PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 64 (78), 1998, 85–97

MICROLOCAL PROPERTIES OF ULTRADISTRIBUTIONS. COMPOSITION AND KERNEL TYPE OPERATORS

Stevan Pilipović

Communicated by Mileva Prvanoviv́

Abstract. The composition of an ultradistribution and a real analytic function as well as kernel type operators are analyzed. The suppleness of various spaces of microfunctions is given.

0. Introduction

The microlocal analysis of various classes of functions, generalized functions and hyperfunctions has a long history. Essentially it is related to the qualitative analysis of solutions of linear and non-linear equations and because of that it is a wast and reach theory.

Hörmander has studied in [9, Section 8] the wave front set of distributions and Roumieu type ultradistributions while Komatsu has investigated in [12] the microlocal properties of sheaves C^{∗} and C_{∗} which correspond to Beurling and Roumieu ultradistribution spaces and the spaces of corresponding ultradifferentiable functions. We refer to Sato, Kawai, Kashiwara [21] for the hyperfunction theory and the theory of microfunctions.

Note that Matsuzawa has developed in [14] the hyperfunction theory and the microlocal analysis by mean of the Gauss kernel. In this context we refer also to papers [4], [5]. The analysis of ultradistribution spaces is given in many papers (among others are [2], [11–13], [15], [18], [20] and [22]).

In this paper we continue our investigations of [18] and give the assertions which are known for distributions but need to be carefully examined for ultradistributions (see also [11] and [19]). In order to make clear our contribution to the theory, we note that uniform estimations related to all the derivatives of a function are the main problems in our proofs.

AMS Subject Classification (1991): 32A40, 46F15

In Sections 1 and 2 we recall the definitions of ultradistributional singular spectrums SS_{∗}, SS^{∗} ([12], [6]) and the definition and the properties of tempered ultradistribution spaces S′^{∗} ([3], [13], [16]). The composition of a u ∈ D′^{∗} with a real analytic function with the singular spectrum disjoint of the set of normals of f is given in Section 3. Note that this composition can not be made by using stationary phase arguments as in the distribution theory [9]. The microlocal analysis of a linear operator with a kernel K ∈ D′^{∗}(X × Y) is given in Section 4. For the decomposition of a microlocal support we refer to [1], [6], and [7]. In Section 5 we recall our definitions of various spaces of microfunctions and the results related to the suppleness of corresponding sheaves.

1. Notation and notions

As usual, M_{p}, p ∈ ℕ_{0}, denotes a sequence of positive numbers with M_{0} = 1. We refer to [11] for the meaning of conditions (M.1), (M.2)’, (M.2), (M.3)’

and (M.3). Also we use the following one [15]:

(M.1)^{∗} M_{p−1}^{∗}M_{p+1}^{∗} ≤ M_{p}^{∗}, p ∈ ℕ, where M_{0}^{∗} = 1, M_{p}^{∗} = M_{p}/p!, p ∈ ℕ.

Let M_{p} satisfy (M.1) and (M.3)’. The associated function M(ρ) and the growth function M̃(ρ) related to M_{p} are defined by

M(ρ) = sup_{p∈ℕ_{0}} ln \frac{ρ^{p}}{M_{p}} , M̃(ρ) = sup_{p∈ℕ_{0}} ln \frac{ρ^{p}}{M_{p}^{∗}} , ρ > 0.

An open set in ℝ^{n} will be always denoted by Ω; K ⊂⊂ Ω means that K is a compact subset of Ω. Recall,

∥φ∥_{K,h,M_{p}} = sup_{x∈K,α∈ℕ_{0}^{n}} \frac{| φ^{(α)}(x) |}{h^{|α|}M_{|α|}}, φ ∈ C^{∞}(Ω).

We use the symbol ∗ for both (M_{p}) and {M_{p}}. For the definitions of E^{∗}(Ω), D_{K}^{∗}(Ω), D^{∗}(Ω) and their strong duals we refer to [11]. Throughout the paper we will assume that (M.1), (M.2)’ and (M.3)’ hold.

Presheaves U → E^{∗}(U), U → D^{′∗}(U), U ⊂ ℝ^{n}, are sheaves. These sheaves are known to be soft, that is, if f is a section on a closed set it can be extended on the whole space.

There exist injections

E^{∗} ↪ B, D^{′∗} ↪ B.

