**Convergent and Divergent Solutions of Singular** **Partial Diﬀerential Equations with Resonance**

**or Small Denominators**

By

MasafumiYoshino*∗*

**Abstract**

We show the solvability and nonsolvability of a singular nonlinear system of partial diﬀerential equations with resonance in a class of functions holomorphic in some neighborhood of the origin. These results are applied to the normal form theory of a singular vector ﬁeld.

**§****1.** **Introduction**

In this paper, we study the solvability and nonsolvability of a singular nonlinear system of partial diﬀerential equations which appear in the normal form theory of vector ﬁelds. It is well-known that under the Poincar´e condition or a Diophantine condition the sytem of equations has a convergent power series solution locally. (cf. Remark 2.) We are interested in the solvability in a class of convergent power series without any Diophantine condition although there are inﬁnite resonances or small denominators. We are also interested in the divergence caused by the presence of a nontrivial Jordan block in the linear part. Because the singular operator which we consider has inﬁnite resonance

Communicated by T. Kawai. Received May 15, 2006. Revised October 6, 2006, November 30, 2006.

2000 Mathematics Subject Classiﬁcation(s): Primary 35C10; Secondary 37F50, 37G05.

Key words and phrases: solvability, resonance, small denominators, divergence, conver- gence.

Partially supported by Grant-in-Aid for Scientiﬁc Research (No. 11640183), Ministry of Education, Science and Culture, Japan.

*∗*Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1
Kagamiyama, Higashi-hiroshima, Hiroshima 739-8526, Japan.

e-mail: yoshino@math.sci.hiroshima-u.ac.jp

or small denominators, a standard energy method or an iterative method does not work due to the presence of high loss of derivatives.

To our best knowledge, few results are known for such operators. One interesting approach for the problem is the geometric viewpoint. To be more precise, let us consider an equation appearing from a Hamiltonian vector ﬁeld.

Clearly, the normalizing transformation satisﬁes an equation with inﬁnite res- onance. (cf. (2.2).) It is well-known that the formal power series solutions of the equation do not converge in general. Due to Ito and Zung, the convergence is equivalent to the existence of a certain number of integrals. (cf. [1] and [5].) We shall give a rather simple wide class of nonlinear perturbations for which one can always ﬁnd a convergent solution, which is diﬀerent from an integrability condition because we put no restriction on the resonance dimen- sion. (cf. [1] and [5].) We also construct a Liouville type linear part and a nonlinear perturbation for which a divergence of a (unique) solution occurs.

(cf. Proposition 3.1.) This especially shows that our suﬃcient condition of a nonlinear perturbation is necessary in general.

We are also interested in the divergence phenomenon caused by the pres- ence of a nontrivial Jordan block of the linear part in a Siegel case. In fact, if this is the case, then the solutions corresponding to the normalizing transformation generally diverge even if we assume a Diophantine condition. (cf. Proposition 3.2.) Theorem 2.1 also gives a convergence criterion for these operators.

This paper is organized as follows. In Section 2, we state convergence results. In Section 3, we study the divergence and Diophantine phenomena. In Section 4, we prepare necessary lemmas. In Section 5 we prove our theorem.

**§****2.** **Convergence Criterions**

Let *x* =* ^{t}*(x

_{1}

*, . . . , x*

*)*

_{n}*∈*C

*,*

^{n}*n*

*≥*2 be the variable in C

*, andR be the set of real numbers. Let Λ be an*

^{n}*n-square constant matrix. LetL*

_{Λ}be the Lie derivative of the linear vector ﬁeld Λx

*·∂*

_{x}(2.1) *L*_{Λ}*v*= [Λx, v] =Λx, ∂_{x}*v −*Λv,
where Λx, ∂_{x}*v*=_{n}

*j=1*(Λx)* _{j}*(∂/∂x

*)v, with (Λx)*

_{j}*being the*

_{j}*j-th component*of Λx. We consider the system of equations

(2.2) *L*_{Λ}*u*=*R(u(x)),*

where *u*=* ^{t}*(u

_{1}

*, u*

_{2}

*, . . . , u*

*) is an unknown vector function and*

_{n}*R(x) =*

*(R*

^{t}_{1}(x), R

_{2}(x), . . . , R

*(x))*

_{n}is holomorphic in some neighborhood of*x*= 0 in C* ^{n}* such that

*R(x) =O(|x|*

^{2}) when

*|x| →*0. The equation (2.2) appears as a linearizing equation of a singular vector ﬁeld. (cf. [4]). Because we can always reduce Λ to a Jordan normal form by the linear change of the unknown functions

*U*=

*Au, we may assume that*Λ is put in a Jordan normal form. Moreover we assume that there exists

*∃τ*

_{0}, 0

*≤τ*

_{0}

*≤π*such that

(2.3) every component of*e*^{−iτ}^{0}Λ is a real number.

It follows that if *λ** _{j}* (j= 1,2, . . . , n) are the eigenvalues of Λ with multiplicity,
then we have

(2.4) *∃* *τ*_{0}*,* 0*≤τ*_{0}*≤π, e*^{−iτ}^{0}*λ*_{j}*∈*R (j= 1,2, . . . , n).

If we set*u(x) =x+v(x),v(x) =O(|x|*^{2}), then*v*satisﬁes the following equation

(2.5) *L*_{Λ}*v*=*R(x*+*v(x)).*

Let Z+ be the set of nonnegative integers, and let Z^{n}_{+}(k) (k *≥*0) be the
*n-product of*Z+, *γ*=* ^{t}*(γ

_{1}

*, γ*

_{2}

*, . . . , γ*

*) such that*

_{n}*|γ|*=

*γ*

_{1}+

*γ*

_{2}+

*· · ·*+

*γ*

_{n}*≥k.*

For*γ∈*Z^{n}_{+}, we set*x** ^{γ}* =

*x*

^{γ}_{1}

^{1}

*· · ·x*

^{γ}

_{n}*. For*

^{n}*k≥*0 and

*n≥*1, we denote byC

^{n}*[[x]]*

_{k}the set of formal power series

*|η|≥k**u*_{η}*x** ^{η}* (u

_{η}*∈*C

*). We also deﬁne the convergent*

^{n}*n-vector power series which vanishes up to the (k−*1)-th derivative byC

^{n}*[x]. We decompose Λ = Λ*

_{k}*+ Λ*

_{S}*, where Λ*

_{N}*and Λ*

_{S}*are the semi-simple and the nilpotent part of Λ, respectively. We denote by*

_{N}*L*

_{Λ}

*the Lie derivative of the linear vector ﬁeld Λ*

_{S}

_{S}*x·∂*

*.*

_{x}For a formal power series *f*(x) =

*γ**f*_{γ}*x** ^{γ}*, we deﬁne the majorant of

*f*,

*M*(f)(x) by

(2.6) *M*(f) :=

*γ*

*|f*_{γ}*|x*^{γ}*.*

For a vector *f* = (f_{1}*, f*_{2}*, . . . , f** _{n}*) we deﬁne

*M*(f) := (M(f

_{1}), M(f

_{2}), . . . ,

*M*(f

*)). For a formal power series with real coeﬃcients*

_{n}*a(x) =*

*γ**a*_{γ}*x** ^{γ}* and

*b(x) =*

*γ**b*_{γ}*x** ^{γ}*, we deﬁne

*ab*if

*a*

_{γ}*≤b*

*for all*

_{γ}*γ∈*Z

^{n}_{+}. We deﬁne (f

_{1}(x), f

_{2}(x), . . . , f

*(x))(g*

_{n}_{1}(x), g

_{2}(x), . . . , g

*(x)) if*

_{n}*f*

*(x)*

_{j}*g*

*(x) for*

_{j}*j*= 1,2, . . . , n.

