or Small Denominators

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(x₁, . . . , x_n) ∈ Cⁿ, n ≥ 2 be the variable in Cⁿ, andR be the set of real numbers. Let Λ be ann-square constant matrix. LetL_Λbe the Lie derivative of the linear vector field Λx·∂_x

(2.1) L_Λv= [Λx, v] =Λx, ∂_xv −Λv, where Λx, ∂_xv=_n

j=1(Λx)_j(∂/∂x_j)v, with (Λx)_j being thej-th component of Λx. We consider the system of equations

(2.2) L_Λu=R(u(x)),

where u=^t(u₁, u₂, . . . , u_n) is an unknown vector function and R(x) =^t(R₁(x), R₂(x), . . . , R_n(x))

is holomorphic in some neighborhood ofx= 0 in Cⁿ such thatR(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∃τ₀, 0≤τ₀≤πsuch that

(2.3) every component ofe^−iτ0Λ 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) ∃ τ₀, 0≤τ₀≤π, e^−iτ0λ_j∈R (j= 1,2, . . . , n).

If we setu(x) =x+v(x),v(x) =O(|x|²), thenvsatisfies the following equation

(2.5) L_Λv=R(x+v(x)).

Let Z+ be the set of nonnegative integers, and let Zⁿ₊(k) (k ≥0) be the n-product ofZ+, γ=^t(γ₁, γ₂, . . . , γ_n) such that|γ|=γ₁+γ₂+· · ·+γ_n ≥k.

Forγ∈Zⁿ₊, we setx^γ =x^γ₁¹· · ·x^γ_nⁿ. Fork≥0 andn≥1, we denote byCⁿ_k[[x]]

the set of formal power series

|η|≥ku_ηx^η (u_η ∈ Cⁿ). We also define the convergentn-vector power series which vanishes up to the (k−1)-th derivative byCⁿ_k[x]. We decompose Λ = Λ_S+ Λ_N, where Λ_S and Λ_N are the semi-simple and the nilpotent part of Λ, respectively. We denote byL_ΛS the Lie derivative of the linear vector field Λ_Sx·∂_x.

For a formal power series f(x) =

γf_γx^γ, we define the majorant off, M(f)(x) by

(2.6) M(f) :=

γ

|f_γ|x^γ.

For a vector f = (f₁, f₂, . . . , f_n) we define M(f) := (M(f₁), M(f₂), . . . , M(f_n)). For a formal power series with real coefficients a(x) =

γa_γx^γ and b(x) =

γb_γx^γ, we defineabifa_γ≤b_γ for allγ∈Zⁿ₊. We define (f₁(x), f₂(x), . . . , f_n(x))(g₁(x), g₂(x), . . . , g_n(x)) iff_j(x)g_j(x) forj= 1,2, . . . , n.

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

t(g₁(x), . . . , g_n(x))∈Cⁿ₂[x] such that

(2.7) (L_ΛS−c)M(g) 0 (resp. (L_ΛS+c)M(g)0)

and thatg(x) is a finite sum of the functionsf =^t(f₁, f₂, . . . , f_n)∈Cⁿ₂[x] with the following expansion at the origin

(2.8) f_j(x) =x_ν

γ

f_j,γx^γ, f_j,γ∈C, j= 1,2, . . . , n,

whereνis such that thej-th and theν-th components of Λ_S belong to the same Jordan block of Λ_N. We can prove thatA_±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 conditionL_ΛSM(R) 0 or L_ΛSM(R)0 is not sufficient. (cf. Proposition 3.1.)