Let N_{p} be a sequence of positive numbers which satisfies (M.1), (M.2)’, (M.3)’ and N_{0} = 1. Then [11]

(N_{p}) ≤ (M_{p}) (resp., {N_{p}} ≤ {M_{p}})

if there are constants L > 0 and C > 0 (resp., for every ϵ > 0 there is C_{ϵ} > 0) such that

N_{p} ≤ CL^{p}M_{p} (resp., N_{p} ≤ C_{ϵ}ϵ^{p}M_{p}), p ∈ ℕ_{0}.

Let f ∈ D′^{†}, where † = N_{p} ≤ M_{p}. Then (x, ω) ∈ S^{∗}Ω = Ω × S^{n−1} (the cosphere bundle) is not in SS_{∗}f (resp., not in SS^{∗}f) iff there exist a neighborhood U ⊂ Ω of x and a conic neighborhood Γ of ω of the form

Γ = {ξ ≠ 0; | ξ/| ξ | − ω |< η}

such that for every ϕ ∈ D^{∗}(U) the following holds:

In the (M_{p})-case, for every ϵ > 0 there is C_{ϵ} > 0 such that

| ϕ̂f(ξ) |≤ C_{ϵ}e^{−M(ϵ|ξ|)}, ξ ∈ Γ

(resp., there are a k > 0 and a C > 0 such that

| ϕ̂f(ξ) |≤ Ce^{M(k|ξ|)}, ξ ∈ Γ.)

In the {M_{p}}-case, there exist a k > 0 and a C > 0 such that

| ϕ̂f(ξ) |≤ Ce^{−M(k|ξ|)}, ξ ∈ Γ

(resp., for every ϵ > 0, there is a C_{ϵ} > 0 such that

| ϕ̂f(ξ) |≤ C_{ϵ}e^{M(ϵ|ξ|)}, ξ ∈ Γ.

Note, the notion SS_{{M_{p}}} is equivalent to Hörmander’s notion W F_{L}.

By using the above definitions of singular spectrums we can formulate definitions (and we will do this) according to which (x, ξ) ∈ Ω × (ℝ^{n} \ {0}) is an element of a singular spectrum defined above if (x, ξ/|ξ|) is an element of the same singular spectrum.

We denote by B(U) the space of Sato’s hyperfunctions on an open set U in ℝ^{n}. It is known that U → B(U) forms a sheaf and that this sheaf is flabby, that is, the restriction mappings S : B(U) → B(U_{1}) (U_{1} ⊂ U) are always surjective (cf. [21]).

For an open set U in a real analytic manifold M, let S^{∗}U be the cosphere bundle of U and V be the complexification of U. Then S_{U}^{∗}V can be identified with \sqrt{−1}S^{∗}U. In the last paragraph we will use the notation accomodated to Kaneko [8]. Recall, a hyperfunction f on U can be written as a formal sum of boundary values of holomorphic functions F_{j}(z) defined on infinitesimal wedges U +\sqrt{−1}Γ_{j}0, where Γ_{j} are open cones in ℝ^{n}, f(x) = ∑_{j=1}^{N} F_{j}(x + \sqrt{−1}Γ_{j}0). It is said that f is microanalytic at the point (x, \sqrt{−1}ξ∞) ∈ \sqrt{−1}S^{∗}U if, for a suitable representation of f (given above) on a neighbourhood of x, Γ_{j} ∩ {y ∈ ℝ^{n}; ⟨ξ, y⟩ > 0} = ∅ holds for all j ∈ {1,…, N}. So, in the last paragraph we will use the identification of \sqrt{−1}S^{∗}U and S^{∗}U and say that (x, \sqrt{−1}ξ∞) ∈ SSu if and only if (x, ξ) ∈ SSu. The same will be used for other types of singular supports. Thus, for an f ∈ D′^{∗}(Ω), (x, ω) ∈ S^{∗}Ω is not in SSf if (x, \sqrt{−1}ω∞) ∈ S^{∗}Ω is not in SS\{f\}, where \{f\} denotes the corresponding hyperfunction. This notion is equal to Hörmander’s W F_{A}f− the analytic wave front set of f [9, Definition 9.3.2., Theorem 9.6.3].

2. Tempered ultradistributions

Tempered ultradistribution spaces are introduced and studied in [13] and [16].

