Abstract. We consider a nonlinear Laplace equation ∆u = f (x, u) in two variables.

POLONICI MATHEMATICI LXVII.1 (1997)

Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables

by M. E. Pli´ s (Krak´ow) and B. Ziemian (Warszawa)

Abstract. We consider a nonlinear Laplace equation ∆u = f (x, u) in two variables.

Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.

0. Introduction. We consider a nonlinear Laplace equation of the form

(1) ∆u =

 ∂

²

∂x

²₁

+ ∂

²

∂x

²₂



u = f (x, u)

where x = (x

₁

, x

₂

) ∈ R

²

. First we are going to construct a formal power series solution of (1) and then prove that every such solution is of the same class as the function f in x. Similar results for some nonlinear ordinary differential equations were proved by Braaksma [Br], following the ideas of J. Ecalle.

We denote by L the image of the positive quadrant R

²₊

= R

+

×R

⁺

under the unitary matrix

^√1₂ ^{1−i 1+i}_{1+i 1−i}

.

Definition 1 ([Zie1]). A function F of the variable z = (z

1

, z

2

) ∈ C

²

is said to be Laplace holomorphic on L if F is holomorphic on some polydisk centered at (0, 0) ∈ C

²

, can be holomorphically continued to some sectorial neighbourhood S = S

1

× S

²

of L with vertex (0, 0), and is of exponential growth on S, i.e. for every closed subsector S

^′

= S

₁^′

× S

2^′

⊂ S there exist constants c = (c

1

, c

2

) and C such that for z ∈ S

^′

,

(2) |F (z

¹

, z

2

)| ≤ Ce

^{c1|z1|+c2|z2|}

.

Definition 2. A function f of the variable x = (x

1

, x

2

) ∈ R

²

is said to be a 1-sum of a formal power series

1991 Mathematics Subject Classification: 35C20, 44A10.

Key words and phrases : Borel resummation, formal solutions, Laplace equation, Supported by KBN grant 2-PO3A-006-08.

[31]

f (x) = b X

∞ k,l=0

g

kl

1

(x

1

+ ix

2

)

^k+1

(ix

1

+ x

2

)

^l+1

if there exists a Laplace holomorphic function F on L such that

f (x) =

\

L

e

^−xz

F (z) dz and

F (z) = X

∞ k,l=0

g

kl

i2

^k+l+1

k!l! (z

1

− iz

²

)

^k

(z

2

− iz

¹

)

^l

near zero. In that case we say that f is 1-resummable.

In this paper we assume that f (x, u) on the right hand side of (1) is the 1-sum in x of a formal power series

f (x, u) = b X

∞ k,l=0

g

_kl

(u) 1

(x

1

+ ix

2

)

^k+1

(ix

1

+ x

2

)

^l+1

with coefficients g

kl

(u) holomorphic for every (k, l) ∈ N

²0

, on some fixed neighbourhood U of zero in C, and g

kl

(0) = 0.

Therefore, if we write f (x, u) = b

X

∞ k,l=0

 X

^∞

j=1

c

^j_kl

u

^j

 1

(x

1

+ ix

2

)

^k+1

(ix

1

+ x

2

)

^l+1

= X

∞ j=1

 X

∞ k,l=0

c

^j_kl

1

(x

1

+ ix

2

)

^k+1

(ix

1

+ x

2

)

^l+1

 u

^j

then we can write f (x, u) = P

_∞

j=1

c

j

(x)u

^j

where c

j

(x) is the 1-sum of the formal series

X

∞ k,l=0

c

^j_k,l

1

(x

1

+ ix

2

)

^k+1

(ix

1

+ x

2

)

^l+1

, and f is holomorphic in u on U . Hence, we have c

_j

(x) =

T

L

e

^−xz

T

_j

(z) dz for some Laplace holomorphic functions T

j

. Moreover, T

j

are holomorphic on the same sector S for all j, and the constants c and C in (2) are independent of j.

Theorem. If T

_j

(0) 6= 0, then there exists a family of 1-resummable solutions of equation (1) of the form

u(x) = X

∞ ν=0

d

ν

1

(x

1

+ ix

2

)

^ν1+1

· 1

(ix

1

+ x

2

)

^ν2+1

.

