**Finitely smooth solutions of nonlinear singular partial differential equa-** **tions**

**Masafumi Yoshino**^{∗}

Graduate School of Sciences, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan Received 15 November 2003, revised 30 November 2003, accepted 2 December 2003

Published online 3 December 2003

**Key words **Fuchsian equations, totally characteristic equations, singular partial differential equations, normal
form, vector fields, Monge Amp`ere equation.

**MSC (2000) **Primary 35G20; Secondary, 37C10, 37J40

*This work is partially supported by Grant-in-Aid for Scientific Research (No. 16654028), Ministry of Education,*
*Science and Culture, Japan. This work was partly done when the author stayed in the university of Cagliari.*

*The author expresses appreciations to Prof. T. Gramchev and Department of Mathematics, Univ. of Cagliari*
*for the invitation and the supports.*

We solve a Fuchsian system of singular nonlinear partial differential equations with resonances. These equa- tions have no smooth solutions in general. We show the solvability in a class of finitely smooth functions.

Typical examples are a homology equation for a vector field and a degenerate Monge-Amp`ere equation.

Copyright line will be provided by the publisher

**1** **Introduction**

This paper is concerned with the solvability of a Fuchsian system of a singular nonlinear partial differential
equations in a bounded domainΩ*⊂***R**^{n} or in**R**^{n}. These equations naturally appear when we solve a class of
Monge-Amp `ere equations or when we linearize a singular vector field by a coordinate change. (See*§*2). As we
can see from the simple example*Lu*:= (*t*_{dt}^{d} *−*1)*u*=*t*, these equations do not have a smooth solution in general.

Indeed, if*u*(*t*) =*c*_{0}+*c*_{1}*t*+*v*(*t*),*v*=*O*(*t*^{2})is a solution, then the relations*L*(*c*_{0}+*c*_{1}*t*) =*−c*_{0}and*Lv*=*O*(*t*^{2})
imply that*u*is not smooth at*t* = 0. In fact, if we allow a singular solution, then we see that*u*=*ct*+*t*log*t*,
(*c*, constant) gives a solution. Here we take the branch of the logarithm such thatlog 1 = 0. If we restrict*t*to
the real line, then*u*gives a H¨older continuous function on the real line. Similar property holds for*Lu*= *x*_{1},
where*L*=*x*_{1}_{∂x}^{∂}

1 +*mx*_{2}_{∂x}^{∂}

2 *−*1,(*m >*1). The equation*Lu*=*x*_{1}has no smooth solution at the origin, while
*u*=*cx*_{1}+*x*_{1}log*x*_{1}, (*c*, constant) is a singular solution. It gives a H¨older continuous function on the real line
for an appropriate choice of the branch oflog*x*_{1}. We also note that this phenomenon is closely related with a
Grobman-Hartman theorem. (cf. Remark 2.9). These examples are known as a so-called totally characteristic
type partial differential equation.(cf. [3]). As to formal solutions of nonlinear first order totally characteristic type
equations we refer [3], and as to singular solutions of nonlinear singular partial differential equations we refer
[19]. We also remark a related work [11] concerning symbolic calculus on manifolds with edges.

The object of this paper is to solve this type of equations in a class of finitely smooth functions. For this purpose we employ a rapidly convergent iteration method in a class of non smooth functions, because the Fuchsian equations have a loss of regularity. We stress that the usual rapidly convergent iteration scheme is not useful in order to solve this type of equations, because one requires high regularity in the iterative scheme, while our solution does not have such smoothness in general. We introduce a partial smoothing operator which preserves the vanishing order of approximate solutions on every coordinate axis. This smoothing operator is useful in the iterative scheme because the Fuchsian partial differential operators which we study in this paper lose derivatives of the transversal direction of every coordinate axis, although they preserve the vanishing order. Concerning the loss of regularity of nonlinear equations (of multiple characteristics) we refer [5], [7] and [18].

*∗* e-mail:yoshino@math.sci.hiroshima-u.ac.jp, Phone: +81 0824 24 7350, Fax: +81 0824 24 0710

This paper is organized as follows. In*§*2 we state the main theorem and we give several consequences and
applications. In*§*3 we prepare lemmas which are necessary for the proof of the main theorem. The proof of the
main theorem is given in*§*4 by using a rapidly convergent iteration method.

**2** **Statement of results**

Let*x* = (*x*_{1}*, . . . , x*_{n}) *∈* R^{n} be the variable inR^{n} (*n* *≥* 2). For a multiinteger*α* = (*α*_{1}*, . . . , α*_{n}) *∈* Z^{n}_{+},
Z_{+}=*{*0*,*1*,*2*, . . .}*we set*|α|*=*α*_{1}+*· · ·*+*α*_{n}. We define

*∂*_{j}=*∂/∂x*_{j}*, δ*_{j} =*x*_{j}*∂*_{j}(*j*= 1*, . . . , n*)*,* *δ*^{α}=*δ*^{α}_{1}^{1}*· · ·δ*^{α}_{n}^{n}*.*
Let*m≥*1,*m≥s≥*0,*N* *≥*1be integers, and let

*p*_{j}(*δ*) =

*|α|≤m*

*a*_{αj}*δ*^{α}*,* (*a*_{αj}*∈*R*, j*= 1*, . . . , N*)

be Fuchsian partial differential operators. Let
*a*_{j}(*x, z*)*, z*= (*z*_{α})_{|α|≤s} *j*= 1*, . . . , N,*

be real-valued*C*^{∞}functions of(*x, z*)*∈*R^{n}*×*Ω, whereΩ*⊂*R^{kN},(*k*= #*{α∈*Z^{n}_{+};*|α| ≤s}*)is a neighborhood
of the origin.

