χj fλj ε and fλj ε ε is a representative of fλj . Moreover, we
where fε =
j=1
set fλj ε = 0 on Ω\Ωλj , so that χj fλj ε is C ∞ on all of Ω. First let K be a
compact subset of Ω. Then Kj = K ∩ supp χj is a compact subset of Ωλj
σ Ω
σ
and fλj ε ε ∈ Em
λj . Then χj fλj ε satisfies Em -estimate on each Kj ,
and χj (x) ≡ 0 on K except for a finite number of j, i.e., ∃N > 0, such that
∞
χj fλj ε (x) =
j=1
N
χj fλj ε (x) ,
∀x ∈ K.
j=1
σ -estimates on K, which means (f ) ∈ E σ (Ω). It
So
χj fλj ε satisfies Em
ε ε
m
remains to show that f/Ωλ = fλ , ∀λ ∈ Λ. Let K be a compact subset of Ωλ ,
N
χj (x) ≡ 1 on a neighborhood Ω of K
choose N > 0 in such a way that
j=1
with Ω compact in Ωλ . For x ∈ K,
fε (x) − fλε (x) =
N
χj (x) fλj ε (x) − fλε (x) .
j=1
Since fλj ε − fλε ∈ N σ Ωλj ∩ Ωλ and Kj = K ∩ supp χj is a compact
N
χj fλj ε − fλε
satisfies the N σ -estimate on
subset of Ω ∩ Ωλj , then
j=1
K. The uniqueness of such f ∈ G σ (Ω) follows from (S1).
Now it is legitimate to introduce the support of f ∈ G σ (Ω) as in the
classical case.
Definition 7. The support of f ∈ G σ (Ω), denoted suppσg f , is the complement of the largest open set U such that f/U = 0.
(Ω) into G σ (Ω) using the
As in [9], we construct the embedding of D3σ−1
σ
sheaf properties of G . First, choose some covering (Ωλ )λ∈Λ of Ω such that
each Ωλ is a compact subset of Ω. Let (ψλ )λ∈Λ be a family of elements of
52
Khaled Benmeriem and Chikh Bouzar
Dσ (Ω) ⊂ D3σ−1 (Ω) with ψλ ≡ 1 in some neighborhood of Ωλ . For each λ
we define
(Ω) −→ G σ (Ωλ )
Jλ : D3σ−1
T −→ [T ]λ = cl (ψλ T ∗ φε )/Ωλ .
ε
σ (Ω ) (see the proof of
∈ Em
One can easily show that (ψλ T ∗ φε )/Ωλ
λ
ε
Theorem 10) and that the family (Jλ (T ))λ∈Λ is coherent, i.e.,
Jλ (T )/Ωλ ∩Ωμ = Jμ (T )/Ωλ ∩Ωμ , ∀λ, μ ∈ Λ.
Then if (χj )∞
j=1 is a smooth partition of unity subordinate to (Ωλ )λ∈Λ , the
preceding theorem allows the embedding
(5.1)
(Ω) −→ G σ (Ω)
Jσ : D3σ−1
⎛
T −→ [T ] = cl ⎝
∞
⎞
χj ψλj T ∗ φε ⎠ .
j=1
(Ω) into G σ (Ω) (see [6] for the case
We can also embed directly D3σ−1
D (Ω) and G(Ω)). Indeed, let ϕ ∈ D σ (B (0, 2)), 0 ≤ ϕ ≤ 1, ϕ ≡ 1 on
B (0, 1) and take φ ∈ S (σ) , define the function ρε by
m x
1
ϕ (x |ln ε|) .
φ
(5.2)
ρε (x) =
ε
ε
It is easy to prove that, ∃c > 0, such that ∀α ∈ Zm
+,
sup |∂ α ρε (x)| ≤ c|α|+1 α!σ ε−m−|α| .
x∈Rm
Define the injective map
(5.3)
(Ω) −→ G σ (Ω)
J : D3σ−1
T −→ [T ] = cl ((T ∗ ρε )ε ) .
(Ω) with J0 .
Proposition 14. The map J coincides on E3σ−1
(Ω), the net (T ∗ (ρε − φε ))ε ∈
Proof. We have to show that for T ∈ E3σ−1
σ
α
N (Ω). For x |ln ε| < 1, we have ∂ (ρε − φε ) (x) = 0, ∀α ∈ Zm
+ . For
x |ln ε| ≥ 1, we have
x
(ϕ (x |ln ε|) − 1)
∂ α (ρε − φε ) (x) = ε−m−|α| ∂ α φ
ε
α
x
−m
∂ α−β ϕ (x |ln ε|) .
ε−|β| |ln ε||α−β| ∂ β φ
+ε
ε
|β|=1
Generalized Gevrey ultradistributions
53
Then, ∃c > 0, ∀b > 0, ∀γ ∈ Zm
+,
|∂ α (ρε − φε ) (x)|
≤b
|γ|+|α|+1 −m−|α|
ε
σ
σ
α! γ!
ε
|x|
|γ|
+ε
× ε−|β| |ln ε||α−β| β!σ (α − β)!σ γ!σ
|γ|+|α|+1 −m−|α|
≤b
ε
σ
σ
α! γ! (ε |ln ε|)
−m
|γ|
α
c|α−β|+1 b|γ|+|β|+1
|β|=1
ε
|x|
|γ|
−m
+ε
α
b|γ|+|β|+1 c|α−β|+1
|β|=1
× ε−|β| |ln ε||α−β| β!σ (α − β)!σ γ!σ (ε |ln ε|)|γ| .
So, ∃c > 0, ∀b > 0, ∀γ ∈ Zm
+,
(5.4) |∂ α (ρε − φε ) (x)| ≤ b|2γ|+|α|+1 c|2γ+α|+1 ε|2γ|−|α|−m |ln ε||2γ| α!σ (2γ)!σ .
As ε ∈ ]0, 1], we have |ln ε|2 ≤ ε−1 , then ∀γ ∈ Zm
+,
|ln ε||2γ| ≤ ε−|γ| ,
then (5.4) gives ∃c > 0, ∀b > 0, ∀γ ∈ Zm
+,
|∂ α (ρε − φε ) (x)| ≤ b|2γ|+|α|+1 c|2γ+α|+1 α!σ (2γ)!σ ε|γ|−|α|−m .
Since
(2γ)!σ ≤ 2σ|2γ| γ!2σ ,
then for N = |γ|, we obtain
|∂ α (ρε − φε ) (x)| ≤ b2N +|α|+1 c2N +|α|+1 22N σ α!σ N !2σ εN −|α|−m
N
≤ 2−N 22σ+1 b2 c2 N !2σ εN × (bc)|α|+1 α!σ ε−|α|−m
− 1
1
≤ c h|α|+1 α!σ ε−|α|−m exp − 22σ+1 b2 c2 2σ ε− 2σ ,
where c = cb
2−N and h = cb. Then for any k take b > 0 and ε
n≥0
sufficiently small such that
− 1
1
1
ε−|α|−m exp − 22σ+1 b2 c2 2σ ε− 2σ ≤ exp −kε− 2σ−1 .
