PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 84(98) (2008), 109–121 DOI: 10.2298/PIM0898109D

A NEW APPROACH TO TEMPERATE GENERALIZED COLOMBEAU FUNCTIONS

Antoine Delcroix

Abstract. A new approach to the algebra G_{τ} of temperate nonlinear generalized functions is proposed, in which G_{τ} is based on the space O_{M} endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of G_{τ}. This is illustrated by the natural introduction of a regularity theory in G_{τ}, of the Fourier transform, with the definition of G_{O′_{C}}, the space of rapidly generalized distributions which is the Fourier image of G_{τ}.

1. Introduction

The theory of generalized functions is nowadays well established. Many applications have been carried out in various fields of mathematics such as partial differential equations, Lie analysis, local and microlocal analysis, probability theory, differential geometry. (See for examples the monographies [1, 2, 7, 15, 16] and the references therein.)

This paper develops some remarks about a new approach to temperate generalized functions. In order to justify the introduction of this new construction, we first recall the main types of special (or simplified) algebras of generalized based on spaces of smooth functions considered in the literature.

The original simplified Colombeau algebra of generalized functions G is based on the space E = C^{∞} of smooth functions and contains the space of Schwartz distributions as a subvector space [1, 7, 15, 16, 18]. The duality in the background of this construction is, of course, (D, D′). As all spaces considered in the sequel, G is a factor space of moderate nets modulo negligible ones, the moderateness and the negligibility being given by the asymptotic behavior of the nets with respect to an asymptotic scale. When an algebra containing the space of tempered distributions is needed, the so-called algebra of temperate generalized functions G_{τ} [1, 7, 17, 18], based on the space O_{M} of slowly increasing smooth functions, is considered. The duality is in this case (S, S′). Note that this construction is not, at first sight, related to the topology of O_{M}. Finally, an algebra based on the space S of rapidly decreasing functions has also been considered [3, 17, 19], with applications (for example) in the field of pseudo differential operators [5, 6] or of microlocal analysis of generalized functions [3, 8, 9, 19]. This algebra G_{S} of rapidly decreasing generalized functions contains as a linear subspace O′_{C}, the space of rapidly decreasing distributions.

2000 Mathematics Subject Classification: Primary 46F30; Secondary 46F05, 46E10, 42A38.

Partially supported by IUFM de Guadeloupe.

The first and the last constructions are based on the natural topology of the underlying space, which can be described by (countable) families of semi-norms. We propose here a new version of the construction of temperate generalized functions based on the usual topology of O_{M}, which therefore fits in the general scheme of construction of Colombeau type algebras. The price to be paid is the non countability of the family of semi-norms defining the topology of O_{M}.

The paper is organized as follows. Section 2 is devoted to a short presentation of the construction of the spaces of Colombeau type generalized functions and of the examples quoted above. In Section 3, we briefly recall the construction of the classical space of temperate generalized functions G_{τ}, develop the new construction and show that it leads to the same space. In Section 4, we turn to the definition of the Fourier transform of elements of G_{τ}. Using the classical theorem asserting that the Fourier image of O_{M} is O′_{C}, we introduce the (new) space G_{O′_{C}} of rapidly generalized distributions which is the Fourier image of G_{τ}. Of course, this Fourier transform will share the classical expected properties. Finally, in Section 5, we introduce the subspace G_{τ}^{∞} of regular elements of G_{τ} and show that the result G_{τ}^{∞} ∩ S′ = O_{M} holds in the spirit of the more classical one G^{∞} ∩ D′ = C^{∞} [16]. (More generally, we could have introduced the notion of R-regularity [3].) We also show, in the spirit of [14], that some subspaces of G of regular temperate elements can be considered leading to the corresponding local analysis of elements of G.

Acknowledgment. I would like to express my gratitude to Prof. Stevan Pilipović for his warm support for more than ten years.

2. Simplified or special algebras of generalized functions

2.1. Colombeau type algebras based on locally convex algebras. Let d be an integer and denote by 𝕂 the field of real or complex numbers. Let E(·) be a presheaf (resp. sheaf) of 𝕂-topological algebras of 𝕂 valued functions over ℝ^{d}. (Thus, the presheaf restriction operator is the usual restriction of 𝕂 valued functions.)

Suppose that, for any open set Ω in ℝ^{d}, the topology of E(Ω) can be described by a family P(Ω) = (p_{i})_{i∈I(Ω)} of semi-norms verifying:

∀i∈I(Ω) ∃(j, k, C)∈I(Ω)×I(Ω)×ℝ_{+}^{∗} ∀f, g ∈E(Ω) p_{i}(fg) ⩽ Cp_{j}(f)p_{k}(g).

Set

M_{(E,P)}(Ω) = {(f_{ε})_{ε} ∈ E(Ω)^{(0,1]} | ∀i∈I ∃m∈ℕ p_{i}(f_{ε}) = o(ε^{−m}) as ε → 0},
N_{(E,P)}(Ω) = {(f_{ε})_{ε} ∈ E(Ω)^{(0,1]} | ∀i∈I ∀m∈ℕ p_{i}(f_{ε}) = o(ε^{m}) as ε → 0}.

(The letter M (resp. N) stands for moderate (resp. negligible). In the sequel, we shall omit the precision “as ε → 0.”)

From [12], it follows that:

Proposition 2.1. (a) Suppose that the following assertion holds:

(1) For any Ω_{1} and Ω_{2}, open subsets of ℝ^{d} with Ω_{1} ⊂ Ω_{2}, we have I(Ω_{1}) ⊂ I(Ω_{2}). Moreover, if ρ_{1}^{2} is the restriction operator E(Ω_{2}) → E(Ω_{1}), then, for each p_{i} ∈ P(Ω_{1}), the semi-norm p̃_{i} = p_{i} ∘ ρ_{1}^{2} extends p_{i} to P(Ω_{2}).