We study the solvability of the system of equations for*u*= (*u*_{1}*, . . . , u*_{N})

*G*_{j}(*u*) :=*p*_{j}(*δ*)*u*_{j}+*a*_{j}(*x, δ*^{α}*u*;*|α| ≤s*) = 0*,* *j*= 1*, . . . , N.* (2.1)
Let*σ*be a nonnegative number, andΓbe a domain ofR^{n}. We define*H*_{σ} *≡H*_{σ,Γ} as the set of holomorphic
(vector) funtions*v*(*ζ*) = (*v*_{1}(*ζ*)*, . . . , v*_{N}(*ζ*))of*ζ*=*η*+*iξ∈*Γ +*i*R^{n}such that

*v*_{σ,Γ}:= sup

*η∈*Γ

R^{n}*ζ*^{σ}*|v*(*ζ*)*|dξ <∞,* (2.2)
where*ζ*= 1 +_{n}

*j*=1*|ζ**j**|*, and*|v*(*ζ*)*|*= (_{N}

*j*=1*|v**j*(*ζ*)*|*^{2})^{1/2}. The space*H*_{σ,Γ}is a Banach space with the norm
(2.2). The fundamental properties of*H*_{σ,Γ}is given in Proposition 3.1 which follows.

Let*f*(*x*)be an integrable*N*- vector function onR^{n}_{+}, R_{+} := *{t* *∈* R;*t* *≥* 0*}* and let*f*ˆ(*ζ*)be the Mellin
transform of*f*

*f*ˆ(*ζ*)*≡M*(*f*)(*ζ*) =

R^{n}_{+}*f*(*x*)*x*^{ζ−e}*dx,* *e*= (1*, . . . ,*1)*, ζ*=*η*+*iξ, η∈*Γ*, ξ∈*R^{n}*,* (2.3)
where*x*^{ζ−e} = *x*^{ζ}_{1}^{1}^{−1}*· · ·x*^{ζ}_{n}^{n}^{−1}, *ζ* = (*ζ*_{1}*, . . . , ζ*_{n}). It is easy to see that*f*ˆ(*ζ*)is analytic if the integral (2.3)
absolutely converges. The inverse Mellin transform is given by

*f*(*x*) =*M*^{−1}( ˆ*f*)(*x*) = (2*π*)^{−n}

R^{n}

*f*ˆ(*η*+*iξ*)*x*^{−η−iξ}*dξ,* (2.4)
where*x*_{j} *>*0 (*j* = 1*, . . . , n*)and*η*is so taken that the integral converges. We note that these formulas follow
from the corresponding ones of the Fourier transform by the change of variables*e*^{θ}^{j} *→x*_{j}.

We define*H**σ,*Γas the inverse Mellin transform of*H*_{σ,Γ}. We note that the Mellin transform gives the one to
one correspondence between the spaces*H**σ,*Γand*H*_{σ,Γ}. For*u∈ H**σ,*Γwe define the norm*u**σ,*Γof*u*by

*u**σ,*Γ:= *M*(*u*)_{σ,Γ}*.*

For an integer*k≥*1we denote by(*H*_{σ,Γ})^{k}the product of*k*copies of*H*_{σ,Γ}. The norm in(*H*_{σ,Γ})^{k} is defined as
the sum of the norm of each component. For simplicity, we denote the norm in(*H**σ,*Γ)^{k} by* · **σ,*Γif there is no
fear of confusion.

Let*p*_{j}(*ζ*) =

*|α|≤m**a*_{αj}(*−ζ*)^{α}be the indicial polynomial associated with*p*_{j}(*δ*), where*ζ*= (*ζ*_{1}*, . . . , ζ*_{n})is
the covariable of*x*in the sense of the Mellin transform. We assume

(*A.*1) There exists a constant*c >*0such that

*|p*_{j}(*η*+*iξ*)*| ≥c*(*|η|*+*|ξ|*)^{s}*,* *∀η∈*Γ*,∀ξ∈*R^{n}*,* *j*= 1*, . . . , N.*

We set*a*(*x, z*) = (*a*_{1}(*x, z*)*, . . . , a*_{N}(*x, z*)). Then we assume that*a*(*x, z*)*∈*(*C*^{∞}(R^{n}*×*Ω))^{N} and
(A.2) *∀α∈*Z^{n}_{+},*∀β∈*Z^{kN}_{+} ,*∃C**αβ* *>*0such that

*|*(*∂/∂z*)^{β}*δ*_{x}^{α}*a*(*x, z*)*| ≤C*_{αβ}*,* *∀*(*x, z*)*∈*R^{n}*×*Ω*.*

Then our main theorem in this paper is the following

**Theorem 2.1 ***Letσ≥mbe an integer. Suppose that*(*A.*1)*hold for some bounded domain*Γ*⊂*R^{n}*containing*
*the origin. Assume (A.2). Then there exist an integerν* =*ν*(*σ*)*≥*0*and anε*=*ε*(*σ*)*>*0*depending onσsuch*
*that, if the following conditions are satisfied*

*a*(*·,*0)_{ν,Γ} *< ε,* *∇**z**a*(*·,*0)_{ν,Γ}*< ε,*

*then Eq. (2.1) has a solutionu∈*(*H*_{σ,Γ})^{N}*.*

Next we study the local solvability. We say that*u∈*(*H**ν,*Γ)^{N} at the origin if there exists a*ψ∈C*^{∞}(R^{n})with
compact support and being identically equal to one in some neighborhood of the origin such that*ψu∈*(*H*_{ν,Γ})^{N}.
For open setsΓ_{1} *⊂*R^{n} andΓ_{2} *⊂*R^{n}the relationΓ_{1} *⊂⊂* Γ_{2} meansΓ_{1} *⊂* Γ_{2}, whereΓ_{1}is the closure ofΓ_{1}.
Then we have

**Theorem 2.2 ***Letσ≥mbe an integer. Suppose that*(*A.*1)*holds for some bounded domain*Γ*containing the*
*origin. Then there exists an integerν* *≥*0*such that, if*

*a*(*x,*0)*∈*(*H**ν,*Γ)^{N} *and* *∇**z**a*(*x,*0)*∈*(*H**ν,*Γ)^{kN} *at the origin,*

*then there exists a solutionu∈*(*H**σ,*Γ^{})^{N} *of (2.1) in some neighborhood of the origin for every*Γ^{}*⊂⊂*Γ*.*

