# On Continuation of Gevrey Class Solutions of Linear Partial

## Transcript Of On Continuation of Gevrey Class Solutions of Linear Partial

J. Math. Sci. Univ. Tokyo 4 (1997), 551–593.

On Continuation of Gevrey Class Solutions of Linear Partial Diﬀerential Equations

By Akira Kaneko1

Dedicated to Professor Hikosaburo KOMATSU for his 60-th anniversary

Abstract. We give a suﬃcient condition for the removability of thin singularities of Gevrey class solutions of linear partial diﬀerential equations. In §1 we give a suﬃcient condition for the removability in the case of equations with constant coeﬃcients. Then in §2 we discuss the necessity of the condition and construct non-trivial solutions with irremovable thin singularities for some class of equations. In §3 we give a suﬃcient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coeﬃcients.

§0. Introduction

In this article, we gather results on continuation to thin singularity (or removability of thin singularities) of Gevrey class solutions to linear partial diﬀerential equations. Some of the results given here are easily derived from Grushin’s pioneering works on continuation of C∞ solutions and from the author’s former works on continuation of regular solutions. But it will be worth gathering them all to an article, because they may not be obvious for the readers who are not specialized in this subject. Moreover it will be adequate to dedicate this to Professor Hikosaburo Komatsu, who devoted his half carreer to the study of ultra-diﬀerentiable functions and ultradistributions.

Here is a brief plan of the present article. The ﬁrst two sections treat equations with constant coeﬃcients. In §1 we give a suﬃcient condition for

1991 Mathematics Subject Classiﬁcation. 35G05, 35B60, 35E20.

1Partially supported by GRANT-IN-AID FOR SCIENTIFIC RESEARCH No. 07404003.

551

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the removability of thin singularities of the Gevrey class solutions. This is a translation of Grushin’s work except for small details. In §2 we discuss the necessity of the condition given in §1. This is to construct non-trivial solutions with irremovable thin singularities under the condition opposite to §1. We generalize the construction of Grushin who gave such a few examples in his work [G2]. As an example, the precise Gevrey index for the threshold of existence of solutions with thin singularity is determined for the Schro¨dinger equation. In §3 we give a suﬃcient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coeﬃcients. This is a modiﬁcation of the author’s work for the removability of thin singularities of real analytic solutions.

For a general survey on this subject, we refer to [Kn11] for results until 1992, and [Kn12], where a list of open problems is gathered. The present article treats some of them concerning Gevrey class solutions.

§1. Continuation of Gevrey class solutions to equations with constant coeﬃcients

Let P (D) be a linear partial diﬀerential operator with constant coef-

ﬁcients, where P (ζ) is a polynomial in n variables ζ = (ζ1, . . . , ζn) and D = (D1, . . . , Dn) with Dj = −i∂/∂xj, j = 1, . . . , n. We deﬁne the two spaces of Gevrey class functions of index s by

(1.1)

E(s)(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∀h > 0, ∃CK,h > 0 sup |Dαf (x)| ≤ CK,hh|α|α!s for ∀α},

x∈K

and

(1.2) E{s}(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∃h = h(K) > 0,

∃C = C(K) > 0, sup |Dαf (x)| ≤ Ch|α|α!s for ∀α}.

x∈K

The ﬁrst space has a simple topology of Fr´echet space and is easier to treat, but it is a little less natural because when s = 1, this corresponds to the space of entire functions. The second space has a very complicated topological structure and does not allow the closed range theorem to hold.

Continuation of Gevrey Class Solutions

553

Hence for this space we cannot utilize fundamental theorems such as the global surjectivity on convex open sets of linear partial diﬀerential operators with constant coeﬃcients or the Fundamental Principle of EhrenpreisPalamodov. But it is more natural because for s = 1 this corresponds to the space of real analytic functions which is localizable along the real axis.

We also set

(1.3)

E1+(Ω) := E(s)(Ω) = E{s}(Ω).

s>1

s>1

This is a very convenient space, still containing enough functions in non-

quasianalytic ultra-diﬀerentiable class. Following the usage of Komatsu, we shall denote in the sequel by E∗(Ω) either of the spaces E(s)(Ω), E{s}(Ω), E1+(Ω) when we can state something commonly to these spaces. Thus E∗ denotes either of these function classes. In the same time, this symbol will denote the corresponding sheaf (that is, the localization) on Rn. As usual we let D∗(Ω) denote the functions of class E∗ with compact support

contained in Ω (together with the obvious topology if the dual space, that is,

the space of ultradistributions of this class, is considered). For a general set L we let E∗(L) denote the functions of class E∗ deﬁned on a neighborhood

of L, with the obvious identiﬁcation in the sense of inductive limit with

respect to the neighborhoods. In general, we denote by EP∗ (Ω) the space of solutions in Ω of the equa-

tion P (D)u = 0 of class E∗. Let K ⊂ Ω denote a thin compact subset.