(b) There exists Rsuch that neither L_ΛSM(R)0 nor L_ΛSM(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 existsRwhich 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 (j = 1,2, . . . , n) satisfy the Poincar´e condition if the convex hull of λ_j (j = 1,2, . . . , n) in the complex plane does not contain the origin 0∈C. We can easily see that the Poincar´e condition is equivalent to the following estimate:

there existC >0 andK >0 independent ofαsuch that

(2.9) |λ, α −λ_j| ≥C|α|, ∀α= (α₁, α₂, . . . , α_n)∈Zⁿ₊, |α| ≥K, forj= 1,2, . . . , n, whereλ= (λ₁, λ₂, . . . , λ_n) andλ, α=_n

i=1λ_iα_i. For α = (α₁, α₂, . . . , α_n) ∈ Zⁿ we define α = _n

i=1|α_i|. We say that λ∈Rⁿ is a Diophantine vector if there exist∃τ >0 and ∃C >0 such that (2.10) |λ, α −λ_j| ≥Cα^−n−τ, ∀α∈Zⁿ, 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 existτ >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_j and define

(2.11) ω_k= inf{|_j(α)|;_j(α)= 0, α∈Zⁿ,2≤ α ≤2^k, j= 1,2, . . . , n}, where k= 1,2, . . .. Then we say that λsatisfies 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∈C²

(3.1) L_Λu=R(x+u), Λ =

1 0

0 −τ

,

whereτ >0 is a Liouville number chosen later andu=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_ΛSM(R) 0 orL_ΛSM(R)0holds 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= [a₁, a₂, . . .],a_j∈N. Namely, if we define the sequence{p_n}and {q_n}by

p_n=a_np_n−1+p_n−2, n≥2, p₀= 0, p₁= 1, (3.2)

q_n=a_nq_n−1+q_n−2, n≥2, q₀= 1, q₁=a₁, (3.3)

then we have p_n/q_n→τ (n→ ∞). Moreover, we have (cf. [3])

(3.4) 1

(a_n+1+ 2)q_n² <

τ−p_n q_n

< 1

a_n+1q²_n, n≥0.

We substitute the expansion ofu= (u₁, u₂) andR_j(x+u), u_j=

|η|≥2

u_j,ηx^η,

R_j(x+u) =

|γ|≥2

R_j,γ(x₁+

u_1,ηx^η)^γ1(x₂+

u_2,ηx^η)^γ2 into the equation (3.1). Then we have the recurrence relations

(η₁−τ η₂−1)u_1,η=R_1,η+P_η,1(u_j,δ,|δ|<|η|, j= 1,2), (3.5)

(η₁−τ η₂+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_k,δ’s with coefficients given by the expansions of R. Now suppose that a₁, a₂, . . . , a_n are given. We determine p_n andq_n by (3.2) and (3.3), and we want to determine u_1,η with η = (p_n + 1, q_n) from (3.5) assuming that 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_η,1(u_j,δ,|δ|<|η|, j= 1,2) is smaller than 2^|η|+1, then we takeR_1,η such that

|R_1,η| = 2·3^|η|. On the other hand, if the absolute value of P_η,1(u_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^|η|. 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 determinea_n+1 such that|u_1,(pn+1,qn)| ≥(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 [a₁, a₂, . . . , a_n+1] avoids the finite number of rational points given in the above if we takea_n+1 sufficiently large. Next we determinep_n+1 andq_n+1 from (3.2) and (3.3). Then we want to determine u_1,η withη= (p_n+1+ 1, q_n+1) from (3.5). We can determine the termsu_j,δ(|δ|<|η|,j= 1,2) ifτavoids a finite number of rational points. This is possible if a_n+2 is sufficiently large. Then we repeat the same argument as in the above. Clearly,R(x) is holomorphic in some neighborhood of the origin.

On the other hand, if we takea_n+1so thata_n+1is larger than polynomial order ofq_n, then it follows from (3.4) thatτ is a Liouville number. Therefore we can determine a Liouville numberτ so that we have a divergent formal power series solutionu=

|η|≥2u_ηx^η.

By the definition of a continued fraction expansion, we havep_n+ 1−τ q_n− 1 >0 or p_n+ 1−τ q_n−1 <0 according asn is odd or even. For simplicity,

we take p_n and q_n for odd 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 η₁ −τ η₂−1 > 0. This proves that L_ΛSM(R) 0. By a similar argument we can also treat the case L_ΛSM(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 numberτsuch 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 inx∈C³

(3.7) L_Λu=R(x+u), Λ =





1 0 0

0 −τ −1

0 0 −τ



,

where τ >0 is an irrational number andu=O(|x|²). Then we have

Proposition 3.2. Let c >0. For every irrational numberτ >0 there exists R ≡ 0 which is not a polynomial such that (L_ΛS −c)M(R) 0 or (L_ΛS+c)M(R)0 holds and that the unique formal power series solution of (3.7)diverges.