**Remark 2.3 **a) Theorem 2.1 and Theorem 2.2 yield the solvability of (2.1) in some neighborhood of the
origin in a class of finitely smooth functions. Indeed, we can solve (2.1) in the sectors*{ε**j**x*_{j} *≥*0;*j*= 1*, . . . , n}*,
(*ε*_{j} =*±*1), after the change of variables*x*_{j} *→ε*_{j}*x*_{j},(*j* = 1*, . . . , n*), because*δ*_{j}is invariant under the change
of variables. By the assumption0 *∈* Γand the definition of*H**ν,*Γ, the solution*u*together with the derivatives
*δ*^{α}*u*,*|α| ≤s*vanishes (to a finite order) on the coordinate planes*x*_{j} = 0 (*j* = 1*, . . . , n*). (See Proposition 3.1.)
Hence, by patching the solutions in these sectors we obtain a finitely smooth solution in some neighborhood of
the origin.

b) (Bifurcation from a resonance) The uniqueness of solutions in Theorem 2.1 and Corollary 2.2 does not always hold if there is a resonance. Indeed, we consider the equation

*p*(*δ*)*u*+*λa*(*x, u*) = 0*,* *a*(*x, u*) =*O*(*|u|*^{2})*,*

where*u*is a scalar unknown function,*λ*is a real parameter, and where*p*(*δ*)is an Fuchisian partial differential
operator similar to*p*_{j}(*δ*)in (2.1). We note that*u≡*0is a trivial solution of the equation. We assume (A.1) for
some domainΓ 0. Then we shall show that the above equation has a non trivial family of solutions*u*=*u*_{λ},
*u*_{λ}=*λu*_{0}+*v*_{λ}for sufficiently small*λ*, where*u*_{0}satisfies*p*(*δ*)*u*_{0}= 0.

First we note that there exists*u*_{0}such that*p*(*δ*)*u*_{0} = 0if there is a resonance. (See also Example 2.8 which
follows.) If we set*v*=*v*_{λ}, then*v*satisfies

*p*(*δ*)*v*+*λa*(*x, λu*_{0}+*v*) = 0*.*

The conditions in Theorem 2.1 read:

*λa*(*·, λu*_{0})*ν,*Γ *< ε* and *λ∇**u**a*(*·, λu*_{0})*ν,*Γ*< ε.*

These conditions are satisfied for sufficiently small *λ*if *a*(*x, λu*_{0}) *∈ H*_{ν,Γ} and*∇*_{u}*a*(*x, λu*_{0}) *∈ H*_{ν,Γ} for all
*λ*close to 0. For example, if the local solvability is concerned, these conditions are verified if *a*(*x, λu*_{0})and

*∇**u**a*(*x, λu*_{0})vanish to some order for all sufficiently small*λ*. (We also refer Lemma 4.2 which follows.)
It follows from Theorem 2.1 or Corollary 2.2 that there exists a solution*v*for sufficiently small*λ*. Moreover,
by the constructions of an approximate sequence*w*_{k}in (4.12), we have*v*= lim_{k}*w*_{k} and

*w*_{1}=*S*_{0}*ρ*_{0}*,* *L*_{0}*ρ*_{0}=*g*_{0}=*−λa*(*x, λu*0)*, . . .*

It follows from the assumption on*a*that the vanishing order of*g*_{0}at the origin is greater than*u*_{0}. Therefore,
we see that the vanishing order of*w*_{1}at the origin is greater than that of*u*_{0}, because*L*^{−1}_{0} and*S*_{0} preserve the
vanishing order. Inductively, we can easily see that the vanishing order at the origin of the solution*v* = lim*w*_{k}
is greater than that of*u*_{0}. It follows that*u*= *λu*_{0}+*v* = 0. Therefore, we have a family of solutions of our
equation.

**Remark 2.4 **The smallness conditions in Theorem 2.1 for the nonlinear part*a*(*x,*0)and*∇**z**a*(*x,*0) of the
equation (2.1) are fulfilled if the following conditions are satisfied

*a*(*x,*0) = 0*,* *∇**z**a*_{j}(*x,*0) = 0*,* *j*= 1*, . . . , n.* (2.5)
On the other hand, the condition (A.2) in Theorem 2.1 is fulfilled if*a*(*x, z*)is independent of*x*or*a*(*x, z*)has a
compact support with respect to*x*.

**Example 2.5 **We give the example which satisfies (A.1). Let

*p*_{2}(*ζ*) :=*ζ*_{1}^{2}*−*
*n*
*j*=2

*c*_{j}*ζ*_{j}^{2}*,* *c*_{j} *>*0*.*

Let*p*_{1}(*ζ*)be a linear function of*ζ*with real coefficients. We set*p*(*ζ*) =*p*_{2}(*ζ*) +*p*_{1}(*ζ*). We assume that
*p*_{1}(*ξ*) +*η· ∇p*_{2}(*ξ*)= 0 for*∀η∈*Γ*,* and*∀ξ∈*R^{n} such that *p*_{2}(*ξ*)*≥*0*,|ξ|*= 1*.*

We want to show that there exists real number*K*such that*p*(*ζ*) +*K*satisfies (A.1) with*s*= 1. We have
*p*(*η*+*iξ*) +*K*=*K−p*_{2}(*ξ*) +*p*(*η*) +*i*(*p*_{1}(*ξ*) +*η· ∇p*_{2}(*ξ*))*.*

Because*η* moves in a bounded set it follows that if*K >* 0is sufficiently large, the zero set of the polynomial
of*ξ*,*p*(*η*+*iξ*) +*K* is contained in the set*p*_{2}(*ξ*)*≥*0,*|ξ| ≥* 1, where*p*is the real part of*p*. On the other
hand, by assumption and the homogeneity, the imaginary part*p*(*η*+*iξ*)does not vanish on the set*p*_{2}(*ξ*)*≥*0,

*|ξ| ≥*1. It follows that*p*(*η*+*iξ*) +*K*= 0for all*η* *∈*Γand*ξ*.

In order to show (A.1) with*s*= 1it is sufficient to consider*ξ*such that*|ξ| ≥N >*0for large*N*. If*ξ*is in a
conical neighborhood of*ξ*_{0}such that*p*_{2}(*ξ*_{0})= 0, we have (A.1) with*s*= 2. If otherwise, the assumption implies
that*p*_{1}(*ξ*) +*η· ∇p*2(*ξ*)= 0. Hence we have

*|p*(*η*+*iξ*)*| ≥ |p*(*η*+*iξ*)*| ≥c|ξ| ≥c*^{}(*|ξ|*+*|η|*)
for some*c >*0and*c*^{}*>*0. This proves (A.1) with*s*= 1.