Let m > 0. The space of smooth functions φ on ℝ^{n} which satisfy

σ_{m,2}(φ) = ( ∑_{α,β∈ℕ_{0}^{n}} ∫_{ℝ^{n}} |\frac{m^{|α+β|}}{M_{|α|}M_{|β|}} (1 + |x|^{2})^{|β|/2}φ^{(α)}(x)|^{2}dx)^{1/2} < ∞,

equipped with the topology induced by the norm σ_{m,2}, is denoted by S_{2}^{M_{p},m}. The strong duals of S^{(M_{p})} = proj lim_{m→∞} S_{2}^{M_{p},m} and S^{{M_{p}}} = ind lim_{m→0} S_{2}^{M_{p},m} are called spaces of tempered ultradistributions of Beurling and Roumieu type. For every fixed p ∈ [1, ∞], the family of norms {σ_{m,2}; m > 0} is equivalent to the family of norms {σ_{m,p}; m > 0} where instead of L^{2} norm we put L^{p} norm.

S^{(M_{p})} and S^{{M_{p}}} are (F S)- and (LS)-spaces, respectively. If (M.2) holds, they are (F N)- and (LN)- spaces, respectively (for these types of spaces, see [8]) and

D^{∗} ↪ S^{∗} ↪ E^{∗}, S^{∗} ↪ S,

where we have also that injections are continuous.

An f ∈ D′^{∗} is in S′^{∗} if and only if there exists a family F_{α,β} ∈ L^{2}(ℝ^{n}), α, β ∈ ℕ_{0}^{n}, such that

f = ∑_{α,β∈ℕ_{0}^{n}}((1 + |x|^{2})^{β/2}F_{α,β})^{(α)} in S′^{∗},

and in case S′^{(M_{p})}, (resp., in the case S′^{{M_{p}}}, for every k > 0) such that

( ∑_{α,β∈ℕ_{0}^{n}} ∫_{ℝ^{n}} |\frac{M_{|α|}M_{|β|}}{k^{|α+β|}} F_{α,β}(x)|^{2})^{1/2} < ∞.

The corresponding structural theorems may be obtained by using the families of norms {ma_{m,p}; m > 0}, p ∈ [1, ∞].

If (M.1), (M.2) and (M.3) are assumed, then we have more precise structural characterisations [16].

The Fourier transformation is an isomorphism of S^{∗} onto itself.

In the next theorem we use the notation N_{p} for the sequence of positive numbers and N(ρ), Ñ(ρ) for the corresponding associated and growth function.

Theorem 1. [17] Assume that (M.1)^{∗}, (M.2) and (M.3)′ hold for N_{p} (N_{0} = 1). Let Γ be an open convex cone in ℝ^{n} and F be an analytic function in

Z = {z ∈ ℂ^{n}; Im z ∈ Γ, | Im z| < a}.

for some a > 0. Moreover, assume

|F (x + \sqrt{−1}y)| ≤ C_{a,b}e^{N(a|x|)+Ñ( \frac{b}{|y|})}, x + \sqrt{−1}y ∈ Z,

for some a > 0, b > 0 and C_{a,b} > 0, in the (N_{p})-case and for every a > 0, b > 0 there exists a C_{a,b} > 0, in the {N_{p}}-case. Then

F (x + \sqrt{−1}y) →^{S′^{†}} F (x + \sqrt{−1}0), y → 0, y ∈ Γ.

In paricular

SS_{†} ⊂ ℝ^{n} × Γ^{0} \ {0}

3. Composition with a real analytic function

We still assume that (M.1)^{∗}, (M.2) and (M.3)′ hold for M_{p}.

Let f be a real analytic mapping X → Y, where X and Y are open sets in ℝ^{m} and ℝ^{n}, respectively. We are going to define the pullbackf^{⋆}u of a u ∈ D′^{†} with suitable properties of its singular spectrums. Denote

N_{f} = {(f(x), η); ^{t}f′(x)η = 0}.

If γ is a closed conic subset of Y × (ℝ^{n} \ {0}) (conic in the second variable), denote

D′_{γ}^{∗}(Y) = {u ∈ D′^{∗}(Y); SSu ⊂ γ},
f^{⋆}γ = {(x,^{t} f′(x)η); (f(x), η) ∈ γ}.

For the composition of an ultradistribution with an analytic function we can not use the asymptotic expansion based on the stationary phase arguments as for distributions. Because of that the first part of the proof of the next theorem is completely different from the corresponding one for distributions [9].

