Gevrey estimates for certain moment partial differential equations
Sławomir Michalik Maria Suwińska

  
  [2010]35C10, 35G10
Gevrey estimates for certain moment pdes]Gevrey estimates for certain moment partial differential equations
  
  Faculty of Mathematics and Natural Sciences,
College of Science

Cardinal Stefan Wyszyński University

Wóycickiego 1/3,
01-938 Warszawa, Poland

ORCiD: 0000-0003-4045-9548
s.michalik@uksw.edu.pl
<http://www.impan.pl/ slawek>

Faculty of Mathematics and Natural Sciences, College of Science
 Cardinal Stefan Wyszyński University

Wóycickiego 1/3, 01-938 Warszawa, Poland
m.suwinska@op.pl

 We consider the Cauchy problem for  inhomogeneous 
 linear moment differential equations with holomorphic
 time dependent coefficients. Using such tools as the
 formal norms, theory of majorants and the properties of
 the Newton polygon, we obtain the Gevrey estimate for
 the formal solution of the equation.


[
    Maria Suwińska
    December 30, 2023
=====================





§ INTRODUCTION


The concept of m-moment differentiation ∂_m,t is a
generalization of, among other differential operators, standard differentiation.
More precisely, if we consider a formal power series û(t)=∑_n=0^∞u_n/m(n)t^n,
where m(n) is a moment function (see: Definition <ref>)
then 

    ∂_m,tû(t)=∂_m,t∑_n=0^∞u_n/m(n)t^n=∑_n=0^∞u_n+1/m(n)t^n.


Moment partial differential equations emerged in <cit.> by W.
Balser and M. Yoshino. Later S. Michalik analysed properties of analytic
solutions of linear moment PDEs with constant coefficients in <cit.>
and <cit.>.

In this paper we consider the initial value problem for a linear moment
partial differential equation in the variables t∈ and z=(z_1,…,z_N)∈^N
of the form 

    {∂_m_0,t^Mu(t,z)+∑_(j,α)∈Λa_j,α(t)∂_m_0,t^j∂_m,z^αu(t,z)     =f(t,z)
    ∂_m_0,t^ju(0,z)     =ϕ_j(z) for 0≤ j<M
    .,

where ∂_m_0,t and ∂_m,z^α:=∂_m_1,z_1^α_1∂_m_2,z_2^α_2…∂_m_N,z_N^α_N
denote moment differential operators and the coefficients a_j,α(t)
do not depend on the variable z.

Our main goal is to find the Gevrey estimate of the formal solution
û(t,z)=∑_n=0^∞u_n(z)t^n of (<ref>)
under a certain set of conditions (see Section <ref>
for more details). To achieve that we generalise some of the results presented
by H. Tahara and H. Yamazawa in <cit.>. In particular, we focus on Theorem 5.1 from the aforementioned paper concerning the Gevrey estimate for solutions of linear PDEs with coefficients depending only on the variable t.
Similar result for the Gevrey order of the formal solutions of the Cauchy problem to linear partial differential equations with variable coefficients was given by A. Yonemura
<cit.>. His result was generalised in many directions.
The extension to different type of Cauchy-Goursat problems for linear equations were given by M. Miyake <cit.>, M. Miyake and Y. Hashimoto <cit.>, M. Miyake and M. Yoshino <cit.>. Similar problems for nonlinear partial differential equations were studied by such authors as
S. Ōuchi <cit.>, R. Gérard and H. Tahara <cit.> and A. Shirai <cit.>.
Throughout this paper the following notation will be used.

By D^N_r we denote the open polydisc in ^N with radius r>0
and a center at the origin, i.e. 

    D^N_r={(z_1,…,z_N)∈^N |z_j|<r for j=1,…,N}.

If N=1 we denote it shortly by D_r.

For any d∈ and ϵ>0 a set 

    S_d(ϵ)={z∈ |z|>0, d-ϵ/2<(z)<d+ϵ/2}

will be called a sector in a direction d with an opening ϵ
on .

If a function f is holomorphic on a set G⊂^N then
we will write that f∈(G).
More generally, if  denotes a complex Banach space with a norm ·_, then
by (G,) we shall denote the set of all -valued  
holomorphic functions on a set G⊆^N.
For more information about functions with values in Banach spaces we refer the reader to <cit.>. 
In the paper, as a Banach space  we will
take mainly the space of complex numbers  (we abbreviate (G,) to (G))
or the space of functions _r:=(D_r^N)∩ C(D_r^N) equipped with the norm
φ__r:=sup_z∈ D_r^N|φ(z)|.

For a function f(t)∈(D_r) we
denote by (f) the order of zero of the function f(t) at t=0.