**Example 2.6 **We write*x*_{1}=*x*,*x*_{2}=*y*, and we consider the Monge-Amp `ere operator
*M*(*u*) :=*u*_{xx}*u*_{yy}*−u*^{2}_{xy}+*kxyu*_{xy}+*cu,* 4*< k <*12*, c∈*C*.*

Let*u*_{0}=*x*^{2}*y*^{2}and set*f*_{0}=*M*(*u*_{0}) = (4*k−*12 +*c*)*x*^{2}*y*^{2}. We want to solve the equation
*M*(*u*_{0}+*v*) =*f*_{0}(*x, y*) +*g*(*x, y*)*,* inR^{2}*,*

where*g*(*x, y*)is a given function. If we define

*Q*= 2*x*^{2}*∂*_{x}^{2}+ 2*y*^{2}*∂*_{y}^{2}+ (*k−*8)*xy∂*_{x}*∂*_{y}+*c,* *M*˜(*u*) =*M*(*u*)*−kxyu*_{xy}*−cu,*

then the equation can be written in the form

*Qv*+ ˜*M*(*v*) =*g.*

In order to write the equation in the form (2.1) we introduce a new unknown function*w*by*v*(*x, y*) =*x*^{2}*y*^{2}*w*(*x, y*).

By simple computations we have
*x*^{−2}*y*^{−2}*M*˜(*x*^{2}*y*^{2}*w*)

= (*x*^{2}*w*_{xx}+ 4*xw*_{x}+ 2*w*)(*y*^{2}*w*_{yy}+ 4*yw*_{y}+ 2*w*)*−*(*xyw*_{xy}+ 2*xw*_{x}+ 2*yw*_{y}+ 4*w*)^{2}*.*
*x*^{−2}*y*^{−2}*Q*(*x*^{2}*y*^{2}*w*)

= 2(*δ*^{2}_{x}+ 4*δ*_{x})*w*+ 2(*δ*_{y}^{2}+ 4*δ*_{y})*w*+ (*k−*8)(*δ*_{x}*δ*_{y}+ 2*δ*_{x}+ 2*δ*_{y})*w*+ (4*k−*24 +*c*)*w,*

where*δ*_{x} =*x∂/∂x*and*δ*_{y} =*y∂/∂y*. This proves that our equation can be written in the form (2.1). We note
that the condition (A.2) is fulfilled. (cf. Remark 2.4).

The indicial polynomial is given by

*p*(*ζ*) := 2(*ζ*_{1}^{2}*−*4*ζ*_{1}) + 2(*ζ*_{2}^{2}*−*4*ζ*_{2}) + (*k−*8)(*ζ*_{1}*ζ*_{2}*−*2*ζ*_{1}*−*2*ζ*_{2}) +*c*+ 4*k−*24*.*

We will show (A.1) with*s*= 2for some bounded domainΓ*⊂*R^{n}*\ {p*(*η*) = 0*}*containing the origin if*c*=*iK*,
*K >*0is sufficiently large . We note that*p*(*ξ*)is elliptic by the condition4*< k <*12. It follows that there exist
*ξ*_{0}*>*0and*α >*0independent of*K*and*η*such that*p*(*η*+*iξ*)*≥α|ξ|*^{2}if*|ξ|> ξ*_{0}and*η∈*Γ. If*|ξ| ≤ξ*_{0}, then
*p*(*η*+*iξ*)does not vanish if*K*is sufficiently large. Therefore we have (A.1) with*s*= 2.

Next we apply our argument to the normal form theory of a singular hyperbolic vector field*χ*=_{n}

*j*=1*X*_{j}(*x*)*∂*_{j},

*∂*_{j} =*∂/∂x*_{j}onR^{n}. We say that*χ*is singular if*X*_{j}(0) = 0 (*j* = 1*, . . . , n*). We set*X* = (*X*_{1}*, . . . , X*_{n}). For the
sake of simplicity, we assume

*X*(*x*) =*x*Λ +*R*(*x*)*,* *R*(*x*) = (*R*_{1}(*x*)*, . . . , R*_{n}(*x*))*,* (2.6)
for a real-valued*C*^{∞}function*R*_{j}(*x*)such that*R*_{j}(0) = 0,*∇R*_{j}(0) = 0, and a diagonal matrixΛ =

diag(*λ*_{1}*,· · ·, λ*_{n}),*λ*_{j} *∈* R. We want to find a change of variables*y* *→x* =*y*+*v*(*y*)which linearizes*χ*. It
follows that*v*satisfies the so-called homology equation

*X*(*y*+*v*(*y*))(1 +*∇v*)^{−1}=*y*Λ*,*

or equivalently,

*Lv*=*R*(*y*+*v*(*y*))*,* *Lv*:=

*n*
*j*=1

*λ*_{j}*δ*_{j}*v−v*Λ*.* (2.7)

We define*p*(*ζ*) =*−*_{n}

*j*=1*ζ*_{j}*λ*_{j}*I−*Λ, where*I*is an identity matrix. Then we have

**Theorem 2.7 ***Suppose that (A.1) is satisfied fors* = 0*and some bounded domain*Γ*containing the origin.*

*Assume (2.6). Letσ≥*1*be an integer. Then there existsν≥*0*such that, if the following conditions are satisfied*
*R∈*(*H**ν,*Γ)^{n}*,* *∇R**j**∈*(*H**ν,*Γ)^{n}^{2} *at the origin* (*j*= 1*, . . . , n*)*,*

*then Eq. (2.7) has a solutionv∈*(*H**σ,*Γ^{})^{n}*for every*Γ^{}*⊂⊂*Γ*.*

**Example 2.8 **We give examples which satisfy (A.1). Suppose that *λ*_{1}*· · ·λ*_{n} = 0. By definition the*k*-th
component of*p*(*ζ*) (*ζ* = *η*+*iξ*)is given by*−*_{n}

*j*=1*η*_{j}*λ*_{j}*−λ*_{k}. Hence the set of*η* such that*p*(*ζ*) = 0
consists of*n*hyperplanes,

*j**η*_{j}*λ*_{j}+*λ*_{k}= 0not passing through the origin. Therefore we have (A.1) with*s*= 0
for some open setΓcontaining the origin. The followings are typical cases which satisfy (A.1).

