Bulletin T.CXXII de l’Académie Serbe des Sciences et des Arts - 2001 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 26

ON THE MICROLOCAL DECOMPOSITION OF ULTRADISTRIBUTIONS AND ULTRADIFFERENTIABLE FUNCTIONS

A. EIDA

(Presented at the 9th Meeting, held on December 22, 2000)

A b s t r a c t. The microlocal decomposition for ultradistributions and ultradifferentiable functions is derived by Bengel-Schapira’s method and these classes of functions are microlocalized as subsheaves C_{M}^{∗}, C_{M}^{d,∗} of the sheaf C_{M} of Sato’s microfunctions on a real analytic manifold M. Moreover, the exactness of the sequences

0 ⟶ A_{M} ⟶ Db_{M}^{∗} ⟶ π_{∗}C_{M}^{∗} ⟶ 0

and

0 ⟶ A_{M} ⟶ Df_{M}^{∗} ⟶ π_{∗}C_{M}^{d,∗} ⟶ 0

is shown and some fundamental properties on C_{M}^{∗}, C_{M}^{d,∗} are described. Here Db_{M}^{∗} is a sheaf of ultradistributions and Df_{M}^{∗} is a sheaf of ultradifferentiable functions. We give some solvability conditions applicable to partial differential equations by operating Aoki’s classes of microdifferential operators of infinite order on these sheaves.

AMS Mathematics Subject Classification (2000): 58J15

Key words: hyperfunctions, ultradistributions, ultradifferentiable functions, microdifferential operators

0. Introduction

The foundation of the microlocal decomposition for distributions dates back to the time when A. Martineau [30](1964) showed that any distribution is represented as a sum of boundary values of holomorphic functions with bounds. This work was generalized by H. Komatsu [24](1973) for ultradistributions. On the other hand, the microlocal point of view was introduced by M. Sato [36] as the notion of singular spectrum. Moreover, the notion of wave front set due to L. Hörmander and that of essential support of Bros-Iagolnitzer came to being at the beginning of 1970’s. Bros-Iagolnitzer [11] conjectured at the same time that analytic singular support for distributions is decomposable. Bengel-Schapira [5](1979) solved it positively by considering Cousin’s problem with bounds in a tuboid. They constructed the subsheaves C^{f}, C^{d} of the sheaf C of microfunctions as J. M. Bony [8] did and initiated the algebro-analytical treatment of distributions. P. Laubin [27](1983) decomposed distributions microlocally by J. Sjöstrand[41]’s method, and J. W. de Roever [34](1984) showed the microlocal decomposition for ultradistributions by a special integral representation. E. Andronikoff [1](1986) reconstructed the sheaf C^{f} by microlocalizing the functor T H(·) by which M. Kashiwara [20] solved the Riemann-Hilbert problem of the holonomic D_{X}-modules.

We study in this paper the microlocalized sheaves for ultradistributions and ultradifferentiable functions by Bengel-Schapira’s methods, and give some applications to algebraic analysis.

1. Ultradistributions and ultradifferentiable functions

1.1. Supple sheaves

First of all we recall the notion of suppleness according to Bengel-Schapira [5].

Definition 1.1.1.Let F be a sheaf of Abelian groups on a topological space X. We say that F is supple if for any open Ω of X, any closed Z, Z_{1}, Z_{2} of Ω with 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) with u = u_{1} + u_{2}.

Proposition 1.1.2. (a) The flabby sheaves are supple. (b) The supple sheaves are soft on a totally paracompact Hausdorff space.

On the microlocal decomposition of ultradistributions … 77

P r o o f. (a) Let F be a flabby sheaf on a topological space X. Let u ∈ Γ(Ω; F) be a section whose support is the union of two closed sets Z_{1} and Z_{2}. Define u_{1} by

u_{1} = \{ \begin{matrix}u & on Ω \ Z_{2}, \\ 0 & on Ω \ Z_{1}. \end{matrix}

Note that this gives a well-defined value 0 on the intersection (Ω \ Z_{1}) ∩ (Ω \ Z_{2}) = Ω \ (Z_{1} ∪ Z_{2}), and hence determines a well-defined section on the union (Ω\Z_{1})∪(Ω\Z_{2}) = Ω\(Z_{1} ∩Z_{2}). By the flabbiness of F, extend this section to Ω, and denote it by u_{1} again. Then supp(u_{1}) ⊂ Z_{1}. Moreover, defining u_{2} := u − u_{1}, we see that supp(u_{2}) ⊂ Z_{2} also holds, and hence the result.

(b) Let F be a supple sheaf on a totally paracompact Hausdorff space X. Let Z be a closed set of X. Since we have

Γ(Z; F) = lim_{\overrightarrow{Ω⊃Z}} Γ(Ω; F),

for any section u ∈ Γ(Z; F), there exists a section ũ ∈ Γ(Ω; F) which represents u. Here the limit on the right hand side is taken with respect to all open sets containing Z. Take an open V of X such that Ω ⊃ \overline{V} ⊃ V ⊃ Z. By the suppleness of F, there exist ũ_{1} ∈ Γ\overline{V} (Ω; F) and ũ_{2} ∈ Γ_{Ω\V} (Ω; F) with ũ = ũ_{1} + ũ_{2}. Extend ũ_{1} to X by 0, and then we see that the restriction of this section to Z coincides with u again. This shows the proposition.

Now we quote the following two theorems due to Bengel-Schapira [5].

Theorem 1.1.3. Let X be a locally compact and σ-compact Hausdorff topological space, and let F be a sheaf of Abelian groups on X. Let A be a subgroup of Γ_{c}(X; F), the space of sections with compact support of F. Assume A to be stable by the decomposition of support, i.e., for any u ∈ A and for any closed subsets Z, Z_{1}, Z_{2} of X with i) Z = Z_{1} ∪ Z_{2}, and ii) supp(u) ⊂ Z, there exists u_{i} ∈ A (i = 1, 2) with u = u_{1} + u_{2} and supp(u_{i}) ⊂ Z_{i}. Then there exists a unique subsheaf G of F such that

a) Γ_{c}(X; G) = A,

b) G is supple. Moreover, for any x ∈ X, we have G_{x} = A_{x}.

Theorem 1.1.4. Let F be a sheaf of Abelian groups on a locally compact and σ-compact Hausdorff space X. Let X = ⋃_{i} Ω_{i} be an open covering of X.

Suppose that for any i the sheaf F | Ω_{i} on Ω_{i} is supple. Then F is supple.

Finally we show

Theorem 1.1.5. Let X be as in Theorem 1.1.4, and let F, F′, F′′ be sheaves of Abelian groups on X. Let F, F′ be supple and assume the sequence

0 ⟶ F ⟶ F′ ⟶ F′′ ⟶ 0

to be exact. Then F′′ is supple.

P r o o f. Recall the fact that

0 ⟶ Γ_{c}(X, F) ⟶ Γ_{c}(X, F′) ⟶ Γ_{c}(X, F′′) ⟶ 0

holds under the assumption.

1.2. Ultradistributions and ultradifferentiable functions

For the theory of ultradistributions and ultradifferentiable functions we review some conditions according to H. Komatsu [24] and others. Note that these conditions were first studied by C. Roumieu [35].

Let M_{p} (p = 0, 1, 2, …) be a sequence of positive numbers. We shall assume the following conditions (M.0) and (M.1): (M.0)

M_{0} = 1,

(M.1)(logarithmic convexity)

M_{p}^{2} ≤ M_{p−1}M_{p+1}, p = 1, 2, ….

Two logarithmic convex sequences M_{p} and N_{p} is defined to be equivalent, M_{p} ∼ N_{p}, if there are numbers A ≥ 1 and H ≥ 1 with

\frac{N_{p}}{AH^{p}} ≤ M_{p} ≤ AH^{p}N_{p}, p = 0, 1, 2, …

Moreover, we shall consider the conditions (M.2) and (M.3) for M_{p} (p = 0, 1, 2, …): (M.2)(stability under ultradifferential operators)

M_{p} ∼ min_{0≤q≤p} M_{q}M_{p−q}, p = 0, 1, 2, …;

(M.3)(strong non − quasi − analyticity) There are a sequence M̃_{p} and a constant δ > 0 such that

M_{p} ∼ M̃_{p} with \frac{M̃_{p−1}}{M̃_{p}} p^{1+δ} ↓ 0, p = 1, 2, ….