We will denote a space of formal power series û(t)=∑_n=0^∞u_nt^n
with coefficients from any Banach space 𝔼 by 𝔼[[t]].

Let ∑_α∈_0^Na_αx^α and
∑_α∈_0^Nb_αx^α
be two formal power series. We write that ∑_α∈_0^Na_αx^α≪∑_α∈_0^Nb_αx^α,
when |a_α|≤ b_α for all α∈_0^N. Then we also call
∑_α∈_0^Nb_αx^α a majorant of ∑_α∈_0^Na_αx^α.



§ MOMENT FUNCTIONS AND MOMENT DIFFERENTIAL OPERATORS


In this section the basic theory of moment functions introduced by W. Balser <cit.> and moment differential operators defined by W. Balser and
M. Yoshino <cit.> will be recalled.

[compare <cit.>] A pair of functions
e_m and E_m is said to be kernel functions of order
k>1/2 if:  


  * e_m∈(S_0(π/k)), e_m(z)/z is integrable at
the origin, e_m(x)∈_+ for x∈_+ and for any ϵ>0
there exist constants A,B>0 such that |e_m(z)|≤ Ae^-(|z|/B)^k
for z∈ S_0(π/k-ε), 


  * E_m∈() and there exist A,B>0 such that |E_m(z)|≤ Ae^B|z|^k
for z∈ and E_m(1/z)/z is integrable at the origin in
S_π(2π-π/k). 


  * The connection between the above functions e_m and E_m
is given by the corresponding moment function m of order
1/k defined as 

    m(u):=∫_0^∞x^u-1e_m(x)dx

for all u such that u≥0, and the kernel function E_m
has the power series expansion 

    E_m(z)=∑_n=0^∞z^n/m(n)    z∈.



  * Additionally we shall assume that the normalisation property
holds for the corresponding moment function m, i.e. that m(0)=1.

 

Please note that for k≤1/2 the sector S_π(2π-π/k)
is not defined. For that case the kernel functions as well as their
corresponding moment function have to be defined separately.

[see <cit.>] For any k>0 we may define a kernel function
e_m of order k if there is n∈ such that nk>1/2
and there exist kernel functions e_m̃ and E_m̃
of order nk satisfying the following condition: 

    e_m(z)=e_m̃(z^1/n)/n.

Then both the kernel function E_m of order k and the corresponding
moment function m of order 1/k are defined by the same formulas
as in Definition <ref>.




 For any k>0 the classical and most important kernel functions of order k and the corresponding moment function of order 1/k are given by
 
  
  * e_m(z)=kz^ke^-z^k;
  
  * m(u)=Γ(1+u/k), where Γ is the Gamma function;
  
  * E_m(z)=∑_j=0^∞z^j/Γ(1+j/k)=:𝐄_1/k(z), where 𝐄_1/k is the
    Mittag-Leffler function of index 1/k.

They are used in the classical theory of k-summability.


Moreover the set of moment functions is closed under multiplication and division. Namely, we have
 Let m_1 and m_2 be two moment
functions of orders s_1 and s_2, respectively. Then m_1· m_2 is a moment function
of order s_1+s_2 and if moreover s_1>s_2, m_1/m_2 is a moment function
of order s_1-s_2.

The above proposition suggests to define moment functions of order zero.

We call m a moment function of order 0,
if there exist moment functions m_1, m_2 of the same order
s such that m=m_1/m_2.


It is worth noting that all moment functions of order s have
the same growth as Γ_s(x):=Γ(1+sx) (see <cit.>), i.e. there exist
positive constants a and A such that 

    a^nΓ_s(n)≤ m(n)≤ A^nΓ_s(n)  for every  
    n∈_0.


We will consider a special class of moment functions:

For any s>0 we say that a moment function m of order s is a regular moment function of order s if there exist constants a,A>0 such that 

    an^s≤m(n)/m(n-1)≤ An^s  for every   n∈.

We denote the set of regular moment functions of order s by _s.


Regular moment functions satisfy the following properties:
  

  * The class of regular moment functions is closed under multiplication and
division. More precisely, if m_1∈_s_1 and m_2∈_s_2 then m_1· m_2∈_s_1+s_2
and if moreover s_1> s_2, m_1/m_2∈_s_1-s_2. 


  * The class of regular moment functions contains classical moment functions Γ_s of order s, i.e. Γ_s∈_s for any s>0.



Let a_1,A_1,a_2,A_2
be positive constants such that for i=1,2 and for every n∈ we have: 

    a_in^s_i≤m_i(n)/m_i(n-1)≤ A_in^s_i.

For any n∈ we have: 

    a_1a_2n^s_1+s_2≤m_1(n)m_2(n)/m_1(n-1)m_2(n-1)≤ A_1A_2n^s_1+s_2.