(*i*)Poincar´e case; i.e.,*λ*_{j}*>*0 (*j*= 1*, . . . , n*).

(*ii*)Nonresonant Siegel case; namely, some*λ*_{j}are positive and others are negative, and*p*(*ζ*) = 0 (*ζ∈*Z^{n}_{+}*,|ζ| ≥*
2)has no solution.

(*iii*)Infinite resonances case; that is,*p*(*ζ*) = 0 (*ζ∈*Z^{n}_{+})has an infinitely many solutions.

The third case contains a volume preserving vector fields, namely_{n}

*j*=1*λ*_{j} = 0. In the case(*i*)the set
*{η* *∈ −*R^{n}_{+};*p*(*η*+*iξ*) = 0 for some*ξ∈*R^{n}*}*

is a compact set not containing the origin. Hence we can takeΓin (A.1) as a bounded domain inR^{n}_{+}*\ {p*(*ζ*) =
0*}*. In the case(*ii*)the intersection of the hyperplanes

*j**η*_{j}*λ*_{j} +*λ*_{k} = 0and*−*R^{n}_{+} is noncompact. Hence
the setΓin (A.1) may be a smaller set. In the case(*iii*)there is an additional restriction toΓdue to an infinite
resonances apart from the ones caused by a Siegel condition. We note that the larger the setΓis, the more regular
the solution is.

**Remark 2.9 **By Remark 2.3 and Theorem 2.7 we can construct a finitely smooth coordinate change which
linearizes*χ*even in the case of resonances. It is natural to ask wether there exists a*C*^{∞}coordinate change which
linearizes*χ*. The answer to this question is not affirmative. Indeed, if the vector field has a resonance,*L*has a
(infinite) kernel. It follows that if (2.7) has a*C*^{∞} solution*v*, then the Taylor expansion of*v*at the origin gives
a formal power series solution of (2.7). Hence the Taylor expansion of*R* satisfies a compatibility condition.

Because we do not assume any compatibility condition a priori, the solution is not smooth in general. We
stress that the regularity of the solution is related with the property of a resonance as we note in the preceeding
example. If we assume the weaker condition*λ*_{1}*· · ·λ*_{n} = 0, the solution is continuous. We remark that this
fact was essentially noted as a Grobman-Hartman theorem for a vector field, which asserts the existence of a
continuous solution of a homology equation (cf. [1], p.127 and p191).

Theorem 2.7 can be extended to a commuting system of hyperbolic singular vector fields onR^{n},
*χ*=*{χ*^{µ};*µ*= 1*, . . . , d},* [*χ*^{µ}*, χ*^{ν}] = 0 for all*ν*and*µ.*

We write *χ*^{µ} = _{n}

*j*=1*X*_{j}^{µ}(*x*)*∂*_{j} and set*X*^{µ} = (*X*_{1}^{µ}*, . . . , X*_{n}^{µ}). For the sake of simplicity we assume that
*X*^{µ}(*x*) =*x*Λ^{µ}+*R*^{µ}(*x*)for some real-valued*C*^{∞}vector function*R*^{µ}such that

*R*^{µ}(0) = 0*,* *∇R*^{µ}(0) = 0*,*
and diagonal matrices

Λ^{µ}=diag(*λ*^{µ}_{1}*,· · ·, λ*^{µ}_{n})*,* *λ*^{µ}_{j} *∈*R*, µ*= 1*, . . . , d.*

We are interested in the simultaneous linearization of*χ*by the change of variables*y* *→* *x*= *y*+*v*(*y*). It
follows that*v*satisfies an overdetermined system of equations

*L*^{µ}*v*=*R*^{µ}(*x*+*v*)*,*

where*L*^{µ} is similarly given by (2.7). Let*C* be a positive cone generated by the vectors(*λ*^{1}_{j}*, . . . , λ*^{d}_{j}) *∈* R^{d},
(*j*= 1*, . . . , n*), namely

*C*:=*{*
*n*
*j*=1

*t*_{j}(*λ*^{1}_{j}*, . . . , λ*^{d}_{j})*∈*R^{d};*t*_{j} *≥*0*,*(*j*= 1*, . . . , n*)*, t*^{2}_{1}+*· · ·*+*t*^{2}_{n}= 0*}.*

We say that*χ* satisfies a simultaneous Poincar´e condition if the cone *C* does not contain the origin. In case
*d*= 1, this condition is equivalent to that the quantity*t*_{1}*λ*^{1}_{1}+*· · ·*+*t*_{n}*λ*^{1}_{n} does not vanish for*t*_{j} *≥*0such that
*t*^{2}_{1}+*· · ·*+*t*^{2}_{n}= 0. The last condition is equivalent to say that*λ*^{1}_{1}*>*0*, . . . , λ*^{1}_{n} *>*0. This is a well-known Poincar´e
condition for a single vector field. We have

**Theorem 2.10 ***Letσ* *≥*1*. Suppose that the simultaneous Poincar´e condition is satisfied. Then there exists*
*ν* *≥*0*such that, if*

*R*^{µ}*∈*(*H**ν,*Γ)^{n} *and* *∇R*^{µ}*∈*(*H**ν,*Γ)^{n}^{2} *at the origin forµ*= 1*, . . . , d,*

*thenχis simultaneously linearized in some neighborhood of the origin by the change of the variablesy* *→x*=
*y*+*v*(*y*)*, withv* *∈*(*H**σ,*Γ^{})^{n}*,∀*Γ^{}*⊂⊂*Γ*.*

**3** **Some lemmas**

In this section we will prepare lemmas which are necessary in the calculus of a class of pseudo-differential operators of totally characteristic type in a Mellin’s sense. We cite [11] concerning symbolic calculus of operators on manifolds with edges.

LetΓbe an open set inR^{n}. First we study fundamental properties of*H*_{s,Γ}(*s∈*R_{+})defined in*§*1.

