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 37 2. Generalized Gevrey ultradistributions 39 3. Generalized point values 43 4. Embedding of Gevrey ultradistributions with compact support 45 5. Sheaf properties of G^{σ} 50 6. Equalities in G^{σ}(Ω) 54 7. Regular generalized Gevrey ultradistributions 55 8. Generalized Gevrey wave front 60 9. Generalized Hörmander’s theorem 69 References 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, Colom-

beau generalized functions, Gevrey wave front, Microlocal analysis, Product of ultradistributions.

37^{ISSN 1076-9803/09}

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 ℝ^{m} as the quotient algebra

G^{σ}(Ω) = \frac{E_{m}^{σ}(Ω)}{N^{σ}(Ω)} ,

where E_{m}^{σ}(Ω) is the space of (f_{ε})_{ε} ∈ C^{∞} (Ω)^{]0,1]} satisfying for every compact subset K of Ω, ∀α ∈ ℤ_{+}^{m}, ∃k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , ∀ε ≤ ε_{0},

sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ c exp (kε^{− \frac{1}{2σ−1}} ) ,

and N^{σ}(Ω) is the space of (f_{ε})_{ε} ∈ C^{∞}(Ω)^{]0,1]} satisfying for every compact subset K of Ω, ∀α ∈ ℤ_{+}^{m}, ∀k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , ∀ε ≤ ε_{0},

sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) .

The functor Ω → G^{σ} (Ω) being a sheaf of differential algebras on ℝ^{m}, we show that G^{σ}(Ω) contains the space of Gevrey ultradistributions of order (3σ − 1), and the following diagram of embeddings is commutative:

\xymatrix{E^{σ}(Ω) \ar[r] \ar[dr] & G^{σ}(Ω) \\  & D′_{3σ−1}(Ω). \ar[u]}

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:

G^{σ,∞}(Ω) ∩ D′_{3σ−1}(Ω) = E^{σ}(Ω).

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 F_{g}^{σ} (f), and further to give its main properties, as W F_{g}^{σ} (T) = W F^{σ} (T), if T ∈ D′_{3σ−1}(Ω) ∩ G^{σ}(Ω), and

W F_{g}^{σ} (P (x, D) f) ⊂ W F_{g}^{σ} (f) ∀f ∈ G^{σ}(Ω),

if P (x, D) = ∑_{|α|≤m} a_{α} (x) D^{α} is a partial differential operator with G^{σ,∞} (Ω)

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 F_{g}^{σ} (f) + W F_{g}^{σ} (g), then

W F_{g}^{σ} (fg) ⊆ (W F_{g}^{σ} (f) + W F_{g}^{σ} (g)) ∪ W F_{g}^{σ} (f) ∪ W F_{g}^{σ} (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 ℝ^{m}.

Definition 1. The space of moderate elements, denoted E_{m}^{σ}(Ω), is the space of (f_{ε})_{ε} ∈ C^{∞}(Ω)^{]0,1]} satisfying for every compact subset K of Ω, ∀α ∈ ℤ_{+}^{m}, ∃k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , ∀ε ≤ ε_{0},

(2.1)  sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ c exp (kε^{− \frac{1}{2σ−1}} ) .

The space of null elements, denoted N^{σ} (Ω), is the space of

(f_{ε})_{ε} ∈ C^{∞}(Ω)^{]0,1]}

satisfying for every compact subset K of Ω, ∀α ∈ ℤ_{+}^{m}, ∀k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1], ∀ε ≤ ε_{0},

(2.2)  sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) .

The main properties of the spaces E_{m}^{σ} (Ω) and N^{σ}(Ω) are given in the following proposition.

Proposition 1. (1) The space of moderate elements E_{m}^{σ} (Ω) is an algebra

stable under derivation.

(2) The space N^{σ}(Ω) is an ideal of E_{m}^{σ}(Ω).

Proof. (1) Let (f_{ε})_{ε} , (g_{ε})_{ε} ∈ E_{m}^{σ}(Ω) and K be a compact subset of Ω. Then ∀β ∈ ℤ_{+}^{m}, ∃k_{1} = k_{1} (β) > 0, ∃c_{1} = c_{1} (β) > 0, ∃ε_{1β} ∈ ]0, 1], ∀ε ≤ ε_{1β},

(2.3)  sup_{x∈K} |∂^{β}f_{ε} (x)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ,

∀β ∈ ℤ_{+}^{m}, ∃k_{2} = k_{2} (β) > 0, ∃c_{2} = c_{2} (β) > 0, ∃ε_{2β} ∈ ]0, 1] , ∀ε ≤ ε_{2β},

(2.4)  sup_{x∈K} |∂^{β}g_{ε} (x)| ≤ c_{2} exp (k_{2}ε^{− \frac{1}{2σ−1}} ) .

Let α ∈ ℤ_{+}^{m}. Then

|∂^{α} (f_{ε}g_{ε}) (x)| ≤ ∑_{β=0}^{α} \binom{α}{β} |∂^{α−β}f_{ε} (x)| |∂^{β}g_{ε} (x)| .

For

k = max {k_{1} (β) : β ≤ α} + max {k_{2} (β) : β ≤ α} ,
ε ≤ min {ε_{1β}, ε_{2β}; |β| ≤ |α|}

and x ∈ K, we have

exp (−kε^{− \frac{1}{2σ−1}} ) |∂^{α} (f_{ε}g_{ε}) (x)| ≤ ∑_{β=0}^{α} \binom{α}{β} exp (−k_{1}ε^{− \frac{1}{2σ−1}} ) |∂^{α−β}f_{ε} (x)|

× exp (−k_{2}ε^{− \frac{1}{2σ−1}} ) |∂^{β}g_{ε} (x)|

≤ ∑_{β=0}^{α} \binom{α}{β}c_{1} (α − β) c_{2} (β) = c (α) ,

i.e., (f_{ε}g_{ε})_{ε} ∈ E_{m}^{σ}(Ω). It is clear, from (2.3) that for every compact subset K of Ω, ∀β ∈ ℤ_{+}^{m}, ∃k_{1} = k_{1} (β + 1) > 0, ∃c_{1} = c_{1} (β + 1) > 0, ∃ε_{1β} ∈ ]0, 1] such that ∀x ∈ K, ∀ε ≤ ε_{1β},

|∂^{β} (∂f_{ε}) (x)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ,

i.e., (∂f_{ε})_{ε} ∈ E_{m}^{σ}(Ω).

(2) If (g_{ε})_{ε} ∈ N^{σ}(Ω), for every compact subset K of Ω, ∀β ∈ ℤ_{+}^{m}, ∀k_{2} > 0, ∃c_{2} = c_{2} (β, k_{2}) > 0, ∃ε_{2β} ∈ ]0, 1],

|∂^{α}g_{ε} (x)| ≤ c_{2} exp (−k_{2}ε^{− \frac{1}{2σ−1}} ) , ∀x ∈ K, ∀ε ≤ ε_{2β}.

Let α ∈ ℤ_{+}^{m} and k > 0. Then

exp (kε^{− \frac{1}{2σ−1}} ) |∂^{α} (f_{ε}g_{ε}) (x)|

≤ exp (kε^{− \frac{1}{2σ−1}} ) ∑_{β=0}^{α} \binom{α}{β} |∂^{α−β}f_{ε} (x)| |∂^{β}g_{ε} (x)| .

Let k_{2} = max {k_{1} (β) ; β ≤ α} + k and ε ≤ min {ε_{1β}, ε_{2β}; β ≤ α}, then ∀x ∈ K,

exp (kε^{− \frac{1}{2σ−1}} ) |∂^{α} (f_{ε}g_{ε}) (x)| ≤ ∑_{β=0}^{α} \binom{α}{β} [exp (−k_{1}ε^{− \frac{1}{2σ−1}} ) |∂^{α−β}f_{ε} (x)|

× exp (k_{2}ε^{− \frac{1}{2σ−1}} ) |∂^{β}g_{ε} (x)|]

≤ ∑_{β=0}^{α} \binom{α}{β}c_{1} (α − β) c_{2} (β, k_{2}) = c (α, k) ,

which shows that (f_{ε}g_{ε})_{ε} ∈ N^{σ}(Ω). □

Remark 1. The algebra of moderate elements E_{m}^{σ} (Ω) is not necessary stable 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) = \frac{E_{m}^{σ}(Ω)}{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) := \frac{E_{m}(Ω)}{N (Ω)}, where E_{m}(Ω) is the space of (f_{ε})_{ε} ∈ C^{∞}(Ω)^{]0,1]} satisfying for every compact subset K of Ω, ∀α ∈ ℤ_{+}^{m}, ∃k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , ∀ε ≤ ε_{0},

sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ cε^{−k},

and N (Ω) is the space of (f_{ε})_{ε} ∈ C^{∞} (Ω)^{]0,1]} satisfying for every compact subset K of Ω, ∀α ∈ ℤ_{+}^{m}, ∀k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , ∀ε ≤ ε_{0},

sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ cε^{k}.

Due to the inequality

exp (−ε^{− \frac{1}{2σ−1}} ) ≤ ε, ∀ε ∈ ]0, 1] ,

we have the strict inclusions N^{σ} (Ω) ⊂ N^{τ} (Ω) ⊂ N (Ω) ⊂ E_{m}(Ω) ⊂ E_{m}^{τ}(Ω) ⊂ E_{m}^{σ}(Ω), with σ < τ.

We have the null characterization of the ideal N^{σ}(Ω).

Proposition 2. Let (u_{ϵ})_{ϵ} ∈ E_{m}^{σ}(Ω), then (u_{ϵ})_{ϵ} ∈ N^{σ} (Ω) if and only if for every compact subset K of Ω, ∀k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1], ∀ε ≤ ε_{0},

(2.5)  sup_{x∈K} |f_{ε} (x)| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) .

Proof. Let (u_{ϵ})_{ε} ∈ E_{m}^{σ}(Ω) satisfy (2.5). We will show that (∂_{i}u_{ϵ})_{ϵ} also satisfy (2.5) when i = 1, …, m, and then it will follow by induction that (u_{ϵ})_{ϵ} ∈ N^{σ}(Ω).

Suppose that u_{ϵ} has real values. In the complex case we do the calculus separately for the real and imaginary part of u_{ϵ}. Let K be a compact subset of Ω. For δ = min (1, dist (K, ∂Ω)), set L = K + \overline{B (0, \frac{δ}{2})}. Then

K ⊂⊂ L ⊂⊂ Ω.

By the moderateness of (u_{ϵ})_{ε}, we have ∃k_{1} > 0, ∃c_{1} > 0, ∃ε_{1} ∈ ]0, 1] , ∀ε ≤ ε_{1}

(2.6)  sup_{x∈L} |∂_{i}^{2}u_{ϵ} (x)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) .

By the assumption (2.5), ∀k > 0, ∃c_{2} > 0, ∃ε_{2} ∈ ]0, 1] , ∀ε ≤ ε_{2}

(2.7)  sup_{x∈L} |u_{ϵ} (x)| ≤ c_{2} exp (− (2k + k_{1}) ε^{− \frac{1}{2σ−1}} ) .

Let x ∈ K, ε sufficiently small and r = exp (− (k + k_{1}) ε^{− \frac{1}{2σ−1}} ) < \frac{δ}{2}. By Taylor’s formula, we have

∂_{i}u_{ϵ} (x) = \frac{(u_{ϵ} (x + re_{i}) − u_{ϵ} (x))}{r} − \frac{1}{2}∂_{i}^{2}u_{ϵ} (x + θre_{i}) r,

where e_{i} is i^{th} vector of the canonical base of ℝ^{m}, hence (x + θre_{i}) ∈ L, and then

|∂_{i}u_{ϵ} (x)| ≤ |u_{ϵ} (x + re_{i}) − u_{ϵ} (x)| r^{−1} + \frac{1}{2} |∂_{i}^{2}u_{ϵ} (x + θre_{i})| r.

From (2.6) and (2.7): |u_{ϵ} (x + re_{i}) − u_{ϵ} (x)| r^{−1} ≤ c_{2} exp (−kε^{− \frac{1}{2σ−1}} ) and

|∂_{i}^{2}u_{ϵ} (x + θre_{i})| r ≤ c_{1} exp (−kε^{− \frac{1}{2σ−1}} ), so

|∂_{i}u_{ϵ} (x)| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) ,

which gives the proof. □

Proposition 3. If P is a polynomial function and f = cl (f_{ε})_{ε} ∈ G^{σ}(Ω), then P (f) = (P (f_{ε}))_{ε} + N^{σ}(Ω) is a well-defined element of G^{σ} (Ω).

Proof. Let (f_{ε})_{ε} ∈ E_{m}^{σ}(Ω), P (ξ) = ∑_{|α|≤m} a_{α}ξ^{α} and K be a compact subset

of Ω. Then we have ∀α ∈ ℤ_{+}^{m}, ∃k = k (α) > 0, ∃c = c (α) > 0, ∃ε_{0} = ε (α) ∈ ]0, 1] , ∀ε ≤ ε_{0},

(2.8)  sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ c exp (kε^{− \frac{1}{2σ−1}} ) .

