FORMAL SYMBOL TYPE SOLUTIONS OF FUCHSIAN MICRODIFFERENTIAL EQUATIONS

KIYOOMI KATAOKA AND YOSHIAKI SATOH

Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914 Japan, Fujitsu Co. Ltd

Abstract. We construct a basis of solutions for a micro-differential equation with Fuchsian singularities in microfunctions with one holomorphic parameter. Solutions have the forms U(w, x, D_{x})f(x) with a formal symbol U. We get m−1-regular formal symbols and one singular formal symbol, which is not holomorphic along the Fuchsian singularities but has boundary values in microfunctions.

1. Introduction

Let X be a complex manifold ℂ_{w} ×ℂ_{z}^{n} and Z, M be its submanifolds

Z = {(w, z) ∈ X; Im z = 0} ≃ Z^{ℝ} ⊃ M = {Im w = 0, Im z = 0}.

Here Z^{ℝ} is the underlying real structure of Z. We denote by (w, z; τ, ζ) the coordinates of T^{∗}X;

w = u + iv ∈ ℂ, z = x + iy ∈ ℂ^{n}, τ ∈ ℂ, ζ = ξ + iη ∈ ℂ^{n},

and (τ, ζ) is identified with the real cotangent vector − Re(τdw +ζ ·dz) of T^{∗}Z^{ℝ}. Then the sheaf CO_{Z} on

T_{Z}^{∗}X = {(w, z; τ, ζ) ∈ T^{∗}X; τ = 0, Im z = 0, Re ζ = 0}

of microfunctions with a holomorphic parameter w is defined by

CO_{Z} := {f(u, v, x) ∈ C_{Z^{ℝ}}; ¯∂_{w}f = 0}.

Here C_{Z^{ℝ}} is the sheaf of usual microfunctions on Z^{ℝ}, and it is wellknown that CO_{Z} is identified with the sheaf C_{Z|X} of relative microfunctions as E_{X}-modules. Setting N = ℝ_{x}^{n} ⊂ ℂ_{z}^{n} = Y, we denote by ρ a projection:

(1.1)  ρ : T_{Z}^{∗}X ∋ (w, x; iη) ↦ (x; iη) ∈ T_{N}^{∗}Y.

Let us consider the following microdifferential equation for a section

f(w, x) of CO_{Z} around p^{∘} = (0, x^{∘}; iη^{∘}) ∈ T_{Z}^{∗}X with η^{∘} ≠ 0:

(1.2) P (w, x, D_{w}, D_{x})f(w, x) := ( ∑_{k=0}^{m} A_{k}(w, x, D_{w}, D_{x})D_{w}^{m−k})f = 0,

2000 Mathematics Subject Classification. Primary 35A27; Secondary 35S99.

where D_{w} = ∂/∂w and D_{x_{j}} = ∂/∂x_{j} (j = 1, ..., n). We suppose that P (w, z, D_{w}, D_{z}) has Fuchsian singularities along {w = φ(x, iη)}; that is, A_{k}(w, z, D_{w}, D_{z})’s are microdifferential operators defined at p^{∘} satisfying the conditions:

(1.3)  \begin{cases}ord(A_{k}) ≤ 0 (k = 0, ..., m), \\ σ_{0}(A_{0})(p^{∘}) = 0, ∂_{w}σ_{0}(A_{0})(p^{∘}) ≠ 0, \\ σ_{0}(A_{1})(p^{∘})/∂_{w}σ_{0}(A_{0})(p^{∘}) ∉ ℤ. \end{cases}

Therefore by the Späth type theorem for E_{X} in S-K-K [12] we can write

(1.4)  A_{0}(w, z, D_{w}, D_{z}) = α(w, z, D_{w}, D_{z})(w − Φ(z, D_{w}, D_{z})).

Here α(w, z, D_{w}, D_{z}), Φ(z, D_{w}, D_{z}) ∈ E_{X}|p^{∘} are operators of order 0 with

(1.5)  \begin{cases}[Φ, D_{w}] = 0, σ_{0}(Φ)(z, 0, ζ) ≡ φ(z, ζ), \\ σ_{0}(α)(p^{∘}) ≠ 0, φ(x^{∘}, iη^{∘}) = 0. \end{cases}

Since the equation (1.2) is microlocally equivalent to D_{w}^{m}f(w, x) = 0 in {w − φ(x, iη) ≠ 0}, the solution sheaf in CO_{Z} of the equation (1.2) is isomorphic to ρ^{−1}C_{N}^{m} on {w − φ(x, iη) ≠ 0}. The subject of this article is to study the behavior of solutions in CO_{Z} around the singular locus K = {w−φ(x, iη) = 0}. Precisely, we have the following (Theorem 5.8):

Theorem. We can construct a system {U^{(ℓ)}(w, z, D_{z}); ℓ = 1, ..., m} of formal symbols of microdifferential operators defined around p^{∘} satisfying the following conditions:

(1) U^{(1)} is a multivalued section of E_{X} over {(w, x; iη) ∈ T_{Z}^{∗}X; 0 <

|w −φ(x, iη)| < r, |x−x^{∘}| < r, |η −η^{∘}| < r} for some small r > 0,

and U^{(ℓ)} ∈ E_{X}|p^{∘} for ℓ = 2, ..., m.

(2) U^{(ℓ)}(w, z, D_{z}) (ℓ = 1, ..., m) commute with w.

(3) P (w, z, D_{w}, D_{z})U^{(ℓ)}(w, z, D_{z}) = 0 (mod E_{X} · D_{w}) for ℓ = 1, ..., m.

(4) ord(U^{(ℓ)}) = 0 for ℓ = 1, ..., m, and holomorphic functions {σ_{0}(U^{(ℓ)})(w, x, iη); ℓ = 1, ..., m} give a complete system of solutions in {w − φ(x, iη) ≠ 0} of the following linear ordinary differential equation :

(1.6)  LU := ( ∑_{k=0}^{m} σ_{0}(A_{k})(w, x, 0, iη)\frac{∂^{m−k}}{∂w^{m−k}} )U = 0

(5) For any microfunction f(x) ∈ C_{N}|_{ρ(p^{∘})}, U^{(1)}(w, x, D_{x})f(x) has a microfunction boundary value at w = φ(x, iη); that is, a microfunction boundary value from any side of any conic and real analytic hypersurface H of T_{Z}^{∗}X passing through K = {w − φ(x, iη) = 0}.

The precise meaning of the condition 5 will be given in Section 3. As a direct consequence, we have a unique decomposition of a solution f(w, x) ∈ CO_{Z} around p^{∘} of (1.2) into a sum:

(1.7)  f(w, x) = ∑_{ℓ=1}^{m} U^{(ℓ)}(w, x, D_{x})f_{ℓ}(x),

where f_{ℓ}(x) ∈ C_{N}|_{ρ(p^{∘})} (ℓ = 1, ..., m) are uniquely determined by f(w, x). Further, we conclude from the condition 5 that any solution f(w, x) has a microfunction boundary value at w = φ(x, iη); this fact will be applied to a construction of microlocal solutions for some differential equations with variable multiplicities discussed in Yamane [15], Kataoka [10].

This article consists of 5 sections as follows:

In Section 2, after giving a brief survey on formal symbols, we transform our P into the normal form under some quantized contact transformation preserving T_{Z}^{∗}X. Further we prepare some estimates for holomorphic solutions of Fuchsian ordinary differential equations, which will be used in Section 4.

Section 3 is devoted to give an elementary proof of the invariance of CO_{Z} under quantized contact transformations preserving T_{Z}^{∗}X. The key theorem is Theorem 3.9 on the structure of holomorphic contact transformations preserving T_{Z}^{∗}X: That is, holomorphic contact transformations preserving T_{Z}^{∗}X are essentially generated by the holomorphic functions of the following type:

z_{n}^{∗} = h_{0}(z, z^{∗}′) + (Ψ(w, z, w^{∗}, z^{∗}′))^{2},

where z^{∗}′ = (z_{1}^{∗}, ..., z_{n−1}^{∗}), and holomorphic functions Ψ(w, z, w^{∗}, z^{∗}′), h_{0}(z, z^{∗}′) satisfy the following conditions:

(1) Ψ(w^{∘}, x^{∘}, w^{∘}^{∗}, x^{∘}^{∗}′) = 0, ∂_{w}Ψ ≠ 0, ∂_{w^{∗}}Ψ ≠ 0.

(2) h_{0} is real-valued for real (z, z^{∗}′), and {x_{n}^{∗} −h_{0}(x, x^{∗}′) = 0} gives a real analytic contact transformation S′ : (x; η) ↦ (x^{∗}; η^{∗}).

At the same time, we justify our definition of microfunction boundary values of sections of CO_{Z} from one side of a conic and real analytic hypersurface H of T_{Z}^{∗}X. As direct consequences, we can reduce our P to the normalized operator obtained in Section 2 for the equation P f(w, x) = 0 in CO_{Z}.

In Section 4, we construct formal symbols U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ) satisfying P U = 0 (mod E_{X} · D_{w}) by successive approximation. The key idea here is in applying some suitable formal norms of Boutet-de-Monvel and Krée’s type to prove the convergence. Once we establish some inequalities on those formal norms, we easily obtain a’pri-o’ri estimates for U. The difficulty appears only when we construct the non-regular type formal symbols related to Fuchsian singularities. To deal with this case, we introduce weighted sup-norms for holomorphic functions with Fuchsian singularities and modifications of the formal norms by these weighted sup-norms.

In Section 5, before we deal with our main theorem, we prove under some growth order conditions near a boundary that a classical formal symbol of pseudo-differential operators has a microfunction boundary value. That is, the following is another main result of this article (Theorem 5.5):

Theorem. Let U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ) be a classical formal symbol of a pseudo-differential operator with order ≤ 0 defined in a conic open set

W_{r} ≡{(w, z; ∗, ζ) ∈ T^{∗}X; Im w > 0, |w| < r, |z| < κ,
|ζ_{j}| < ρ|ζ_{n}| (1 ≤ ∀j ≤ n − 1), | Re ζ_{n}| < δ Im ζ_{n}}

for some r, κ, ρ, δ > 0 (δ < 1). We suppose that U_{j} ∈ O(W_{r}) (∀j ≤ 0) and that there exists some constants C, μ > 0 satisfying the following inequalities:

|U_{−p}(w, z, ζ)| ≤ C^{p+1}p!| Im w|^{−p−μ}|ζ|^{−p} on W_{r} (∀p ≥ 0).

Then for any microfunction f(x) ∈ C_{N}|_{(0;idx_{n})}, a section U(w, x, D_{x})f(x) ∈ Γ ({w ∈ ℂ; Im w > 0, |w| < r} × {(0; idx_{n})}; CO_{Z}) has a microfunction boundary value at (0, 0; idx_{n}) from Im w > 0.

Further, we show by a counter-example that the growth condition above is the best possible in some sense.

2. Preliminaries

2.1. Formal symbols and quantized contact transformations.

Definition 2.1. A microdifferential operator Q(w, z, D_{w}, D_{z}) ∈ E_{X} at

q^{∘} = (w^{∘}, z^{∘}; τ^{∘}, ζ^{∘}) ∈ T^{∗}X of order ≤ m(∈ ℤ) is identified with a formal sum

Q(w, z, D_{w}, D_{z}) = ∑_{j=−∞}^{m} Q_{j}(w, z, D_{w}, D_{z}).

of holomorphic functions {Q_{j}(w, z, τ, ζ)}_{j=−∞}^{m} satisfying the following: There exist a conic neighborhood W of q^{∘} in T^{∗}X and a positive constant C such that each Q_{j}(z, x, ζ, ξ) is holomorphic in W, and homogeneous of degree j with respect to (τ, ζ) ∈ ℂ × ℂ^{n}, and that we have the following estimates on W:

(2.1)  |Q_{j}(w, z, τ, ζ)| ≤ (m − j)!C^{1+m−j}(|τ| + |ζ|)^{j} (∀j ≤ m).

The formal sum ∑_{j=−∞}^{m} Q_{j}(w, z, τ, ζ) is called the formal symbol of Q(w, z, D_{w}, D_{z}). The composition of two formal symbols ∑_{j=−∞}^{m′} Q′_{j}, ∑_{j=−∞}^{m′′} Q′′_{j} is defined by a formal symbol ∑_{j=−∞}^{m′+m′′} Q′′′_{j} with

(2.2)  Q′′′_{j}(w, z, τ, ζ) = ∑_{j′+j′′−k−|α|=j} \frac{1}{k!α!} \frac{∂^{k+|α|}Q′_{j′}}{∂τ^{k}∂ζ^{α}} \frac{∂^{k+|α|}Q′′_{j′′}}{∂w^{k}∂z^{α}} ,

where k ∈ ℕ ∪ {0}, α = (α_{1}, …, α_{n}) ∈ (ℕ ∪ {0})^{n}. Note that the summation is performed on some finite terms for each j and that {Q′′′_{j}}_{j} satisfy some estimates like (2.1). It is well-known that this composition rule is associative and gives the operator product Q′(w, z, D_{w}, D_{z}) × Q′′(w, z, D_{w}, D_{z}) ([6, 5], [12]).

Remark 2.2. This definition naturally extends to the classical definition of formal symbols of pseudo-differential operators Q ∈ E_{X}^{ℝ} due to Boutet-de-Monvel and Krée [6, 5]. That is, a formal sum ∑_{j=−∞}^{m} Q_{j}

is said to be a classical formal symbol at q^{∘} = (w^{∘}, z^{∘}; τ^{∘}, ζ^{∘}) ∈ T^{∗}X of pseudo-differential operators of order ≤ m(∈ ℤ) if there exist a conic neighborhood W of q^{∘} in T^{∗}X and a positive constant C such that each Q_{j}(z, x, ζ, ξ) is holomorphic in W satisfying (2.1). Then the most of arguments for formal symbols of E_{X} extend to the arguments for classical formal symbols of E_{X}^{ℝ}. Only one difference is the non-uniqueness of expressions; we cannot determine the j-th order term for a pseudodifferential operator with finite order in general. The modern definition by Aoki [2] of formal symbols of pseudo-differential operators is much simpler, and easy to handle. However, we have an advantage to use the classical formal symbols when we employ the formal norms due to Boutet-de-Monvel and Krée [6]. Indeed we employ some variants of formal norms in the construction of formal symbol type solutions.

Before constructing the solutions of (1.2) we reduce P to a simpler microdifferential operator by using some quantized contact transformation preserving T_{Z}^{∗}X.

Proposition 2.3. Let W^{∗}(w, z, τ, ζ) be a holomorphic function defined at p^{∘} of homogeneous degree 0 with respect to (τ, ζ) satisfying

W^{∗}(p^{∘}) = 0, ∂_{w}W^{∗}(p^{∘}) ≠ 0.

Then there exists a holomorphic contact transformation:

(2.3) S : \begin{cases}w^{∗} = W^{∗}(w, z, τ, ζ), & τ^{∗} = τ^{∗}(w, z, τ, ζ) \\ z_{k}^{∗} = z_{k}^{∗}(w, z, τ, ζ), & ζ_{k}^{∗} = ζ_{k}^{∗}(w, z, τ, ζ) (k = 1, …, n) \end{cases}

defined in a neighborhood of p^{∘} satisfying

(2.4)  \begin{cases}τ^{∗}|_{τ=0} = 0, & ∂_{τ}τ^{∗}(p^{∘}) ≠ 0, \\ z_{k}^{∗}|_{τ=0} = z_{k}, & ζ_{k}^{∗}|_{τ=0} = ζ_{k} (k = 1, …, n). \end{cases}

Proof. Solve the following Cauchy problem for χ = χ(w, z, τ^{∗}, ζ^{∗})

\begin{cases}∂_{τ^{∗}}χ = W^{∗}(w, z, ∂_{w}χ, ∂_{z}χ), \\ χ|_{τ^{∗}=0} = z · ζ^{∗}. \end{cases}

Then the unique holomorphic solution χ at p^{∘} is of homogeneous degree 0 with respect to (τ^{∗}, ζ^{∗}), and generates the desired contact transformation:

S : \begin{cases}w^{∗} = ∂_{τ^{∗}}χ = W^{∗}, & τ = ∂_{w}χ, \\ z_{k}^{∗} = ∂_{ζ_{k}^{∗}}χ, & ζ_{k} = ∂_{z_{k}}χ (k = 1, …, n). \end{cases}

QED

Applying Proposition 2.3 to W^{∗} = w − σ_{0}(Φ)(z, τ, ζ) for Φ at (1.4), we get a holomorphic contact transformation S satisfying (2.3). Since the solution χ has the form χ = wτ^{∗}+z·ζ^{∗}+ψ(z, τ^{∗}, ζ^{∗}) in this case, we know that τ^{∗} = τ. Therefore by the theory of quantizations of contact transformations due to S-K-K [12] we have an isomorphism between sheaves of rings

S : S^{−1}E_{X} ⟶^{∼} E_{X}

such that

S(D_{w^{∗}}) = D_{w}, S(w^{∗}) = w − Φ(z, D_{w}, D_{z}).

Thus we obtain

S^{−1}(P) = α^{∗}(w^{∗}, z^{∗}, D_{w^{∗}}, D_{z^{∗}})w^{∗}D_{w^{∗}}^{m} + ∑_{k=1}^{m} A_{k}^{∗}(w^{∗}, z^{∗}, D_{w^{∗}}, D_{z^{∗}})D_{w^{∗}}^{m−k}.

Here, α^{∗} = S^{−1}(α), A_{k}^{∗} = S^{−1}(A_{k}) ∈ E_{X}|_{S(p^{∘})} (k = 1, …, m) are op-

erators of order ≤ 0, and α^{∗} is an elliptic operator at S(p^{∘}) of order 0. Note that S preserves T_{Z}^{∗}X = {(w, z; τ, ζ) ∈ T^{∗}X; τ = 0, Im z = 0, Re ζ = 0}; that is, in a neighborhood of p^{∘} we have S(T_{Z}^{∗}X) ⊂ T_{Z}^{∗}X. Further the ordinary differential operator L at (1.6) associated with P is transformed into

L^{∗} = ∑_{k=0}^{m} σ_{0}(A_{k}^{∗})(\frac{∂}{∂w^{∗}} )^{m−k} = ∑_{k=0}^{m} (σ_{0}(A_{k}) ∘ S^{−1})(\frac{∂}{∂w^{∗}} )^{m−k}.

Therefore the conditions (1.3) are also satisfied for S^{−1}(P) because

(σ_{0}(A_{1}^{∗})/∂_{w^{∗}}σ_{0}(A_{0}^{∗}))(S(p^{∘})) = σ_{0}(A_{1})(p^{∘})/σ_{0}(α^{∗})(S(p^{∘}))

= σ_{0}(A_{1})(p^{∘})/σ_{0}(α)(p^{∘}) = (σ_{0}(A_{1})/∂_{w}σ_{0}(A_{0}))(p^{∘}).