Proof. LetKbe such thatK > c+ 2. We denote by [c] the largest integer which does not exceedc. Then we define

(3.8) R₁(x) =x^[c]+2₁ R˜₁(x₁), R₂(x) =

max{c,2}≤i−τ j<K

xⁱ₁x^j₂, R₃(x)≡0,

where ˜R₁ is holomorphic at the origin such that ˜R₁ 0. We can easily see that (L_ΛS −c)M(R) 0.

We will construct the solution u= (u₁, u₂, u₃) of (3.7). We set u₁(x) = x₁w₁(x₁). Then it follows from the first equation of (3.7) that w₁ satisfies x₁∂₁w₁=x^[c]+1₁ (1 +w₁)^[c]+2R˜₁(x₁(1 +w₁)). By the elementary computations, we can easily show that the equation has a holomorphic solution w₁(x₁) such that w₁ 0. Nextu₃ satisfies (x₁∂₁−τ x₂∂₂−τ x₃∂₃−x₃∂₂+τ)u₃ = 0. By the irrationality ofτ, we haveu₃= 0.

Next, by the second equation of (3.7)u₂ satisfies

(3.9) (x₁∂₁−τ x₂∂₂−τ x₃∂₃−x₃∂₂+τ)u₂=R₂(x₁(1 +w₁), x₂+u₂).

We denote by g(x) the right-hand side of (3.9). By the expansions u₂(x) =

αu_2,αx^α andg(x) =

αg_αx^α, we define the vectorsU andGby (3.10) U :=^t(u_{2,(α1,N−,)})^N₌₀, G:=^t(g_(α1,N−,))^N₌₀,

where we may assume that N +α₁ ≥ 2, N, α₁ ∈ 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 +α₁ ≥ 2, N, α₁ ∈ 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 forU. By substituting the expansions of g(x) andu₂(x) into (3.9), we have

(3.11) (α₁−τ N+τ)U− M_NU =G, where MN is given by

(3.12) MN =















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,

andM0= 0. By inductive arguments we get (3.13) u_{2,(α1,N−,)}=

r=0

1

(α₁−τ N +τ)^r+1

(N−+r)!

(N−)! g_{(α1,N−+r,−r)} for = 0,1, . . . , N, because α₁−τ N+τ = 0 by the assumptionα₁+N ≥2 and the irrationality ofτ.

We set L := x₁∂₁−τ x₂∂₂−τ x₃∂₃. By the definition of R₂, we obtain (L −c)M(R₂) 0. Because the order ofx₁w₁,u₂orR₂is equal to or greater than 2 by the constructions of w₁ and u₂ or the definition of R₂, it follows that, in the Taylor expansion of the right-hand side of (3.9) the terms x^α (α₁+N = 2, α₂+α₃=N) appear only in the expansion ofR₂(x₁, x₂). Hence, we see that (L −c)M(u₂) 0 up to the terms of order 2. Now, suppose that (L −c)M(u₂) 0 holds up to order k. Then, we want to show that (L−c)M(g) 0 holds up to order at leastk+1. Indeed, in view of the definition ofgwe may consider the terms of order less than or equal tok+1 which appear in xⁱ₁(1 +w₁)ⁱ(x₂+u₂)^j. Let us consider the term_i

ν

_j

µ

xⁱ₁w^ν₁x^j−µ₂ u^µ₂, (ν ≥0

and 0≤µ≤j). Then, by the definition of g we have (L −c)M(xⁱ₁x^j−µ₂ ) 0 because i−τ(j−µ)> i−τ j ≥c. On the other hand the terms of order less than or equal tok+ 1 appearing inu^µ₂ satisfies (L −c)M(·) 0. Similarly we have (L −c)M(w₁^ν) 0. Hence we have the assertion. It follows from (3.9) that (L −c)M(u₂) 0 holds up to order at least k+ 1. By induction, we obtain (L −c)M(u₂) 0. Becauseτ >0, it follows that (L+τ−c)M(u₂) (L −c)M(u₂) 0. On the other hand, we have that (L −1−c)M(u₁) 0, because the order ofu₁=u₁(x₁) is greater than [c] + 2 by the construction. We also have (L+τ−c)M(u₃) = 0 0. Therefore we have (L_ΛS −c)M(u) 0.