This means that

(3) u(x) =

\

L

e

^−xz

T (z) dz

with T being a Laplace holomorphic function on L. Moreover , every formal solution b u of (1) of the above form is 1-resummable.

The proof will be divided into three parts.

1. Convolution equation. Applying ∆ to u in the form (3) we arrive at the complex symbol of ∆ as the complex polynomial

P (z

1

, z

2

) = z

₁²

+ z

²₂

=

 1 + i

√ 2 z

1

+ 1 − i

√ 2 z

2

 1 − i

√ 2 z

1

+ 1 + i

√ 2 z

2

 . In the new variables

ζ

1

= 1 + i

√ 2 z

1

+ 1 − i

√ 2 z

2

, ζ

2

= 1 − i

√ 2 z

1

+ 1 + i

√ 2 z

2

,

P becomes the polynomial e P (ζ

1

, ζ

2

) = ζ

1

· ζ

²

, and after changing variables on the left hand side of (1) we get

∆u(x

1

, x

2

) = (P (z

1

, z

2

)T )[e

^{−x1z1−x2z2}

]

= ( e P (ζ

1

, ζ

2

) e T )[e

^{−x1(1−i2√2ζ1+21+i√2ζ2)−x2(21+i√2ζ1+21−i√2ζ2)}

]

= ( e P (ζ

1

, ζ

2

) e T )[e

^{−(21−i√2x1+21+i√2x2)ζ1−(21+i√2x1+21−i√2x2)ζ2}

].

So we are looking for a solution u(y e

1

, y

2

) = e T [e

^{−y1ζ1−y2ζ2}

] = u

 1 − i 2 √

2 x

1

+ 1 + i 2 √

2 x

2

, 1 + i 2 √

2 x

1

+ 1 − i 2 √

2 x

2



of the convolution equation

(4) ζ

₁

ζ

₂

T = f e

^∗

T e

where f

^∗

T = e P

_∞

j=1

T e

j

∗ e T

^∗j

with e T

^∗j

denoting the jth convolution power of e T , i.e. T

^∗j

= T ∗ ... ∗ T (j times). From now on we write T instead of e T . We can assume that T

1

(0) = 1, for otherwise we modify slightly the change of variables after dividing equation (1) by T

1

(0).

Since our existence proof for the solution of (4) essentially follows that of Braaksma [Br], we shall consider T having the formal expansion

(5) T =

X

∞ k,l=0

d

_kl

ζ e

₁^k

ζ e

₂^l

with e ζ

^p

= ζ

^p

/Γ (p + 1). Then due to the convolution formula

ζ e

^l

∗ e ζ

^k

= e ζ

^l+k+1

we find f

^∗

T =

X

∞ j=1

X

∞ m1,m2=0

c

^j_m1m2

ζ e

₁^m1

ζ e

₂^m2

∗  X

^∞

k,l=0

d

kl

ζ e

₁^k

ζ e

₂^l



_∗j

= X

∞ j=1

X

∞ m1,m2=0

c

^j_m1m2

ζ e

₁^m1

ζ e

₂^m2

∗

X

∞ ν1+...+νj=0

d

ν1

. . . d

νj

e ζ

^{ν1+...+νj+j−1}

= X

∞ j=1

X

∞ m+ν1+...+νj=0

c

^j_m

d

_ν1

. . . d

_νj

ζ e

^{m+ν1+...+νj+j}

= X

∞ k=0



^k+1

X

j=1

X

m+ν1+...+νj=k+1−j

c

^j_m

d

ν1

. . . d

νj

 ζ e

^k+1

for k, m, ν

j

∈ N

²0

, j = (j, j), 1 = (1, 1), k = min{k

¹

, k

2

}.

Inserting this in (4) we find

(6) d

k

(k + 1) =

k+1

X

j=1

X

m+ν1+...+νj=k+1−j

c

^j_m

d

ν1

. . . d

νj

, since

ζ · e ζ

^p

= (p + 1)e ζ

^p+1

.