Then we have ∀k > 0, ∃h > 0, ∃ε0 ∈ ]0, 1] such that ∀ε ≤ ε0 ,
(5.5)
|∂ α (ρε − φε ) (x)|
1
1
≤ h|α|+1 α!σ exp −kε− 2σ ≤ h|α|+1 α!σ exp −kε− 2σ−1 .
54
Khaled Benmeriem and Chikh Bouzar
(Ω) ⊂ Eσ (Ω), ∃L a compact subset of Ω such that ∀h > 0, ∃c > 0,
As E3σ−1
and
h|α| α
.
(ρ
−
φ
)
(x
−
y)
∂
|T (y) , (ρε − φε ) (x − y)| ≤ c sup
ε
ε
y
σ
α!
α∈Zm
,y∈L
+
So for x ∈ K, y ∈ L and by (5.5), we obtain
1
|(T ∗ (ρε − φε )) (x)| ≤ c exp −kε− 2σ−1 ,
which proves that (T ∗ (ρε − φε ))ε ∈ N σ (Ω).
The sheaf properties of G σ and the proof of Proposition 14 show that the
embedding Jσ coincides with the embedding J. Summing up, we have the
following commutative diagram
E σ (Ω)
/ G σ (Ω) .
8
r
rr
r
r
rr
rrr
(Ω)
D3σ−1
Definition 8. The space of elements of G σ (Ω) with compact support is
denoted GCσ (Ω).
As in the case of Colombeau generalized functions, it is not difficult to
prove the following result.
Proposition 15. The space GCσ (Ω) is the space of elements f of G σ (Ω)
satisfying: there exists a representative (fε )ε∈]0,1] and a compact subset K
of Ω such that supp fε ⊂ K, ∀ε ∈ ]0, 1].
6. Equalities in G σ (Ω)
In G σ (Ω), we have the strong equality, denoted =, between two elements
f = [(fε )ε ] and g = [(gε )ε ], which means that
(fε − gε )ε ∈ N σ (Ω).
One can easily
check that if K is a compact subset of Ω and f = [(fε )ε ] ∈
G σ (Ω), then K fε (x) dx ε defines an element of C σ .
t
We define the equality in the sense of ultradistributions, denoted ∼, where
t ∈ [σ, 3σ − 1], by
t
(fε (x) − gε (x)) ϕ (x) dx ∈ N0t , ∀ϕ ∈ Dt (Ω),
f ∼ g ⇐⇒
ε
and we say that f equals g in the sense of ultradistributions.
We say that f = [(fε )ε ] is associated to g = [(gε )ε ], denoted f ≈ g, if
lim (fε − gε ) (x) ψ (x) dx = 0, ∀ψ ∈ D3σ−1 (Ω).
ε→0
Generalized Gevrey ultradistributions
55
In particular, we say that f = [(fε )ε ] ∈ G σ (Ω) is associated to the Gevrey
(Ω), denoted f ≈ T , if
ultradistribution T ∈ E3σ−1
lim fε (x) ψ (x) dx = T, ψ , ∀ψ ∈ D3σ−1 (Ω).
ε→0
The main relationship between these inequalities is given by the following
results. The proof is left to the reader.
(Ω), and t ∈ [σ, 3σ − 1].
Proposition 16. Let f, g ∈ G σ (Ω), T ∈ E3σ−1
Then:
t
σ
(1) f = g =⇒ f ∼ g =⇒ f ∼ g =⇒ f ≈ g.
(Ω) .
(2) T ≈ 0 in G σ (Ω) =⇒ T = 0 in E3σ−1
7. Regular generalized Gevrey ultradistributions
To develop a local and a microlocal analysis with respect to a “good
space of regular elements” one needs first to define these regular elements,
the notion of singular support and its microlocalization with respect to the
class of regular elements.
σ,∞
(Ω), is the space
Definition 9. The space of regular elements, denoted Em
]0,1]
∞
satisfying, for every compact subset K of Ω, ∃k > 0,
of (fε )ε ∈ (C (Ω))
∃c > 0, ∃ε0 ∈ ]0, 1] , ∀α ∈ Zm
+ , ∀ε ≤ ε0 ,
1
sup |∂ α fε (x)| ≤ c|α|+1 α!σ exp kε− 2σ−1 .
x∈K
σ,∞
(Ω) is an algebra stable under the
Proposition 17.
(1) The space Em
action of σ-ultradifferential operators.
σ,∞
σ,∞
(Ω) is an ideal of Em
(Ω).
(2) The space N σ,∞ (Ω) := N σ (Ω) ∩ Em
σ,∞
(Ω) and K be a compact subset of Ω.
Proof. (1) Let (fε )ε , (gε )ε ∈ Em
Then ∃k1 > 0, ∃c1 > 0, ∃ε1 ∈ ]0, 1] such that ∀x ∈ K, ∀α ∈ Zm
+ , ∀ε ≤ ε1 ,
1
|α|+1 σ
α! exp k1 ε− 2σ−1 .
|∂ α fε (x)| ≤ c1
We have also ∃k2 > 0, ∃c2 > 0, ∃ε2 ∈ ]0, 1] such that ∀x ∈ K, ∀α ∈
Zm
+ , ∀ε ≤ ε2 ,
1
|α|+1 σ
α! exp k2 ε− 2σ−1 .
|∂ α gε (x)| ≤ c2
Let α ∈ Zm
+ . Then
α 1 α
1
1
α−β
β
α
|∂
(f
g
)
(x)|
≤
f
(x)
g
(x)
.
∂
∂
ε
ε
ε
ε
σ
σ
σ
α!
β!
β (α − β)!
β=0
56
Khaled Benmeriem and Chikh Bouzar
Let ε ≤ min (ε1 , ε2 ) and k = k1 + k2 . Then we have ∀α ∈ Zm
+ , ∀x ∈ K,
1
1
|∂ α (fε gε ) (x)|
exp −kε− 2σ−1
σ
α!
1
1
− 2σ−1
α exp −k1 ε− 2σ−1 exp
−k
ε
2
α
α−β
β
f
(x)
g
(x)
≤
∂
∂ ε
ε
(α − β)!σ
β!σ
β
β=0
α α |α−β| |β|
c
c2
≤
β 1
β=0
|α|
≤2
(c1 + c2 )|α| ,
σ,∞
(Ω).
i.e., (fε )ε (gε )ε ∈ Em
Now let P (D) =
aγ Dγ be a σ-ultradifferential operator, then ∀h >
0, ∃b > 0, such that
1
1
|∂ α (P (D) fε (x))|
exp −k1 ε− 2σ−1
α!σ
h|γ| 1 1
b σ σ ∂ α+γ fε (x)
≤ exp −k1 ε− 2σ−1
γ! α!
m
γ∈Z+
2σ|α+γ| h|γ| 1
∂ α+γ fε (x)
≤ b exp −k1 ε− 2σ−1
σ
(α + γ)!