We strengthen this reduction as follows:

Lemma 2.4. Set K = {σ_{0}(A_{0}) = 0} ∩ T_{Z}^{∗}X = {(w, x; iη) ∈ T_{Z}^{∗}X; w = φ(x, iη)}. Let H be any conic and real analytic hypersurface in T_{Z}^{∗}X passing through K. Then there exist a holomorphic contact transformation S and a quantization S of S defined in a neighborhood p^{∘} such that

(2.5) \begin{cases}S(K) = \{w^{∗} = 0\} ∩ T_{Z}^{∗}X & ⊂ S(H) = \{Im w^{∗} = 0\} ∩ T_{Z}^{∗}X, \\ S(T_{Z}^{∗}X) & ⊂ T_{Z}^{∗}X \end{cases}

and that

S^{−1}(D_{w}) ∈ E_{X} · D_{w^{∗}}, (2.6)

S^{−1}(P) = α^{∗}w^{∗}D_{w^{∗}}^{m} + ∑_{k=1}^{m} A_{k}^{∗}D_{w^{∗}}^{m−k}. (2.7)

Here, α^{∗}, A_{k}^{∗} ∈ E_{X}|_{S(p^{∘})} (k = 1, …, m) are operators of order ≤ 0,

and α^{∗} is an elliptic operator at S(p^{∘}) of order 0. Further S^{−1}(P) also satisfies the conditions (1.3) at S(p^{∘}).

Remark 2.5. As a direct consequence of (2.6) we have the following equivalence for a microdifferential operator U ∈ E_{X}|p^{∘} of order ≤ 0:

P U = 0 (mod E_{X} · D_{w}) ⟺ S^{−1}(P)U^{∗} = 0 (mod E_{X} · D_{w^{∗}})

with U^{∗} = S^{−1}(U). In particular, we get

σ_{0}(U)(w, x, 0, iη) = σ_{0}(U^{∗})(w^{∗}, x^{∗}, 0, iη^{∗})

under the correspondence S : (w, x; 0, iη) ↦ (w^{∗}, x^{∗}, 0, iη^{∗}) and the ordinary differential equations:

L(σ_{0}(U)|_{τ=0}) = 0, L^{∗}(σ_{0}(U^{∗})|_{τ^{∗}=0}) = 0.

Proof. By the arguments above we may suppose that A_{0} = α · w with an elliptic operator α(w, z, D_{w}, D_{z}) ∈ E_{X}|p^{∘} of order 0. Hence K = {w = 0} ∩ T_{Z}^{∗}X and H is written locally as

H = {w = T (t, x, η); t ∈ ℝ}

in T_{Z}^{∗}X. Here T (t, x, η) is some ℂ-valued real analytic function defined at (0, x^{∘}, η^{∘}) of homogeneous degree 0 with respect to η such that

T (0, x, η) ≡ 0 and ∂_{t}T (0, x^{∘}, η^{∘}) ≠ 0.

Find a holomorphic function F (w, z, ζ) of homogeneous degree 0 with respect to ζ satisfying T (F (w, x, iη), x, η) ≡ w by the implicit function theorem. Then we apply Proposition 2.3 to W^{∗} = F (w, z, ζ). Hence we get a holomorphic contact transformation S satisfying (2.5) and

w^{∗} = F (w, z, ζ) = β(w, z, ζ)w, τ^{∗} = γ(w, z, τ, ζ)τ

with some non-vanishing holomorphic functions β, γ of homogeneous degree 0 with respect to (τ, ζ). Choose a quantization S : S^{−1}E_{X} →^{∼} E_{X} of S such that

(2.8)  \begin{cases}S(w^{∗}) = w · β(w, z, D_{z}) + δ(w, z, D_{w}, D_{z}), \\ S(D_{w^{∗}}) = γ(w, z, D_{w}, D_{z})D_{w}, \end{cases}

where δ ∈ E_{X}|p^{∘} is an operator of order ≤ −1. Then, we have

S^{−1}(P) = α^{∗}λ^{∗}(G^{∗}D_{w^{∗}})^{m} + ∑_{k=1}^{m} A_{k}^{∗}(G^{∗}D_{w^{∗}})^{m−k},

where α^{∗} = S^{−1}(α), λ^{∗} = S^{−1}(w), G^{∗} = S^{−1}(γ^{−1}), A_{k}^{∗} = S^{−1}(A_{k}) (k = 1, …, m) are operators of order ≤ 0. Write

(G^{∗}D_{w^{∗}})^{j} = (G^{∗})^{j}D_{w^{∗}}^{j} + ∑_{ℓ=0}^{j−1} G_{jℓ}D_{w^{∗}}^{ℓ}

with some G_{jℓ} ∈ E_{X}|_{S(p^{∘})} of order ≤ 0. Therefore we have

S^{−1}(P) = ∑_{k=0}^{m} A_{k}^{∗′}D_{w^{∗}}^{m−k}

with

A_{0}^{∗′} = α^{∗}λ^{∗}(G^{∗})^{m},
A_{k}^{∗′} = A_{k}^{∗}(G^{∗})^{m−k} + ∑_{j=1}^{k−1} A_{j}^{∗}G_{m−j,m−k} + α^{∗}λ^{∗}G_{m,m−k}

for k = 1, …, m. Since σ_{0}(A_{0}^{∗′}) = w^{∗}σ_{0}(α^{∗})σ_{0}(S^{−1}(β))^{−1}σ_{0}(G^{∗})^{m}, we can write A_{0}^{∗′} as follows:

A_{0}^{∗′} = α^{∗′}(w^{∗}, z^{∗}, D_{w^{∗}}, D_{z^{∗}})(w^{∗} + Ψ(z^{∗}, D_{w^{∗}}, D_{z^{∗}})),

where α^{∗′}, Ψ ∈ E_{X}|_{S(p^{∘})}, ord(α^{∗′}) = 0, ord(Ψ) ≤ −1 and α^{∗′} is an elliptic operator. Find an elliptic operator κ(z^{∗}, D_{w^{∗}}, D_{z^{∗}}) ∈ E_{X}|_{S(p^{∘})} of order 0 satisfying

κ^{−1}(w^{∗} + Ψ(z^{∗}, D_{w^{∗}}, D_{z^{∗}}))κ = w^{∗},

and define a modification S′ of S:

(2.9)  S′(Q) := S(κQκ^{−1}).

Then, S′ is also a quantization of S and S′^{−1}(P) gives the normalized form (2.7) of P. Further, since σ_{0}(A_{1}^{∗′}) = σ_{0}(A_{1}^{∗})σ_{0}(G^{∗})^{m−1} + w^{∗}σ(α^{∗})σ_{0}(S^{−1}(β))^{−1}σ_{0}(G_{m,m−1}), we have

(σ_{0}(A_{1}^{∗′})/∂_{w^{∗}}σ_{0}(A_{0}^{∗′}))(S(p^{∘}))

= (σ_{0}(A_{1}^{∗})/(σ_{0}(α^{∗})σ_{0}(S^{−1}(β))^{−1}σ_{0}(G^{∗})))(S(p^{∘}))

= (σ_{0}(A_{1})σ_{0}(β)σ_{0}(γ)/σ_{0}(α))(p^{∘}) = (σ_{0}(A_{1})/∂_{w}σ_{0}(A_{0}))(p^{∘}).

We used at the last step that (σ_{0}(β)σ_{0}(γ))(p^{∘}) = 1, which is a conclusion from the commutation relation [D_{w^{∗}}, w^{∗}] = 1 for the equations (2.8). Therefore S′^{−1}(P) also satisfies the conditions (1.3). This completes the proof. QED

2.2. Fuchsian ordinary differential operators. For an ε > 0 we set D, Ω ⊂ ℂ as follows:

(2.10)  D = {w ∈ ℂ; |w| ≤ 1},
(2.11)  Ω = {z ∈ ℂ; 0 < |w| ≤ 1, | arg w| ≤ π − ε}.

Let L be an m-th order ordinary differential operator of the form

L = ∑_{k=0}^{m} a_{k}(w)∂_{w}^{m−k},

where a_{0}(w) = w and each a_{k}(w) is holomorphic in a neighborhood of D. Then, we obtain estimations for solutions of

LU = f

for two cases: Holomorphic functions f(w) on D and also on Ω.

Definition 2.6. For a holomorphic function U(w) in a neighborhood of D, we define two norms as follows:

∥U∥ = sup_{D} |U(w)|, ∥U∥′ = sup_{w∈D,0≤j≤m} |U^{(j)}(w)| (2.12)

and define another two norms with weight μ ∈ ℝ by

(2.13)  ∥U∥_{μ} = sup_{w∈Ω} |w|^{μ}|U(w)|,
(2.14)  ∥U∥′_{μ} = sup_{w∈Ω,0≤j≤m} |w|^{μ−m+1+j}|U^{(j)}(w)|

for a holomorphic function U(w) defined in a neighborhood of Ω .

Lemma 2.7. We suppose that a_{1}(0) ≠ 0, −1, −2, . . .. Set

(2.15)  M = 1 + sup_{w∈D} ∑_{k=1}^{m} |a_{k}(w)| < +∞,
(2.16)  δ = min{|p + a_{1}(0)|; p = 0, 1, 2, …} > 0.

Then we have a positive constant C depending only on M and δ, which satisfies the following estimations:

(1) Regular case: For a f(w) ∈ O(D), any solution U(w) ∈ O(D) of LU = f satisfies

(2.17)  ∥U∥′ ≤ C{∥f∥ + |U(0)| + ⋯ + |U^{(m−2)}(0)|}.

(2) Non-regular case : For a f(w) ∈ O(Ω), any solution U(w) ∈ O(Ω) of LU = f satisfies

(2.18)  ∥U∥′_{μ} ≤ C{∥f∥_{μ} + |U(1)| + ⋯ + |U^{(m−1)}(1)|}

with ∀μ ≥ M + m + 1.

Remark 2.8. It is well known by the theory of Fuchsian differential equations that under the assumption a_{1}(0) ≠ 0, −1, −2, …, there exists a unique solution for any given (U(0), …, U^{(m−2)}(0)) or (U(1), ..., U^{(m−1)}(1)) for both cases.

Proof. Put an m × m-matrix

A(w) = (\begin{matrix}0 & w & 0 & ⋯ & 0 \\ ⋮ & ... & ... & ... & ⋮ \\ ⋮ & & ... & ... & 0 \\ 0 & ⋯ & ⋯ & 0 & w \\ −a_{m}(w) & ⋯ & ⋯ & ⋯ & −a_{1}(w) \end{matrix}) ,

and two m-dimensional vectors

X(w) = ( U(w) ; U′(w) ; ⋮ ; U^{(m−1)}(w)) , B(w) = ( 0 ; ⋮ ; 0 ; f(w)) .

Then, LU = f reduces to

\frac{dX(w)}{dw} = \frac{1}{w}A(w)X(w) + \frac{1}{w}B(w).

Hence,

(2.19)  X(w) = X(w_{0}) + ∫_{w_{0}}^{w} \frac{1}{s}A(s)X(s)ds + ∫_{w_{0}}^{w} \frac{1}{s}B(s)ds.

Firstly we consider the non-regular case. We introduce the following norms for m × m matrix X = (X_{jk})_{j,k=1}^{m} and m-vector B = (B_{j})_{j=1}^{m}:

|X| ≡ max_{j=1,...,m} ( ∑_{k=1}^{m} |X_{jk}|), |B| ≡ max_{j=1,...,m} |B_{j}|.

Therefore we have |A(w)| ≤ M on D. We put w = e^{iθ} and w_{0} = 1 in (2.19) and we get the following integral inequality for θ ∈ [0, π − ε]:

(2.20)  |X(e^{iθ})| ≤ |X(1)| + ∫_{0}^{θ} M|X(e^{iφ})|dφ + ∫_{0}^{θ} |f(e^{iφ})|dφ
≤ |X(1)| + π∥f∥_{μ} + ∫_{0}^{θ} M|X(e^{iφ})|dφ.

Further putting w = re^{iθ} and w_{0} = e^{iθ} we get the following for |θ| ≤ π − ε and r ∈ (0, 1]:

(2.21) |X(re^{iθ})| ≤ |X(e^{iθ})| + ∫_{r}^{1} \frac{M}{s} |X(se^{iθ})|ds + ∫_{r}^{1} \frac{|f(se^{iθ})|}{s}ds

≤ |X(e^{iθ})| + \frac{r^{−μ} − 1}{μ}∥f∥_{μ} + ∫_{r}^{1} \frac{M}{s} |X(se^{iθ})|ds.

Now we apply Gronwall’s lemma to (2.21) for μ ≥ M + m + 1:

|X(re^{iθ})| ≤ |X(e^{iθ})| + \frac{r^{−μ} − 1}{μ}∥f∥_{μ}

+ ∫_{r}^{1} \{|X(e^{iθ})| + \frac{t^{−μ} − 1}{μ}∥f∥_{μ}\}\frac{M}{t} exp( ∫_{r}^{t} \frac{M}{s} ds)dt

≤ r^{−M}|X(e^{iθ})| + \frac{r^{−μ}}{μ − M} ∥f∥_{μ} ≤ r^{−M}|X(e^{iθ})| + r^{−μ}∥f∥_{μ}.

In the same way, we obtain from (2.20) that

|X(e^{iθ})| ≤ |X(1)| + π∥f∥_{μ} + ∫_{0}^{θ}{|X(1)| + π∥f∥_{μ}}Me^{M(θ−φ)}dφ
≤ e^{Mπ}{|X(1)| + π∥f∥_{μ}}.

The last inequality holds for |θ| ≤ π − ε. Therefore we have

|X(re^{iθ})| ≤ r^{−M}e^{Mπ}|X(1)| + (r^{−M}πe^{Mπ} + r^{−μ})∥f∥_{μ}
≤ r^{−μ}(1 + πe^{Mπ})(∥f∥_{μ} + |X(1)|).

Thus we obtain the inequalities:

|U^{(m)}(w)| = |w|^{−1} · | − a_{1}(w)U^{(m−1)}(w) − ⋯ − a_{m}(w)U(w) + f(w)|
≤ |w|^{−μ−1}M(1 + πe^{Mπ})(∥f∥_{μ} + |X(1)|),

and

(2.22) |U^{(m−1)}(w)| ≤ |X(w)| ≤ |w|^{−μ}(1 + πe^{Mπ})(∥f∥_{μ} + |X(1)|).

Further by using (2.22) and some integral expressions we get recursively the estimates for |w|^{μ+j−m+1}U^{(j)}(w) (j = m−1, …, 0). Hence we have the estimate (2.18).

To deal with the regular case we expand A(w), B(w), X(w) into power series:

A(w) = ∑_{p=0}^{∞} A_{p}w^{p}, B(w) = ∑_{p=0}^{∞} B_{p}w^{p}, X(w) = ∑_{p=0}^{∞} X_{p}w^{p}.

Hence we have the following equations for the coefficients:

(2.23)  \begin{cases}a_{1}(0)U^{(m−1)}(0) + ⋯ + a_{m}(0)U^{(0)}(0) = f(0), \\ (p − A_{0})X_{p} = ∑_{q=1}^{p} A_{q}X_{p−q} + B_{p} (p ≥ 1). \end{cases}

By the Cauchy estimates we obtain |A_{p}| ≤ M, |B_{p}| ≤ ∥f∥ (p ≥ 0). Further (p − A_{0})^{−1} is given by

(p − A_{0})^{−1} = \frac{1}{p(p + a_{1}(0))} (\begin{matrix}p + a_{1}(0) & 0 & ⋯ & ⋯ & 0 \\ 0 & ... & ... & & ⋮ \\ ⋮ & ... & ... & ... & ⋮ \\ 0 & ⋯ & 0 & p + a_{1}(0) & 0 \\ −a_{m}(0) & ⋯ & ⋯ & −a_{2}(0) & p \end{matrix}) .

for p ≥ 1. Therefore we have

|X_{p}| ≤ |(p − A_{0})^{−1}|( ∑_{q=1}^{p} |A_{q}||X_{p−q}| + |B_{p}|)

≤ max \{\frac{1}{p}, \frac{p + |a_{2}(0)| + ⋯ + |a_{m}(0)|}{p|p + a_{1}(0)|}\}( ∑_{q=1}^{p} M|X_{p−q}| + ∥f∥)

≤ K(M ∑_{q=0}^{p−1} |X_{q}| + ∥f∥)

with K = max{1, M/δ}. Thus, adding ∑_{q=0}^{p−1} |X_{q}| to both sides we get

∑_{q=0}^{p} |X_{q}| ≤ (KM + 1) ∑_{q=0}^{p−1} |X_{q}| + K∥f∥

≤ \frac{(KM + 1)^{p} − 1}{M}∥f∥ + (KM + 1)^{p}|X_{0}|.

Since we obtain from the first equation of (2.23) that |U^{(m−1)}(0)| ≤ (1/δ)(M|U^{(m−2)}(0)| + ⋯ + M|U^{(0)}(0)| + ∥f∥), we have

|X_{0}| ≤ K(|U^{(m−2)}(0)| + ⋯ + |U^{(0)}(0)| + ∥f∥).

Consequently

|X_{p}| ≤ (KM + 1)^{p}\{(\frac{1}{M} + K)∥f∥ + K(|U(0)| + ⋯ + |U^{(m−2)}(0)|)\}

for ∀p ≥ 0, and so we have

sup \{|X(z)|; |z| ≤ \frac{1}{2(KM + 1)}\} ≤ 2(K + \frac{1}{M} )(∥f∥ + ∑_{j=0}^{m−2} |U^{(j)}(0)|).

Putting r_{0} = 1/{2(MK + 1)} < 1, we get an integral inequality similar to (2.21) for r ≥ r_{0} :

|X(re^{iθ})| ≤ |X(r_{0}e^{iθ})| + ∥f∥ log \frac{1}{r_{0}} + ∫_{r_{0}}^{r} \frac{M}{s} |X(se^{iθ})|ds.

Hence by Gronwall’s lemma we obtain for r ∈ [r_{0}, 1] that

|X(re^{iθ})| ≤ (\frac{r}{r_{0}} )^{M}\{2(K + \frac{1}{M}) + log \frac{1}{r_{0}} \}(∥f∥ + ∑_{j=0}^{m−2} |U^{(j)}(0)|)

Therefore,

sup_{D} |X(w)| ≤ (\frac{1}{r_{0}} )^{M}\{2(K + \frac{1}{M}) + log \frac{1}{r_{0}} \}(∥f∥ + ∑_{j=0}^{m−2} |U^{(j)}(0)|)

Note that

sup_{w∈D} |U^{(m)}(w)| = sup_{|w|=1} |\frac{− ∑_{j=1}^{m} a_{j}(w)U^{(m−j)}(w) + f(w)}{w}|

≤ M sup_{w∈D} |X(w)| + ∥f∥.

Therefore since M ≥ 1,

∥U∥′ ≤ M sup_{w∈D} |X(w)| + ∥f∥ ≤ C(∥f∥ + ∑_{j=0}^{m−2} |U^{(j)}(0)|)

with

C = M\{2(KM + 1)\}^{M}[2(K + \frac{1}{M} ) + log \{2(KM + 1)\}] + 1

and

K = max{1, M/δ}.

This completes the proof of Lemma 2.7. QED

3. Quantized contact transformations for sheaf CO_{Z}

Before introducing quantized contact transformations for sheaf CO_{Z}, we investigate the structure of holomorphic contact transformations preserving T_{Z}^{∗}X.

Definition 3.1. Let S : T^{∗}X →^{∼} T^{∗}X be a holomorphic contact transformation defined in a neighborhood of p^{∘} ∈ T^{∗}X \ X. Then the antigraph of S is defined by the following G:

G = {(w, z, w^{∗}, z^{∗}; τ, ζ, −τ^{∗}, −ζ^{∗}) ∈ T^{∗}(X × X)},
π : T^{∗}(X × X) ⊃ G → π(G) ⊂ X × X.