We shall possibly replace (M.2) and (M.3) by the following weaker conditions: (M.2)′(stability under differential operators)

M_{p+1} ∼ M_{p}, p = 0, 1, 2, …;

(M.3)′ (non − quasi − analyticity)

∑_{p=1}^{∞} \frac{M_{p−1}}{M_{p}} < ∞.

We note that if s > 1 the Gevrey sequence

M_{p} = (p!)^{s} or p^{ps} or Γ(1 + ps)

satisfies the above conditions (M.0), (M.1), (M.2), and (M.3).

Definition 1.2.1. We assume that M_{p} satisfies (M.1) and (M.3)′. Then ∗ will stand for either (M_{p}) or {M_{p}}. Let M be a real analytic manifold, U be an open subset of M, and φ(x) ∈ C^{∞}(U). Then we set

∥φ∥_{K,h,M_{p}} = sup_{x∈K,α≥0} \frac{| D^{α}φ(x) |}{h^{|α|}M_{|α|}}

for a compact subset K of U and a positive number h. We define the space Df_{M}^{∗}(U) of ultradifferential functions of class ∗ on U by

i) in case ∗ = (M_{p}),

Df_{M}^{(M_{p})}(U) = {φ ∈ C^{∞}(U) | for ∀K ⊂⊂ U and ∀h > 0 ∥φ∥_{K,h,M_{p}} < ∞},

ii) in case ∗ = {M_{p}},

Df_{M}^{{M_{p}}}(U) = {φ ∈ C^{∞}(U) | for ∀K ⊂⊂ U ∃h > 0 ∥φ∥_{K,h,M_{p}} < ∞}

We denote by Df_{M,c}^{∗}(U) the space of ultradifferentiable functions of class ∗ with compact support on U. Refer to H. Komatsu [24] for its topology. We also define the space Db_{M}^{∗} of ultradistributions of class ∗ by the strong dual space of Df_{M,c}^{∗}(U). Note that the presheaves

U ⟼ Df_{M}^{∗}(U),
U ⟼ Db_{M}^{∗}(U)

become sheaves. We denote thus by Df_{M}^{∗} the sheaf of ultradifferentiable functions of class ∗ on M, and by Db_{M}^{∗} the sheaf of ultradistributions of class ∗ on M. We always consider Df_{M}^{∗} and Db_{M}^{∗} as subsheaves of the sheaf B_{M} of Sato’s hyperfunctions on M.

We introduce an order to the set of all (M_{p})’s and {M_{p}}’s satisfying the conditions as follows.

i) (M_{p}) < (N_{p}), {M_{p}} < {N_{p}} if there are constants L and C such that

M_{p} ≤ CL^{p}N_{p}, p = 0, 1, 2, ….

ii) {M_{p}} < (N_{p}) if for any ϵ > 0 there is a constant C_{ϵ} such that

M_{p} ≤ C_{ϵ}ϵ^{p}N_{p}, p = 0, 1, 2, ….

iii) (M_{p}) < {M_{p}}.

We may consider Db_{M}^{(∞)} and Df_{M}^{(∞)} as the sheaves of distributions and differentiable functions, respectively. The order is considered as iv) (M_{p}), {M_{p}} < (∞).

Definition 1.2.2. Let M_{p} satisfy (M.1), (M.2) and (M.3). Then we define the associated function M(ρ) (resp. the growth function M^{⋆}(ρ)) of M_{p} by

M(ρ) = sup_{p} log \frac{ρ^{p}M_{0}}{M_{p}} (resp. M^{⋆}(ρ) = sup_{p} log \frac{ρ^{p}p!M_{0}}{M_{p}} ) .

Next we recall the following theorem by H. Komatsu [24].

Theorem 1.2.3.Suppose that the sequence M_{p} p = 0, 1, 2, … satisfy (M.1), (M.2) and (M.3). Let V be a Stein open set in ℂ^{n} and Γ be an open convex cone in ℝ^{n}. We set

V_{Γ} = (ℝ^{n} + \sqrt{−1}Γ) ∩ V, and U = ℝ^{n} ∩ V.

Let F (x+\sqrt{−1}y) be a holomorphic function on V_{Γ} and assume the condition:

For any compact set K in U and closed subcone Γ′ ⊂ Γ, there are constants L and C (resp. for any L > 0 there is a constant C) satisfying the estimate

sup_{x∈K} | F (x + \sqrt{−1}y) |≤ C exp M^{⋆} (\frac{L}{| y |})

for y ∈ Γ with | y | sufficiently small.

Then there is an ultradistribution F (x + \sqrt{−1}Γ 0) ∈ Db^{(M_{p})}(U) (resp. Db^{\{M_{p}\}}(U)) which is the boundary value of F (x + \sqrt{−1}y) in the sense that

F (x + \sqrt{−1}y) ⟶ F (x + \sqrt{−1}Γ 0) in Db^{(M_{p})}(U) (resp. Db^{\{M_{p}\}}(U))

as y tends to 0 in a closed subcone Γ′ of Γ. Moreover, the boundary value F (x + \sqrt{−1}Γ 0) coincides with that in the sense of hyperfunction.

1.3. ∗^{−1}-Singular spectrum and ∗-singular spectrum

Hereafter ∗ stands for either (M_{p}) or {M_{p}}. In this section we will give the definition of singular spectrum of various classes.

Definition 1.3.1. Let u(x) be a hyperfunction with compact support. we define the Fourier transform ũ(ξ) of u by

ũ(ξ) = ∫ u(x)e^{−\sqrt{−1}xξ} dx.

Let us give a well-known theorem about the estimate of ũ(ξ). Refer to H. Komatsu [24] for more details.

Theorem 1.3.2. Suppose that M_{p} satisfies (M.1), (M.2)′ and (M.3)′. Let u(x) be a hyperfunction with compact support on M. Then u(x) is in Db_{M}^{∗}(M) (Df_{M}^{∗}(M)) if and only if

i) in case Db^{(M_{p})}, there exist positive numbers L and C with

| ũ(ξ) |≤ C exp M(L | ξ |),

ii) in case Db^{{M_{p}}}, for any positive ϵ there exists a positive number C_{ϵ} with

| ũ(ξ) |≤ C_{ϵ} exp M(ϵ | ξ |),

iii) in case Df^{(M_{p})}, for any positive ϵ there exists a positive number C_{ϵ} with

| ũ(ξ) |≤ C exp {−M(ϵ | ξ |)},

iv) in case Df^{{M_{p}}}, there exist positive numbers L and C with

| ũ(ξ) |≤ C exp {−M(L | ξ |)}.

Definition 1.3.3. Let M be an open set of ℝ^{n}, and denote by \sqrt{−1}S^{∗}M the pure imaginary cotangential sphere bundle of M. Let u(x) ∈ Db_{M}^{†}(M) with compact support. We introduce ∗^{−1}-singular spectrum (resp. ∗-singular spectrum) SS^{∗^{−1}}(u) (resp. SS^{∗}(u)) of u in \sqrt{−1}S^{∗}M when † < ∗. Let q^{∘} = ( x^{∘}, \sqrt{−1} ξ^{∘}∞) ∈ \sqrt{−1}S^{∗}M. Then q^{∘} ∉ SS^{∗^{−1}}(u) (resp. SS^{∗}(u)) if

i) in case ∗ = (M_{p}), there exist a neighborhood U of x^{∘} and a conic neighborhood Ξ of ξ^{∘} such that for any ϕ ∈ Df^{†}(U) there exist L > 0 and C > 0 (resp. for any ϕ ∈ Df^{†}(U) and any ϵ > 0 there exists C_{ϵ} > 0) satisfying

| ϕ̃u(ξ) |≤ C exp M(L | ξ |), ξ ∈ Ξ,
(resp. | ϕ̃u(ξ) |≤ C_{ϵ} exp {−M(ϵ | ξ |)}, ξ ∈ Ξ, )

ii) in case ∗ = {M_{p}}, there exist a neighborhood U of x^{∘} and a conic neighborhood Ξ of ξ^{∘} such that for any ϕ ∈ Df^{†}(U) and any ϵ > 0 there exists C_{ϵ} > 0 (resp. for any ϕ ∈ Df^{†}(U) there exist L > 0 and C > 0) satisfying

| ϕ̃u(ξ) |≤ C_{ϵ} exp M(ϵ | ξ |), ξ ∈ Ξ,
(resp. | ϕ̃u(ξ) |≤ C exp {−M(L | ξ |)}, ξ ∈ Ξ.)