γ∈Zm
+
|α+γ|
2σ|α+γ| h|γ| c1
,
≤b
γ∈Zm
+
hence, for 2σ hc1 ≤ 12 , we have
1
1
|∂ α (P (D) fε (x))| ≤ c (2σ c1 )|α| ,
exp −kε− 2σ−1
α!σ
σ,∞
(Ω).
which shows that (P (D) fε )ε ∈ Em
σ,∞
σ,∞
σ
σ (Ω) and N σ (Ω) is
(Ω) = N (Ω) ∩ Em
(Ω) ⊂ Em
(2) The fact that N
σ,∞
σ
σ,∞
(Ω) is an ideal of Em
(Ω).
an ideal of Em (Ω) implies that N
Now, we define the Gevrey regular elements of G σ (Ω).
Definition 10. The algebra of regular generalized Gevrey ultradistributions
of order σ > 1, denoted G σ,∞ (Ω), is the quotient algebra
G σ,∞ (Ω) =
σ,∞
Em
(Ω)
.
σ,∞
N
(Ω)
It is clear that E σ (Ω) → G σ,∞ (Ω) and that G σ,∞ is subpresheaf of G σ .
Proposition 18. G σ,∞ is a subsheaf of G σ .
Proof. Let Ω be a nonvoid open subset of Rm and (Ωλ )λ∈Λ be an open
covering of Ω. We have to show the properties:
Generalized Gevrey ultradistributions
57
(S1) If f, g ∈ G σ,∞ (Ω) such that f/Ωλ = g/Ωλ , ∀λ ∈ Λ, then f = g.
(S2) If for each λ ∈ Λ, we have fλ ∈ G σ,∞ (Ωλ ), such that
fλ/Ωλ ∩Ωμ = fμ/Ωλ ∩Ωμ
for all λ, μ ∈ Λ with Ωλ ∩ Ωμ = φ,
then there exists a unique f ∈ G σ,∞ (Ω) with f/Ωλ = fλ , ∀λ ∈ Λ.
σ
The property (S1) is evident. To show (S2), let (χj )∞
j=1 be a E -partition
of unity subordinate to the covering (Ωλ )λ∈Λ . Define
f := (fε )ε + N σ,∞ (Ω),
where fε =
∞
j=1
χj fλj ε and fλj ε ε is a representative of fλj . Moreover, we
set fλj ε = 0 on Ω\Ωλj , so that χj fλj ε is C ∞ on all of Ω.
is a compact
Let K be a compact
of Ω. ThenKj = K
subset
∩ supp χj σ,∞
σ,∞ -estimates
Ωλj , and χj fλj ε satisfies Em
subset of Ωλj , fλj ε ε ∈ Em
on each Kj . We have χj (x) ≡ 0 on K except for a finite number of j, i.e.,
∃N > 0, such that
∞
So
χj fλj ε (x) =
j=1
χj fλj ε satisfies
N
χj fλj ε (x) ,
∀x ∈ K.
j=1
σ,∞
-estimates
Em
on all K, which means
σ,∞
(Ω).
(fε )ε ∈ Em
It remains now to show that f/Ωλ = fλ , ∀λ ∈ Λ. Let K be a compact subset
N
χj (x) ≡ 1 on a neighborhood
of Ωλ , choose N > 0 in such a way that
j=1
Ω of K with Ω a compact subset of Ωλ . For x ∈ K,
fε (x) − fλε (x) =
N
χj (x) fλj ε (x) − fλε (x) .
j=1
Since fλj ε − fλε ∈ N σ,∞ Ωλj ∩ Ωλ and Kj = K ∩ supp χj is a compact
N
σ,∞
χj fλj ε − fλε
estimates
satisfies the N σ and Em
subset of Ω ∩ Ωλj ,
j=1
on K. The uniqueness of such f ∈ G σ,∞ (Ω) follows from (S1).
The above proposition motivates the following definition.
Definition 11. We define the G σ,∞ -singular support of a generalized Gevrey
ultradistribution f ∈ G σ (Ω), denoted σ-singsuppg (f ), as the complement of
the largest open set Ω such that f ∈ G σ,∞ (Ω ).
The following result is a Paley–Wiener type characterization of G σ,∞ (Ω).
58
Khaled Benmeriem and Chikh Bouzar
Proposition 19. Let f = cl (fε )ε ∈ GCσ (Ω). Then f is regular if and only
if ∃k1 > 0, ∃k2 > 0, ∃c > 0, ∃ε1 > 0, ∀ε ≤ ε1 , such that
1
1
(7.1)
|F (fε ) (ξ)| ≤ c exp k1 ε− 2σ−1 − k2 |ξ| σ , ∀ξ ∈ Rm ,
where F (fε ) denote Fourier transform of fε .
Proof. Suppose that f = cl (fε )ε ∈ GCσ (Ω) ∩ G σ,∞ (Ω). Then ∃k1 > 0, ∃c1 >
0, ∃ε1 > 0, ∀α ∈ Zm
+ , ∀x ∈ K, ∀ε ≤ ε1 , supp fε ⊂ K, such that
1
|α|+1 σ
α! exp k1 ε− 2σ−1 .
|∂ α fε | ≤ c1
Consequently we have, ∀α ∈ Zm
+,
|ξ α | |F (fε ) (ξ)| ≤ exp (−ixξ) ∂ α fε (x) dx .
Then ∃c > 0, ∀ε ≤ ε1 ,
1
|ξ||α| |F (fε ) (ξ)| ≤ c|α|+1 α!σ exp k1 ε− 2σ−1 .
For α ∈ Zm
+ , ∃N ∈ Z+ such that
N
N
≤ |α| <
+ 1,
σ
σ
so
1
N
|ξ| σ |F (fε ) (ξ)| ≤ c|α|+1 |α||α|σ exp k1 ε− 2σ−1
1
≤ cN +1 N N exp k1 ε− 2σ−1 .
Hence ∃c > 0, ∀N ∈ Z+ ,
N
1
|F (fε ) (ξ)| ≤ cN +1 |ξ|− σ N ! exp k1 ε− 2σ−1 ,
which gives
|F (fε ) (ξ)| exp
1
1
|ξ| σ
2c
1
2−N ,
≤ c exp k1 ε− 2σ−1
1
1
1
− 2σ−1
σ
|ξ|
−
,
|F (fε ) (ξ)| ≤ c exp k1 ε
2c
i.e., we have (7.1).
Suppose now that (7.1) is valid. Then ∀ε ≤ ε0 ,
1
1
− 2σ−1
α
|ξ α | exp −k2 |ξ| σ dξ.
|∂ fε (x)| ≤ c1 exp k1 ε
or
Due to the inequality tN ≤ N ! exp (t) , ∀t > 0, then ∃c2 = c (k2 ) such that
1
k2
|α|
|ξ α | exp − |ξ| σ ≤ c2 α!σ .
2
Generalized Gevrey ultradistributions
Then
α
1
− 2σ−1
|∂ fε (x)| ≤ c1 exp k1 ε
|α|
c2 α!σ
1
k2
exp − |ξ| σ
2
59
dξ
1
≤ c|α|+1 α!σ exp k1 ε− 2σ−1 ,
1
where c = max c1 exp − k22 |ξ| σ dξ, c2 , i.e., f ∈ G σ,∞ (Ω).