In particular, we can take d

00

arbitrarily (since c

¹₀₀

= 1), d

10

= c

¹₀₁

d

00

, d

01

= c

¹₀₁

d

00

,

2d

20

= c

¹₂₀

d

00

+ c

¹₁₀

d

10

, 2d

02

= c

¹₀₂

d

00

+ c

¹₀₁

d

01

, 3d

11

= c

¹₁₁

d

00

+ c

¹₁₀

d

01

+ c

¹₀₁

d

10

+ c

²₀₀

d

00

, . . .

We are going to prove that T defined formally by (5) with coefficients d

ν

satisfying the recurrence (6) is a holomorphic function of exponential growth in some sector S.

Before starting the resummation proof for the expansion (5), we consider the resummation problem with respect to one variable. Therefore, let us write (5) in the form

T = X

∞ k=0

T

1k

(ζ

1

)e ζ

₂^k

where T

_1k

(ζ

₁

) = P

_∞

l=0

d

_lk

ζ e

₁^l

.

In a way similar to that of deriving (6), we find that T

1k

satisfy the

convolution equation

ζ

1

(k + 1)T

1k

=

k+1

X

j=1

X

m+ν1+...+νj=k+1−j

T

_m^j

∗ T

^1ν1

∗ . . . ∗ T

^1νj

for k, m, ν

_j

∈ N

0

, where T

_m^j

= P

_∞

p=0

c

^j_pm

ζ e

₁^p

. For k = 0 this gives (7) ζ

₁

T

₁₀

= T

₀¹

∗ T

10

,

which is equivalent to the equation

(7

^′

) d

dt u

0

= c

¹₀

(t)u

0

, in the variable t =

₂^1−i√₂

x

1

+

₂^1+i√₂

x

2

.

For k = 1 we get

2ζ

₁

T

₁₁

= T

₀¹

∗ T

11

+ T

₁¹

∗ T

10

+ T

₀²

∗ T

10^∗2

or equivalently 2 d

dt u

1

= c

¹₀

(t)u

1

+ c

¹₁

(t)u

0

+ c

²₀

(t)(u

0

)

²

.

We easily see that the jth equation is linear in u

j

with u

0

, . . . , u

_j−1

regarded as coefficients. Since the solutions of linear equations with re- summable coefficients are resummable themselves (cf. [Br], [Zie1]), we see that all T

1k

are Laplace holomorphic functions. The same is also true for

T

l2

(ζ

2

) = X

∞ j=0

d

lj

ζ e

₂^j

.

Now we pass to the proof of the convergence of the formal series (5) with d

ν

satisfying (6). Since the series (5) satisfies (4), for a fixed N ∈ N

⁰

the series

T

N

= X

∞ l,j=N +1

d

lj

ζ e

₁^l

ζ e

₂^j

= T − S

^N

satisfies the equation

ζ

1

ζ

2

T

N

= G

^N

(ζ, T

N

) = X

∞ j=1

X

j k=0

 j k



T

j

∗ T

N^∗k

∗ S

N^∗(j−k)

− ζ

¹

ζ

2

S

N

= X

∞ k=0

 X

∞ j=k j≥1

 j k



T

_j

∗ S

N^∗(j−k)



∗ T

N^∗k

− ζ

1

ζ

₂

S

_N

.

We write

(8) G

^N

(ζ, ψ) =

X

∞ k=0

g

k

(ζ) ∗ ψ

^∗k

where g

0

= P

_∞

j=1

T

j

∗ S

N^∗j

− ζ

¹

ζ

2

S

N

, and g

k

= P

_∞

j=k j k

T

j

∗ S

N^∗(j−k)

for k > 0. The series g

_k

are convergent near (0, 0) due to the remarks about the resummation problem with respect to one variable and the fact that the series P

_∞

j=k j k

T

j

(ζ)u

^j−k

is convergent near 0. Moreover, we can see that for every subsector S

^′

⊂ S there exist K and c = (c

1

, c

₂

) such that for ζ ∈ S

^′

, (9)

 |g

0

(ζ)| ≤ K|ζ

1

|

^{N +1}

|ζ

2

|

^{N +1}

e

^{c1|ζ1|+c2|ζ2|}

,

|g

^k

(ζ)| ≤ Ke

^{c1|ζ1|+c2|ζ2|}

for k ≥ 1.