Then M_{(E,P)}(·) is a presheaf of 𝕂-algebras and N_{(E,P)}(·) a presheaf of ideals of M_{(E,P)}(·).

(b) Suppose that E(·) is a sheaf of 𝕂-topological algebras and that assumption (1) and the following hold:

(2) For any family (Ω_{h})_{h∈H} of open sets in ℝ^{d} with Ω = ⋃_{h∈H} Ω_{h} and for any p ∈ P(Ω), there exist a finite subfamily (Ω_{j})_{1⩽j⩽n} and corresponding semi-norms p_{j} ∈ P(Ω_{j}) such that, for any u ∈ E(Ω),

p(u) ⩽ C max_{1⩽j⩽n} p_{j}(u|_{Ω_{j}}), C > 0.

Then M_{(E,P)}(·) is a sheaf of 𝕂-algebras and N_{(E,P)}(·) a presheaf of ideals of M_{(E,P)}(·).

Definition 2.1. For any Ω open subset of ℝ^{d}, the Colombeau type algebra associated to E(Ω) is the factor algebra G(Ω) = M_{(E,P)}(Ω)/N_{(E,P)}(Ω).

Proposition 2.2. [1, 12] (a) Under assumption (1), G(·) is a presheaf of algebras.

(b) In addition, suppose that assumption (2) is fulfilled. Then, the localization principle (F_{1}) holds for G(·): (F_{1}) Let (Ω_{h})_{h∈H} be a family of open sets in ℝ^{d} with Ω = ⋃_{h∈H} Ω_{h}. Consider u, v ∈ G(Ω) such that all restrictions u|_{Ω_{h}} and v|_{Ω_{h}} (h ∈ H) coincide. Then u = v.

(c) Moreover, if E(·) is a fine sheaf of algebras, G(·) is also a fine sheaf of algebras.

There is a natural presheaf (resp. sheaf) embedding of E(·) into G(·) defined by

(2.1)  σ_{E,G}(Ω) : E(Ω) → G(Ω), f ↦ (f)_{ε} + N_{(E,P)}(Ω).

The presheaf (rest. sheaf) G(·) turns out to be a presheaf (rest. sheaf) of modules on the factor ring ℂ̃ = M(ℂ)/N (ℂ) with

M(𝕂) = {(r_{ε})_{ε} ∈ 𝕂^{(0,1]} | ∃m ∈ ℕ |r_{ε}| = o(ε^{−m})},
N (𝕂) = {(r_{ε})_{ε} ∈ 𝕂^{(0,1]} | ∀m ∈ ℕ |r_{ε}| = o(ε^{m})},

with 𝕂 = ℂ or 𝕂 = ℝ, ℝ_{+}. Moreover, for the cases under consideration in this paper, E(·) is a subpresheaf (rest. subsheaf) of the sheaf C^{∞}(·) of smooth functions.

Then, one easily checks that, for α ∈ ℕ^{d}, a presheaf family of differential operators ∂^{α}f is defined component-wise on G(Ω) by

∂^{α}f = (∂^{α}f_{ε})_{ε} + N_{(E,P)}(Ω) with (f_{ε})_{ε} ∈ f.

The family of differential operators (∂^{α})_{α∈ℕ^{d}} satisfies the usual rules (such as the Leibniz rule) and G(·) turns out to be a presheaf (rest. sheaf) of differential algebras. The embedding defined by (2.1) turns out to be an embedding of differential algebras.

2.2. Examples. Example 2.1. Take E(·) = C^{∞}(·). For any Ω open subset of ℝ^{d}, C^{∞}(Ω) is endowed with the family of semi-norms P(Ω) = (p_{K,l})_{K⋐Ω,l∈ℕ} defined by

p_{K,l}(f) = sup_{|α|⩽l, x∈K} |∂^{α}f(x)|,

where the notation K ⋐ Ω means that the set K is a compact set included in Ω. We set

M_{C^{∞}}(·) = M_{(C^{∞},P)}(·), N_{C^{∞}}(·) = N_{(C^{∞},P)}(·).

The sheaf G(·) = M_{C^{∞}}(·)/N_{C^{∞}}(·) is the sheaf of special or simplified Colombeau algebras of generalized functions [1, 7, 16, 18].

Example 2.2. Take for E(·) the presheaf H^{∞}(·) = D_{L_{2}}(·), with

H^{∞}(Ω) = ⋂_{m∈ℕ} H^{m}(Ω), H^{m}(Ω) = W^{m,2}(Ω).

From Sobolev inequalities, it follows that H^{∞}(Ω) is continuously embedded into C^{∞}(Ω). We may suppose a priori that elements of H^{∞}(Ω) are C^{∞}. H^{∞}(Ω) is endowed with the family of norms P_{L_{2}}(Ω) = (∥ · ∥_{m,Ω})_{m∈ℕ} defined by

∥f∥_{m,Ω} = sup_{|α|⩽m} ∥∂^{α}f∥_{L^{2}(Ω)} .

We set

M_{H}(·) = M_{(H^{∞},P_{L_{2}} )}(·), N_{H}(·) = N_{(H^{∞},P_{L_{2}} )}(·).

The presheaf G_{H}(·) = M_{H}(·)/N_{H}(·) is a presheaf of Sobolev generalized functions.