Let *c >* 0 be a constant. Let *A*+ (resp. *A**−*) be the set of *g(x) =*

*t*(g_{1}(x), . . . , g* _{n}*(x))

*∈*C

^{n}_{2}[x] such that

(2.7) (L_{Λ}_{S}*−c)M*(g) 0 (resp. (L_{Λ}* _{S}*+

*c)M*(g)0)

and that*g(x) is a ﬁnite sum of the functionsf* =* ^{t}*(f

_{1}

*, f*

_{2}

*, . . . , f*

*)*

_{n}*∈*C

^{n}_{2}[x] with the following expansion at the origin

(2.8) *f** _{j}*(x) =

*x*

_{ν}*γ*

*f*_{j,γ}*x*^{γ}*,* *f*_{j,γ}*∈*C*,* *j*= 1,2, . . . , n,

where*ν*is such that the*j-th and theν-th components of Λ** _{S}* belong to the same
Jordan block of Λ

*. We can prove that*

_{N}*A*

*are linear spaces. (cf. Lemma 4.3.) Then we have*

_{±}**Theorem 2.1.** *Suppose that* (2.3) *holds.* *Let* *R(x)* *∈ A*_{±}*.* *Assume*
*that* *R(x)* *is a polynomial with degree< c*+ 1 *if* Λ* _{N}* = 0. Then, (2.5)

*has a*

*holomorphic solution in some neighborhood of the originx*= 0.

*Remark* 1. If we drop the conditions of Theorem 2.1, then we encounter
the divergence caused by small denominators and the presence of a Jordan
block. More precisely, we have

(a) If *c* = 0, then Theorem 2.1 does not hold in general because of small
denominators. Namely, the condition*L*_{Λ}_{S}*M*(R) 0 or *L*_{Λ}_{S}*M*(R)0 is not
suﬃcient. (cf. Proposition 3.1.)

(b) There exists *R*such that neither *L*_{Λ}_{S}*M*(R)0 nor *L*_{Λ}_{S}*M*(R) 0 holds
for which Theorem 2.1 does not hold. This follows from Proposition 3.1 in view
of the arbitrariness of *R** ^{}* in Proposition 3.1.

(c) If Λ is not semi-simple, then there exists*R*which is not a polynomial such
that Theorem 2.1 does not hold. This follows from Proposition 3.2.

Finally we note that because (2.5) has inﬁnite resonance in general, the uniqueness of a solution in Theorem 2.1 does not hold in general.

*Remark* 2. We will brieﬂy review the notions used in this paper. Let
*λ** _{j}* (j = 1,2, . . . , n) be the eigenvalues of Λ with multiplicity. We say that

*λ*

*(j = 1,2, . . . , n) satisfy the Poincar´e condition if the convex hull of*

_{j}*λ*

*(j = 1,2, . . . , n) in the complex plane does not contain the origin 0*

_{j}*∈*C. We can easily see that the Poincar´e condition is equivalent to the following estimate:

there exist*C >*0 and*K >*0 independent of*α*such that

(2.9) *|λ, α −λ*_{j}*| ≥C|α|,* *∀α*= (α_{1}*, α*_{2}*, . . . , α** _{n}*)

*∈*Z

^{n}_{+}

*,*

*|α| ≥K,*for

*j*= 1,2, . . . , n, where

*λ*= (λ

_{1}

*, λ*

_{2}

*, . . . , λ*

*) and*

_{n}*λ, α*=

_{n}*i=1**λ*_{i}*α** _{i}*.
For

*α*= (α

_{1}

*, α*

_{2}

*, . . . , α*

*)*

_{n}*∈*Z

*we deﬁne*

^{n}*α*=

_{n}*i=1**|α*_{i}*|*. We say that
*λ∈*R* ^{n}* is a Diophantine vector if there exist

*∃τ >*0 and

*∃C >*0 such that (2.10)

*|λ, α −λ*

_{j}*| ≥Cα*

^{−n−τ}*,*

*∀α∈*Z

^{n}*,*

*α ≥*2, j= 1,2, . . . , n.

We call (2.10) a Diophantine condition. The vector which does not satisfy a
Diophantine condition is called a Liouville vector. By considering the special
case of (2.10) we have the classical deﬁnition of a Diophantine number. We
say that a number *t∈*R*\*Qis a Diophantine number if there exist*τ >*0 and
*C >* 0 such that for every *p, q* *∈* Z, *q >* 0, one has *|t−*^{p}_{q}*| ≥* *Cq** ^{−2−τ}*. The
Liouville numbers are the complement of Diophantine numbers in R

*\*Q.

Finally, we give the deﬁnition of a Brjuno type Diophantine condition. We
set * _{j}*(α) =

*λ, α −λ*

*and deﬁne*

_{j}(2.11) *ω** _{k}*= inf

*{|*

*(α)*

_{j}*|*;

*(α)= 0, α*

_{j}*∈*Z

^{n}*,*2

*≤ α ≤*2

^{k}*, j*= 1,2, . . . , n

*},*where

*k*= 1,2, . . .. Then we say that

*λ*satisﬁes the Brjuno type Diophantine condition if

(2.12) *−*

*k≥0*

ln(ω* _{k+1}*)

2^{k}*<*+*∞.*

We can easily see that a Diophantine vector satisﬁes (2.12).