Here (w^{∗}, z^{∗}; τ^{∗}, ζ^{∗}) = S((w, z; τ, ζ)) and (w, z; τ, ζ) moves over a neighborhood of p^{∘}, and π denotes the natural projection. It is clear that G becomes a conic and complex Lagrangian submanifold of T^{∗}(X × X).

S is said to be of generic type iff the projection π(G) becomes a complex hypersurface of X × X; more precisely, there exists either one of coordinates z_{1}^{∗}, …, z_{n}^{∗}, for example z_{n}^{∗}, such that we have

(3.1)  π(G) = {g ≡ z_{n}^{∗} − h(w, z, w^{∗}, z^{∗}′) = 0}

with some holomorphic function h. Here z^{∗}′ = (z_{1}^{∗}, …, z_{n−1}^{∗}).

Remark 3.2. We can get such an expression (3.1) if the complex submanifold G with dimension 2n + 2 has (w, z, w^{∗}, z^{∗}′, ζ_{n}^{∗}) as a local

coordinate system in a neighborhood of (w^{∘}, z^{∘}, w^{∘}^{∗}, z^{∘}^{∗}; τ^{∘}, ζ^{∘}, −τ^{∘}^{∗}, −ζ^{∘}^{∗}). It is the most important in this case that G coincides with the conormal (line) bundle of π(G):

G = T_{π(G)}^{∗}(X × X).

That is, we have the equations:

(3.2) \begin{cases}z_{n}^{∗} = h(w, z, w^{∗}, z^{∗}′) \\ τ^{∗} = −ζ_{n}^{∗}∂_{w^{∗}}h, & ζ_{k}^{∗} = −ζ_{n}^{∗}∂_{z_{k}^{∗}}h (k = 1, …, n − 1), \\ τ = ζ_{n}^{∗}∂_{w}h, & ζ_{j} = ζ_{n}^{∗}∂_{z_{j}}h (j = 1, …, n). \end{cases}

Further, a holomorphic function h(w, z, w^{∗}, z^{∗}′) in (3.1) induces a local contact transformation iff

(3.3)  det (\begin{matrix}∂_{w}∂_{w^{∗}}h, & ∂_{w}∂_{z^{∗}′}h, & ∂_{w}h \\ ∂_{z}∂_{w^{∗}}h, & ∂_{z}∂_{z^{∗}′}h, & ∂_{z}h \end{matrix}) ≠ 0.

A contact transformation preserving T_{Z}^{∗}X also preserves {τ = 0}. As for these transformations we have the following lemma:

Lemma 3.3. Let S : T^{∗}X ∋ (w, z; τ, ζ) ↦ (w^{∗}, z^{∗}; τ^{∗}, ζ^{∗}) ∈ T^{∗}X be a holomorphic contact transformation defined in a neighborhood p^{∘} =

(w^{∘}, z^{∘}; 0, ζ^{∘}) with ζ^{∘} ≠ 0. We suppose that S preserves {τ = 0}; that is, S({τ = 0}) ⊂ {τ^{∗} = 0}. Let S be given by the holomorphic functions

\begin{cases}w^{∗} = W (w, z, τ, ζ), & τ^{∗} = T (w, z, τ, ζ), \\ z_{j}^{∗} = Z_{j}(w, z, τ, ζ), & ζ_{j} = Ξ_{j}(w, z, τ, ζ) (j = 1, ..., n). \end{cases}

Then, the specializations Z_{j}(w, z, 0, ζ), Ξ_{j}(w, z, 0, ζ) do not depend on w for j = 1, ..., n. Hence S induces a holomorphic contact transformation

(3.4)  S′ : T^{∗}Y ∋ (z; ζ) ↦ (Z(∗, z, 0, ζ); Ξ(∗, z, 0, ζ)) ∈ T^{∗}Y.

Proof. By the assumption we have

∑_{j=1}^{n} dΞ_{j}(w, z, 0, ζ) ∧ dZ_{j}(w, z, 0, ζ) = ∑_{j=1}^{n} dζ_{j} ∧ dz_{j}.

Therefore we obtain a system of equations for k = 1, ..., n:

∑_{j=1}^{n} (\frac{∂Z_{j}}{∂z_{k}} \frac{∂Ξ_{j}}{∂w} − \frac{∂Ξ_{j}}{∂z_{k}} \frac{∂Z_{j}}{∂w} )|_{τ=0} = 0,

∑_{j=1}^{n} (\frac{∂Z_{j}}{∂ζ_{k}} \frac{∂Ξ_{j}}{∂w} − \frac{∂Ξ_{j}}{∂ζ_{k}} \frac{∂Z_{j}}{∂w} )|_{τ=0} = 0.

Since the (2n) × (2n)-matrix

(\begin{matrix}∂Z_{j}/∂z_{k}, & ∂Ξ_{j′}/∂z_{k} \\ ∂Z_{j}/∂ζ_{k′}, & ∂Ξ_{j′}/∂ζ_{k′} \end{matrix})_{kk′,jj′}

is non-singular, we have that

\frac{∂Ξ_{k}}{∂w} (w, z, 0, ζ) = 0, \frac{∂Z_{k}}{∂w} (w, z, 0, ζ) = 0 (k = 1, ..., n).

QED

Example 3.4. Let α ∈ ℂ be a non-zero constant. Set 2 generating functions g_{0}, g_{1} by

(3.5) g_{0} ≡ z_{n}^{∗} − z_{n} + ∑_{j=1}^{n−1} z_{j}z_{j}^{∗} + \frac{1}{2α}(w^{∗} − w)^{2}  (n ≥ 1),

(3.6) g_{1} ≡ z_{n}^{∗} − z_{n} + ∑_{j=2}^{n−1} z_{j}z_{j}^{∗} + (z_{1}^{∗} − z_{1})(w^{∗} − w)  (n ≥ 2).

Then the contact transformations S_{0}, S_{1} corresponding to {g_{0} = 0}, {g_{1} = 0} respectively are given as follows:

(3.7) S_{0} : \begin{cases}τ^{∗} = τ, & w^{∗} = w + α(τ/ζ_{n}), \\ ζ_{j}^{∗} = ζ_{n}z_{j}, & z_{j}^{∗} = −ζ_{j}/ζ_{n}  (j = 1, …, n − 1), \\ ζ_{n}^{∗} = ζ_{n}, & z_{n}^{∗} = z_{n} + (∑_{k=1}^{n−1} ζ_{k}z_{k})/ζ_{n} − \frac{α}{2}(τ/ζ_{n})^{2}. \end{cases}

(3.8) S_{1} : \begin{cases}τ^{∗} = τ, & w^{∗} = w + (ζ_{1}/ζ_{n}), \\ ζ_{1}^{∗} = ζ_{1}, & z_{1}^{∗} = z_{1} + (τ/ζ_{n}), \\ ζ_{j}^{∗} = ζ_{n}z_{j}, & z_{j}^{∗} = −ζ_{j}/ζ_{n}  (j = 2, …, n − 1), \\ ζ_{n}^{∗} = ζ_{n}, & z_{n}^{∗} = z_{n} + (∑_{k=2}^{n−1} ζ_{k}z_{k})/ζ_{n} − (τζ_{1}/ζ_{n}^{2}). \end{cases}

It is clear that S_{0}, S_{1} are holomorphic contact transformations of generic type preserving T_{Z}^{∗}X. Further if α ∈ ℝ\{0}, S_{0}, S_{1} also preserve T_{M}^{∗}X; that is, real contact transformations.

As we see in the next theorem, S_{0} is a typical example of the generic and normal case, and S_{1} is a typical example of the generic and nonnormal case.

Theorem 3.5. Let S : T^{∗}X →^{∼} T^{∗}X be a holomorphic contact transformation defined in a neighborhood of p^{∘} = (w^{∘}, x^{∘}; 0, iη^{∘}) ∈ T_{Z}^{∗}X with η^{∘} ≠ 0 preserving T_{Z}^{∗}X. We assume that S is of generic type and that the anti-graph is given by the conormal bundle of {z_{n}^{∗} = h(w, z, w^{∗}, z^{∗}′)} like (3.1). Set S(p^{∘}) = p^{∘}^{∗} = (w^{∘}^{∗}, x^{∘}^{∗}; 0, iη^{∘}^{∗}). We suppose η^{∘}_{n}^{∗} ≠ 0 and the following condition (the normal case condition):

∂_{w^{∗}}^{2}h(w^{∘}, x^{∘}, w^{∘}^{∗}, x^{∘}^{∗}′) ≠ 0.

Then h has a form:

(3.9)  h = h_{0}(z, z^{∗}′) + Ψ(w, z, w^{∗}, z^{∗}′)^{2},

where holomorphic functions Ψ(w, z, w^{∗}, z^{∗}′), h_{0}(z, z^{∗}′) satisfy the following conditions:

(1) Ψ(w^{∘}, x^{∘}, w^{∘}^{∗}, x^{∘}^{∗}′) = 0, ∂_{w}Ψ ≠ 0, ∂_{w^{∗}}Ψ ≠ 0.

(2) h_{0} is real-valued for real (z, z^{∗}′), and {x_{n}^{∗} −h_{0}(x, x^{∗}′) = 0} gives a real analytic contact transformation S′ : (x; η) ↦ (x^{∗}; η^{∗}), which is the induced transformation in the sense of Lemma 3.3.

Conversely, if Ψ, h_{0} satisfy these conditions, then the hypersurface {z_{n}^{∗} = h_{0}(z, z^{∗}′) + Ψ(w, z, w^{∗}, z^{∗}′)^{2}} generates a holomorphic contact transformation preserving T_{Z}^{∗}X.

Proof. Since S preserves T_{Z}^{∗}X, we have a nowhere-vanishing holomorphic function ϕ(w, z, w^{∗}, z^{∗}′) satisfying

(3.10)  ∂_{w}h = ϕ∂_{w^{∗}}h.

Here we note that ∂_{w}h = ∂_{w^{∗}}h = 0 at (w^{∘}, x^{∘}, w^{∘}^{∗}, x^{∘}^{∗}′). By the assumption ∂_{w^{∗}}^{2}h(w^{∘}, x^{∘}, w^{∘}^{∗}, x^{∘}^{∗}′) ≠ 0 we can find a holomorphic function w^{∗} = ψ(w, z, z^{∗}′) satisfying

∂_{w^{∗}}h|_{w^{∗}=ψ} = 0, ψ(w^{∘}, z^{∘}, z^{∘}^{∗}′) = w^{∘}^{∗}.

Thus, by expanding h into a power series of w^{∗} − ψ(w, z, z^{∗}′), we know that any branch

Ψ(w, z, w^{∗}, z^{∗}′) ≡ ±\sqrt{h − (h|_{w^{∗}=ψ})}

is a holomorphic function satisfying ∂_{w^{∗}}Ψ ≠ 0 at (w^{∘}, z^{∘}, w^{∘}^{∗}, z^{∘}^{∗}′). We note here that the critical value

h_{0} ≡ h(w, z, Ψ(w, z, z^{∗}′), z^{∗}′)

does not depend on w. Because

∂_{w}h_{0} = (∂_{w}h)|_{w^{∗}=Ψ} + (∂_{w^{∗}}h)|_{w^{∗}=Ψ} · ∂_{w}Ψ
= (ϕ|_{w^{∗}=Ψ} + ∂_{w}Ψ)(∂_{w^{∗}}h)|_{w^{∗}=Ψ} = 0.

Therefore we can write h in the form (3.9). Further, on {Ψ = 0} we know that the determinant of the matrix (3.3) is equal to

Ψ_{w}Ψ_{w^{∗}} det (∂_{z}∂_{z^{∗}′}h_{0}, ∂_{z}h_{0}) .

Hence we directly obtain the conditions 1, 2 and the converse statement. QED

Lemma 3.6. Let S : T^{∗}X → T^{∗}X be any holomorphic contact transformation defined in a neighborhood of p^{∘} = (w^{∘}, x^{∘}; 0, iη^{∘}) ∈ T_{Z}^{∗}X with η^{∘} ≠ 0 preserving T_{Z}^{∗}X. Suppose that the induced transformation S′ : T^{∗}Y → T^{∗}Y for S is equal to the identity map. Then for the contact transformation S_{0} in (3.7) with a sufficiently large positive number α, the composition S_{0} ∘ S becomes a holomorphic contact transformation preserving T_{Z}^{∗}X of the generic and normal case type in the sense of Theorem 3.5.

Proof. We may assume that η^{∘}_{n} ≠ 0. We write

S : (w, z; τ, ζ) ↦ (w^{∗}, z^{∗}; τ^{∗}, ζ^{∗}),
S_{0} : (w^{∗}, z^{∗}; τ^{∗}, ζ^{∗}) ↦ (w^{∗∗}, z^{∗∗}; τ^{∗∗}, ζ^{∗∗}).

In order to get an expression {z_{n}^{∗∗} − h(w, z, w^{∗∗}, z^{∗∗}′) = 0} for the projection of the anti-graph of S_{0} ∘ S, It is sufficient to show that

(3.11)  I ≡ det (\begin{matrix}∂_{τ}w^{∗∗}, & ∂_{τ}z^{∗∗}′, & ∂_{τ}ζ_{n}^{∗∗} \\ ∂_{ζ}w^{∗∗}, & ∂_{ζ}z^{∗∗}′, & ∂_{ζ}ζ_{n}^{∗∗} \end{matrix}) ≠ 0

at p^{∘}. Here z^{∗∗}′ = (z_{1}^{∗∗}, ..., z_{n−1}^{∗∗}) is a row vector, and ∂_{ζ} denotes the column vector of the gradient here. For any function F (w^{∗}, z^{∗}, τ^{∗}, ζ^{∗}) we have that

∂_{ζ_{j}}(F ∘ S) = ∂_{ζ_{j}}w^{∗} · ∂_{w^{∗}}F + ∂_{ζ_{j}^{∗}}F (j = 1, ..., n)

on {τ = 0}. Therefore at p^{∘} we have

I = det (\begin{matrix}∂_{τ}w^{∗} + (α∂_{τ}τ^{∗})/ζ_{n}^{∗}, & ∂_{τ}ζ^{∗}∂_{ζ^{∗}}(−ζ^{∗}′/ζ_{n}^{∗}), & ∂_{τ}ζ_{n}^{∗} \\ ∂_{ζ}w^{∗}, & ∂_{ζ^{∗}}(−ζ^{∗}′/ζ_{n}^{∗}), & ∂_{ζ^{∗}}ζ_{n}^{∗} \end{matrix})

= det (\begin{matrix}∂_{τ}w^{∗} + (α∂_{τ}τ^{∗})/ζ_{n}^{∗} − ∂_{ζ_{n}}w^{∗}∂_{τ}ζ_{n}^{∗}, & −(∂_{τ}ζ^{∗}′)/ζ_{n}^{∗} \\ ∂_{ζ′}w^{∗}, & −E_{n−1}/ζ_{n}^{∗} \end{matrix})

= (−ζ_{n}^{∗})^{1−n}(∂_{τ}w^{∗} + (α∂_{τ}τ^{∗})/ζ_{n}^{∗} − ∑_{j=1}^{n} ∂_{ζ_{j}}w^{∗}∂_{τ}ζ_{j}^{∗}).

Here E_{n−1} is the identity matrix of size n−1. Since ∂_{τ}τ^{∗}(p^{∘}) ≠ 0, we have I ≠ 0 for any sufficiently large α > 0. Note that ∂_{w^{∗∗}}^{2}h(w^{∘}, x^{∘}, w^{∘}^{∗∗}, x^{∘}^{∗∗}′) ≠ 0 is equivalent to ˜∂_{w^{∗∗}}τ^{∗∗} ≠ 0, where ˜∂ means the differentiation in the coordinates (w, z, w^{∗∗}, z^{∗∗}′, ζ_{n}^{∗∗}). On the other hand we have

∂_{τ}τ^{∗∗} = ∂_{τ}w^{∗∗} ˜∂_{w^{∗∗}}τ^{∗∗} + ∑_{j=1}^{n−1} ∂_{τ}z_{j}^{∗∗} ˜∂_{z_{j}^{∗∗}}τ^{∗∗} + ∂_{τ}ζ_{n}^{∗∗} ˜∂_{ζ_{n}^{∗∗}}τ^{∗∗},

∂_{ζ}τ^{∗∗} = ∂_{ζ}w^{∗∗} ˜∂_{w^{∗∗}}τ^{∗∗} + ∑_{j=1}^{n−1} ∂_{ζ}z_{j}^{∗∗} ˜∂_{z_{j}^{∗∗}}τ^{∗∗} + ∂_{ζ}ζ_{n}^{∗∗} ˜∂_{ζ_{n}^{∗∗}}τ^{∗∗}.

Hence we obtain

˜∂_{w^{∗∗}}τ^{∗∗} = \frac{1}{I} det (\begin{matrix}∂_{τ}τ^{∗∗}, & ∂_{τ}z^{∗∗}′, & ∂_{τ}ζ_{n}^{∗∗} \\ ∂_{ζ}τ^{∗∗}, & ∂_{ζ}z^{∗∗}′, & ∂_{ζ}ζ_{n}^{∗∗} \end{matrix}) .

Since ∂_{ζ}τ^{∗∗} = 0 on {τ = 0}, we get

˜∂_{w^{∗∗}}τ^{∗∗} = \frac{∂_{τ}τ^{∗∗}}{I} det (∂_{ζ^{∗}}(−ζ^{∗}′/ζ_{n}^{∗}), ∂_{ζ^{∗}}ζ_{n}^{∗}) = \frac{∂_{τ}τ^{∗∗}}{I(−ζ_{n}^{∗})^{n−1}} ≠ 0

at p^{∘}. This completes the proof. QED

Theorem 3.7. Let S be a holomorphic contact transformation defined in a neighborhood of p^{∘} ∈ T_{Z}^{∗}X \ Z preserving T_{Z}^{∗}X, and S be any quantization of S. Then, there exists a sheaf isomorphism

T_{S} : S^{−1}CO_{Z} ⟶^{∼} CO_{Z}

satisfying

T_{S}(Qf) = S(Q)T_{S}(f)

at any point q near p^{∘} for any germs f ∈ CO_{Z}|_{S(q)}, Q ∈ E_{X}|_{S(q)}. Further such a T_{S} is determined up to a constant by S.

Remark 3.8. This result is proven in a general situation in [7, 8]. We introduce here a more elementary proof based on Theorem 4.2.17 of [9] for clarifying a definition given later.

Proof. We have only to prove this theorem for some quantization S of S because other quantizations are written as the composition of S and some inner automorphisms similar to (2.9). Further by the preceding lemma we can reduce S to the following 3 cases:

(1) S is of the generic and normal case type.

(2) S = S_{0}^{−1}, where S_{0} is the one at (3.7).