Note that if ∗ = {p!}, SS^{∗} coincides with the singular spectrum S.S. that M. Sato et al. [37] introduced. We denote it by SS hereafter. We remark that L. Hörmander [18] introduced the wave front set of several classes WF_{L} which is the original idea of our SS^{∗}.

1.4. Operations for ultradistributions

Hereafter we always assume that the sequence M_{p} satisfies (M.1), (M.2) and (M.3). We devote this section to product, integration, and restriction of ultradistributions.

We recall some theorems by M. Sato et al. [37]. Note that we define ŜS as follows.

ŜS(u) = \{(x, \sqrt{−1}ξ) ∈ \sqrt{−1}T^{∗}M | x ∈ suppu and ξ = 0\}

∪\{(x, \sqrt{−1}ξ) ∈ \sqrt{−1}T^{∗}M | ξ ≠ 0 and (x, \sqrt{−1}ξ∞) ∈ SS(u)\}.

Theorem 1.4.1. Let M_{1} and M_{2} be real analytic manifolds, and set M = M_{1} × M_{2}. For u_{1}(x_{1}) ∈ Db^{∗}(M_{1}) (resp. Df^{∗}(M_{1})) and u_{2}(x_{2}) ∈ Db^{∗}(M_{2}) (resp. Df^{∗}(M_{2})), we can define canonically the product u(x_{1}, x_{2}) = u_{1}(x_{1})u_{2}(x_{2}) so that

ŜS(u) ⊂ ŜS(u_{1}) × ŜS(u_{2}).

P r o o f. If we consult Theorem 2.4.1 in Chapter I of M. Sato et al. [37], we obtain the desired result.

Theorem 1.4.2. Let u(x) and v(x) be ultradistributions of class ∗ on a real analytic manifold M such that SS^{∗}(u)∩(SS^{∗}(v))^{a} = ∅. Then the product u(x)v(x) ∈ Db^{∗}(M) exists with the property

SS^{∗}(uv) ⊂ \{(x, \sqrt{−1}(θξ_{1} + (1 − θ)ξ_{2})∞) |

(x, \sqrt{−1}ξ_{1}∞) ∈ SS^{∗}(u), (x, \sqrt{−1}ξ_{2}∞) ∈ SS^{∗}(v), 0 ≤ θ ≤ 1\},

where a : \sqrt{−1}S^{∗}M ⟶ \sqrt{−1}S^{∗}M is the antipodal mapping (i.e., a(x, \sqrt{−1}ξ∞) = (x, −\sqrt{−1}ξ∞)), and (SS^{∗}(v))^{a} denotes the image of SS^{∗}(v) by a.

P r o o f. The theorem can be reduced to the case where u and v have small compact support in ℝ^{n}. Moreover we may assume from the beginning that there exist an open set U of ℝ^{n} and closed convex cones Ξ_{1} and Ξ_{2} of ℝ^{n} such that

SS^{∗}(u_{1}) ⊂ U × \sqrt{−1}Ξ̂_{1}∞,
SS^{∗}(u_{2}) ⊂ U × \sqrt{−1}Ξ̂_{2}∞,

where Ξ̂_{i} is the image of Ξ_{i} by the natural projection from ℝ^{n} \ {0} to S^{n−1} (i = 1, 2). Take a closed cone Ξ of ℝ^{n} with Ξ ∩ ch(Ξ_{1} ∪ Ξ_{2}) = ∅. Here ch(·) means the convex hull. Then

Ξ ∩ Ξ_{2} = ∅, (Ξ − Ξ_{2}) ∩ Ξ_{1} = ∅.

Let ξ ∈ Ξ, η ∈ Ξ_{2}. Since we have | ξ − η |≥ c(| ξ | + | η |) for some c > 0, we obtain for any ϵ > 0 and for some L > 0 the estimate

| ũ(ξ − η)ṽ(η) | ≤ C_{ϵ} exp{−M(ϵ | ξ − η |)}C exp M(L | η |)
≤ C_{ϵ}C exp{−M(cϵ | ξ | +cϵ | η |) + M(L | η |)}.

Here, by Proposition 3.6 of H. Komatsu [24],

−M(ρ_{1} + ρ_{2}) ≤ −M(\frac{ρ_{1}}{H}) − M(\frac{ρ_{2}}{H}) + log(AM_{0})

holds for some H ≥ 1. We obtain thereby

| ũ(ξ−η)ṽ(η) |≤ C_{ϵ}CA exp \{−M(\frac{cϵ}{H} | ξ |)\} exp \{−M(\frac{cϵ}{H} | η |) + M(L | η |)\} .

Hence we have

| ∫_{Ξ_{2}} ũ(ξ − η)ṽ(η) dη |≤ C_{ϵ} exp{−M(ϵ | ξ |)}

for any ϵ > 0 and ξ ∈ Ξ.

Next let ξ ∈ Ξ, η ∉ Ξ_{2}. Take a small ϵ′ > 0 and suppose that

| ξ |≥ 1, | η |≤ ϵ′ | ξ | .

Then we have ξ − η ∉ Ξ_{1} and | ξ − η |≥ (1 − ϵ′) | ξ |≥ (1 − 2ϵ′) | ξ | + | η |, which leads to the estimate

| ũ(ξ − η)ṽ(η) |≤ C_{ϵ,ϵ′′} exp \{−M(\frac{ϵ(1 − 2ϵ′)}{H} | ξ |)\}

· exp \{−M(\frac{ϵ}{H} | η |) − M(ϵ′′ | η |)\}

for any ϵ > 0 and ϵ′′ > 0.

Finally let ξ ∈ Ξ, η ∉ Ξ_{2} and suppose that

| η |≥ ϵ′ | ξ | .

Then

| η |≥ (| η | +ϵ′ | ξ |)/2,

| ũ(ξ − η)ṽ(η) | ≤ C_{ϵ′′} exp \{M(L | ξ − eta |) − M(\frac{ϵ′′}{2} | η | +\frac{ϵ′ϵ′′}{2} | ξ |)\}

≤ C_{ϵ′′}A exp \{−M(\frac{ϵ′ϵ′′}{2H} | ξ |)\} exp \{M((1 + \frac{1}{ϵ′})L | η |) − M(\frac{ϵ′′}{2H} | η |)\}

hold for any ϵ′′ > 0. Hence we deduce

| ∫_{ℝ^{n}\Ξ_{2}} ũ(ξ − η)ṽ(η) dη |≤ C_{ϵ} exp{−M(ϵ | ξ |)}

for any ϵ > 0 and ξ ∈ Ξ. Thus if we define the product of u and v by

u(x)v(x) = \frac{1}{(2π)^{n}} ∫_{ℝ^{n}×ℝ^{n}} ũ(ξ − η)ṽ(η)e^{\sqrt{−1}xξ} dηdξ,

we will obtain the desired result. Note that it is easy to show uv ∈ Db^{∗}(ℝ^{n}).

Theorem 1.4.3.Let M and N be real analytic manifolds, and let f : M × N ⟶ N be the natural projection. If f |_{suppu} is a proper map for u(t, x) an ultradistribution of class ∗ on M × N, then the integration of u(t, x) along the fiber

v(x) = ∫_{f^{−1}(x)} u(t, x) dt

can be defined. Moreover, we have

SS^{∗}(v) ⊂ ϖ(SS^{∗}(u) ∩ M × \sqrt{−1}S^{∗}N),

where ϖ denotes the natural projection from M × \sqrt{−1}S^{∗}N to \sqrt{−1}S^{∗}N.

P r o o f. We define the integration of ultradistributions as that of hyperfunctions. Refer to M. Sato et al. [37] for more details. We can obtain the estimate of SS^{∗} immediately from Fourier transform of v.

Corollary 1.4.4. Let u(x) and v(x) be ultradistributions of class ∗ on a real analytic manifold M, either of which has compact support. Then the convolution u(x) ∗ v(x) ∈ Db^{∗}(M) exists with the property

SS^{∗}(u∗v) ⊂ \{(x+y, \sqrt{−1}ξ∞) | (x, \sqrt{−1}ξ∞) ∈ SS^{∗}(u), (y, \sqrt{−1}ξ∞) ∈ SS^{∗}(v)\}.

Let u be an ultradistribution of class ∗ on an open set U of ℝ^{n} with (x, ±\sqrt{−1}(1, 0, …, 0)∞) ∉ SS^{∗}(u) for any x ∈ U. Then we say u contains x_{1} as ∗-ultradifferentiable parameter on U.