For the following example, we set for f ∈ C^{∞}(Ω), r ∈ ℤ and l ∈ ℕ,

p_{r,l}(f) = sup_{x∈Ω, |α|⩽l}⟨x⟩^{r}|∂^{α}f(x)| with ⟨x⟩ = (1 + |x|^{2})^{1/2}.

Example 2.3. Take E(·) = S(·), the presheaf of rapidly decreasing smooth functions. For any Ω open subset of ℝ^{d}, the topology of S(Ω) is described by the family of semi-norms P_{S}(Ω) = (p_{q,l})_{(q,l)∈ℕ^{2}}. We set

M_{S}(·) = M_{(S,P_{S})}(·), N_{S}(·) = N_{(S,P_{S})}(·).

The presheaf G_{S}(·) = M_{S}(·)/N_{S}(·) is the presheaf of algebras of rapidly decreasing generalized functions [3, 5, 6, 17, 19].

Remark 2.1. More general constructions can be given, for example if E(Ω) is a projective or inductive limit of topological algebras. We refer the reader to [4] for these cases.

2.3. Topology on G(·) . We follow [4] and use the notations of Subsection 2.1. Set, for (f_{ε})_{ε}, (g_{ε})_{ε} ∈ E(Ω)^{(0,1]} and i ∈ I(Ω),

∥f_{ε}∥_{i} = lim sup_{ε→0} p_{i}(f_{ε})^{| ln ε|^{−1}}

and d_{i}(f_{ε}, g_{ε}) = ∥f_{ε} − g_{ε}∥_{i}. We get [4, Proposition-definition 2]

M_{(E,P)}(Ω) = {(f_{ε})_{ε} ∈ E(Ω)^{(0,1]} | ∀i∈I ∥f_{ε}∥_{i} < +∞},
N_{(E,P)}(Ω) = {(f_{ε})_{ε} ∈ E(Ω)^{(0,1]} | ∀i∈I ∥f_{ε}∥_{i} = 0}.

The family (d_{i})_{i∈I(Ω)} defines a family of ultrapseudometrics on M_{(E,P)}(Ω), inducing on M_{(E,P)}(Ω) the structure of a topological ring such that the intersection of neighborhoods of 0 is equal to N_{(E,P)}(Ω). Thus, this topology transfers to the factor space G(Ω) which turns out to be a topological ring, and a topological algebra over the factor ring ℂ̃ = M(ℂ)/N (ℂ). (By setting ∥r_{ε}∥′ = lim sup_{ε→0} |r_{ε}|^{| ln ε|^{−1}}, one easily get that M(ℂ) (resp. N (ℂ)) = {(r_{ε})_{ε} ∈ 𝕂^{(0,1]} | ∥r_{ε}∥′ < +∞ (resp. = 0)}. This topological structure turns ℂ̃ into a topological ring. This topology coincides with the sharp topology, usually defined in terms of valuations [17, 18].

3. Temperate generalized functions

3.1. Classical construction [7, 15, 17]. We recall that

O_{M}(Ω) = {f ∈ C^{∞}(Ω) | ∀l ∈ℕ ∃q ∈ℕ p_{−q,l}(f) < +∞}.

Define

M_{τ} (Ω) = {(f_{ε})_{ε} ∈ O_{M}(Ω)^{(0,1]} | ∀l ∈ℕ ∃q ∈ℕ ∃m∈ℕ p_{−q,l}(f_{ε}) = o(ε^{−m})},
N_{τ} (Ω) = {(f_{ε})_{ε} ∈ O_{M}(Ω)^{(0,1]} | ∀l ∈ℕ ∃q ∈ℕ ∀m∈ℕ p_{−q,l}(f_{ε}) = o(ε^{m})}.

One can show that M_{τ} (Ω) is a subalgebra of O_{M}(Ω)^{(0,1]} and N_{τ} (Ω) an ideal of M_{τ} (Ω). The algebra G_{τ} (Ω) = M_{τ} (Ω)/N_{τ} (Ω) is called the algebra of tempered generalized functions.

3.2. New construction. The topology of O_{M}(Ω) may be described by the non-countable family of semi-norms P_{O_{M}} (Ω) = (ν_{φ,l})_{(φ,l)∈S(Ω)×ℕ} defined by

ν_{φ,l}(f) = sup_{x∈Ω,|α|⩽l} |φ(x)∂^{α}f(x)|.

Proposition 3.1. O_{M}(Ω) endowed with the family P_{O_{M}} (Ω) is a topological algebra.

This result is classical. For the continuity of the product, one establishes the property

∀(φ, l)∈S(Ω)×ℕ ∃ψ ∈S(Ω) ∃C >0 ∀(f, g)∈O_{M}(Ω)^{2} ν_{φ,l}(fg) ⩽ Cν_{ψ,l}(f)ν_{ψ,l}(g),

which is a consequence of the following:

Lemma 3.1. For any ψ ∈ C^{0}(Ω) with positive values such that, for any q > 0, p_{q,0}(ψ) < +∞ there exists φ ∈ S(Ω) such that ψ ⩽ φ.

With the previous notations, we set

M_{O_{M}} (Ω) = M_{(O_{M} ,P_{O_{M}})}(Ω)

= {(f_{ε})_{ε} ∈ O_{M}(Ω)^{(0,1]} | ∀φ∈S(Ω) ∀l ∈ℕ ∃m∈ℕ ν_{φ,l}(f_{ε}) = o(ε^{−m})},

N_{O_{M}} (Ω) = N_{(O_{M} ,P_{O_{M}})}(Ω)

= {(f_{ε})_{ε} ∈ O_{M}(Ω)^{(0,1]} | ∀φ∈S(Ω) ∀l ∈ℕ ∀m∈ℕ ν_{φ,l}(f_{ε}) = o(ε^{m})}.