(3) S is induced by a (tangential) real analytic contact transforma-

tion S′ : T_{N}^{∗}Y → T_{N}^{∗}Y; that is, w^{∗} ≡ w, τ^{∗} ≡ τ. The third case is a trivial case. Further the second case belongs to the first case because S_{0}^{−1} is generated by

g_{1} = z_{n}^{∗} − z_{n} − ∑_{j=1}^{n−1} z_{j}z_{j}^{∗} − \frac{1}{2α}(w^{∗} − w)^{2}. (3.12)

Hence we have only to deal with the first case. Using the form (3.9) for h, we consider the following integral transformation

(3.13) (T f)(w, x) = ∫ δ(x_{n}^{∗} − h(w, x, w^{∗}, x^{∗}′))f(w^{∗}, x^{∗})dw^{∗}dx^{∗}

for a section f(w^{∗}, x^{∗}) of CO_{Z}. This integral has no meaning because the values of h are not limited to real numbers when w, x, w^{∗}, x_{1}^{∗}, …, x_{n−1}^{∗} move. However we can formally modify this integral as follows:

(3.14) T f = ∫ f(w^{∗}, x^{∗}′, h(w, x, w^{∗}, x^{∗}′))dw^{∗}dx^{∗}′
= ∫ f(w^{∗}, x^{∗}′, h_{0}(x, x^{∗}′) + Ψ(w, x, w^{∗}, x^{∗}′)^{2})dw^{∗}dx^{∗}′

= ∫ f(W^{∗}(w, x, t, x^{∗}′), x^{∗}′, h_{0}(x, x^{∗}′) + t^{2})∂_{t}W^{∗}dtdx^{∗}′,

where W^{∗}(w, z, t, z^{∗}′) is a holomorphic function satisfying

t = Ψ(w, z, w^{∗}, z^{∗}′)|_{w^{∗}=W^{∗}}, W^{∗}(w^{∘}, x^{∘}, 0, x^{∘}^{∗}′) = w^{∘}^{∗}.

Then the last integral of (3.14) has a meaning as an integral for microfunctions of (Re w, Im w, x, t, x^{∗}′) with respect to (t, x^{∗}′). Further it is clear that this integral becomes a section of CO_{Z} as a microfunction of (Re w, Im w, x). Then, by the well-known method in [12], we can show the following: There exists a quantization S of S such that this integral transformation T induces a desired sheaf isomorphism

T_{S} : S^{−1}CO_{Z} →^{∼} CO_{Z} for S. QED

We give here the precise meaning concerning boundary values of sections of CO_{Z}. Let K be a real analytic submanifold of T_{Z}^{∗}X with codimension 2, and H be a real analytic hypersurface in T_{Z}^{∗}X passing through K given as follows:

K = {(w, x; iη) ∈ T_{Z}^{∗}X; w = ψ(x, η)}
⊂ H = {(u + iv, x; iη) ∈ T_{Z}^{∗}X; Φ(u, v, x, η) = 0}.

Here ψ(x, η) is a complex valued analytic function of (x, η) with homogeneous degree 0 with respect to η, and Φ(u, v, x, η) is a real-valued analytic function of (u, v, x, η) of homogeneous degree 0 with respect to η satisfying the following:

∇Φ ≠ 0 on Φ = 0, Φ ∘ ψ = 0.

By Lemma 2.4, we can choose a holomorphic contact transformation S defined in a neighborhood p^{∘} ∈ K such that

(3.15) \begin{cases}S(K) = \{w^{∗} = 0\} ∩ T_{Z}^{∗}X & ⊂ S(H) = \{Im w^{∗} = 0\} ∩ T_{Z}^{∗}X, \\ S(T_{Z}^{∗}X) & ⊂ T_{Z}^{∗}X. \end{cases}

Set σ = the signature of S^{∗}(d Im w^{∗})/dΦ, where S^{∗}(ω) denotes the pull-back of a differential form ω by S. We denote by π : T^{∗}X → X the canonical projection, and by BO_{Z} = CO_{Z}|_{Z} the sheaf on Z of hyperfunctions with a holomorphic parameter w.

Definition 3.9. Let p^{∘} = (w^{∘}, x^{∘}; iη^{∘}) be a point of K, and f(w, x) be a section of CO_{Z} on {Φ > 0} ∩ U with a conic neighborhood U ⊂ T_{Z}^{∗}X of p^{∘}. Then, f(w, x) is said to have a boundary value at p^{∘} from Φ > 0 if there exist a small neighborhood U′ of p^{∘} and a section F (w^{∗}, x^{∗}) ∈ Γ ({σ Im w^{∗} > 0} ∩ π(S(U′)); BO_{Z}) satisfying

(T_{S}^{−1}f)(w^{∗}, x^{∗}) = [F (w^{∗}, x^{∗})]

as sections of Γ ({σ Im w^{∗} > 0}∩S(U′); CO_{Z}). Here T_{S} is a quantization of S introduced in Theorem 3.7.

Though the boundary value [F (u^{∗} + iσ0, x^{∗})] itself depends on a choice of T_{S}, this definition neither depends on a choice of S nor T_{S} (shown as below).

Remark 3.10. A germ of CO_{Z} is represented by a germ of BO_{Z}. However it is well-known that a section of CO_{Z} cannot be represented globally by a section of BO_{Z} in general. Indeed, the cohomological boundary value (T_{S}^{−1}f)(u^{∗} +iσ0, x^{∗}) defines a second hyperfunction on Σ = {(w^{∗}, x^{∗}; iη^{∗}) ∈ T_{Z}^{∗}X; Im w^{∗} = 0}. On the other hand the sheaf B_{Σ}^{2} of second hyperfunctions is essentially larger than the sheaf C_{M}|_{Σ}. Here M = {(w, z) ∈ X; Im w = 0, Im z = 0}. Hence the definition above is equivalent to the following:

(T_{S}^{−1}f)(u^{∗} + iσ0, x^{∗}) ∈ C_{M}|_{S(p^{∘})}.

Further this boundary value is equal to [F (u^{∗} + iσ0, x^{∗})] as a microfunction of (u^{∗}, x^{∗}) at S(p^{∘}). The uniqueness of this boundary value [F (u^{∗} + iσ0, x^{∗})] ∈ C_{M}|_{S(p^{∘})} for a section (T_{S}^{−1}f)(w^{∗}, x^{∗}) is justified by Schapira’s N-regularity property of \overline{∂}_{w}-operator. We refer to [3, 4] as for the second microlocal analysis, and to [13] as for the N-regularity of \overline{∂}_{w}-operator. Further as for a self-contained proof of the equivalent fact, see Proposition 4.1.11 of [9].

A holomorphic contact transformation S generated by (3.9) preserves H = {Im w = 0} iff the holomorphic function Ψ(w, z, w^{∗}, z^{∗}′) is realvalued on {Im w = Im w^{∗} = 0, Im z = 0, Im z^{∗}′ = 0}. Hence the explicit formula in (3.14) for T together with the partial flabbiness of BO_{Z} leads to the following lemma, which is also due to Theorem 4.2.17 of [9].

Lemma 3.11. Let H = {(w, x; iη) ∈ T_{Z}^{∗}X; Im w = 0}, and S : T^{∗}X →^{∼} T^{∗}X be a holomorphic contact transformation defined in a neighborhood of p^{∘} ∈ H. We assume that S preserves T_{Z}^{∗}X, H respectively. Let F_{±}(w^{∗}, x^{∗}) be sections of BO_{Z} on {± Im w^{∗} > 0} ∩ π(U) for a neighborhood U in T_{Z}^{∗}X of p^{∘}, respectively. Then for any quantization T_{S} : S^{−1}CO_{Z} →^{∼} CO_{Z} of S, each (T_{S}[F_{±}])(w, x) has a boundary value at S^{−1}(p^{∘}) from ± Im w > 0. That is, there exist a neighborhood U′ of S^{−1}(p^{∘}) and sections G_{±}(w, x) ∈ Γ ({± Im w > 0} ∩ π(U′); BO_{Z}) such that

(T_{S}[F_{±}])(w, x) = [G_{±}(w, x)] on {± Im w > 0} ∩ U′.

By Lemma 2.4 and this lemma, we can reduce our equation P f = 0 to the case

(3.16)  P = wD_{w}^{m} + ∑_{k=1}^{m} A_{k}(w, z, D_{w}, D_{z})D_{w}^{m−k}

with some operators A_{k} ∈ E_{X}|p^{∘} (k = 1, …, m) of order ≤ 0.

4. Construction of solutions of formal symbol type

4.1. An iteration scheme. As seen in Section 3 we can assume P has the form (3.16); that is, A_{0}(w, z, D_{w}, D_{z}) ≡ w. Write

A_{k}(w, z, D_{w}, D_{z}) = ∑_{j=−∞}^{0} A_{jk}(w, z, D_{w}, D_{z}),

where A_{jk}(w, z, τ, ζ) is the j-th order part of A_{k} with homogeneous degree j in (τ, ζ) for each k. Hence the ordinary differential operator defined at (1.6) is given by

L = w∂_{w}^{m} + ∑_{k=1}^{m} A_{k}^{0}(w, z, ζ)∂_{w}^{m−k}.

Here A_{k}^{0}(w, z, ζ) ≡ A_{0,k}(w, z, 0, ζ) satisfying

A_{1}^{0}(0, x^{∘}, iη^{∘}) ∉ ℤ.

Definition 4.1. For a formal symbol U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ) of order ≤ 0 at p^{∘}, we define linear operators L, L by

(4.1) LU = ∑_{j=−∞}^{0}(LU_{j})(w, z, ζ),

(4.2) LU = ∑_{j=−∞}^{0} ( ∑_{\begin{matrix}−|α|+q=j \\ 0≤k≤m \end{matrix}} \frac{1}{α!}∂_{ζ}^{α}A_{k}^{0}(w, z, ζ)∂_{w}^{m−k}∂_{z}^{α}U_{q}(w, z, ζ)).

Here L operates on each holomorphic function U_{j} as an ordinary differential operator with parameters (z, ζ). It is easy to see that the results of these operations also become formal symbols of type V = ∑_{j=−∞}^{0} V_{j}(w, z, ζ). Indeed, LU coincides with the operator composition ( mod. E_{X}D_{w}):

( ∑_{k=0}^{m} A_{k}^{0}(w, z, D_{z})D_{w}^{m−k})U(w, z, D_{z}) ≡ (LU)(w, z, D_{z}).

Further let R be a microdifferential operator of the form

(4.3)  \begin{cases}R(w, z, D_{w}, D_{z}) & = ∑_{k=0}^{m} R_{k}(w, z, D_{w}, D_{z})D_{w}^{m−k}, \\ R_{k}(w, z, τ, ζ) & = ∑_{j=−∞}^{−1} R_{jk}(w, z, τ, ζ). \end{cases}

Here each R_{k} is a formal symbol of order ≤ −1 defined at p^{∘}. Then we define an operator R∘ by

R∘U = ∑_{j=−∞}^{−1} ( ∑_{\begin{matrix}−|α|−s+ℓ+q=j \\ 0≤k≤m \end{matrix}} \frac{(∂_{τ}^{s}∂_{ζ}^{α}R_{ℓk})(w, z, 0, ζ)}{s!α!}∂_{w}^{m−k+s}∂_{z}^{α}U_{q}(w, z, ζ)).

Indeed R ∘ U becomes a formal symbol of type V = ∑_{j=−∞}^{−1} V_{j}(w, z, ζ) of order ≤ −1 satisfying

R ∘ U ≡ R(w, z, D_{w}, D_{z})U(w, z, D_{z}) (mod. E_{X}D_{w}).

Then, our successive approximation process for formal symbols U_{k} = ∑_{j=−∞}^{0} U_{jk}(w, z, ζ) (k = 0, 1, 2, …) is formulated as follows:

(4.4)  \begin{cases}LU_{0} = 0, \\ LU_{k+1} = \{(L − L) − R∘\}U_{k} (k = 0, 1, 2, …). \end{cases}

Indeed, if ∑_{k=0}^{∞} U_{k} converges as a formal symbol, the sum U(w, z, ζ) satisfies the following equation (mod. E_{X}D_{w}):

(4.5) ( ∑_{k=0}^{m} A_{k}^{0}(w, z, D_{z})D_{w}^{m−k} + R(w, z, D_{w}, D_{z}))U(w, z, D_{z}) ≡ 0.

Further since we have

(4.6)  ord((L − L)U) ≤ ord(U) − 1, ord(R ∘ U) ≤ ord(U) − 1,

we can choose U_{k}’s satisfying ord(U_{k}) ≤ −k (∀k ≥ 0). That is, the j-degree component of ∑_{k=0}^{∞} U_{k} is determined only by U_{0}, …, U_{|j|}.

We set (4.7)

R_{jk}^{A} = \begin{cases}A_{−1,k}(w, z, τ, ζ) + (A_{0,k+1}(w, z, τ, ζ) − A_{0,k+1}(w, z, 0, ζ))/τ \\ (j = −1), \\ A_{jk}(w, z, τ, ζ) (j ≤ −2). \end{cases}

Then R^{A} = ∑_{k=0}^{m} ∑_{j=−∞}^{−1} R_{jk}^{A}(w, z, D_{w}, D_{z})D_{w}^{m−k} is a microdifferential operator at p^{∘} satisfying the conditions in (4.3). Further if we set R = R^{A} in our successive approximation process (4.4), the corresponding equation (4.5) is just equal to P (w, z, D_{w}, D_{z})U(w, z, D_{z}) ≡ 0. Consequently our program reduces to a construction of ‘convergent series’ ∑_{k=0}^{∞} U_{k} of formal symbols satisfying (4.4).

4.2. Formal norms. Boutet-de-Monvel and Krée introduced so-called a formal norm N(Q; t) for a formal symbol Q of analytic pseudodifferential operators [6]. N(Q; t) is a formal power series of a variable t with real non-negative coefficients depending Q. In particular the following properties are the most important for Q_{1}, Q_{2} ∈ E_{X} of order ≤ 0:

N(Q_{1} + Q_{2}; t) ≪ N(Q_{1}; t) + N(Q_{2}; t),
N(Q_{1}Q_{2}; t) ≪ N(Q_{1}; t)N(Q_{2}; t),

where F_{1}(t) ≪ F_{2}(t) means that F_{2}(t) is a majorant series for F_{1}(t). To show the convergence of our formal symbols, we introduce some variants of formal norms for U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ). We may assume that each A_{k}(w, z, D_{w}, D_{z}) is defined in a neighborhood of

D_{ν} = \{(w, z; τ, ζ) ∈ T^{∗}X; |w| ≤ 1 + ν, |z − x^{∘}| ≤ ν, \frac{|τ|}{|ζ|} + |\frac{ζ}{|ζ|} − \frac{iη^{∘}}{|η^{∘}|}| ≤ ν\}

with some small ν > 0.

Definition 4.2. (Regular type). When each component U_{j}(w, z, ζ) of U = ∑_{j=−∞}^{0} U_{j} is holomorphic in D_{ν}, we define a formal power series N_{m′}(U; t) in t with parameters z, ζ for each m′ = 0, 1, 2, … by

N_{m′}(U; t) ≡ ∑_{p,α,β,ℓ} \frac{p!t^{2p+ℓ+|α+β|}|ζ|^{p+|β|}}{(p + ℓ + |α|)!(p + |β|)!} max_{0≤k≤m′} ∥∂_{w}^{k+ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}∥.

Here ∥ · ∥ is the sup-norm in w ∈ D introduced at (2.12). Indeed, if U is a section of Γ (D × {(x^{∘}; iη^{∘})}; E_{X}), N_{m′}(U; t) has a convergent majorant series independent of (z, ζ). Conversely, if the formal norm N_{m′}(U; t) for a set of homogeneous holomorphic functions U_{j}(w, z, ζ) ∈ O(D_{ν}) has a convergent majorant series independent of (z, ζ), then U = ∑_{j−∞}^{0} U_{j}(w, z, D_{z}) becomes a section of Γ (D × {(x^{∘}; iη^{∘})}; E_{X}).

Definition 4.3. (Non-regular type). When each component U_{j}(w, z, ζ) of U is holomorphic in a neighborhood of {w ∈ Ω} ∩ D_{ν} with

Ω = {w ∈ ℂ; 0 < |w| ≤ 1, | arg w| ≤ π − ε},

we define a formal power series N_{m′}^{μ}(U; t) in t with parameters z, ζ for each m′ = 0, 1, 2, … and a positive constant μ by

(4.8) N_{m′}^{μ}(U; t) ≡ ∑_{p,α,β,ℓ} \frac{p!t^{2p+ℓ+|α+β|}}{(p + ℓ + |α|)!(p + |β|)!}

× |ζ|^{p+|β|} max_{0≤k≤m′} ∥∂_{w}^{k+ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}∥_{μ+k+ℓ+|α+β|+p−κ(m′)}

with κ(0) = 0, κ(m′) = m′ − 1 (∀ m′ ≥ 1). Here ∥ · ∥_{μ} is the sup-norm in w ∈ Ω with some weight introduced at (2.13). Further, when each component U_{j}(w, z, ζ) ≡ U_{j}(z, ζ) is not depending on w, we define

(4.9)  K(U; t) ≡ ∑_{p,α,β} \frac{p!t^{2p+|α+β|}}{(p + |α|)!(p + |β|)!}|ζ|^{p+|β|}|∂_{z}^{α}∂_{ζ}^{β}U_{−p}|.

In the approximation process, we need some a’pri-o’ri estimates for N_{m}(U_{k}; t) or N_{m}^{μ}(U_{k}; t). In the next subsections we get our main estimates by these formal norms.

4.3. Estimates for L. Let us consider the following equation for formal symbols U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ), F = ∑_{j=−∞}^{0} F_{j}(w, z, ζ):

LU = F ⟺ LU_{j} = F_{j} (j = 0, −1, −2, …).

We estimate N_{m}(U; t) by N_{0}(F; t) and ∑_{k=0}^{m−2} K(∂_{w}^{k}U(0, z, ζ); t) for regular type formal symbols. Further, we estimate N_{m}^{μ}(U; t) by N_{0}^{μ}(F; t) and ∑_{k=0}^{m−1} K(∂_{w}^{k}U(1, z, ζ); t) for non-regular type formal symbols.

To derive such estimates we apply ∂_{w}^{ℓ}∂_{z}^{α}∂_{ζ}^{β} to both sides of LU_{−p} = F_{−p}. Then we obtain

L(∂_{w}^{ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}) = ∂_{w}^{ℓ}∂_{z}^{α}∂_{ζ}^{β}F_{−p}

− ∑_{\begin{matrix}ℓ′,α′,β′ \\ ℓ′′,α′′,β′′ \end{matrix}} ∑_{k=0}^{m} \binom{ℓ}{ℓ′}\binom{α}{α′}\binom{β}{β′}∂_{w}^{ℓ′}∂_{z}^{α′}∂_{ζ}^{β′}A_{k}^{0} · ∂_{w}^{ℓ′′+m−k}∂_{z}^{α′′}∂_{ζ}^{β′′}U_{−p}

(ℓ = ℓ′ + ℓ′′, α = α′ + α′′, β = β′ + β′′, (ℓ′, α′, β′) ≠ 0) . Here we employ Lemma 2.7. For a sufficiently small ν > 0 we set

M_{ν} = 1 + sup_{(w,z;0,ζ)∈D_{ν}} ∑_{k=1}^{m} |A_{k}^{0}(w, z, ζ)| < +∞

and

δ_{ν} = inf{|p + A_{1}^{0}(0, z, ζ)|; p = 0, 1, 2, …, (z; ζ) ∈ V_{ν}} > 0

with

V_{ν} = {(z; ζ) ∈ ℂ^{n} × ℂ^{n}; |z − x^{∘}| ≤ ν, |ζ/|ζ| − iη^{∘}/|η^{∘}|| ≤ ν}.

Then there exists a positive constant C_{0} depending only on M_{ν} and δ_{ν}, which satisfies the estimates (2.17), (2.18) for

L = ∑_{k=0}^{m} A_{k}^{0}(w, z, ζ)∂_{w}^{m−k}.