Corollary 1.4.5. Let u(x) be an ultradistribution of class ∗ in a neighborhood U of the origin in ℝ^{n}, and assume that it contains x_{1} as ∗-ultra-differentiable parameter. Then the restriction u |_{x_{1}=0} can be defined as

u |_{x_{1}=0}= ∫ δ(x_{1})u(x) dx_{1}.

Moreover, we have the estimate

SS^{∗}(u |_{x_{1}=0}) ⊂ π(SS^{∗}(u) ∩ \{x_{1} = 0\} × \sqrt{−1}S^{n−1}∞),

where π is the natural projection from \sqrt{−1}S^{∗}U to \{ξ_{1} = 0\}.

2. Microlocal decomposition

2.1. The sheaves C_{M}^{∗} and C_{M}^{d,∗}

We construct the sheaves C_{M}^{∗} and C_{M}^{d,∗} in order to decompose analytic microlocal singularities of ultradistributions and ultradifferentiable functions. As we know well, the sheaf C_{M} of microfunctions on a real analytic manifold M enjoys the exact sequence

0 ⟶ A_{M} ⟶ B_{M} ⟶ π_{∗}C_{M} ⟶ 0,

which is due to M. Sato et al. [37]. Here A_{M} denotes the sheaf of real analytic functions on M, and π is the natural projection from \sqrt{−1}S^{∗}M to M. Moreover there exists a canonical surjective spectrum map

Sp_{M} : π^{−1}B_{M} ⟶ C_{M},

by which we define the singular spectrum SS(u) of u ∈ B_{M} by

SS(u) = supp(Sp_{M}(u)).

The injection

Db_{M}^{∗} ↪ B_{M}
(resp. Df_{M}^{∗} ↪ B_{M})

induces a sheaf homomorphism

π^{−1}Db_{M}^{∗} ⟶ C_{M}
(resp. π^{−1}Df_{M}^{∗} ⟶ C_{M}).

Then we define a subsheaf C_{M}^{∗} (resp. C_{M}^{d,∗}) of C_{M} on \sqrt{−1}S^{∗}M as the image of the above morphism and call it the sheaf of microfunctions of class ∗ (resp. d, ∗). Explicitly the sheaf C_{M}^{∗} (resp. C_{M}^{d,∗}) coincides with the sheaf associated to the presheaf

Ω ⟼ Γ(π(Ω); Db_{M}^{∗})/{u ∈ Γ(π(Ω); Db_{M}^{∗}) | SS(u) ∩ Ω = ∅}.
(resp. Ω ⟼ Γ(π(Ω); Df_{M}^{∗})/{u ∈ Γ(π(Ω); Df_{M}^{∗}) | SS(u) ∩ Ω = ∅}.)

Furthermore, we have a canonical exact sequence

0 ⟶ A_{M} ⟶ Db_{M}^{∗} ⟶ π_{∗}C_{M}^{∗} ⟶ 0

(resp. 0 ⟶ A_{M} ⟶ Df_{M}^{∗} ⟶ π_{∗}C_{M}^{d,∗} ⟶ 0).

Note that if † < ∗, we have canonical injections

C_{M}^{d,†} ↪ C_{M}^{d,∗} ↪ C_{M}^{d,(∞)} ↪ C_{M}^{(∞)} ↪ C_{M}^{∗} ↪ C_{M}^{†} ↪ C_{M}.

2.2. Curvilinear expansion

As we know well, the δ-function on ℝ^{n} can be written as

δ(x) = \frac{1}{(2π)^{n}} ∫_{ℝ^{n}} e^{\sqrt{−1}ξx} dξ.

If we change the variables ξ into ξ + \sqrt{−1} | ξ | x, we derive the following representation due to Bros-Iagolnitzer:

δ(x) = \frac{1}{(2π)^{n}} ∫_{ℝ^{n}} (1 + \sqrt{−1}\frac{ξ}{| ξ |}x) e^{\sqrt{−1}ξx−|ξ|x^{2}} dξ.

However if we introduce an extra variable u as in Melin-Sjöstrand [32], we also have

δ(x − x′) = \frac{1}{(2π\sqrt{π})^{n}} ∫_{ℝ^{n}×ℝ^{n}}(2 | ξ |)^{\frac{n}{2}} \{1 + \frac{1}{2} \frac{ξ}{| ξ |}(x − x′)\}
×e^{\sqrt{−1}ξ(x−x′)−|ξ|\{(x−u)^{2}+(x′−u)^{2}\}} dudξ.

If we introduce the polar coordinates ξ = rω here and integrate it with respect to the radius r, we naturally deduce the formula

δ(x − x′) = \frac{2^{n}Γ(n + \frac{n}{2})}{(−2π\sqrt{−1})^{n+ \frac{n}{2}}} ·

· ∫_{ℝ^{n}×S^{n−1}} \frac{\{1 + \frac{1}{2} \sqrt{−1}ω(x − x′)\} dudω}{[ω(x − x′) + \sqrt{−1}\{(x − u)^{2} + (x′ − u)^{2}\}]^{n+ \frac{n}{2}}} .

This formula was obtained by P. Laubin [27]. In this situation we have the following theorem.

Theorem 2.2.1. Let Z be a closed subset of ℝ^{n} × S^{n−1} and let

W_{Z}(x, x′) = \frac{2^{n}Γ(n + \frac{n}{2})}{(−2π\sqrt{−1})^{n+ \frac{n}{2}}} ∫_{Z} \frac{\{1 + \frac{1}{2} \sqrt{−1}ω(x − x′)\} dudω}{[ω(x − x′) + \sqrt{−1}\{(x − u)^{2} + (x′ − u)^{2}\}]^{n+ \frac{n}{2}}} .

Then W_{Z} is a distribution and we have the estimate

SS(W_{Z}) ⊂ \{(x, x′, \sqrt{−1}(ξ, ξ′)∞) | x = x′, ξ = −ξ′, (x, ξ) ∈ Z\}.

Refer to M. Kashiwara et al. [21] for details about the above estimate for SS(W_{Z}).

2.3. Microlocal decomposition of singularities

Now we give our main theorem. Let M be a real analytic manifold in this section.

Theorem 2.3.1. The sheaf C_{M}^{∗} (resp. C_{M}^{d,∗}) is supple.

Before entering the proof, we prepare the following lemma.

Lemma 2.3.2. Let G be a proper open convex subset of the sphere S^{n−1}. Let u be an element of (Db^{∗}/A)(ℝ^{n}) (resp. (Df^{∗}/A)(ℝ^{n})) whose singular spectrum SS(u) is a compact set K of ℝ^{n} × \sqrt{−1}G∞. Let K_{1} and K_{2} be two compact sets of ℝ^{n} × \sqrt{−1}G∞ with K = K_{1} ∪ K_{2}. Then there exists u_{i} ∈ (Db^{∗}/A)(ℝ^{n}) (resp. (Df^{∗}/A)(ℝ^{n})) (i = 1, 2) with u = u_{1} + u_{2} and SS(u_{i}) ⊂ K_{i} (i = 1, 2).

P r o o f. We identify ℝ^{n} × \sqrt{−1}G∞ with ℝ^{n} × G.

a) Let u be an ultradistribution of class ∗ on ℝ^{n} whose singular spectrum SS(u) is a compact set of K of ℝ^{n} × G. We take closed sets Z_{1}, Z_{2} of ℝ^{n} × S^{n−1} such that Z_{1} ∪ Z_{2} = ℝ^{n} × S^{n−1}, K ∩ Z_{1} ⊂ K_{1}, K ∩ Z_{2} ⊂ K_{2} and m(Z_{1} ∩ Z_{2}) = 0 where m implies (2n − 1) dimensional Lebesgue measure on ℝ^{n} × S^{n−1}. Then, by Theorems 1.4.2 and 1.4.3, U_{Z_{1}}(x) and U_{Z_{2}}(x) defined by

U_{Z_{i}}(x) = ∫_{ℝ^{n}} W_{Z_{i}}(x, x′)u(x′) dx′ (i = 1, 2)

are ultradistributions of class ∗ on ℝ^{n} with the estimates

SS(U_{Z_{i}}) ⊂ Z_{i} ∩ SS(u) ⊂ K_{i} (i = 1, 2).

Finally remark that

u(x) = U_{Z_{1}}(x) + U_{Z_{2}}(x).

b) Let u be an ultradifferentiable function of class ∗ on ℝ^{n} whose singular spectrum SS(u) is a compact set K of ℝ^{n} × G. Then we can define U_{Z_{1}} and U_{Z_{2}} in the same way as in the part a) and we obtain the same estimates. Moreover, by Theorems 1.4.2 and 1.4.3, we have

SS^{∗}(U_{Z_{1}}) = SS^{∗}(U_{Z_{2}}) = ∅,

which shows the lemma.