Next we will show thatu₂ 0. Indeed, by the relation (L −c)M(u₂) 0 we see that the support of the Taylor expansion ofu₂ satisfies thatα₁−τ α₂− τ α₃ ≥ c > 0. Because R 0, it follows from (3.13) that the coefficients in Taylor expansion of u₂ of homogeneous order 2 are nonnegative. Namely, we have u₂ 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) thatu₂ 0 up to at least order 3. By inductive argument, we have u₂ 0.

We will show the divergence. By the definition of g, we can write g_α =

˜

g_α+h_α, where ˜g_α comes from R₂(x) and h_α comes from terms containing w₁ and u₂. By the assumption and what we have proved in the above, we have ˜g_α ≥ 0 and h_α ≥ 0. Because ˜g_(α1,N,0) = 1, it follows from (3.13) that u_2,(α1,0,N)≥N!(α₁−τ N+τ)^−N−1. Henceu₂diverges. This ends the proof.

§4. Preliminary Lemmas

In order to prove lemmas, we use subspaces of A_±. Letf ∈ A₋ be given by (2.8). Forρ >0, we introduce the norm off by

(4.1) fρ:=

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 allf such thatfρ<∞is denoted by A_−,ρ. We similarly define A_+,ρ. If we make the change of the variables x_j →εx_j, then we may assume that 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·_ρ, and denote it by·if there is no fear of confusion.

Let the operatorsQ_± on the spacesA∓ be defined by Q_±V(x) =−

_±∞

0

e^−tΛV(e^tΛx)dt, V ∈ A_∓, (4.2)

if the right-hand side integral converges. We denote by A⁰_∓ the subset of ele- ments ofA∓ which are polynomials inx. Then we have

Lemma 4.1. Suppose that (2.3)holds. Moreover, assume thatΛ_N = 0.

Then, Q_± is a continuous linear operator on A∓ intoA∓ such thatL_ΛQ_±V = V for every V ∈ A∓. Moreover, there exists c₁ > 0 such that Q_±Vρ ≤ c₁Vρ for all V ∈ A_∓. If Λ_N = 0, then Q_± is a linear operator onA⁰_∓ into A⁰_∓.

Proof. Because the proof is similar, we prove the lemma for Q₊. By multiplying (2.5) with e^−iτ0, we may assume that all components of Λ are real. For V(x) = ^t(V₁(x), V₂(x), . . . , V_n(x)) ∈ A−, let V_j(x) =

γx^γV_j,γ (j= 1,2, . . . , n) be the Taylor expansion ofV_j(x). We setλ=^t(λ₁, λ₂, . . . , λ_n).

We write Λ = Λ_S+Λ_N, where Λ_Sand Λ_N are the semi-simple and the nilpotent parts of Λ, respectively. Because [Λ_S,Λ_N] = 0, we have

(4.3) e^−tΛV(e^tΛx) =e^−tΛNe^−tΛSV(e^tΛx).

Since (e^tΛx)^γ = (e^tΛNx)^γe^tλ,γ, it follows that the j-th component of e^−tΛSV(e^tΛx) is given by

γ

e^−tλj(e^tΛx)^γV_j,γ =

γ

(e^tΛNx)^γe^{tλ,γ−tλj}V_j,γ. (4.4)

On the other hand, it follows from (2.7) that, for every γ ∈ Zⁿ₊(2) and j = 1,2, . . . , n, we haveλ, γ −λ_j ≤ −c <0 ifV_j,γ = 0. Hence we obtain

(4.5) exp(tλ, 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_N = 0, then we see that the growth of terms appearing in (e^tΛNx)^γ is at mostt^|γ|(−1), where ≥2 is the maximal size of the Jordan block of Λ_N. Because we assume thatV is a polynomial in case Λ_N = 0, it follows from (4.3), (4.4) and (4.5) that the integral in (4.2) converges and it is a polynomial ofx.