**§****3.** **Divergence and Diophantine Phenomena**

In this section we study divergence caused by small denominators and the
presence of a Jordan block. We consider in *x∈*C^{2}

(3.1) *L*_{Λ}*u*=*R(x*+*u),* Λ =

1 0

0 *−τ*

*,*

where*τ >*0 is a Liouville number chosen later and*u*=*O(|x|*^{2}). Then we have
**Proposition 3.1.** *For every* *R** ^{}*(x)

*holomorphic in some neighborhood*

*of the origin, there exist a Liouville number*

*τ >*0

*and a holomorphic pertur-*

*bation*

*R*

*(x)*

^{}*≡*0

*such thatL*

_{Λ}

_{S}*M*(R

*) 0*

^{}*orL*

_{Λ}

_{S}*M*(R

*)0*

^{}*holds and that*

*the unique formal power series solution of*(3.1)

*with*

*R*=

*R*

*+*

^{}*R*

^{}*diverges.*

*Proof.* We construct an irrational number *τ* by the continued fraction
expansion *τ*= [a_{1}*, a*_{2}*, . . .],a*_{j}*∈*N. Namely, if we deﬁne the sequence*{p*_{n}*}*and
*{q*_{n}*}*by

*p** _{n}*=

*a*

_{n}*p*

*+*

_{n−1}*p*

_{n−2}*,*

*n≥*2, p

_{0}= 0, p

_{1}= 1, (3.2)

*q** _{n}*=

*a*

_{n}*q*

*+*

_{n−1}*q*

_{n−2}*,*

*n≥*2, q

_{0}= 1, q

_{1}=

*a*

_{1}

*,*(3.3)

then we have *p*_{n}*/q*_{n}*→τ* (n*→ ∞*). Moreover, we have (cf. [3])

(3.4) 1

(a* _{n+1}*+ 2)q

_{n}^{2}

*<*

*τ−p*_{n}*q*_{n}

*<* 1

*a*_{n+1}*q*^{2}_{n}*,* *n≥*0.

We substitute the expansion of*u*= (u_{1}*, u*_{2}) and*R** _{j}*(x+

*u),*

*u*

*=*

_{j}*|η|≥2*

*u*_{j,η}*x*^{η}*,*

*R** _{j}*(x+

*u) =*

*|γ|≥2*

*R** _{j,γ}*(x

_{1}+

*u*_{1,η}*x** ^{η}*)

^{γ}^{1}(x

_{2}+

*u*_{2,η}*x** ^{η}*)

^{γ}^{2}into the equation (3.1). Then we have the recurrence relations

(η_{1}*−τ η*_{2}*−*1)u_{1,η}=*R*_{1,η}+*P** _{η,1}*(u

_{j,δ}*,|δ|<|η|, j*= 1,2), (3.5)

(η_{1}*−τ η*_{2}+*τ)u*_{2,η}=*R*_{2,η}+*P** _{η,2}*(u

_{j,δ}*,|δ|<|η|, j*= 1,2), (3.6)

where *P** _{η,j}* is a polynomial of

*u*

*’s with coeﬃcients given by the expansions of*

_{k,δ}*R. Now suppose that*

*a*

_{1}

*, a*

_{2}

*, . . . , a*

*are given. We determine*

_{n}*p*

*and*

_{n}*q*

*by (3.2) and (3.3), and we want to determine*

_{n}*u*

_{1,η}with

*η*= (p

*+ 1, q*

_{n}*) from (3.5) assuming that*

_{n}*u*

*(*

_{j,δ}*|δ|*

*<|η|*,

*j*= 1,2) are already determined. This is possible if

*τ*avoids a ﬁnite number of rational points. If the absolute value of

*P*

*(u*

_{η,1}

_{j,δ}*,|δ|<|η|, j*= 1,2) is smaller than 2

*, then we take*

^{|η|+1}*R*

_{1,η}such that

*|R*_{1,η}*|* = 2*·*3* ^{|η|}*. On the other hand, if the absolute value of

*P*

*(u*

_{η,1}

_{j,δ}*,|δ|*

*<*

*|η|, j* = 1,2) is larger than 2* ^{|η|+1}*, then we take

*R*

_{1,η}such that

*|R*

_{1,η}

*|*= 2

*. It follows that the absolute value of the right-hand side of (3.5) is larger than 2*

^{|η|}*. In view of (3.4), we determine*

^{|η|}*a*

*such that*

_{n+1}*|u*

_{1,(p}

_{n}_{+1,q}

_{n}_{)}

*| ≥*(p

*+*

_{n}*q*

*)!.*

_{n}This is possible if we take *a** _{n+1}* suﬃciently large. Moreover, by the deﬁnition
of continued fractions we see that the approximant [a

_{1}

*, a*

_{2}

*, . . . , a*

*] avoids the ﬁnite number of rational points given in the above if we take*

_{n+1}*a*

*suﬃciently large. Next we determine*

_{n+1}*p*

*and*

_{n+1}*q*

*from (3.2) and (3.3). Then we want to determine*

_{n+1}*u*

_{1,η}with

*η*= (p

*+ 1, q*

_{n+1}*) from (3.5). We can determine the terms*

_{n+1}*u*

*(*

_{j,δ}*|δ|<|η|*,

*j*= 1,2) if

*τ*avoids a ﬁnite number of rational points. This is possible if

*a*

*is suﬃciently large. Then we repeat the same argument as in the above. Clearly,*

_{n+2}*R(x) is holomorphic in some neighborhood of the origin.*

On the other hand, if we take*a** _{n+1}*so that

*a*

*is larger than polynomial order of*

_{n+1}*q*

*, then it follows from (3.4) that*

_{n}*τ*is a Liouville number. Therefore we can determine a Liouville number

*τ*so that we have a divergent formal power series solution

*u*=

*|η|≥2**u*_{η}*x** ^{η}*.

By the deﬁnition of a continued fraction expansion, we have*p** _{n}*+ 1

*−τ q*

_{n}*−*1

*>*0 or

*p*

*+ 1*

_{n}*−τ q*

_{n}*−*1

*<*0 according as

*n*is odd or even. For simplicity,

we take *p** _{n}* and

*q*

*for odd*

_{n}*n. If we take*

*R*

^{}_{1,η}appropriately, then we have

*R*

_{1,η}= 0 and the support of

*R*

^{}_{1,η}is contained in

*η*

_{1}

*−τ η*

_{2}

*−*1

*>*0. This proves that

*L*

_{Λ}

_{S}*M*(R

*) 0. By a similar argument we can also treat the case*

^{}*L*

_{Λ}

_{S}*M*(R

*)0. This completes the proof.*

^{}*Remark* 3. We know that almost all nonlinear perturbations of a Liou-
ville type linear operator has a divergent solution. (cf. [2]) Our result shows
that for any nonlinear perturbation there exists a Liouville number*τ*such that
the divergence occurs if we add a limited type of nonlinear perturbations.

Next we study the divergence caused by the presence of a nontrivial Jordan
block even if a Diophantine condition is veriﬁed. We consider in*x∈*C^{3}

(3.7) *L*_{Λ}*u*=*R(x*+*u),* Λ =

1 0 0

0 *−τ* *−*1

0 0 *−τ*

*,*

where *τ >*0 is an irrational number and*u*=*O(|x|*^{2}). Then we have

**Proposition 3.2.** *Let* *c >*0. For every irrational number*τ >*0 *there*
*exists* *R* *≡* 0 *which is not a polynomial such that* (L_{Λ}_{S}*−c)M*(R) 0 *or*
(L_{Λ}* _{S}*+

*c)M*(R)0

*holds and that the unique formal power series solution of*(3.7)

*diverges.*

*Proof.* Let*K*be such that*K > c+ 2. We denote by [c] the largest integer*
which does not exceed*c. Then we deﬁne*

(3.8) *R*_{1}(x) =*x*^{[c]+2}_{1} *R*˜_{1}(x_{1}), R_{2}(x) =

max{c,2}≤i−τ j<K

*x*^{i}_{1}*x*^{j}_{2}*, R*_{3}(x)*≡*0,

where ˜*R*_{1} is holomorphic at the origin such that ˜*R*_{1} 0. We can easily see
that (L_{Λ}_{S}*−c)M*(R) 0.