Proposition 3.2. We have M_{O_{M}} (ℝ^{d}) = M_{τ} (ℝ^{d}) and N_{O_{M}} (ℝ^{d}) = N_{τ} (ℝ^{d}).

Proof. From the definitions, we immediately get that M_{τ} (ℝ^{d}) ⊂ M_{O_{M}} (ℝ^{d}) (resp. N_{τ} (ℝ^{d}) ⊂ N_{O_{M}} (ℝ^{d}).) For the inverse inclusions, we begin by proving that, for (f_{ε})_{ε} ∈ M_{O_{M}} (ℝ^{d}), (f_{ε})_{ε} ∈ M_{τ} (ℝ^{d}) if, and only if, (f_{ε})_{ε} satisfies the following characteristic property

(3.1) ∀α∈ℕ^{d} ∃q ∈ℕ ∃m∈ℕ ∃ε_{0} ∈(0, 1] ∃r >0 ∀ε∈(0, ε_{0}] ∀x∉B(0, r)

⟨x⟩^{−q}|∂^{α}f_{ε}(x)| ⩽ ε^{−m}.

Indeed, we can easily see that if (f_{ε})_{ε} ∈ M_{τ} (ℝ^{d}), the property (3.1) holds even if (f_{ε})_{ε} ∉ M_{O_{M}} (ℝ^{d}). Conversely suppose that (f_{ε})_{ε} ∈ M_{O_{M}} (ℝ^{d}) and that (3.1) holds. Fix α ∈ ℕ^{d}. There exist q ∈ ℕ, m ∈ ℕ, ε_{0} ∈ (0, 1], r > 0 such that (3.1) holds. Let us show that ⟨x⟩^{−q} |∂^{α}f_{ε}(x)| ⩽ ε^{−m′} for some m′ ∈ ℕ, ε small enough and all x ∈ B(0, r). Consider φ ∈ D(ℝ^{d}) with 0 ⩽ φ ⩽ 1 and φ ≡ 1 on B(0, r). According to the definition of M_{O_{M}} (ℝ^{d}), used with l = |α|, there exists ε′_{0} ∈ (0, 1] such that

∀ε∈(0, ε′_{0}] ∀x∈B(0, r) ⟨x⟩^{−q} |∂^{α}f_{ε}(x)| ⩽ |∂^{α}f_{ε}(x)| ⩽ ν_{φ,l}(f_{ε}) ⩽ ε^{−m′}.

Taking ε_{1} = min(ε_{0}, ε′_{0}), m_{1} = max(m, m′), we obtain that

∀ε∈(0, ε_{1}] ∀x∈ℝ^{d}, ⟨x⟩^{−q} |∂^{α}f_{ε}(x)| ⩽ ε^{−m_{1}}.

From this last property, a classical argument shows that

p_{−q,l}(f_{ε}) = sup_{x∈ℝ^{d}, |α|⩽l}⟨x⟩^{−q} |∂^{α}f_{ε}(x)| = o (ε^{−M})

provided M is chosen big enough. Thus (f_{ε})_{ε} ∈ M_{τ} (ℝ^{d}).

Let us return to the proof of the inclusion M_{O_{M}} (ℝ^{d}) ⊂ M_{τ} (ℝ^{d}). Take (f_{ε})_{ε} ∈ M_{O_{M}} (Ω) and suppose that (3.1) does not hold. There exist α ∈ ℕ^{d} for which we can built by induction a sequence (ε_{q})_{q⩾0} with ε_{q} ⟶^{q→+∞} 0 and a sequence (x_{q})_{q⩾0} with |x_{q+1}| ⩾ |x_{q}| + 2 such that

⟨x_{q}⟩^{−q} |∂^{α}f_{ε_{q}}(x_{q})| > ε_{q}^{−q}.

Consider θ ∈ D (ℝ^{d}) with supp θ ⊂ B(0, 1), 0 ⩽ θ ⩽ 1 and, say, θ(0) = 1. Set

φ(x) = ∑_{q=0}^{+∞}⟨x_{q}⟩^{−q}θ(x − x_{q}).

Following [11], it can be verified that φ belongs to S(ℝ^{d}). (Note that

supp(x ↦ θ(x − x_{q})) ∩ supp(x ↦ θ(x − x_{q′})) = ∅ for q ≠ q′,

justifying the choice of (x_{q})_{q⩾0}.) We have

φ(x_{q}) |∂^{α}f_{ε_{q}}(x_{q})| > ε_{q}^{−q}θ(0) = ε_{q}^{−q}.

Thus, for all q ∈ ℕ, ν_{φ,|α|}(f_{ε_{q}}) > ε_{q}^{−q}, with ε_{q} ⟶^{q→+∞} 0 in contradiction with the definition of M_{O_{M}} (Ω). Finally (f_{ε})_{ε} ∈ M_{τ} (ℝ^{d}). The proof of the inclusion N_{O_{M}} (ℝ^{d}) ⊂ N_{τ} (ℝ^{d}) is quite similar. □

Corollary 3.1. We have G_{τ} (ℝ^{d}) = M_{O_{M}} (ℝ^{d})/N_{O_{M}} (ℝ^{d}).

Remark 3.1. (i) Following subsection 2.3, G_{τ} (ℝ^{d}) is naturally equipped with a topological structure, given by the non countable family of ultrapseudometrics (d_{φ,l})_{(φ,l)∈S(Ω)×ℕ} defined by

d_{φ,l}(f, g) = lim sup_{ε→0} ν_{φ,l}(f_{ε} − g_{ε})^{| ln ε|^{−1}}, where (f_{ε})_{ε} ∈ f, (g_{ε})_{ε} ∈ g.