Because the substitutionx^γ→(e^tΛx)^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_ΛS +c)M(Q₊V) 0. We note that every monomial x^δ which appears in (e^tΛNx)^γ satisfies λ, γ= λ, δ. Indeed, the mapx→e^tΛNxinduces a linear upper (lower) triangular transformation among the components of x corresponding to the same Jordan block. Because λ_i’s coincide with each other for such components, we have the assertion. In view of the definition of 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₊:A⁰₋→ A⁰₋ is well-defined if Λ_N = 0.

Next we will show thatL_ΛQ₊V =V forV ∈ A₋. For everyV ∈ A₋ we will prove

(4.6) e^−tΛΛx, ∂_xV(e^tΛx) =e^−tΛ d

dtV(e^tΛx), t≥0,

in some neighborhood of the originx= 0 independent oft, 0≤t <∞. Indeed, by the relation ∂_xV(e^tΛx) = (∇V)(e^tΛx)e^tΛ we have

Λx, ∂_xV(e^tΛx) = (∇V)(e^tΛx)e^tΛΛx (4.7)

= (∇V)(e^tΛx)d

dte^tΛx= d

dtV(e^tΛx).

This proves (4.6) for each t≥0 andxin some neighborhood of the origin. If Λ_N = 0, then we have

e^−tΛd

dtV(e^tΛx) =

γ

λ, γe^{tλ,γ−ΛSt}V_γx^γ, where V(x) =

γV_γx^γ. Because e^{tλ,γ−ΛSt} ≤ 1 for all t≥0, we see that the right-hand side is holomorphic in some neighborhood ofx= 0 independent of t. By an analytic continuation, (4.6) holds for allxin some neighborhood of the origin independent oft,t≥0. This proves (4.6).

By (4.6) we have

L_ΛQ₊V = (Λx, ∂_x −Λ)Q₊V (4.8)

=− _∞

0

e^−tΛΛx, ∂_xV(e^tΛx)dt+ Λ _∞

0

e^−tΛV(e^tΛx)dt

=− _∞

0

e^−tΛd

dtV(e^tΛx)dt+ Λ _∞

0

e^−tΛV(e^tΛx)dt

=− _∞

0

d dt

e^−tΛV(e^tΛx)

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(tλ, γ −tλ_j)dtV_j,γ=

γ

x^γ(λ, γ −λ_j)⁻¹V_j,γ. Therefore, there existsc₁>0 independent ofV such thatQ₊Vρ≤c₁Vρ. This ends the proof.

For the later use, we give several lemmas.

Lemma 4.2. If f ∈C¹[x],g∈C¹[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 that0 f g and c >0. If (L_ΛS +c)g0, then (L_ΛS+c)f 0. Similarly, if (L_ΛS −c)g 0, then(L_ΛS −c)f 0.

Proof. Suppose that (L_ΛS +c)g 0. Iff =

f_γx^γ and g = g_γx^γ, then (λ, γ+c)g_γ ≤ 0 and 0 ≤ f_γ ≤ g_γ. If g_γ = 0, then we have f_γ = 0 and hence (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_ΛS+c)f 0. We can prove the latter half similarly.

Remark 4. The spacesA_± are linear spaces. Indeed, let f, g∈ A₋ and α∈C. The condition (2.8) is easily verified forf+gorαf. We setL:=L_ΛS+c (c >0). ThenLM(f)0 andLM(g)0 imply thatL(M(f) +M(g))0.

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

Lemma 4.4. Let ρ > 0. For u, v ∈ Cⁿ[x] such that uρ < ∞ and vρ<∞, we haveu·vρ ≤ uρvρ.

This is clear from the definition.