Theorem 2. The pullback f^{⋆}u can be defined in one and only one way for all u ∈ D′^{∗}(Y) with N_{f} ∩ SSu = ∅ such that f^{⋆}u = u ∘ f if u ∈ C^{∞}(Y) and for any closed conic subset γ of Y × (ℝ^{n} \ {0}) with γ ∩ N_{f} = ∅, u ∈ D′_{γ}^{∗}(Y) implies

(1)  SS(f^{⋆}u) ⊂ f^{⋆}(SSu) and SS_{∗}(f^{⋆}u) ⊂ f^{⋆}(SS_{∗}u).

Proof. Let Γ be an open convex cone in ℝ^{n} such that (Y × Γ^{0}) ∩ N_{f} = ∅. Let u ∈ D′_{Γ}^{∗}(Y). We define f^{⋆}u as an element from D′^{∗}(X) as follows.

For every x_{0} ∈ X choose h_{0} ∈ ℝ^{m} such that f′(x_{0})h_{0} ∈ Γ. There exist an open set O_{x_{0}}, x ∈ O_{x_{0}} ⊂ X, an open convex cone Γ_{h_{0}} and ϵ_{0} > 0 such that

f(x + \sqrt{−1}h) ∈ Y + \sqrt{−1}Γ, h ∈ Γ_{h_{0}}, |h| < ϵ_{0}.

Theorem 3.9. in [12] and Theorem 1, imply that the following limit exists in D′^{∗}(O_{x_{0}})

(2)  f^{⋆}u|_{O_{x_{0}}} = lim_{h∈Γ_{h_{0}}^{h→0}} Φ(f(· + \sqrt{−1}h)),

where Φ(z) ∈ O^{∗}(Y + \sqrt{−1}Γ) and u = Φ(· + \sqrt{−1}Γ0).

The family of ultradistributions {f^{⋆}u|_{O_{x_{0}}} ; x_{0} ∈ X} has the property:

If O_{x_{0}} ∩O_{x_{1}} ≠ ∅, x_{0}, x_{1} ∈ X then f^{⋆}u|_{O_{x_{0}}} and f^{⋆}u|_{O_{x_{1}}} coincide on O_{x_{0}} ∩O_{x_{1}}. Let us prove that for every x ∈ O_{x_{0}} ∩ O_{x_{1}} there is an O_{x} ⊂ O_{x_{0}} ∩ O_{x_{1}} and an h such that f^{⋆}u|_{O_{x_{0}}} and f^{⋆}u|_{O_{x_{1}}} coincide with f^{⋆}u|_{O_{x}} on O_{x}. Since f′(x)h_{0} and f′(x)h_{1} belong to Γ, this holds for f′(x)((1−t)h_{0} +th_{1}), for every t ∈ [0, 1]. By the compactness of [0,1], there exist O_{x} and h such that the quoted assertion holds.

Thus, there exists an element in D′^{∗}(X), denoted by f^{⋆}u, which coincides with f^{⋆}u|_{O_{x_{0}}} for every x_{0}.

More generally, let u ∈ D′^{∗}(Y) be such that SSu ∩ N_{f} = ∅. There exist open convex cones Γ_{j}, j = 1,…, s, such that

SSu ⊂ ⋃_{i=1}^{s} Y × Γ_{j}^{0} and ⋃_{j=1}^{s}(Y × Γ_{j}^{0}) ∩ N_{f} = ∅.

Theorem 3.9. in [12] implies that there are holomorphic functions Φ_{j} ∈ O^{∗}(Y + \sqrt{−1}Γ_{j}), j = 1,…, s, such that

u_{j} = Φ_{j}(· + \sqrt{−1}Γ_{j}0), SSu_{j} ⊂ Y × Γ_{j}, j = 1,…, s,

(in the sense of convergence in D′^{∗}(Y)) and u = ∑_{j=1}^{s} u_{j}.

We define

(3)  f^{⋆}u = ∑_{j=1}^{s} f^{⋆}u_{j}.

In order to prove that this definition does not depend on the decomposition of u, assume that G_{i}, i = 1,…, p, are convex open cones such that

⋃_{i=1}^{p}(Y × G_{i}^{0}) ∩ N_{f} = ∅, ⋃_{i=1}^{p} Y × G_{i}^{0} ⊃ SSu, v_{i} = F_{i}(· + \sqrt{−1}G_{i}0)

and u = ∑_{i=1}^{p} v_{i}. We have to prove

(4)  ∑_{j=1}^{s} f^{⋆}u_{j} = ∑_{i=1}^{p} f^{⋆}v_{i}.