In particular we have the following estimates:

|∂_{w}^{ℓ}∂_{z}^{α}∂_{ζ}^{β}A_{k}^{0}(w, z, ζ)| ≤ ℓ!α!β!(2n/ν)^{ℓ+|α|+|β|}|ζ|^{−|β|}M_{ν}

for |w| ≤ 1, (z, ζ) ∈ V_{ν/2}. Hereafter we fix a (z, ζ) ∈ V_{ν/2} and set

C_{1} = max{M_{ν}, 2n/ν}.

(1) Regular type case: Firstly for m′ = m we consider

max_{0≤k≤m} ∥∂_{w}^{k+ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}∥

≤ C_{0}(∥∂_{w}^{ℓ}∂_{z}^{α}∂_{ζ}^{β}F_{−p}∥ + ∑_{k=0}^{m−2} |∂_{w}^{k+ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}(0, z, ζ)|

+ ∑_{(ℓ′,α′,β′)≠0} \frac{(m + 1)ℓ!α!β!C_{1}^{ℓ′+|α′|+|β′|+1}|ζ|^{−|β′|}}{ℓ′′!α′′!β′′!} max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′}∂_{ζ}^{β′′}U_{−p}∥).

Then, we obtain

N_{m}(U; t) ≪ C_{0}{N_{0}(F; t) + ∑_{k=0}^{m−2} K(∂_{w}^{k}U(0, z, ζ); t)

+ ∑_{\begin{matrix}p,α,β \\ ℓ>0 \end{matrix}} \frac{p!t^{2p+ℓ+|α+β|}|ζ|^{p+|β|}(m − 1)}{(p + ℓ + |α|)!(p + |β|)!} max_{0≤k≤m−2} |∂_{w}^{k+ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}|_{w=0}|

+ ∑_{\begin{matrix}p,ℓ,α,β \\ (ℓ′,α′,β′)≠0 \end{matrix}} \frac{p!t^{2p+ℓ+|α+β|}|ζ|^{p+|β|}}{(p + ℓ + |α|)!(p + |β|)!}

× \frac{(m + 1)ℓ!α!β!C_{1}^{ℓ′+|α′|+|β′|+1}|ζ|^{−|β′|}}{ℓ′′!α′′!β′′!} max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′}∂_{ζ}^{β′′}U_{−p}∥\}

≪ C_{0}{N_{0}(F; t) + ∑_{k=0}^{m−2} K(∂_{w}^{k}U(0, z, ζ); t) + (m − 1)t · N_{m}(U; t)

+ ∑_{\begin{matrix}p,ℓ,α,β \\ (ℓ′,α′,β′)≠0 \end{matrix}} \frac{(m + 1)C_{1}ℓ!α!β!(p + ℓ′′ + |α′′|)!(p + |β′′|)!}{ℓ′′!α′′!β′′!(p + ℓ + |α|)!(p + |β|)!}(C_{1}t)^{ℓ′+|α′+β′|}

× \frac{p!t^{2p+ℓ′′+|α′′+β′′|}|ζ|^{p+|β′′|}}{(p + ℓ′′ + |α′′|)!(p + |β′′|)!} max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′}∂_{ζ}^{β′′}U_{−p}∥\}.

We recall here an inequality for r ≥ r′ ≥ 0, s ≥ s′ ≥ 0:

(4.10)  \binom{r}{r′}\binom{s}{s′} ≤ \binom{r + s}{r′ + s′}.

Therefore

\frac{ℓ!α!β!(p + ℓ′′ + |α′′|)!(p + |β′′|)!}{ℓ′′!α′′!β′′!(p + ℓ + |α|)!(p + |β|)!}

= \frac{ℓ′!α′!β′!}{(ℓ′ + |α′|)!|β′|!}\binom{ℓ}{ℓ′}\binom{α}{α′}\binom{β}{β′}\binom{p + ℓ + |α|}{ℓ′ + |α′|}^{−1}\binom{p + |β|}{|β′|}^{−1} ≤ 1.

Hence we have

N_{m}(U; t) ≪ C_{0}{N_{0}(F; t) + ∑_{k=0}^{m−2} K(∂_{w}^{k}U(0, z, ζ); t)

+ (m − 1)tN_{m}(U; t) + (m + 1)C_{1}N_{m}(U; t) ∑_{(l′,α′,β′)≠0}(C_{1}t)^{ℓ′+|α′+β′|}}.

Set

ψ(t) ≡ ∑_{(ℓ′,α′,β′)≠0}(C_{1}t)^{ℓ′+|α′+β′|},
Φ(t) ≡ \frac{C_{0}}{1 − (m + 1)\{C_{0}C_{1}ψ(t) + C_{0}t\}}.

Since ψ(0) = 0, we get the following proposition:

Proposition 4.4. If each component of F and U is holomorphic on a neighborhood of {|w| ≤ 1} with respect to w, we have

N_{m}(U; t) ≪ Φ(t){N_{0}(F; t) + ∑_{k=0}^{m−2} K(∂_{w}^{k}U(0, z, ζ); t)}.

Here Φ(t) is a convergent power series of t with non-negative coefficients independent of F, U.

(2) Non-regular type case: For m′ = m ≥ 1 and μ ≥ M_{ν} + m + 1, we obtain

max_{0≤k≤m} ∥∂_{w}^{k+ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}∥_{μ+k+ℓ+p+|α+β|−m+1}

≤ C_{0}{∥∂_{w}^{ℓ}∂_{z}^{α}∂_{ζ}^{β}F_{−p}∥_{μ+ℓ+|α+β|+p} + ∑_{k=0}^{m−1} |∂_{w}^{k+ℓ}∂_{z}^{α}∂_{ζ}^{β}U_{−p}(1, z, ζ)|

+ ∑_{(ℓ′,α′,β′)≠0} \frac{(m + 1)ℓ!α!β!C_{1}^{ℓ′+|α′|+|β′|+1}|ζ|^{−|β′|}}{ℓ′′!α′′!β′′!}

× max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′}∂_{ζ}^{β′′}U_{−p}∥_{μ+ℓ+|α+β|+p}}.

Note that μ + ℓ + |α + β| + p ≥ μ + k + ℓ′′ + |α′′ + β′′| + p − m + 1 in the last term because ℓ′ + α′ + β′ ≥ 1. Hence we obtain

max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′}∂_{ζ}^{β′′}U_{−p}∥_{μ+ℓ+|α+β|+p}

≤ max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′}∂_{ζ}^{β′′}U_{−p}∥_{μ+k+ℓ′′+|α′′+β′′|+p−m+1}.

In the same way as the regular type case (replace max_{0≤k≤m} ∥ · ∥ by the norms above), we obtain the following proposition:

Proposition 4.5. If each component of F and U is holomorphic on a neighborhood of Ω with respect to w, for ∀μ ≥ M_{ν} + m + 1 we have

N_{m}^{μ}(U; t) ≪ Φ(t){N_{0}^{μ}(F; t) + ∑_{k=0}^{m−1} K(∂_{w}^{k}U(1, z, ζ); t)}.

Here Φ(t) is a convergent power series of t with non-negative coefficients independent of F, U.

4.4. Estimates for L − L. We estimate N_{0}((L − L)U; t) by N_{m}(U; t) and N_{0}^{μ}((L − L)U; t) by N_{m}^{μ}(U; t) respectively.

Proposition 4.6. Set a convergent power series of t with positive coefficients with value 0 at t = 0:

ψ_{1}(t) ≡ mC_{1} ∑_{ℓ′=0}^{∞}(C_{1}t)^{ℓ′} ∑_{α′≥0}(C_{1}t)^{|α′|} ∑_{β′≥0}(C_{1}t)^{|β′|} ∑_{|γ|≥1}(C_{1}t)^{|γ|}.

(1) Regular type case: If each component of U is holomorphic on a neighborhood of {|w| ≤ 1} with respect to w, we have

(4.11)  N_{0}((L − L)U; t) ≪ ψ_{1}(t)N_{m}(U; t).

(2) Non-regular type case: If each component of U is holomorphic on a neighborhood of Ω with respect to w, for ∀μ ≥ 0 we have

(4.12)  N_{0}^{μ}((L − L)U; t) ≪ ψ_{1}(t)N_{m}^{μ}(U; t).

Proof. (1) Regular type case:

N_{0}((L − L)U; t) = ∑_{p,α,β,ℓ} \frac{p!t^{2p+ℓ+|α+β|}}{(p + ℓ + |α|)!(p + |β|)!}|ζ|^{p+|β|}

× ∥∂_{w}^{ℓ}∂_{z}^{α}∂_{ζ}^{β}( ∑_{p=|γ|+q,|γ|>0,1≤k≤m} \frac{1}{γ!}∂_{ζ}^{γ}A_{k}^{0} · ∂_{w}^{m−k}∂_{z}^{γ}U_{−q})∥

≪ ∑_{ℓ′,ℓ′′,α′,α′′,β′,β′′,q,|γ|>0} \frac{p!t^{2p+ℓ+|α+β|}}{(p + ℓ + |α|)!(p + |β|)!} \frac{1}{γ!}\binom{ℓ}{ℓ′}\binom{α}{α′}\binom{β}{β′}m

× ℓ′!α′!(β′ + γ)!C_{1}^{|α′+β′+γ|+ℓ′+1}|ζ|^{q+|β′′|} max_{0≤k≤m−1} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′+γ}∂_{ζ}^{β′′}U_{−q}∥

≪ ∑_{ℓ′,ℓ′′,α′,α′′,β′,β′′,q,|γ|>0,α′′^{∗}=α′′+γ}(∗)mC_{1}(C_{1}t)^{ℓ′+|α′+β′+γ|}

× \frac{q!t^{2q+ℓ′′+|α′′^{∗}+β′′|}|ζ|^{q+|β′′|}}{(q + ℓ′′ + |α′′^{∗}|)!(q + |β′′|)!} max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′^{∗}}∂_{ζ}^{β′′}U_{−q}∥.

Here (∗) is given by

\frac{p!(p + ℓ′′ + |α′′|)!(q + |β′′|)!ℓ!α!β!(β′ + γ)!}{(p + ℓ + |α|)!(p + |β|)!γ!q!ℓ′′!α′′!β′!β′′!}

= \frac{ℓ′!α′!β′!}{(ℓ′ + |α′|)!|β′|!}\binom{β′ + γ}{β′}\binom{p}{|γ|}\binom{β}{β′}\binom{ℓ}{ℓ′}\binom{α}{α′}

× \binom{|β′ + γ|}{|β′|}^{−1}\binom{p + |β|}{|γ + β′|}^{−1}\binom{p + ℓ + |α|}{ℓ′ + |α′|}^{−1} ≤ 1,

where we used the inequality (4.10). Therefore, we obtain the estimate (4.11) for the regular type case.

(2) Non-regular type case: In the same way as above, we have

N_{0}^{μ}((L − L)U; t) ≪ ∑_{ℓ′,ℓ′′,α′,α′′,β′,β′′,q,|γ|>0} \frac{p!t^{2p+ℓ+|α+β|}}{(p + ℓ + |α|)!(p + |β|)!}

× \frac{1}{γ!}\binom{ℓ}{ℓ′}\binom{α}{α′}\binom{β}{β′}mℓ′!α′!(β′ + γ)!C_{1}^{|α′+β′+γ|+ℓ′+1}

× |ζ|^{q+|β′′|} max_{0≤k≤m−1} ∥∂_{w}^{k+ℓ′′}∂_{z}^{α′′+γ}∂_{ζ}^{β′′}U_{−q}∥_{μ+p+ℓ+|α+β|}.

Since μ + p + ℓ + |α + β| ≥ μ + k + ℓ′′ + |α′′ + γ| + |β′′| + q − m + 1, we have

max_{0≤k≤m−1} ∥ · ∥_{μ+p+ℓ+|α+β|} ≤ max_{0≤k≤m} ∥ · ∥_{μ+k+ℓ′′+|α′′+γ|+|β′′|+q−(m−1)}.

Therefore the same argument as in the regular type case leads to the conclusion (4.12). QED

4.5. Estimates for R ∘ ∗. We estimate N_{0}(R ∘ U; t), N_{0}^{μ}(R ∘ U; t) by N_{m}(U; t), N_{m}^{μ}(U; t) respectively. Here R = ∑_{k=0}^{m} R_{k}(w, z, D_{w}, D_{z})D_{w}^{m−k} with ord(R_{k}) ≤ −1(k = 0, 1, …, m) is supposed to be a microdifferential operator defined in D_{ν} for some ν > 0. Therefore we have a constant C_{2} > 0 satisfying the following estimates for each j-th degree component R_{jk} of R_{k} on D_{ν/2}:

|∂_{w}^{ℓ}∂_{z}^{α}∂_{τ}^{s}∂_{ζ}^{β}R_{−p,k}| ≤ C_{2}^{1+p+ℓ+s+|α+β|}ℓ!α!s!β!p!|ζ|^{−p−s−|β|}.

Proposition 4.7. Set a convergent power series of t with positive coefficients with value 0 at t = 0:

ψ_{2}(t) ≡ (m + 1) ∑_{\begin{matrix}ℓ′,α′,β′,γ≥0 \\ r≥1,s≥0 \end{matrix}} C_{2}^{1−r}(C_{2}t)^{s+2r+ℓ′+|γ+α′+β′|}.

(1) Regular type case: If each component of U is holomorphic on a neighborhood of {|w| ≤ 1} with respect to w, we have

(4.13)  N_{0}((R ∘ U; t) ≪ ψ_{2}(t)N_{m}(U; t).

(2) Non-regular type case: If each component of U is holomorphic on a neighborhood of Ω with respect to w, for ∀μ ≥ 0 we have

(4.14)  N_{0}^{μ}(R ∘ U; t) ≪ ψ_{2}(t)N_{m}^{μ}(U; t).

Proof. Firstly we consider the regular type case:

N_{0}(R ∘ U; t) ≪ ∑_{p,α,β,ℓ} \frac{p!t^{2p+ℓ+|α+β|}|ζ|^{p+|β|}}{(p + ℓ + |α|)!(p + |β|)!}\binom{ℓ}{ℓ′}\binom{α}{α′}\binom{β}{β′}

× ∑_{\begin{matrix}|γ|+s+r+q=p \\ 0≤k≤m \end{matrix}} ∥\frac{(∂_{w}^{ℓ′}∂_{z}^{α′}∂_{τ}^{s}∂_{ζ}^{β′+γ}R_{−r,k})|_{τ=0}}{s!γ!} ∂_{w}^{m−k+s+ℓ′′}∂_{z}^{α′′+γ}∂_{ζ}^{β′′}U_{−q}∥

≪ ∑_{\begin{matrix}|γ|+s+r+q=p \\ r≥1,p,α,β,ℓ, \end{matrix}} \frac{p!t^{2p+ℓ+|α+β|}|ζ|^{p+|β|}}{(p + ℓ + |α|)!(p + |β|)!}\binom{ℓ}{ℓ′}\binom{α}{α′}\binom{β}{β′}

× \frac{(C_{2}^{1+r+s+ℓ′+|α′+β′+γ|}r!ℓ′!α′!s!(β′ + γ)!|ζ|^{−s−r−|β′+γ|}}{s!γ!}

× (m + 1) max_{0≤k≤m} ∥∂_{w}^{k+s+ℓ′′}∂_{z}^{α′′+γ}∂_{ζ}^{β′′}U_{−q}∥

≪ ∑_{\begin{matrix}|γ|+s+r+q=p \\ r≥1,p,α,β,ℓ \end{matrix}}(∗)(m + 1)C_{2}^{1−r}(C_{2}t)^{s+2r+ℓ′+|γ+α′+β′|}

× \frac{q!t^{2q+ℓ′′^{∗}+|α′′^{∗}+β′′|}|ζ|^{q+|β′′|}}{(q + ℓ′′^{∗} + |α′′^{∗}|)!(q + |β′′|)!} max_{0≤k≤m} ∥∂_{w}^{k+ℓ′′^{∗}}∂_{z}^{α′′^{∗}}∂_{ζ}^{β′′}U_{−q}∥.

Here ℓ′′^{∗} = ℓ′′ + s, α′′^{∗} = α′′ + γ and (∗) is given by

\frac{p!(q + s + ℓ′′ + |α′′ + γ|)!(q + |β′′|)!r!ℓ!α!β!(β′ + γ)!}{(p + ℓ + |α|)!(p + |β|)!q!γ!ℓ′′!α′′!β′!β′′!}

= \frac{r!ℓ′!α′!β′!}{(r + ℓ′ + |α′|)!|β′|!}\binom{ℓ}{ℓ′}\binom{α}{α′}\binom{p}{|γ| + s + r}\binom{β}{β′}\binom{β′ + γ}{β′}

× \binom{p + ℓ + |α|}{r + ℓ′ + |α′|}^{−1}\binom{p + |β|}{|γ| + s + r + |β′|}^{−1}\binom{s + r + |β′ + γ|}{|β′|}^{−1} ≤ 1,

where we used the inequality (4.10). Therefore we obtain the estimate (4.13).

Secondly we consider the non-regular type case. The proof for the regular type case is also available in this case if we check the following:

max_{0≤k≤m} ∥∂_{w}^{k+s+ℓ′′}∂_{z}^{α′′+γ}∂_{ζ}^{β′′}U_{−q}∥_{μ+ℓ+|α+β|+p}

≤ max_{0≤k≤m} ∥∂_{w}^{k+s+ℓ′′}∂_{z}^{α′′+γ}∂_{ζ}^{β′′}U_{−q}∥_{μ+k+s+ℓ′′+|α′′+γ+β′′|+q−m+1}.

Indeed this inequality holds since (μ + ℓ + |α + β| + p) − (μ + k + s + ℓ′′ + |α′′ + γ + β′′| + q − m + 1) = ℓ′ + |α′ + β′| + r − k + m − 1 ≥ 0 (use r ≥ 1). This completes the proof. QED

4.6. A construction of solutions. We recall our approximation process:

(4.15)  \begin{cases}LU_{0} = 0, \\ LU_{k+1} = \{(L − L) − R∘\}U_{k} (k = 0, 1, 2, …). \end{cases}

We assume that the coefficients A_{k}^{0}(w, z, ζ) (k = 0, 1, …, m; A_{0}^{0} ≡ w) of L and L are holomorphic functions defined in D_{ν} of homogeneous degree 0 with respect to ζ for some ν > 0. Further we also assume that the microdifferential operator R = ∑_{k=0}^{m} R_{k}(w, z, D_{w}, D_{z})D_{w}^{m−k} with ord(R_{k}) ≤ −1 is defined in D_{ν}.

Theorem 4.8. Let U_{0} ≡ U_{00} be any holomorphic solution of LU_{0} = 0 in D_{ν}(⊃ {|w| ≤ 1} × {(x^{∘}; iη^{∘})}) with homogeneous degree 0 concerning ζ. We can choose a series U_{k} (k = 1, 2, …,) of solutions of (4.15) such that each U_{k} = ∑_{j=−∞}^{0} U_{jk} is a formal symbol defined in a neighborhood of D_{ν/2} satisfying

∂_{w}^{ℓ}U_{k}|_{w=0} = 0 (ℓ = 0, 1, …, m − 2, k ≥ 1).

Then U ≡ ∑_{k=0}^{∞} U_{k} converges in N_{m}(·; t)-norm uniformly on {(z; ζ) ∈ V_{ν/2}}, and it gives a solution of

LU + R ∘ U = 0.

Further ord(U_{k}) ≤ −k (∀k ≥ 0). In particular, σ_{0}(U) = U_{0}.