Here “thin” means that the interior is void. We assume that it is contained

in a hyperplane, say ν · x = 0. (This follows automatically for convex thin

set, as we mainly consider in the sequel.) We study the continuation of solutions of P (D)u = 0 in E∗(Ω \ K) to solutions in E∗(Ω).

Proposition 1.1. The canonical map induced by the canonical restriction from Ω to Ω \ K:

EP∗ (Ω) → EP∗ (Ω \ K)

is injective. In other words, there are no solutions of P (D)u = 0 with compact support.

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Actually, any element in the kernel of the above map would be a solution of P (D)u = 0 with compact support. But via the Fourier transform we would then obtain P (ζ)u = 0, where u is entire, whence u ≡ 0.

Thus the quotient space

(1.4)

EP∗ (Ω \ K)/EP∗ (Ω)

will represent the obstruction for continuation of solutions of this class to

K. The reason why we restrict K to thin sets is obvious from the non-

quasianalyticity of the function class under consideration: If K had a nonvoid interior, then choosing f ∈ D∗(Int K) \ P (D)D∗(Int K) and a solution u ∈ E∗(Rn) of P (D)u = f , u|Ω\K would present a non-trivial element of (1.4).

First we shall show that the obstruction space (1.4) depends only on K

and not on Ω. For this purpose we recall the notion of local cohomology groups with coeﬃcients in the solution sheaf EP∗ of class E∗ of the equation P u = 0.

Proposition 1.2. We have the following isomorphism

(1.5)

EP∗ (Ω \ K)/EP∗ (Ω) =∼ HK1 (Ω, EP∗ ).

More generally, for any set L containing K in its interior, we have

(1.5bis)

EP∗ (L \ K)/EP∗ (L) =∼ HK1 (L, EP∗ ).

The quotient space in (1.5) or (1.5bis) is determined by K only and does not depend on the choice of the neighborhoods.

Proof. Recall the following fundamental exact sequence of local cohomology groups:

(1.6)

0 → ΓK (Ω, EP∗ ) → Γ (Ω, EP∗ ) → Γ (Ω \ K, EP∗ ) → HK1 (Ω, EP∗ ) → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → H2(Ω, EP∗ ) = 0.

Continuation of Gevrey Class Solutions

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Here we have ΓK(Ω, EP∗ ) = 0 by Proposition 1.1. (The fact H2(Ω, EP∗ ) = 0 follows from the resolution (1.7) as will be discussed below.) We shall show

that the mapping

H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ )

is always injective. Then we will obtain the isomorphism (1.5). Incidentally, we obtain the exact sequence

(1.7)

0 → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → 0.

Recall now the following exact sequence of sheaves

(1.8)

0 → EP∗ → E ∗ P→(D) E ∗ → 0.

Here the surjectivity in the last arrow, that is, the local solvability in this class, is an easy consequence of the existence of a fundamental solution for a single linear partial diﬀerential operator P with constant coeﬃcients. Taking the fundamental exact sequence of global cohomology groups on an open set Ω, we obtain

(1.9)

0 → Γ (Ω, EP∗ ) → Γ (Ω, E∗) P−(→D) Γ (Ω, E∗) → H1(Ω, EP∗ ) → H1(Ω, E∗) = 0.

The fact H1(Ω, E∗) = 0 is obvious because the sheaf E∗ is ﬁne. (From this H2(Ω, EP∗ ) = 0 also follows.) Thus it suﬃces to show that the natural mapping induced from the restriction

E∗(Ω)/P (D)E∗(Ω) → E∗(Ω \ K)/P (D)E∗(Ω \ K)

is injective. Suppose that u ∈ E∗(Ω) represents an element mapped to 0. This implies that there exists v ∈ E∗(Ω \ K) such that

u|Ω\K = P (D)v.

Employing partitions of unity, construct h ∈ E∗(Ω) and w ∈ E∗(Rn \ K) such that v = h − w on Ω \ K. We can obviously choose w in such a way

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Akira Kaneko

that w ≡ 0 outside a ball of some radius R > 0, by cutting v smoothly in the class E∗ on a neighborhood of K. Put

P (D)w on Rn \ K, g=

P (D)h − u on Ω.

This deﬁnition is consistent on the common domain:

P (D)w = P (D)h − P (D)v = P (D)h − u on Ω \ K.