Let β ∈ ℤ_{+}^{m}, so

|∂^{β}P (f_{ε}) (x)| ≤ ∑_{|α|≤m} |a_{α}| |∂^{β}f_{ε}^{α} (x)| .

By the Leibniz formula and (2.8), we obtain

|∂^{β}P (f_{ε}) (x)| ≤ ∑_{\begin{matrix}|α|≤m \\ γ≤β \end{matrix}} c_{α,γ} (exp (k_{α,γ}ε^{− \frac{1}{2σ−1}} ))^{n_{α,γ}} ,

where c_{α,γ} > 0 and n_{α,γ} ∈ ℤ_{+}. Hence

|∂^{β}P (f_{ε}) (x)| ≤ c exp (kε^{− \frac{1}{2σ−1}} ) .

One can easily cheek that if (f_{ε})_{ε} ∈ N^{σ}(Ω), then (P (f_{ε}))_{ε} ∈ N^{σ}(Ω). □

The space of functions slowly increasing, denoted O_{M} (𝕂^{m}), is the space of C^{∞}-functions all derivatives growing at most like some power of |x|, as |x| → +∞, where 𝕂^{m} ≃ ℝ^{m} or ℝ^{2m}.

Corollary 4. If v ∈ O_{M} (𝕂^{m}) and f = (f_{1}, f_{2}, …, f_{m}) ∈ G^{σ}(Ω)^{m}, then v ∘ f := (v ∘ f_{ε})_{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

C^{σ} = \frac{E_{0}^{σ}}{N_{0}^{σ}} ,

where

E_{0}^{σ} = {(a_{ε})_{ε} ∈ ℂ^{]0,1]}; ∃k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , such that

∀ε ≤ ε_{0}, |a_{ε}| ≤ c exp (kε^{− \frac{1}{2σ−1}} )\}

and

N_{0}^{σ} = {(a_{ε})_{ε} ∈ ℂ^{]0,1]}; ∀k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , such that

∀ε ≤ ε_{0}, |a_{ε}| ≤ c exp (−kε^{− \frac{1}{2σ−1}} )\}.

It is not difficult to see that E_{0}^{σ} is an algebra and N_{0}^{σ} is an ideal of E_{0}^{σ}. 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 (u_{ε}(x))_{ε} is an element of C^{σ} independent of the representative (u_{ε})_{ε} of u.

A generalized Gevrey ultradistribution is not defined by its point values. We give here an example of a generalized Gevrey ultradistribution f = [(f_{ε})_{ε}] ∉ N^{σ} (ℝ), but [(f_{ε} (x))_{ε}] ∈ N_{0}^{σ} for every x ∈ ℝ. Let φ ∈ D (ℝ) such that φ (0) ≠ 0. For ε ∈ ]0, 1], define

f_{ε} (x) = x exp (−ε^{− \frac{1}{2σ−1}} ) φ (\frac{x}{ε}) , x ∈ ℝ.

It is clear that (f_{ε})_{ε} ∈ E_{m}^{σ} (ℝ). Let K be a compact neighborhood of 0, then

sup_{K} |f′ (x)| ≥ |f′_{ε} (0)| = exp (−ε^{− \frac{1}{2σ−1}} ) |φ (0)| ,

which shows that (f_{ε})_{ε} ∉ N^{σ} (ℝ). For any x_{0} ∈ ℝ, there exists ε_{0} such that φ (\frac{x_{0}}{ε} ) = 0, ∀ε ≤ ε_{0}, i.e., f (x_{0}) ∈ N_{0}^{σ}.

In order to give a solution to this situation, set

(3.1) Ω_{M}^{σ} =

\{(x_{ε})_{ε} ∈ Ω^{]0,1]} : ∃k > 0, ∃c > 0, ∃ε_{0} > 0, ∀ε ≤ ε_{0}, |x_{ε}| ≤ ce^{kε^{− \frac{1}{2σ−1}}} \} .

Define in Ω_{M}^{σ} the equivalence relation ∼ by

(3.2) x_{ε} ∼ y_{ε}

⟺ ∀k > 0, ∃c > 0, ∃ε_{0} > 0, ∀ε ≤ ε_{0}, |x_{ε} − y_{ε}| ≤ ce^{−kε^{− \frac{1}{2σ−1}}} .

Definition 3. The set Ω̃^{σ} = Ω_{M}^{σ}/ ∼ is called the set of generalized Gevrey points. The set of compactly supported Gevrey points is defined by

(3.3) Ω̃_{c}^{σ} =

{x̃ = [(x_{ε})_{ε}] ∈ Ω̃^{σ} : ∃K a compact set of Ω, ∃ε_{0} > 0, ∀ε ≤ ε_{0}, x_{ε} ∈ K} .

Remark 3. It is easy to see that the Ω̃_{c}^{σ}-property does not depend on the choice of the representative.

Proposition 6. Let f ∈ G^{σ}(Ω) and x̃ = [(x_{ε})_{ε}] ∈ Ω̃_{c}^{σ}, then the generalized Gevrey point value of f at x̃, i.e.,

f \binom{}{x} = [(f_{ε} (x_{ε}))_{ε}]

is a well-defined element of the algebra of generalized Gevrey complex numbers C^{σ}.

Proof. Let f ∈ G^{σ}(Ω) and x̃ = [(x_{ε})_{ε}] ∈ Ω̃_{c}^{σ}, there exists a compact subset K of Ω such that x_{ε} ∈ K for ε small, then ∃k > 0, ∃c > 0, ∃ε_{0} > 0, ∀ε ≤ ε_{0},

|f_{ε} (x_{ε})| ≤ sup_{x∈K} |f_{ε} (x)| ≤ c exp (kε^{− \frac{1}{2σ−1}} ) .

Therefore (f_{ε} (x_{ε}))_{ε} ∈ E_{0}^{σ}, and it is clear that if f ∈ N^{σ}(Ω), then (f_{ε} (x_{ε}))_{ε} ∈ N_{0}^{σ}, i.e., f \binom{}{x} does not depend on the choice of the representative (f_{ε})_{ε}.

Let now x̃ = [(x_{ε})_{ε}] ∼ ỹ = [(y_{ε})_{ε}], then ∀k > 0, ∃c > 0, ∃ε_{0} > 0, ∀ε ≤ ε_{0},

|x_{ε} − y_{ε}| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) .

Since (f_{ε})_{ε} ∈ E^{σ}(Ω), so for every compact subset K of Ω, ∀j ∈ {1, m} , ∃k_{j} > 0, ∃c_{j} > 0, ∃ε_{j} > 0, ∀ε ≤ ε_{j},

sup_{x∈K} |\frac{∂}{∂x_{j}} f_{ε} (x)| ≤ c_{j} exp (k_{j}ε^{− \frac{1}{2σ−1}} ) .

We have

|f_{ε} (x_{ε}) − f_{ε} (y_{ε})| ≤ |x_{ε} − y_{ε}| ∑_{j=1}^{m} ∫_{0}^{1} |(\frac{∂}{∂x_{j}} f_{ε}) (x_{ε} + t (y_{ε} − x_{ε}))| dt,

and x_{ε} + t (y_{ε} − x_{ε}) remains within some compact subset K of Ω for ε ≤ ε′. Let k′ > 0, then for k + k′ = sup_{j} k_{j} and ε ≤ min (ε′, ε_{0}, ε_{j} : i = 1, m), we

have

|f_{ε} (x_{ε}) − f_{ε} (y_{ε})| ≤ c exp (−k′ε^{− \frac{1}{2σ−1}} ),

which gives (f_{ε} (x_{ε}) − f_{ε} (y_{ε}))_{ε} ∈ N_{0}^{σ}. □

The characterization of nullity of f ∈ G^{σ}(Ω) is given by the following theorem.

Theorem 7. Let f ∈ G^{σ} (Ω). Then

f = 0 in G^{σ}(Ω) ⟺ f(x̃) = 0 in C^{σ} for all x̃ ∈ Ω̃_{c}^{σ}.

Proof. It is easy to see that if f ∈ N^{σ} (Ω), then f \binom{}{x} ∈ N_{0}^{σ}, ∀x̃ ∈ Ω̃_{c}^{σ}. Suppose that f ≠ 0 in G^{σ}(Ω). Then by the characterization of N^{σ}(Ω) we have, there exists a compact subset K of Ω, ∃k > 0, ∀c > 0, ∀ε_{0} > 0, ∃ε ≤ ε_{0},

sup_{K} |f_{ε} (x)| > c exp (−kε^{− \frac{1}{2σ−1}} ) .

So there exists a sequence ε_{m} ↘ 0 and x_{m} ∈ K such that ∀m ∈ ℤ^{+},

(3.4)  |f_{ε_{m}} (x_{m})| > exp (−kε_{m}^{− \frac{1}{2σ−1}} ) .

For ε > 0 we set x_{ε} = x_{m} when ε_{m+1} < ε ≤ ε_{m}. We have (x_{ε})_{ε} ∈ Ω_{M}^{σ} with values in K, so x̃ = [(x_{ε})_{ε}] ∈ Ω̃_{c}^{σ} and (3.4) means that (f_{ε} (x_{ε}))_{ε} ∉ N_{0}^{σ}, i.e.,

f \binom{}{x} ≠ 0 in C^{σ}. □

4. Embedding of Gevrey ultradistributions with compact support

We recall some definitions and results on Gevrey ultradistributions (see [15], [16] or [21]).

Definition 4. A function f ∈ E^{σ}(Ω), if f ∈ C^{∞}(Ω) and for every compact subset K of Ω, ∃c > 0, ∀α ∈ ℤ_{+}^{m},

sup_{x∈K} |∂^{α}f (x)| ≤ c^{|α|+1} (α!)^{σ} .

Obviously we have E^{t}(Ω) ⊂ E^{σ} (Ω) if 1 ≤ t ≤ σ. It is well-known that E^{1}(Ω) = A(Ω) is the space of all real analytic functions in Ω. Denote by D^{σ}(Ω) the space E^{σ}(Ω) ∩ C_{0}^{∞}(Ω). Then D^{σ}(Ω) is nontrivial if and only if σ > 1. The topological dual of D^{σ}(Ω), denoted D′_{σ}(Ω), is called the space of Gevrey ultradistributions of order σ. The space E′_{σ}(Ω) is the topological dual of E^{σ}(Ω) and is identified with the space of Gevrey ultradistributions with compact support.

Definition 5. A differential operator of infinite order

P (D) = ∑_{γ∈ℤ_{+}^{m}} a_{γ}D^{γ}

is called a σ-ultradifferential operator, if for every h > 0 there exist c > 0 such that ∀γ ∈ ℤ_{+}^{m},

(4.1)  |a_{γ}| ≤ c\frac{h^{|γ|}}{(γ!)^{σ}} .

The importance of σ-ultradifferential operators lies in the following result.

Proposition 8. Let T ∈ E′_{σ}(Ω), σ > 1 and supp T ⊂ K. Then there exist a σ-ultradifferential operator

P (D) = ∑_{γ∈ℤ_{+}^{m}} a_{γ}D^{γ},

M > 0 and continuous functions f_{γ} ∈ C_{0} (K) such that

sup_{γ∈ℤ_{+}^{m},x∈K} |f_{γ} (x)| ≤ M,

T = ∑_{γ∈ℤ_{+}^{m}} a_{γ}D^{γ}f_{γ}.

The space S^{(σ)}(ℝ^{m}), σ > 1 (see [10]) is the space of functions φ ∈ C^{∞} (ℝ^{m}) such that ∀b > 0, we have

(4.2)  ∥φ∥_{b,σ} = sup_{α,β∈ℤ_{+}^{m}} ∫ \frac{|x|^{|β|}}{b^{|α+β|}α!^{σ}β!^{σ}} |∂^{α}φ (x)| dx < ∞.

Lemma 9. There exists ϕ ∈ S^{(σ)} (ℝ^{m}) satisfying

∫ ϕ (x) dx = 1 and ∫ x^{α}ϕ (x) dx = 0, ∀α ∈ ℤ_{+}^{m}\{0}.

Proof. For an example of function ϕ ∈ S^{(σ)} satisfying these conditions, take the Fourier transform of a function of the class D^{(σ)} (ℝ^{m}) equal 1 in a neighborhood of the origin. Here D^{(σ)} (ℝ^{m}) denotes the projective Gevrey space of order σ, i.e., D^{(σ)} (ℝ^{m}) = E^{(σ)} (ℝ^{m}) ∩ C_{0}^{∞} (ℝ^{m}), where f ∈ E^{(σ)} (ℝ^{m}), if f ∈ C^{∞} (ℝ^{m}) and for every compact subset K of ℝ^{m}, ∀b > 0, ∃c > 0, ∀α ∈ ℤ_{+}^{m},

sup_{x∈K} |∂^{α}f (x)| ≤ cb^{|α|} (α!)^{σ} .  □

Definition 6. The net ϕ_{ε} = 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

(4.3)  I : E^{σ}(Ω) ⟶ G^{σ}(Ω)
f ⟼ [f] = cl (f_{ε}) ,

where f_{ε} = f , ∀ε ∈ ]0, 1].

