37 N.M.TriONTHEGEVREYANALYTICITYOFSOLUTIONSOFSEMILINEARPERTURBATIONSOFPOWERSOFTHEMIZOHATAOPERATOR Rend.Sem.Mat.Univ.Pol.TorinoVol.57,1(1999)

N.M. Tri

ON THE GEVREY ANALYTICITY OF SOLUTIONS OF SEMILINEAR

PERTURBATIONS OF POWERS OF THE MIZOHATA OPERATOR

Abstract. We consider the Gevrey hypoellipticity of nonlinear equations with mul- tiple characteristics, consisting of powers of the Mizohata operator and a Gevrey analytic nonlinear perturbation. The main theorem states that under some con- ditions on the perturbation every H^s solution (with s ≥ s₀) of the equation is a Gevrey function. To prove the result we do not use the technique of cut-off functions, instead we have to control the behavior of the solutions through a fun- damental solution. We combine some ideas of Grushin and Friedman.

1. Introduction

Since a Hilbert’s conjecture the analyticity of solutions of partial differential equations has at- tracted considerable interest of specialists. The conjecture stated that every solution of an an- alytic nonlinear elliptic equation is analytic (see [1]). This conjecture caused considerable re- search by several authors. In 1904 Bernstein solved the problem for second order nonlinear elliptic equations in two variables [2]. For equations with an arbitrary number of variables the problem was established by Hopf [3], Giraud [4]. For a general elliptic system it was proved by Petrowskii [5], Morrey [6], and Friedman [7]. The last author also obtained results on the Gevrey analyticity of solutions. At present the results on the analyticity and Gevrey analyticity of solutions of linear partial differential equations have gone beyond the frame of the elliptic theory. A linear partial differential equation is called Gevrey hypoelliptic if every its solution is Gevrey analytic (in this context we can assume a solution in a very large space such as the space of distributions), provided the data is Gevrey analytic. Recently the class of linear Gevrey hypoelliptic equations with multiple characteristics has drawn much interest of mathematicians (see [8], and therein references). In this paper we consider the Gevrey analyticity of solutions in the Sobolev spaces of the nonlinear equation with multiple characteristics:

M_2k^hu+ϕ x₁,x₂,u, . . . ,∂^l1+l2u

∂^l_x¹₁∂^l_x²₂

!

l1+l₂≤(h−1)

=0,

where u = u(x₁,x₂), M_2k = _∂^∂_x

1 +i x^2k_{1 ∂x}^∂

2, the Mizohata operator inR². The Gevrey hy- poellipticity for the equation was obtained in [9] in the linear case. We will prove a theorem for a so-called Grushin type perturbationϕ(see the notations in §2 below). The analyticity of

∗Dedicated to my mother on her sixtieth birthday.

37

C^∞solutions was considered in [10]. The paper is organized as follows. In §2 we introduce notations used in the paper and state some auxiliary lemmas. In §3 we state the main theorem and prove a theorem on C^∞regularity of the solutions. In §4 we establish the Gevrey analyticity of the solutions.

2. Some notations and lemmas

First let us define the function

F_2k^h(x₁,x₂,y₁,y₂)= 1 2π(h−1)!

(x₁−y₁)^h−1 x₁^2k+1−y^2k+1₁

2k+1 +i(x₂−y₂) .

For j=1, . . . ,h−1 we have (see [11]) M_2k^j F_2k^h = 1

2π(h− j−1)!

(x₁−y₁)^h−j−1 x₁^2k+1−y₁^2k+1

2k+1 +i(x₂−y₂)

and M_2k^h F_2k^h =δ(x−y) ,

where x=(x₁,x₂), y=(y₁,y₂)andδ(·)is the Dirac function. We will denote bya bounded domain inR²with piece-wise smooth boundary. We now state a lemma on the representation formula. Its proof can be found in [10].