Thus g becomes a well deﬁned element of E∗(Rn), which has compact support by the choice of w as above. Let f ∈ E∗(Rn) be a solution of P (D)f = g. (We can simply obtain such f by convoluting g with the distributional fundamental solution of P (D) which obviously preserves the Gevrey regularity.) Then, on Ω we have

u = P (D)(h − f ).

Hence, it represents 0 in E∗(Ω)/P (D)E∗(Ω). The proof for general neighborhood L of K is just similar. Now that

the isomorphism (1.5) is established, the ﬁnal conclusion follows from the excision theorem of local cohomology groups.

Remark. 1) We cannot expect H1(Ω, EP∗ ) = 0, unless we have the global surjectivity of P (D) on Ω in this function class. This follows from the exact sequence (1.8). To have this surjectivity for open Ω, we ﬁrst of all need to assume that Ω is convex. Then it is valid for the class E(s) (see e.g. Bjo¨rck [Bj1]), but still not in general for E{s} (see e.g. Cattabriga [C1]). The above method of argument was ﬁrst introduced by [Kn6] for the real analytic solutions, to which the global surjectivity is neither available. Note that the above Proposition (or the sequence (1.7)) implies that in such a situation, the obstruction for the global surjectivity is concentrated on the neighborhood of ∂Ω.

2) We have an alternative choice of neighborhoods of K for which the global surjectivity holds. It is to take compact neighborhoods L ⊃⊃ K. In this case the surjectivity of P (D) : E∗(L) → E∗(L) holds irrespective of the convexity of L, because of the non-quasianalytic property of our class. This

Continuation of Gevrey Class Solutions

557

fact was implicitly employed by some of the proofs for the corresponding assertions in the author’s former publications.

In view of the above Proposition, we can always assume Ω to be convex, thus allowing to apply the Fourier analysis. Sometimes the choice of convex compact L simpliﬁes the situation further.

Next, we shall show that the obstruction (1.4) can be decomposed via the irreducible components of P (D):

Corollary 1.3. Let Q be any factor of P . Then we have a canonical injection

(1.10a)

H

1 K

(Ω

,

EQ∗

)

→

H

1 K

(Ω

,

EP∗

).

Conversely, let P = Qm1 1 · · · QmNN be the decomposition of P (ζ) into diﬀerent irreducible components with their multiplicities counted. Then we have a

(non-canonical ) injection

(1.10b)

N

H

1 K

(Ω

,

EP∗

)

→

[HK1 (Ω, EQ∗ j )]mj .

j=1

Hence we have HK1 (Ω, EP∗ ) = 0 if and only if HK1 (Ω, EQ∗ j ) = 0 for j = 1, . . . , N .

Proof. Let P = QR be a decomposition of polynomial. (We do not assume that Q, R are mutually prime.) Note that we have the following exact sequence of sheaves similar to (1.8):

(1.11)

0 → EQ∗ → EP∗ Q→(D) ER∗ → 0.

As a matter of fact, the exactness is obvious except for the surjectivity of the last arrow. But any solution u ∈ E∗ of Q(D)u = f for f ∈ ER∗ will satisfy P (D)u = R(D)(Q(D)u) = R(D)f = 0. Taking the fundamental

exact sequence of the relative cohomology groups for an open neighborhood

Ω ⊃ K, we obtain from (1.11) the following exact sequence:

(1.12)

0 → ΓK (Ω, EQ∗ ) → ΓK (Ω, EP∗ ) → ΓK (Ω, ER∗ )

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Akira Kaneko

→ HK1 (Ω, EQ∗ ) → HK1 (Ω, EP∗ ) → HK1 (Ω, ER∗ ) → HK2 (Ω, EQ∗ ) → HK2 (Ω, EP∗ ).

Here the terms in the ﬁrst row vanish because of the absence of solutions with compact support. Hence the existence of canonical inclusion mapping (1.10a) follows. Since E∗ is not ﬂabby, the second degree relative cohomology groups do not vanish even for single equations. But we have at least the injection mapping

H

1 K

(Ω

,

EP∗

)/H

1 K

(Ω

,

EQ∗

)

→

H

1 K

(Ω

,

ER∗

),

whence in view of the complete reducibility of the vector spaces, we have a (non-canonical) injection mapping

(1.13)

H

1 K

(Ω

,

EP∗

)

→

H

1 K

(Ω

,

EQ∗

)

⊕

H

1 K

(Ω

,

ER∗

).

Repeating this argument for Q, R, we ﬁnally obtain an injection like (1.10a).

Remark. In the preprint version of this article, we gave a proof for the

assertion that (1.10b) is an algebraic isomorphism, which was wrong as the

referee kindly pointed out. Here we give another proof of the isomorphism

for curiosity’s sake, although it will not be useful because we cannot give a

canonical mapping.