The following proposition gives the natural embedding of Gevrey ultradistributions into G^{σ} (Ω).

Theorem 10. The map

(4.4)  J_{0} : E′_{3σ−1}(Ω) ⟶ G^{σ}(Ω)
T ⟼ [T] = cl ((T ∗ ϕ_{ε})_{/Ω})_{ε}

is an embedding.

Proof. Let T ∈ E′_{3σ−1}(Ω) with supp T ⊂ K. Then there exists an (3σ − 1)- ultradifferential operator P (D) = ∑_{γ∈ℤ_{+}^{m}} a_{γ}D^{γ} and continuous functions f_{γ}

with supp f_{γ} ⊂ K, ∀γ ∈ ℤ_{+}^{m}, and sup_{γ∈ℤ_{+}^{m},x∈K} |f_{γ} (x)| ≤ M, such that

T = ∑_{γ∈ℤ_{+}^{m}} a_{γ}D^{γ}f_{γ}.

We have

T ∗ ϕ_{ε} (x) = ∑_{γ∈ℤ_{+}^{m}} a_{γ} \frac{(−1)^{|γ|}}{ε^{|γ|}} ∫ f_{γ} (x + εy) D^{γ}ϕ (y) dy.

Let α ∈ ℤ_{+}^{m}. Then

|∂^{α} (T ∗ ϕ_{ε} (x))| ≤ ∑_{γ∈ℤ_{+}^{m}} a_{γ} \frac{1}{ε^{|γ+α|}} ∫ |f_{γ} (x + εy)| |D^{γ+α}ϕ (y)| dy.

From (4.1) and the inequality

(4.5)  (β + α)!^{t} ≤ 2^{t|β+α|}α!^{t}β!^{t}, ∀t ≥ 1,

we have, ∀h > 0, ∃c > 0, such that

|∂^{α} (T ∗ ϕ_{ε} (x))| ≤ ∑_{γ∈ℤ_{+}^{m}} c\frac{h^{|γ|}}{γ!^{3σ−1}} \frac{1}{ε^{|γ+α|}} ∫ |f_{γ} (x + εy)| |D^{γ+α}ϕ (y)| dy

≤ ∑_{γ∈ℤ_{+}^{m}} cα!^{3σ−1} \frac{2^{(3σ−1)|γ+α|}h^{|γ|}}{(γ + α)!^{2σ−1}} \frac{1}{ε^{|γ+α|}} b^{|γ+α|}

× ∫ |f_{γ} (x + εy)| \frac{|D^{γ+α}ϕ (y)|}{b^{|γ+α|} (γ + α)!^{σ}} dy.

Then for h > \frac{1}{2},

\frac{1}{α!^{3σ−1}} |∂^{α} (T ∗ ϕ_{ε} (x))| ≤ ∥ϕ∥_{b,σ} Mc ∑_{γ∈ℤ_{+}^{m}} 2^{−|γ|} \frac{(2^{3σ}bh)^{|γ+α|}}{(γ + α)!^{2σ−1}} \frac{1}{ε^{|γ+α|}}

≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ,

i.e.,

(4.6)  |∂^{α} (T ∗ ϕ_{ε} (x))| ≤ c (α) exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ,

where k_{1} = (2σ − 1) (2^{3σ}bh)^{\frac{1}{2σ−1}} .

Suppose that (T ∗ ϕ_{ε})_{ε} ∈ N^{σ}(Ω). Then for every compact subset L of Ω, ∃c > 0, ∀k > 0, ∃ε_{0} ∈ ]0, 1],

(4.7)  |T ∗ ϕ_{ε} (x)| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) , ∀x ∈ L, ∀ε ≤ ε_{0}.

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_{ε→0} ∫ (T ∗ ϕ_{ε}) (x) χ (x) ψ (x) dx.

Consequently, from (4.7), we obtain

|∫ (T ∗ ϕ_{ε}) (x) χ (x) ψ (x) dx| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) , ∀ε ≤ ε_{0},

which gives ⟨T, ψ⟩ = 0. □

Remark 4. We have c (α) = α!^{3σ−1} ∥ϕ∥_{b,σ} Mc in (4.6).

In order to show the commutativity of the following diagram of embeddings

\xymatrix{D^{σ}(Ω) \ar[r] \ar[dr] & G^{σ} (Ω) \\  & E′_{3σ−1}(Ω), \ar[u]}

we have to prove the following fundamental result.

Generalized Gevrey ultradistributions 49

Proposition 11. Let f ∈ D^{σ}(Ω) and (ϕ_{ε})_{ε} be a net of mollifiers. Then

(f − (f ∗ ϕ_{ε})_{/Ω})_{ε} ∈ N^{σ} (Ω) .

Proof. Let f ∈ D^{σ}(Ω). Then there exists a constant c > 0, such that

|∂^{α}f (x)| ≤ c^{|α|+1}α!^{σ}, ∀α ∈ ℤ_{+}^{m}, ∀x ∈ Ω.

Let α ∈ ℤ_{+}^{m}. Taylor’s formula and the properties of ϕ_{ε} give

∂^{α} (f ∗ ϕ_{ε} − f) (x) = ∑_{|β|=N} ∫ \frac{(εy)^{β}}{β!} ∂^{α+β}f (ξ) ϕ (y) dy,

where x ≤ ξ ≤ x + εy. Consequently, for b > 0, we have

|∂^{α} (f ∗ ϕ_{ε} − f) (x)| ≤ ε^{N} ∑_{|β|=N} ∫ \frac{|y|^{N}}{β!} |∂^{α+β}f (ξ)| |ϕ (y)| dy

≤ α!^{σ}ε^{N} ∑_{|β|=N} β!^{2σ−1}2^{σ|α+β|}b^{|β|} ∫ \frac{|∂^{α+β}f (ξ)|}{(α + β)!^{σ}}

× \frac{|y|^{|β|}}{b^{|β|}β!^{σ}} |ϕ (y)| dy.

Let k > 0 and T > 0. Then

|∂^{α} (f ∗ ϕ_{ε} − f) (x)| ≤ α!^{σ} (εN^{2σ−1})^{N} (k^{2σ−1}T )^{−N}

× ∑_{|β|=N} ∫ 2^{σ|α+β|} (k^{2σ−1}bT )^{|β|} \frac{|∂^{α+β}f (ξ)|}{(α + β)!^{σ}}

× \frac{|y|^{|β|}}{b^{|β|}β!^{σ}} |ϕ (y)| dy

≤ α!^{σ} (εN^{2σ−1})^{N} (k^{2σ−1}T )^{−N}

× c ∥ϕ∥_{b,σ} (2^{σ}c)^{|α|} ∑_{|β|=N} (2^{σ}k^{2σ−1}bT )^{|β|} c^{|β|},

hence, taking 2^{σ}k^{2σ−1}bT c ≤ \frac{1}{2a}, with a > 1, we obtain

(4.8) |∂^{α} (f ∗ ϕ_{ε} − f) (x)| ≤ α!^{σ} (εN^{2σ−1})^{N} (k^{2σ−1}T )^{−N}

× c ∥ϕ∥_{b,σ} (2^{σ}c)^{|α|} a^{−N} ∑_{|β|=N} (\frac{1}{2})^{|β|}

≤ ∥ϕ∥_{b,σ} c^{|α|+1}α!^{σ} (εN^{2σ−1})^{N} (k^{2σ−1}T )^{−N} a^{−N}.

Let ε_{0} ∈ ]0, 1] such that ε0^{\frac{1}{2σ−1}} \frac{ln a}{k} < 1 and take T > 2^{2σ−1}. Then

(T^{\frac{1}{2σ−1}} − 1) > 1 > \frac{ln a}{k}ε^{\frac{1}{2σ−1}} , ∀ε ≤ ε_{0}.

In particular, we have

(\frac{ln a}{k}ε^{\frac{1}{2σ−1}} )^{−1} T^{\frac{1}{2σ−1}} − (\frac{ln a}{k}ε^{\frac{1}{2σ−1}} )^{−1} > 1.

Then, there exists N = N (ε) ∈ ℤ^{+}, such that

(\frac{ln a}{k}ε^{\frac{1}{2σ−1}} )^{−1} < N < (\frac{ln a}{k}ε^{\frac{1}{2σ−1}} )^{−1} T^{\frac{1}{2σ−1}} ,

i.e.,

(4.9)  1 ≤ \frac{ln a}{k}ε^{\frac{1}{2σ−1}} N ≤ T^{\frac{1}{2σ−1}} ,

which gives

a^{−N} ≤ exp (−kε^{− \frac{1}{2σ−1}} ) and \frac{εN^{2σ−1}}{k^{2σ−1}T} ≤ (\frac{1}{ln a})^{2σ−1} < 1,

if we choose ln a > 1. Finally, from (4.8), we have

(4.10)  |∂^{α} (f ∗ ϕ_{ε} − f) (x)| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) ,

i.e., f ∗ ϕ_{ε} − 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, ∃ε_{0} ∈ ]0, 1] , ∀α ∈ ℤ_{+}^{m}, ∀ε ≤ ε_{0},

(4.11)  sup_{x∈K} |∂^{α} (f ∗ ϕ_{ε} − f) (x)| ≤ c^{|α|+1}α!^{σ} exp (−kε^{− \frac{1}{2σ−1}} ) .

5. Sheaf properties of G^{σ}

Let Ω′ be an open subset of Ω and let f = (f_{ε})_{ε} + N^{σ}(Ω) ∈ G^{σ}(Ω), the restriction of f to Ω′, denoted f_{/Ω′}, is defined as

(f_{ε/Ω′})_{ε} + N^{σ} (Ω′) ∈ G^{σ} (Ω′) .

Theorem 13. The functor Ω → G^{σ} (Ω) is a sheaf of differential algebras on ℝ^{m}.

Proof. Let Ω be a nonvoid open of ℝ^{m} and (Ω_{λ})_{λ∈Λ} be an open covering of Ω. We have to show the properties (S1) If f, g ∈ G^{σ}(Ω) such that f_{/Ω_{λ}} = g_{/Ω_{λ}}, ∀λ ∈ Λ, then f = g. (S2) If for each λ ∈ Λ, we have f_{λ} ∈ G^{σ} (Ω_{λ}), such that

f_{λ/Ω_{λ}∩Ω_{μ}} = f_{μ/Ω_{λ}∩Ω_{μ}} for all λ, μ ∈ Λ with Ω_{λ} ∩ Ω_{μ} ≠ ϕ,

then there exists a unique f ∈ G^{σ} (Ω) with f_{/Ω_{λ}} = f_{λ}, ∀λ ∈ Λ.

To show (S1), take K a compact subset of Ω. Then there exist compact sets K_{1}, K_{2}, …, K_{m} and indices λ_{1}, λ_{2}, …, λ_{m} ∈ Λ such that

K ⊂ ⋃_{i=1}^{m} K_{i} and K_{i} ⊂ Ω_{λ_{i}},

where (f_{ε} − g_{ε})_{ε} satisfies the N^{σ}-estimate on each K_{i}. Then it satisfies the N^{σ}-estimate on K which means (f_{ε} − g_{ε})_{ε} ∈ N^{σ}(Ω).

To show (S2), let (χ_{j})_{j=1}^{∞} be a C^{∞}-partition of unity subordinate to the covering (Ω_{λ})_{λ∈Λ}. Set

f := (f_{ε})_{ε} + N^{σ}(Ω),

where f_{ε} = ∑_{j=1}^{∞} χ_{j}f_{λ_{j}ε} and (f_{λ_{j}ε})_{ε} is a representative of f_{λ_{j}}. Moreover, we

set f_{λ_{j}ε} = 0 on Ω\Ω_{λ_{j}}, so that χ_{j}f_{λ_{j}ε} is C^{∞} on all of Ω. First let K be a compact subset of Ω. Then K_{j} = K ∩ supp χ_{j} is a compact subset of Ω_{λ_{j}} and (f_{λ_{j}ε})_{ε} ∈ E_{m}^{σ} (Ω_{λ_{j}}). Then (χ_{j}f_{λ_{j}ε}) satisfies E_{m}^{σ}-estimate on each K_{j}, and χ_{j} (x) ≡ 0 on K except for a finite number of j, i.e., ∃N > 0, such that

∑_{j=1}^{∞} χ_{j}f_{λ_{j}ε} (x) = ∑_{j=1}^{N} χ_{j}f_{λ_{j}ε} (x) , ∀x ∈ K.

So (∑ χ_{j}f_{λ_{j}ε}) satisfies E_{m}^{σ}-estimates on K, which means (f_{ε})_{ε} ∈ E_{m}^{σ}(Ω). It remains to show that f_{/Ω_{λ}} = f_{λ}, ∀λ ∈ Λ. Let K be a compact subset of Ω_{λ},

choose N > 0 in such a way that ∑_{j=1}^{N} χ_{j} (x) ≡ 1 on a neighborhood Ω′ of K

with \overline{Ω′} compact in Ω_{λ}. For x ∈ K,

f_{ε} (x) − f_{λε} (x) = ∑_{j=1}^{N} χ_{j} (x) (f_{λ_{j}ε} (x) − f_{λε} (x)) .