P r o o f o f T h e o r e m 2.3.1. Put

A = \{u ∈ (Db^{∗}/A)(ℝ^{n}) | SS(u) ⊂ ℝ^{n} × \sqrt{−1}G∞, SS(u) is compact\}

(resp. A = \{u ∈ (Df^{∗}/A)(ℝ^{n}) | SS(u) ⊂ ℝ^{n}×\sqrt{−1}G∞, SS(u) is compact\}).

By Lemma 2.3.2 and Theorem 1.1.3 there exists a unique subsheaf G of the sheaf C |_{ℝ^{n}×\sqrt{−1}G∞} such that

a) Γ_{c}(ℝ^{n} × \sqrt{−1}G∞, G) = A,
b) G is supple.

Then it is easy to show that

G = C^{∗} |_{ℝ^{n}×\sqrt{−1}G∞} (resp. G = C^{d,∗} |_{ℝ^{n}×\sqrt{−1}G∞}).

Thus, by Theorem 1.1.4, the theorem is proved.

Note that the suppleness of C_{M}^{∗} was shown by J. W. de Roever [34] for the first time.

Corollary 2.3.3. Let u ∈ Db^{∗}(M) (resp. Df^{∗}(M)) with SS(u) = F.

Let F = ⋃_{i=1}^{p} F_{i} be a closed covering of F in \sqrt{−1}S^{∗}M. Then there exists

u_{i} ∈ Db^{∗}(M) (resp. Df^{∗}(M)) with SS(u_{i}) ⊂ F_{i} and u = ∑_{i=1}^{p} u_{i}.

Corollary 2.3.4 (“Edge of the Wedge theorem”) Let u_{i} (i = 1, …, p) be ultradistributions of class ∗ (resp. ultradifferentiable functions of class

∗) on M with ∑_{i=1}^{p} u_{i} = 0. Let F_{i} = SS(u_{i}) ⊂ \sqrt{−1}S^{∗}M. Then there exist

ultradistributions of class ∗ (resp. ultradifferentiable functions of class ∗) u_{ij}(i, j = 1, …, p) such that

u_{i} = ∑_{j≠i} u_{ij} for any i,

SS(u_{ij}) ⊂ F_{i} ∩ F_{j} for any i, j.

P r o o f. We prove this by induction on p. When p = 2, this corollary is trivial. Now suppose that the statement is true in the case of p − 1. Since

u_{p} = − ∑_{j=1}^{p−1} u_{j},

we have

SS(u_{p}) ⊂ ⋃_{j=1}^{p−1}(F_{p} ∩ F_{j}).

By Corollary 2.3.3 there exist v_{pj} ∈ Db^{∗}(M) (resp. Df^{∗}(M)) (j = 1, …, p− 1) such that

u_{p} = ∑_{j=1}^{p−1} v_{pj}, SS(v_{pj}) ⊂ F_{p} ∩ F_{j}.

Then

∑_{j=1}^{p−1}(u_{j} + v_{pj}) = 0, SS(u_{j} + v_{pj}) ⊂ F_{j}.

Thus by the hypothesis of the induction there exist v_{jk} ∈ Db^{∗}(M) (resp. Df^{∗}(M)) such that

u_{j} + v_{pj} = ∑_{{k=1 \atop k≠j}}^{p−1} v_{jk} j = 1, …, p − 1,

SS(v_{jk}) ⊂ F_{j} ∩ F_{k} j, k = 1, …, p − 1, j ≠ k.

Putting v_{jp} = −v_{pj} and transposing this to the right-hand side, we complete the proof for the case of p.

Definition 2.3.5. We define sheaves C_{M}^{†,∗}, C_{M}^{†/∗}, C_{M}^{d,†,∗} on \sqrt{−1}S^{∗}M by the following exact sequences. i)

0 ⟶ C_{M}^{d,∗} ⟶ C_{M}^{†} ⟶ C_{M}^{†,∗} ⟶ 0.

ii)

0 ⟶ C_{M}^{∗} ⟶ C_{M}^{†} ⟶ C_{M}^{†/∗} ⟶ 0

when ∗ > †. iii)

0 ⟶ C_{M}^{d,∗} ⟶ C_{M}^{d,†} ⟶ C_{M}^{d,†,∗} ⟶ 0

when ∗ < †.

Theorem 2.3.6. i) The sheaf C_{M}^{†,∗} is supple and we have the exact sequence

0 ⟶ Df_{M}^{∗} ⟶ Db_{M}^{†} ⟶ π_{∗}C_{M}^{†,∗} ⟶ 0.

ii) The sheaf C_{M}^{†/∗} is supple and we have the exact sequence

0 ⟶ Db_{M}^{∗} ⟶ Db_{M}^{†} ⟶ π_{∗}C_{M}^{†/∗} ⟶ 0

when ∗ > †. iii) The sheaf C_{M}^{d,†,∗} is supple and we have the exact sequence

0 ⟶ Df_{M}^{∗} ⟶ Df_{M}^{†} ⟶ π_{∗}C_{M}^{d,†,∗} ⟶ 0

when ∗ < †.

P r o o f. i) The suppleness of C_{M}^{†,∗} is directly derived from the definition and Theorem 1.1.5. When we construct a diagram

0  0
↓  ↓
0 → A_{M} → A_{M} → 0
↓ ↓ ↓
0 → Df_{M}^{∗} → Db_{M}^{†} → π_{∗}C_{M}^{†,∗} → 0
↓ ↓ ↓
0 → π_{∗}C_{M}^{d,∗} → π_{∗}C_{M}^{†} → π_{∗}C_{M}^{†,∗} → 0
↓  ↓  ↓
0  0  0

we can easily see the proof by the nine lemmata. The parts ii) and iii) follow from a similar argument as in i).

Theorem 2.3.7. Let Ω be an open subset in \sqrt{−1}S^{∗}M and let u ∈ Db_{M}^{∗}(π(Ω)). Then the following conditions i), ii) are equivalent:

i) SS^{∗}(u) ∩ Ω = ∅, ii) Sp(u) |_{Ω}∈ C_{M}^{d,∗}(Ω).

P r o o f. Let i) hold. For any closed set Z in Ω, we put

v_{Z}(x) = ∫_{π(Ω)} W_{Z}(x, x′)u(x′) dx′,

where W_{Z} is the same as that of Theorem 2.2.1. Then, by Theorems 1.4.2 and 1.4.3, SS^{∗}(v_{Z}) = ∅. Hence v_{Z} ∈ Df_{M}^{∗}(π(Ω)). Moreover,

Sp(u − v_{Z}) | Z^{∘} = 0.

Then we have

Sp(u) | Z^{∘} ∈ C_{M}^{d,∗}( Z^{∘})

for any closed Z in Ω. Therefore we obtain ii). Next let ii) hold. Take q^{∘} ∈ SS^{∗}(u)∩Ω. By the softness of C_{M}^{d,∗} and the exact sequence

0 ⟶ A_{M}(π(Ω)) ⟶ Df_{M}^{∗}(π(Ω)) ⟶ C_{M}^{d,∗}(π^{−1}(π(Ω))) ⟶ 0,

for any closed set Z in Ω, there exists v_{Z} ∈ Df_{M}^{∗}(π(Ω)) so that

Sp(u − v_{Z}) |_{Z}= 0.

Take a closed set Z so that q^{∘} ∈ Z. Since q^{∘} ∉ SS(u − v_{Z}), we have q^{∘} ∉ SS^{∗}(u − v_{Z}). Moreover, q^{∘} ∉ SS^{∗}(v_{Z}), which leads to q^{∘} ∉ SS^{∗}(u).

Moreover, we have the following theorem in the same way as above.

Theorem 2.3.8. Let Ω be an open subset in \sqrt{−1}S^{∗}M and let u ∈ Db_{M}^{†}(π(Ω)) († < ∗). Then the following conditions i), ii) are equivalent:

i) SS^{∗^{−1}}(u) ∩ Ω = ∅, ii) Sp(u) |_{Ω}∈ C_{M}^{∗}(Ω).