LEMMA4.1. Assume that u∈C^h()¯ then we have

(1)

u(x)= Z



(−1)^hF_2k^h(x,y)M_2k^hu(y)d y₁d y₂

+ Z

∂



 h−1X

j=0

(−1)^jM_2k^j u M_2k^h−j−1F_2k^h(x,y)



(n₁+i y₁^2kn₂)ds.

For reason of convenience we shall use the following Heaviside function θ (z)=

1 if z≥0, 0 if z<0.

We then have the following formula_dt^dmm(zⁿ)=n. . . (n−m+1)θ (n−m+1)z^n−m. We will use∂₁^α,∂₂^β,γ∂_α,β,∂_1y^α ,∂_2y^β andγ∂_α,β^y instead of_∂^∂_x^αα

1

, ^∂β

∂x₂^β, x₁^{γ ∂α+β}

∂x₁^α∂x₂^β, _∂y^∂αα 1

, ^∂β

∂y₂^β and y^γ₁ ^∂α+β

∂y₁^α∂y₂^β, respectively. Throughout the paper we use the following notation n!=

n! if n≥1,

1 if n≤0, and Cⁿ=

Cⁿ if n≥1, 1 if n≤0.

Furthermore all constants C_iwhich appear in the paper are taken such that they are greater than 1. Put h(2k+1)=r₀. For any integer r≥0 let0_rdenote the set of pair of multi-indices(α, β) such that0r =0_r¹∪0_r²where

0_r¹= {(α, β):α≤r₀,2α+β≤r}, 0_r²= {(α, β):α≥r₀, α+β≤r−r₀}.

For a pair(α, β)we denote by(α, β)^∗ the minimum of r such that (α, β) ∈ 0r. For any nonnegative integer t we define the following sieves

4_t = {(α, β, γ ):α+β≤t,2kt≥γ≥α+(2k+1)β−t}, 4⁰_t = {(α, β, γ ):(α, β, γ )∈4_t, γ =α+(2k+1)β−t}. Later on we will use the following properties of4t:

4tcontains a finite number of elements (less than 2k(t+1)³elements).

If(α, β, γ )∈4_t,β≥1,γ ≥1, then(α, β−1, γ−1)∈4_t−1.

For every(α, β, γ )∈ 4t,(α₀, β₀, γ₀)∈4t₀ we can expressγ∂_α,β(γ₀∂_α0,β0)as a linear combination of_γ0∂_α0,β⁰, where(α⁰, β⁰, γ⁰)∈4t+t₀.

For every (α, β, γ ) ∈ 4t, (α₁, β₁) ∈ 0r and a nonnegative integer m we can rewrite γ−m∂_{α+α1−m,β+β1} asγ−m∂_α2,β2(∂₁^α3∂₂^β3)where(α₂, β₂, γ−m)∈4tand(α₃, β₃)∈0r−m (see [12]).

For any nonnegative integer r let us define the norm

|u, |_r = max (α₁,β₁)∈0r

∂₁^α1∂₂^β1u, 

+ max

(α1,β1)∈0r α1≥1,β1≥1

max x∈ ¯ ∂₁^h

∂₁^α1∂₂^β1u(x), where|w, | =P

(α,β,γ )∈4_h−1max_{x∈ ¯}|_γ∂_α,βw(x)|.

For any nonnegative integer l letH^l_loc()denote the space of all u such that for any compact K inwe haveP

(α,β,γ )∈4lk_γ∂α,βuk_L2(K)<∞. We note the following properties ofH^l_loc():

H_loc^l ()⊂H^l_loc()where H_loc^l ()stands for the standard Sobolev spaces.

H^4k+2_loc ()⊂H_loc² ()⊂C().

If u∈H^l_loc()and(α, β, γ )∈4t, t≤l then_γ∂_α,βu∈H^l−t_loc().

The following lemma is due to Grushin (see [12])

LEMMA4.2. Assume that u∈D⁰()and M_2k^hu∈H^l_loc()then u∈H^l+h_loc().