We will construct the solution *u*= (u_{1}*, u*_{2}*, u*_{3}) of (3.7). We set *u*_{1}(x) =
*x*_{1}*w*_{1}(x_{1}). Then it follows from the ﬁrst equation of (3.7) that *w*_{1} satisﬁes
*x*_{1}*∂*_{1}*w*_{1}=*x*^{[c]+1}_{1} (1 +*w*_{1})^{[c]+2}*R*˜_{1}(x_{1}(1 +*w*_{1})). By the elementary computations,
we can easily show that the equation has a holomorphic solution *w*_{1}(x_{1}) such
that *w*_{1} 0. Next*u*_{3} satisﬁes (x_{1}*∂*_{1}*−τ x*_{2}*∂*_{2}*−τ x*_{3}*∂*_{3}*−x*_{3}*∂*_{2}+*τ*)u_{3} = 0. By
the irrationality of*τ*, we have*u*_{3}= 0.

Next, by the second equation of (3.7)*u*_{2} satisﬁes

(3.9) (x_{1}*∂*_{1}*−τ x*_{2}*∂*_{2}*−τ x*_{3}*∂*_{3}*−x*_{3}*∂*_{2}+*τ)u*_{2}=*R*_{2}(x_{1}(1 +*w*_{1}), x_{2}+*u*_{2}).

We denote by *g(x) the right-hand side of (3.9). By the expansions* *u*_{2}(x) =

*α**u*_{2,α}*x** ^{α}* and

*g(x) =*

*α**g*_{α}*x** ^{α}*, we deﬁne the vectors

*U*and

*G*by (3.10)

*U*:=

*(u*

^{t}_{2,(α}

_{1}

_{,N}*)*

_{−,)}

^{N}

_{=0}*, G*:=

*(g*

^{t}_{(α}

_{1}

_{,N}*)*

_{−,)}

^{N}

_{=0}*,*

where we may assume that *N* +*α*_{1} *≥* 2, N, α_{1} *∈* Z_{+}. Indeed, by (3.8) and
(3.9) one may assume that the order of *g(x) is greater than 2. Hence in the*
deﬁnition of *G* in (3.10) we may assume that *N* +*α*_{1} *≥* 2, N, α_{1} *∈* Z+. On
the other hand, because the diﬀerential operator in the left-hand side of (3.9)
preserves homogeneous polynomials, we may assume the conditions for*U*. By
substituting the expansions of *g(x) andu*_{2}(x) into (3.9), we have

(3.11) (α_{1}*−τ N*+*τ)U− M*_{N}*U* =*G,*
where *M**N* is given by

(3.12) *M**N* =

0 0 0 *. . .* 0 0 0

*N* 0 0 *. . .* 0 0 0

0 *N−*1 0 *. . .* 0 0 0
0 0 *N−*2 *. . .* 0 0 0
... ... ... . .. ... ... ...

0 0 0 *. . .* 2 0 0

0 0 0 *. . .* 0 1 0

*,* *N≥*1,

and*M*0= 0. By inductive arguments we get
(3.13) *u*_{2,(α}_{1}* _{,N−,)}*=

*r=0*

1

(α_{1}*−τ N* +*τ)*^{r+1}

(N*−*+*r)!*

(N*−)!* *g*_{(α}_{1}* _{,N−+r,−r)}*
for = 0,1, . . . , N, because

*α*

_{1}

*−τ N*+

*τ*= 0 by the assumption

*α*

_{1}+

*N*

*≥*2 and the irrationality of

*τ*.

We set *L* := *x*_{1}*∂*_{1}*−τ x*_{2}*∂*_{2}*−τ x*_{3}*∂*_{3}. By the deﬁnition of *R*_{2}, we obtain
(*L −c)M*(R_{2}) 0. Because the order of*x*_{1}*w*_{1},*u*_{2}or*R*_{2}is equal to or greater
than 2 by the constructions of *w*_{1} and *u*_{2} or the deﬁnition of *R*_{2}, it follows
that, in the Taylor expansion of the right-hand side of (3.9) the terms *x** ^{α}*
(α

_{1}+

*N*= 2, α

_{2}+

*α*

_{3}=

*N*) appear only in the expansion of

*R*

_{2}(x

_{1}

*, x*

_{2}). Hence, we see that (

*L −c)M*(u

_{2}) 0 up to the terms of order 2. Now, suppose that (

*L −c)M*(u

_{2}) 0 holds up to order

*k. Then, we want to show that*(

*L−c)M*(g) 0 holds up to order at least

*k+1. Indeed, in view of the deﬁnition*of

*g*we may consider the terms of order less than or equal to

*k*+1 which appear in

*x*

^{i}_{1}(1 +

*w*

_{1})

*(x*

^{i}_{2}+

*u*

_{2})

*. Let us consider the term*

^{j}

_{i}*ν*

_{j}

*µ*

*x*^{i}_{1}*w*^{ν}_{1}*x*^{j−µ}_{2} *u*^{µ}_{2}, (ν *≥*0

and 0*≤µ≤j). Then, by the deﬁnition of* *g* we have (*L −c)M*(x^{i}_{1}*x*^{j−µ}_{2} ) 0
because *i−τ(j−µ)> i−τ j* *≥c. On the other hand the terms of order less*
than or equal to*k*+ 1 appearing in*u*^{µ}_{2} satisﬁes (*L −c)M*(*·*) 0. Similarly we
have (*L −c)M*(w_{1}* ^{ν}*) 0. Hence we have the assertion. It follows from (3.9)
that (

*L −c)M*(u

_{2}) 0 holds up to order at least

*k*+ 1. By induction, we obtain (

*L −c)M*(u

_{2}) 0. Because

*τ >*0, it follows that (

*L*+

*τ−c)M*(u

_{2}) (

*L −c)M*(u

_{2}) 0. On the other hand, we have that (

*L −*1

*−c)M*(u

_{1}) 0, because the order of

*u*

_{1}=

*u*

_{1}(x

_{1}) is greater than [c] + 2 by the construction. We also have (

*L*+

*τ−c)M*(u

_{3}) = 0 0. Therefore we have (L

_{Λ}

_{S}*−c)M*(u) 0.