For p = (p

1

, p

2

), p

i

> 0, s = (s

1

, s

2

) ∈ R

²

, we denote by W

s

(p) the space of functions ψ holomorphic in the polydisc {|ζ

¹

| ≤ p

¹

, |ζ

²

| ≤ p

²

} and such that

kψk

^s,p

= sup

|ζⁱ|≤pⁱ

|ζ

^−s

ψ(ζ)| < ∞.

Observe that for ζ ∈ {|ζ

ⁱ

| ≤ p

ⁱ

} and s

¹

> −1, s

²

> −1,

|ψ

^∗m

(ζ)| ≤ kψk

^ms,p

Γ (s

1

+ 1)

^m

Γ (s

2

+ 1)

^m

Γ (m(s

1

+ 1))Γ (m(s

2

+ 1)) (10)

× |ζ

¹

|

^m(s1+1)−1

|ζ

²

|

^m(s2+1)−1

.

Therefore by the properties of the Γ -function the function (8) makes sense for ψ ∈ W

^s

(p) if s is large enough.

Consider the operator (11) Rψ(ζ) = 1

ζ g

0

(ζ) + 1

ζ (g

1

∗ ψ)(ζ) + X

∞ m=2

1

ζ (g

m

∗ ψ

^∗m

)(ζ).

Denoting the summands by R

0

ψ, R

lin

ψ and Qψ respectively, for ψ ∈ W

_{N −1}

(p) and ζ ∈ {|ζ

ⁱ

| ≤ p

ⁱ

, i = 1, 2} we get the estimates

(12)

 

 

 

 

 



 

 

 

 

 

|R

⁰

ψ(ζ)| ≤ K|ζ

¹

|

^N

|ζ

²

|

^N

,

|R

^lin

ψ(ζ)| ≤ Kkψk

N −1,p

|ζ

¹

|

^{N −1}

|ζ

²

|

^{N −1}

 Γ (N ) Γ (N + 1)



2

= K

N

²

kψk

N −1,p

|ζ

1

ζ

₂

|

^{N −1}

,

|Qψ(ζ)| ≤ K

 X

∞ m=2

kψk

^m_{N −1,p}

Γ (N )

^2m

Γ (mN )

²

|ζ

¹

ζ

2

|

^{(m−1)N −1}



|ζ

¹

ζ

2

|

^N

. Set M =

¹_ζ

g

₀

(ζ)

N −1,p

. If kψk

N −1,p

≤ 2M then by choosing p small and N large we may have (by (12))

kQψk

N −1,p

≤

¹₃

M and kR

lin

ψk ≤

¹₃

M.

Therefore the operator R acts in the space

B

_{N −1,p}

= {ψ ∈ W

N −1

(p) : kψk

N −1,p

≤ 2M}.

Observe that for ψ, ψ + χ ∈ B

N −1,p

we have

|(ψ + χ)

^∗m

(ζ) − ψ

^∗m

(ζ)| =    

X

m l=1

 m l



(ψ

^∗(m−l)

∗ χ

^∗l

)(ζ)    

≤ X

m

l=1

 m l



kψk

^m−l_{N −1,p}

kχk

^lN −1,p

Γ (N )

^2m

Γ (mN )

²

|ζ

¹

ζ

2

|

^{mN −1}

≤ Γ (N )

^2m

Γ (mN )

²

|ζ

1

ζ

₂

|

^{mN −1}

X

m l=1

 m l



kψk

^m−l_{N −1,p}

kχk

^l_{N −1,p}

. We have

X

m l=1

 m l



kψk

^m−l_{N −1,p}

kχk

^lN −1,p

≤ kχk

N −1,p

X

m l=1

 m l



(2M )

^m−l

(4M )

^l−1

≤ kχk

N −1,p

4M (2M + 4M )

^m

= (6M )

^m

4M kχk

N −1,p

since kχk ≤ kψ + χk + kψk ≤ 4M. Hence

|(ψ + χ)

^∗m

(ζ) − ψ

^∗m

(ζ)| ≤ (6M Γ (N )

²

)

^m

4M Γ (mN )

²

kχk

N −1,p

(|ζ

¹

ζ

2

|)

^{mN −1}

, and 

   1

ζ (g

m

∗ ((ψ + χ)