Since (f_{λ_{j}ε} − f_{λε}) ∈ N^{σ} (Ω_{λ_{j}} ∩ Ω_{λ}) and K_{j} = K ∩ supp χ_{j} is a compact

subset of Ω ∩ Ω_{λ_{j}}, then ( ∑_{j=1}^{N} χ_{j} (f_{λ_{j}ε} − f_{λε})) satisfies the N^{σ}-estimate on

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 supp_{g}^{σ} f, is the complement of the largest open set U such that f_{/U} = 0.

As in [9], we construct the embedding of D′_{3σ−1}(Ω) into G^{σ}(Ω) using the sheaf properties of G^{σ}. First, choose some covering (Ω_{λ})_{λ∈Λ} of Ω such that each \overline{Ω_{λ}} is a compact subset of Ω. Let (ψ_{λ})_{λ∈Λ} be a family of elements of D^{σ}(Ω) ⊂ D^{3σ−1}(Ω) with ψ_{λ} ≡ 1 in some neighborhood of \overline{Ω_{λ}}. For each λ we define

J_{λ} : D′_{3σ−1}(Ω) ⟶ G^{σ} (Ω_{λ})
T ⟼ [T]_{λ} = cl ((ψ_{λ}T ∗ ϕ_{ε})_{/Ω_{λ}})_{ε} .

One can easily show that ((ψ_{λ}T ∗ ϕ_{ε})_{/Ω_{λ}})_{ε} ∈ E_{m}^{σ} (Ω_{λ}) (see the proof of Theorem 10) and that the family (J_{λ} (T))_{λ∈Λ} is coherent, i.e.,

J_{λ} (T)_{/Ω_{λ}∩Ω_{μ}} = J_{μ} (T)_{/Ω_{λ}∩Ω_{μ}} , ∀λ, μ ∈ Λ.

Then if (χ_{j})_{j=1}^{∞} is a smooth partition of unity subordinate to (Ω_{λ})_{λ∈Λ}, the preceding theorem allows the embedding

(5.1)  J_{σ} : D′_{3σ−1}(Ω) ⟶ G^{σ}(Ω)
T ⟼ [T] = cl (∑_{j=1}^{∞} χ_{j} (ψ_{λ_{j}}T ∗ ϕ_{ε})) .

We can also embed directly D′_{3σ−1} (Ω) into G^{σ}(Ω) (see [6] for the case 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 ρ_{ε} by

(5.2)  ρ_{ε} (x) = (\frac{1}{ε})^{m} ϕ (\frac{x}{ε}) φ (x |ln ε|) .

It is easy to prove that, ∃c > 0, such that ∀α ∈ ℤ_{+}^{m},

sup_{x∈ℝ^{m}} |∂^{α}ρ_{ε} (x)| ≤ c^{|α|+1}α!^{σ}ε^{−m−|α|}.

Define the injective map

(5.3)  J : D′_{3σ−1}(Ω) ⟶ G^{σ}(Ω)
T ⟼ [T] = cl ((T ∗ ρ_{ε})_{ε}) .

Proposition 14. The map J coincides on E′_{3σ−1}(Ω) with J_{0}.

Proof. We have to show that for T ∈ E′_{3σ−1} (Ω), the net (T ∗ (ρ_{ε} − ϕ_{ε}))_{ε} ∈ N^{σ} (Ω). For x |ln ε| < 1, we have ∂^{α} (ρ_{ε} − ϕ_{ε}) (x) = 0, ∀α ∈ ℤ_{+}^{m}. For x |ln ε| ≥ 1, we have

∂^{α} (ρ_{ε} − ϕ_{ε}) (x) = ε^{−m−|α|}∂^{α}ϕ (\frac{x}{ε}) (φ (x |ln ε|) − 1)

+ ε^{−m} ∑_{|β|=1}^{α} ε^{−|β|} |ln ε|^{|α−β|} ∂^{β}ϕ (\frac{x}{ε}) ∂^{α−β}φ (x |ln ε|) .

Generalized Gevrey ultradistributions 53

Then, ∃c > 0, ∀b > 0, ∀γ ∈ ℤ_{+}^{m}, |∂^{α} (ρ_{ε} − ϕ_{ε}) (x)|

≤ b^{|γ|+|α|+1}ε^{−m−|α|}α!^{σ}γ!^{σ} (\frac{ε}{|x|})^{|γ|} + ε^{−m} ∑_{|β|=1}^{α} c^{|α−β|+1}b^{|γ|+|β|+1}

× ε^{−|β|} |ln ε|^{|α−β|} β!^{σ} (α − β)!^{σ}γ!^{σ} (\frac{ε}{|x|})^{|γ|}

≤ b^{|γ|+|α|+1}ε^{−m−|α|}α!^{σ}γ!^{σ} (ε |ln ε|)^{|γ|} + ε^{−m} ∑_{|β|=1}^{α} b^{|γ|+|β|+1}c^{|α−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 ∈ ℤ_{+}^{m},

(5.4) |∂^{α} (ρ_{ε} − ϕ_{ε}) (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 ∈ ℤ_{+}^{m},

|ln ε|^{|2γ|} ≤ ε^{−|γ|},

then (5.4) gives ∃c > 0, ∀b > 0, ∀γ ∈ ℤ_{+}^{m},

|∂^{α} (ρ_{ε} − ϕ_{ε}) (x)| ≤ b^{|2γ|+|α|+1}c^{|2γ+α|+1}α!^{σ} (2γ)!^{σ}ε^{|γ|−|α|−m}. Since

(2γ)!^{σ} ≤ 2^{σ|2γ|}γ!^{2σ},

then for N = |γ|, we obtain

|∂^{α} (ρ_{ε} − ϕ_{ε}) (x)| ≤ b^{2N+|α|+1}c^{2N+|α|+1}2^{2Nσ}α!^{σ}N!^{2σ}ε^{N−|α|−m}

≤ 2^{−N} (2^{2σ+1}b^{2}c^{2})^{N} N!^{2σ}ε^{N} × (bc)^{|α|+1} α!^{σ}ε^{−|α|−m} ≤ c′h^{|α|+1}α!^{σ}ε^{−|α|−m} exp (− (2^{2σ+1}b^{2}c^{2})^{− \frac{1}{2σ}} ε^{− \frac{1}{2σ}} ) ,

where c′ = cb ∑_{n≥0} 2^{−N} and h = cb. Then for any k take b > 0 and ε sufficiently small such that

ε^{−|α|−m} exp (− (2^{2σ+1}b^{2}c^{2})^{− \frac{1}{2σ}} ε^{− \frac{1}{2σ}} ) ≤ exp (−kε^{− \frac{1}{2σ−1}} ) . Then we have ∀k > 0, ∃h > 0, ∃ε_{0} ∈ ]0, 1] such that ∀ε ≤ ε_{0},

(5.5) |∂^{α} (ρ_{ε} − ϕ_{ε}) (x)|

≤ h^{|α|+1}α!^{σ} exp (−kε^{− \frac{1}{2σ}} ) ≤ h^{|α|+1}α!^{σ} exp (−kε^{− \frac{1}{2σ−1}} ) .

As E′_{3σ−1}(Ω) ⊂ E′_{σ}(Ω), ∃L a compact subset of Ω such that ∀h > 0, ∃c > 0, and

|⟨T (y) , (ρ_{ε} − ϕ_{ε}) (x − y)⟩| ≤ c sup_{α∈ℤ_{+}^{m},y∈L} \frac{h^{|α|}}{α!^{σ}} |∂_{y}^{α} (ρ_{ε} − ϕ_{ε}) (x − y)| .

So for x ∈ K, y ∈ L and by (5.5), we obtain

|(T ∗ (ρ_{ε} − ϕ_{ε})) (x)| ≤ c exp (−kε^{− \frac{1}{2σ−1}} ) ,

which proves that (T ∗ (ρ_{ε} − ϕ_{ε}))_{ε} ∈ N^{σ}(Ω). □

The sheaf properties of G^{σ} and the proof of Proposition 14 show that the embedding J_{σ} coincides with the embedding J. Summing up, we have the following commutative diagram

\xymatrix{E^{σ}(Ω) \ar[r] \ar[d] & G^{σ} (Ω) . \\ D′_{3σ−1}(Ω) \ar@{-->}[ur]}

Definition 8. The space of elements of G^{σ} (Ω) with compact support is denoted G_{C}^{σ}(Ω).

As in the case of Colombeau generalized functions, it is not difficult to prove the following result.

Proposition 15. The space G_{C}^{σ}(Ω) is the space of elements f of G^{σ} (Ω) satisfying: there exists a representative (f_{ε})_{ε∈]0,1]} and a compact subset K of Ω such that supp f_{ε} ⊂ K, ∀ε ∈ ]0, 1].

6. Equalities in G^{σ}(Ω)

In G^{σ}(Ω), we have the strong equality, denoted =, between two elements f = [(f_{ε})_{ε}] and g = [(g_{ε})_{ε}], which means that

(f_{ε} − g_{ε})_{ε} ∈ N^{σ}(Ω).

One can easily check that if K is a compact subset of Ω and f = [(f_{ε})_{ε}] ∈ G^{σ}(Ω), then (∫_{K} f_{ε} (x) dx)_{ε}defines an element of C^{σ}.

We define the equality in the sense of ultradistributions, denoted ∼^{t}, where t ∈ [σ, 3σ − 1], by

f ∼^{t} g ⟺ (∫ (f_{ε} (x) − g_{ε} (x)) φ (x) dx)_{ε} ∈ N_{0}^{t}, ∀φ ∈ D^{t}(Ω),

and we say that f equals g in the sense of ultradistributions.

We say that f = [(f_{ε})_{ε}] is associated to g = [(g_{ε})_{ε}], denoted f ≈ g, if

lim_{ε→0} ∫ (f_{ε} − g_{ε}) (x) ψ (x) dx = 0, ∀ψ ∈ D^{3σ−1}(Ω).

In particular, we say that f = [(f_{ε})_{ε}] ∈ G^{σ} (Ω) is associated to the Gevrey ultradistribution T ∈ E′_{3σ−1}(Ω), denoted f ≈ T, if

lim_{ε→0} ∫ f_{ε} (x) ψ (x) dx = ⟨T, ψ⟩ , ∀ψ ∈ D^{3σ−1}(Ω).

The main relationship between these inequalities is given by the following results. The proof is left to the reader.

Proposition 16. Let f, g ∈ G^{σ}(Ω), T ∈ E′_{3σ−1}(Ω), and t ∈ [σ, 3σ − 1]. Then:

(1) f = g ⟹ f ∼^{t} g ⟹ f ∼^{σ} g ⟹ f ≈ g.

(2) T ≈ 0 in G^{σ}(Ω) ⟹ T = 0 in E′_{3σ−1} (Ω) .

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.

Definition 9. The space of regular elements, denoted E_{m}^{σ,∞}(Ω), is the space of (f_{ε})_{ε} ∈ (C^{∞}(Ω))^{]0,1]} satisfying, for every compact subset K of Ω, ∃k > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , ∀α ∈ ℤ_{+}^{m}, ∀ε ≤ ε_{0},

sup_{x∈K} |∂^{α}f_{ε} (x)| ≤ c^{|α|+1}α!^{σ} exp (kε^{− \frac{1}{2σ−1}} ) .

Proposition 17. (1) The space E_{m}^{σ,∞} (Ω) is an algebra stable under the

action of σ-ultradifferential operators.

(2) The space N^{σ,∞}(Ω) := N^{σ}(Ω) ∩ E_{m}^{σ,∞}(Ω) is an ideal of E_{m}^{σ,∞}(Ω).

Proof. (1) Let (f_{ε})_{ε} , (g_{ε})_{ε} ∈ E_{m}^{σ,∞}(Ω) and K be a compact subset of Ω. Then ∃k_{1} > 0, ∃c_{1} > 0, ∃ε_{1} ∈ ]0, 1] such that ∀x ∈ K, ∀α ∈ ℤ_{+}^{m}, ∀ε ≤ ε_{1},

|∂^{α}f_{ε} (x)| ≤ c_{1}^{|α|+1}α!^{σ} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) .

We have also ∃k_{2} > 0, ∃c_{2} > 0, ∃ε_{2} ∈ ]0, 1] such that ∀x ∈ K, ∀α ∈ ℤ_{+}^{m}, ∀ε ≤ ε_{2},

|∂^{α}g_{ε} (x)| ≤ c_{2}^{|α|+1}α!^{σ} exp (k_{2}ε^{− \frac{1}{2σ−1}} ) .