Next we will show that*u*_{2} 0. Indeed, by the relation (*L −c)M*(u_{2}) 0
we see that the support of the Taylor expansion of*u*_{2} satisﬁes that*α*_{1}*−τ α*_{2}*−*
*τ α*_{3} *≥* *c >* 0. Because *R* 0, it follows from (3.13) that the coeﬃcients in
Taylor expansion of *u*_{2} of homogeneous order 2 are nonnegative. Namely, we
have *u*_{2} 0 up to order 2. Hence, we have *g(x)* 0 up to at least order 3,
because *R* 0. It follows from (3.13) that*u*_{2} 0 up to at least order 3. By
inductive argument, we have *u*_{2} 0.

We will show the divergence. By the deﬁnition of *g, we can write* *g** _{α}* =

˜

*g** _{α}*+

*h*

*, where ˜*

_{α}*g*

*comes from*

_{α}*R*

_{2}(x) and

*h*

*comes from terms containing*

_{α}*w*

_{1}and

*u*

_{2}. By the assumption and what we have proved in the above, we have ˜

*g*

_{α}*≥*0 and

*h*

_{α}*≥*0. Because ˜

*g*

_{(α}

_{1}

*= 1, it follows from (3.13) that*

_{,N,0)}*u*

_{2,(α}

_{1}

_{,0,N)}*≥N!(α*

_{1}

*−τ N*+τ)

*. Hence*

^{−N−1}*u*

_{2}diverges. This ends the proof.

**§****4.** **Preliminary Lemmas**

In order to prove lemmas, we use subspaces of *A** _{±}*. Let

*f*

*∈ A*

*be given by (2.8). For*

_{−}*ρ >*0, we introduce the norm of

*f*by

(4.1) *f**ρ*:=

*n*
*j=1*

*M*(f* _{j}*)(ρ, . . . , ρ) =

*n*

*j=1*

*γ*

*|f*_{j,γ}*|ρ*^{|γ|+1}*,*

if the right-hand side is ﬁnite. The set of all*f* such that*f**ρ**<∞*is denoted
by *A** _{−,ρ}*. We similarly deﬁne

*A*

_{+,ρ}. If we make the change of the variables

*x*

_{j}*→εx*

*, then we may assume that*

_{j}*ρ >*1 in the above deﬁnition. Hence we assume

*ρ >*1 in the following. For the sake of simplicity, we sometimes omit the suﬃx of the norm

*·*

*, and denote it by*

_{ρ}*·*if there is no fear of confusion.

Let the operators*Q** _{±}* on the spaces

*A*

*∓*be deﬁned by

*Q*

_{±}*V*(x) =

*−*

_{±∞}

0

*e*^{−tΛ}*V*(e^{tΛ}*x)dt,* *V* *∈ A*_{∓}*,*
(4.2)

if the right-hand side integral converges. We denote by *A*^{0}* _{∓}* the subset of ele-
ments of

*A*

*∓*which are polynomials in

*x. Then we have*

**Lemma 4.1.** *Suppose that* (2.3)*holds. Moreover, assume that*Λ* _{N}* = 0.

*Then,* *Q*_{±}*is a continuous linear operator on* *A**∓* *intoA**∓* *such thatL*_{Λ}*Q*_{±}*V* =
*V* *for every* *V* *∈ A**∓**. Moreover, there exists* *c*_{1} *>* 0 *such that* *Q*_{±}*V**ρ* *≤*
*c*_{1}*V**ρ* *for all* *V* *∈ A*_{∓}*. If* Λ* _{N}* = 0, then

*Q*

_{±}*is a linear operator onA*

^{0}

_{∓}*into*

*A*

^{0}

_{∓}*.*

*Proof.* Because the proof is similar, we prove the lemma for *Q*_{+}. By
multiplying (2.5) with *e*^{−iτ}^{0}, we may assume that all components of Λ are
real. For *V*(x) = * ^{t}*(V

_{1}(x), V

_{2}(x), . . . , V

*(x))*

_{n}*∈ A*

*−*, let

*V*

*(x) =*

_{j}*γ**x*^{γ}*V** _{j,γ}*
(j= 1,2, . . . , n) be the Taylor expansion of

*V*

*(x). We set*

_{j}*λ*=

*(λ*

^{t}_{1}

*, λ*

_{2}

*, . . . , λ*

*).*

_{n}We write Λ = Λ* _{S}*+Λ

*, where Λ*

_{N}*and Λ*

_{S}*are the semi-simple and the nilpotent parts of Λ, respectively. Because [Λ*

_{N}

_{S}*,*Λ

*] = 0, we have*

_{N}(4.3) *e*^{−tΛ}*V*(e^{tΛ}*x) =e*^{−tΛ}^{N}*e*^{−tΛ}^{S}*V*(e^{tΛ}*x).*

Since (e^{tΛ}*x)** ^{γ}* = (e

^{tΛ}

^{N}*x)*

^{γ}*e*

*, it follows that the*

^{tλ,γ}*j-th component of*

*e*

^{−tΛ}

^{S}*V*(e

^{tΛ}*x) is given by*

*γ*

*e*^{−tλ}* ^{j}*(e

^{tΛ}*x)*

^{γ}*V*

*=*

_{j,γ}*γ*

(e^{tΛ}^{N}*x)*^{γ}*e*^{tλ,γ−tλ}^{j}*V*_{j,γ}*.*
(4.4)

On the other hand, it follows from (2.7) that, for every *γ* *∈* Z^{n}_{+}(2) and *j* =
1,2, . . . , n, we have*λ, γ −λ*_{j}*≤ −c <*0 if*V** _{j,γ}* = 0. Hence we obtain

(4.5) exp(t*λ, γ −tλ** _{j}*)

*≤e*

^{−ct}*,*

*∀t≥*0.

It follows that for each *t≥*0, the sum (4.4) converges.

If Λ is semi-simple, i.e., Λ* _{N}* = 0, then it follows from (4.4) and (4.5) that
the integral (4.2) converges. If Λ

*= 0, then we see that the growth of terms appearing in (e*

_{N}

^{tΛ}

^{N}*x)*

*is at most*

^{γ}*t*

*, where*

^{|γ|(−1)}*≥*2 is the maximal size of the Jordan block of Λ

*. Because we assume that*

_{N}*V*is a polynomial in case Λ

*= 0, it follows from (4.3), (4.4) and (4.5) that the integral in (4.2) converges and it is a polynomial of*

_{N}*x.*

Because the substitution*x*^{γ}*→*(e^{tΛ}*x)** ^{γ}* preserves the property (2.8), we see
that

*Q*

_{+}

*V*is a ﬁnite sum of vector functions whose components satisfy (2.8).