These theorems imply that we can generalize the definitions of SS^{∗} and SS^{∗^{−1}} even for hyperfunctions. Therefore we have

Dfinition 2.3.9.Let u ∈ B_{M}(M). We define SS^{∗}(u) (resp. SS^{∗^{−1}}(u)) of u in \sqrt{−1}S^{∗}M. Let q^{∘} = ( x^{∘}, \sqrt{−1} ξ^{∘}∞) ∈ \sqrt{−1}S^{∗}M. Then q^{∘} ∉ SS^{∗}(u) (resp. SS^{∗^{−1}}(u)) if Sp(u) q^{∘} ∈ C_{M, q^{∘}}^{d,∗} (resp. C_{M, q^{∘}}^{∗}).

2.4. Operations for the sheaves C_{M}^{∗} and C_{M}^{d,∗}

We study in this section some operations of C_{M}^{∗} and C_{M}^{d,∗} induced from those of Sato’s microfunctions.

Definition 2.4.1.Let M_{1} and M_{2} be real analytic manifolds and let M = M_{1}×M_{2}. Let (\sqrt{−1}S^{∗}M)′ = \sqrt{−1}S^{∗}M −\sqrt{−1}S^{∗}M_{1}×M_{2}−M_{1}×\sqrt{−1}S^{∗}M_{2}, and define

p_{1} : (\sqrt{−1}S^{∗}M)′ ⟶ \sqrt{−1}S^{∗}M_{1} and p_{2} : (\sqrt{−1}S^{∗}M)′ ⟶ \sqrt{−1}S^{∗}M_{2}

by

p_{1}((x_{1}, x_{2}), \sqrt{−1}(ξ_{1}, ξ_{2})∞) = (x_{1}, \sqrt{−1}ξ_{1}∞)

and

p_{2}((x_{1}, x_{2}), \sqrt{−1}(ξ_{1}, ξ_{2})∞) = (x_{2}, \sqrt{−1}ξ_{2}∞),

respectively.

Theorem 2.4.2. There exist canonical sheaf homomorphisms

p_{1}^{−1}C_{M_{1}}^{∗} × p_{2}^{−1}C_{M_{2}}^{∗} ⟶ C_{M}^{∗} |_{(\sqrt{−1}S^{∗}M)′},
p_{1}^{−1}C_{M_{1}}^{d,∗} × p_{2}^{−1}C_{M_{2}}^{d,∗} ⟶ C_{M}^{d,∗} |_{(\sqrt{−1}S^{∗}M)′} .
(u_{1}(x_{1}), u_{2}(x_{2})) ⟼ u_{1}(x_{1})u_{2}(x_{2})

P r o o f. We only treat the case of C^{∗} because the other case is the same. Let (x_{ν}, \sqrt{−1}ξ_{ν}∞) ∈ \sqrt{−1}S^{∗}M_{ν} and let u_{ν} ∈ C_{M_{ν},(x_{ν},\sqrt{−1}ξ_{ν}∞)}^{∗} (ν = 1, 2). There are f_{1} ∈ Db_{M_{1},x_{1}}^{∗} and f_{2} ∈ Db_{M_{2},x_{2}}^{∗} such that u_{1} = Sp(f_{1}) and u_{2} = Sp(f_{2}) since we have

π^{−1}Db_{M_{ν}}^{∗} ⟶ C_{M_{ν}}^{∗} ⟶ 0.

Then, by Theorem 1.4.1, we can define u_{1}(x_{1})u_{2}(x_{2}) = Sp(f_{1}f_{2}) ∈ C_{M,((x_{1},x_{2}),\sqrt{−1}(ξ_{1},ξ_{2})∞)}^{∗}. We can show that this definition is independent of the choice of f_{1} and f_{2} in the same way as in the case of C_{M}.

Definition 2.4.3. Let N and M be real analytic manifolds, and let f : N ⟶ M be a real analytic map. We associate to f natural maps

\sqrt{−1}S^{∗}N ⟵ ρ N ×_{M} \sqrt{−1}S^{∗}M \ \sqrt{−1}S_{N}^{∗}M ⟶ ϖ \sqrt{−1}S^{∗}M

where

ρ((y, \sqrt{−1}ξ∞)) = (y, \sqrt{−1}f^{∗}(ξ)∞),
ϖ((y, \sqrt{−1}ξ∞)) = (f(y), \sqrt{−1}ξ∞).

We have the following theorems. Refer to M. Kashiwara et al. [21] for more details.

Theorem 2.4.4. Let N be a submanifold of M. Then there exist sheaf homomorphisms

ρ_{!}ϖ^{−1}C_{M}^{∗} ⟶ C_{N}^{∗},
ρ_{!}ϖ^{−1}C_{M}^{d,∗} ⟶ C_{N}^{d,∗}.

Let M be a real analytic manifold, and let Δ_{M} be a diagonal set of M × M. Then define

N = Δ_{M} ×_{M × M} (\sqrt{−1}S^{∗}(M × M)) − Δ_{M} ×_{M × M} (M × \sqrt{−1}S^{∗}M)

−Δ_{M} ×_{M × M} (\sqrt{−1}S_{M}^{∗}M × M) − \sqrt{−1}S_{M}^{∗}(M × M).

For a point x^{⋆} = (x, x, \sqrt{−1}(ξ_{1}, ξ_{2})∞) ∈ N, with ξ_{1} ≠ 0, ξ_{2} ≠ 0, and ξ_{1}+ξ_{2} ≠ 0, we let p_{1}(x^{⋆}) = (x, \sqrt{−1}ξ_{1}∞) ∈ \sqrt{−1}S^{∗}M, p_{2}(x^{⋆}) = (x, \sqrt{−1}ξ_{2}∞) ∈ \sqrt{−1}S^{∗}M, and q(x^{⋆}) = (x, \sqrt{−1}(ξ_{1} + ξ_{2})∞) ∈ \sqrt{−1}S^{∗}M.

Theorem 2.4.5. Under the above notation there exist sheaf homomorphisms

q_{!}(p_{1}^{−1}C_{M}^{∗} × p_{2}^{−1}C_{M}^{∗}) ⟶ C_{M}^{∗},
q_{!}(p_{1}^{−1}C_{M}^{d,∗} × p_{2}^{−1}C_{M}^{d,∗}) ⟶ C_{M}^{d,∗}.
(u(x), v(x)) ⟼ u(x)v(x)

Theorem 2.4.6. There exists a sheaf homomorphism

F q_{!}(p_{1}^{−1}C_{M}^{∗} × p_{2}^{−1}C_{M}^{d,∗}) ⟶ C_{M}^{d,∗}.
(u(x), v(x)) ⟼ u(x)v(x)

P r o o f. If we recall the exact sequences

π^{−1}Df_{M}^{∗} ⟶ C_{M}^{d,∗} ⟶ 0, π^{−1}Db_{M}^{∗} ⟶ C_{M}^{∗} ⟶ 0

and Theorem 1.4.2, the compatibility between the product of hyperfunctions and that of microfunctions shows the theorem.

Theorem 2.4.7. Let M and N be real analytic manifolds, and let ϖ̃ be the natural projection from M ×\sqrt{−1}S^{∗}N to \sqrt{−1}S^{∗}N, and let U be an open subset of \sqrt{−1}S^{∗}N. If for u(t, x) ∈ C_{M×N}^{∗}(ϖ̃^{−1}(U)) (resp. C_{M×N}^{d,∗}(ϖ̃^{−1}(U))) ϖ̃ |_{supp_{u(t,x)}} is a proper map, then the integration v(x) = ∫_{f^{−1}(x)} u(t, x) dt is well-defined as a microfunction of class ∗ (resp. d, ∗). Thus there exists a homomorphism

ϖ̃_{!}(C_{M×N}^{∗} |_{M×\sqrt{−1}S^{∗}N} ⊗v_{M}) ⟶ C_{N}^{∗}
(resp. ϖ̃_{!}(C_{M×N}^{d,∗} |_{M×\sqrt{−1}S^{∗}N} ⊗v_{M}) ⟶ C_{N}^{d,∗})

where v_{M} is the sheaf of real analytic volume elements on M.