Seeing as m_1· m_2 is a moment function of order s_1+s_2,
this proves the first part of (<ref>). In the case
of m_1/m_2, which is a moment function of order s_1-s_2,
the proof is similar and it will be omitted.

In order to show (<ref>) we can use the Stirling formula (see <cit.>): 

    √(2π)x^x-1/2e^-x≤Γ(x)≤√(2π)e^1/12xx^x-1/2e^-x<√(2π)x^x-1/2e^-x+1 for x≥1.

Then for any n∈ we receive: 

    Γ(1+ns)/Γ(1+ns-s)   ≤ e^-s+1(1+ns/1+ns-s)^1+ns-s-1/2(1+ns)^s
       ≤ e^-s+1(1+s/1+ns-s)^1+ns-s√(1+ns-s/1+ns)(1+1/s)^ss^sn^s
       ≤(1+1/s)^ses^sn^s.


In a similar
way we obtain the inequalities: 

    Γ(1+ns)/Γ(1+ns-s)   ≥ e^-s-1(1+ns/1+ns-s)^ns-s+1/2(1+ns)^s
       ≥ e^-s-1s^sn^s.
 


One more property of the Gamma function will be used extensively in
subsequent sections of this paper:


For any s≥0 there exist constants c̃_̃s̃ and C̃_s
such that for all x≥ s we have

    c̃_s(1+x)^s≤Γ(1+x)/Γ(1+x-s)≤C̃_s(1+x)^s.




As before, we use the Stirling formula (<ref>) to receive:

    Γ(1+x)/Γ(1+x-s)≤(1+x/1+x-s)^1/2+x-se^-s+1(1+x)^s≤ e(1+x)^s.

Similarly, we show the second inequality:

    Γ(1+x)/Γ(1+x-s)≥(1+x/1+x-s)^1/2+x-se^-s-1(1+x)^s≥ e^-s-1(1+x)^s.



Using moment functions, we can also define a moment Borel transform.


Let m be a moment function of order s. Then
we define an m-moment Borel transform as an operator _m,t𝔼[[t]]→𝔼[[t]]
given by the formula: 

    _m,t(∑_n=0^∞u_nt^n):=∑_n=0^∞u_n/m(n)t^n.




Assume that m is a moment function of order
s≥ 0. Then û∈𝔼[[t]] is a formal power series of
Gevrey order s if there exists r>0 such that _m,tû∈(D_r,).
We denote the space of all such power series by 𝔼[[t]]_s.




 Observe that by (<ref>)  the formal series û=∑_n=0^∞u_nt^n∈𝔼[[t]] is of Gevrey order s
 if and only if there exist B,C>0 such that
 
    u_n_≤ BC^nΓ_s(n)  for   n∈_0.

 For this reason any formal power series of Gevrey order 0 is convergent.
 
 It also means that the definition of formal power series of Gevrey order s does not depend on the choice of a moment function m of order s.
 

[see <cit.>] Let m be a moment function. Then
we define an m-differential operator ∂_m,t𝔼[[t]]→𝔼[[t]]
by the formula: 

    ∂_m,t(∑_n=0^∞u_n/m(n)t^n):=∑_n=0^∞u_n+1/m(n)t^n.




   Below we present most important examples of moment differential operators.
     Other examples, including also integro-differential operators, can be found in
     <cit.>.
      
     If m(u)=Γ_1(u) then the operator ∂_m,x coincides with the usual differentiation ∂_x.
     
     More generally, if s>0 and m(u)=Γ_s(u) then the operator ∂_m,x satisfies
     (∂_m,xu)(x^s)=∂^s_x(u(x^s)),
     where ∂^s_x denotes the Caputo fractional derivative of order s
     defined by
       
    ∂^s_x(∑_j=0^∞u_j/Γ_s(j)x^sj):=
         ∑_j=0^∞u_j+1/Γ_s(j)x^sj.

     

Immediately by the definitions, we obtain the following commutation formula between moment differential operators and moment Borel transforms



Let m and m' be moment functions. Then
the operators _m',t, ∂_m,t[[t]]→[[t]] satisfy the commutation formula

    _m',t∂_m,tû=∂_mm',t_m',tû  for any  û∈[[t]].




We estimate moment derivatives of holomorphic functions as follows

 Let
f be a function holomorphic on D_R⊂ and let m
be a moment function of order s. Then for any positive
r<r'<R there
exist a constant h>0 such that for all α∈_0

    sup_z∈ D_r|∂_m,z^αf(z)|≤sup_z∈ D_r'|f(z)|h^αΓ(1+sα).



Proposition <ref> can be generalized to the multidimensional case.