Remark 5. It is easy to see if f = cl (fε )ε ∈ GCσ (Ω), then ∃k1 > 0, ∃c >
0, ∃ε0 > 0, ∀k2 > 0, ∀ε ≤ ε0 ,
1
1
(7.2)
|F (fε ) (ξ)| ≤ c exp k1 ε− 2σ−1 + k2 |ξ| σ ,∀ξ ∈ Rm .
The algebra G σ,∞ (Ω) plays the same role as the Oberguggenberger subalgebra of regular elements G ∞ (Ω) in the Colombeau algebra G(Ω) (see [19]).
Theorem 20. We have
(Ω) = E σ (Ω) .
G σ,∞ (Ω) ∩ D3σ−1
(Ω). For any fixed x0 ∈ Ω we take ψ ∈
Proof. Let S ∈ G σ,∞ (Ω) ∩ D3σ−1
3σ−1
(Ω) with ψ ≡ 1 on neighborhood U of x0 . Then T = ψS ∈ E3σ−1
(Ω).
D
σ
Let φε be a net of mollifiers with φ̌ = φ and let χ ∈ D (Ω) such that χ ≡ 1 on
K = supp ψ. As [T ] ∈ G σ,∞ (Ω), ∃k1 > 0, ∃k2 > 0, ∃c1 > 0, ∃ε1 > 0, ∀ε ≤ ε1 ,
|F (χ (T ∗ φε )) (ξ)| ≤ c1 ek1 ε
1
− 2σ−1
1
−k2 |ξ| σ
.
Thus
|F (χ (T ∗ φε )) (ξ) − F (T ) (ξ)|
= |F (χ (T ∗ φε )) (ξ) − F (χT ) (ξ)|
= T (x) , χ (x) e−iξx ∗ φε (x) − χ (x) e−iξx .
(Ω) ⊂ Eσ (Ω), ∃L a compact subset of Ω such that ∀h > 0, ∃c > 0,
As E3σ−1
and
|F (χ (T ∗ φε )) (ξ) − F (T ) (ξ)|
h|α| α −iξx
−iξx
∗
φ
(x)
−
χ
(x)
e
χ
(x)
e
∂
≤ c sup
.
ε
x
σ
α!
α∈Zm
,x∈L
+
We have e−iξ χ ∈ Dσ (Ω). From Corollary 12, ∀k3 > 0, ∃c2 > 0, ∃η > 0, ∀ε ≤
η,
|α|
− 1
c2 α −iξx
−iξx −k3 ε 2σ−1
∗
φ
(x)
−
χ
(x)
e
e
,
χ
(x)
e
≤
c
sup
∂
ε
2
α!σ x
α∈Zm
+ ,x∈L
so there exists c = c (k3 ) > 0, such that
|F (T ) (ξ) − F (χ (T ∗ φε )) (ξ)| ≤ c e−k3 ε
1
− 2σ−1
.
60
Khaled Benmeriem and Chikh Bouzar
Let ε ≤ min (η, ε1 ). Then
|F (T ) (ξ)| ≤ |F (T ) (ξ) − F (χ (T ∗ φε )) (ξ)| + |F (χ (T ∗ φε )) (ξ)|
≤ c e−k3 ε
1
− 2σ−1
Take c = max (c , c1 ), ε =
+ c1 ek1 ε
k1
(k2 − r) |ξ|
Then ∃δ > 0, ∃c > 0 such that
1
− 2σ−1
1
σ
1
−k2 |ξ| σ
.
2σ−1
, r ∈ ]0, k2 [ and k3 =
k1 r
.
k2 − r
1
|F (T ) (ξ)| ≤ ce−δ|ξ| σ ,
which means T = ψS ∈ E σ (Ω). As ψ ≡ 1 on the neighborhood U of x0 ,
S ∈ E σ (U ). Consequently S ∈ E σ (Ω), which proves
(Ω) ⊂ E σ (Ω) .
G σ,∞ (Ω) ∩ D3σ−1
(Ω) and E σ (Ω) ⊂ G σ,∞ (Ω), so
We have E σ (Ω) ⊂ E 3σ−1 (Ω) ⊂ D3σ−1
(Ω) .
E σ (Ω) ⊂ G σ,∞ (Ω) ∩ D3σ−1
Consequently we have
(Ω) = E σ (Ω) .
G σ,∞ (Ω) ∩ D3σ−1
8. Generalized Gevrey wave front
The aim of this section is to introduce the generalized Gevrey wave front
of a generalized Gevrey ultradistribution and to give its main properties.
Definition 12. We define σg (f ) ⊂ Rm \ {0} , f ∈ GCσ (Ω), as the complement of the set of points having a conic neighborhood Γ such that ∃k1 >
0, ∃k2 > 0, ∃c > 0, ∃ε0 ∈ ]0, 1] , ∀ξ ∈ Γ, ∀ε ≤ ε0 ,
1
1
(8.1)
|F (fε ) (ξ)| ≤ c exp k1 ε− 2σ−1 − k2 |ξ| σ .
The following essential properties of σg (f ) are sufficient to define later
the generalized Gevrey wave front of generalized Gevrey ultradistributions.
Proposition 21. For every f ∈ GCσ (Ω), we have:
(1) The set σg (f ) is a closed cone.
σ
(f ) = ∅ ⇐⇒ f ∈ G σ,∞ (Ω).
(2)
σ
gσ
σ
(3)
g (ψf ) ⊂
g (f ) , ∀ψ ∈ E (Ω).
Proof. One can easily, from the definition and Proposition 19, prove the
assertions (1) and (2).
σ
/
Let us suppose that ξ0 ∈
g (f ), then ∃Γ a conic neighborhood of
ξ0 , ∃k1 > 0, ∃k2 > 0, ∃c1 > 0, ∃ε1 ∈ ]0, 1], ∀ξ ∈ Γ, ∀ε ≤ ε1 ,
1
1
(8.2)
|F (fε ) (ξ)| ≤ c1 exp k1 ε− 2σ−1 − k2 |ξ| σ .
Generalized Gevrey ultradistributions
61
Let χ ∈ D σ (Ω), χ ≡ 1 on neighborhood of supp f , so χψ ∈ D σ (Ω), hence
∃k3 > 0, ∃c2 > 0, ∀ξ ∈ Rm ,
1
(8.3)
|F (χψ) (ξ)| ≤ c2 exp −k3 |ξ| σ .
Let Λ be a conic neighborhood of ξ0 such that Λ ⊂ Γ. We have, for a fixed
ξ ∈ Λ,
F (ψfε ) (ξ) = F (χψfε ) (ξ)
F (fε ) (η) F (χψ) (ξ − η) dη
=
A
F (fε ) (η) F (χψ) (ξ − η) dη,
+
B
where
1
1
1
,
A = η ∈ Rm ; |ξ − η| σ ≤ δ |ξ| σ + |η| σ
1
1
1
.