^∗m

− ψ

^∗m

))(ζ)    

≤ K

4M

 (6M Γ (N )

²

)

^m

Γ (mN )

²

|ζ

1

ζ

₂

|

^{(m−1)N −1}



|ζ

1

ζ

₂

|

^N

kχk

N −1,p

. From this and from (12), we derive that

kR(ψ + χ) − Rψk

N −1,p

≤ kR

^lin

ψk

N −1,p

+ kQ(ψ + χ) − Qψk

≤

¹3

kψk

N −1,p

+ K

^′

kχk

N −1,p

≤

²3

kχk

N −1,p

provided p is small enough. Therefore, for p small and N large, the operator R is a contraction on B

_{N −1,p}

. Hence we get a unique function ψ

N

solving the nonlinear convolution equation

(13) ζ

1

ζ

2

ψ

N

= G

^N

(ζ, ψ

N

), ψ

N

∈ B

N −1,p

.

From the construction of G

^N

it follows that for every N (sufficiently large) the function ψ

N

+ S

N

satisfies the equation (4), hence the kth Taylor coefficient of ψ

N

(at 0) must satisfy (6) (for k

i

≥ N + 1), so T defined formally by (5) and (6) converges on {|ζ

ⁱ

| ≤ p

ⁱ

}.

2. Analytic continuation of solutions. Define S(r) = {ζ ∈ C : |ζ| ≤ r}

∩ S

¹

(see Introduction) and let p be such that the solution ψ

N

of (13) is

holomorphic in the interior of S

²

(p) = S(p) × S(p). We shall extend this

solution to a unique solution on some complex neighbourhood of R

²₊

.

Choose δ, p

1

∈ R

⁺

, δ < p

1

< p. Define S

0

= S(p

1

) × S(p),

S

₊

= {ζ ∈ C

²

: (ζ

₁

− p

1

, ζ

₂

) ∈ S(δ) × S(p) or ζ

1

= p

₁

}, S

¹

= S

0

∪ S

⁺

.

Then S

0

∩ S

⁺

= {p

¹

} × S(p).

Let W

0

denote the space of functions on S

¹

which are continuous on S

¹

\ (S

⁰

∩ S

⁺

) and analytic in its interior. Next define e ψ ∈ W

⁰

by setting ψ = ψ e

N

on S

0

and e ψ ≡ 0 on S

⁺

. Introduce the space

V

_{N −1}

(δ) = {φ ∈ C

⁰

(S

+

) ∩ O(intS

⁺

) : sup

ζ∈S+

|ζ

2^{−N +1}

φ(ζ)| < ∞}.

For φ ∈ V

N −1

(δ) define φ

0

∈ W

⁰

by extending φ by zero on S

0

. Then (φ

0

∗ φ

⁰

)(ζ) =

\

C(ζ)

φ

0

(ζ − γ)φ

⁰

(γ) dγ ≡ 0

where C(ζ) = C(ζ

1

)×C(ζ

²

), C(ζ

i

) is a path from 0 to ζ

i

. Hence also φ

^∗m₀

≡ 0 for m ≥ 2. Clearly, e ψ

^∗m

= ψ

_N^∗m

on S

0

for all m. Therefore ( e ψ + φ)

^∗m

= ψ b

^∗m

+ m b ψ

^∗(m−1)

∗ φ

⁰

.

Consequently, for G(ζ, ψ) = G

^N

(ζ, ψ) given by (8) we have G(ζ, e ψ + φ

0

) = G(ζ, e ψ) + (B ∗ φ

⁰

)(ζ) where

B(ζ) = g

1

(ζ) + X

∞ m=2

m(g

m

∗ e ψ

^∗(m−1)

)(ζ).

Thus the equation

ζ( e ψ + φ

₀

) = G(ζ, e ψ + φ

₀

) gives rise to a linear convolution equation

(14) φ

₀

= χ + 1

ζ (B ∗ φ

0

)(ζ) for φ

0

∈ V

N −1

(δ), with χ(ζ) =

¹_ζ

G(ζ, e ψ) − e ψ.