Let m_1,…,m_N be moment functions of orders s_1,…,s_N, respectively, with s=(s_1,…,s_N), and suppose that f∈(D^N_R) for certain R>0. Then for any 0<r<r'<R there
exist constant h̃>0 such that for all α∈^N_0

    sup_z∈ D^N_r|∂_m_1,z_1^α_1…∂_m_N,z_N^α_Nf(z)|≤sup_z∈ D^N_r'|f(z)|h̃^|α|Γ(1+s·α).



After applying N times Proposition <ref> we receive:

    sup_z∈ D^N_r|∂_m_1,z_1^α_1…∂_m_N,z_N^α_Nf(z)|≤sup_z∈ D^N_r'|f(z)| h^|α|Γ(1+s_1α_1)…Γ(1+s_Nα_N).

Moreover, let us note that for any complex u, w such that u≥ 0 and w ≥ 0 we have Γ(u)Γ(w)≤Γ(u+w). Using that fact we receive:

    Γ(1+s_1α_1)…Γ(1+s_Nα_N)≤Γ(N+s·α).

Using properties of the Gamma function we obtain the following:

    Γ(N+s·α)   =(N-1+s·α)…(1+s·α)Γ(1+s·α)
       ≤ e^(N-1)(N-2)/2e^(N-2)s·αΓ(1+s·α).

Let us now use a notation s̅=max_1≤ j≤ Ns_j. Then

    Γ(N+s·α)≤ e^(N-1)(N-2)/2e^(N-2)s̅|α|Γ(1+s·α).

Since we may assume that |α|≥ 1, It is therefore enough to take h̃=he^(N-1)(N-2)/2e^(N-2)s̅.




§ NEWTON POLYGON



The Newton polygon for linear partial differential operators with variable coefficients was introduced by
Yonemura <cit.>, who also described the Gevrey order of solution in terms
 of its Newton polygon. In this way he generalized the previous results of <cit.> given for ordinary differential equations.
 
 On the other hand the Newton polygon for linear moment partial differential
 operators with constant coefficients in two variables  (t,z) was introduced by the first author in <cit.>.
 
In this section we extend the notion of the Newton polygon to the case of linear moment partial differential operators with variable coefficients and with multidimensional spatial variable z∈^N.

 
 

Let m_0,m_1,…,m_N be moment functions of positive orders s_0,s_1,…,s_N respectively.
 We assume that m=(m_1,…,m_N), s=(s_1,…,s_N), t∈, z=(z_1,…,z_N)∈^N, α=(α_1,…,α_N)∈_0^N,
 Σ⊆_0 is a set of indices (finite or infinite), J⊂_0 and A⊂_0^N are finite set of indices,
 and the moment operator is given by

    P(t,z,∂_m_0,t,∂_m,z)=
     ∑_(σ,j,α)∈Σ× J× Aa_σ jα(z)
    t^σ∂_m_0,t^j∂_m,z^α,

 where we use the multidimensional notation ∂_m,z^α:=∂_m_1,z_1^α_1⋯∂_m_N,z_N^α_N.
 
 
  The Newton polygon for the operator P given by (<ref>)
  is defined as the convex hull of the union
 of sets
 Q(js_0+α· s,σ-j), where 
 (σ,j,α)∈Σ× J× A, that is
  
    N(P,s_0,s)=conv{Q(js_0+α· s,σ-j) (σ,j,α)∈Σ× J× A, a_σ jα(z)≢0},

 where Q(a,b) denotes the second quadrant of ^2 translated by the vector (a,b) (i.e. Q(a,b):={(x,y) x≤ a, y≥ b} for any (a,b)∈^2) and α· s=α_1s_1+⋯+α_Ns_N is the scalar product of α and s.
 

The definition given above is a fairly general one, covering a wide variety of operators. However, from now on we will focus exclusively on the case when coefficients of the moment operator P do not depend on the variable z.

An example of a Newton polygon can be seen on Figure <ref>.



§ FORMAL NORMS

In this section we introduce a useful tool, which allows us to keep together estimations of all Γ_s-moment derivatives of a given holomorphic function.

For f(z)∈(D^N_R) let us define
the formal norm of f by the formula: 

    f(z)_ρ:=∑_α∈_0^N|∂_Γ_s,z^αf(z)|/Γ(1+s·α)ρ^α

where s=(s_1,…,s_N), ∂^α_Γ_s,z=∂^α_1_Γ_s_1,z_1…∂^α_N_Γ_s_N,z_N,  ρ=(ρ_1,…,ρ_N)∈^N and
ρ^α=ρ_1^α_1⋯ρ_N^α_N.