Next we define a space generalizing the space of analytic functions (see for example [7]).

Let Ln and L¯n be two sequences of positive numbers, satisfying the monotonicity condition i

n

L_iL_n−i ≤ ALn(i = 1,2. . .; n = 1,2. . .), where A is a positive constant. A function F(x, v), defined for x = (x₁,x₂)and forv = (v₁, . . . , v_µ)in aµ-dimensional open set E , is said to belong to the class C{L_n−a;| ¯Ln−a;E}(a is an integer) if and only if F(x, v)is infinitely differentiable and to every pair of compact subsets₀⊂and E₀⊂E there corre- spond constants A₁and A₂such that for x∈₀andv∈E₀

∂^j+kF(x, v)

∂x₁^j1∂x₂^j2∂v^k₁¹. . . ∂v_µ^kµ

≤A₁A₂^j+kL_j−aL¯_k−a

j₁+ j₂= j,X

k_i =k; j,k=0,1,2. . .

.

We use the notation L_−i = 1 (i = 1,2, . . .). If F(x, v) = f(x), we simply write f(x) ∈ C{L_n−a;}. Note that C{n!;}, (C{n!^s;}) is the space of all analytic functions (s-Gevrey functions), respectively, in. Extensive treatments of non-quasi analytic functions (in particular, the Gevrey functions) can be found in [13, 14]. Finally we would like to mention the following lemma of Friedman [7].

LEMMA4.3. There exists a constant C₁such that if g(z)is a positive monotone decreasing function, defined in the interval 0≤z≤1 and satisfying

g(z)≤ 1

8^12kg z 1−6^k n

!!

+ C

z^n−r0−1 (n≥r₀+2,C>0) , then g(z) <CC₁/z^n−r0−1.

3. The main theorem

We will consider the following problem

(2) M_2k^hu+ϕ(x₁,x₂,u, . . . ,_γ∂_α,βu)_{(α,β,γ )∈4h−1}=0 in  . We now state our main

THEOREM4.1. Let s≥4k²+6k+h+1. Assume that u is aH^s_loc()solution of the equation (2) andϕ ∈C{L_n−a−2;|L_n−a−2;E}for every a ∈ [0,r₀]. Then u ∈ C{L_n−r0−2;}. In particular, ifϕis a s-Gevrey function then so is u.

The proof of this theorem consists of Theorem 4.2 and Theorem 4.3.

THEOREM4.2. Let s ≥ 4k²+6k+h+1. Assume that u is aH^s_loc()solution of the equation (2) andϕ∈C^∞. Then u is a C^∞()function.

Proof. Note that M_2k^h is elliptic onR², except the line(x₁,x₂)= (0,x₂). Therefore from the elliptic theory we have already u∈C^∞(∩R²\(0,x₂)).

LEMMA4.4. Let s ≥ 4k²+6k+h+1. Assume that u ∈ H^s_loc()andϕ ∈ C^∞then ϕ(x₁,x₂,u, . . . ,_γ∂_α,βu)_{(α,β,γ )∈4h−1}∈H^s−h+1_loc ().

Proof. It is sufficient to prove thatγ₁∂_α1,β1ϕ(x₁,x₂,u, . . . ,γ∂_α,βu)∈ L²_loc()for every(α₁, β₁, γ₁)∈4_s−h+1. Let us denote(u, . . . ,_γ∂_α,βu)_{(α,β,γ )∈4h−1}by(w₁, w₂, . . . , w_µ)withµ≤ 2kh³. Since s ≥ 4k²+6k+h+1 it follows thatw₁, . . . , wµ ∈ C(). It is easy to verify that∂₁^α1∂₂^β1ϕ(x₁,x₂,u, . . . ,_γ∂_α,βu)is a linear combination with positive coefficients of terms of the form