To prove an abstract isomorphism, it suﬃces to show that both sides of

(1.10b) have algebraic dimension (over C always) of the same cardinality.

Note that in view of Corollary 1.3 the algebraic dimension of each side of

(1.10b) is estimated by a ﬁnite multiple of the other’s. Thus it suﬃces to

show

that

each

H

1 K

(Ω

,

EQ∗

)

is

either

0

or

is

inﬁnite

dimensional.

Suppose

that it has a non-zero ﬁnite dimension, and let u ∈ EQ∗ (Ω \ K) represent

a non-trivial element. Choose R which is irreducible and not contained

in the factors of Q. Then R(D)ju, j = 0, 1, 2, . . . will deﬁne elements of

HK1 (Ω, EQ∗ ) of which a ﬁnite number are linearly dependent, say

m

S(D)u := cjR(D)ju = v,

j=0

Continuation of Gevrey Class Solutions

559

v ∈ EP∗ (Ω), cj ∈ C , j = 0, 1, 2, . . . , m, cm = 0.

The simultaneous equation

S(D)w = v,

Q(D)w = 0

has a solution w ∈ E∗(Ω) as long as we shrink Ω a little for the fear of the

case of ∗ = {s} type space. (We neglect to introduce a new notation for the shrinked domain.) Then u − w ∈ E∗(Ω \ K) will satisfy

Q(D)(u − w) = 0,

S(D)(u − w) = 0.

Obviously, Q and S are primary to each other. Hence they deﬁne an overde-

termined system, and by Ehrenpreis-Malgrange’s classical theorem the so-

lution u − w can be continued to K. (Though the theorem may not have been written down for the class E∗, it is easy to modify their theory to

this case. A more easy-going way is that if ever we have a continuation as a C∞-solution, we can show that it is in class E∗ via the propagation of E∗ regularity for solutions of, say, Q(D)u = 0 up to K. This propagation

theorem can be shown by a standard argument employing a cut-oﬀ func-

tion in this class and a distribution fundamental solution of Q(D) by which the convolution preserves the E∗ regularity.) Thus u − w, hence u, can be

continued to a solution of Q(D)u = 0 near K, and irrespective of the fact of shrinking Ω, we conclude that u ∈ EQ∗ (Ω) for the original Ω, which is a contradiction.

Here we recall the notion of irregularity of a characteristic direction. We adopt the following deﬁnition. Let P (ζ) be an irreducible polynomial of order m such that Pm(ν) = 0, where Pm denotes the principal part. Consider

Q(s, t) := P (tξ + sν) = q0(ξ)tm + q1(s; ξ)tm−1 + · · · + qm(s; ξ)

as a polynomial of the two variables (s, t). For generically ﬁxed ξ, let κ be the minimum value of the leading powers of the Puiseux expansions of the roots of Q for t in terms of s representing irreducible germs of N (Q) passing through the point (∞, 0) at inﬁnity. Then we set µ = (1 − κ)−1

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Akira Kaneko

and call it the multiplicity of ν. A more exact deﬁnition may be given via the Newton polygon, transforming the point at inﬁnity to the origin: Set σ = 1/s, τ = t/s and let

q(σ, τ ) = s−mQ(s, t) s=1/σ,t=τ/σ.

Factorize it as a polynomial of τ with coeﬃcients in Oσ,0 at 0. Then µ is the inverse of the minimum value with respect to the irreducible factors of the leading powers of the Puiseux expansions of the roots of them.

Notice that the irregularity employed here is the mildest one, in comparison with the strongest one which is usually used e.g. to deﬁne the hyperbolicity.

Our main result here is the following

Theorem 1.4. Let K be a compact set contained in a hyperplane ν · x = 0. Assume further that every irreducible component of P (ζ) has ν as characteristic direction of irregularity ≤ µ. Then

EP(s)(Ω \ K)/EP(s)(Ω) = 0, if s ≤ µ/(µ − 1),

EP{s}(Ω \ K)/EP{s}(Ω) = 0, if s < µ/(µ − 1).

Corollary 1.5. Assume that every irreducible component of P (ζ) is non-elliptic. Then the isolated singularities of solutions of class E1+ are

removable, that is,

(1.14)

EP1+(Ω \ {0})/EP1+(Ω) = 0.

Remark that this suﬃcient condition on P (D) is the same as the one for the removablilty of isolated singularities of real analytic solutions given in [Kn1]–[Kn2].