Let α ∈ ℤ_{+}^{m}. Then

\frac{1}{α!^{σ}} |∂^{α} (f_{ε}g_{ε}) (x)| ≤ ∑_{β=0}^{α} \binom{α}{β}\frac{1}{(α − β)!^{σ}} |∂^{α−β}f_{ε} (x)| \frac{1}{β!^{σ}} |∂^{β}g_{ε} (x)| .

Let ε ≤ min (ε_{1}, ε_{2}) and k = k_{1} + k_{2}. Then we have ∀α ∈ ℤ_{+}^{m}, ∀x ∈ K,

exp (−kε^{− \frac{1}{2σ−1}} ) \frac{1}{α!^{σ}} |∂^{α} (f_{ε}g_{ε}) (x)|

≤ ∑_{β=0}^{α} \binom{α}{β}\frac{exp (−k_{1}ε^{− \frac{1}{2σ−1}} )}{(α − β)!^{σ}} |∂^{α−β}f_{ε} (x)| \frac{exp (−k_{2}ε^{− \frac{1}{2σ−1}} )}{β!^{σ}} |∂^{β}g_{ε} (x)|

≤ ∑_{β=0}^{α} \binom{α}{β}c_{1}^{|α−β|}c_{2}^{|β|}

≤ 2^{|α|} (c_{1} + c_{2})^{|α|} ,

i.e., (f_{ε})_{ε} (g_{ε})_{ε} ∈ E_{m}^{σ,∞}(Ω).

Now let P (D) = ∑ a_{γ}D^{γ} be a σ-ultradifferential operator, then ∀h > 0, ∃b > 0, such that

exp (−k_{1}ε^{− \frac{1}{2σ−1}} ) \frac{1}{α!^{σ}} |∂^{α} (P (D) f_{ε} (x))|

≤ exp (−k_{1}ε^{− \frac{1}{2σ−1}} ) ∑_{γ∈ℤ_{+}^{m}} b\frac{h^{|γ|}}{γ!^{σ}} \frac{1}{α!^{σ}} |∂^{α+γ}f_{ε} (x)|

≤ b exp (−k_{1}ε^{− \frac{1}{2σ−1}} ) ∑_{γ∈ℤ_{+}^{m}} \frac{2^{σ|α+γ|}h^{|γ|}}{(α + γ)!^{σ}} |∂^{α+γ}f_{ε} (x)|

≤ b ∑_{γ∈ℤ_{+}^{m}} 2^{σ|α+γ|}h^{|γ|}c_{1}^{|α+γ|},

hence, for 2^{σ}hc_{1} ≤ \frac{1}{2}, we have

exp (−kε^{− \frac{1}{2σ−1}} ) \frac{1}{α!^{σ}} |∂^{α} (P (D) f_{ε} (x))| ≤ c′ (2^{σ}c_{1})^{|α|} ,

which shows that (P (D) f_{ε})_{ε} ∈ E_{m}^{σ,∞} (Ω).

(2) The fact that N^{σ,∞}(Ω) = N^{σ}(Ω) ∩ E_{m}^{σ,∞}(Ω) ⊂ E_{m}^{σ}(Ω) and N^{σ}(Ω) is an ideal of E_{m}^{σ}(Ω) implies that N^{σ,∞}(Ω) is an ideal of E_{m}^{σ,∞}(Ω). □

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) = \frac{E_{m}^{σ,∞}(Ω)}{N^{σ,∞}(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 ℝ^{m} and (Ω_{λ})_{λ∈Λ} be an open covering of Ω. We have to show the properties:

(S1) If f, g ∈ G^{σ,∞}(Ω) such that f_{/Ω_{λ}} = g_{/Ω_{λ}}, ∀λ ∈ Λ, then f = g. (S2) If for each λ ∈ Λ, we have f_{λ} ∈ G^{σ,∞} (Ω_{λ}), such that

f_{λ/Ω_{λ}∩Ω_{μ}} = f_{μ/Ω_{λ}∩Ω_{μ}} for all λ, μ ∈ Λ with Ω_{λ} ∩ Ω_{μ} ≠ ϕ,

then there exists a unique f ∈ G^{σ,∞}(Ω) with f_{/Ω_{λ}} = f_{λ}, ∀λ ∈ Λ. The property (S1) is evident. To show (S2), let (χ_{j})_{j=1}^{∞} be a E^{σ}-partition

of unity subordinate to the covering (Ω_{λ})_{λ∈Λ}. Define

f := (f_{ε})_{ε} + N^{σ,∞}(Ω),

where f_{ε} = ∑_{j=1}^{∞} χ_{j}f_{λ_{j}ε} and (f_{λ_{j}ε})_{ε} is a representative of f_{λ_{j}}. Moreover, we

set f_{λ_{j}ε} = 0 on Ω\Ω_{λ_{j}}, so that χ_{j}f_{λ_{j}ε} is C^{∞} on all of Ω.

Let K be a compact subset of Ω. Then K_{j} = K ∩ supp χ_{j} is a compact subset of Ω_{λ_{j}}, (f_{λ_{j}ε})_{ε} ∈ E_{m}^{σ,∞} (Ω_{λ_{j}}), and (χ_{j}f_{λ_{j}ε}) satisfies E_{m}^{σ,∞}-estimates on each K_{j}. We have χ_{j} (x) ≡ 0 on K except for a finite number of j, i.e., ∃N > 0, such that

∑_{j=1}^{∞} χ_{j}f_{λ_{j}ε} (x) = ∑_{j=1}^{N} χ_{j}f_{λ_{j}ε} (x) , ∀x ∈ K.

So (∑ χ_{j}f_{λ_{j}ε}) satisfies E_{m}^{σ,∞}-estimates on all K, which means

(f_{ε})_{ε} ∈ E_{m}^{σ,∞}(Ω).

It remains now to show that f_{/Ω_{λ}} = f_{λ}, ∀λ ∈ Λ. Let K be a compact subset

of Ω_{λ}, choose N > 0 in such a way that ∑_{j=1}^{N} χ_{j} (x) ≡ 1 on a neighborhood

Ω′ of K with \overline{Ω′} a compact subset of Ω_{λ}. For x ∈ K,

f_{ε} (x) − f_{λε} (x) = ∑_{j=1}^{N} χ_{j} (x) (f_{λ_{j}ε} (x) − f_{λε} (x)) .

Since (f_{λ_{j}ε} − f_{λε}) ∈ N^{σ,∞} (Ω_{λ_{j}} ∩ Ω_{λ}) and K_{j} = K ∩ supp χ_{j} is a compact

subset of Ω∩Ω_{λ_{j}}, ( ∑_{j=1}^{N} χ_{j} (f_{λ_{j}ε} − f_{λε})) satisfies the N^{σ} and E_{m}^{σ,∞} estimates

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-singsupp_{g} (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).

Proposition 19. Let f = cl (f_{ε})_{ε} ∈ G_{C}^{σ}(Ω). Then f is regular if and only if ∃k_{1} > 0, ∃k_{2} > 0, ∃c > 0, ∃ε_{1} > 0, ∀ε ≤ ε_{1}, such that

(7.1)  |F (f_{ε}) (ξ)| ≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} ) , ∀ξ ∈ ℝ^{m},

where F (f_{ε}) denote Fourier transform of f_{ε}.

Proof. Suppose that f = cl (f_{ε})_{ε} ∈ G_{C}^{σ}(Ω) ∩ G^{σ,∞}(Ω). Then ∃k_{1} > 0, ∃c_{1} > 0, ∃ε_{1} > 0, ∀α ∈ ℤ_{+}^{m}, ∀x ∈ K, ∀ε ≤ ε_{1}, supp f_{ε} ⊂ K, such that

|∂^{α}f_{ε}| ≤ c_{1}^{|α|+1}α!^{σ} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) .

Consequently we have, ∀α ∈ ℤ_{+}^{m},

|ξ^{α}| |F (f_{ε}) (ξ)| ≤ |∫ exp (−ixξ) ∂^{α}f_{ε} (x) dx| .

Then ∃c > 0, ∀ε ≤ ε_{1},

|ξ|^{|α|} |F (f_{ε}) (ξ)| ≤ c^{|α|+1}α!^{σ} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) .

For α ∈ ℤ_{+}^{m}, ∃N ∈ ℤ_{+} such that

\frac{N}{σ} ≤ |α| < \frac{N}{σ} + 1,

so

|ξ|^{\frac{N}{σ}} |F (f_{ε}) (ξ)| ≤ c^{|α|+1} |α|^{|α|σ} exp (k_{1}ε^{− \frac{1}{2σ−1}} )
≤ c^{N+1}N^{N} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) .

Hence ∃c > 0, ∀N ∈ ℤ^{+},

|F (f_{ε}) (ξ)| ≤ c^{N+1} |ξ|^{− \frac{N}{σ}} N! exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ,

which gives

|F (f_{ε}) (ξ)| exp (\frac{1}{2c} |ξ|^{\frac{1}{σ}} ) ≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ∑ 2^{−N},

or

|F (f_{ε}) (ξ)| ≤ c′ exp (k_{1}ε^{− \frac{1}{2σ−1}} − \frac{1}{2c} |ξ|^{\frac{1}{σ}} ) ,

i.e., we have (7.1).

Suppose now that (7.1) is valid. Then ∀ε ≤ ε_{0},

|∂^{α}f_{ε} (x)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ∫ |ξ^{α}| exp (−k_{2} |ξ|^{\frac{1}{σ}} ) dξ.

Due to the inequality t^{N} ≤ N! exp (t) , ∀t > 0, then ∃c_{2} = c (k_{2}) such that

|ξ^{α}| exp (−\frac{k_{2}}{2} |ξ|^{\frac{1}{σ}} ) ≤ c_{2}^{|α|}α!^{σ}.

Then

|∂^{α}f_{ε} (x)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) c_{2}^{|α|}α!^{σ} ∫ exp (−\frac{k_{2}}{2} |ξ|^{\frac{1}{σ}} ) dξ

≤ c^{|α|+1}α!^{σ} exp (k_{1}ε^{− \frac{1}{2σ−1}} ) ,

where c = max (c_{1} ∫ exp (−\frac{k_{2}}{2} |ξ|^{\frac{1}{σ}} ) dξ, c_{2}), i.e., f ∈ G^{σ,∞}(Ω). □

Remark 5. It is easy to see if f = cl (f_{ε})_{ε} ∈ G_{C}^{σ}(Ω), then ∃k_{1} > 0, ∃c > 0, ∃ε_{0} > 0, ∀k_{2} > 0, ∀ε ≤ ε_{0},

(7.2)  |F (f_{ε}) (ξ)| ≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} + k_{2} |ξ|^{\frac{1}{σ}} ) ,∀ξ ∈ ℝ^{m}.

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

G^{σ,∞}(Ω) ∩ D′_{3σ−1}(Ω) = E^{σ} (Ω) .

Proof. Let S ∈ G^{σ,∞}(Ω) ∩ D′_{3σ−1}(Ω). For any fixed x_{0} ∈ Ω we take ψ ∈ D^{3σ−1} (Ω) with ψ ≡ 1 on neighborhood U of x_{0}. Then T = ψS ∈ E′_{3σ−1}(Ω). Let ϕ_{ε} be a net of mollifiers with ϕ̌ = ϕ and let χ ∈ D^{σ} (Ω) such that χ ≡ 1 on K = supp ψ. As [T] ∈ G^{σ,∞}(Ω), ∃k_{1} > 0, ∃k_{2} > 0, ∃c_{1} > 0, ∃ε_{1} > 0, ∀ε ≤ ε_{1},

|F (χ (T ∗ ϕ_{ε})) (ξ)| ≤ c_{1}e^{k_{1}ε^{− \frac{1}{2σ−1}} −k_{2}|ξ|^{\frac{1}{σ}}} .

Thus

|F (χ (T ∗ ϕ_{ε})) (ξ) − F (T) (ξ)|
= |F (χ (T ∗ ϕ_{ε})) (ξ) − F (χT) (ξ)|
= |⟨T (x) , (χ (x) e^{−iξx}) ∗ ϕ_{ε} (x) − (χ (x) e^{−iξx})⟩| .

As E′_{3σ−1}(Ω) ⊂ E′_{σ}(Ω), ∃L a compact subset of Ω such that ∀h > 0, ∃c > 0, and

|F (χ (T ∗ ϕ_{ε})) (ξ) − F (T) (ξ)|

≤ c sup_{α∈ℤ_{+}^{m},x∈L} \frac{h^{|α|}}{α!^{σ}} |(∂_{x}^{α} (χ (x) e^{−iξx} ∗ ϕ_{ε} (x) − χ (x) e^{−iξx}))| .

We have e^{−iξ}χ ∈ D^{σ}(Ω). From Corollary 12, ∀k_{3} > 0, ∃c_{2} > 0, ∃η > 0, ∀ε ≤ η,

sup_{α∈ℤ_{+}^{m},x∈L} \frac{c_{2}^{|α|}}{α!^{σ}} |∂_{x}^{α} (χ (x) e^{−iξx} ∗ ϕ_{ε} (x) − χ (x) e^{−iξx})| ≤ c_{2}e^{−k_{3}ε^{− \frac{1}{2σ−1}}} ,

so there exists c′ = c′ (k_{3}) > 0, such that

|F (T) (ξ) − F (χ (T ∗ ϕ_{ε})) (ξ)| ≤ c′e^{−k_{3}ε^{− \frac{1}{2σ−1}}} .