(3) ∂^kϕ

∂x₁^k1∂x₂^k2∂w₁^k3. . . ∂w_µ^kµ+2 Yµ

j=1 Y

(α1,j,β1,j)

∂₁^α1,j∂₂^β1,jw_j

ζ (α_1,j,β_1,j) ,

where k=k₁+k₂+ · · · +k_µ+2≤α₁+β₁;ζ (α_1,j, β_1,j)may be a multivalued function of α_1,j, β_1,j;α_1,j, β_1,jmay be a multivalued function of j , and

X

j

α_1,j·ζ (α_1,j, β_1,j) ≤ α₁, X

j

β_1,j·ζ (α_1,j, β_1,j) ≤ β₁.

Hence x^γ₁¹∂₁^α1∂₂^β1ϕ(x₁,x₂,u, . . . ,γ∂_α,βu)is a linear combination with positive coefficients of terms of the form

∂^kϕ

∂x₁^k1∂x₂^k2∂w₁^k3. . . ∂w^k_µ^µ+2 x₁^γ1

Yµ

j=1 Y

(α1,j,β1,j)

∂₁^α1,j∂₂^β1,jw_j

ζ (α_1,j,β_1,j) .

Therefore the theorem is proved if we can show this general terms are in L²_loc(). If all ζ (α_1,j, β_1,j)vanish then it is immediate that∂^kϕ/∂x₁^k1∂x₂^k2∂w₁^k3. . . ∂wµ^kµ+2 ∈ C, sinceϕ ∈ C^∞,w₁, . . . , wµ ∈C(). Then we can assume that there exists at least one ofζ (α_1,j, β_1,j) that differs from 0. Choose j₀such that there existsα_1,j0, β_1,j0withζ (α_1,j0, β_1,j0)≥1 and

α_1,j0+(2k+1)β_1,j0 = max

j=1,...,µ ζ (α1,j,β1,j)≥1

α_1,j+(2k+1)β_1,j.

Consider the following possibilities

I)ζ (α_1,j0, β_1,j0)≥2. We then haveα_1,j+β_1,j ≤l−(h−1)−(4k+2). Indeed, if j 6= j₀ andα_1,j+β_1,j >l−(h−1)−(4k+2)thenα_1,j0+β_1,j0 ≥2k. Therefore

l−(h−1)−(4k+2) < α_1,j+β_1,j≤α_1,j+(2k+1)β_1,j

≤α_1,j0+(2k+1)β_1,j0 ≤(2k+1)(α_1,j0+β_1,j0)

≤2k(2k+1) . Thus l< (2k+2)(2k+1)+(h−1), a contradiction.

If j= j₀andα_1,j0+β_1,j0 >l−(h−1)−(4k+2)then we have

l−(h−1)≥α₁+β₁≥2(α_1,j0+β_1,j0) >2(l−(h−1)−(4k+2)) . Therefore l< (h+1)+4(2k+1), a contradiction.

Next define

γ (α_1,j, β_1,j)=max{0, α_1,j+(2k+1)β_1,j+(h−1)+(4k+2)−l}.

We claim thatγ (α_1,j, β_1,j)≤2k(l−(h−1)−(4k+2)). Indeed, if j6= j₀andγ (α_1,j, β_1,j) >

2k(l−(h−1)−(4k+2))then

(2k+1)(l−(h−1)) ≥ α₁+(2k+1)β₁

≥ (α_1,j+2α_1,j0)+(2k+1)(β_1,j0+2β_1,j0)

> 3(2k+1)(l−(h−1)−(4k+2)) . Thus l< (h−1)+3(2k+1), a contradiction.

If j= j₀andγ (α_1,j0, β_1,j0) >2k(l−(h−1)−(4k+2))then it follows that (2k+1)(l−(h−1))≥α₁+(2k+1)β₁≥2(α_1,j0+(2k+1)β_1,j0)

>2(2k+1)(l−(h−1)−(4k+2)) . Thus l< (h−1)+4(2k+1), a contradiction.