(ii) According to Proposition 2.1, G_{τ} (·) is a presheaf of algebras. However, the localization principle (F_{1}) does not hold for G_{τ} (·) as shown by the following example.

Example 3.1. We adapt a classical example, which was first used to show that G_{τ} (·) is not a subpresheaf of G(·) [17]. Consider Ψ ∈ D(ℝ) such that 0 ⩽ Ψ ⩽ 1 and, say, Ψ(0) = 1. Set f_{ε}(·) = Ψ(· − | ln ε|^{1/2}). Obviously (f_{ε})_{ε} ∈ M_{O_{M}} (ℝ),

defining f ∈ G_{τ} (ℝ). Consider Ω_{h} = ]−h, h[ for h ∈ ℕ^{∗}. As | ln ε|^{1/2} ⟶^{ε→0} +∞, we have f|_{Ω_{h}} = 0. However, f_{ε}(| ln ε|^{1/2}) = Ψ(0) = 1. Take φ ∈ S(ℝ) defined by φ(x) = exp(−x^{2}). We have φ(| ln ε|^{1/2})f_{ε}(| ln ε|^{1/2}) = ε. Thus ν_{φ,0}(f_{ε}) ⩾ ε and (f_{ε})_{ε} ∉ N_{O_{M}} (ℝ). Therefore f is non equal to 0 on ℝ = ⋃_{h∈ℕ} Ω_{h}.

We set

N_{O_{M} ,0} = {(f_{ε})_{ε} ∈M_{O_{M}} (Ω) | ∀φ∈S(Ω) ∀m∈ℕ : ν_{φ,0}(f_{ε}) = o(ε^{m})} .

We have the same result as theorem 1.2.27 in [7] concerning N_{τ} (·) (the proof is similar):

Proposition 3.3. If the open set Ω is a product of d intervals, then N_{O_{M}} (Ω) is equal to N_{O_{M} ,0}(Ω) ∩ M_{O_{M}} (Ω).

This result renders easier the proof of:

Proposition 3.4. [7, 18] Consider ρ ∈ S(ℝ^{d}) such that

(3.2)  ∫ ρ(x) dx = 1, ∀α∈ℕ^{d} ∖ {0}, ∫ x^{α}ρ(x) dx = 0.

Set

(3.3)  ρ_{ε}(x) = ε^{−d}ρ (x/ε) .

(i) The map

σ_{τ} : O_{M}(ℝ^{d}) → G_{τ} (ℝ^{d}), u ↦ (u)_{ε} + N_{O_{M}} (ℝ^{d})

is an embedding of differential algebras.

(ii) The map

ι_{τ} : S′(ℝ^{d}) → G_{τ} (ℝ^{d}), T ↦ (T ∗ ρ_{ε})_{ε} + N_{O_{M}} (ℝ^{d})

is an embedding of differential vector spaces.

(iii) Moreover, ι_{τ} |_{O_{M} (ℝ^{d})} = σ_{τ}, which means that the following diagram is commutative:

(3.4)  \begin{matrix}O_{M}(ℝ^{d}) & ⟶^{σ_{τ}} & G_{τ} (ℝ^{d}). \\ ↘ & ^{ι_{τ}}↗ \\ S′(ℝ^{d}) \end{matrix}

(The arrow without label is the classical embedding of O_{M}(ℝ^{d}) into S′(ℝ^{d}).)

The assertion (iii) is an improvement of the classical one which only gives ι_{τ} |_{O_{M} (ℝ^{d})} = σ_{τ} |_{O_{C}(ℝ^{d})}.

Proof. The assertion (i) is an application of the general principle recalled in subsection 2.1 to the case of O_{M}(·). We refer the reader to [7, 18] for the proof of the assertion (ii) which uses mainly the structure of elements of S′(ℝ^{d}). We shall prove the assertion (iii) in the case d = 1; the general case only differs by more complicate algebraic expressions. Let f be in O_{M}(ℝ) and set Δ = ι_{τ} (f) − σ_{τ} (f). One representative of Δ is given by (Δ_{ε} : ℝ → M_{O_{M}} (ℝ))_{ε} with

Δ_{ε}(y) = (f ∗ θ_{ε})(y) − f(y) = ∫ f(y − x)ρ_{ε}(x) dx − f(y)
= ∫ (f(y − x) − f(y)) ρ_{ε}(x) dx = ∫ (f(y − εu) − f(y)) ρ(u) du

since ∫ ρ_{ε}(x) dx = 1. Let k be a positive integer. Taylor’s formula gives

f(y − εu) − f(y) = ∑_{i=1}^{k} \frac{(−εu)^{i}}{i!} f^{(i)}(y) + \frac{(−εu)^{k}}{k!} ∫_{0}^{1} f^{(k+1)}(y − εuv)(1 − v)^{k}dv.

Using ∫ x^{i}ρ_{ε}(x) dx = 0, for i ∈ {1, …, k}, we get

Δ_{ε}(y) = ∫ \frac{(−εu)^{k}}{k!} ∫_{0}^{1} f^{(k+1)}(y − εuv)(1 − v)^{k}dv ρ(u) du.

As f ∈ O_{M}(ℝ), there exists p ∈ ℕ and C_{3} > 0 such that |f^{(k+1)}(ξ)| ⩽ C_{3}(1 + |ξ|)^{p}. Thus

∀(u, y)∈ℝ^{2} ∀v ∈[0, 1] ∀ε∈(0, 1] |f^{(k+1)}(y − εuv)| ⩽ C_{3}(1 + |y|)^{p} (1 + |u|)^{p}.