B = η ∈ Rm ; |ξ − η| σ > δ |ξ| σ + |η| σ
|ξ|
We choose δ sufficiently small that A ⊂ Γ and σ < |η| < 2σ |ξ| , ∀η ∈ A.
2
Then for ε ≤ ε1 ,
F (fε ) (η) F (χψ) (ξ − η) dη (8.4)
A
1
1
1
k2
− 2σ−1
|ξ| σ exp −k3 |ξ − η| σ dη −
≤ c1 c2 exp k1 ε
2
A
1
k2 σ1
− 2σ−1
|ξ|
−
.
≤ c exp k1 ε
2
As f ∈ GCσ (Ω), from (7.2), ∃c3 > 0, ∃μ1 > 0, ∃ε2 ∈ ]0, 1] , ∀μ2 > 0, ∀ξ ∈
Rm , ∀ε ≤ ε2 , such that
1
1
|F (fε ) (ξ)| ≤ c3 exp μ1 ε− 2σ−1 + μ2 |ξ| σ ,
hence, for ε ≤ min (ε1 , ε2 ), we have
F (fε ) (η) F (χψ) (ξ − η) dη B
1
1
1
exp μ2 |η| σ − k3 |ξ − η| σ dη
≤ c2 c3 exp μ1 ε− 2σ−1
B
1
1
1
1
exp μ2 |η| σ − k3 δ |ξ| σ + |η| σ
dη.
≤ c exp μ1 ε− 2σ−1
B
Then, taking μ2 < k3 δ, we obtain
1
1
F (fε ) (η) F (χψ) (ξ − η) dη ≤ c exp μ1 ε− 2σ−1 − k3 δ |ξ| σ .
(8.5)
B
62
Khaled Benmeriem and Chikh Bouzar
/
Consequently, (8.4) and (8.5) give ξ0 ∈
σ
g
(ψf ).
Definition 13. Let f ∈ G σ (Ω) and x0 ∈ Ω. The cone of σ-singular directions of f at x0 , denoted σg,x0 (f ), is
σ
(f )
(8.6)
g,x0
σ
(φf ) : φ ∈ Dσ (Ω) and φ ≡ 1 on a neighborhood of x0 .
=
g
Lemma 22. Let f ∈ G σ (Ω), then
σ
(f ) = ∅ ⇐⇒ x0 ∈
/ σ- singsuppg (f ) .
g,x0
/ σ-singsuppg (f ), i.e., ∃U ⊂ Ω an open neighborhood of x0
Proof. Let x0 ∈
such that f ∈ G σ,∞ (U ). Let φ ∈ Dσ (U ) such that φ ≡ 1 on
a neighborhood
of x0 . Then φf ∈ G σ,∞ (Ω). Hence, from Proposition 21, σg (φf ) = ∅, i.e.,
σ
g,x0 (f ) = ∅. Suppose now σg,x0 (f ) = ∅. Let r > 0 such that B (x0 , 2r) ⊂ Ω and let
ψ ∈ D σ (B (x0 , 2r)) such that 0 ≤ ψ ≤ 1 and ψ ≡ 1 on B (x0, r). Let
ψj (x) =
⊂ Ω and
ψ 3j (x − x0 ) + x0 . Then it is clear that supp (ψj ) ⊂ B x0 , 2r
j
3
r
σ
ψj ≡ 1 on B x0 , 3j . We have ∀φ ∈ D (Ω) with φ ≡ 1 on a neighborhood
U of x0 , ∃j ∈ Z+ such that supp (ψj ) ⊂ U , then ψj fε = ψj φfε and from
Proposition 21, we have
σ
σ
(ψj f ) ⊂
(φf ) ,
g
which gives
(8.7)
g
σ
j∈Z+
g
(ψj f ) = ∅.
σ
σ
We have ψj ≡ 1 on supp (ψj+1 ), then
g (ψj+1 f ) ⊂
g (ψj f ), so from
+
σ,∞ (Ω),
(8.7), there exists
n ∈r Z sufficiently large such that (ψn f ) ∈ G
σ,∞
/ σ-singsuppg (f ).
B x0 , 3n , which means. x0 ∈
then f ∈ G
Now, we are ready to give the definition of the generalized Gevrey wave
front.
/ W Fgσ (f ) ⊂ Ω×Rm\ {0} if ξ0 ∈
/ σg,x0 (f ),
Definition 14. A point (x0 , ξ0 ) ∈
i.e., there exists φ ∈ D σ (Ω), φ (x) = 1 neighborhood of x0 , and conic
neighborhood Γ of ξ0 , ∃k1 > 0, ∃k2 > 0, ∃c > 0, ∃ε0 ∈ ]0, 1], such that
∀ξ ∈ Γ, ∀ε ≤ ε0 ,
1
1
|F (φfε ) (ξ)| ≤ c exp k1 ε− 2σ−1 − k2 |ξ| σ .
The main properties of the generalized Gevrey wave front W Fgσ are subsumed in the following proposition.
Proposition 23. Let f ∈ G σ (Ω). Then:
Generalized Gevrey ultradistributions
(1)
(2)
(3)
(4)
63
The projection of W Fgσ (f ) on Ω is σ- singsuppg (f ).
If f ∈ GCσ (Ω), then the projection of W Fgσ (f ) on Rm \ {0} is σg (f ).
σ
α
σ
∀α ∈ Zm
+ , W Fg (∂ f ) ⊂ W Fg (f ).
∀g ∈ G σ,∞ (Ω), W Fgσ (gf ) ⊂ W Fgσ (f ).
Proof. (1) and (2) hold from the definition, Proposition 21 and Lemma 22.
/ W Fgσ (f ), then ∃φ ∈ Dσ (Ω), φ ≡ 1 on a neighborhood
(3) Let (x0 , ξ0 ) ∈
U of x0 , there exist a conic neighborhood Γ of ξ0 , ∃k1 > 0, ∃k2 > 0, ∃c1 >
0, ∃ε0 ∈ ]0, 1], such that ∀ξ ∈ Γ, ∀ε ≤ ε0 ,
1
1
− 2σ−1
σ
(8.8)
|F (φfε ) (ξ)| ≤ c1 exp k1 ε
− k2 |ξ| .
We have, for ψ ∈ D σ (U ) such that ψ (x0 ) = 1,
|F (ψ∂fε ) (ξ)| = |F (∂ (ψfε )) (ξ) − F ((∂ψ) fε ) (ξ)|
≤ |ξ| |F (ψφfε ) (ξ)| + |F ((∂ψ) φfε ) (ξ)| .
As W Fgσ (ψf ) ⊂ W Fgσ (f ), (8.8) holds for both
|F (ψφfε ) (ξ)| and |F ((∂ψ) φfε ) (ξ)| .