§5. Proof of Theorem 2.1

Proof of Theorem 2.1. We will prove the theorem in the caseR ∈ 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 AandQinstead ofA∓,ρandQ_±,ρ, respectively. Similarly, we sometimes omit the suffix of · 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) V⁰=R(x), V¹=R(x+QV⁰)−R(x),

(5.3) V^j+1=R(x+QV⁰+· · ·+QV^j)−R(x+QV⁰+· · ·+QV^j−1), j= 1,2, . . . In order to show that V^j’s are well-defined we first consider the case Λ_N = 0.

By Lemma 4.1 and the assumption, we see that V⁰ and QV⁰ are polynomi- als. Hence, by (5.2) V¹ is a polynomial. Inductively, we see that V^j’s are polynomials. We will show that V^j ∈ A⁰. For this purpose, we will prove that R(x+QV)∈ A⁰ ifV ∈ A⁰. Indeed, if we can prove this, then we have R(x+QV⁰)∈ A⁰. It follows thatV¹ =R(x+QV⁰)−R(x)∈ A⁰. Next we have V⁰+V¹ ∈ A⁰, and thus V² ∈ A⁰. By the induction we have V^j ∈ A⁰ (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)^γi, where (QV)_i is thei-th component of QV. Because QV ∈ A⁰, (QV)_k is the sum of the functionsh_µwithh_µbeing divisible byx_µwherex_kandx_µbelong to the same Jordan block. IfR_j(x) is divisible byx_k withx_k andx_j belonging to the same Jordan block, then it follows thatR_j(x+QV) is the sum of functions divisible by somex_νwithx_ν andx_jbelonging to the same Jordan block. Hence (2.8) holds.

Next we will show that (L_ΛS +c)M(R(·+QV)) 0. Let x^η be any monomial appearing in the right-hand side of (5.4). Because (L_ΛS+c)M(R) 0 by the assumption, it follows that theγ’s in (5.4) satisfy thatλ, γ−λ_j ≤ −c.

In view of the relation QV ∈ A⁰, (QV)_i can be expanded in the power series ofx^δ such thatλ, δ −λ_i≤ −c. If we expand (x_i+ (QV)_i)^γi in the right-hand side of (5.4) into the power series of x, then we see that x^η appears if some x_i in x^γ is replaced byx^δ appearing in (QV)_i a finite number of times. Ifx^γ turns into x^η by the one substitution, then we have

λ, η −λ_j =λ, γ −λ_j−λ_i+λ, δ ≤ −2c <−c.

By the similar argument, we have the estimateλ, η −λ_j≤ −cin the general case. Hence R(x+QV) satisfies (L_ΛS +c)M(R(·+QV)) 0. This proves that R(x+QV)∈ A⁰.

Next we consider the case Λ_N = 0. LetV ∈ A. BecauseQis continuous by Lemma 4.1,R(x+QV) is well-defined in some neighborhood of the origin if Vis sufficiently small. We will estimateR(·+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+c₁V_ρ)^|γ|.

IfVρ < εfor sufficiently smallεsuch thatc₁ε < ρ, then the right-hand side of (5.5) is bounded by εbecause R2ρ < ε. On the other hand, by the same argument as in the case Λ_N = 0, we can prove thatR(x+QV)∈ A. Therefore, one can defineV^j ∈ A(j= 0,1, . . .) by (5.2) and (5.3) inductively.

Next we will prove the convergence of{V^j}inA. For this purpose we will show that there exist constantsc₀≥0 andK₀≥0 independent ofj such that, forj= 0,1,2, . . .,

V^j ≤c^j₀ε^j+1, (5.6)

QV^j ≤K₀V^j. (5.7)

Clearly we haveV⁰=R< εby 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_N = 0. Letd₀be the degree ofR. By Lemma 4.1 we haveQV⁰∈ A⁰. By the definition we have

QV⁰=− _∞

0

e^−tΛV⁰(e^tΛx)dt=− _∞

0

e^−tΛNe^−tΛSV⁰(e^tΛx)dt.

(5.8)

We set

(5.9) W(t, x)≡(W₁(t, x), . . . , W_n(t, x)) :=e^−tΛSV⁰(e^tΛx).