As ρ is rapidly decreasing, the integral ∫ |u|^{k}(1 + |u|)^{p}ρ(u) du converges and

|Δ_{ε}(y)| ⩽ \frac{ε^{k}}{k!} C_{3}(1 + |y|)^{p} ∫ |u|^{k}(1 + |u|)^{p}ρ(u) du ⩽ ε^{k}C_{4}(1 + |y|)^{p}.

Consider φ ∈ S(ℝ). The function (1 + | · |)^{p} |φ(·)| is bounded. Thus

sup_{y∈ℝ} |φ(y)Δ_{ε}(y)| = o(ε^{k}) as ε → 0.

As (Δ_{ε})_{ε} ∈ M_{O_{M}} (ℝ) and sup_{y∈ℝ} |φ(y)Δ_{ε}(y)| = o(ε^{k}), we can conclude without estimating the derivatives that (Δ_{ε})_{ε} ∈ N_{O_{M}} (ℝ) by using Proposition 3.3. □

4. Fourier transform and space of rapidly decreasing generalized distributions

There is no need to recall the importance of spectral analysis, based on the Fourier transform in the theories of distributions [10] and Colombeau generalized functions (See, for example, [3, 8, 9, 19]). In this section, we first define in a new way the Fourier transform of elements of G_{τ} (ℝ^{d}) in relationship with a (new) space of generalized distributions.

The Fourier transform of elements of G_{τ} (Ω) has already been defined with the help of ad hoc cutoff functions [17, 18]. More precisely, one sets

∀u∈G_{τ} (Ω), F(u) = ∫ e^{−ıxy}u_{ε}(y)ρ̂(εy) dy + N_{τ} (ℝ^{d}) with (u_{ε})_{ε} ∈ u,

where ρ ∈ S(ℝ^{d}) satisfies (3.2) so that ρ̂(εy) ⟶^{ε→0} 1. One shows that this definition makes sense for F(u) does not depend on the chosen representative (u_{ε})_{ε} ∈ u. Analogously, one defines F^{−1}. However, this Fourier transform lacks some expected properties such as F^{−1} ∘ F = id. (The reader will find a complete discussion on this subject in [18].)

Recalling that O_{M}(ℝ^{d}) is the Fourier image of O′_{C}(ℝ^{d}) (and reciprocally), we prefer here to construct the Fourier transform starting from this fact since G_{τ} (ℝ^{d}) is directly built on O_{M}(ℝ^{d}). In other words, we consider G_{τ} (ℝ^{d}) as a space of multiplicators and we introduce a space of convolutors, both of them being linked as usual by the Fourier Transform and its inverse.

Set

M_{O′_{C}} (ℝ^{d}) = {(T_{ε})_{ε} ∈ O′_{C}(ℝ^{d})^{(0,1]} | (F^{−1}(T_{ε}))_{ε} ∈ M_{O_{M}} (ℝ^{d})} ,
N_{O′_{C}} (ℝ^{d}) = {(T_{ε})_{ε} ∈ O′_{C}(ℝ^{d})^{(0,1]} | (F^{−1}(T_{ε}))_{ε} ∈ N_{O_{M}} (ℝ^{d})} .

From the linearity of F^{−1} and the linear properties of the spaces M_{O_{M}} (ℝ^{d}) and N_{O_{M}} (ℝ^{d}), we immediately get that M_{O′_{C}} (ℝ^{d}) is a ℂ̃-submodule (resp. ℂ-subvector space) of O′_{C}(ℝ^{d})^{(0,1]} and N_{O′_{C}} (ℝ^{d}) a ℂ̃-submodule (resp. ℂ-subvector space) of M_{O′_{C}} (ℝ^{d}).

Definition 4.1. The factor space G_{O′_{C}} (ℝ^{d}) = M_{O′_{C}} (ℝ^{d})/N_{O′_{C}} (ℝ^{d}) is called the space of rapidly decreasing generalized distributions.

With this previous material, the Fourier transform of elements of G_{τ} (ℝ^{d}) (we keep the notation F) F : G_{τ} (ℝ^{d}) → G_{O′_{C}} (ℝ^{d}) is well defined by

∀u∈G_{τ} (ℝ^{d}), F(u) = F(u_{ε}) + N_{O′_{C}} (ℝ^{d}) with (u_{ε})_{ε} ∈ u.

The inverse Fourier transform from G_{O′_{C}} (ℝ^{d}) into G_{τ} (ℝ^{d}) is defined analogously. This Fourier transform has the expected classical properties as they only have to be verified component-wise.

Proposition 4.1. (i) The map

σ_{O′_{C}} : O′_{C}(ℝ^{d}) → G_{O′_{C}} (ℝ^{d}), u ↦ (u)_{ε} + N_{O′_{C}} (ℝ^{d})

is an embedding of ℂ-vector spaces.

(ii) Take, as in Proposition 3.4, ρ ∈ S(ℝ^{d}) satisfying (3.2) and (ρ_{ε})_{ε} defined by (3.3). The map

ι_{O′_{C}} : S′_{C}(ℝ^{d}) → G_{O′_{C}} (ℝ^{d}), T ↦ (T ρ̂(ε·))_{ε} + N_{O′_{C}} (ℝ^{d})

is an embedding of ℂ-vector spaces.