Although the proof of Theorem 1.4 is almost a literal translation of Grushin’s original article [G2] for the removability of isolated singularities of C∞ solutions, we shall reproduce it here in detail, because it is nevertheless

On Continuation of Gevrey Class Solutions of Linear Partial Diﬀerential Equations

By Akira Kaneko1

Dedicated to Professor Hikosaburo KOMATSU for his 60-th anniversary

Abstract. We give a suﬃcient condition for the removability of thin singularities of Gevrey class solutions of linear partial diﬀerential equations. In §1 we give a suﬃcient condition for the removability in the case of equations with constant coeﬃcients. Then in §2 we discuss the necessity of the condition and construct non-trivial solutions with irremovable thin singularities for some class of equations. In §3 we give a suﬃcient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coeﬃcients.

§0. Introduction

In this article, we gather results on continuation to thin singularity (or removability of thin singularities) of Gevrey class solutions to linear partial diﬀerential equations. Some of the results given here are easily derived from Grushin’s pioneering works on continuation of C∞ solutions and from the author’s former works on continuation of regular solutions. But it will be worth gathering them all to an article, because they may not be obvious for the readers who are not specialized in this subject. Moreover it will be adequate to dedicate this to Professor Hikosaburo Komatsu, who devoted his half carreer to the study of ultra-diﬀerentiable functions and ultradistributions.

Here is a brief plan of the present article. The ﬁrst two sections treat equations with constant coeﬃcients. In §1 we give a suﬃcient condition for

1991 Mathematics Subject Classiﬁcation. 35G05, 35B60, 35E20.

1Partially supported by GRANT-IN-AID FOR SCIENTIFIC RESEARCH No. 07404003.

551

552

Akira Kaneko

the removability of thin singularities of the Gevrey class solutions. This is a translation of Grushin’s work except for small details. In §2 we discuss the necessity of the condition given in §1. This is to construct non-trivial solutions with irremovable thin singularities under the condition opposite to §1. We generalize the construction of Grushin who gave such a few examples in his work [G2]. As an example, the precise Gevrey index for the threshold of existence of solutions with thin singularity is determined for the Schro¨dinger equation. In §3 we give a suﬃcient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coeﬃcients. This is a modiﬁcation of the author’s work for the removability of thin singularities of real analytic solutions.

For a general survey on this subject, we refer to [Kn11] for results until 1992, and [Kn12], where a list of open problems is gathered. The present article treats some of them concerning Gevrey class solutions.

§1. Continuation of Gevrey class solutions to equations with constant coeﬃcients

Let P (D) be a linear partial diﬀerential operator with constant coef-

ﬁcients, where P (ζ) is a polynomial in n variables ζ = (ζ1, . . . , ζn) and D = (D1, . . . , Dn) with Dj = −i∂/∂xj, j = 1, . . . , n. We deﬁne the two spaces of Gevrey class functions of index s by

(1.1)

E(s)(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∀h > 0, ∃CK,h > 0 sup |Dαf (x)| ≤ CK,hh|α|α!s for ∀α},

x∈K

and

(1.2) E{s}(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∃h = h(K) > 0,

∃C = C(K) > 0, sup |Dαf (x)| ≤ Ch|α|α!s for ∀α}.

x∈K

The ﬁrst space has a simple topology of Fr´echet space and is easier to treat, but it is a little less natural because when s = 1, this corresponds to the space of entire functions. The second space has a very complicated topological structure and does not allow the closed range theorem to hold.

Continuation of Gevrey Class Solutions

553

Hence for this space we cannot utilize fundamental theorems such as the global surjectivity on convex open sets of linear partial diﬀerential operators with constant coeﬃcients or the Fundamental Principle of EhrenpreisPalamodov. But it is more natural because for s = 1 this corresponds to the space of real analytic functions which is localizable along the real axis.

We also set

(1.3)

E1+(Ω) := E(s)(Ω) = E{s}(Ω).

s>1

s>1

This is a very convenient space, still containing enough functions in non-

quasianalytic ultra-diﬀerentiable class. Following the usage of Komatsu, we shall denote in the sequel by E∗(Ω) either of the spaces E(s)(Ω), E{s}(Ω), E1+(Ω) when we can state something commonly to these spaces. Thus E∗ denotes either of these function classes. In the same time, this symbol will denote the corresponding sheaf (that is, the localization) on Rn. As usual we let D∗(Ω) denote the functions of class E∗ with compact support

contained in Ω (together with the obvious topology if the dual space, that is,

the space of ultradistributions of this class, is considered). For a general set L we let E∗(L) denote the functions of class E∗ deﬁned on a neighborhood

of L, with the obvious identiﬁcation in the sense of inductive limit with

respect to the neighborhoods. In general, we denote by EP∗ (Ω) the space of solutions in Ω of the equa-

tion P (D)u = 0 of class E∗. Let K ⊂ Ω denote a thin compact subset.