2.5. Some classes of microdifferential operators

Let X be an open set in ℂ^{n} and q^{∘} be a point in the cotangent bundle T^{∗}X of X. Let us denote by E_{X}^{∞} (resp. E_{X}) the sheaf on T^{∗}X of rings of microdifferential operators of infinite (resp. finite) order constructed by M. Sato et al. [37]. A microdifferential operator of infinite order P is represented as the infinite formal sum

P = ∑_{j∈ℤ} P_{j}(z, D_{z}),

where P_{j}(z, ζ) is a holomorphic function defined in a neighborhood Ω of q^{∘}, homogeneous of degree j with respect to ζ, and {P_{j}(z, ζ)} satisfies the following conditions i), ii).

i) For any ϵ > 0 and any compact set K of Ω, there is a constant C_{ϵ,K} > 0 such that

| P_{j}(z, ζ) |≤ C_{ϵ,K} \frac{ϵ^{j}}{j!} for j ≥ 0, (z, ζ) ∈ K.

ii) For any compact set K of Ω, there is a constant R_{K} > 0 such that

| P_{j}(z, ζ) |≤ (−j)!R_{K}^{−j} for j < 0, (z, ζ) ∈ K.

Definition 2.5.1. Let Ω be an open subset of T^{∗}X and let P ∈ E_{X}^{∞}(Ω). Then we say that P belongs to E_{X}^{∞,(M_{p})}(Ω) (resp. E_{X}^{∞,{M_{p}}}(Ω)) if and only if, for any compact set K of Ω, there are constants h > 0, C > 0 (resp. for any h > 0 and any compact set K of Ω, there is a constant C > 0) such that

| P_{j}(z, ζ) |≤ C \frac{h^{j}}{M_{j}} for j > 0, (z, ζ) ∈ K.

Note that T. Aoki [3] constructed E_{(ρ)}^{∞} and E_{{ρ}}^{∞} which coincide with our

E_{X}^{∞,(p!^{\frac{1}{ρ}})} and E_{X}^{∞,\{p!^{\frac{1}{ρ}} \}}, respectively.

We can prove the following theorem in the same way as in T. Aoki [3].

Theorem 2.5.2. Let ∗ = (M_{p}) or {M_{p}}.

i) E_{X}^{∞,∗}(Ω) forms a subring of E_{X}^{∞}(Ω). ii) E_{X}^{∞,∗}(Ω) is closed under the adjoint operation.

Note that if ∗ < †

E_{X}^{∞,†} ↪ E_{X}^{∞,∗}.

Now let M be an open

E_{X}^{∞,†} ↪ E_{X}^{∞,∗}.

Now let M be an open subset of ℝ^{n} with complexification X. Then we have

Theorem 2.5.3. C_{M}^{∗} and C_{M}^{d,∗} are left E_{X}^{∞,†} modules if ∗ < †.

P r o o f. First we study the case of C_{M}^{(M_{p})}. Let P (z, D) = ∑_{j∈ℤ} p_{j}(z, D) ∈

E_{X}^{∞,(M_{p})}. For any u ∈ C_{M}^{(M_{p})}, P (x, D)u(x) is defined by

\{ \begin{matrix}P (x, D)u(x) & = ∫ K(x, x′)u(x′) dx′, \\ K(x, x′) & = ∫ K(x, x′, ξ)ω(ξ), \end{matrix}

where K(x, x′, ξ) is a boundary value of

∑_{j∈ℤ} P_{j}(z, \sqrt{−1}ζ)Φ_{j+n}(\sqrt{−1} < z − z′, ζ >)

from Im(< z − z′, ζ >) > 0 and

ω(ξ) = ∑_{i=1}^{n}(−1)^{i−1}ξ_{i}dξ_{1} ∧ … ∧ dξ_{i−1} ∧ dξ_{i+1} ∧ … ∧ dξ_{n}.

On the microlocal decomposition of ultradistributions … 97

Here

Φ_{m}(τ) = \frac{Γ(m)}{(−τ)^{m}} (m ≥ 1),

Φ_{−m}(τ) = \frac{−1}{m!}τ^{m}\{log (−τ) + (γ − 1 − \frac{1}{2} − … − \frac{1}{m})\} (m ≥ 0),

where γ is the Euler constant. Then we have only to show K(x, x′, ξ) ∈ C_{M×M×S^{n−1}}^{(M_{p})} in order to prove P (x, D)u(x) ∈ C_{M}^{(M_{p})} since we know C^{(M_{p})} is closed under integration by Theorem 2.4.7. Thus we estimate ∑_{j=0}^{∞} P_{j}(z, \sqrt{−1}ζ)Φ_{n+j}(\sqrt{−1}τ). For any compact set K there exist C_{K} > 0 and h > 0 such that

| P_{j}(z, \sqrt{−1}ζ) |≤ \frac{C_{K}h^{j}}{M_{j}} (j ≥ 0).

Hence

| P_{j}(z, \sqrt{−1}ζ)Γ(n + j)(−\sqrt{−1}τ)^{−(n+j)} |≤ \frac{C}{| τ |^{n}} \frac{Γ(n + j)}{M_{j}} (\frac{h}{| τ |})^{j} ,

| ∑_{j=0}^{∞} P_{j}(z, \sqrt{−1}ζ)Φ_{n+j}(\sqrt{−1}τ) |≤ C′ exp M^{⋆} (\frac{h′}{| τ |}).

for positive h′ and C′. Therefore, by Theorem 1.2.3, K(x, x′, ξ) ∈ C_{M×M×S^{n−1}}^{(M_{p})} is shown. Next we study the case of C_{M}^{d,(M_{p})}. Since we have

SS^{(M_{p})}(K(x, x′)) ⊂ SS(K(x, x′))
⊂ \{(x, x′, \sqrt{−1}(ξ, η)∞) | x = x′, ξ = −η\},

we obtain

SS^{(M_{p})}(P u) ⊂ SS^{(M_{p})}(u)

by Theorems 2.4.5 and 2.4.7. Therefore P (x, D)u(x) ∈ C_{M}^{d,(M_{p})} if u ∈ C_{M}^{d,(M_{p})}. The other cases are almost the same.

3. Microlocal solvability 3.1. Quantized contact transformations

We give some preliminaries to state our theorems. We have the following theorem due to M. Sato et al. [37].

Theorem 3.1.1. Let M and N be real analytic manifolds of dimension n. Assume that a real-valued real analytic function Ω(x, y) defined on M ×N satisfies the conditions

i) H = {(x, y) ∈ M ×N | Ω(x, y) = 0} is non-singular; i.e., d_{(x,y)}Ω(x, y) ≠ 0 holds on H, ii) We have on H,

det ( \begin{matrix}0 & d_{y}Ω \\ d_{x}Ω & d_{x}d_{y}Ω \end{matrix} ) ≠ 0.

Then, for any P (x, D_{x}) ∈ E_{M}^{∞}, Q(y, D_{y}) ∈ E_{N}^{∞} is uniquely determined such that

∫ P (x, D_{x})δ(Ω(x, y))u(y) dy = ∫ δ(Ω(x, y))Q(y, D_{y})u(y) dy

holds for any microfunction u(y). Conversely, if Q is given, then P is uniquely determined so that the above holds. Moreover, the correspondence P ⟼ Q^{∗} gives rise to the sheaf homomorphism, the quantized contact transformation,

q ∘ p^{−1}E_{M}^{∞} ⟶ ∼ E_{N}^{∞}

where

p : \sqrt{−1}S_{M}^{∗}(M × N) ⟶ \sqrt{−1}S^{∗}M,
q : \sqrt{−1}S_{M}^{∗}(M × N) ⟶ \sqrt{−1}S^{∗}N

are locally isomorphic projections.

Note that this theorem is proved by showing

p^{−1}E_{M}^{∞} ⟶ ∼ E_{M×N}^{∞}δ(Ω(x, y)) = E_{M×N}^{∞}/I

where

I = {P (x, y, D_{x}, D_{y}) ∈ E_{M×N}^{∞} |
P (x, y, D_{x}, D_{y})δ(Ω(x, y)) = 0}.

The following division theorem is proved in the same way as in T. Aoki [3].

Theorem 3.1.2. Let P (x, D_{x}) be a microdifferential operator of finite order m defined in a neighborhood ω of ( x^{∘}, \sqrt{−1} ξ^{∘}dx) = (0, \sqrt{−1}(1, 0, …, 0)dx) ∈ \sqrt{−1}T^{∗}M. Denote its principal symbol by σ(P)(x, \sqrt{−1}ξ) and assume that σ(P)(0, \sqrt{−1}(1, 0, …, ξ_{n}))/ξ_{n}^{p} is holomorphic and never vanishes in a neighborhood of ξ_{n} = 0. Then we can find a neighborhood ω′ of ( x^{∘}, \sqrt{−1} ξ^{∘}dx) such that any S(x, D) ∈ E_{M}^{∞,∗}(ω′) can be written uniquely in the form

S(x, D) = Q(x, D)P (x, D) + R(x, D)

where

Q(x, D), R(x, D) ∈ E_{M}^{∞,∗}(ω′)

and R(x, D) has the form

∑_{j=0}^{p−1} R^{(j)}(x, D′)D_{n}^{j}.