Let ε ≤ min (η, ε_{1}). Then

|F (T) (ξ)| ≤ |F (T) (ξ) − F (χ (T ∗ ϕ_{ε})) (ξ)| + |F (χ (T ∗ ϕ_{ε})) (ξ)|
≤ c′e^{−k_{3}ε^{− \frac{1}{2σ−1}}} + c_{1}e^{k_{1}ε^{− \frac{1}{2σ−1}} −k_{2}|ξ|^{\frac{1}{σ}}} .

Take c = max (c′, c_{1}), ε = (\frac{k_{1}}{(k_{2} − r) |ξ|^{\frac{1}{σ}}} )^{2σ−1}, r ∈ ]0, k_{2}[ and k_{3} = \frac{k_{1}r}{k_{2} − r}.

Then ∃δ > 0, ∃c > 0 such that

|F (T) (ξ)| ≤ ce^{−δ|ξ|^{\frac{1}{σ}}} ,

which means T = ψS ∈ E^{σ}(Ω). As ψ ≡ 1 on the neighborhood U of x_{0}, S ∈ E^{σ} (U). Consequently S ∈ E^{σ} (Ω), which proves

G^{σ,∞}(Ω) ∩ D′_{3σ−1}(Ω) ⊂ E^{σ} (Ω) .

We have E^{σ}(Ω) ⊂ E^{3σ−1} (Ω) ⊂ D′_{3σ−1}(Ω) and E^{σ}(Ω) ⊂ G^{σ,∞}(Ω), so

E^{σ}(Ω) ⊂ G^{σ,∞}(Ω) ∩ D′_{3σ−1} (Ω) .

Consequently we have

G^{σ,∞}(Ω) ∩ D′_{3σ−1}(Ω) = E^{σ} (Ω) .  □

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 ∑_{g}^{σ} (f) ⊂ ℝ^{m}\ {0} , f ∈ G_{C}^{σ}(Ω), as the complement of the set of points having a conic neighborhood Γ such that ∃k_{1} > 0, ∃k_{2} > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1] , ∀ξ ∈ Γ, ∀ε ≤ ε_{0},

(8.1)  |F (f_{ε}) (ξ)| ≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} ) .

The following essential properties of ∑_{g}^{σ} (f) are sufficient to define later the generalized Gevrey wave front of generalized Gevrey ultradistributions.

Proposition 21. For every f ∈ G_{C}^{σ} (Ω), we have:

(1) The set ∑_{g}^{σ} (f) is a closed cone.

(2) ∑_{g}^{σ} (f) = ∅ ⟺ f ∈ G^{σ,∞} (Ω).

(3) ∑_{g}^{σ} (ψf) ⊂ ∑_{g}^{σ} (f) , ∀ψ ∈ E^{σ}(Ω).

Proof. One can easily, from the definition and Proposition 19, prove the assertions (1) and (2).

Let us suppose that ξ_{0} ∉ ∑_{g}^{σ} (f), then ∃Γ a conic neighborhood of ξ_{0}, ∃k_{1} > 0, ∃k_{2} > 0, ∃c_{1} > 0, ∃ε_{1} ∈ ]0, 1], ∀ξ ∈ Γ, ∀ε ≤ ε_{1},

(8.2)  |F (f_{ε}) (ξ)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} ) .

Let χ ∈ D^{σ}(Ω), χ ≡ 1 on neighborhood of supp f, so χψ ∈ D^{σ}(Ω), hence ∃k_{3} > 0, ∃c_{2} > 0, ∀ξ ∈ ℝ^{m},

(8.3)  |F (χψ) (ξ)| ≤ c_{2} exp (−k_{3} |ξ|^{\frac{1}{σ}} ) .

Let Λ be a conic neighborhood of ξ_{0} such that \overline{Λ} ⊂ Γ. We have, for a fixed ξ ∈ Λ,

F (ψf_{ε}) (ξ) = F (χψf_{ε}) (ξ)
= ∫_{A} F (f_{ε}) (η) F (χψ) (ξ − η) dη
+ ∫_{B} F (f_{ε}) (η) F (χψ) (ξ − η) dη,

where

A = \{η ∈ ℝ^{m}; |ξ − η|^{\frac{1}{σ}} ≤ δ (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} )\} ,
B = \{η ∈ ℝ^{m}; |ξ − η|^{\frac{1}{σ}} > δ (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} )\} .

We choose δ sufficiently small that A ⊂ Γ and \frac{|ξ|}{2^{σ}} < |η| < 2^{σ} |ξ| , ∀η ∈ A. Then for ε ≤ ε_{1},

|∫_{A} F (f_{ε}) (η) F (χψ) (ξ − η) dη| (8.4)

≤ c_{1}c_{2} exp (k_{1}ε^{− \frac{1}{2σ−1}} − \frac{k_{2}}{2} |ξ|^{\frac{1}{σ}} ) |∫_{A} exp (−k_{3} |ξ − η|^{\frac{1}{σ}} ) dη|

≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} − \frac{k_{2}}{2} |ξ|^{\frac{1}{σ}} ) .

As f ∈ G_{C}^{σ}(Ω), from (7.2), ∃c_{3} > 0, ∃μ_{1} > 0, ∃ε_{2} ∈ ]0, 1] , ∀μ_{2} > 0, ∀ξ ∈ ℝ^{m}, ∀ε ≤ ε_{2}, such that

|F (f_{ε}) (ξ)| ≤ c_{3} exp (μ_{1}ε^{− \frac{1}{2σ−1}} + μ_{2} |ξ|^{\frac{1}{σ}} ) ,

hence, for ε ≤ min (ε_{1}, ε_{2}), we have

|∫_{B} F (f_{ε}) (η) F (χψ) (ξ − η) dη|
≤ c_{2}c_{3} exp (μ_{1}ε^{− \frac{1}{2σ−1}} ) ∫_{B} exp (μ_{2} |η|^{\frac{1}{σ}} − k_{3} |ξ − η|^{\frac{1}{σ}} ) dη
≤ c exp (μ_{1}ε^{− \frac{1}{2σ−1}} ) ∫_{B} exp (μ_{2} |η|^{\frac{1}{σ}} − k_{3}δ (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} )) dη.

Then, taking μ_{2} < k_{3}δ, we obtain

(8.5) |∫_{B} F (f_{ε}) (η) F (χψ) (ξ − η) dη| ≤ c exp (μ_{1}ε^{− \frac{1}{2σ−1}} − k_{3}δ |ξ|^{\frac{1}{σ}} ) .

Consequently, (8.4) and (8.5) give ξ_{0} ∉ ∑_{g}^{σ} (ψf). □

Definition 13. Let f ∈ G^{σ}(Ω) and x_{0} ∈ Ω. The cone of σ-singular directions of f at x_{0}, denoted ∑_{g,x_{0}}^{σ} (f), is

(8.6) ∑_{g,x_{0}}^{σ}(f)

= ⋂ {∑_{g}^{σ} (ϕf) : ϕ ∈ D^{σ} (Ω) and ϕ ≡ 1 on a neighborhood of x_{0}} .

Lemma 22. Let f ∈ G^{σ} (Ω), then

∑_{g,x_{0}}^{σ} (f) = ∅ ⟺ x_{0} ∉ σ- singsupp_{g} (f) .

Proof. Let x_{0} ∉ σ-singsupp_{g} (f), i.e., ∃U ⊂ Ω an open neighborhood of x_{0} such that f ∈ G^{σ,∞} (U). Let ϕ ∈ D^{σ}(U) such that ϕ ≡ 1 on a neighborhood of x_{0}. Then ϕf ∈ G^{σ,∞}(Ω). Hence, from Proposition 21, ∑_{g}^{σ} (ϕf) = ∅, i.e., ∑_{g,x_{0}}^{σ} (f) = ∅.

Suppose now ∑_{g,x_{0}}^{σ} (f) = ∅. Let r > 0 such that B (x_{0}, 2r) ⊂ Ω and let ψ ∈ D^{σ} (B (x_{0}, 2r)) such that 0 ≤ ψ ≤ 1 and ψ ≡ 1 on B (x_{0}, r). Let ψ_{j} (x) = ψ (3^{j}(x − x_{0}) + x_{0}). Then it is clear that supp (ψ_{j}) ⊂ B (x_{0}, \frac{2r}{3^{j}} ) ⊂ Ω and ψ_{j} ≡ 1 on B (x_{0}, \frac{r}{3^{j}} ). We have ∀ϕ ∈ D^{σ} (Ω) with ϕ ≡ 1 on a neighborhood U of x_{0}, ∃j ∈ ℤ^{+} such that supp (ψ_{j}) ⊂ U, then ψ_{j}f_{ε} = ψ_{j}ϕf_{ε} and from Proposition 21, we have

∑_{g}^{σ} (ψ_{j}f) ⊂ ∑_{g}^{σ} (ϕf) ,

which gives

(8.7)  ⋂_{j∈ℤ^{+}} (∑_{g}^{σ} (ψ_{j}f)) = ∅.

We have ψ_{j} ≡ 1 on supp (ψ_{j+1}), then ∑_{g}^{σ} (ψ_{j+1}f) ⊂ ∑_{g}^{σ} (ψ_{j}f), so from (8.7), there exists n ∈ ℤ^{+} sufficiently large such that (ψ_{n}f) ∈ G^{σ,∞} (Ω), then f ∈ G^{σ,∞} (B (x_{0}, \frac{r}{3^{n}} )), which means. x_{0} ∉ σ-singsupp_{g} (f). □

Now, we are ready to give the definition of the generalized Gevrey wave front.

Definition 14. A point (x_{0}, ξ_{0}) ∉ W F_{g}^{σ} (f) ⊂ Ω×ℝ^{m}\ {0} if ξ_{0} ∉ ∑_{g,x_{0}}^{σ} (f), i.e., there exists ϕ ∈ D^{σ}(Ω), ϕ (x) = 1 neighborhood of x_{0}, and conic neighborhood Γ of ξ_{0}, ∃k_{1} > 0, ∃k_{2} > 0, ∃c > 0, ∃ε_{0} ∈ ]0, 1], such that ∀ξ ∈ Γ, ∀ε ≤ ε_{0},

|F (ϕf_{ε}) (ξ)| ≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} ) .

The main properties of the generalized Gevrey wave front W F_{g}^{σ} are subsumed in the following proposition.

Proposition 23. Let f ∈ G^{σ}(Ω). Then:

(1) The projection of W F_{g}^{σ} (f) on Ω is σ- singsupp_{g} (f).

(2) If f ∈ G_{C}^{σ}(Ω), then the projection of W F_{g}^{σ} (f) on ℝ^{m}\ {0} is ∑_{g}^{σ} (f).

(3) ∀α ∈ ℤ_{+}^{m}, W F_{g}^{σ} (∂^{α}f) ⊂ W F_{g}^{σ} (f).

(4) ∀g ∈ G^{σ,∞} (Ω), W F_{g}^{σ} (gf) ⊂ W F_{g}^{σ} (f).

Proof. (1) and (2) hold from the definition, Proposition 21 and Lemma 22.

(3) Let (x_{0}, ξ_{0}) ∉ W F_{g}^{σ} (f), then ∃ϕ ∈ D^{σ}(Ω), ϕ ≡ 1 on a neighborhood U of x_{0}, there exist a conic neighborhood Γ of ξ_{0}, ∃k_{1} > 0, ∃k_{2} > 0, ∃c_{1} > 0, ∃ε_{0} ∈ ]0, 1], such that ∀ξ ∈ Γ, ∀ε ≤ ε_{0},

(8.8)  |F (ϕf_{ε}) (ξ)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} ) .

We have, for ψ ∈ D^{σ} (U) such that ψ (x_{0}) = 1,

|F (ψ∂f_{ε}) (ξ)| = |F (∂ (ψf_{ε})) (ξ) − F ((∂ψ) f_{ε}) (ξ)|
≤ |ξ| |F (ψϕf_{ε}) (ξ)| + |F ((∂ψ) ϕf_{ε}) (ξ)| .

As W F_{g}^{σ} (ψf) ⊂ W F_{g}^{σ} (f), (8.8) holds for both

|F (ψϕf_{ε}) (ξ)| and |F ((∂ψ) ϕf_{ε}) (ξ)| .

So

|ξ| |F (ψϕf_{ε}) (ξ)| ≤ c |ξ| exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} )
≤ c′ exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{3} |ξ|^{\frac{1}{σ}} ) ,

with c′ > 0, k_{3} > 0 such that |ξ| ≤ c′ exp (k_{2} − k_{3}) |ξ|^{\frac{1}{σ}}. Hence (8.8) holds for |F (ψ∂f_{ε}) (ξ)|, which proves (x_{0}, ξ_{0}) ∉ W F_{g}^{σ} (∂f).