Here “thin” means that the interior is void. We assume that it is contained

in a hyperplane, say ν · x = 0. (This follows automatically for convex thin

set, as we mainly consider in the sequel.) We study the continuation of solutions of P (D)u = 0 in E∗(Ω \ K) to solutions in E∗(Ω).

Proposition 1.1. The canonical map induced by the canonical restriction from Ω to Ω \ K:

EP∗ (Ω) → EP∗ (Ω \ K)

is injective. In other words, there are no solutions of P (D)u = 0 with compact support.

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Akira Kaneko

Actually, any element in the kernel of the above map would be a solution of P (D)u = 0 with compact support. But via the Fourier transform we would then obtain P (ζ)u = 0, where u is entire, whence u ≡ 0.

Thus the quotient space

(1.4)

EP∗ (Ω \ K)/EP∗ (Ω)

will represent the obstruction for continuation of solutions of this class to

K. The reason why we restrict K to thin sets is obvious from the non-

quasianalyticity of the function class under consideration: If K had a nonvoid interior, then choosing f ∈ D∗(Int K) \ P (D)D∗(Int K) and a solution u ∈ E∗(Rn) of P (D)u = f , u|Ω\K would present a non-trivial element of (1.4).

First we shall show that the obstruction space (1.4) depends only on K

and not on Ω. For this purpose we recall the notion of local cohomology groups with coeﬃcients in the solution sheaf EP∗ of class E∗ of the equation P u = 0.

Proposition 1.2. We have the following isomorphism

(1.5)

EP∗ (Ω \ K)/EP∗ (Ω) =∼ HK1 (Ω, EP∗ ).

More generally, for any set L containing K in its interior, we have

(1.5bis)

EP∗ (L \ K)/EP∗ (L) =∼ HK1 (L, EP∗ ).

The quotient space in (1.5) or (1.5bis) is determined by K only and does not depend on the choice of the neighborhoods.

Proof. Recall the following fundamental exact sequence of local cohomology groups:

(1.6)

0 → ΓK (Ω, EP∗ ) → Γ (Ω, EP∗ ) → Γ (Ω \ K, EP∗ ) → HK1 (Ω, EP∗ ) → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → H2(Ω, EP∗ ) = 0.

Continuation of Gevrey Class Solutions

555

Here we have ΓK(Ω, EP∗ ) = 0 by Proposition 1.1. (The fact H2(Ω, EP∗ ) = 0 follows from the resolution (1.7) as will be discussed below.) We shall show

that the mapping

H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ )

is always injective. Then we will obtain the isomorphism (1.5). Incidentally, we obtain the exact sequence

(1.7)

0 → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → 0.

Recall now the following exact sequence of sheaves

(1.8)

0 → EP∗ → E ∗ P→(D) E ∗ → 0.

Here the surjectivity in the last arrow, that is, the local solvability in this class, is an easy consequence of the existence of a fundamental solution for a single linear partial diﬀerential operator P with constant coeﬃcients. Taking the fundamental exact sequence of global cohomology groups on an open set Ω, we obtain

(1.9)

0 → Γ (Ω, EP∗ ) → Γ (Ω, E∗) P−(→D) Γ (Ω, E∗) → H1(Ω, EP∗ ) → H1(Ω, E∗) = 0.

The fact H1(Ω, E∗) = 0 is obvious because the sheaf E∗ is ﬁne. (From this H2(Ω, EP∗ ) = 0 also follows.) Thus it suﬃces to show that the natural mapping induced from the restriction

E∗(Ω)/P (D)E∗(Ω) → E∗(Ω \ K)/P (D)E∗(Ω \ K)

is injective. Suppose that u ∈ E∗(Ω) represents an element mapped to 0. This implies that there exists v ∈ E∗(Ω \ K) such that

u|Ω\K = P (D)v.

Employing partitions of unity, construct h ∈ E∗(Ω) and w ∈ E∗(Rn \ K) such that v = h − w on Ω \ K. We can obviously choose w in such a way

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Akira Kaneko

that w ≡ 0 outside a ball of some radius R > 0, by cutting v smoothly in the class E∗ on a neighborhood of K. Put

P (D)w on Rn \ K, g=

P (D)h − u on Ω.

This deﬁnition is consistent on the common domain:

P (D)w = P (D)h − P (D)v = P (D)h − u on Ω \ K.