So
1
1
|ξ| |F (ψφfε ) (ξ)| ≤ c |ξ| exp k1 ε− 2σ−1 − k2 |ξ| σ
1
1
≤ c exp k1 ε− 2σ−1 − k3 |ξ| σ ,
1
with c > 0, k3 > 0 such that |ξ| ≤ c exp (k2 − k3 ) |ξ| σ . Hence (8.8) holds
/ W Fgσ (∂f ).
for |F (ψ∂fε ) (ξ)|, which proves (x0 , ξ0 ) ∈
σ
/ W Fg (f ). Then ∃φ ∈ Dσ (Ω), φ ≡ 1 on a neighborhood
(4) Let (x0 , ξ0 ) ∈
U of x0 , there exists a conic neighborhood Γ of ξ0 , ∃k1 > 0, ∃k2 > 0, ∃c1 >
0, ∃ε0 ∈ ]0, 1], such that ∀ξ ∈ Γ, ∀ε ≤ ε0 ,
1
1
|F (φfε ) (ξ)| ≤ c1 exp k1 ε− 2σ−1 − k2 |ξ| σ .
Let ψ ∈ D σ (Ω) and ψ ≡ 1 on supp φ. Then F (φgε fε ) = F (ψgε ) ∗ F (φfε ).
We have ψg ∈ G σ,∞ (Ω) ∩ GCσ (Ω), then ∃c2 > 0, ∃k3 > 0, ∃k4 > 0, ∃ε1 >
0, ∀ξ ∈ Rm , ∀ε ≤ ε1 ,
1
1
|F (ψgε ) (ξ)| ≤ c2 exp k3 ε− 2σ−1 − k4 |ξ| σ ,
so
F (φgε fε ) (ξ) =
F (φfε ) (η) F (ψgε ) (η − ξ) dη
F (φfε ) (η) F (ψgε ) (η − ξ) dη,
+
A
B
64
Khaled Benmeriem and Chikh Bouzar
where A and B are the same as in the proof of Proposition 21. By (7.2), we
have ∃c > 0, ∃μ1 > 0, ∀μ2 > 0, ∃ε2 > 0, ∀ξ ∈ Rm , ∀ε ≤ ε2 ,
1
1
|F (φfε ) (ξ)| ≤ c exp μ1 ε− 2σ−1 + μ2 |ξ| σ .
The same steps as in Proposition 21 finish the proof.
aα (x) Dα be a partial differential operCorollary 24. Let P (x, D) =
|α|≤m
ator with G σ,∞ (Ω) coefficients, then
W Fgσ (P (x, D) f ) ⊂ W Fgσ (f )
∀f ∈ G σ (Ω).
Remark 6. The reverse inclusion will give a generalized Gevrey microlocal
hypoellipticity of linear partial differential operators with regular Gevrey
generalized coefficients. The case of generalized G ∞ -microlocal hypoellipticity in Colombeau algebra has been studied recently in [14]. The generalized
G ∞ -microlocal hypoellipticity of microelliptic generalized pseudodifferential
operators has been tackled in [8], and lastly in [7] this result wasextended to
.
the level of basic functionals in the dual of Colombeau algebra L GC (Ω), C
We need the following lemma to show the relationship between W Fgσ (T )
(Ω).
and W F σ (T ), when T ∈ D3σ−1
Lemma 25. Let ϕ ∈ D σ (B (0, 2)), 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 on B (0, 1), and
let φ ∈ S (σ) . Then ∃c > 0, ∃ν > 0, ∃ε0 > 0, ∀ε ∈ ]0, ε0 ] , ∀ξ ∈ Rm ,
1
σ
1
|ρε (ξ)| ≤ cε−m e−νε σ |ξ| ,
m where ρε (x) = 1ε φ xε ϕ (x |ln ε|), and ρ denotes the Fourier transform
of ρ.
Proof. We have, for ε sufficiently small,
εm ≤ |ln ε|−m ≤ 1.
Let ξ ∈ Rm . Then
−m
ρε (ξ) = |ln ε|
A
φ (ε (ξ − η)) ϕ
η
|ln ε|
B
where
1
1
A = η; |ξ − η| σ ≤ δ σ
1
1
B = η; |ξ − η| σ > δ σ
dη+
η
φ (ε (ξ − η)) ϕ
|ln ε|
1
1
|ξ| σ + |η| σ
,
1
1
|ξ| σ + |η| σ
.
dη
,
Generalized Gevrey ultradistributions
65
|ξ|
< |η| < 2σ |ξ| , ∀η ∈ A. Since
2σ
φ ∈ S (σ) , ϕ ∈ Dσ (Ω), then ∃k1 , k2 > 0, ∃c1 , c2 > 0, ∀ξ ∈ Rm ,
1
1
and |ϕ
(ξ)| ≤ c1 exp −k2 |ξ| σ .
φ (ξ) ≤ c2 exp −k1 |ξ| σ
We choose δ sufficiently small such that
So
η
dη I1 = |ln ε|
φ (ε (ξ − η)) ϕ
|ln ε|
A
1
1
1
1
|ln ε|− σ |ξ| σ
exp −k1 ε σ |ξ − η| σ dη.
≤ c1 c2 exp −k2
2
−m Let z = ε (η − ξ). Then
1
1
k2
exp − |ln ε|− σ |ξ| σ
I1 ≤ cε
2
1
1
≤ cε−m exp −vε σ |ξ| σ .
−m
1
exp −k1 |z| σ dz
For I2 , we have
η
dη I2 = |ln ε|
φ (ε (ξ − η)) ϕ
|ln ε|
B
1
1
1
1
exp −k1 ε σ |ξ − η| σ − k2 |ln ε|− σ |η| σ dη
≤ c1 c2
B
1
1
1
1
1
1
exp −k1 δε σ |η| σ − k2 |ln ε|− σ |η| σ dη
≤ c1 c2 exp −k1 δε σ |ξ| σ
B
1
1
1
1
exp −kε σ |η| σ dη
≤ c1 c2 exp −k1 δε σ |ξ| σ
B
1
1
−m
exp −vε σ |ξ| σ .
≤ cε
−m Consequently, ∃c > 0, ∃v > 0, ∃ε0 > 0, , ∀ε ≤ ε0 such that
1
1
|ρε (ξ)| ≤ cε−m exp −vε σ |ξ| σ , ∀ξ ∈ Rm .
We have the following important result.
(Ω) ∩ G σ (Ω). Then W Fgσ (T ) = W F σ (T ).
Theorem 26. Let T ∈ D3σ−1
(Ω) ⊂ Eσ (Ω) and ψ ∈ Dσ (Ω). We have
Proof. Let S ∈ E3σ−1
|F (ψ (S ∗ φε )) (ξ) − F (ψS) (ξ)|
= S (x) , ψ (x) e−iξx ∗ φ̆ε (x) − ψ (x) e−iξx .
66
Khaled Benmeriem and Chikh Bouzar
Then ∃L a compact subset of Ω such that ∀h > 0, ∃c > 0,
|F (ψ (S ∗ φε )) (ξ) − F (ψS) (ξ)|
h|α| α −iξx
−iξx
∗
φ̆
(x)
−
ψ
(x)
e
ψ
(x)
e
∂
≤ c sup
.
ε
x
σ
m
α!