From all above arguments we deduce that(α_1,j, β_1,j, γ (α_1,j, β_1,j))∈4l−(h−1)−(4k+2). Next we claim thatP

γ (α_1,j, β_1,j)ζ (α_1,j, β_1,j)≤γ₁. Indeed, ifP

γ (α_1,j, β_1,j)ζ (α_1,j, β_1,j) >

γ₁then we deduce that

α₁+(2k+1)β₁−2(l−(h−1)−(4k+2))≥X

γ (α_1,j, β_1,j)ζ (α_1,j, β_1,j) > γ₁

≥α₁+(2k+1)β₁−(l−(h−1)) . Therefore l< (h−1)+4(2k+1), a contradiction.

Now we have x^γ₁¹

Yµ

j=1 Y

(α_1,j,β_1,j)

∂₁^α1,j∂₂^β1,jw_jζ (α1,j,β1,j)

=x₁^γ1 Yµ

j=1 Y

(α_1,j,β_1,j)

x₁^{γ (α1,j,β1,j)}∂₁^α1,j∂₂^β1,jw_jζ (α1,j,β1,j)

∈C()

since x^{γ (α}₁ ^1,j,β1,j)∂₁^α1,j∂₂^β1,jw_j ∈H^4k+2_loc ()⊂C().

II)ζ (α_1,j0, β_1,j0)=1 andζ (α_1,j, β_1,j)=0 for j6= j₀. We have

x₁^γ1 Yµ

j=1 Y

(α1,j,β1,j)

∂₁^α1,j∂₂^β1,jw_j

ζ (α_1,j,β_1,j)

=x^γ₁¹∂₁^α1,j0∂₂^β1,j0w_j0 ∈L²_loc() .

III)ζ (α_1,j0, β_1,j0)=1 and there exists j₁6=j₀such thatζ (α_1,j1, β_1,j1)6=0. Define

¯

γ (α_1,j0, β_1,j0)=max{0, α_1,j0+(2k+1)β_1,j0+(h−1)−l}.

As in part I) we can prove(α_1,j, β_1,j, γ (α_1,j, β_1,j)) ∈ 4l−(h−1)−(4k+2) for j 6= j₀ and (α_1,j0, β_1,j0,γ (α¯ _1,j0, β_1,j0))∈ 4_l−(h−1). Therefore x^{γ (α}₁ ^1,j,β1,j)∂₁^α1,j∂₂^β1,jw_j ∈ H^4k+2_loc ()

⊂C()for j6= j₀and x₁^{γ (α¯ 1,j0,β1,j0)}∂₁^α1,j0∂₂^β1,j0w_j0∈L²_loc(). We also haveP

j6=j₀γ (α_1,j, β_1,j)ζ (α_1,j, β_1,j)+ ¯γ (α_1,j0, β_1,j0)≤γ₁as in part I). Now the desired result follows from the decomposition of the general terms.

(continuing the proof of Theorem 4.2) u∈ H_loc^l (), l≥4k²+6k+h+1H⇒u∈H^l_loc(), l≥4k²+6k+h+1H⇒ϕ(x₁,x₂,u, . . . ,γ∂_α,βu)∈H^l−h+1_loc ()(by Lemma 4.4). Therefore by Lemma 4.2 we deduce that u ∈H^l+1_loc(). Repeat the argument again and again we finally arrive at u∈H^l+t_loc()for every positive t , i.e.

u∈ ∩_lH^l_loc()=C^∞() .

REMARK4.1. Theorem 4.2 can be extended to the so-called Grushin type operators (non- linear version).