The above definition is a generalization
of a concept used in <cit.>. This section is devoted mainly to
presenting properties characterizing formal norms defined by (<ref>),
which in many cases are very similar to the results from <cit.>.

For any a≥0 and ρ∈^N let us introduce the following formal power series: 

    Θ^(a)(ρ):=∑_α∈_0^NΓ(1+s·α+a)/Γ(1+s·α)ρ^α.


They satisfy



Let b≥ 0. Then for any a>0 we have

    Θ^(b)(ρ)≪ e(e/1+a+b)^aΘ^(a+b)(ρ).

 

The proof of this fact is identical to
the one presented in <cit.> and it will be omitted.


Using (<ref>) we may estimate the formal norm of holomorphic function in the following way:



 If f∈(D_R^N) then for every r<r'<R there exists h>0
 such that
 sup_z∈ D^N_rf(z)_ρ≪ CΘ^(0)(hρ),
 where C=sup_z∈ D^N_r'|f(z)|.



 It is sufficient to use (<ref>), (<ref>) and Proposition <ref>. 




Let f∈(D^N_R) be a function such that 
sup_z∈ D^N_rf(z)_ρ≪ CΘ^(a)(hρ) for certain constants r∈(0,R), C,h>0 and a≥ 0. Then for any β∈_0^N we have sup_z∈ D^N_r∂^β_Γ_sf(z)_ρ≪ Ch^|β|Θ^(a+s·β)(hρ).


Since f(z)_ρ≪ CΘ^(a)(hρ), for any α∈_0^N and z∈ D_r^N, it follows that

    sup_z∈ D^N_r|∂_Γ_s,z^αf(z)|/Γ(1+s·α)≤Ch^|α|Γ(1+s·α+a)/Γ(1+s·α),

which means that sup_z∈ D^N_r|∂_Γ_s,z^αf(z)|≤ Ch^|α|Γ(1+s·α+a). Seeing as the second inequality holds for all α∈_0^N, it is also true for α̅=α+β. Hence, for all α∈_0^N we have:

    sup_z∈ D^N_r|∂_Γ_s,z^α+βf(z)|/Γ(1+s·α)≤Ch^|α|+|β|Γ(1+s·α+s·β+a)/Γ(1+s·α).





Let us consider a function f(t,z)=∑_n=0^∞f_n(z)t^n∈_R[[t]]_s.
For any r<R there exist constants A,B,h>0 such that for any n∈_0

    sup_z∈ D^N_rf_n(z)_ρ≪ AB^nΘ^(0)(hρ)n!^s.




Since f is of a Gevrey order s, for
every r<r'<R there exist certain positive constants A and B such that we get
sup_z∈ D^N_r'|f_n(z)|≤ AB^n n!^s
for any n∈_0. Using it and Lemma <ref>, we receive: 

    sup_z∈ D^N_rf_n(z)_ρ≪sup_z∈ D^N_r'|f_n(z)|Θ^(0)(hρ)≪ AB^nn!^sΘ^(0)(hρ).





§ MAIN PROBLEM


 

Let us consider the moment differential operator

    P(∂_m_0,t,∂_m_1,z_1,…,∂_m_N,z_N)=∂_m_0,t^M+∑_(j,α)∈Λa_j,α(t)∂_m_0,t^j∂_m,z^α,

where m_0,m_1,…,m_N are moment functions of positive orders s_0,s_1,…,s_N respectively, m=(m_1,…,m_N), s=(s_1,…,s_N), and a_j,α(t) are holomorphic functions in
a neighborhood of the origin.
Then the Newton polygon for (<ref>) is given
by the set 

    N(P,s_0,s)=conv{ Q(Ms_0,-M)∪⋃_(j,α)∈ΛQ(s_0j+s·α, (a_j,α)-j)}.


For any i=0,1,…,p+1 let us denote the slope of the segment
Γ_i by k_i with all k_i≥ 0. It is easy to observe
that 0=k_0<k_1<…<k_p+1=∞. If p>0 as well, then
we can also calculate the value of k_1 by the formula

    1/k_1=max_(j,α)∈Λ{s_0(j-M)+s·α/(a_j,α)-j+M}.

Hence, in the general case, for q_j,α:=(a_j,α)-j+M, we get

    1/k_1=max{ 0, max_(j,α)∈Λ{s_0(j-M)+s·α/q_j,α}}.


Let us return to our main equation (<ref>). We shall further
assume that: 

    ϕ_j∈_R̃ (j=0,…,M-1) and f(t,z)∈_R̃[[t]]_1/k_1 for certain R̃>0,


    the set Λ⊂_0×_0^N is finite,


    a_j,α(t)∈[[t]]_1/k_1for any (j,α)∈Λ.