α∈Z+ ,x∈L
We have e−iξ ψ ∈ Dσ (Ω). Corollary 12 gives, ∀k0 > 0, ∃c2 > 0, ∃η > 0, ∀ε ≤
η,
(8.9)
|α|
c2 α −iξx
−iξx ∗
φ̆
(x)
−
ψ
(x)
e
ψ
(x)
e
∂
ε
α!σ x
α∈Zm
+ ,x∈L
sup
≤ c2 e−k0 ε
so ∀k0 >
(8.10)
0, ∃c
1
− 2σ−1
,
> 0, ∃η > 0, ∀ε ≤ η, such that
|F (ψS) (ξ) − F (ψ (S ∗ φε )) (ξ)| ≤ c e−k0 ε
1
− 2σ−1
.
(Ω) ∩ G σ (Ω) and (x0 , ξ0 ) ∈
/ W Fgσ (T ). Then there exist
Let T ∈ D3σ−1
σ
χ ∈ D (Ω), χ (x) = 1 in a neighborhood of x0 , and a conic neighborhood Γ
of ξ0 , ∃k1 > 0, ∃k2 > 0, ∃c1 > 0, ∃ε0 ∈ ]0, 1[, such that ∀ξ ∈ Γ, ∀ε ≤ ε0 ,
(8.11)
|F (χ (T ∗ ρε )) (ξ)| ≤ c1 ek1 ε
1
− 2σ−1
1
−k2 |ξ| σ
.
of x0 such that for sufficiently
Let ψ ∈ D σ (Ω) equal 1 in neighborhood
2
small ε we have χ ≡ 1 on supp ψ + B 0, |ln ε| , and let ϕ ∈ Dσ (B (0, 2)),
0 ≤ ϕ ≤ 1 and ϕ ≡ 1 on B (0, 1). Then there exists ε0 < 1, such that
∀ε < ε0 ,
ψ (T ∗ ρε ) (x) = ψ (χT ∗ ρε ) (x) ,
x
1
. As χT ∈ E3σ−1
(Ω), then, from Propowhere ρε (x) = m ϕ (x |ln ε|) φ
ε
ε
sition 14,
ψ (T ∗ ρε ) (x) = ψ (χT ∗ ρε ) (x) = ψ (χT ∗ φε ) (x) .
Let ε ≤ min (η, ε0 ) and ξ ∈ Γ. We have
|F (ψT ) (ξ)| ≤ |F (ψT ) (ξ) − F (ψ (T ∗ ρε )) (ξ)| + |F (χ (T ∗ ρε )) (ξ)|
≤ |F (ψχT ) (ξ) − F (ψ (χT ∗ φε )) (ξ)| + |F (χ (T ∗ ρε )) (ξ)| .
Then by (8.10)and (8.11), we obtain
|F (ψT ) (ξ)| ≤ c e−k0 ε
Take c = max (c , c1 ), ε =
1
− 2σ−1
k1
(k2 − r) |ξ|
Then ∃δ > 0, ∃c > 0, such that
+ c1 ek1 ε
2σ−1
1
− 2σ−1
1
−k2 |ξ| σ
.
, r ∈ ]0, k2 [ , k0 =
1
σ
1
σ
|F (χT ) (ξ)| ≤ ce−δ|ξ| ,
k1 r
.
k2 − r
Generalized Gevrey ultradistributions
67
/ W F σ (T ), i.e., W F σ (T ) ⊂ W Fgσ (T ).
which proves that (x0 , ξ0 ) ∈
σ
/ W F (T ), then there exist χ ∈ D σ (Ω) , χ (x) = 1 in
Suppose (x0 , ξ0 ) ∈
a neighborhood of x0 , a conic neighborhood Γ of ξ0 , ∃λ > 0, ∃c1 > 0, such
that ∀ξ ∈ Γ,
1
σ
|F (χT ) (ξ)| ≤ c1 e−λ|ξ| .
(8.12)
of x0 such that for sufficiently
Let also ψ ∈ D σ (Ω) equal 1 in neighborhood
2
small ε we have χ ≡ 1 on supp ψ + B 0, |ln ε| . Then there exists ε0 < 1,
such that ∀ε < ε0 ,
ψ (T ∗ ρε ) (x) = ψ (χT ∗ ρε ) (x) .
We have
F (ψ (T ∗ ρε )) (ξ) =
F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη.
Let Λ be a conic neighborhood of ξ0 such that Λ ⊂ Γ. For a fixed ξ ∈ Λ, we
have
F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη
F (ψ (χT ∗ ρε )) (ξ) =
A
F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη,
+
B
where
1
1
1
,
A = η; |ξ − η| σ ≤ δ |ξ| σ + |η| σ
1
1
1
.
B = η; |ξ − η| σ > δ |ξ| σ + |η| σ
We choose δ sufficiently small such that A ⊂ Γ and
|ξ|
< |η| < 2σ |ξ|. Since
2σ
ψ ∈ D σ (Ω), then ∃μ > 0, ∃c2 > 0, ∀ξ ∈ Rm ,
1
|F (ψ) (ξ)| ≤ c2 exp −μ |ξ| σ .
Then ∃c > 0, ∃ε0 ∈ ]0, 1[ , ∀ε ≤ ε0 ,
(8.13) F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη A
1
λ 1 .
σ
σ
exp
−μ
|η
−
ξ|
)
(η)
dη
F
(ρ
≤ c exp − |ξ|
ε
2
A
From Lemma 25, ∃c3 > 0, ∃ν > 0, ∃ε0 > 0, such that ∀ε ∈ ]0, ε0 ],
1
1
σ
|F (ρε ) (ξ)| ≤ c3 ε−m e−νε σ |ξ| , ∀ξ ∈ Rm .
68
Khaled Benmeriem and Chikh Bouzar
Then ∃c > 0, such that
F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη A
1
1
1
λ 1
≤ cε−m exp − |ξ| σ
exp −μ |η − ξ| σ exp −νε σ |η| σ dη.
2
A
We have ∃k > 0, ∀ε ∈ ]0, ε0 ],
1
1
1
(8.14)
ε−m exp −νε σ |η| σ ≤ exp kε− 2σ−1 ,
so
(8.15)
F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη A
1
− 2σ−1
≤ c exp kε
λ 1
− |ξ| σ
2
.
(Ω) ⊂ Eσ (Ω), then ∀l > 0, ∃c > 0, ∀ξ ∈ Rm ,
As χT ∈ E3σ−1
1
|F (χT ) (ξ)| ≤ c exp l |ξ| σ ,
hence, we have
F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη B
1
1
exp l |η| σ − μ |η − ξ| σ |F (ρε )| dη
≤c
B
1
1
1
1
−m
σ
exp −μδ |ξ|
exp (l − μδ) |η| σ − νε σ |η| σ dη.
≤cε
B
Then, taking l − μδ = −a < 0 and using (8.14), we obtain for a constant
c > 0,
(8.16) F (ψ) (ξ − η) F (χT ) (η) F (ρε ) (η) dη B
1
1
≤ c exp kε− 2σ−1 − μδ |ξ| σ .