Next we will show that (L_{Λ}* _{S}* +

*c)M*(Q

_{+}

*V*) 0. We note that every monomial

*x*

*which appears in (e*

^{δ}

^{tΛ}

^{N}*x)*

*satisﬁes*

^{γ}*λ, γ*=

*λ, δ*. Indeed, the map

*x→e*

^{tΛ}

^{N}*x*induces a linear upper (lower) triangular transformation among the components of

*x*corresponding to the same Jordan block. Because

*λ*

*’s coincide with each other for such components, we have the assertion. In view of the deﬁnition of*

_{i}*Q*

_{+}, the condition

*λ, γ −λ*

_{j}*≤ −c*is preserved by the

operator *Q*_{+}. Hence we have *Q*_{+}*V* *∈ A**−*. Therefore *Q*_{+} : *A**−* *→ A**−* is
well-deﬁned if Λ* _{N}* = 0. We remark that the above argument also shows that

*Q*

_{+}:

*A*

^{0}

_{−}*→ A*

^{0}

*is well-deﬁned if Λ*

_{−}*= 0.*

_{N}Next we will show that*L*_{Λ}*Q*_{+}*V* =*V* for*V* *∈ A** _{−}*. For every

*V*

*∈ A*

*we will prove*

_{−}(4.6) *e** ^{−tΛ}*Λx, ∂

_{x}*V*(e

^{tΛ}*x) =e*

^{−tΛ}*d*

*dtV*(e^{tΛ}*x),* *t≥*0,

in some neighborhood of the origin*x*= 0 independent of*t, 0≤t <∞*. Indeed,
by the relation *∂*_{x}*V*(e^{tΛ}*x) = (∇V*)(e^{tΛ}*x)e** ^{tΛ}* we have

Λx, ∂_{x}*V*(e^{tΛ}*x) = (∇V*)(e^{tΛ}*x)e** ^{tΛ}*Λx
(4.7)

= (*∇V*)(e^{tΛ}*x)d*

*dte*^{tΛ}*x*= *d*

*dtV*(e^{tΛ}*x).*

This proves (4.6) for each *t≥*0 and*x*in some neighborhood of the origin. If
Λ* _{N}* = 0, then we have

*e*^{−tΛ}*d*

*dtV*(e^{tΛ}*x) =*

*γ*

*λ, γe*^{tλ,γ−Λ}^{S}^{t}*V*_{γ}*x*^{γ}*,*
where *V*(x) =

*γ**V*_{γ}*x** ^{γ}*. Because

*e*

^{tλ,γ−Λ}

^{S}

^{t}*≤*1 for all

*t≥*0, we see that the right-hand side is holomorphic in some neighborhood of

*x*= 0 independent of

*t. By an analytic continuation, (4.6) holds for allx*in some neighborhood of the origin independent of

*t,t≥*0. This proves (4.6).

By (4.6) we have

*L*_{Λ}*Q*_{+}*V* = (Λx, ∂_{x}* −*Λ)Q_{+}*V*
(4.8)

=*−*
_{∞}

0

*e** ^{−tΛ}*Λx, ∂

_{x}*V*(e

^{tΛ}*x)dt*+ Λ

_{∞}0

*e*^{−tΛ}*V*(e^{tΛ}*x)dt*

=*−*
_{∞}

0

*e*^{−tΛ}*d*

*dtV*(e^{tΛ}*x)dt*+ Λ
_{∞}

0

*e*^{−tΛ}*V*(e^{tΛ}*x)dt*

=*−*
_{∞}

0

*d*
*dt*

*e*^{−tΛ}*V*(e^{tΛ}*x)*

*dt*=*V*(x).

Finally we shall prove the estimate. If Λ is semi-simple, then we have
*Q*_{+}*V** _{j}*(x) =

*−*

*γ*

*x*^{γ}_{∞}

0

exp(t*λ, γ −tλ** _{j}*)dtV

*=*

_{j,γ}*γ*

*x** ^{γ}*(

*λ, γ −λ*

*)*

_{j}

^{−1}*V*

_{j,γ}*.*Therefore, there exists

*c*

_{1}

*>*0 independent of

*V*such that

*Q*

_{+}

*V*

*ρ*

*≤c*

_{1}

*V*

*ρ*. This ends the proof.

For the later use, we give several lemmas.

**Lemma 4.2.** *If* *f* *∈*C^{1}[x],*g∈*C^{1}[x] *andc* *is a complex number, then*
*M*(f g)*M*(f)M(g),*M*(f+*g)M*(f) +*M*(g) *andM*(cf)* |c|M*(f).

The proof is clear from the deﬁnition.

**Lemma 4.3.** *Assume that*0 *f* *g* *and* *c >*0. If (L_{Λ}* _{S}* +

*c)g*0,

*then*(L

_{Λ}

*+*

_{S}*c)f*0. Similarly, if (L

_{Λ}

_{S}*−c)g*0, then(L

_{Λ}

_{S}*−c)f*0.

*Proof.* Suppose that (L_{Λ}* _{S}* +

*c)g*0. If

*f*=

*f*_{γ}*x** ^{γ}* and

*g*=

*g*

_{γ}*x*

*, then (*

^{γ}*λ, γ*+

*c)g*

_{γ}*≤*0 and 0

*≤*

*f*

_{γ}*≤*

*g*

*. If*

_{γ}*g*

*= 0, then we have*

_{γ}*f*

*= 0 and hence (*

_{γ}*λ, γ*+

*c)f*

_{γ}*≤*0. On the other hand, if

*λ, γ*+

*c*

*≤*0, then (

*λ, γ*+

*c)f*

_{γ}*≤*0. This proves that (L

_{Λ}

*+*

_{S}*c)f*0. We can prove the latter half similarly.

*Remark* 4. The spaces*A** _{±}* are linear spaces. Indeed, let

*f, g∈ A*

*and*

_{−}*α∈*C. The condition (2.8) is easily veriﬁed for

*f*+gor

*αf*. We set

*L*:=

*L*

_{Λ}

*+c (c >0). Then*

_{S}*LM*(f)0 and

*LM*(g)0 imply that

*L*(M(f) +

*M*(g))0.

Because*M*(f +*g)M*(f) +*M*(g) by Lemma 4.2, it follows from Lemma 4.3
that *LM*(f +*g)*0. This proves that*A**−* is a linear space. The proof is the
same for *A*_{+}.

**Lemma 4.4.** *Let* *ρ >* 0. For *u, v* *∈* C* ^{n}*[x]

*such that*

*u*

*ρ*

*<*

*∞*

*and*

*v*

*ρ*

*<∞, we haveu·v*

*ρ*

*≤ u*

*ρ*

*v*

*ρ*

*.*

This is clear from the deﬁnition.

**§****5.** **Proof of Theorem 2.1**

*Proof of Theorem* 2.1. We will prove the theorem in the case*R* *∈ A** _{−}*.
The proof is the same in the case

*R*

*∈ A*+. If there is no fear of confusion, we omit the suﬃces and we simply denote

*A*and

*Q*instead of

*A*

*∓,ρ*and

*Q*

*, respectively. Similarly, we sometimes omit the suﬃx of*

_{±,ρ}*·*

*ρ*and write

*·*instead of

*·*

*ρ*.