We will also assume 
that 

    (a_j,α)≥max{0, j-M+1} for any (j,α)∈Λ

in order to ensure that (<ref>) has a unique formal solution.

Additionally we assume that 

    m_0 is a regular moment function.

We can then change the form of (<ref>) using the composition of Borel transforms
of order 0 with respect to z_1,…,z_N given by: 

    _Γ_s/m,z:=_Γ_s_1/m_1,z_1…_Γ_s_N/m_N,z_N.

Since (<ref>) is a linear equation with coefficients
which do not depend on z, by Proposition <ref> we receive an equation equivalent to (<ref>):

    {∂_m_0,t^Mv(t,z)+∑_(j,α)∈Λa_j,α(t)∂_m_0,t^j∂_Γ_s,z^αv(t,z)     =g(t,z)
    ∂_m_0,t^jv(0,z)     =ψ_j(z) for 0≤ j<M
    .,

where v=_Γ_s/m,zu, g=_Γ_s/m,zf and ψ_j=_Γ_s/m,zϕ_j
for j=0,1,…,M-1. By Remark <ref>
there exists R>0 such that
ψ_j∈_R (j=0,…,M-1) and 
g∈_R[[t]]_1/k_1.

First we show that the formal solution of (<ref>) is of a Gevrey order 1/k_1. To this end we will use the formal norms and their properties.

 Let v̂(t,z)=∑_n=0^∞v_n(z)t^n
be a formal solution of (<ref>). Then for every r<R there exist constants C,H,h>0
such that for any n∈_0 and z∈ D_r we have: 

    v_n(z)_ρ≪CH^n/n!^Ms_0Θ^(dn)(hρ),

where d=Ms_0+1/k_1.


Before we can move on to the proof of this fact, we will present an additional technical lemma:

Let M∈. Then the following formula holds true:

    (n-M)!/n!≤(M/n)^M for all n≥ M



We shall prove this fact using induction with respect to M.

Firstly, let us assume that M=1. After substituting 1 for M in (<ref>) we receive an inequality

    1/n≤1/n,

which is obviously true for all n∈.

Now suppose that

    (n-K)!/n!≤(K/n)^K for all n≥ K

holds true for any K≤ M. We shall prove that it is true for K=M+1 as well. Let us take n≥ M+1. Then:

    (n-M-1)!/n!=(n-M)!/n!(n-M)≤1/n-M(M/n)^M.

It is enough to show that

    1/n-M(M/n)^M≤(M+1/n)^MM/n,

which is equivalent to the inequality

    n[M(M+1/M)^M-1]≥ M^2(M+1/M)^M.

Seeing as n≥ M+1, to prove the last formula it is enough to show that

    M^2(M+1/M)^M/M(M+1/M)^M-1≤ M+1.

This last inequality is equivalent to the following one:

    M(M+1/M)^M-M-1≥ 0,

which is true for all M≥ 1, because (M+1/M)^M≥M+1/M.



For n≤ M-1 we have v_n(z)=ψ_n(z)/m_0(n)
and (<ref>) holds. Assume then that (<ref>) (with n replaced by i)
is true for all i≤ n-1. We shall show the same for n. To
this end, first we note that if v(t,z)=∑_n=0^∞v_n(z)t^n
then: 

    ∑_n=0^∞v_n(z)∂_m_0,t^Mt^n+∑_(j,α)∈Λa_j,α(t)∑_n=0^∞(∂_Γ_s,z^αv_n(z))∂_m_0,t^jt^n=g(t,z).

After differentiating with respect to t and multiplying both sides
of the equation by t^M we receive a formula: 

    ∑_n=0^∞v_n(z)m_0(n)/m_0(n-M)t^n+∑_(j,α)∈Λt^M-ja_j,α(t)∑_n=0^∞∂_Γ_s,z^αv_n(z)m_0(n)/m_0(n-j)t^n=t^Mg(t,z).

Seeing as t^Mg(t,z)=∑_n=M^∞g_n(z)t^n and t^M-ja_j,α(t)=∑_p=q_j,α^∞c_j,α,pt^p for every (j,α)∈Λ, we receive for all
n≥ M: 

    v_n(z)=m_0(n-M)/m_0(n)[g_n(z)-∑_(j,α)∈Λ∑_p=q_j,α^nc_j,α,pm_0(n-p)/m_0(n-p-j)∂_Γ_s,z^αv_n-p(z)],

assuming the convention that whenever n-p-j≤0, the term 
m_0(n-p)/m_0(n-p-j)
is equal to 0. Please note that from (<ref>) it follows that q_j,α≥ 1 for any (j,α)∈Λ. Consequently v_n given by (<ref>) are coefficients of a unique formal solution of (<ref>).