The proof of (i) is immediate, whereas (ii) is obtained by “taking the Fourier transform image of the diagram (3.4)” in Proposition 3.4. In fact, the following diagram is commutative

O_{M}(ℝ^{d}) ⟶^{σ_{τ}}  G_{τ} (ℝ^{d})
↘ ↗^{ι_{τ}}
S′(ℝ^{d})
F ↓↑^{F^{−1}}  ^{F} ↓↑^{F^{−1}}  ^{F} ↓↑^{F^{−1}}
S′(ℝ^{d})
↗  _{ι_{O′_{C}}}↘
O′_{C}(ℝ^{d}) ⟶_{σ_{O′_{C}}}  G_{O′_{C}} (ℝ^{d})

(The arrows without labels are the usual embeddings of O_{M}(ℝ^{d}) and O′_{C}(ℝ^{d}) into S′(ℝ^{d}).)

Remark 4.1. Following ideas of Jean-André Marti (private communication), the Fourier transform in G_{τ} (ℝ^{d}) can be used to define Sobolev type subspaces of G_{τ} (ℝ^{d}). More precisely, we say that (u_{ε})_{ε} ∈ M_{O_{M}} (ℝ^{d}) is of H^{s} type if, for all ε ∈ (0, 1], ⟨·⟩^{s}(·)û_{ε}(·) ∈ L^{2}(ℝ^{d}) and (∥⟨·⟩^{s}û_{ε}(·)∥_{L^{2}})_{ε} ∈ M(ℝ). One shows that the space H^{s}(ℝ^{d}) is embedded into G_{τ}^{(s)}(ℝ^{d}) through ι_{τ} defined in Proposition 3.4. This will be used in a forthcoming paper to introduce a H^{s} local and microlocal analysis in spaces of generalized functions.

5. Introduction to regularity theory

5.1. The spaces M_{τ}^{∞}(Ω) and G_{τ}^{∞}(Ω). In analogy to the definition of G^{∞} [7, 16], we set

M_{O_{M}}^{∞} (Ω) = {(f_{ε})_{ε} ∈ O_{M}(Ω)^{(0,1]} | ∀φ∈S(Ω) ∃m∈ℕ ∀l ∈ℕ ν_{φ,l}(f_{ε}) = o(ε^{−m})}.

It is easy to check that M_{O_{M}}^{∞} (·) is a subpreasheaf of algebras of M_{O_{M}} (·). From this, we get:

Proposition 5.1. G_{τ}^{∞}(·) = M_{O_{M}}^{∞} (·)/N_{O_{M}}^{∞} (·) is a subpresheaf of differential algebras of G_{τ} (·).

Going further with the above mentioned analogy, we recall that G^{∞}(ℝ^{d}) ∩ D′(ℝ^{d}) = C^{∞}(ℝ^{d}) [16]. This result can be interpreted as follows: The subsheaf G^{∞} of regular sections of G is such that the sheaf embedding G^{∞} → G is the natural extension of the classical one C^{∞} → D′. We have here the same situation (modulo the fact G_{τ} (·) is only a presheaf) that given by:

Proposition 5.2. G_{τ}^{∞}(ℝ^{d}) ∩ S′(ℝ^{d}) = O_{M}(ℝ^{d}).

The result should be understood as follows. For u ∈ S′(ℝ^{d}), if ι_{τ} (u) is in G_{τ}^{∞}(ℝ^{d}), then u is in O_{M}(ℝ^{d}).

Proof. Take u ∈ S′(ℝ^{d}) such that ι_{τ} (u) is in G_{τ}^{∞}(ℝ^{d}). Then (u ∗ ρ_{ε})_{ε} is in M_{O_{M}}^{∞} (ℝ^{d}). Recall that

u ∈ O_{M}(ℝ^{d}) ⇔ û ∈ O′_{C}(ℝ^{d}) ⇔ ∀ψ ∈ S(ℝ^{d}), û ∗ ψ ∈ S(ℝ^{d}).

So consider ψ ∈ S(ℝ^{d}). We are going to show that ⟨·⟩^{m}û ∗ ψ(·) is bounded for all m ∈ ℕ. We have

(5.1)  û ∗ ψ = (û(1 − ρ̂_{ε})) ∗ ψ + (û ρ̂_{ε}) ∗ ψ.

Recalling that ρ_{ε} = ε^{−d}ρ(·/ε), we easily get that ρ̂_{ε}(·) = ρ̂(ε·). Note also that ρ̂(0) = ∫ ρ(x) dx = 1. Thus

1 − ρ̂_{ε}(x) = −ε ∫_{0}^{1} ∇ρ̂(εxt) · x dt = εB(ε, x).

As ρ̂ is rapidly decreasing, there exists C > 0 such that |∇ρ̂(εxt) · x| ⩽ C⟨x⟩ for all (ε, x, t) ∈ (0, 1] × ℝ^{d} × [0, 1]. The same holds for the derivatives with respect to x and, thus, for the function B and its derivatives. From this, for example by using the structure of elements of S′(ℝ^{d}), it can be shown that û(1 − ρ̂_{ε})) ∗ ψ satisfies

(5.2)  ∀x∈ℝ^{d} |((û(1 − ρ̂_{ε})) ∗ ψ)(x)| ⩽ C_{0}ε⟨x⟩^{q},

for some C_{0} > 0 and q not depending on ε.

Consider l ∈ ℕ and β ∈ ℕ^{d} with |β| = l. We have, for all x ∈ ℝ^{d},

(ıx)^{β}((û ρ̂_{ε}) ∗ ψ)(x) = (ıx)^{β}F ((u ∗ ρ_{ε}) F^{−1}(ψ)) (x)
= F ((∂^{β}(u ∗ ρ_{ε})) F^{−1}(ψ)) (x).