(4) Let (x_{0}, ξ_{0}) ∉ W F_{g}^{σ} (f). Then ∃ϕ ∈ D^{σ}(Ω), ϕ ≡ 1 on a neighborhood U of x_{0}, there exists a conic neighborhood Γ of ξ_{0}, ∃k_{1} > 0, ∃k_{2} > 0, ∃c_{1} > 0, ∃ε_{0} ∈ ]0, 1], such that ∀ξ ∈ Γ, ∀ε ≤ ε_{0},

|F (ϕf_{ε}) (ξ)| ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} ) .

Let ψ ∈ D^{σ}(Ω) and ψ ≡ 1 on supp ϕ. Then F (ϕg_{ε}f_{ε}) = F (ψg_{ε}) ∗ F (ϕf_{ε}). We have ψg ∈ G^{σ,∞}(Ω) ∩ G_{C}^{σ}(Ω), then ∃c_{2} > 0, ∃k_{3} > 0, ∃k_{4} > 0, ∃ε_{1} > 0, ∀ξ ∈ ℝ^{m}, ∀ε ≤ ε_{1},

|F (ψg_{ε}) (ξ)| ≤ c_{2} exp (k_{3}ε^{− \frac{1}{2σ−1}} − k_{4} |ξ|^{\frac{1}{σ}} ) ,

so

F (ϕg_{ε}f_{ε}) (ξ) = ∫_{A} F (ϕf_{ε}) (η) F (ψg_{ε}) (η − ξ) dη
+ ∫_{B} F (ϕf_{ε}) (η) F (ψg_{ε}) (η − ξ) dη,

where A and B are the same as in the proof of Proposition 21. By (7.2), we have ∃c > 0, ∃μ_{1} > 0, ∀μ_{2} > 0, ∃ε_{2} > 0, ∀ξ ∈ ℝ^{m}, ∀ε ≤ ε_{2},

|F (ϕf_{ε}) (ξ)| ≤ c exp (μ_{1}ε^{− \frac{1}{2σ−1}} + μ_{2} |ξ|^{\frac{1}{σ}} ) .

The same steps as in Proposition 21 finish the proof. □

Corollary 24. Let P (x, D) = ∑_{|α|≤m} a_{α} (x) D^{α} be a partial differential oper-

ator with G^{σ,∞}(Ω) coefficients, then

W F_{g}^{σ} (P (x, D) f) ⊂ W F_{g}^{σ} (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 was extended to the level of basic functionals in the dual of Colombeau algebra L (G_{C}(Ω), ℂ̃).

We need the following lemma to show the relationship between W F_{g}^{σ} (T) and W F^{σ} (T), when T ∈ D′_{3σ−1} (Ω).

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, ∃ε_{0} > 0, ∀ε ∈ ]0, ε_{0}] , ∀ξ ∈ ℝ^{m},

|ρ̂_{ε} (ξ)| ≤ cε^{−m}e^{−νε^{\frac{1}{σ}} |ξ|^{\frac{1}{σ}}} ,

where ρ_{ε} (x) = (\frac{1}{ε})^{m} ϕ (\frac{x}{ε}) φ (x |ln ε|), and ρ̂ denotes the Fourier transform of ρ.

Proof. We have, for ε sufficiently small,

ε^{m} ≤ |ln ε|^{−m} ≤ 1.

Let ξ ∈ ℝ^{m}. Then

ρ̂_{ε} (ξ) = |ln ε|^{−m} [∫_{A} ϕ̂(ε (ξ − η)) φ̂ (\frac{η}{|ln ε|}) dη+

∫_{B} ϕ̂(ε (ξ − η)) φ̂ (\frac{η}{|ln ε|}) dη ],

where

A = \{η; |ξ − η|^{\frac{1}{σ}} ≤ δ^{\frac{1}{σ}} (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} )\} ,
B = \{η; |ξ − η|^{\frac{1}{σ}} > δ^{\frac{1}{σ}} (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} )\} .

Generalized Gevrey ultradistributions 65

We choose δ sufficiently small such that \frac{|ξ|}{2^{σ}} < |η| < 2^{σ} |ξ| , ∀η ∈ A. Since ϕ ∈ S^{(σ)}, φ ∈ D^{σ}(Ω), then ∃k_{1}, k_{2} > 0, ∃c_{1}, c_{2} > 0, ∀ξ ∈ ℝ^{m},

|ϕ̂(ξ)| ≤ c_{2} exp (−k_{1} |ξ|^{\frac{1}{σ}} ) and |φ̂ (ξ)| ≤ c_{1} exp (−k_{2} |ξ|^{\frac{1}{σ}} ) . So

I_{1} = |ln ε|^{−m} |∫_{A} ϕ̂(ε (ξ − η)) φ̂ (\frac{η}{|ln ε|}) dη|

≤ c_{1}c_{2} exp (−k_{2} \frac{|ln ε|^{− \frac{1}{σ}} |ξ|^{\frac{1}{σ}}}{2} ) ∫ exp (−k_{1}ε^{\frac{1}{σ}} |ξ − η|^{\frac{1}{σ}} ) dη.

Let z = ε (η − ξ). Then

I_{1} ≤ cε^{−m} exp (−\frac{k_{2}}{2} |ln ε|^{− \frac{1}{σ}} |ξ|^{\frac{1}{σ}} ) ∫ exp (−k_{1} |z|^{\frac{1}{σ}} ) dz
≤ cε^{−m} exp (−vε^{\frac{1}{σ}} |ξ|^{\frac{1}{σ}} ) .

For I_{2}, we have

I_{2} = |ln ε|^{−m} |∫_{B} ϕ̂(ε (ξ − η)) φ̂ (\frac{η}{|ln ε|}) dη|
≤ c_{1}c_{2} ∫_{B} exp (−k_{1}ε^{\frac{1}{σ}} |ξ − η|^{\frac{1}{σ}} − k_{2} |ln ε|^{− \frac{1}{σ}} |η|^{\frac{1}{σ}} ) dη

≤ c_{1}c_{2} exp (−k_{1}δε^{\frac{1}{σ}} |ξ|^{\frac{1}{σ}} ) ∫_{B} exp (−k_{1}δε^{\frac{1}{σ}} |η|^{\frac{1}{σ}} − k_{2} |ln ε|^{− \frac{1}{σ}} |η|^{\frac{1}{σ}} ) dη

≤ c_{1}c_{2} exp (−k_{1}δε^{\frac{1}{σ}} |ξ|^{\frac{1}{σ}} ) ∫_{B} exp (−kε^{\frac{1}{σ}} |η|^{\frac{1}{σ}} ) dη
≤ cε^{−m} exp (−vε^{\frac{1}{σ}} |ξ|^{\frac{1}{σ}} ) .

Consequently, ∃c > 0, ∃v > 0, ∃ε_{0} > 0, , ∀ε ≤ ε_{0} such that

|ρ̂_{ε} (ξ)| ≤ cε^{−m} exp (−vε^{\frac{1}{σ}} |ξ|^{\frac{1}{σ}} ) , ∀ξ ∈ ℝ^{m}.  □

We have the following important result. Theorem 26. Let T ∈ D′_{3σ−1}(Ω) ∩ G^{σ}(Ω). Then W F_{g}^{σ} (T) = W F^{σ} (T). Proof. Let S ∈ E′_{3σ−1}(Ω) ⊂ E′_{σ}(Ω) and ψ ∈ D^{σ}(Ω). We have

|F (ψ (S ∗ ϕ_{ε})) (ξ) − F (ψS) (ξ)| = |⟨S (x) , (ψ (x) e^{−iξx}) ∗ ϕ̆_{ε} (x) − (ψ (x) e^{−iξx})⟩| .

Then ∃L a compact subset of Ω such that ∀h > 0, ∃c > 0,

|F (ψ (S ∗ ϕ_{ε})) (ξ) − F (ψS) (ξ)|

≤ c sup_{α∈ℤ_{+}^{m},x∈L} \frac{h^{|α|}}{α!^{σ}} |(∂_{x}^{α} (ψ (x) e^{−iξx} ∗ ϕ̆_{ε} (x) − ψ (x) e^{−iξx}))| .

We have e^{−iξ}ψ ∈ D^{σ}(Ω). Corollary 12 gives, ∀k_{0} > 0, ∃c_{2} > 0, ∃η > 0, ∀ε ≤ η,

(8.9) sup_{α∈ℤ_{+}^{m},x∈L} \frac{c_{2}^{|α|}}{α!^{σ}} |∂_{x}^{α} (ψ (x) e^{−iξx} ∗ ϕ̆_{ε} (x) − ψ (x) e^{−iξx})|

≤ c_{2}e^{−k_{0}ε^{− \frac{1}{2σ−1}}} ,

so ∀k_{0} > 0, ∃c′ > 0, ∃η > 0, ∀ε ≤ η, such that

(8.10)  |F (ψS) (ξ) − F (ψ (S ∗ ϕ_{ε})) (ξ)| ≤ c′e^{−k_{0}ε^{− \frac{1}{2σ−1}}} .

Let T ∈ D′_{3σ−1}(Ω) ∩ G^{σ}(Ω) and (x_{0}, ξ_{0}) ∉ W F_{g}^{σ} (T). Then there exist χ ∈ D^{σ}(Ω), χ (x) = 1 in a neighborhood of x_{0}, and a conic neighborhood Γ of ξ_{0}, ∃k_{1} > 0, ∃k_{2} > 0, ∃c_{1} > 0, ∃ε_{0} ∈ ]0, 1[, such that ∀ξ ∈ Γ, ∀ε ≤ ε_{0},

(8.11)  |F (χ (T ∗ ρ_{ε})) (ξ)| ≤ c_{1}e^{k_{1}ε^{− \frac{1}{2σ−1}} −k_{2}|ξ|^{\frac{1}{σ}}} .

Let ψ ∈ D^{σ}(Ω) equal 1 in neighborhood of x_{0} such that for sufficiently small ε we have χ ≡ 1 on supp ψ + B (0, \frac{2}{|ln ε|}), and let φ ∈ D^{σ} (B (0, 2)), 0 ≤ φ ≤ 1 and φ ≡ 1 on B (0, 1). Then there exists ε_{0} < 1, such that ∀ε < ε_{0},

ψ (T ∗ ρ_{ε}) (x) = ψ (χT ∗ ρ_{ε}) (x) ,

where ρ_{ε} (x) = \frac{1}{ε^{m}} φ (x |ln ε|) ϕ (\frac{x}{ε}). As χT ∈ E′_{3σ−1}(Ω), then, from Proposition 14,

ψ (T ∗ ρ_{ε}) (x) = ψ (χT ∗ ρ_{ε}) (x) = ψ (χT ∗ ϕ_{ε}) (x) .

Let ε ≤ min (η, ε_{0}) and ξ ∈ Γ. We have

|F (ψT) (ξ)| ≤ |F (ψT) (ξ) − F (ψ (T ∗ ρ_{ε})) (ξ)| + |F (χ (T ∗ ρ_{ε})) (ξ)|

≤ |F (ψχT) (ξ) − F (ψ (χT ∗ ϕ_{ε})) (ξ)| + |F (χ (T ∗ ρ_{ε})) (ξ)| .

Then by (8.10)and (8.11), we obtain

|F (ψT) (ξ)| ≤ c′e^{−k_{0}ε^{− \frac{1}{2σ−1}}} + c_{1}e^{k_{1}ε^{− \frac{1}{2σ−1}} −k_{2}|ξ|^{\frac{1}{σ}}} .

Take c = max (c′, c_{1}), ε = (\frac{k_{1}}{(k_{2} − r) |ξ|^{\frac{1}{σ}}} )^{2σ−1} , r ∈ ]0, k_{2}[ , k_{0} = \frac{k_{1}r}{k_{2} − r}.

Then ∃δ > 0, ∃c > 0, such that

|F (χT) (ξ)| ≤ ce^{−δ|ξ|^{\frac{1}{σ}}} ,

which proves that (x_{0}, ξ_{0}) ∉ W F^{σ} (T), i.e., W F^{σ} (T) ⊂ W F_{g}^{σ} (T).

Suppose (x_{0}, ξ_{0}) ∉ W F^{σ} (T), then there exist χ ∈ D^{σ} (Ω) , χ (x) = 1 in a neighborhood of x_{0}, a conic neighborhood Γ of ξ_{0}, ∃λ > 0, ∃c_{1} > 0, such that ∀ξ ∈ Γ,

(8.12)  |F (χT) (ξ)| ≤ c_{1}e^{−λ|ξ|^{\frac{1}{σ}}} .

Let also ψ ∈ D^{σ}(Ω) equal 1 in neighborhood of x_{0} such that for sufficiently small ε we have χ ≡ 1 on supp ψ + B (0, \frac{2}{|ln ε|}). Then there exists ε_{0} < 1, such that ∀ε < ε_{0},

ψ (T ∗ ρ_{ε}) (x) = ψ (χT ∗ ρ_{ε}) (x) .

We have

F (ψ (T ∗ ρ_{ε})) (ξ) = ∫ F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη.