Because  a_j,α(t)∈[[t]]_1/k_1 for all (j,α)∈Λ,
there exist constants A_j,α,B>0 such that
|c_j,α,p|≤ A_j,αB^p(p-q_j,α)!^1/k_1
for any p≥ q_j,α. Additionally, since t^Mg(t,z)∈_R[[t]]_1/k_1, by Lemma <ref> we may
assume that there exist also constants K,h>0
such that g_n(z)_ρ≪ KB^nΘ^(0)(hρ)n!^1/k_1  for any z∈ D_r^N and n∈_0.
So for any z∈ D_r^N we can estimate: 

    v_n(z)_ρ≪(n-M)!^s_0/a^Mn!^s_0[KB^nΘ^(0)(hρ)n!^1/k_1+
    ∑_(j,α)∈Λ∑_p=q_j,α^nA_j,αA^jB^p(p-q_j,α)!^1/k_1(n-p)!^s_0/(n-p-j)!^s_0∂_Γ_s,z^αv_n-p(z)_ρ].

By Lemma <ref> and by the inductive assumption we get

    ∂_Γ_s,z^αv_n-p(z)_ρ≪CH^n-ph^|α|/(n-p)!^Ms_0Θ^(d(n-p)+s·α)(hρ)  for any   z∈ D^N_r.
 Hence, continuing our estimation we see that

    v_n(z)_ρ≪(n-M)!^s_0/a^Mn!^s_0[KB^nΘ^(0)(hρ)n!^1/k_1+
    ∑_(j,α)∈Λ∑_p=q_j,α^nA_j,αA^jB^p(p-q_j,α)!^1/k_1(n-p)!^s_0CH^n-ph^|α|/(n-p-j)!^s_0(n-p)!^Ms_0Θ^(d(n-p)+s·α)(hρ)].


Using Lemma <ref> we receive: 

    v_n(z)_ρ≪M^Ms_0/a^Mn^Ms_0[KB^nΘ^(0)(hρ)n!^1/k_1+I],

where 

    I=∑_(j,α)∈Λ∑_p=q_j,α^nA_j,αA^jB^p(p-q_j,α)!^1/k_1(n-p)!^s_0CH^n-ph^|α|/(n-p-j)!^s_0(n-p)!^Ms_0Θ^(d(n-p)+s·α)(hρ).


Furthermore, let us note that by Lemma <ref>: 

    KM^Ms_0B^n/a^Mn^Ms_0Θ^(0)(hρ)n!^1/k_1    =KM^Ms_0B^n/a^M(n^s_0)^Mn-M/(n^s_0)^MnΘ^(0)(hρ)n!^1/k_1
       ≪KM^Ms_0B^n(n^n)^Ms_0+1/k_1/a^Mn!^Ms_0Θ^(0)(hρ)
        =KM^Ms_0B^n/a^Mn!^Ms_0n^dnΘ^(0)(hρ)
       ≪KM^Ms_0B^n/a^Mn!^Ms_0e(en/1+dn)^dnΘ^(dn)(hρ)
       ≪KM^Ms_0e/a^Mn!^Ms_0(e^dB/d^d)^nΘ^(dn)(hρ).


It is enough to choose C≥2KM^Ms_0e/a^M and
H≥e^dB/d^d, for 1/2CH^n/n!^Ms_0Θ^(dn)(hρ)
to be a majorant of KM^Ms_0B^n/a^Mn^Ms_0Θ^(0)(hρ)n!^1/k_1.

Moreover, note that (n-p)!/(n-p-j)!≤ n^j and 1/(n-p)!≤n^p/n!,
and because of these facts we receive the inequality: 

    (n-p)!^s_0/(n-p-j)!^s_01/(n-p)!^Ms_0≤n^s_0(Mp+j)/n!^Ms_0.

Analogously for q_j,α≤ p≤ n we estimate (p-q_j,α)!^1/k_1≤ n^(p-q_j,α)/k_1.

Using these inequalities and the fact that s_0(j-M)≤q_j,α/k_1-s·α
for any (j,α)∈Λ, we conclude that: 

    1/n^Ms_0I    ≪CH^n/n!^Ms_0∑_(j,α)∈Λ∑_p=q_j,α^nA_j,αA^jh^|α|B^p/H^pn^s_0(Mp+j-M)+(p-q_j,α)/k_1
                               ×Θ^(d(n-p)+s·α)(hρ)
       ≪CH^n/n!^Ms_0∑_(j,α)∈Λ∑_p=q_j,α^nA_j,αA^jh^|α|B^p/H^pn^dp-s·αΘ^(d(n-p)+s·α)(hρ).