For ζ ∈ S

+

and φ ∈ V

N −1

(δ) we have  

  1

ζ (B ∗ φ

⁰

)(ζ)     =

    1 ζ

ζ\1

p1

ζ\2

0

B(ζ

1

− η

¹

, ζ

2

− η

²

)φ(η

1

, η

2

) dη

1

dη

2

   

=     1 ζ

ζ2

\

0

h

^ζ1−p\1

0

B(ζ

1

− p

¹

− γ

¹

, ζ

2

− η

²

)φ(γ

1

+ p

1

, η

2

) dγ

1

i dη

2

   

≤ 1

|ζ| kφk

N −1

  

ζ2

\

0

η

^{N −1}₂

h

^ζ1−p\1

0

B(τ, ζ

₂

− η

2

) dτ i dη

₂



 

with kφk

N −1

= sup

_ζ∈S+

|ζ

2^{−N +1}

φ(ζ)|. Now, from the definition of B, we see that for τ ∈ S(δ) × S(p),

|B(τ)|≤ C

 1 +

X

∞ m=2

mkψ

N

k

^m−1_{N −1,p}

Γ (N )

^2(m−1)

Γ ((m − 1)N)

²

(|τ

1

| · |τ

2

|)

^(m−1)N



< M.

Thus for ζ ∈ S

⁺

(and consequently for |ζ

¹

| ≥ p

¹

) we have  

  1

ζ (B ∗ φ

⁰

)(ζ)  

  ≤ Kkφk

^{N −1}

|ζ

2^{N −1}

| with K = M δ N p

₁

.

Hence if we take δ < N p

1

/M , then the operator φ →

¹_ζ

B ∗φ

⁰

is a contraction in the space V

_{N −1}

(δ). Thus there exists a unique solution φ ∈ V

N −1

(δ) satisfying (14). Hence φ = ψ

_N

on the interior of S

²

(p) ∩ S

+

and it is clear that φ extends ψ

N

to S

+

.

A repeated application of this procedure yields an extension of ψ

N

to some region U × S(p), where U is a sectorial neighbourhood of R

⁺

in C.

By interchanging variables and proceeding by the same method we get an extension of ψ

N

to some region S(p) × V with V being a sectorial neigh- bourhood of R

+

in C. Finally, in the same way we obtain an extension of ψ

N

to some sector U × V .

3. Exponential estimation. It follows from the results on analytic continuation of the solution of (13) that there exists a function ψ, holomor- phic in some sector S containing R

²₊

, satisfying (13) and such that ζ

^{−N +1}

ψ is locally bounded. We shall prove a global exponential estimate: for every closed subsector S

^′

⊂ S,

|ψ(ζ

¹

, ζ

2

)| ≤ K|ζ

^{N −1}

|e

^{c1|ζ1|+c2|ζ2|}

for ζ ∈ S

^′

, with appropriate constants K and c

1

, c

2

. The proof is again a two-dimensional variant of the reasoning given in [Br].

For p > 0 define

M (p) = sup{|ζ

1^{−N +1}

ψ(ζ

₁

, ζ

₂

)| : 0 < |ζ

1

| < 1, |ζ

2

| = p, ζ ∈ S

^′

}.

It follows from the local estimates for ψ that M (p) makes sense for each fixed p > 0. Then for 0 < |ζ

¹

| < 1, |ζ

²

| = p, ζ ∈ S

^′

,

|ψ(ζ

1

, ζ

₂

)| ≤ M(p)|ζ

1^{N −1}

|, and as in (10),

|ψ

^∗m

(ζ

1

, ζ

2

)| ≤ M

^∗m

(p)

 Γ (N )

^m

Γ (mN ) |ζ

1^{(m−1)N −1}

|



|ζ

¹

|

^N

. Then, by (9), we find that for any bc

2

> c

₂

,

    1

ζ (g

m

∗ ψ

^∗m

)(ζ)  

  ≤ Ke

^ˆc2p

∗ q

^m

M

^∗m

(p)|ζ

¹

|

^N

where q is a sufficiently small constant such that

 Γ (N )

^m

Γ (mN ) |ζ

1^{(m−1)N −1}

|



1/m

≤ q for m ∈ N.