**Proposition 3.1** (1)*Lets* *≥* 0*be an integer and letu*ˆ *∈* *H*_{s,Γ}*. Then the inverse Mellin transformu*(*x*) =
*M*^{−1}(ˆ*u*)(*x*)*ofu*ˆ*is a bounded continuous function on* R^{n}_{+}*such that for everyα,|α| ≤sandη∈*Γ*, the function*
*x*^{η}*δ*^{α}*u*(*x*)*is continuous and satisfies*

*x*^{η}*δ*^{α}*u*(*x*) *→* 0 *as* *x*_{j} *→*0*,* *j*= 1*, . . . , n,* (3.1)

*x*^{η}*δ*^{α}*u*(*x*) *→* 0 *as* *x*_{j} *→*+*∞,* *j*= 1*, . . . , n.* (3.2)
*Moreover, for every*Γ^{}*⊂⊂*Γ*there existsc >*0*independent ofu*ˆ*such that*

sup

*x∈*R^{n}_{+}*,|α|≤s,η∈*Γ^{}

*|x*^{η}*δ*^{α}*u*(*x*)*| ≤cu*ˆ_{s,Γ}*,* *∀u*ˆ*∈H*_{s,Γ}*.* (3.3)

(2)*Lets≥*0*be an integer and letu*(*x*)*be any bounded continuous function on* R^{n}_{+}*satisfying (3.1) and (3.2)*
*for everyη* *∈*Γ*. Then the Mellin transformu*(*ζ*) =ˆ *M*(*u*)(*ζ*)*ofuexists andu*(*ζ*)ˆ *is holomorphic in*Γ +*i*R^{n}*.*
*Moreover, for every*Γ^{}*⊂⊂*Γ^{}*⊂⊂*Γ*there existC >*0*such that*

*ζ*^{s}*|u*(*ζ*)ˆ *| ≤C* sup

*x∈*R^{n}_{+}*,|α|≤s,η∈*Γ^{}*|x*^{η}*δ*^{α}*u*(*x*)*|,* *∀ζ,ζ∈*Γ^{} (3.4)
*whereζ*= 1 +_{n}

*j*=1*|ζ**j**|.*

(3) *H*_{s,Γ}*is a Banach space with the norm (2.2).*

*Proof. *We will prove (1). The inverse Mellin transform of*u*ˆexists because*u*ˆ*∈H*_{s,Γ}. Moreover we have
*x*^{η}*δ*^{α}*u*(*x*) = (2*πi*)^{−n}

R^{n}

(*−ζ*)^{α}*u*(*ζ*)*x*ˆ ^{η−ζ}*dξ,* *η* *∈*Γ*,ζ∈*Γ*.* (3.5)
We take*η*and*ζ*in (3.5) such that*η*_{j}*− ζ*_{j} *>*0if*x*_{j} *<*1,*η*_{j}*− ζ*_{j} *<*0if*x*_{j} *≥*1. We easily see that (3.1)
and (3.2) hold. The estimate (3.3) follows from (3.5) because*|x*^{η−ζ}*|*is bounded by some constant.

We prove (2). The conditions (3.1) and (3.2) with*α*= 0imply that the Mellin transform*M*(*u*)(*ζ*)exists and
it is holomorphic inΓ +*i*R^{n}. In order to show (3.4), we first note that the right-hand side of (3.4) is finite by
(3.1) and (3.2). It follows from (3.1) and (3.2) that, for*|α| ≤s*

*ζ*^{α}*u*(*ζ*) =ˆ

*u*(*x*)*ζ*^{α}*x*^{ζ−e}*dx*=

*u*(*x*)(*∂*_{x}*·x*)^{α}*x*^{ζ−e}*dx*=

R^{n}_{+}*δ*^{α}*u*(*x*)*x*^{ζ−e}*dx.* (3.6)
Let*τ*_{j}(*j*= 1*, . . . , n*)be such that*τ*_{j} = 1or*τ*_{j}=*−*1and define*τ* = (*τ*_{1}*, . . . , τ*_{n}). We define*S*_{τ}by

*S*_{τ} =*{x*= (*x*_{1}*, . . . , x*_{n})*∈*R^{n}_{+}; 0*≤x*^{τ}_{j}^{j} *≤*1*}.*

By (3.6), there exists*C*^{} *>*0independent of*ζ*such that, if*ζ∈*Γ^{}
*ζ*^{s}*|u*(*ζ*)ˆ *| ≤C*^{} sup

*|α|≤s*

R^{n}_{+}*x*^{ζ−e}*δ*^{α}*u*(*x*)*dx*

*≤C*^{} sup

*|α|≤s*

*τ*

*S**τ*

*x*^{ζ−e}*δ*^{α}*u*(*x*)*dx*

*.* (3.7)

By assumption, for each*S*_{τ}we take an*η*=*η*(*τ*) = (*η*_{1}*, . . . , η*_{n})and a small*ε*_{1}*>*0such that*ζ*_{j}*−η**j**> ε*_{1}*>*0
if*τ*_{j} = 1, and*ζ*_{j}*−η*_{j} *≤ −ε*1if*τ*_{j} =*−*1. For a givenΓ^{},Γ^{} *⊂⊂*Γ^{} *⊂⊂*Γwe can choose*ε*_{1}so small that
*η∈*Γ^{}.

Therefore, there exists*C*^{}*>*0independent of*ζ*such that

*S**τ*

*x*^{ζ}^{−e}*δ*^{α}*u*(*x*)*dx*
=

*S**τ*

*x*^{ζ−e−η}*x*^{η}*δ*^{α}*u*(*x*)*dx*

*≤* sup

*x∈*R^{n}_{+}*|x*^{η}*δ*^{α}*u*(*x*)*|*

*S**τ*

*x*^{}^{ζ−e−η}*dx*

*≤C*^{} sup

*x∈*R^{n}_{+}*|x*^{η}*δ*^{α}*u*(*x*)*|.* (3.8)

Hence there exists*C >*0such that
*ζ*^{s}*|u*(*ζ*)ˆ *| ≤C* sup

*x∈*R^{n}_{+}*,η∈*Γ^{}*,|α|≤s**|x*^{η}*δ*^{α}*u*(*x*)*|.*
This proves (3.4).