4. Gevrey analyticity of the solutions

PROPOSITION4.1. Assume that

ϕ(x₁,x₂,u, . . . ,γ∂_α,βu)_{(α,β,γ )∈4h−1}∈C{L_n−a−2;|L_n−a−2;E}

for every a∈[0,r₀]. Then there exist constantsHe₀,He₁,C₂,C₃such that for every H₀≥ He₀, H₁≥He₁, H₁≥C₂H₀^2r0+3if

|u, |_q≤ H₀H₁^(q−r0−2)L_q−r0−2, 0≤q≤N+1, r₀+2≤N then

max x∈ ¯

∂₁^α1∂₂^β1ϕ(x₁,x₂,u, . . . ,_γ∂_α,βu)

≤C₃H₀H₁^N−r0−1L_N−r0−1 for every(α₁, β₁)∈0_N+1.

Proof. ∂₁^α1∂₂^β1ϕ(x₁,x₂,u, . . . ,γ∂_α,βu)as in Lemma 4.4 is a linear combination with positive coefficients of terms of the form

∂^kϕ

∂x^k₁¹∂x^k₂²∂w^k₁³. . . ∂w^k_µ^µ+2 Yµ

j=1 Y

(α_1,j,β_1,j)

∂₁^α1,j∂₂^β1,jw_j

ζ (α1,j,β1,j) .

Substitutingw_jby one of the termsγ∂_α,βu with(α, β, γ )∈4_h−1we obtain

∂₁^α1,j∂₂^β1,j(γ∂_α,βu)= α_1,j X

m=0 m

α_1,j

γ· · ·(γ−m+1)θ (γ−m+1)_(γ−m)∂_{α+α1,j−m,β+β1,j}u.

We can decompose_(γ−m)∂_{α+α1,j−m,β+β1,j}u into_(γ−m)∂_α2,β2

∂₁^α3∂₂^β3u

with(α₂, β₂, γ − m)∈4h−1and(α₃, β₃)∈0_{(α1,j,β1,j)}^∗_−m. Put S = N+1−α₁−β₁. Define R =r₀−S.

It is easy to see that R is a nonnegative integer and less than r₀. Sinceα_1,j ≤ α₁we deduce that(α_1,j, β_1,j)∈0_(α1,j+β_1,j+S). Using the monotonicity condition on Lnand the inductive assumption we have

m α_1,j

γ· · ·(γ−m+1)θ (γ−m+1)_(γ−m)∂α+α_1,j−m,β+β_1,ju

≤ m

α_1,j

γ· · ·(γ−m+1)θ (γ−m+1)H₀H₁^{α1,j+β1,j−m−R−2}L_{α1,j+β1,j−m−R−2}

≤C₄H₀H₁^{α1,j+β1,j−m−R−2}L_{α1,j+β1,j−R−2}. Therefore we deduce that

∂₁^α1,j∂₂^β1,j(_γ∂_α,βu)

≤

α_1,j X

m=0 m

α_1,j

γ· · ·(γ−m+1)θ (γ−m+1)_(γ−m)∂_{α+α1,j−m,β+β1,j}u

≤C₅H₀H₁^{α1,j+β1,j−R−2}L_{α1,j+β1,j−R−2}.

and the general terms can be estimated by Yµ

j=1

C₅H₀H₁^{α1,j+β1,j−R−2}L_{α1,j+β1,j−R−2}.

Sinceϕ∈C{Ln−a;|Ln−a;E}, there exist constants C₆,C₇such that

∂^kϕ

∂x₁^q1∂x₂^q2∂w^r₁¹. . . ∂w^r_µ^µ

≤C₆C₇^q−RL_q−R−2C₇^rL_r−R−2 (q=q₁+q₂,r =r₁+ · · · +rµ) . Now we take X(ξ, v)=X₁(v)·X₂(ξ )where

X₁(v)=C₆ Xp

i=0

C₈ⁱL_i−R−2vⁱ

i ! , X₂(ξ )= Xp

i=0

C₇^i−RL_i−R−2ξⁱ

i ! .

and

v(ξ )=C₅H₀ Xp

i=1

H₁^i−R−2L_i−R−2ξⁱ

i ! .