Let Λ be a conic neighborhood of ξ_{0} such that \overline{Λ} ⊂ Γ. For a fixed ξ ∈ Λ, we have

F (ψ (χT ∗ ρ_{ε})) (ξ) = ∫_{A} F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη
+ ∫_{B} F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη,

where

A = \{η; |ξ − η|^{\frac{1}{σ}} ≤ δ (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} )\} ,
B = \{η; |ξ − η|^{\frac{1}{σ}} > δ (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} )\} .

We choose δ sufficiently small such that A ⊂ Γ and \frac{|ξ|}{2^{σ}} < |η| < 2^{σ} |ξ|. Since ψ ∈ D^{σ} (Ω), then ∃μ > 0, ∃c_{2} > 0, ∀ξ ∈ ℝ^{m},

|F (ψ) (ξ)| ≤ c_{2} exp (−μ |ξ|^{\frac{1}{σ}} ) .

Then ∃c > 0, ∃ε_{0} ∈ ]0, 1[ , ∀ε ≤ ε_{0},

(8.13) |∫_{A} F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη|

≤ c exp (−\frac{λ}{2} |ξ|^{\frac{1}{σ}} ) |∫_{A} exp (−μ |η − ξ|^{\frac{1}{σ}} ) F (ρ_{ε}) (η) dη| .

From Lemma 25, ∃c_{3} > 0, ∃ν > 0, ∃ε_{0} > 0, such that ∀ε ∈ ]0, ε_{0}],

|F (ρ_{ε}) (ξ)| ≤ c_{3}ε^{−m}e^{−νε^{\frac{1}{σ}} |ξ|^{\frac{1}{σ}}} , ∀ξ ∈ ℝ^{m}.

Then ∃c > 0, such that

|∫_{A} F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη|

≤ cε^{−m} exp (−\frac{λ}{2} |ξ|^{\frac{1}{σ}} ) ∫_{A} exp (−μ |η − ξ|^{\frac{1}{σ}} ) exp (−νε^{\frac{1}{σ}} |η|^{\frac{1}{σ}} ) dη.

We have ∃k > 0, ∀ε ∈ ]0, ε_{0}],

(8.14)  ε^{−m} exp (−νε^{\frac{1}{σ}} |η|^{\frac{1}{σ}} ) ≤ exp (kε^{− \frac{1}{2σ−1}} ) ,

so

(8.15) |∫_{A} F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη|

≤ c exp (kε^{− \frac{1}{2σ−1}} − \frac{λ}{2} |ξ|^{\frac{1}{σ}} ) .

As χT ∈ E′_{3σ−1}(Ω) ⊂ E′_{σ}(Ω), then ∀l > 0, ∃c > 0, ∀ξ ∈ ℝ^{m},

|F (χT) (ξ)| ≤ c exp (l |ξ|^{\frac{1}{σ}} ) ,

hence, we have

|∫_{B} F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη|
≤ c ∫_{B} exp (l |η|^{\frac{1}{σ}} − μ |η − ξ|^{\frac{1}{σ}} ) |F (ρ_{ε})| dη
≤ c′ε^{−m} exp (−μδ |ξ|^{\frac{1}{σ}} ) ∫_{B} exp ((l − μδ) |η|^{\frac{1}{σ}} − νε^{\frac{1}{σ}} |η|^{\frac{1}{σ}} ) dη.

Then, taking l − μδ = −a < 0 and using (8.14), we obtain for a constant c > 0,

(8.16) |∫_{B} F (ψ) (ξ − η) F (χT) (η) F (ρ_{ε}) (η) dη|
≤ c exp (kε^{− \frac{1}{2σ−1}} − μδ |ξ|^{\frac{1}{σ}} ) .

Consequently, (8.15) and (8.16) give ∃c > 0, ∃k_{1} > 0, ∃k_{2} > 0,

(8.17)  |F (ψ (T ∗ ρ_{ε})) (ξ)| ≤ c exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ|^{\frac{1}{σ}} ) ,

which gives that (x_{0}, ξ_{0}) ∉ W F_{g}^{σ} (T), so W F_{g}^{σ} (T) ⊂ W F^{σ} (T) which ends the proof. □

9. Generalized Hörmander’s theorem

To extend the generalized Hörmander’s result on the wave front set of the product, define W F_{g}^{σ} (f) + W F_{g}^{σ} (g), where f, g ∈ G^{σ}(Ω), as the set

{(x, ξ + η) ; (x, ξ) ∈ W F_{g}^{σ} (f) , (x, η) ∈ W F_{g}^{σ} (g)} .

We recall the following fundamental lemma (see [13] for the proof).

Lemma 27. Let ∑_{1}, ∑_{2} be closed cones in ℝ^{m}\ {0}, with 0 ∉ ∑_{1} + ∑_{2}. Then:

(1) \overline{∑_{1} + ∑_{2}}^{ℝ^{m}\\{0\}} = (∑_{1} + ∑_{2}) ∪ ∑_{1} ∪ ∑_{2}.

(2) For any open conic neighborhood Γ of ∑_{1} + ∑_{2} in ℝ^{m}\ {0}, one can find open conic neighborhoods of Γ_{1}, Γ_{2} in ℝ^{m}\ {0} of, respectively, ∑_{1}, ∑_{2}, such that

Γ_{1} + Γ_{2} ⊂ Γ.

The principal result of this section is the following theorem.

Theorem 28. Let f, g ∈ G^{σ}(Ω), such that ∀x ∈ Ω,

(9.1)  (x, 0) ∉ W F_{g}^{σ} (f) + W F_{g}^{σ} (g) .

Then

(9.2)  W F_{g}^{σ} (fg) ⊆ (W F_{g}^{σ} (f) + W F_{g}^{σ} (g)) ∪ W F_{g}^{σ} (f) ∪ W F_{g}^{σ} (g) .

Proof. Let (x_{0}, ξ_{0}) ∉ (W F_{g}^{σ} (f) + W F_{g}^{σ} (g)) ∪ W F_{g}^{σ} (f) ∪ W F_{g}^{σ} (g). Then

∃ϕ ∈ D^{σ} (Ω), ϕ (x_{0}) = 1, ξ_{0} ∉ (∑_{g}^{σ} (ϕf) + ∑_{g}^{σ} (ϕg)) ∪ ∑_{g}^{σ} (ϕf) ∪ ∑_{g}^{σ} (ϕg). From (9.1) we have 0 ∉ ∑_{g}^{σ} (ϕf) + ∑_{g}^{σ} (ϕg), so by Lemma 27 (1), we have

ξ_{0} ∉ (∑_{g}^{σ} (ϕf) + ∑_{g}^{σ} (ϕg)) ∪ ∑_{g}^{σ} (ϕf) ∪ ∑_{g}^{σ} (ϕg)
= \overline{∑_{g}^{σ} (ϕf) + ∑_{g}^{σ} (ϕg)}^{ℝ^{m}\\{0\}}.

Let Γ_{0} be an open conic neighborhood of ∑_{g}^{σ} (ϕf) + ∑_{g}^{σ} (ϕg) in ℝ^{m}\ {0} such that ξ_{0} ∉ \overline{Γ}_{0}. Then from Lemma 27 (2) there exist open cones Γ_{1} and Γ_{2} in ℝ^{m}\ {0} such that

∑_{g}^{σ} (ϕf) ⊂ Γ_{1}, ∑_{g}^{σ} (ϕg) ⊂ Γ_{2} and Γ_{1} + Γ_{2} ⊂ Γ_{0}.

Define Γ = ℝ^{m}\\overline{Γ}_{0}, so

(9.3)  Γ ∩ Γ_{2} = ∅ and (Γ − Γ_{2}) ∩ Γ_{1} = ∅.

70 Khaled Benmeriem and Chikh Bouzar Let ξ ∈ Γ and ε ∈ ]0, 1].

F (ϕf_{ε}ϕg_{ε}) (ξ) = (F (ϕf_{ε}) ∗ F (ϕg_{ε})) (ξ)
= ∫_{Γ_{2}} F (ϕf_{ε}) (ξ − η) F (ϕg_{ε}) (η) dη
+ ∫_{Γ_{2}^{c}} F (ϕf_{ε}) (ξ − η) F (ϕg_{ε}) (η) dη
= I_{1} (ξ) + I_{2} (ξ) .

From (9.3), ∃c_{1} > 0, ∃k_{1}, k_{2} > 0, ∃ε_{1} > 0 such that ∀ε ≤ ε_{1}, ∀η ∈ Γ_{2},

F (ϕf_{ε}) (ξ − η) ≤ c_{1} exp (k_{1}ε^{− \frac{1}{2σ−1}} − k_{2} |ξ − η|^{\frac{1}{σ}} ) ,

and from (7.2) ∃c_{2} > 0, ∃k_{3} > 0, ∀k_{4} > 0, ∃ε_{2} > 0, ∀η ∈ ℝ^{m}, ∀ε ≤ ε_{2},

|F (ϕg_{ε}) (η)| ≤ c_{2} exp (k_{3}ε^{− \frac{1}{2σ−1}} + k_{4} |η|^{\frac{1}{σ}} ) .

Let γ > 0 sufficiently small such that |ξ − η|^{\frac{1}{σ}} ≥ γ (|ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}} ) , ∀η ∈ Γ_{2}. Hence for ε ≤ min (ε_{1}, ε_{2}),

|I_{1} (ξ)| ≤ c_{1}c_{2} exp ((k_{1} + k_{3}) ε^{− \frac{1}{2σ−1}} − k_{2}γ |ξ|^{\frac{1}{σ}} )
× ∫ exp (−k_{2}γ |η|^{\frac{1}{σ}} + k_{4} |η|^{\frac{1}{σ}} ) dη.

Take k_{4} < k_{2}γ. Then

(9.4)  |I_{1} (ξ)| ≤ c′ exp (k′_{1}ε^{− \frac{1}{2σ−1}} − k′_{2} |ξ|^{\frac{1}{σ}} ) .

Let r > 0,

I_{2} (ξ) = ∫_{Γ_{2}^{c}∩{|η|≤r|ξ|}} F (ϕf_{ε}) (ξ − η) F (ϕg_{ε}) (η) dη
+ ∫_{Γ_{2}^{c}∩{|η|≥r|ξ|}} F (ϕf_{ε}) (ξ − η) F (ϕg_{ε}) (η) dη
= I_{21} (ξ) + I_{22} (ξ) .

Choose r sufficiently small such that \{|η|^{\frac{1}{σ}} ≤ r |ξ|^{\frac{1}{σ}} \} ⟹ ξ − η ∉ Γ_{1}. Then |ξ − η|^{\frac{1}{σ}} ≥ (1 − r) |ξ|^{\frac{1}{σ}} ≥ (1 − 2r) |ξ|^{\frac{1}{σ}} + |η|^{\frac{1}{σ}}, so ∃c_{3} > 0, ∃λ_{1}, λ_{2}, λ_{3} > 0, ∃ε_{3} > 0 such that ∀ε ≤ ε_{3},

|I_{21} (ξ)| ≤ c_{3} exp (λ_{1}ε^{− \frac{1}{2σ−1}} ) ∫ exp (−λ_{2} |ξ − η|^{\frac{1}{σ}} − λ_{3} |η|^{\frac{1}{σ}} ) dη
≤ c_{3} exp (λ_{1}ε^{− \frac{1}{2σ−1}} − λ′_{2} |ξ|^{\frac{1}{σ}} ) ∫ exp (−λ′_{3} |η|^{\frac{1}{σ}} ) dη
≤ c′_{3} exp (λ_{1}ε^{− \frac{1}{2σ−1}} − λ′_{2} |ξ|^{\frac{1}{σ}} ) .

Generalized Gevrey ultradistributions 71

If |η|^{\frac{1}{σ}} ≥ r |ξ|^{\frac{1}{σ}}, we have |η|^{\frac{1}{σ}} ≥ \frac{|η|^{\frac{1}{σ}} + r |ξ|^{\frac{1}{σ}}}{2} , and then ∃c_{4} > 0, ∃μ_{1}, μ_{3} >

0, ∀μ_{2} > 0, ∃ε_{4} > 0 such that ∀ε ≤ ε_{4},

|I_{21} (ξ)| ≤ c_{4} exp (μ_{1}ε^{− \frac{1}{2σ−1}} ) ∫ exp (μ_{2} |ξ − η|^{\frac{1}{σ}} − μ_{3} |η|^{\frac{1}{σ}} ) dη

≤ c_{4} exp (μ_{1}ε^{− \frac{1}{2σ−1}} ) ∫ exp (μ_{2} |ξ − η|^{\frac{1}{σ}} − μ′_{3} |η|^{\frac{1}{σ}} − μ′_{3}r |ξ|^{\frac{1}{σ}} ) dη,

if we take μ_{2} sufficiently small we obtain

|I_{21} (ξ)| ≤ c′_{4} exp (k′_{3}ε^{− \frac{1}{2σ−1}} − μ′′_{3} |ξ|^{\frac{1}{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

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