In the identity

(5)  ∑_{j=1}^{s} Φ_{j}(· + \sqrt{−1}Γ_{j}0) = ∑_{i=1}^{p} F_{i}(· + \sqrt{−1}G_{i}0)

we may assume that Γ_{j}, j = 1,…, s, are pairwise disjoint sets because if for some j_{1} and j_{2}, Γ_{j_{1}} ∩ Γ_{j_{2}} ≠ ∅, then the left hand side sum is reduced by

Φ_{j_{1}}(· + \sqrt{−1}Γ_{j_{1}}0) + Φ_{j_{2}}(· + \sqrt{−1}Γ_{j_{2}}0) = (Φ_{j_{1}} + Φ_{j_{2}})(· + \sqrt{−1}(Γ_{j_{1}} ∩ Γ_{j_{2}})0).

In the same way we may assume that G_{i}, i = 1,…, p, are pairwise disjoint sets.

Denote by Λ_{j} a subset of {1,…, p} which consists of i for which Γ_{j} ∩ G_{i} ≠ ∅, j = 1,…, s. By using “Edge of the Wedge theorem” (see [12]) we have:

Λ_{j} = ∅ ⇒ Φ_{j}(· + \sqrt{−1}Γ_{j}0) is real analytic on Y.

Put d_{j} = \overline{\overline{Λ}}_{j}. For every i ∈ \{1,…, p\} decompose F_{i}(· + \sqrt{−1}G_{i}0)

F_{i}(· + \sqrt{−1}G_{i}0) = ∑_{j∈V_{i}} F_{ij}(· + \sqrt{−1}(G_{i} ∩ Γ_{j})0),

where V_{i} is the subset of {1,…, s} which consists of j for which Γ_{j} ∩ G_{i} ≠ ∅, i = 1,…, p. Again,

V_{i} = ∅ ⇒ F_{i}(· + \sqrt{−1}G_{i}0) is real analytic on Y.

So, we rewrite (5) as

∑_{j=1}^{s} d_{j}(\frac{1}{d_{j}} Φ_{j}(· + \sqrt{−1}Γ_{j}0)) + φ = ∑_{j∈V_{i}≠∅} F_{ij}(· + \sqrt{−1}(G_{i} ∩ Γ_{j})0) + ψ

where φ and ψ are real analytic and both sums have the same number of terms. Note, if j ∈ V_{i_{1}} ∩ V_{i_{2}}, then (G_{i_{1}} ∩ Γ_{j_{2}}) ∩ (G_{i_{2}} ∩ Γ_{j_{2}}) = ∅ and

\frac{1}{d_{j}} Φ_{j}(· + \sqrt{−1}Γ_{j}0) − F_{i_{1}j}(· + \sqrt{−1}(Γ_{j} ∩ G_{i_{1}})0)

is real analytic.

Thus, the change of variable defined by (3) on both side of (4) gives the same element in D′^{∗}(X).

For the proof of (1) we use the notation from the beginning of this proof. From (2) it follows that

SS(f^{⋆}u)|_{x_{0}} ⊂ {(x_{0}, ξ); ⟨h, ξ⟩ ≥ 0, h ∈ Γ_{h_{0}}},

where SS(f^{⋆}u)|_{x_{0}} is a subset of SS(f^{⋆}u) with the first projection x_{0}. From

tf′(x_{0})Γ^{0} = {ξ; < h, ξ >≥ 0 if f′(x_{0})h ∈ Γ} ([9, p. 296]),

and f′(x_{0})Γ_{h_{0}} ⊂ Γ it follows

SS(f^{⋆}u)|_{x_{0}} ⊂ {(x_{0}, ^{t}f′(x_{0})η); η ∈ Γ^{0} \ {0}}.

By using the decomposition u = Σu_{j} in the neighborhood of f(x_{0}) given in [18], with cones Γ_{j} such that Γ_{j}^{0} are small enough, and Theorem 1, the proof of (1) follows.

4. The analysis of some operators

The six theorems which are to follow are well-known for distributions [9]. Their proofs are the same as for distributions.

Theorem 3. If u ∈ D^{′∗}(X) and v ∈ D^{′∗}(Y), then

i) SS_{∗}(u⊗v) ⊂ SS_{∗}u×SS_{∗}v ∪(supp u×{0})×SS_{∗}v ∪SS_{∗}u×(supp v ×{0}). ii) SS(u ⊗ v) ⊂ SSu × SSv ∪ (supp u × {0}) × SSv ∪ SSu × (supp v × {0}).