Thus g becomes a well deﬁned element of E∗(Rn), which has compact support by the choice of w as above. Let f ∈ E∗(Rn) be a solution of P (D)f = g. (We can simply obtain such f by convoluting g with the distributional fundamental solution of P (D) which obviously preserves the Gevrey regularity.) Then, on Ω we have

u = P (D)(h − f ).

Hence, it represents 0 in E∗(Ω)/P (D)E∗(Ω). The proof for general neighborhood L of K is just similar. Now that

the isomorphism (1.5) is established, the ﬁnal conclusion follows from the excision theorem of local cohomology groups.

Remark. 1) We cannot expect H1(Ω, EP∗ ) = 0, unless we have the global surjectivity of P (D) on Ω in this function class. This follows from the exact sequence (1.8). To have this surjectivity for open Ω, we ﬁrst of all need to assume that Ω is convex. Then it is valid for the class E(s) (see e.g. Bjo¨rck [Bj1]), but still not in general for E{s} (see e.g. Cattabriga [C1]). The above method of argument was ﬁrst introduced by [Kn6] for the real analytic solutions, to which the global surjectivity is neither available. Note that the above Proposition (or the sequence (1.7)) implies that in such a situation, the obstruction for the global surjectivity is concentrated on the neighborhood of ∂Ω.

2) We have an alternative choice of neighborhoods of K for which the global surjectivity holds. It is to take compact neighborhoods L ⊃⊃ K. In this case the surjectivity of P (D) : E∗(L) → E∗(L) holds irrespective of the convexity of L, because of the non-quasianalytic property of our class. This

Continuation of Gevrey Class Solutions

557

fact was implicitly employed by some of the proofs for the corresponding assertions in the author’s former publications.

In view of the above Proposition, we can always assume Ω to be convex, thus allowing to apply the Fourier analysis. Sometimes the choice of convex compact L simpliﬁes the situation further.

Next, we shall show that the obstruction (1.4) can be decomposed via the irreducible components of P (D):

Corollary 1.3. Let Q be any factor of P . Then we have a canonical injection

(1.10a)

H

1 K

(Ω

,

EQ∗

)

→

H

1 K

(Ω

,

EP∗

).

Conversely, let P = Qm1 1 · · · QmNN be the decomposition of P (ζ) into diﬀerent irreducible components with their multiplicities counted. Then we have a

(non-canonical ) injection

(1.10b)

N

H

1 K

(Ω

,

EP∗

)

→

[HK1 (Ω, EQ∗ j )]mj .

j=1

Hence we have HK1 (Ω, EP∗ ) = 0 if and only if HK1 (Ω, EQ∗ j ) = 0 for j = 1, . . . , N .

Proof. Let P = QR be a decomposition of polynomial. (We do not assume that Q, R are mutually prime.) Note that we have the following exact sequence of sheaves similar to (1.8):

(1.11)

0 → EQ∗ → EP∗ Q→(D) ER∗ → 0.

As a matter of fact, the exactness is obvious except for the surjectivity of the last arrow. But any solution u ∈ E∗ of Q(D)u = f for f ∈ ER∗ will satisfy P (D)u = R(D)(Q(D)u) = R(D)f = 0. Taking the fundamental

exact sequence of the relative cohomology groups for an open neighborhood

Ω ⊃ K, we obtain from (1.11) the following exact sequence:

(1.12)

0 → ΓK (Ω, EQ∗ ) → ΓK (Ω, EP∗ ) → ΓK (Ω, ER∗ )

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Akira Kaneko

→ HK1 (Ω, EQ∗ ) → HK1 (Ω, EP∗ ) → HK1 (Ω, ER∗ ) → HK2 (Ω, EQ∗ ) → HK2 (Ω, EP∗ ).

Here the terms in the ﬁrst row vanish because of the absence of solutions with compact support. Hence the existence of canonical inclusion mapping (1.10a) follows. Since E∗ is not ﬂabby, the second degree relative cohomology groups do not vanish even for single equations. But we have at least the injection mapping

H

1 K

(Ω

,

EP∗

)/H

1 K

(Ω

,

EQ∗

)

→

H

1 K

(Ω

,

ER∗

),

whence in view of the complete reducibility of the vector spaces, we have a (non-canonical) injection mapping

(1.13)

H

1 K

(Ω

,

EP∗

)

→

H

1 K

(Ω

,

EQ∗

)

⊕

H

1 K

(Ω

,

ER∗

).

Repeating this argument for Q, R, we ﬁnally obtain an injection like (1.10a).

Remark. In the preprint version of this article, we gave a proof for the

assertion that (1.10b) is an algebraic isomorphism, which was wrong as the

referee kindly pointed out. Here we give another proof of the isomorphism

for curiosity’s sake, although it will not be useful because we cannot give a

canonical mapping.