In order to solve (2.5) we set *v* =*QV*. By Lemma 4.1, Eq. (2.5) can be
written in the form

(5.1) *V* =*R(x*+*QV*),

if Λ* _{N}* = 0. In view of this we will solve (5.1). We deﬁne the sequence

*{V*

^{j}*}*

*j*by (5.2)

*V*

^{0}=

*R(x), V*

^{1}=

*R(x*+

*QV*

^{0})

*−R(x),*

(5.3) *V** ^{j+1}*=

*R(x+QV*

^{0}+

*· · ·*+QV

*)*

^{j}*−R(x+QV*

^{0}+

*· · ·*+QV

*), j= 1,2, . . . In order to show that*

^{j−1}*V*

*’s are well-deﬁned we ﬁrst consider the case Λ*

^{j}*= 0.*

_{N}By Lemma 4.1 and the assumption, we see that *V*^{0} and *QV*^{0} are polynomi-
als. Hence, by (5.2) *V*^{1} is a polynomial. Inductively, we see that *V** ^{j}*’s are
polynomials. We will show that

*V*

^{j}*∈ A*

^{0}. For this purpose, we will prove that

*R(x*+

*QV*)

*∈ A*

^{0}if

*V*

*∈ A*

^{0}. Indeed, if we can prove this, then we have

*R(x*+

*QV*

^{0})

*∈ A*

^{0}. It follows that

*V*

^{1}=

*R(x*+

*QV*

^{0})

*−R(x)∈ A*

^{0}. Next we have

*V*

^{0}+

*V*

^{1}

*∈ A*

^{0}, and thus

*V*

^{2}

*∈ A*

^{0}. By the induction we have

*V*

^{j}*∈ A*

^{0}(j= 0,1,2, . . .).

In order to show (2.8) we write
(5.4) *R** _{j}*(x+

*QV*) =

*γ*

*R** _{j,γ}*(x+

*QV*)

*=*

^{γ}*γ*

*R*_{j,γ}

*i*

(x* _{i}*+ (QV)

*)*

_{i}

^{γ}

^{i}*,*where (QV)

*is the*

_{i}*i-th component of*

*QV*. Because

*QV*

*∈ A*

^{0}, (QV)

*is the sum of the functions*

_{k}*h*

*with*

_{µ}*h*

*being divisible by*

_{µ}*x*

*where*

_{µ}*x*

*and*

_{k}*x*

*belong to the same Jordan block. If*

_{µ}*R*

*(x) is divisible by*

_{j}*x*

*with*

_{k}*x*

*and*

_{k}*x*

*belonging to the same Jordan block, then it follows that*

_{j}*R*

*(x+*

_{j}*QV*) is the sum of functions divisible by some

*x*

*with*

_{ν}*x*

*and*

_{ν}*x*

*belonging to the same Jordan block. Hence (2.8) holds.*

_{j}Next we will show that (L_{Λ}* _{S}* +

*c)M*(R(

*·*+

*QV*)) 0. Let

*x*

*be any monomial appearing in the right-hand side of (5.4). Because (L*

^{η}_{Λ}

*+c)M(R) 0 by the assumption, it follows that the*

_{S}*γ’s in (5.4) satisfy thatλ, γ−λ*

_{j}*≤ −c.*

In view of the relation *QV* *∈ A*^{0}, (QV)* _{i}* can be expanded in the power series
of

*x*

*such that*

^{δ}*λ, δ −λ*

_{i}*≤ −c. If we expand (x*

*+ (QV)*

_{i}*)*

_{i}

^{γ}*in the right-hand side of (5.4) into the power series of*

^{i}*x, then we see that*

*x*

*appears if some*

^{η}*x*

*in*

_{i}*x*

*is replaced by*

^{γ}*x*

*appearing in (QV)*

^{δ}*a ﬁnite number of times. If*

_{i}*x*

*turns into*

^{γ}*x*

*by the one substitution, then we have*

^{η}*λ, η −λ** _{j}* =

*λ, γ −λ*

_{j}*−λ*

*+*

_{i}*λ, δ ≤ −*2c <

*−c.*

By the similar argument, we have the estimate*λ, η −λ*_{j}*≤ −c*in the general
case. Hence *R(x*+*QV*) satisﬁes (L_{Λ}* _{S}* +

*c)M*(R(

*·*+

*QV*)) 0. This proves that

*R(x*+

*QV*)

*∈ A*

^{0}.

Next we consider the case Λ* _{N}* = 0. Let

*V*

*∈ A*. Because

*Q*is continuous by Lemma 4.1,

*R(x*+

*QV*) is well-deﬁned in some neighborhood of the origin if

*V*is suﬃciently small. We will estimate

*R(·*+

*QV*)

*ρ*. By making the scale change of the variables, if necessary, one may assume

*R*

_{2ρ}

*< ε. By Lemma*4.1, Lemma 4.4 and (4.1) we have

(5.5) *R(·*+*QV*)_{ρ}*≤*

*γ,j*

*|R*_{j,γ}*|*(ρ+*c*_{1}*V** _{ρ}*)

^{|γ|}*.*

If*V**ρ* *< ε*for suﬃciently small*ε*such that*c*_{1}*ε < ρ, then the right-hand side*
of (5.5) is bounded by *ε*because *R*2ρ *< ε. On the other hand, by the same*
argument as in the case Λ* _{N}* = 0, we can prove that

*R(x+QV*)

*∈ A*. Therefore, one can deﬁne

*V*

^{j}*∈ A*(j= 0,1, . . .) by (5.2) and (5.3) inductively.

Next we will prove the convergence of*{V*^{j}*}*in*A*. For this purpose we will
show that there exist constants*c*_{0}*≥*0 and*K*_{0}*≥*0 independent of*j* such that,
for*j*= 0,1,2, . . .,

*V*^{j}* ≤c*^{j}_{0}*ε*^{j+1}*,*
(5.6)

*QV*^{j}* ≤K*_{0}*V*^{j}*.*
(5.7)

Clearly we have*V*^{0}=*R< ε*by the deﬁnition. Next we will show (5.7) for
*j* = 0. If Λ* _{N}* = 0, then the estimate follows from Lemma 4.1. Hence we may
assume Λ

*= 0. Let*

_{N}*d*

_{0}be the degree of

*R. By Lemma 4.1 we haveQV*

^{0}

*∈ A*

^{0}. By the deﬁnition we have

*QV*^{0}=*−*
_{∞}

0

*e*^{−tΛ}*V*^{0}(e^{tΛ}*x)dt*=*−*
_{∞}

0

*e*^{−tΛ}^{N}*e*^{−tΛ}^{S}*V*^{0}(e^{tΛ}*x)dt.*

(5.8)

We set

(5.9) *W*(t, x)*≡*(W_{1}(t, x), . . . , W* _{n}*(t, x)) :=

*e*

^{−tΛ}

^{S}*V*

^{0}(e

^{tΛ}*x).*

Then the components of the ﬁrst Jordan block of *e*^{−tΛ}^{N}*W* are given by

*W*_{1}*−tW*_{2}+*t*^{2}

2*W*_{3}*−t*^{3}

3!*W*_{4}+*· · ·*
(5.10)