The above theorem implies that, for an arbitrary A(x, y, D_{x}, D_{y}) ∈ E_{M×N}^{∞,∗}, Q_{j} ∈ E_{M×N}^{∞,∗} and Ã(x, D_{x}) ∈ E_{M}^{∞,∗} can be chosen so that, determining Ã uniquely,

A = ∑_{j=1}^{2n} Q_{j}R_{j} + Ã

may hold, where R_{j}(x, y, D_{x}, D_{y}) are properly chosen generators for

J = {P (x, y, D_{x}, D_{y}) ∈ E_{M×N}^{∞,∗} |
P (x, y, D_{x}, D_{y})δ(Ω(x, y)) = 0}.

This shows that we have an isomorphism

E_{M×N}^{∞,∗}/J ⟶ ∼ p^{−1}E_{M}^{∞,∗}.

Therefore we have

Theorem 3.1.3 Under the same assumptions in Theorem 3.1.1. Then we have a quantized contact transformation

q ∘ p^{−1}E_{M}^{∞,∗} ⟶ ∼ E_{N}^{∞,∗}.

100 A. Eida

Moreover, we have the following theorem.

Theorem 3.1.4. Under the same assumptions of Theorem 3.1.1, we have isomorphisms

C_{M}^{∗} ⟶ C_{N}^{∗},
C_{M}^{d,∗} ⟶ C_{N}^{d,∗}.
u(y) ⟼ ∫ δ(Ω(x, y))u(y) dy

P r o o f. It is easy to see

δ(Ω(x, y)) ∈ C_{M×N}^{(∞)}

and

SS(δ(Ω(x, y))) ⊂ \{(x, y, \sqrt{−1}(ξ, η)∞) | Ω(x, y) = 0,
(ξ, η) = cd_{(x,y)}Ω(x, y) (c > 0)}.

In the case of C^{∗}, by Theorems 2.4.5 and 2.4.7, we have

∫ δ(Ω(x, y))u(y) dy ∈ C_{N}^{∗}

if u(y) ∈ C_{M}^{∗}. In the case of C^{d,∗}, we first remark that for u ∈ C_{M}^{∗}, we have

SS^{∗}(∫ δ(Ω(x, y))u(y) dy) ⊂ \{(x, \sqrt{−1}ξ∞) | ∃η_{1} ∃η_{2} 0 ≤ ∃θ ≤ 1 (x, y, \sqrt{−1}(ξ, η_{1})∞) ∈ SS^{∗}(δ(Ω(x, y))), (y, \sqrt{−1}η_{2}∞) ∈ SS^{∗}(u), θη_{1}+(1−θ)η_{2}=0\}. Thus if u ∈ C_{M}^{d,∗}, we deduce

SS^{∗}(∫ δ(Ω(x, y)u(y) dy) = ∅.

This shows

∫ δ(Ω(x, y))u(y) dy ∈ C_{N}^{d,∗}.

The bijectivity directly results from the isomorphism from C_{M} to C_{N}.

3.2. Irregularity for microdifferential operators

Let us recall the definition of the irregularity for a microdifferential operator of finite order due to T. Aoki [4]. Let X be an open subset of ℂ^{n}, and q^{∘} = (z, ζdz) ∈ T^{∗}X. Let P be a microdifferential operator of finite order in a neighborhood of q^{∘} with principal symbol p = σ(P) satisfying p( q^{∘}) = 0. Assume that P has constant multiplicity d in the neighborhood of q^{∘} in the characteristic variety char(P) of P. Then there are holomorphic functions l and k in the neighborhood of q^{∘}, homogeneous with respect to ζ such that

p(z, ζ) = l(z, ζ)k(z, ζ)^{d},
char(P) = {k(z, ζ) = 0},
d_{(z,ζ)}k(z, ζ) ≠ 0 on char(P) ∩ {ζ ≠ 0}
l(z, ζ) ≠ 0 on char(P) ∩ {ζ ≠ 0}.

Let K be a microdifferential operator whose principal symbol is k. Then P can be written in the form

P (z, D) = ∑_{i∈I} L_{i}(z, D)K(z, D)^{i},

where

i) I is a subset of {0, 1, 2, …, d}, ii) L_{i} is a microdifferential operator whose principal symbol σ(L_{i}) does not vanish identically on char(P), iii) the function ω : I ⟶ ℤ defined by ω(i) = ord(L_{i}K^{i}) strictly increases with i.

Definition 3.2.1. The rational number

σ = max_{i∈I\\{d\}} \{1, \frac{d − i}{ω(d) − ω(i)}\}

is said to be the irregularity of the microdifferential operator P at q^{∘}.

T. Aoki [4] showed that the function ω : I ⟶ ℤ does not depend on the choice of K and that a quantized contact transformation preserves the irregularity. Then we have the following theorem due to T. Aoki [3].

Theorem 3.2.2. Let q^{∘} = (0, dz_{n}) ∈ T^{∗}X, and let P be a microdifferential operator of finite order which satisfies the conditions i) p = σ(P) = ζ_{1}^{d}, ii) the irregularity of P at q^{∘} is equal to σ,

Then there exists an invertible operator Q ∈ E_{X}^{∞,(p!^{\frac{σ}{σ−1}} )} satisfying

QP Q^{−1} = D_{1}^{d}.

Note that in the above theorem Q^{−1} ∈ E_{X}^{∞,(p!^{\frac{σ}{σ−1}} )}.

3.3. Microlocal solvability

Let M be an open subset in ℝ^{n}, and X be a complexification of M in ℂ^{n}. We take a coordinate system of T_{M}^{∗}X as (x, \sqrt{−1}ξdx). We set q^{∘} = (0, \sqrt{−1}dx_{n}). Let P be a microdifferential operator of finite order in a neighborhood Ω of q^{∘}. We shall assume

i) p can be written as

p(x, ξ) = l(x, ξ)k(x, ξ)^{d}

with holomorphic functions l and k in Ω homogeneous with respect to ξ, ii) l( q^{∘}) ≠ 0, k( q^{∘}) = 0, iii) k is real valued on T_{M}^{∗}X, homogeneous of order 1 with ξ, dk( q^{∘}) ≠ 0, iv) {k(x, ξ) = 0} is regular involutive near q^{∘} in Ω,

v) the irregularity of P at q^{∘} is σ, vi) ∗ < (p!^{\frac{σ}{σ−1}} ).

Theorem 3.3.1. We assume the above conditions i)-vi). Then

P : C_{M, q^{∘}}^{∗} ⟶ C_{M, q^{∘}}^{∗},
P : C_{M, q^{∘}}^{d,∗} ⟶ C_{M, q^{∘}}^{d,∗}

are surjective.

P r o o f. We find a real contact transformation ϕ satisfying

k ∘ ϕ = ξ_{1}, ϕ( q^{∘}) = q^{∘}.

Then we have real quantized transformations

E_{X, q^{∘}} ⟶ ∼ E_{X, q^{∘}},
C_{M, q^{∘}}^{∗} ⟶ ∼ C_{M, q^{∘}}^{∗},

C_{M, q^{∘}}^{d,∗} ⟶ ∼ C_{M, q^{∘}}^{d,∗}

associated with ϕ. Since a quantized transformation preserves the irregularity of P, we may assume from the beginning

k = ξ_{1}.

Moreover by deviding an elliptic factor of P, we have only to study the operator

P = D_{1}^{d} + (lower).

Then by Theorem 3.2.2, we can find an invertible Q ∈ E_{X}^{∞,(p!^{\frac{σ}{σ−1}} )} which satisfies

QP Q^{−1} = D_{1}^{d}.

We remark Q and Q^{−1} act on C_{M, q^{∘}}^{∗} and C_{M, q^{∘}}^{d,∗}. Thus the surjectivity of

P : C_{M, q^{∘}}^{∗} ⟶ C_{M, q^{∘}}^{∗},

P : C_{M, q^{∘}}^{d,∗} ⟶ C_{M, q^{∘}}^{d,∗}

follows from that of D_{1}^{d}.

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Faculty of Engineering Science Tokyo University of Technology 1404-1 Katakura, Hachioji Tokyo 192-8580 JAPAN

E-mail adress : eida@cc.teu.ac.jp