Hence by Lemma <ref>: 

    M^Ms_0/a^Mn^Ms_0I    ≪CH^n/n!^Ms_0∑_(j,α)∈Λ∑_p=q_j,α^nA_j,αA^jh^|α|B^p/a^MM^-Ms_0H^pe(en/1+dn)^dp-s·αΘ^(dn)(hρ)
       ≪CH^n/n!^Ms_0∑_(j,α)∈Λ∑_p=q_j,α^∞eA_j,αA^jh^|α|/a^MM^-Ms_0(d/e)^s·α(e^dB/Hd^d)^pΘ^(dn)(hρ)
       ≪CH^n/n!^Ms_0Θ^(dn)(hρ)∑_(j,α)∈ΛeA_j,αA^jh^|α|/a^MM^-Ms_0(d/e)^s·α(e^dB/Hd^d)^q_j,α/1-e^dB/Hd^d.

which can be bounded from
above by 1/2CH^n/n!^Ms_0Θ^(dn)(hρ)
for sufficiently large H.




Let v̂(t,z)=∑_n=0^∞v_n(z)t^n
be a formal solution of (<ref>). Then v̂(t,z) is of Gevrey order 1/k_1 with respect to t, i.e. for any r<R there exist constants
C̃,H̃>0 such that 

    sup_z∈ D_r^N|v_n(z)|≤C̃H̃^nn!^1/k_1  for any   n∈_0.



Let us note that |v_n(z)|=v_n(z)_0, where 0 is the zero vector in ^N. 
Then we can use Lemma <ref> to conclude that: 

    |v_n(z)|    ≤CH^n/n!^Ms_0Θ^(dn)(0)=CH^n/n!^Ms_0Γ(1+dn)
       ≤CH^n/n!^Ms_0BL^nn!^d=C̃H̃^nn!^1/k_1,

which finishes the proof.


We would like to find the similar result for the formal
solution û(t,z) of (<ref>).
To this end we need the stronger version of Proposition <ref>, where the series v_n(z)=∑_|l|=0^∞v_nlz^l, with l=(l_1,…,l_N)∈_0^N, is replaced by its majoring
series M[v_n](z):=∑_|l|=0^∞|v_nl|z^l.

 
 Let v̂(t,z)=∑_n=0^∞v_n(z)t^n
be a formal solution of (<ref>). Then for every r<R there exist constants C,H>0
such that: 

    sup_z∈ D_r^N|M[v_n](z)|≤ CH^nn!^1/k_1  for any   n∈_0,

where d=Ms_0+1/k_1.


 It is sufficient to repeat the proofs of Lemma <ref> and Proposition <ref> with ψ_n(z) replaced by
 M[ψ_n](z) and g_n(z) replaced by M[g_n](z).
 Using the estimation
 
    M[v_n](z)≪m_0(n-M)/m_0(n)[M[g_n](z)
    
    
    +∑_(j,α)∈Λ∑_p=q_j,α^n|c_j,α,p|m_0(n-p)/m_0(n-p-j)∂_Γ_s,z^αM[v_n-p](z)]

 instead of (<ref>) and repeating the proofs we get the assertion.


Now we are ready to prove the main result of the paper:

Let û(t,z)=∑_n=0^∞u_n(z)t^n
be a formal solution of (<ref>). We also assume that the conditions (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) are satisfied. Then û(t,z) is of Gevrey order 1/k_1 with respect to t, i.e. there exists R̃>0 such that for any r<R̃ there exist constants
C,H>0 satisfying 

    sup_z∈ D_r^N|u_n(z)|≤ CH^nn!^1/k_1  for any   n∈_0.



Since ∑_n=0^∞v_n(z)t^n=v̂(t,z)=_Γ_s/m,zû(t,z)=
∑_n=0^∞_Γ_s/m,zu_n(z)t^n, where _Γ_s/m,z is the composition of
Borel transforms of order zero with respect to (z_1,…,z_N) given by (<ref>), we see that v_n(z)=_Γ_s/m,zu_n(z) for every n∈_0.
By (<ref>) there exists B>0 such that

    Γ_s(l)/m(l)=Γ_s_1(l_1)/m_1(l_1)×⋯×Γ_s_N(l_N)/m_N(l_N)≤ B^|l|  for every   l∈_0^N.
 
Hence we get

    u_n(z)≪ M[u_n](z)≪∑_|l|=0^∞|u_nl|B^|l|/Γ_s(l)/m(l)z^l= M[v_n](Bz)

and using Proposition <ref> we conclude that
there exist constants
C,H>0 such that 

    |u_n(z)|≤ |M[v_n](Bz)|≤
    CH^nn!^1/k_1

for every z∈ D^N_r with r<R/B,
which gives the assertion with R̃=R/B.



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