By comparing terms of the form (3) with the corresponding terms in_dξ^dppX(ξ, v)it follows that

∂₁^α1∂₂^β1ϕ(x₁,x₂,u, . . . ,γ∂_α,βu) _x=x

0

≤ d^p dξ^pX(ξ, v)

_ξ=0.

Next we introduce the following notation: v(ξ ) h(ξ )if and only ifv^(j)(0) ≤ h^(j)(0)for 1≤ j≤ p. It is not difficult to check that there exists a constant C₉(independent of p) such that

v²(ξ )(C₅H₀)²C₉ Xp

i=2

H₁^i−R−3L_i−R−3ξⁱ (i−1)! . And by induction we have

v^j(ξ )(C₅H₀)^jC₉^j−1 Xp

i=j

H₁^{i−j−R−1}L_{i−j−R−1}ξⁱ (i−j+1)! .

Next, it is easy to verify that X₁(0) = 0, ^{d X}_dξ^{1(ξ )}

ξ=0 ≤ C₅C₆C₈H₀, and ^{djX2(ξ )} dξ^j

ξ=0 = C₇^j−RL_j−R−2.

We now compute ^{djX1(ξ )} dξ^j

ξ=0 when 2 ≤ j ≤ p. If 2 ≤ j ≤ 2R + 4 we can al- ways choose a constant C₁₀ such that ^{djX1(ξ )}

dξ^j

ξ=0 ≤ C₁₀H₀H₁^j−R−2L_j−R−2, provided H₁≥(C₅C₈C₉H₀)^2r0+3.

If j≥2R+5, we have

(4)

d^jX₁(ξ ) dξ^j

ξ=0

≤C₆



 R+2X

i=1

Cⁱ₈(C₅H₀)ⁱC₉ⁱ⁻¹H₁^{j−i−R−1}L_{j−i−R−1}j ! i !(j−i+1)!

+ j−R−2

X

i=R+3

C₈ⁱ(C₅H₀)ⁱCⁱ⁻¹₉ H₁^{j−i−R−1}L_i−R−2L_{j−i−R−1}j ! i !(j−i+1)!

+ Xj

i=j−R−1

C₈ⁱ(C₅H₀)ⁱC₉ⁱ⁻¹L_i−R−2j ! i !(j−i+1)!



.

The first sum in (4) is estimated by

(5) C₁₀H₀H₁^j−R−2L_j−R−2

provided H₁≥(C₅C₈C₉H₀)^2r0+3as for 2≤ j≤2R+4.

By using the monotonicity condition on L_nthe second sum in (4) is estimated by (6)

C₆ j−R−2X

i=R+3

Cⁱ₈(C₅H₀)ⁱC₉ⁱ⁻¹H₁^{j−i−R−1}L_i−R−2L_{j−i−R−1}j ! i !(j−i+1)!

≤ C₁₁H₀H₁^j−R−2j !L_j−2R−3 (j−2R−3)!

j−R−2X

i=R+3

1 i· · ·(i−R−1)

1

(j−i+1)· · ·(j−i−R)

≤C₁₂H₀H₁^j−R−2L_j−R−2, provided H₁≥C₅C₈C₉H₀. For the third sum we see that

(7)

C₆ Xj

i=j−R−1

Cⁱ₈(C₅H₀)ⁱC₉ⁱ⁻¹L_i−R−2j ! i !(j−i+1)!

≤C₅C₆C₈H₀H₁^j−R−2j ! Xj

i=j−R−1

L_i−R−2 i !(j−i+1)!

≤C₁₃H₀H₁^j−R−2L_j−R−2, if H₁≥(C₅C₈C₉H₀)².