We will prove (3). In order to show that*H*_{s,Γ} is complete, suppose that *w*ˆ_{n}*−w*ˆ_{m s,Γ} *→* 0 (*m, n* *→ ∞*).

It follows from (3.3) and (3.4) that *{w*ˆ_{n}(*ζ*)*}* converges compactly uniformly in*ζ* *∈* Γ to a function*w*(*ζ*)
holomorphic in*ζ* *∈* Γ +*i*R^{n}. Let*η* *∈* Γbe arbitrarily taken and fixed. By assumption, for every*ε >* 0there
exists*N* *≥*1such that

*ζ*^{s}*|w*ˆ_{n}(*ζ*)*−w*ˆ_{m}(*ζ*)*|dξ < ε,* *∀n, m≥N.*

It follows that, for any compact set*K⊂*R^{n}we have

*K*

*ζ*^{s}*|w*ˆ_{n}(*ζ*)*−w*ˆ_{m}(*ζ*)*|dξ < ε,* *∀n, m≥N.*

We let*m* *→ ∞*. Then we have

*K**ζ*^{s}*|w*ˆ_{n}(*ζ*)*−w*(*ζ*)ˆ *|dξ* *≤* *ε*for all*n* *≥* *N*. Letting *K* *↑* R^{n} we obtain

R^{n}*ζ*^{s}*|w*ˆ_{n}(*ζ*)*−w*(*ζ*)ˆ *|dξ* *≤* *ε*for all*n* *≥* *N*. By taking the supremum with respect to*η* *∈* Γ, we see that
*w*ˆ_{n}*−w*ˆ*∈H*_{s,Γ}and*{w*ˆ_{n}*}*converges to*w*ˆin*H*_{s,Γ}.*2*

Now we define a smoothing operator in*H*_{s,Γ}. Let*φ* *∈C*^{∞}(R^{n}),0 *≤* *φ* *≤*1 be a smooth function with a
compact support such that*φ≡*1in some neighborhood of the origin*x*= 0and

R^{n}*φ*(*σ*)*dσ* = 1. Let*N* *≥*1,
*≥*1be integers and let*τ*be an odd integer,2*τ≥*. We set*ψ*_{N}(*ζ*) := exp(*N*^{−2τ}_{n}

*j*=1*ζ*_{j}^{2τ})and define
*χ*^{}_{N}(*ζ*) :=

R^{n}

*φ*(*σ*)

*ψ*_{N}(*ζ*)

*e*^{−σζ/N}*−*
*ν*=1

*−σζ*
*N*

*ν* 1
*ν*!

+ (1*−ψ*_{N}(*ζ*))*e*^{−σζ/N}

*dσ.* (3.9)

The function*χ*^{}_{N}(*ζ*)is an entire function of*ζ*inC^{n}such that*χ*^{}_{N}(*ζ*) =*χ*^{}_{N}(¯*ζ*). We define a smoothing operator
*S*_{N} by

*S*_{N}*v*:=*M*^{−1}(*χ*^{}_{N+1}(*ζ*)ˆ*v*(*ζ*))*,* *v∈ H*_{s,Γ} (3.10)

where*v*(*ζ*)ˆ is the Mellin transform of*v*and*M*^{−1}denotes the inverse Mellin transform. Then we have
**Proposition 3.2 ***Let*Γ*be a bounded domain. ThenS*_{N} *has the following properties.*

(1) *For every*0*≤s≤rsuch thatr−sis an integer, there existsC*_{r}*>*0*such that*
*S**N**v**r,*Γ*≤C*_{r}(*N*+ 1)^{r−s}*v**s,*Γ*,* *v∈ H**s,*Γ*.*

(2) *For every*0*≤s≤rsuch thatr−s≤is an integer, there existsC*_{r}*>*0*such that*
(*I−S*_{N})*v**s,*Γ*≤C*_{r}(*N*+ 1)^{s−r}*v**r,*Γ*.*

(3) *S*_{N} *maps a real-valued function to a real-valued function.*

*Proof. Proof of (1)*. In view of the definition of the norm*S*_{N}*v*_{r,Γ}we consider

*ζ*^{r}*|χ*^{}_{N}_{+1}(*ζ*)ˆ*v*(*ζ*)*|dξ,* *ζ*=*η*+*iξ, η∈*Γ*.* (3.11)

Writing*ζ*^{r}=*ζ*^{r−s}*ζ*^{s}and recalling that*r−s*is a nonnegative integer we have that*ζ*^{r−s}= (1+

*|ζ*_{j}*|*)^{r−s}
is a polynomial of*|ζ**j**|*. Hence we will estimate*|ζ*^{α}*χ*^{}_{N}_{+1}(*ζ*)*|*(*|α| ≤r−s*). In view of (3.9) we consider

*φ*(*σ*)(1*−ψ*_{N+1}(*ζ*))*ζ*^{α}*e*^{−σζ/(N}^{+1)}*dσ*

=

*φ*(*σ*)(1*−ψ*_{N+1}(*ζ*))(*−*(*N* + 1)*∂*_{σ})^{α}*e*^{−σζ/(N+1)}*dσ*

= (*N*+ 1)^{|α|}

*∂*_{σ}^{α}*φ*(*σ*)(1*−ψ*_{N}_{+1}(*ζ*))*e*^{−σζ/(N}^{+1)}*dσ.* (3.12)

In order to estimate the right-hand side, we note

*ψ*_{N+1}(*ζ*) = exp

⎛

⎝ 1
(*N*+ 1)^{2τ}

*n*
*j*=1

(*η*_{j}^{2}+ 2*iη*_{j}*ξ*_{j}*−ξ*^{2}_{j})^{τ}

⎞

⎠*.*

Because*τ*is an odd integer,*ψ*_{N}_{+1}(*η*+*iξ*)tends to zero for*N* = 1*,*2*, . . .* when*ξ*tends to infinity for a bounded
*η*. Similarly,*e*^{−σζ/(N+1)}is bounded for*N* = 1*,*2*, . . .* when*ξ* *→ ∞*and*η*is bounded. Hence the term (3.12)
can be estimated by*C*_{r}(*N*+ 1)^{|α|}*≤C*_{r}(*N*+ 1)^{r−s}for some constant*C*_{r}*>*0.