Theorem 4. If u, v ∈ D′^{∗}(X) and there are no points (x, ξ) such that

(x, ξ) ∈ SS_{∗}u and (x, −ξ) ∈ SS_{∗}v,

then by the pullback of the mapping

δ : X → X × X, δ(x) = (x, x)

is defined the product

uv = δ^{⋆}(u(x) ⊗ v(y))

D′^{∗} and

SS_{∗}uv ⊂ {(x, ξ + η); (x, ξ) ∈ SS_{∗}u or ξ = 0,
(x, η) ∈ SS_{∗}v or η = 0}.

The same holds for SSuv.

Theorem 5. Let u ∈ E′^{∗}(ℝ^{n}), x′ = (x_{1},…, x_{p}), x′′ = (x_{p+1},…, x_{n}) and let

u_{1}(x′) = ∫ u(x′, x′′)dx′′

be defined by

⟨u_{1}, ϕ⟩ = ⟨u(x′, x′′), ϕ(x′) ⊗ 1_{x′′}⟩, ϕ ∈ D^{∗}(ℝ^{n}).

Then

SS_{∗}u_{1} ⊂ {(x′, ξ′); ((x′, x′′), (ξ′, 0)) ∈ SS_{∗}u, for some x′′}.

The same holds for SSu_{1}.

Proof. Since the proof of the theorem is slightly different from the proof of Theorem 8.4.5 in [9], we give details which are important in our assertion. Let |ω| = 1, suppu = Q, ϕ ∈ D^{∗}(ℝ^{p}). Put

ψ_{δ}(x′′) = κ(δx′′), δ > 0, x′′ ∈ ℝ^{n−p},

where κ ∈ D^{∗}(ℝ^{n−p}) is equal to 1 in the unit ball B(0, 1). Take R such that Q ⊂ B(0, R), and η ∈ D^{∗}(ℝ^{n}) such that η = 1 in B(0, 2R).

Note that

⟨u(t), K(· + \sqrt{−1}ω − t)⟩

is analytic and exponentially decreasing if |x′′| > 2R, and ω belongs to a suitable neighbourhood of {ω; |ω| ≤ 1}. By letting δ → 0 in

⟨U(x + \sqrt{−1}ω), ϕ(x′)ψ_{δ}(x′′)⟩ = ⟨⟨u(t), K(x + \sqrt{−1}ω − t)⟩, η(x)ϕ(x′)ψ_{δ}(x′′)⟩

+ ⟨⟨u(t), K(x + \sqrt{−1}ω − t)⟩, (1 − η(x))ϕ′(x′)ψ_{δ}(x′′)⟩,

it follows

⟨u_{1}, ϕ⟩ = ∫_{|ω|=1}⟨U(· + \sqrt{−1}ω), ϕ(x′) ⊗ 1_{x′′}⟩dω =

= ∫_{|ω|=1}⟨U_{1}(· + \sqrt{−1}ω′), ϕ⟩dω,

where

U_{1}(z′) = ∫ U(z′, x′′)dx′′ = ∫ U(z′, x′′ + iy′′)ϕx′′, | Im z′|^{2} + |y′′|^{2} < 1

is an analytic function when | Im z′| < 1 which is bounded by Ce^{M̃( \frac{k}{1−|y|})} for some k > 0 and C > 0 in (M_{p})-case (respectively, for every k > 0 there is C in {M_{p}}- case). Now, by similar arguments as in the proof of Theorem 8.5.4 in [9], we finish the proof by using a corresponding result of [17].

Theorem 6. Assume K ∈ D′(X × Y), where X ⊂ ℝ^{n}, Y ⊂ ℝ^{m} are open, and π_{1} : supp K → X is proper (π_{1} is the first projection). If u ∈ E^{∗}(Y), then

SS_{∗}Ku ⊂ {(x, ξ); (x, y, ξ, 0) ∈ SS_{∗}K for some y ∈ supp u},

where K is the linear operator with the kernel K. The same holds for SSKu.

Theorem 7. If u ∈ E′^{∗} and SS_{∗}u ∩ SS′_{∗}(K)_{Y} = ∅, then
SS_{∗}Ku ⊂ SS_{∗}(K)_{X} ∪ (SS′_{∗}(K) ∘ SS_{∗}u),

where

SS′_{∗}(K) = {(x, y, ξ, η); (x, y, ξ, −η) ∈ SS_{∗}K},
SS_{∗}(K)_{X} = {(x, ξ); (x, y, ξ, 0) ∈ SS_{∗}K for some y ∈ Y },
SS′_{∗}(K)_{Y} = {(y, η); (x, y, 0, −η) ∈ SS_{∗}K for some x ∈ X}.