Applying the definition of M_{O_{M}}^{∞} (ℝ^{d}) for (ρ_{ε} ∗ u)_{ε} with φ = ⟨·⟩^{(d+1)/2}F^{−1}(ψ), we get the existence of N (only depending on (ρ_{ε} ∗ u)_{ε} and ψ) and C_{1} > 0 such that

∀y ∈ℝ^{d} |(∂^{β}(u ∗ ρ_{ε}))(y)F^{−1}(ψ)(y)| ⩽ C_{1}⟨y⟩^{−(d+1)/2}ε^{−N} for ε small enough.

Thus, we get the existence of C_{2} > 0 such that

|(ıx)^{β}((û ρ̂_{ε}) ∗ ψ)(x)| = |F((∂^{β}(u ∗ ρ_{ε})) F^{−1}(ψ))(x)| ⩽ C_{2} ε^{−N}

for ε small enough and all x ∈ ℝ^{d}. Using a classical argument, we get a constant C_{3} > 0 such that

∀x∈ℝ^{d} ⟨x⟩^{l}|(û ρ̂_{ε}) ∗ ψ)(x)| ⩽ C_{3} ε^{−N} for ε small enough.

Fix m ∈ ℕ and take l = m + (m + q)N. Writing the previous inequality in the form ⟨x⟩^{m} |(û ρ̂_{ε}) ∗ ψ)(x)| ⩽ C_{3}(ε⟨x⟩^{m+q})^{−N}, using (5.2) and finally inserting these intermediates steps in (5.1), we get

∀x∈ℝ^{d} |⟨x⟩^{m}(û ∗ ψ)(x)| = C(ε⟨x⟩^{m+q} + (ε⟨x⟩^{m+q})^{−N}) = T (ε⟨x⟩^{m+q})

for ε smaller than some ε_{0} and some C > 0. Thus, for x such that ⟨x⟩^{m+q} ⩾ ε_{0}^{−1}, take ε_{x} such that ε_{x} = ⟨x⟩^{−m−q} to obtain that |⟨x⟩^{m}(û ∗ ψ)(x)| ⩽ T (1). From this, it follows that the function |⟨·⟩^{m}(û ∗ ψ)| is bounded on ℝ^{d}, as claimed. □

5.2. Regularities for temperate generalized functions. As in the presheaf G_{τ} (·) the localization principle (F_{1}) is not fulfilled, we are not in the situation to apply the results of [14] concerning singular supports and their properties. Indeed, following the notations of the quoted paper, we need a presheaf A(·) (of vector spaces, of algebras,…) with localization principle and a subpresheaf B(·) of A(·) to define the B-singular support of a section u ∈ A(Ω). Thus, as it is done in [13] for the definition of the presheaf G^{L}(·), we shall start from the sheaf G(·) and define some regular subpresheaves of it. More precisely, for the two cases B(·) = G_{τ} (·), G_{τ}^{∞}(·), we set N_{O_{M} ,∗}^{♯}(·) = N (·) ∩ M_{O_{M}}^{♯} (·), where the symbol ♯ means successively the blank character and ∞. According to the results recalled in Section 2 and to the inclusion M_{O_{M}} (ℝ^{d}) ⊂ M_{C^{∞}}(ℝ^{d})), G_{τ,∗}^{♯}(·) = M_{O_{M}}^{♯} (·)/N_{O_{M} ,∗}^{♯}(·) is a subpresheaf of G(·). Using the framework and the results of [14], we say that the elements of G_{τ,∗}^{♯}(Ω) are G_{τ}^{♯}-regular elements of G(Ω). For u ∈ G(Ω), we can define O_{τ}^{♯} (u), the set of all x ∈ Ω such that u is G_{τ}^{♯}-regular at x, that is

O_{τ}^{♯} (u) = {x∈Ω ∃V ∈V_{x} : u|_{V} ∈ G_{O_{M,∗}}^{♯}(V)}

(V_{x} being the family of all open neighborhoods of x.) The G_{τ}^{♯}-singular support of u is the well defined set S_{τ}^{♯}(u) = singsupp_{G_{τ}^{♯}} u = Ω ∖ O_{τ}^{♯} (u) and has the following properties [14]:

Proposition 5.3. Consider u, v ∈ G(Ω), α in ℕ^{d} and g in G_{τ,∗}^{♯}(Ω). We have:

(i) S_{τ}^{♯}(u + v) ⊂ S_{τ}^{♯}(u) ∪ S_{τ}^{♯}(v);  (iii) S_{τ}^{♯}(∂^{α}u) ⊂ S_{τ}^{♯}(u);
(ii)  S_{τ}^{♯}(uv) ⊂ S_{τ}^{♯}(u) ∪ S_{τ}^{♯}(v);  (iv) S_{τ}^{♯}(gu) ⊂ S_{τ}^{♯}(u).

From these properties, one easily gets:

Corollary 5.1. Let P (∂) = ∑_{|α|⩽m} C_{α}∂^{α} be a differential polynomial with coefficients in G_{τ,∗}(Ω) (resp. G_{τ,∗}^{∞}(Ω)). For any u ∈ G(Ω), we have

S_{τ} (P (∂)u) ⊂ S_{τ} (u) (resp. S_{τ}^{∞}(P (∂)u) ⊂ S_{τ}^{∞}(u)).

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Laboratoire Analyse, Optimisation, Contrôle Faculté des sciences – Université des Antilles et de la Guyane Pointe à Pitre Guadeloupe (France)

antoine.delcroix@univ-ag.fr