Proof. As seen at (2.23), under the condition A_{1}^{0}(0, x^{∘}, iη^{∘}) ≠ 0, −1, −2, ⋯ we can construct U_{k} successively at least in a neighborhood of w = 0. Then each U_{k} extends analytically to D_{ν/2} because it satisfies a linear ordinary differential equation. Then by Propositons 4.4, 4.6, 4.7 we obtain the following estimates:

N_{m}(U_{k+1}; t)

≪ Φ(t){N_{0}(R ∘ U_{k}; t) + N_{0}((L − L)U_{k}; t) + ∑_{j=0}^{m−2} K(∂_{w}^{j}U_{k+1}|_{w=0}; t)}

≪ Φ(t){(ψ_{1}(t) + ψ_{2}(t))N_{m}(U_{k}; t)}

≪ ⋯ ≪ {Φ(t)(ψ_{1}(t) + ψ_{2}(t))}^{k+1}N_{m}(U_{0}; t).

Therefore ∑_{k=0}^{∞} N_{m}(U_{k}; t) has a convergent majorant series

[1 − Φ(t){ψ_{1}(t) + ψ_{2}(t)}]^{−1}N_{m}(U_{0}; t).

This completes the proof. QED

Theorem 4.9. Let U_{0} ≡ U_{00} be any holomorphic solution of LU_{0} = 0 in {w ∈ Ω} ∩ D_{ν}(⊃ {w ∈ Ω} × {(x^{∘}; iη^{∘})}) with homogeneous degree 0 concerning ζ. We can choose a series U_{k} (k = 1, 2, …,) of solutions of (4.15) such that each U_{k} = ∑_{j=−∞}^{0} U_{jk} is a formal symbol defined in a neighborhood of {w ∈ Ω} ∩ D_{ν/2} satisfying

∂_{w}^{ℓ}U_{k}|_{w=1} = 0 (ℓ = 0, 1, …, m − 1, k ≥ 1).

Then U ≡ ∑_{k=0}^{∞} U_{k} converges in N_{m}^{μ}(·; t)-norm uniformly on {(z; ζ) ∈ V_{ν/2}} for any sufficiently large μ, and it gives a solution of

LU + R ∘ U = 0.

Further ord(U_{k}) ≤ −k (∀k ≥ 0). In particular, σ_{0}(U) = U_{0}.

Proof. We can construct U_{k} successively in a neighborhood of w = 1, then each U_{k} extends analytically to {w ∈ Ω} ∩ D_{ν/2}. By using Propositions 4.5, 4.6, 4.7 we obtain the following estimates in the same way as above:

N_{m}^{μ}(U_{k+1}; t) ≪ {Φ(t)(ψ_{1}(t) + ψ_{2}(t))}^{k+1}N_{m}^{μ}(U_{0}; t),

∑_{k=0}^{∞} N_{m}^{μ}(U_{k}; t) ≪ [1 − Φ(t){ψ_{1}(t) + ψ_{2}(t)}]^{−1}N_{m}^{μ}(U_{0}; t).

This completes the proof. QED

5. Boundary values and the main theorems

To state the main theorems, we prepare some theorem concerning boundary values for operators obtained in Section 4.

The non-regular type solution constructed in Theorem 4.9 is written as

U = ∑_{j=−∞}^{0} U_{j}(w, z, D_{z}).

If A_{1}^{0}(0, z, ζ) ≢ any integer, we can take a singular solution at w = 0 of LU_{0} = 0 as the principal symbol U_{0} of U; that is, a solution of the form

U_{0}(w, z, ζ) = w^{m−1−A_{1}^{0}(0,z,ζ)}(1 + ∑_{ℓ=1}^{∞} c_{ℓ}(z, ζ)w^{ℓ}),

where A_{1}^{0}(0, z, ζ), c_{ℓ}(z, ζ) (ℓ ≥ 1) are holomorphic functions of homogeneous degree 0 with respect to ζ. Then the lower order terms also have stronger singularities at w = 0 in general. Since N_{m}^{μ}(U; t) has a convergent majorant series, we have the estimates

|U_{−p}| ≤ C^{p+1}p!|w|^{−μ−p}|ζ|^{−p} (w ∈ Ω, p ≥ 0),

for some fixed μ, C > 0. We show that the operators of this type have boundary values on any conic and real analytic hypersurface H passing through w = 0. To simplify the situation, by using Lemma 2.4 we reduce H to H = {Im w = 0} under some quantized contact transformation preserving CO_{Z}.

The main idea of the proof is in decomposing the kernel function of U into 2 kernel functions with double phases. We prepare an elementary inequality about integrations of holomorphic functions of w:

Lemma 5.1. Let f(w) be a holomorphic function defined in a neighborhood of G = {w ∈ ℂ; r_{0} < Im w ≤ r_{1}, | Re w| < r_{1}} satisfying an estimate

|f(w)| ≤ C| Im w|^{−μ} (∀w ∈ G)

for some constants C, μ, r_{1} > 0 and r_{0} (0 ≤ r_{0} < r_{1} ≤ 1). Choose an positive integer p as μ < p ≤ μ + 1. Then the p + 1-times integration

g_{p}(w) ≡ ∫_{ir_{1}}^{w} \frac{(w − w′)^{p}}{p!} f(w′)dw′

is holomorphic in a neighborhood of G, continuous up to Im w = r_{0}, and satisfies

|g_{p}(w)| ≤ C(2^{p} + 1)/p! (∀w ∈ G).

Proof.

|g_{p}(w)| ≤ | ∫_{ir_{1}}^{u+ir_{1}} \frac{(w − w′)^{p}}{p!} f(w′)dw′| + | ∫_{u+ir_{1}}^{u+iv} \frac{(w − w′)^{p}}{p!} f(w′)dw′|

≤ r_{1} \frac{(2r_{1})^{p}}{p!} Cr_{1}^{−μ} + ∫_{v}^{r_{1}} \frac{|v′ − v|^{p}}{p!} C|v′|^{−μ}dv′

≤ \frac{(2r_{1})^{p}}{p!} Cr_{1}^{1−μ} + ∫_{v}^{r_{1}} \frac{C}{p!}|v′|^{p−μ}dv′ ≤ C \frac{(2^{p} + 1)}{p!}.

QED

Theorem 5.2. Let U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ) be a classical formal symbol of a pseudo-differential operator with order ≤ 0 defined in a conic open set

(5.1)  W_{r} ≡{(w, z; ∗, ζ) ∈ T^{∗}X; Im w > 0, |w| < r, |z| < κ,
|ζ_{j}| < ρ|ζ_{n}| (1 ≤ ∀j ≤ n − 1), | Re ζ_{n}| < δ Im ζ_{n}}

for some r, κ, ρ, δ > 0 (δ < 1). We suppose that U_{j} ∈ O(W_{r}) (∀j ≤ 0) and that there exists some constants C, μ > 0 satisfying the following inequalities:

(5.2)  |U_{−p}(w, z, ζ)| ≤ C^{p+1}p!| Im w|^{−p−μ}|ζ|^{−p} on W_{r} (∀p ≥ 0).

Then, for a sufficiently large number

(5.3)  λ > max{1, 2560C/r}

we have 2 holomorphic functions E^{(k)}(w, z, z − z^{∗}, s) (k = 1, 2) defined in

(5.4) W^{(1)} ≡{(w, z, z − z^{∗}, s) ∈ ℂ × ℂ^{n} × ℂ^{n} × ℂ; |w| < r/40,
max{0, −80 Im w/λr} < Im s ≤ λ^{−1}, | Re s| < λ^{−1},
|z| < κ, |z_{j} − z_{j}^{∗}| > ρ^{−1}|z_{n} − z_{n}^{∗}| (j = 1, …, n − 1)},

and

(5.5) W^{(2)} ≡{(w, z, z − z^{∗}, s) ∈ ℂ × ℂ^{n} × ℂ^{n} × ℂ; | Re s| < λ^{−1},
|w − ir/80| < r/320, −(320λ)^{−1} < Im s ≤ λ^{−1},
|z| < κ, |z_{j} − z_{j}^{∗}| > ρ^{−1}|z_{n} − z_{n}^{∗}| (j = 1, …, n − 1)}.

respectively satisfying the following:

∑_{k=1}^{2} E^{(k)}(w, z, z − z^{∗}, s) = ∑_{|α′|≥0,p≥0} \frac{α′!}{|α′|!}( ∏_{j=1}^{n−1} \frac{(z_{n} − z_{n}^{∗})^{α_{j}}}{(z_{j}^{∗} − z_{j})^{α_{j}+1}} ) (5.6)

× ∫_{i/λ}^{s} \frac{(s − s^{∗})^{p+ν+3}}{(p + ν + 3)!} ds^{∗} ∫_{λ}^{∞} U_{−p,α′}(w, z, it) · (it)^{p−2}e^{its^{∗}} \frac{dt}{2π}

on W^{(1)} ∩ W^{(2)} ≡ W^{(3)}:

(5.7) W^{(3)} ={(w, z, z − z^{∗}, s) ∈ ℂ × ℂ^{n} × ℂ^{n} × ℂ;

0 < Im s ≤ λ^{−1}, | Re s| < λ^{−1}, |w − ir/80| < r/320, |z| < κ, |z_{j} − z_{j}^{∗}| > ρ^{−1}|z_{n} − z_{n}^{∗}| (j = 1, …, n − 1)}.

Here we expand each U_{−p} in W_{r} as follows:

U_{−p}(w, z, ζ) = ∑_{α′≥0} U_{−p,α′}(w, z, ζ_{n})(ζ′/ζ_{n})^{α′},

with ζ′/ζ_{n} = (ζ_{1}/ζ_{n}, …, ζ_{n−1}/ζ_{n}). Further ν is some positive integer. Proof. Each U_{−p,α′}(w, z, ζ_{n}) is holomorphic in W_{r}^{(1)} ≡ {(w, z; ζ_{n}) ∈ ℂ^{n+1} × ℂ; Im w > 0, |w| < r, |z| < κ, | Re ζ_{n}| < δ Im ζ_{n}} and satisfies

(5.8)  |U_{−p,α′}(w, z, ζ_{n})| ≤ p!C^{p+1}ρ^{−|α′|}| Im w|^{−p−μ}|ζ_{n}|^{−p}

on W_{r}^{(1)}. By the preceding lemma, we get holomorphic functions V_{p,α′}(w, z, ζ_{n}) ∈ O(W_{r/2}^{(1)}) satisfying

(5.9)  \begin{cases}∂_{w}^{p+ν+1}V_{p,α′}(w, z, ζ_{n}) = U_{−p,α′}(w, z, ζ_{n}), \\ |V_{p,α′}(w, z, ζ_{n})| ≤ C^{p+1}2^{p+ν+1}ρ^{−|α′|}|ζ_{n}|^{−p} \end{cases}

on W_{r/2}^{(1)} for all p, α′. Here ν is the integer satisfying μ < ν ≤ μ + 1. Take a conformal mapping w̃ = φ(w):

φ : {w ∈ ℂ; Im w > 0, |w| < r/2} ˜⟶ {w̃ ∈ ℂ; |w̃| < 1} such that φ(0) = 1; for example,

(5.10)  φ(w) = \{i(\frac{r + 2w}{r − 2w})^{2} + 1\}/\{(\frac{r + 2w}{r − 2w})^{2} + i\}
= Φ( − 2i(\sqrt{2} − 1)w/r)Φ(2i(\sqrt{2} + 1)w/r),

where Φ(t) = (1 + t)(1 − t)^{−1}. Then by expanding V_{p,α′}(φ^{−1}(w̃), z, ζ_{n}) into a power series of w̃, we have expansions

(5.11) V_{p,α′}(w, z, ζ_{n}) = ∑_{ℓ=0}^{∞} V_{p,α′,ℓ}(z, ζ_{n})φ(w)^{ℓ} (∀(w, z, ζ_{n}) ∈ W_{r/2}^{(1)}).

Here V_{p,α′,ℓ}(z, ζ_{n})’s are holomorphic functions in

W^{(2)} ≡ {ζ_{n} ∈ ℂ; |z| < κ, | Re ζ_{n}| < δ Im ζ_{n}}

satisfying

|V_{p,α′,ℓ}(z, ζ_{n})| ≤ 2^{ν}(2C)^{p+1}ρ^{−|α′|}|ζ_{n}|^{−p}

on W^{(2)} for all p, α′, ℓ. Therefore we have

(5.12)  U_{−p} = ∑_{α′,ℓ} ∂_{w}^{p+ν+1}{φ(w)^{ℓ}} · (ζ′/ζ_{n})^{α′}V_{p,α′,ℓ}(z, ζ_{n}).

Now for a large positive constant λ > 1 we introduce 2 kernel functions for V_{p,α′,ℓ}(z, ζ_{n})ζ_{n}^{p−2}:

A_{p,α′,ℓ}^{(1)}(z, s) ≡ ∫_{λ(ℓ+1)}^{∞} V_{p,α′,ℓ}(z, it) · (it)^{p−2}e^{its} \frac{dt}{2π} , (5.13)

A_{p,α′,ℓ}^{(2)}(z, s) ≡ ∫_{λ}^{λ(ℓ+1)} V_{p,α′,ℓ}(z, it) · (it)^{p−2}e^{its} \frac{dt}{2π} , (5.14)

which are holomorphic functions defined in

W^{(3)} ≡ {s ∈ ℂ; |z| < κ, Im s > −δ| Re s| }

with the estimates:

|A_{p,α′,ℓ}^{(1)}(z, s)| ≤ 2^{ν}(2C)^{p+1}ρ^{−|α′|}e^{−λ(ℓ+1) Im s}/π (5.15)

on W^{(3)} for all p, α′, ℓ. Further A_{p,α′,ℓ}^{(2)}(z, s) are entire functions satisfying

(5.16)  |A_{p,α′,ℓ}^{(2)}(z, s)| ≤ 2^{ν}(2C)^{p+1}ρ^{−|α′|}e^{λ(ℓ+1)(− Im s)_{+}}/π

on {s ∈ ℂ} for all p, α′, ℓ. Here (t)_{+} = t (∀t ≥ 0), = 0 (∀t < 0). Therefore for each k = 1, 2 and each p, α′ the series

A_{p,α′}^{(k)}(w, z, s) ≡ ∑_{ℓ=0}^{∞} A_{p,α′,ℓ}^{(k)}(z, s)φ(w)^{ℓ} (5.17)

converges locally uniformly in

(5.18)  \begin{cases}\{|φ(w)| < e^{λ Im s}\}  (∀s ∈ W^{(3)}) & for k = 1, \\ \{|φ(w)|e^{λ(− Im s)_{+}} < 1\} (∀s ∈ ℂ) & for k = 2. \end{cases}

We note that φ(w) at (5.10) is holomorphic in |w| < (\sqrt{2} − 1)r/2 and

(5.19) log |Φ(t)| = \frac{1}{2} log (Φ(\frac{2 Re t}{1 + |t|^{2}} )) = ∑_{ℓ=0}^{∞} \frac{1}{2ℓ + 1}(\frac{2 Re t}{1 + |t|^{2}} )^{2ℓ+1}
≤ 4(Re t)_{+} − (− Re t)_{+}

for ∀t (|t| ≤ 1/4). Consequently if |w| < (\sqrt{2} − 1)r/8, we have

log |φ(w)| ≤ (6\sqrt{2} + 10)(− Im w)_{+}/r − (10 − 6\sqrt{2})(Im w)_{+}/r.

Hence A_{p,α′}^{(1)}(w, z, s)’s are holomorphic in \{(w, z, s) ∈ ℂ × W^{(3)}; |w| < (\sqrt{2} − 1)r/8, |z| < κ, (5.20)

(10 + 6\sqrt{2})(− Im w)_{+} < λr Im s/2\}

⊃{(w, z, s) ∈ ℂ^{n+1} × ℂ; |w| < r/20, |z| < κ,

(− Im w)_{+} < \frac{λr}{40} Im s\} ≡ W_{λ}^{(4)}

because |φ(w)|e^{−λ Im s} ≤ e^{−λ Im s/2} < 1 on W_{λ}^{(4)}. At the same time we have the following estimates:

|A_{p,α′}^{(1)}(w, z, s)| ≤ \frac{2^{ν}(2C)^{p+1}}{πρ^{|α′|}(1 − e^{−λ Im s/2})} ≤ \frac{e2^{ν+1}(2C)^{p+1}}{πλρ^{|α′|}Im s}

on W_{λ}^{(4)} ∩ {0 < Im s ≤ 1/λ}. Consequently we obtain

|∂_{w}^{p+ν+1}A_{p,α′}^{(1)}(w, z, s)| ≤ (p + ν + 1)!\frac{re2^{ν+1}(2C)^{p+1}}{80πρ^{|α′|}} (\frac{80}{λr Im s})^{p+ν+2}

on

(5.21) W_{λ}^{(5)} ≡ {(w, z, s) ∈ ℂ^{n+1} × ℂ; |w| < r/40, |z| < κ,
max{0, −80 Im w/(λr)} < Im s ≤ 1/λ, | Re s| < λ^{−1}}.

Further A_{p,α′}^{(2)}(w, z, s)’s are holomorphic in \{(w, z, s) ∈ ℂ^{n+1} × ℂ; |w| < (\sqrt{2} − 1)r/8, |z| < κ, (5.22)

λ(− Im s)_{+} − (10 − 6\sqrt{2}) Im w/(2r) < 0\}
⊃{(w, z, s) ∈ ℂ^{n+1} × ℂ; |w| < r/20, |z| < κ,
Im w > 0, − Im w/(2λr) < Im s} ≡ W_{λ}^{(6)}

and satisfy the following estimates

|A_{p,α′}^{(2)}(w, z, s)| ≤ \frac{2^{ν}(2C)^{p+1}}{πρ^{|α′|}(1 − e^{− Im w/(2r)})} ≤ \frac{re2^{ν+1}(2C)^{p+1}}{πρ^{|α′|}Im w}

on W_{λ}^{(6)} because |φ(w)|e^{λ(− Im s)_{+}} ≤ e^{− Im w/(2r)} < 1 on W_{λ}^{(6)}. Consequently, setting

(5.23) W_{λ}^{(6)} ⊃ W_{λ}^{(7)} ≡ {(w, z, s) ∈ ℂ^{n+1} × ℂ; |z| < κ,
|w − ir/80| < r/320, Im s > −(320λ)^{−1}},

we get the following estimates:

|∂_{w}^{p+ν+1}A_{p,α′}^{(2)}(w, z, s)| ≤ (p + ν + 1)!\frac{re2^{ν+1}(2C)^{p+1}}{πρ^{|α′|}(r/320)^{p+ν+2}} (5.24)

on W_{λ}^{(7)}. Fixing the initial point s = i/λ, we apply again the preceding lemma to ∂_{w}^{p+ν+1}A_{p,α′}^{(k)}(w, z, s) for k = 1, 2. That is, we have holomorphic functions E_{p,α′}^{(1)}(w, z, s) ∈ O(W_{λ}^{(5)}) satisfying

(5.25)  \begin{cases}∂_{s}^{p+ν+4}E_{p,α′}^{(1)}(w, z, s) = ∂_{w}^{p+ν+1}A_{p,α′}^{(1)}(w, z, s), \\ |E_{p,α′}^{(1)}(w, z, s)| ≤ \frac{2^{p+ν+4}re2^{ν+1}(2C)^{p+1}}{80πρ^{|α′|}} (\frac{80}{λr})^{p+ν+2} \end{cases}

on W_{λ}^{(5)} for all p, α′, and E_{p,α′}^{(2)}(w, z, s) ∈ O(W_{λ}^{(7)} ∩ {|s − i/λ| < 2/λ}) satisfying

(5.26)  \begin{cases}∂_{s}^{p+ν+4}E_{p,α′}^{(2)}(w, z, s) = ∂_{w}^{p+ν+1}A_{p,α′}^{(2)}(w, z, s), \\ |E_{p,α′}^{(2)}(w, z, s)| ≤ \frac{re2^{ν+1}(2C)^{p+1}}{πρ^{|α′|}(r/320)^{p+ν+2}} (2/λ)^{p+ν+4} \end{cases}

on W_{λ}^{(7)} ∩ {|s − i/λ| < 2/λ}) for all p, α′. Choose λ (> 1) as

(5.27)  1280C/(λr) ≤ 1/2.