Consequently, (8.15) and (8.16) give ∃c > 0, ∃k1 > 0, ∃k2 > 0,
1
1
(8.17)
|F (ψ (T ∗ ρε )) (ξ)| ≤ c exp k1 ε− 2σ−1 − k2 |ξ| σ ,
/ W Fgσ (T ), so W Fgσ (T ) ⊂ W F σ (T ) which ends
which gives that (x0 , ξ0 ) ∈
the proof.
Generalized Gevrey ultradistributions
69
9. Generalized Hörmander’s theorem
To extend the generalized Hörmander’s result on the wave front set of the
product, define W Fgσ (f ) + W Fgσ (g), where f, g ∈ G σ (Ω), as the set
"
!
(x, ξ + η) ; (x, ξ) ∈ W Fgσ (f ) , (x, η) ∈ W Fgσ (g) .
We recall the following fundamental lemma (see [13] for the proof).
/ 1 + 2.
Lemma 27. Let 1 , 2 be closed cones in Rm \ {0}, with 0 ∈
Then:
Rm \{0}
= ( 1 + 2 ) ∪ 1 ∪ 2 . (1)
1+
2
(2) For any open conic neighborhood Γ of 1 + 2 in Rm \ {0}, one can
find
open conic neighborhoods of Γ1 , Γ2 in Rm \ {0} of, respectively,
1,
2 , such that
Γ1 + Γ2 ⊂ Γ.
The principal result of this section is the following theorem.
Theorem 28. Let f, g ∈ G σ (Ω), such that ∀x ∈ Ω,
(x, 0) ∈
/ W Fgσ (f ) + W Fgσ (g) .
(9.1)
Then
W Fgσ (f g) ⊆ W Fgσ (f ) + W Fgσ (g) ∪ W Fgσ (f ) ∪ W Fgσ (g) .
/ W Fgσ (f ) + W Fgσ (g) ∪ W Fgσ (f ) ∪ W Fgσ (g). Then
Proof. Let (x0 , ξ0 ) ∈
σ
σ
σ
σ
/
∃φ ∈ D σ (Ω), φ (x0 ) = 1, ξ0 ∈
g (φf ) +
g (φg) ∪
g (φf ) ∪
g (φg).
σ
σ
From (9.1) we have 0 ∈
/ g (φf ) + g (φg), so by Lemma 27 (1), we have
(9.2)
/
ξ0 ∈
=
σ
g
σ
g
(φf ) +
(φf ) +
σ
g
σ
g
σ
σ
(φg) ∪
(φf ) ∪
(φg)
g
g
Rm \{0}
(φg)
.
Let Γ0 be an open conic neighborhood of σg (φf ) + σg (φg) in Rm \ {0}
/ Γ0 . Then from Lemma 27 (2) there exist open cones Γ1 and
such that ξ0 ∈
Γ2 in Rm \ {0} such that
σ
σ
(φf ) ⊂ Γ1 ,
(φg) ⊂ Γ2 and Γ1 + Γ2 ⊂ Γ0 .
g
g
Define Γ = Rm \Γ0 , so
(9.3)
Γ ∩ Γ2 = ∅
and
(Γ − Γ2 ) ∩ Γ1 = ∅.
70
Khaled Benmeriem and Chikh Bouzar
Let ξ ∈ Γ and ε ∈ ]0, 1].
F (φfε φgε ) (ξ) = (F (φfε ) ∗ F (φgε )) (ξ)
F (φfε ) (ξ − η) F (φgε ) (η) dη
=
Γ2
F (φfε ) (ξ − η) F (φgε ) (η) dη
+
Γc2
= I1 (ξ) + I2 (ξ) .
From (9.3), ∃c1 > 0, ∃k1 , k2 > 0, ∃ε1 > 0 such that ∀ε ≤ ε1 , ∀η ∈ Γ2 ,
1
1
F (φfε ) (ξ − η) ≤ c1 exp k1 ε− 2σ−1 − k2 |ξ − η| σ ,
and from (7.2) ∃c2 > 0, ∃k3 > 0, ∀k4 > 0, ∃ε2 > 0, ∀η ∈ Rm , ∀ε ≤ ε2 ,
1
1
|F (φgε ) (η)| ≤ c2 exp k3 ε− 2σ−1 + k4 |η| σ .
1
1
1
Let γ > 0 sufficiently small such that |ξ − η| σ ≥ γ |ξ| σ + |η| σ , ∀η ∈ Γ2 .
Hence for ε ≤ min (ε1 , ε2 ),
1
1
|I1 (ξ)| ≤ c1 c2 exp (k1 + k3 ) ε− 2σ−1 − k2 γ |ξ| σ
1
1
× exp −k2 γ |η| σ + k4 |η| σ dη.
Take k4 < k2 γ. Then
1
1
|I1 (ξ)| ≤ c exp k1 ε− 2σ−1 − k2 |ξ| σ .
(9.4)
Let r > 0,
I2 (ξ) =
Γc2 ∩{|η|≤r|ξ|}
F (φfε ) (ξ − η) F (φgε ) (η) dη
+
Γc2 ∩{|η|≥r|ξ|}
F (φfε ) (ξ − η) F (φgε ) (η) dη
= I21 (ξ) + I22 (ξ) .
Choose r sufficiently small such that
1
1
1
1
/ Γ1 . Then
|η| σ ≤ r |ξ| σ =⇒ ξ − η ∈
1
1
|ξ − η| σ ≥ (1 − r) |ξ| σ ≥ (1 − 2r) |ξ| σ + |η| σ , so ∃c3 > 0, ∃λ1 , λ2 , λ3 >
0, ∃ε3 > 0 such that ∀ε ≤ ε3 ,
1
1
1
− 2σ−1
exp −λ2 |ξ − η| σ − λ3 |η| σ dη
|I21 (ξ)| ≤ c3 exp λ1 ε
1
1
1
− 2σ−1
σ
− λ2 |ξ|
exp −λ3 |η| σ dη
≤ c3 exp λ1 ε
1
1
≤ c3 exp λ1 ε− 2σ−1 − λ2 |ξ| σ .
Generalized Gevrey ultradistributions
1
71
1
|η| σ + r |ξ| σ
, and then ∃c4 > 0, ∃μ1 , μ3 >
If |η| ≥ r |ξ| , we have |η| ≥
2
0, ∀μ2 > 0, ∃ε4 > 0 such that ∀ε ≤ ε4 ,
1
1
1
− 2σ−1
|I21 (ξ)| ≤ c4 exp μ1 ε
exp μ2 |ξ − η| σ − μ3 |η| σ dη
1
1
1
1
exp μ2 |ξ − η| σ − μ3 |η| σ − μ3 r |ξ| σ dη,
≤ c4 exp μ1 ε− 2σ−1
1
σ
1
σ
1
σ
if we take μ2 sufficiently small we obtain
1
1
|I21 (ξ)| ≤ c4 exp k3 ε− 2σ−1 − μ3 |ξ| σ ,
which finishes the proof.
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Centre Universitaire de Mascara. Algeria
benmeriemkhaled@yahoo.com
Department of Mathematics, Oran-Essenia University. Algeria
bouzar@yahoo.com bouzar@univ-oran.dz
This paper is available via http://nyjm.albany.edu/j/2009/15-3.html.