The same holds for SSKu.

Now, one can easily prove the following assertion.

Theorem 8. If u ∈ D′^{∗} and v ∈ E′^{∗}, then

SS_{∗}(u ∗ v) ⊂ {(x + y, ξ); (x, ξ) ∈ SS_{∗}u, (y, ξ) ∈ SS_{∗}v}.

The same holds for SS(u ∗ v).

Remark. Theorem 8 has applications in the theory of pseudo-differential operators. For example, for a pseudo-differential operator P it follows that SS_{∗}P u ⊂ SS_{∗}u, u ∈ D′^{∗}.

If P (D) is an ultradifferential operator with constant coefficients of ∗-class [12], then Theorem 8 implies SS_{∗}P (D)u ⊂ SS_{∗}u, u ∈ D′. The same theorem implies that the Poisson transform of a u ∈ D′, if it exists, does not enlarge the SS_{∗}u because SSP is empty (cf. [18]).

5. Quotient sheaves of microfunctions and flabbiness

The results of this paragraph are from [6] and [7].

Let π : \sqrt{−}S^{∗}ℝ^{n} → ℝ^{n} be the canonical projection. The sheaf of microfunctions on \sqrt{−1}S^{∗}ℝ^{n} is the associated sheaf of the presheaf

\sqrt{−1}S^{∗}ℝ^{n} ⊃ Ω → Γ(π(Ω); B)/\{u ∈ Γ(π(Ω); B)|SSu ∩ Ω = ∅\}.

This sheaf is denoted by C. It enjoys the exact sequence

0 → A → B → π_{∗}C → 0, [21]

where A denotes the sheaf of real analytic functions on ℝ^{n}. Moreover, there exists a canonical surjective spectrum map

Sp : π^{−1}B → C.

Then, for u ∈ B, SS(u) = supp(Sp(u)). The injection

D^{′∗} ↪ B (respectively, E^{∗} ↪ B)

induces a sheaf homomorphism

π^{−1}D^{′∗} → C (respectively π^{−1}E^{∗} → C).

We define a subsheaf C^{∗} (respectively, C^{d,∗}) of C as the image of the above morphism and call it the sheaf of microfunctions of class ∗ (respectively d, ∗). Furthermore, we have a canonical exact sequence

0 → A → D^{′∗} → π_{∗}C^{∗} → 0
(resp. 0 → A → E^{′∗} → π_{∗}C^{d,∗} → 0).

We have given in Section 1 an order to the set of sequences satisfying conditions (M.1), (M.2) and (M.3)’. If † ≤ ∗, we have canonical injections

C^{d,†} ↪ C^{d,∗} ↪ C^{∗} ↪ C^{†} ↪ C.

From now on we use the notation C^{1} = C and 1 ≤ ∗ for any ∗.

We define sheaves C^{†,∗}, C^{†/∗}, C^{d,†,∗} on \sqrt{−1}S^{∗}ℝ^{n} by the following exact sequences:

i) 0 → C^{d,∗} → C^{†} → C^{†,∗} → 0, when † ≤ ∗, ii) 0 → C^{∗} → C^{†} → C^{†/∗} → 0, when † ≤ ∗, iii) 0 → C^{d,∗} → C^{d,†} → C^{d,†,∗} → 0, when ∗ ≤ †. The canonical surjective spectrum map Sp induces the following surjective

spectrum maps:

Sp^{1,∗} : π^{−1}B → C^{1,∗},
Sp^{1/∗} : π^{−1}B → C^{1/∗}.

Let u ∈ B. We can also define the singular spectrum of class ∗, SS_{∗}(u), and that of class 1/∗, SS^{∗}(u), in the following way:

SS_{∗}u = supp(Sp^{1,∗}u),
SS^{∗}u = supp(Sp^{1/∗}u).

Let F be a sheaf of Abelian groups on a topological space X. Recall [1], F is supple if for any open set Ω of X, any closed Z, Z_{1}, Z_{2} of Ω such that Z = Z_{1} ∪ Z_{2}, and any section u ∈ Γ_{Z}(Ω, F), there exists u_{i} ∈ Γ_{Z_{i}}(Ω, F) (i = 1, 2) such that u = u_{1} + u_{2}. Clearly, if a sheaf is flabby, then it is supple.