By (4), (5), (6), (7) and taking H₁≥(C₅C₇C₈C₉H₀)^2r0+3=C₂H₀^2r0+3we obtain d^pX(ξ, v)

dξ^p

ξ=0

≤C₁₄ Xp

j=0 j

p

H₀H₁^j−R−2L_j−R−2C₇^p−j−RL_{p−j−R−2}

≤C₁₅H₀H₁^p−R−2L_p−R−2. Hence

max x∈ ¯

∂₁^α1∂₂^β1ϕ(x₁,x₂,u, . . . ,γ∂_α,βu)

≤C₁₅H₀H₁^p−R−2L_p−R−2

≤C₃H₀H₁^N−r0−1L_N−r0−1.

REMARK4.2. The constantsHe₀,He₁,C₂,C₃depend onincreasingly in the sense that with the sameHe₀,He₁,C₂,C₃Proposition 4.1 remains valid if we substituteby any⁰⊂.

COROLLARY4.1. Under the same hypotheses of Proposition 4.1 with q≤N+1 replaced by q≤N , then

max x∈ ¯

∂₁^α1∂₂^β1ϕ(x₁,x₂,u, . . . ,γ∂_α,βu) ≤C₃

|u, |_N+1+H₀H₁^N−r0−1L_N−r0−1

for every(α₁, β₁)∈0_N+1.

Proof. Indeed, as in the proof of Proposition 4.1 all typical terms, except _∂w^∂ϕ

j0∂₁^α1∂₂^β1w_j0 can be estimated by| · |_N. Replacingw_j0by one ofγ∂_α,βu, we have

∂ϕ

∂w_j0∂₁^α1∂₂^β1w_j0= ∂ϕ

∂w_j0 ^γ∂_{α+α1,β+β1}u + ∂ϕ

∂w_j0 α₁ X

m=1 m

α₁

γ· · ·(γ−m+1)θ (γ−m+1)_(γ−m)∂_{α+α1−m,β+β1}u. The first summand is estimated by C₃|u, |_N+1. The second sum is majorized as in Proposition 4.1.

THEOREM 4.3. Let u be a C^∞ solution of the equation (2) and ϕ ∈ C{L_n−a−2;|L_n−a−2;E} for every a ∈ [0,r₀]. Then u ∈ C{L_n−r0−2;}. In particu- lar, ifϕis a s-Gevrey function then so is u.

Proof. It suffices to consider the case(0,0)∈. Let us define a distanceρ((y₁,y₂), (x₁,x₂))= max

|x₁^2k+1−y₁^2k+1|

2k+1 ,|x₂−y₂|

. For two sets S₁,S₂the distance between them is defined as ρ(S₁,S₂)=inf_{x∈S1,y∈S2}ρ(x,y). Let V^T be the cube with edges of size (in theρmetric) 2T , which are parallel to the coordinate axes and centered at(0,0). Denote by V_δ^Tthe subcube which is homothetic with V^T and such that the distance between its boundary and the boundary of V^T isδ. We shall prove by induction that if T is small enough then there exist constants H₀,H₁ with H₁≥C₂H₀^2r0+3such that

(8)

u,V_δ^T

n≤H₀ for 0≤n≤max{r₀+2,6^k+1}

and (9)

u,V_δ^T

n≤H₀

H₁ δ

n−r₀−2

L_n−r0−2 for n≥max{r₀+2,6^k+1}

andδsufficiently small. Hence the Gevrey analyticity of u follows. (8) follows easily from the C^∞smoothness assumption on u. Assume that (9) holds for n = N . We shall prove it for n= N+1. Putδ⁰=δ(1−1/N). Fix x ∈V_δ^T and then defineσ =ρ(x, ∂V^T)andσ˜ =σ/N.

Let V_σ˜(x)denote the cube with center at(x)and edges of length 2˜σ which are parallel to the co- ordinate axes, and S_σ˜(x)the boundary of V_σ˜(x). Let E₁,E₃(E₂,E₄) be edges of S_σ˜(x)which are parallel to O x₁(O x₂) respectively. We have to estimate max_x∈VT

δ

γ∂_α,β

∂₁^α1∂₂^β1u(x)