*W*_{2}*−tW*_{3}+*t*^{2}

2!*W*_{4}*− · · ·*

*· · ·*

There are ﬁnite number of similar terms corresponding to every Jordan block
of*e*^{−tΛ}^{N}*W*. Hence we can easily see that*QV*^{0}*ρ*is bounded by the following

quantity

*n−1*

*k=0*

_{∞}

0

(*−t)*^{k}

*k!* *W** _{k+1}*(t, x)dt
+

*n−2*

*k=0*

_{∞}

0

(*−t)*^{k}

*k!* *W** _{k+2}*(t, x)dt
+

*· · ·*(5.11)

*≤*

*n−1*

*k=0*

_{∞}

0

*t*^{k}

*k!W** _{k+1}*(t,

*·*)

*dt*+

*n−2*

*k=0*

_{∞}

0

*t*^{k}

*k!W** _{k+2}*(t,

*·*)

*dt*+

*· · ·*

=
_{∞}

0 *W*_{1}(t,*·*)*dt*+
_{∞}

0 *W*_{2}(t,*·*)(1 +*t)dt*
+

_{∞}

0 *W*_{3}(t,*·*)

1 +*t*+*t*^{2}
2

*dt*+*· · ·*

*≤*
*n*
*j=1*

_{∞}

0

*B*_{0}(t)*W** _{j}*(t,

*·*)

*dt,*

where *B*_{0}(t) = _{n−1}

*ν=0**t*^{ν}*/ν!. We write* *V*^{0} = (V_{1}^{0}*, V*_{2}^{0}*, . . . , V*_{n}^{0}), and expand *V*_{j}^{0}
(1*≤j≤n) into the Taylor seriesV*_{j}^{0}=

*γ**V*_{j,γ}^{0} *x** ^{γ}*. Then, by (5.9) we have

*W*

*(t, x) =*

_{j}*e*

^{−tλ}

^{j}*V*

_{j}^{0}(e

^{tΛ}*x) =e*

^{−tλ}

^{j}*γ*

*V*_{j,γ}^{0} (e^{tΛ}*x)** ^{γ}*
(5.12)

=

*γ*

*e*^{tλ,γ−tλ}^{j}*V*_{j,γ}^{0} (e^{tΛ}^{N}*x)*^{γ}*.*
Because the sum with respect to*γ* is ﬁnite, we have
(5.13)

*n*
*j=1*

_{∞}

0

*B*_{0}(t)*W** _{j}*(t,

*·*)

*dt≤*

*j,γ*

*|V*_{j,γ}^{0} *|*

*B*_{0}(t)e^{tλ,γ−tλ}* ^{j}*(e

^{tΛ}

^{N}*x)*

^{γ}

_{ρ}*dt.*

On the other hand, we have

(5.14) (e^{tΛ}^{N}*x)*^{γ}*ρ**≤ρ** ^{|γ|}*(e

^{tΛ}

^{N}*e)*

^{γ}*≤ρ*

^{|γ|}*|γ|≤d*0

(e^{tΛ}^{N}*e)*^{γ}*,*

where *e* = (1,1, . . . ,1). In terms of the estimate *λ, γ −* *λ*_{j}*≤ −c* (j =
1,2, . . . , n) and (5.14), the right-hand side of (5.13) can be estimated in the
following way

*≤*

*j,γ*

*ρ*^{|γ|}*|V*_{j,γ}^{0} *|*

*B*_{0}(t)e^{−ct}

*|δ|≤d*0

(e^{tΛ}^{N}*e)*^{δ}*dt*
(5.15)

*≤K(d*_{0})

*j,γ*

*ρ*^{|γ|}*|V*_{j,γ}^{0} *|*=*K(d*_{0})*V*^{0}_{ρ}*,*

where *K(d*_{0}) =

*B*_{0}(t)e^{−ct}

*|δ|≤d*0(e^{tΛ}^{N}*e)*^{δ}*dt. Therefore we haveQV*^{0}*ρ* *≤*
*K(d*_{0})*V*^{0}*ρ*. If we set*K*_{0}= max*{K(d*_{0}), c_{1}*}*with*c*_{1}given by Lemma 4.1, then
we have (5.7) for*j* = 0.

Next we will prove (5.6) for*j*= 1. It follows from (5.2) that

(5.16) *V*^{1}* ≤ QV*^{0}

_{1}

0 *∇R(·*+*τ QV*^{0})*dτ.*

In order to estimate *∇R(·*+*τ QV*^{0})we make the same argument as in (5.5).

Indeed, if we have the estimate

(5.17)

*γ*

*|R*_{j,γ}*||γ|*(2ρ)^{|γ|}*< c*_{2}*ε,* *j*= 1,2, . . . , n,

for some constant*c*_{2}*>*0 independent of*ε, then we obtain*
(5.18) *∇R(·*+*τ QV*^{0})*< c*_{2}*ε,* *∀τ,*0*≤τ≤*1,

if *K(d*_{0})ε < ρ. The estimate (5.17) follows from the assumption*R*_{2ρ} *< ε*if
we replace *ρ >*1 with 1 *< ρ*^{}*< ρ. For the sake of simplicity we assume that*
(5.17) holds in the following.

Therefore we get, from (5.16) that
(5.19) *V*^{1}* ≤K(d*_{0})c_{2}*ε*^{2}

_{1}

0

*dτ* =*c*_{0}*ε*^{2}*,*
where *c*_{0}=*K(d*_{0})c_{2}.

Next we will estimate*QV*^{1}. In view of Lemma 4.1 we may assume that
Λ* _{N}* = 0. We write

*V*

^{1}= (V

_{1}

^{1}

*, V*

_{2}

^{1}

*, . . . , V*

_{n}^{1}) and consider the Taylor expansion

*V*

_{j}^{1}(x) =

*γ**V*_{j,γ}^{1} *x** ^{γ}*. Here the sum is a ﬁnite one. We will show that for every

*γ*such that

*V*

_{j,γ}^{1}= 0 we have

(5.20) *λ, γ −λ*_{j}*≤ −* *c*

*d*_{0}*−*1(*|γ| −*1).

Noting that*V*_{j}^{1}=*R** _{j}*(x+QR)

*−R*

*(x), we ﬁrst consider*

_{j}*R*

*(x). Because*

_{j}*R∈ A*

^{0}, we have

*λ, γ−λ*

_{j}*≤ −c*(j = 1,2, . . . , n) for every

*x*

*in the expansion of*

^{γ}*R*

*(x).*

_{j}Because *|γ| ≤d*_{0}, we have

(5.21) *λ, γ −λ*_{j}*≤ −c*=*−cd*_{0}*−*1

*d*_{0}*−*1 *≤ −* *c*

*d*_{0}*−*1(*|γ| −*1), *j*= 1,2, . . . , n.

Next we will prove (5.20) for *R** _{j}*(x+

*QR). We note that*

*QR*satisﬁes (5.20) because

*QR*

*∈ A*

^{0}and

*QR*is the polynomial of degree

*d*

_{0}. We set