Theorem 9. [6] a) The quotient sheaf D^{′∗}/E^{∗} is supple.

b) The sheaves C^{1,∗} and C^{1/∗} are flabby and the following sequences are exact

0 → E^{∗} → B → π_{∗}C^{1,∗} → 0,
0 → D^{′∗} → B → π_{∗}C^{1/∗} → 0.

c) Let 1 ≤ † ≤ ∗. The sheaves C^{†,∗} and C^{†/∗} are supple and the sequences

0 → E^{∗} → D^{′†} → π_{∗}C^{†,∗} → 0,
0 → D^{′∗} → D^{′†} → π_{∗}C^{†/∗} → 0,

are exact.

d) Let 1 ≤ ∗ ≤ †. The sheaf C^{d,†,∗} is supple and the sequence

0 → D^{∗} → E^{†} → π_{∗}C^{d,†,∗} → 0

is exact.

References

[1] G. Bengel, P. Schapira, Décomposition microlocale analitique des distributions, Ann. Inst. Fourier, Grenoble 29(3) (1979), 101–124.

[2] R. W. Braun, R. Meise, B. A. Taylor, Ultradifferentiable functions and Fouirer analysis, Result Math. 17 (1990), 206–237.

[3] R. D. Carmichael, S. Pilipović, On the convolution and the Laplace transformation in the Spaces of Beurling - Gevrey Tempered Ultradistributions, Math. Nachr. 158 (1992), 119–132.

[4] J. Chung, S.-Y. Chung, D. Kim, Representation of quasianalytic ultradistributions, Ark. Mat. 31 (1993), 51–60.

[5] J. Chung, S.-Y. Chung, D. Kim, A characterisation for Fourier hyperfunctions, Publ. RIMS, Kyoto Univ. 30 (1994), 203–208.

[6] A. Eida, S. Pilipović, Suppleness of Some Quotient Spaces of Microfunctions, Preprint.

[7] A. Eida, S. Pilipović, On the microlocal decomposition of some classes of hyperfunctions, Math. Proc. Cambridge. Phil. Soc., to appear.

[8] K. Floret, J. Wloka, Einfürung in die Theorie der lokalkonvexen Räume, Springer-Verlag, Lec. Not. Math. 59, Berlin - Heidelberg - New York, 1968.

[9] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, 1983.

[10] A. Kaneko, Introductionà la Théorie des Hyperfonctions I, II, Cours de DEA, Université de Grenoble, 1978.

[11] H. Komatsu, Ultradistributions, I – III, J. Fac. Sci. Univ. Tokyo, Sect. IA 20 (1973), 25–105; 24 (1977), 607–628; 29 (1982), 653-717.

[12] H. Komatsu, Microlocal Analysis in Gevrey Classes and in Convex Domains, Springer, Lec. Not. Math. 1726 (1989), 426–493.

[13] D. Kovačević, S. Pilipović, Structural properties of the space of tempered ultradistributions, Proc. Conf. “Complex Analysis and Generalized Functions”, Varna 1991, 169–184.

[14] T. Matsuzawa, Foundation of a calculus approach to hyperfunctions, Lecture Notes, 1992, Preprint (see: A calculus approach to hyperfunctions III, Nagoya Math. J. 118 (1990), 133– 153.

[15] H. J. Petzsche, Generalized functions and the boundary values of holomorphic functions, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 31 (1984), 391–431.

[16] S. Pilipović, Characterization of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191–1206.

[17] S. Pilipović, Structural Theorems for Ultradistributions, Ann. Polon. Math. 340 (1994), 223– 235.

[18] S. Pilipović, Microlocal analysis of ultradistributions, Proc. Amer. Math. Soc. 126 (1998), 105–113.

[19] L. Rodino, Linear partial differential operators in Gevrey spaces, World Scientific, Singapore, 1993 (to appear).

[20] J. W. de Roever, Hyperfunctional singular support of ultradistributions, J. Fasc. Sci. Univ. Tokyo, Sec IA Math. 31 (1985), 585–631.

[21] M. Sato, T. Kawai, M. Kashiwara, Microfunctions and Pseudo-differential Equations, Springer-Verlag, Lect. Not. Math. 287 (1973), 265–529.

[22] L. Zanghirati, Pseudodifferential operators of infinite order and Gevrey classes, Ann. Univ. Ferrara Ser. VII 31(1985), 197–219.

Institut za Matematiku (Received 11 06 1998) Trg D. Obradovića 4 21000 Novi Sad Yugoslavia