Then we can introduce 2 kernel functions E^{(k)}(w, z, z−z^{∗}, s) for k = 1, 2 as follows:

(5.28)  E^{(k)} ≡ ∑_{|α′|≥0,p≥0} \frac{α′!}{|α′|!}E_{p,α′}^{(k)}(w, z, s) ∏_{j=1}^{n−1} \frac{(z_{n} − z_{n}^{∗})^{α_{j}}}{(z_{j}^{∗} − z_{j})^{α_{j}+1}} .

Here, E^{(k)} are holomorphic in W^{(k)} at (5.4), (5.5), respectively for k = 1, 2. On the other hand, from (5.13), (5.14) and (5.11) we obtain the following:

(5.29) ∑_{k=1}^{2} ∂_{w}^{p+ν+1}A_{p,α′}^{(k)}(w, z, s) = ∂_{w}^{p+ν+1}( ∑_{ℓ=0}^{∞} ∑_{k=1}^{2} A_{p,α′,ℓ}^{(k)}(z, s)φ(w)^{ℓ})

= ∂_{w}^{p+ν+1}( ∫_{λ}^{∞} V_{p,α′}(w, z, it) · (it)^{p−2}e^{its} \frac{dt}{2π} )

= ∫_{λ}^{∞} U_{−p,α′}(w, z, it) · (it)^{p−2}e^{its} \frac{dt}{2π}

for any (w, z, s) ∈ W_{λ}^{(5)} ∩ W_{λ}^{(7)} ∩ {|s − iλ^{−1}| < 2λ^{−1}}

={(w, z, s) ∈ ℂ^{n+1} × ℂ; |w − ir/80| < r/320,
|z| < κ, 0 < Im s ≤ λ^{−1}, | Re s| < λ^{−1}}.

Hence we have our conclusion (5.6). QED

By using the expressions of the kernel functions obtained in the preceding theorem, we prove that U(w, x, D_{x})f(x) has a boundary value at w = 0 from Im w > 0 for any microfunction f(x). To do so, we introduce actions of E^{(∗)}(w, z, z − z^{∗}, s) on holomorphic functions F (z) similar to the Bony-Schapira’s actions of microdifferential operators on holomorphic functions [4].

Definition 5.3. We inherit the notation from the preceding theorem. Let F (z) be a holomorphic function defined in

Ω ≡{z ∈ ℂ^{n}; |z′| < r′, | Re z_{n}| < (3λ)^{−1} + r′, r′ > Im z_{n} > k| Im z′|} (5.30)
∪ ∪_{σ=±1}{z ∈ ℂ^{n}; |z′| < r′, |z_{n} − σ/(3λ)| < r′}

for positive small constants r′ (r′ < 1/(3λ)), and k (< ρ/(2n)). Let E(w, z, z − z^{∗}, s) be a holomorphic kernel function defined in W^{(1)} at (5.4). For a sufficiently small ε > 0, we define a holomorphic function (E ∗ F)_{λ,ε}(w, z) depending on λ, ε by

∫_{iε}^{z_{n}} dz_{n}^{∗} ∫_{γ} dz^{∗}′ ∫_{−1/(3λ)}^{1/(3λ)} E(w, z, z − z^{∗}, z_{n}^{∗} − z_{n}^{∗∗})F (z^{∗}′, z_{n}^{∗∗})dz_{n}^{∗∗}. (5.31)

Here the path for z_{n}^{∗} is the line segment

z_{n}^{∗}(t) = z_{n} + t(iε − z_{n}) (0 ≤ t ≤ 1)

combining z_{n} with iε, γ = {z_{j}^{∗} = z_{j} +R(z_{n}, z_{n}^{∗}(t))e^{iθ_{j}} (0 ≤ θ_{j} ≤ 2π); j = 1, …, n − 1} with some R(z_{n}, z_{n}^{∗}(t)) > ρ^{−1}|z_{n} − z_{n}^{∗}(t)| = tρ^{−1}|z_{n} − iε|. Further the path for z_{n}^{∗∗} is the line graph passing through

−(3λ)^{−1}, −(3λ)^{−1} + ih Im z_{n}^{∗}(t), (3λ)^{−1} + ih Im z_{n}^{∗}(t), (3λ)^{−1}

for a constant h (1/2 < h < 1). That is,

z_{n}^{∗∗}(θ; t) = \begin{cases}−1/(3λ) + 3ihθ Im z_{n}^{∗}(t) & (0 ≤ θ ≤ 1/3), \\ (2θ − 1)/λ + ih Im z_{n}^{∗}(t) & (1/3 ≤ θ ≤ 2/3), \\ 1/(3λ) + 3ih(1 − θ) Im z_{n}^{∗}(t) & (2/3 ≤ θ ≤ 1) \end{cases}

Indeed this integral is well-defined if Im w > 0, |w| < r/40, |z| < κ, | Re z_{n}| < (3λ)^{−1}, 0 < Im z_{n} ≤ ε < r′ and the following sets are contained in Ω :

{(z_{1} + tω_{1}, …, z_{n−1} + tω_{n−1}, q + ih(Im z_{n} + t(ε − Im z_{n}))); 0 ≤ t ≤ 1,

|q| ≤ (3λ)^{−1} (q ∈ ℝ), |ω_{1}| ≤ ρ^{−1}|z_{n} − iε|, …, |ω_{n−1}| ≤ ρ^{−1}|z_{n} − iε|},

{(z_{1} + tω_{1}, …, z_{n−1} + tω_{n−1}, ±(3λ)^{−1} + iq(Im z_{n} + t(ε − Im z_{n})));

0 ≤ t ≤ 1, 0 ≤ q ≤ h, |ω_{1}| ≤ ρ^{−1}|z_{n} − iε|, …, |ω_{n−1}| ≤ ρ^{−1}|z_{n} − iε|}.

The former set is contained in Ω if ε < r′ and

| Re z_{n}| < (2h − 1)(ε − Im z_{n}), Im z_{n} > (k/h)| Im z′|,
|z′| + (n/ρ)|z_{n} − iε| < r′.

The latter set is contained in Ω if | Im z_{n}| ≤ ε < r′ and

|z′| + (n/ρ)|z_{n} − iε| < r′.

Hence we obtain the following lemma:

Lemma 5.4. Let ε (> 0) be smaller than min{κ/2, (1 + 2n/ρ)^{−1}r′}. Then (E ∗ F)_{λ,ε}(w, z) is holomorphic in

{ Im w > 0, |w| < r/40, |z′| < ε, |z_{n}| < ε, (5.32)

Im z_{n} > k| Im z′|, | Re z_{n}| < ε − Im z_{n}}

Further let E′(w, z, z−z^{∗}, s) be a holomorphic kernel function defined in W^{(2)} at (5.5). Then (E′ ∗ F)_{λ,ε}(w, z) is holomorphic in a neighborhood of

{|w − ir/80| < r/320, z = 0}.

Proof. We have only to prove the latter statement. In this case we modify the paths of integrations as follows:

z_{n}^{∗}(t) = z_{n} + t(iε − z_{n}) (0 ≤ t ≤ 1),

z_{n}^{∗∗}(θ; t) = \begin{cases}−1/(3λ) + 3iθψ(t) & (0 ≤ θ ≤ 1/3), \\ (2θ − 1)/λ + iψ(t) & (1/3 ≤ θ ≤ 2/3), \\ 1/(3λ) + 3i(1 − θ)ψ(t) & (2/3 ≤ θ ≤ 1), \end{cases}

where ψ(t) = max{Im z_{n}^{∗}(t), ϵ} with some small ϵ > 0. If we choose ϵ < min{(640λ)^{−1}, ε}, for any z_{n} = iy_{n} (y_{n} ∈ (−ϵ, ε)) and t ∈ [0, 1] we have

ψ(t) = max\{y_{n} + t(ε − y_{n}), ϵ\} > \frac{t}{2}(ε − y_{n}) ≥ \frac{nk}{ρ} t(ε − y_{n}).

Therefore (E′ ∗ F)_{λ,ε}(w, z) is holomorphic in a neighborhood of

{|w − ir/80| < r/320, z′ = 0, Re z_{n} = 0, −ϵ < Im z_{n} < ε}.

This completes the proof. QED

The following is our secondary main result. K. Uchikoshi [14] used a similar method (Bronshtein’s method) of considering boundary values of holomorphic pseudodifferential operators for constructing fundamental solutions of weakly hyperbolic microdifferential operators. However the situations are different from each other, and the proofs and results are completely independent.

The most part of the proof is devoted to the proof of the compatibility of actions of pseudo-differential operators.

Theorem 5.5. Let U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ) be the classical formal symbol of the pseudo-differential operator treated in Theorem 5.2. Then for any microfunction f(x) ∈ C_{N}|_{(0;idx_{n})}, a section U(w, x, D_{x})f(x) ∈ Γ ({w ∈ ℂ; Im w > 0, |w| < r}×{(0; idx_{n})}; CO_{Z}) has a boundary value at (0, 0; idx_{n}) from Im w > 0 in the sense of Definition 3.9.

Proof. Choose a large positive number λ as indicated in Theorem 5.2. Let f(x) be any germ of C_{N} at (0; idx_{n}). Then by the flabbiness of the sheaf of microfunctions, we can take a defining function F (z) ∈ O(Ω) for (2πi)^{1−n}∂_{x_{n}}^{ν+7}f(x) with a sufficiently small r′ > 0, where Ω at (5.30) satisfies a tighter condition (because 0 < δ < 1):

(5.33)  0 < k < min{1, ρδ/(8n)}.

Choose a small positive number ε as indicated in Lemma 5.4:

(5.34)  0 < ε < min{κ/2, (1 + 2n/ρ)^{−1}r′},

which will be replaced by a tighter condition. Then, by Lemma 5.4 we conclude that the boundary value g(w, x) ≡ (E^{(1)} ∗ F)_{λ,ε}(w, x′, x_{n} + i0) is a section of

Γ ({Im w > 0, |w| < r/40, |x| < ε}; BO_{Z}).

On the other hand, by the latter result in Lemma 5.4 we have [g(w, x)] = [((E^{(1)} + E^{(2)}) ∗ F)_{λ,ε}(w, x′, x_{n} + i0)] as a section of CO_{Z} over {|w − ir/80| < r/320} × {(0; idx_{n})}. Therefore considering Theorem 5.2 and the unique continuation property of sections of CO_{Z}, we have only to show that U(w, x, D_{x})f(x) = [(E^{(0)} ∗ F)_{λ,ε}(w, x′, x_{n} + i0)] on a neighborhood of {|w − ir/80| < r/320} × {(0; idx_{n})}. Here

E^{(0)}(w, z, z − z^{∗}, s) ≡ ∑_{|α′|≥0,p≥0} \frac{α′!}{|α′|!}( ∏_{j=1}^{n−1} \frac{(z_{n} − z_{n}^{∗})^{α_{j}}}{(z_{j}^{∗} − z_{j})^{α_{j}+1}} )

× ∫_{i/λ}^{s} \frac{(s − s^{∗})^{p+ν+3}}{(p + ν + 3)!} ds^{∗} ∫_{λ}^{∞} U_{−p,α′}(w, z, it) · (it)^{p−2}e^{its^{∗}} \frac{dt}{2π} .

Set

(5.35)  K_{p,α′}(w, z, s) ≡ ∫_{λ}^{∞} U_{−p,α′}(w, z, it) · (it)^{p−2}e^{its} \frac{dt}{2π} .

Noting the estimates (5.8), we obtain the following: K_{p,α′}(w, z, s)’s are holomorphic in

(5.36)  W′ ≡ {|w − ir/80| < r/320, |z| < κ, Im s > −δ| Re s|},

and continuous up to s = 0 with estimates

|K_{p,α′}(w, z, s)| ≤ p!C^{p+1}ρ^{−|α′|}(160/r)^{p+μ}/π on W′.

Hence for ℓ = 0, 1, 2, …

(5.37)  K_{p,α′}^{(ℓ)}(w, z, s) ≡ ∫_{0}^{s} \frac{(s − s′)^{ℓ}}{ℓ!} K_{p,α′}(w, z, s′)ds′

are holomorphic functions in W′ with estimates

(5.38)  |K_{p,α′}^{(ℓ)}(w, z, s)| ≤ \frac{p!|s|^{ℓ+1}}{π(ℓ + 1)!ρ^{|α′|}} C^{p+1}(160/r)^{p+μ} on W′.

Therefore we have

E^{(0)}(w, z, z − z^{∗}, s) = ∑_{|α′|≥0,p≥0} \frac{α′!}{|α′|!}( ∏_{j=1}^{n−1} \frac{(z_{n} − z_{n}^{∗})^{α_{j}}}{(z_{j}^{∗} − z_{j})^{α_{j}+1}} ) (5.39)

× (K_{p,α′}^{(p+ν+3)}(w, z, s) − ∫_{0}^{i/λ} \frac{(s − s^{∗})^{p+ν+3}}{(p + ν + 3)!} K_{p,α′}(w, z, s^{∗})ds^{∗}).

Since λ^{−1} < r/(2560C), the first term of (5.39) is holomorphic in

(5.40) W^{(4)} ≡ {(w, z, z − z^{∗}, s) ∈ ℂ × ℂ^{n} × ℂ^{n} × ℂ; |s| < 16λ^{−1},
|w − ir/80| < r/320, Im s > −δ| Re s|, |z| < κ,
|z_{j} − z_{j}^{∗}| > ρ^{−1}|z_{n} − z_{n}^{∗}| (j = 1, …, n − 1)}.

Further the second term of (5.39) is holomorphic in

(5.41)  W^{(5)} ≡ {(w, z, z − z^{∗}, s) ∈ ℂ × ℂ^{n} × ℂ^{n} × ℂ;
|w − ir/80| < r/320, |s| < 15λ^{−1}, |z| < κ,
|z_{j} − z_{j}^{∗}| > ρ^{−1}|z_{n} − z_{n}^{∗}| (j = 1, …, n − 1)}.

Since W^{(5)} includes W^{(2)} at (5.5), the contribution of the second term to [(E^{(0)} ∗ F)_{λ,ε}(w, x′, x_{n} + i0)] is 0. Thus, our problem reduces to the action of the following operator on F (z): (5.42)

G(w, z, z − z^{∗}, s) = ∑_{|α′|≥0,p≥0} \frac{α′!}{|α′|!}( ∏_{j=1}^{n−1} \frac{(z_{n} − z_{n}^{∗})^{α_{j}}}{(z_{j}^{∗} − z_{j})^{α_{j}+1}} )K_{p,α′}^{(p+ν+3)}(w, z, s),

which is holomorphic in W^{(4)} and continuous up to s = 0. For any small positive numbers c < (3λ)^{−1} and δ′ (δ/2 ≤ δ′ < δ), we put

(G ∗ F)_{c,δ′,ε}(w, z) = (5.43)

∫_{iε}^{z_{n}} dz_{n}^{∗} ∫_{γ} dz^{∗}′ ∫_{−c(1−iδ′)}^{c(1+iδ′)} G(w, z, z − z^{∗}, z_{n}^{∗} − z_{n}^{∗∗})F (z^{∗}′, z_{n}^{∗∗})dz_{n}^{∗∗}.

Here the path z_{n}^{∗} = z_{n} + t(iε − z_{n}) (0 ≤ t ≤ 1) for z_{n}^{∗} and γ are chosen in the same way with Definition 5.3. Further the path z_{n}^{∗∗} = z_{n}^{∗∗}(θ; t) for z_{n}^{∗∗} is the line graph passing through

−c(1 − iδ′), z_{n}^{∗}, c(1 + iδ′)

That is, for −1 ≤ θ ≤ 1, 0 ≤ t ≤ 1 we have

(5.44)  z_{n}^{∗∗}(θ; t) = (1 − |θ|)(z_{n} + t(iε − z_{n})) + cθ + icδ′|θ|
= cθ + (1 − |θ|)(1 − t) Re z_{n}
+ i(cδ′|θ| + (1 − |θ|)(εt + (1 − t) Im z_{n})).

This integral is well-defined if |w−ir/80| < r/320, |z| < κ, |z_{n}|+ε+2c < 16λ^{−1}, δ| Re z_{n}| − Im z_{n} < (δ − δ′)c and the following set is contained in Ω :

{(z_{1} + tω_{1}, …, z_{n−1} + tω_{n−1}, z_{n}^{∗∗}(θ; t)); −1 ≤ θ ≤ 1,
0 ≤ t ≤ 1, |ω_{1}| ≤ ρ^{−1}|z_{n} − iε|, …, |ω_{n−1}| ≤ ρ^{−1}|z_{n} − iε|},

Indeed, this set is contained in Ω if |z′| + (n/ρ)|z_{n} − iε| < r′, | Re z_{n}| + c < (3λ)^{−1} and if for any θ ∈ [−1, 1], t ∈ [0, 1] we have

k| Im z′| + \frac{nkt}{ρ}|z_{n} − iε| < cδ′|θ| + (1 − |θ|)(εt + (1 − t) Im z_{n}) < r′

⟺ \begin{cases}k| Im z′| + nkt|z_{n} − iε|/ρ < Im z_{n} + t(ε − Im z_{n}) (∀t ∈ [0, 1]), \\ k| Im z′| + nk|z_{n} − iε|/ρ < cδ′, \\ cδ′|θ| + (1 − |θ|)(εt + (1 − t) Im z_{n}) < r′ (∀t, ∀|θ| ∈ [0, 1]). \end{cases}

Hence, these conditions are all satisfied under (5.33),(5.34) and δ/2 ≤ δ′ < δ if

ε + c < min{(3λ)^{−1}, r′}, ε < (ρ/n + 2)^{−1}c,
|w − ir/80| < r/320, |z′| < ε, |z_{n}| < ε,
k| Im z′| < Im z_{n}, | Re z_{n}| < ε − Im z_{n}.

Replace the condition (5.34) by

(5.45)  c < \frac{min\{(3λ)^{−1}, r′\}}{2 + 3n/ρ}, ε < min \{\frac{κ}{2}, \frac{r′}{1 + 2n/ρ}, \frac{c}{ρ/n + 2}\}.

Therefore (G ∗ F)_{c,δ′,ϵ}(w, z) is holomorphic in

{|w − ir/80| < r/320, |z′| < ε, |z_{n}| < ε,
k| Im z′| < Im z_{n}, | Re z_{n}| < ε − Im z_{n}}.