Then the components of the first Jordan block of e^−tΛNW are given by

W₁−tW₂+t²

2W₃−t³

3!W₄+· · · (5.10)

W₂−tW₃+t²

2!W₄− · · ·

· · ·

There are finite number of similar terms corresponding to every Jordan block ofe^−tΛNW. Hence we can easily see thatQV⁰ρis 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 W₁(t,·)dt+ _∞

0 W₂(t,·)(1 +t)dt +

_∞

0 W₃(t,·)

1 +t+t² 2

dt+· · ·

≤ n j=1

_∞

0

B₀(t)W_j(t,·)dt,

where B₀(t) = _n−1

ν=0t^ν/ν!. We write V⁰ = (V₁⁰, V₂⁰, . . . , V_n⁰), and expand V_j⁰ (1≤j≤n) into the Taylor seriesV_j⁰=

γV_j,γ⁰ x^γ. Then, by (5.9) we have W_j(t, x) =e^−tλjV_j⁰(e^tΛx) =e^−tλj

γ

V_j,γ⁰ (e^tΛx)^γ (5.12)

=

γ

e^{tλ,γ−tλj}V_j,γ⁰ (e^tΛNx)^γ. Because the sum with respect toγ is finite, we have (5.13)

n j=1

_∞

0

B₀(t)W_j(t,·)dt≤

j,γ

|V_j,γ⁰ |

B₀(t)e^{tλ,γ−tλj}(e^tΛNx)^γ_ρdt.

On the other hand, we have

(5.14) (e^tΛNx)^γρ≤ρ^|γ|(e^tΛNe)^γ ≤ρ^|γ|

|γ|≤d0

(e^tΛNe)^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,γ⁰ |

B₀(t)e^−ct

|δ|≤d0

(e^tΛNe)^δdt (5.15)

≤K(d₀)

j,γ

ρ^|γ||V_j,γ⁰ |=K(d₀)V⁰_ρ,

where K(d₀) =

B₀(t)e^−ct

|δ|≤d0(e^tΛNe)^δdt. Therefore we haveQV⁰ρ ≤ K(d₀)V⁰ρ. If we setK₀= max{K(d₀), c₁}withc₁given by Lemma 4.1, then we have (5.7) forj = 0.

Next we will prove (5.6) forj= 1. It follows from (5.2) that

(5.16) V¹ ≤ QV⁰

₁

0 ∇R(·+τ QV⁰)dτ.

In order to estimate ∇R(·+τ QV⁰)we make the same argument as in (5.5).

Indeed, if we have the estimate

(5.17)

γ

|R_j,γ||γ|(2ρ)^|γ|< c₂ε, j= 1,2, . . . , n,

for some constantc₂>0 independent ofε, then we obtain (5.18) ∇R(·+τ QV⁰)< c₂ε, ∀τ,0≤τ≤1,

if K(d₀)ε < ρ. The estimate (5.17) follows from the assumptionR_2ρ < εif 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) V¹ ≤K(d₀)c₂ε²

₁

0

dτ =c₀ε², where c₀=K(d₀)c₂.

Next we will estimateQV¹. In view of Lemma 4.1 we may assume that Λ_N = 0. We write V¹ = (V₁¹, V₂¹, . . . , V_n¹) and consider the Taylor expansion V_j¹(x) =

γV_j,γ¹ x^γ. Here the sum is a finite one. We will show that for every γ such thatV_j,γ¹ = 0 we have

(5.20) λ, γ −λ_j ≤ − c

d₀−1(|γ| −1).

Noting thatV_j¹=R_j(x+QR)−R_j(x), we first considerR_j(x). BecauseR∈ A⁰, we haveλ, γ−λ_j≤ −c(j = 1,2, . . . , n) for everyx^γin the expansion ofR_j(x).

Because |γ| ≤d₀, we have

(5.21) λ, γ −λ_j ≤ −c=−cd₀−1

d₀−1 ≤ − c

d₀−1(|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 ∈ A⁰ and QR is the polynomial of degree d₀. We set