We consider the term

*I*:=

*φ*(*σ*)*ψ*_{N}_{+1}(*ζ*)

*e*^{−σζ/(N}^{+1)}*−*
*ν*=1

*−* *σζ*
*N*+ 1

*ν* 1
*ν*!

*ζ*^{α}*dσ.*

By setting*t*= (*t*_{1}*, . . . , t*_{n}) =*ζ/*(*N*+ 1)we have

*I*= (*N*+ 1)^{|α|}

*φ*(*σ*)*ψ*_{N}_{+1}(*t*(*N*+ 1))(*e*^{−tσ}*−*
*ν*=1

(*−σt*)^{ν}(*ν*!)^{−1})*t*^{α}*dσ.*

Because*ψ*_{N}_{+1}(*t*(*N*+ 1))is exponentially decreasing to zero when*t* *→ ∞*for*N* = 1*,*2*, . . .*, the integrand
is uniformly bounded for*ξ* *∈* R^{n} and*N* = 0*,*1*,*2*, . . .*. Therefore we see that*|ζ*^{α}*χ*^{}_{N}_{+1}(*ζ*)*|*(*|α| ≤* *r−s*)is
bounded by*C*_{r}^{}(*N*+ 1)^{r−s}for some constant*C*_{r}^{} *>*0which is uniform in*η∈*Γ,*N* = 0*,*1*,*2*, . . .* and*ξ∈*R^{n},

*|ξ| → ∞*. It follows that*|ζ*^{r−s}*χ*^{}_{N}_{+1}(*ζ*)*|*is bounded by*C*_{r}(*N*+ 1)^{r−s}for some constant*C*_{r} *>*0 which is
uniform in*η* *∈*Γ,*ξ∈*R^{n}and*N* = 0*,*1*,*2*, . . .* By (3.11) we obtain (1).

*Proof of (2)*. By (3.10) we have

(*I−S*_{N})*v**s,*Γ= (*I−χ*^{}_{N+1})ˆ*v*_{s,Γ}*.*

For the sake of simplicity we set*s−r*=*a≤*0. Recalling that

*φ*(*σ*)*dσ*= 1and*r−s≤*we have

*χ*^{}_{N}_{+1}(*ζ*)*−*1 =

*φ*(*σ*)*ψ*_{N+1}(*ζ*)

*e*^{−σζ/(N+1)}*−*
*ν*=1

*−* *σζ*
*N*+ 1

*ν* 1
*ν*!

*dσ* (3.13)

+

*φ*(*σ*)(1*−ψ*_{N+1}(*ζ*))*e*^{−σζ/(N}^{+1)}*dσ−*

*φ*(*σ*)*dσ*

=

*φ*(*σ*)*ψ*_{N+1}(*ζ*)

*e*^{−σζ/(N+1)}*−*^{}

*ν*=0

*−* *σζ*
*N*+ 1

*ν* 1
*ν*!

*dσ*
+

*φ*(*σ*)(1*−ψ*_{N+1}(*ζ*))*e*^{−σζ/(N}^{+1)}*dσ−*

*φ*(*σ*)*dσ*+

*φ*(*σ*)*ψ*_{N+1}(*ζ*)*dσ*

=

*φ*(*σ*)*ψ*_{N+1}(*ζ*)

*e*^{−σζ/(N+1)}*−*
*ν*=0

*−* *σζ*
*N*+ 1

*ν* 1
*ν*!

*dσ*
+

*φ*(*σ*)(1*−ψ*_{N+1}(*ζ*))(*e*^{−σζ/(N}^{+1)}*−*1)*dσ≡I*_{1}+*I*_{2}*.*
By the definition of the norm we consider

(*χ*^{}_{N}_{+1}(*ζ*)*−*1)*ζ*^{a}=*ζ*^{a}*I*_{1}+*ζ*^{a}*I*_{2}*.*

As to the term*ζ*^{a}*I*_{1}, we set*ζ*=*t*(*N*+ 1). Then the integrand is equal to
*φ*(*σ*)*N t*+*t*^{a}*ψ*_{N+1}(*tN* +*t*)(*e*^{−tσ}*−*

*ν*=0

(*−σt*)^{ν}(*ν*!)^{−1})*.*

We note that

*N t*+*t*= 1 + (*N*+ 1)

*j*

*|t*_{j}*|.*
If

*j**|t*_{j}*| ≥ε >*0for some*ε*, we have*N t*+*t ≥*(*N*+ 1)*ε*. Hence it follows that*N t*+*t*^{a}*≤*(*N*+ 1)^{a}*ε*^{a}.
Because*ψ*_{N}_{+1}(*tN* +*t*)is an exponentially decreasing function of(*t*_{j})^{2τ} when*t* *→ ∞*the integrand is
bounded by*C*(*N*+ 1)^{a}for some*C >*0independent of*t*.

Next we consider the case

*j**|t*_{j}*|< ε*. Because*N t*+*t ≥*(*N*+ 1)

*j**|t*_{j}*|*we have
*N t*+*t*^{a} *≤*(*N*+ 1)^{a}(

*j*

*|t**j**|*)^{a}*.*

Hence we have

(*|t*1*|*+*· · ·*+*|t**n**|*)^{a}(*e*^{−tσ}*−*
*ν*=0

(*−σt*)^{ν}(*ν*!)^{−1}) = (*|t*1*|*+*· · ·*+*|t**n**|*)^{a} ^{∞}

*ν*=+1

(*−σt*)^{ν}(*ν*!)^{−1}*.* (3.14)

Noting that

*|σt| ≤*(*|t*_{1}*|*+*· · ·*+*|t**n**|*)(*|σ*_{1}*|*+*· · ·*+*|σ**n**|*)

and*−a* *≤* , the right-hand side of (3.14) is bounded by some constant independent of*t*. Hence*ζ*^{a}*I*_{1} is
estimated by*C*(*N*+ 1)^{a}for some*C >*0independent of*ζ*.

We will estimate*ζ*^{a}*I*_{2}. By setting*ζ*=*t*(*N*+ 1)we consider the term
*J* *≡*(1*−*exp(

*t*^{2τ}_{j} ))(*e*^{−tσ}*−*1)*.*