We claim here that

[(G ∗ F)_{λ,ε}(w, x′, x_{n} + i0)] = [(G ∗ F)_{c,δ′,ε}(w, x′, x_{n} + i0)]

on a neighborhood of {|w − ir/80| < r/320} × {(0; idx_{n})}. To prove this, it is sufficient to show that the functions

∫_{iε}^{z_{n}} dz_{n}^{∗} ∫_{γ} dz^{∗}′ ∫_{cσ+icδ′}^{σ(3λ)^{−1}} G(w, z, z − z^{∗}, z_{n}^{∗} − z_{n}^{∗∗})F (z^{∗}′, z_{n}^{∗∗})dz_{n}^{∗∗}

extend holomorphically to {|w − ir/80| < r/320} × {z = 0} for all σ = ±1. Take the line graph Γ_{σ} passing through

cσ + icδ′, σ(3λ)^{−1} + icδ′, σ(3λ)^{−1}

as the paths of integration. Indeed, these integrals are well-defined for (w, z) ∈ {|w − ir/80| < r/320} × {z = 0} because Ω includes

{(tω_{1}, …,tω_{n−1}, z_{n}^{∗∗}); z_{n}^{∗∗} ∈ Γ_{σ}, 0 ≤ t ≤ 1,
|ω_{1}| ≤ ρ^{−1}ε, …, |ω_{n−1}| ≤ ρ^{−1}ε}

under the conditions (5.45). This proves our claim above.

As a last step of the proof, we eliminate the variable z_{n}^{∗} from the integral expression (5.43). When we restrict the variables (w, z) to a neighborhood of {|w − ir/80| < r/320} × {z′ = 0, z_{n} = 3iε/4}, we can deform the path z_{n}^{∗∗}(θ; t) at (5.44) for z_{n}^{∗∗} to the line graph passing through

−c + icδ′, iε/2, c + icδ′.

Indeed, this deformation is possible if ε + 2c < 16λ^{−1} and the following set is contained in Ω :

{(ω_{1}, …, ω_{n−1}, cθ + i(cδ′|θ| + (1 − |θ|)ε/2));
− 1 ≤ θ ≤ 1, |ω_{1}| ≤ ρ^{−1}ε/4, …, |ω_{n−1}| ≤ ρ^{−1}ε/4}.

Hence, all of these conditions are satisfied under (5.45) because nε/(4ρ) < r′ and

nkε/(4ρ) < cδ′|θ| + (1 − |θ|)ε/2 < r′ (∀|θ| ∈ [0, 1])

⟺ \begin{cases}nkε/(4ρ) < min\{cδ′, ε/2\}, \\ max\{cδ′, ε/2\} < r′. \end{cases}

Since this new path for z_{n}^{∗∗} does not depend on z_{n}^{∗}, we can exchange the order of integration in the integral (G ∗ F)_{c,δ′,ε}(w, z) : That is, for any (w, z) as above, we have

(5.46) (G ∗ F)_{c,δ′,ε}(w, z)
= ∫_{−c(1−iδ′)}^{c(1+iδ′)} dτ ∫_{γ} dz^{∗}′F (z^{∗}′, τ) ∫_{iε−τ}^{z_{n}−τ} G(w, z, z′ − z^{∗}′, z_{n} − τ − s, s)ds

= ∫_{−c(1−iδ′)}^{c(1+iδ′)} dτ ∫_{γ} F (z^{∗}′, τ)(H(w, z′ − z^{∗}′, z_{n} − τ, z_{n} − τ)
− H(w, z′ − z^{∗}′, z_{n} − τ, iε − τ))dz^{∗}′

Here

H(w, z′ − z^{∗}′, s_{1}, s_{2}) ≡ ∫_{0}^{s_{2}} G(w, z, z′ − z^{∗}′, s_{1} − s, s)ds (5.47) is holomorphic in

W^{(6)} ≡ {(w, z, z′ − z^{∗}′, s_{1}, s_{2}) ∈ ℂ × ℂ^{n} × ℂ^{n−1} × ℂ^{2}; |s_{2}| < 16λ^{−1},
|w − ir/80| < r/320, Im s_{2} > −δ| Re s_{2}|, |z| < κ,
|z_{j} − z_{j}^{∗}| > ρ^{−1} max{|s_{1} − s_{2}|, |s_{1}|} (j = 1, …, n − 1)}.

Therefore the second term of (5.46) extends holomorphically to

{|w − ir/80| < r/320} × {(z′, z_{n}); z′ = 0, z_{n} = it (0 ≤ t ≤ 3ε/4)}

if the following sets are contained in Ω for all t ∈ [0, 3ε/4]:

{(ω_{1}, …, ω_{n−1}, cθ + i(cδ′|θ| + (1 − |θ|)ε/2)); −1 ≤ θ ≤ 1,

max_{1≤j≤n−1} |ω_{j}| ≤ \frac{max\{|t − ε|, c|θ| + |t − (cδ′|θ| + (1 − |θ|)ε/2|\}}{ρ}\}

⊂ {(ω_{1}, …, ω_{n−1}, cθ + i(cδ′|θ| + (1 − |θ|)ε/2));

− 1 ≤ θ ≤ 1, max_{1≤j≤n−1} |ω_{j}| ≤ \frac{ε + 2c|θ|}{ρ}\}

Hence these sets are contained in Ω if n(ε + 2c)/ρ < r′ and

nk(ε + 2c|θ|)/ρ < cδ′|θ| + (1 − |θ|)ε/2 < r′ (∀|θ| ∈ [0, 1])

⟺ \begin{cases}nkε/ρ < ε/2, nk(ε + 2c)/ρ < cδ′, \\ max\{cδ′, ε/2\} < r′. \end{cases}

Indeed all these conditions are fullfilled under the conditions (5.45); we essentially required the tight condition (5.33) for nk(ε + 2c)/ρ < cδ′.

Further we deform the path of integration for the first term of (5.46) to the line graph passing through

−c + icδ′, z_{n}, c + icδ′.

Then we can extend the first term of (5.46) holomorphically to

{|w − ir/80| < r/320, |z′| < ε, |z_{n}| < ε,
k| Im z′| < Im z_{n}, | Re z_{n}| < ε − Im z_{n}}

by the same estimates of domains with ones for (G ∗ F)_{c,δ′,ε}(w, z); indeed, the present estimates for domains exactly correspond to the case t = 0 at (5.43). Consequently the boundary value of

∫_{−c(1−iδ′)}^{c(1+iδ′)} dτ ∫_{γ} H(w, z′ − z^{∗}′, z_{n} − τ, z_{n} − τ)F (z^{∗}′, τ)dz^{∗}′

coincides with [(E^{(0)} ∗ F)_{λ,ε}(w, x′, x_{n} + i0)] as a section of CO_{Z} in a neighborhood of {|w − ir/80| < r/320} × {(0; idx_{n})}. On the other hand it is clear that

H(w, z, z − z^{∗}′, s, s) = ∑_{|α′|≥0,p≥0} K_{p,α′}^{(p+|α′|+ν+4)}(w, z, s) ∏_{j=1}^{n−1} \frac{α_{j}!}{(z_{j}^{∗} − z_{j})^{α_{j}+1}}

is the kernel function for (2πi)^{n−1}U(w, z, D_{z})D_{z_{n}}^{−ν−7} (see [1]). Thus we have

[(E^{(0)} ∗ F)_{λ,ε}(w, x′, x_{n} + i0)] =

(2πi)^{n−1}U(w, x, D_{x})D_{x_{n}}^{−ν−7}((2πi)^{1−n}D_{x_{n}}^{ν+7}f(x)) = U(w, x, D_{x})f(x)

on a neighborhood of {|w − ir/80| < r/320} × {(0; idx_{n})}. This completes the proof. QED

Remark 5.6. The growth order condition (5.2) for the lower order terms of ∑_{j=−∞}^{0} U_{j}(w, z, ζ) is the best possible in the following sense: For any constant k (1 < k < 2) there exists a classical formal symbol U = ∑_{j=−∞}^{0} U_{j}(w, z, ζ) satisfying the following 1∼3:

(1) U_{j} ∈ O(W_{r}) (∀j ≤ 0).

(2) For some constants C, μ > 0 we have

|U_{−p}(w, z, ζ)| ≤ C^{p+1}p!| Im w|^{−kp−μ}|ζ|^{−p} on W_{r} (∀p ≥ 0).

(3) U(w, x, D_{x})δ(x_{n}) does not have a boundary value at (0, 0; idx_{n})

in microfunctions of (Re w, x) from Im w > 0. Indeed, we can give an explicit example as follows:

(5.48)  U_{−p}(w, z, ζ) ≡ p!{−i(w/i)^{k}}^{−p−1}ζ_{n}^{−p−1},

where p = 0, 1, 2, … and | arg(w/i)| < π/2. It is easy to see that the above conditions 1, 2 are satisfied and that

U(w, x, D_{x})δ(x_{n}) = [ ∑_{p=0}^{∞} −\frac{1}{2πi}\{−i(w/i)^{k}\}^{−p−1}z_{n}^{p} log z_{n}]

= [\frac{1}{2πi} · \frac{log z_{n}}{z_{n} + i(w/i)^{k}} ].

Here the equalities are valid for sections of CO_{Z} over {Im w > ε} × {(0; idx_{n})} with any small positive ε. Then by the following lemma we get the above condition 3 for U.

The following example is a variant of the example in [11] of second hyperfunctions:

Lemma 5.7. Let k be a constant satisfying 1 < k < 2. Then the

microfunction f(w, x) = \frac{log(x + i0)}{x + i(w/i)^{k}} extends to \{(w, x; iηdx) ∈ ℂ ×

T_{ℝ}^{∗}ℂ; Im w > 0, η > 0} as a microfunction with holomorphic parameter w. However f(w, x) never has a microfunction boundary value at (0, 0; idx) from Im w > 0.

Proof. Consider a holomorphic function

F_{1}(w, z) = \frac{log z − log\{(w/i)^{k}/i\}}{z + i(w/i)^{k}}

defined in {(w, z) ∈ ℂ^{2}; Im z > 0, π(1 − k^{−1})/2 < arg w < π}, where −π < arg{(w/i)^{k}/i} < π(k − 1)/2 < π/2. Further set

F_{2}(w, z) = F_{1}(w, z) + \frac{−2πi}{z + i(w/i)^{k}} .

Then F_{2}(w, z) ∈ O({(w, z) ∈ ℂ^{2}; Im z > 0, 0 < arg w < π(1+k^{−1})/2}). Hence the extension of f(w, x) is given by

f(w, x) = \begin{cases}[F_{1}(w, x + i0)] (π(1 − k^{−1})/2 < arg w < π), \\ [F_{2}(w, x + i0)] (0 < arg w < π(1 + k^{−1})/2) \end{cases}

as microfunctions with holomorphic parameter w. We suppose here that f(w, x) has a microfunction boundary value at (0, 0; idx) from Im w > 0. Therefore, we have a holomorphic function G(w, z) defined in {Im w > 0, Im z > 0, |w| + |z| < ε} with some ε > 0 satisfying

f(w, x) = [G(w, x + i0)]

as sections of microfunctions with holomorphic parameter w in

{(w, x; iηdx) ∈ ℂ × T_{ℝ}^{∗}ℂ; |w| + |x| < ε, Im w > 0, η > 0}.

Since

0 = {x + i(w/i)^{k}}(f(w, x) − [G(w, x + i0)])
= [log(x + i0) − {x + i(w/i)^{k}}G(w, x + i0)],

we conclude that

A(w, z) ≡ log z − {z + i(w/i)^{k}}G(w, z)
∈ O({Im w > 0, Im z > 0, |w| + |z| < ε})

extends holomorphically to {|w| + |z| < ε′, Im z = 0, Im w > 0} with some smaller ε′ > 0. Therefore by Kashiwara’s theorem on the local version of Bochner’s tube theorem we can extend A(w, z) to a holomorphic function Ã(w, z) in

Ω = {|w| + |z| < ε′′, | Im z| < ε′′ Im w}

for some smaller ε′′ > 0. Set

P (r, θ) = (re^{iθ}, r^{k}e^{i(kθ−π(k+1)/2)}) ∈ {(w, z) ∈ ℂ^{2}; z + i(w/i)^{k} = 0}

for r > 0, 0 < θ < π. We note that P (r, θ) ∈ Ω for any θ ∈ (0, π) with any sufficiently small r > 0 because k > 1. Therefore H(w) ≡ Ã(w, −i(w/i)^{k}) is a holomorphic function in

W = {w ∈ ℂ; 0 < arg w < π, 0 < |w| < φ(arg w)}

with a positive valued continuous function φ(θ) on (0, π). On the other hand, we have that

H(re^{iθ}) = A(re^{iθ}, r^{k}e^{ik{θ−π(1+k^{−1})/2}}) = k log r + ik{θ − π(1 + k^{−1})/2}

for π(1 + k^{−1})/2 < θ < π, and that

H(re^{iθ}) = A(re^{iθ}, r^{k}e^{ik{θ+π(3k^{−1}−1)/2}}) = k log r + ik{θ + π(3k^{−1} − 1)/2}

for 0 < θ < π(1 − k^{−1})/2. That is, H(w) − k log w ∈ O(W) coincides with 2 different constants −π(k + 1)i/2 and −π(k − 3)i/2 in the above domains, respectively. This contradicts with the connectedness of W. Thus f(w, x) never have a microfunction boundary value at (0, 0; idx) from Im w > 0. QED

Here we return to our original subject for solving a microdifferential equation (1.2) in CO_{Z}.

Theorem 5.8. We have a system {U^{(ℓ)}(w, z, D_{z}); ℓ = 1, ..., m} of formal symbols of microdifferential operators defined around p^{∘} satisfying the conditions 1 ∼ 5 in Introduction:

(1) U^{(1)} is a multivalued section of E_{X} over {(w, x; iη) ∈ T_{Z}^{∗}X; 0 <

|w −φ(x, iη)| < r, |x−x^{∘}| < r, |η −η^{∘}| < r} for some small r > 0,

and U^{(ℓ)} ∈ E_{X}|p^{∘} for ℓ = 2, ..., m.

(2) U^{(ℓ)}(w, z, D_{z}) (ℓ = 1, ..., m) commute with w.

(3) P (w, z, D_{w}, D_{z})U^{(ℓ)}(w, z, D_{z}) = 0 (mod E_{X} · D_{w}) for ℓ = 1, ..., m.

(4) ord(U^{(ℓ)}) = 0 for ℓ = 1, ..., m, and holomorphic functions {σ_{0}(U^{(ℓ)})(w, x, iη); ℓ = 1, ..., m} give a complete system of solutions in {w − φ(x, iη) ≠ 0} of the following linear ordinary differential equation :

LU := ( ∑_{k=0}^{m} σ_{0}(A_{k})(w, x, 0, iη)\frac{∂^{m−k}}{∂w^{m−k}} )U = 0

(5) For any microfunction f(x) ∈ C_{N}|_{ρ(p^{∘})}, U^{(1)}(w, x, D_{x})f(x) has a microfunction boundary value at w = φ(x, iη); that is, a microfunction boundary value from any side of any conic and real analytic hypersurface H of T_{Z}^{∗}X passing through K = {w − φ(x, iη) = 0}.

Proof. Let H be any conic and real analytic hypersurface passing through K = {w − φ(x, iη) = 0} in T_{Z}^{∗}X. Then, by Lemma 2.4, we can reduce the triple (K, H, P) to the case that K = {w = 0}, H = {Im w = 0} and P with A_{0}(w, z, D_{w}, D_{z}) ≡ w under a quantized contact transformation S, which preserves CO_{Z} and satisfies

S^{−1}(D_{w}) ∈ E_{X} · D_{w^{∗}}.

Further by taking the multiple of w by some non-zero constant as a new variable, we can suppose the coefficients A_{k}(w, z, D_{w}, D_{z}) (k = 1, ..., m) are defined in some neighborhood of {|w| ≤ 1} × {p^{∘}}. Therefore under the assumption σ_{0}(A_{1})(p^{∘}) ∉ ℤ we can construct a system {U_{0}^{(ℓ)}; ℓ = 1, ..., n} of solutions of LU = 0 satisfying the following:

(1) U_{0}^{(1)}(w, z, ζ) has the following form in a neighborhood of w = 0:

U_{0}^{(1)}(w, z, ζ) = w^{m−1−A_{1}^{0}(0,z,ζ)}(1 + ∑_{k=1}^{∞} c_{k}(z, ζ)w^{k}),

where A_{1}^{0}(0, z, ζ) = σ_{0}(A_{1})(0, z, 0, ζ), c_{k}(z, ζ) are holomorphic functions of homogeneous degree 0 with respect to ζ.

(2) U_{0}^{(ℓ)}(w, z, ζ) (2 ≤ ℓ ≤ m) are holomorphic solutions in a neighborhood of w = 0 such that

∂_{w}^{k}U_{0}^{(ℓ)}(0, z, ζ) = δ_{k,ℓ−2} (0 ≤ ∀k ≤ m − 2).

It is clear that these solutions are of homogeneous degree 0 with respect to ζ, and that they uniquely extend to the solutions defined in a neighborhood of {|w| ≤ 1} × {p^{∘}} (for ℓ = 2, ..., m), or a neighborhood of the universal covering of {0 < |w| ≤ 1}×{p^{∘}} (for ℓ = 1). Therefore by Theorems 4.8, 4.9 we get the formal symbols {∑_{j=−∞}^{0} U_{j}^{(ℓ)}; ℓ = 1, ..., m} of microdifferential operators satisfying the conditions 1 ∼ 4. Further by Theorem 5.5 we also get the condition 5 for U^{(1)}.

On the other hand, as for the relationship with the original equation, we know the following: Let S be any quantized contact transformation satisfying S^{−1}(D_{w}) ∈ E_{X} ·D_{w^{∗}}. If the original operator P is transformed into P^{∗} = S^{−1}(P), for the solutions U^{(ℓ)} of P^{∗}U = 0 mod E_{X} · D_{w^{∗}} S(U^{(ℓ)}) also become solutions of P U = 0 mod E_{X} · D_{w}. Considering Remark 2.5, we complete the proof. QED

As a direct corollary, under the same notation with the preceding theorem we have the following:

Theorem 5.9. Let q = (w′, x^{∘}; iη^{∘}) (w′ ≠ w^{∘}) be a point of T_{Z}^{∗}X sufficiently close to p^{∘}, and f(w, x) ∈ CO_{Z}|_{q} be a solution of P f = 0 in a neighborhood of q. Then there exists uniquely an m-vector

(f_{1}(x), ..., f_{m}(x)) ∈ C_{N}^{m}|_{(x^{∘};iη^{∘})}

of microfunctions of x such that f(w, x) = ∑_{ℓ=1}^{m} U^{(ℓ)}(w, x, D_{x})f_{ℓ}(x) in a neighborhood of q. In particular, f(w, x) extends uniquely to a multivalued CO_{Z}-soltuion of P f = 0 around p^{∘} with microfunction boundary values at p^{∘}.

Proof. We have only to show the following: The matrix

(∂_{w}^{k−1}U^{(ℓ)}(w′, x, D_{x}))_{k,ℓ=1,...,m}

of microdifferential operators in E_{Y} is invertible at q as a morphism C_{N}^{m} → C_{N}^{m}. Further this invertibility reduces to that of the matrix

(∂_{w}^{k−1}σ_{0}(U^{(ℓ)})(w′, x, iη))_{k,ℓ=1,...,m}.

Indeed, since σ_{0}(U^{(ℓ)})(w, x, iη) (ℓ = 1, ..., m) are linearly independent solutions of LU = 0 from each other, we have the invertibility of this matrix. Thus the proof is completed. QED

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