To prove an abstract isomorphism, it suﬃces to show that both sides of

(1.10b) have algebraic dimension (over C always) of the same cardinality.

Note that in view of Corollary 1.3 the algebraic dimension of each side of

(1.10b) is estimated by a ﬁnite multiple of the other’s. Thus it suﬃces to

show

that

each

H

1 K

(Ω

,

EQ∗

)

is

either

0

or

is

inﬁnite

dimensional.

Suppose

that it has a non-zero ﬁnite dimension, and let u ∈ EQ∗ (Ω \ K) represent

a non-trivial element. Choose R which is irreducible and not contained

in the factors of Q. Then R(D)ju, j = 0, 1, 2, . . . will deﬁne elements of

HK1 (Ω, EQ∗ ) of which a ﬁnite number are linearly dependent, say

m

S(D)u := cjR(D)ju = v,

j=0

Continuation of Gevrey Class Solutions

559

v ∈ EP∗ (Ω), cj ∈ C , j = 0, 1, 2, . . . , m, cm = 0.

The simultaneous equation

S(D)w = v,

Q(D)w = 0

has a solution w ∈ E∗(Ω) as long as we shrink Ω a little for the fear of the

case of ∗ = {s} type space. (We neglect to introduce a new notation for the shrinked domain.) Then u − w ∈ E∗(Ω \ K) will satisfy

Q(D)(u − w) = 0,

S(D)(u − w) = 0.

Obviously, Q and S are primary to each other. Hence they deﬁne an overde-

termined system, and by Ehrenpreis-Malgrange’s classical theorem the so-

lution u − w can be continued to K. (Though the theorem may not have been written down for the class E∗, it is easy to modify their theory to

this case. A more easy-going way is that if ever we have a continuation as a C∞-solution, we can show that it is in class E∗ via the propagation of E∗ regularity for solutions of, say, Q(D)u = 0 up to K. This propagation

theorem can be shown by a standard argument employing a cut-oﬀ func-

tion in this class and a distribution fundamental solution of Q(D) by which the convolution preserves the E∗ regularity.) Thus u − w, hence u, can be

continued to a solution of Q(D)u = 0 near K, and irrespective of the fact of shrinking Ω, we conclude that u ∈ EQ∗ (Ω) for the original Ω, which is a contradiction.

Here we recall the notion of irregularity of a characteristic direction. We adopt the following deﬁnition. Let P (ζ) be an irreducible polynomial of order m such that Pm(ν) = 0, where Pm denotes the principal part. Consider

Q(s, t) := P (tξ + sν) = q0(ξ)tm + q1(s; ξ)tm−1 + · · · + qm(s; ξ)

as a polynomial of the two variables (s, t). For generically ﬁxed ξ, let κ be the minimum value of the leading powers of the Puiseux expansions of the roots of Q for t in terms of s representing irreducible germs of N (Q) passing through the point (∞, 0) at inﬁnity. Then we set µ = (1 − κ)−1

560

Akira Kaneko

and call it the multiplicity of ν. A more exact deﬁnition may be given via the Newton polygon, transforming the point at inﬁnity to the origin: Set σ = 1/s, τ = t/s and let

q(σ, τ ) = s−mQ(s, t) s=1/σ,t=τ/σ.

Factorize it as a polynomial of τ with coeﬃcients in Oσ,0 at 0. Then µ is the inverse of the minimum value with respect to the irreducible factors of the leading powers of the Puiseux expansions of the roots of them.

Notice that the irregularity employed here is the mildest one, in comparison with the strongest one which is usually used e.g. to deﬁne the hyperbolicity.

Our main result here is the following

Theorem 1.4. Let K be a compact set contained in a hyperplane ν · x = 0. Assume further that every irreducible component of P (ζ) has ν as characteristic direction of irregularity ≤ µ. Then

EP(s)(Ω \ K)/EP(s)(Ω) = 0, if s ≤ µ/(µ − 1),

EP{s}(Ω \ K)/EP{s}(Ω) = 0, if s < µ/(µ − 1).

Corollary 1.5. Assume that every irreducible component of P (ζ) is non-elliptic. Then the isolated singularities of solutions of class E1+ are

removable, that is,

(1.14)

EP1+(Ω \ {0})/EP1+(Ω) = 0.

Remark that this suﬃcient condition on P (D) is the same as the one for the removablilty of isolated singularities of real analytic solutions given in [Kn1]–[Kn2].

Although the proof of Theorem 1.4 is almost a literal translation of Grushin’s original article [G2] for the removability of isolated singularities of C∞ solutions, we shall reproduce it here in detail, because it is nevertheless