Therefore we have for 0 < |ζ

¹

| < 1, |ζ

²

| = p, ζ ∈ S

^′

,

|ψ(ζ)| = |Rψ(ζ)| ≤ K|ζ

¹

|

^N

e

^cˆ2p

+ K|ζ

¹

|

^N

 e

^ˆc2p

∗

X

∞ m=1

q

^m

M

^∗m

(p)  and for all p > 0,

(15) |ζ

1^{−N +1}

ψ(ζ)| ≤ e Ke

^ˆc2p

+ e K  e

^ˆc2p

∗

X

∞ m=1

q

^m

M

^∗m

(p)  .

Denoting the right hand side of (15) by SM we get M (p) ≤ SM(p) for p > 0.

Consider the equation

(16) N (p) = SN (p).

Under the Laplace transformation v(s) = LN(s) =

∞\

0

e

^−ps

N (p) dp equation (16) becomes

v(s) = K e

s − bc

²

+ K e s − bc

²

·

X

∞ m=1

(qv(s))

^m

= K e

s − bc

²

· 1 1 − qv(s) or equivalently

qv

²

− v + K e

s − bc

²

= 0.

This equation has a unique solution analytic in 1/s at infinity, of the form v(s) = K e

s + X

∞

l=1

b

l

s

^l+1

for s large enough with coefficients b

_l

∈ R. Hence

N (p) = e K + X

∞

l=1

b

l

l! p

^l

is a solution of (16) real-valued for p > 0 and of exponential growth: N (p) ≤

Ke be

^cˆ2p

with some b K < ∞. e

Since M (0) = 0 and N (0) = e K > 0, and therefore M (p) ≤ N(p), it follows from the definition of M that for ζ ∈ S

^′

∩ {|ζ| ≤ 1, |ζ

²

| ≥ 1}, (17) |ψ(ζ)| ≤ be K(|ζ

¹

| · |ζ

²

|)

^{N −1}

e

^ˆc2|ζ2|

.

By the same method we get for ζ ∈ S

^′

∩ {|ζ

¹

| ≥ 1, |ζ

²

| ≤ 1}, (17

^′

) |ψ(ζ)| ≤ be K(|ζ

¹

| · |ζ

²

|)

^{N −1}

e

^¯c1|ζ1|

.

Now we pass to the global estimate on S

^′

. By (9) we get for ¯ c

i

> c

i

(i = 1, 2),

|Rψ(ζ)| ≤ K



e

^{c¯1|ζ1|+¯c2|ζ2|}

+ 1

|ζ

1

| · |ζ

2

|



e

^{c¯1|ζ1|+¯c2|ζ2|}

∗ X

∞ m=1

|ψ|

^∗m

(ζ) 

. Using this for |ζ

1

| ≥ 1, |ζ

2

| ≥ 1, since ψ = Rψ, we get

|ψ(ζ

¹

, ζ

2

)| ≤ e K 

e

^h¯c,|ζ|i

+ e

^h¯c,|ζ|i

∗ X

∞ m=1

|ψ|

^∗m

(ζ) 

As above, under the two-dimensional Laplace transformation we are led to considering the equation

v(s

1

, s

2

) = K e

(s

₁

− ¯ c

₁

)(s

₂

− ¯c

2

) · 1 1 − v(s

1

, s

₂

)

with v = Lψ. Again we prove that it has a solution v analytic in (1/s

¹

, 1/s

2

) at infinity, so ψ satisfies the exponential growth condition.

References

[Br] B. B r a a k s m a, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble) 42 (1992), 517–541.

[Sz-Zie] Z. S z m y d t and B. Z i e m i a n, The Mellin Transformation and Fuchsian Type PDEs, Kluwer, 1992.

[Zie1] B. Z i e m i a n, Generalized analytic functions with applications to singular ordi- nary and partial differential equations, Dissertationes Math. 354 (1996).

[Zie2] —, Leray residue formula and asymptotics of solutions to constant coefficient PDEs, Topol. Methods Nonlinear Anal. 3 (1994), 257–293.

Institute of Mathematics Institute of Mathematics

Krak´ ow Pedagogical University Polish Academy of Sciences

Podchor¸a˙zych 2 Sniadeckich 8 ´

30-084 Krak´ ow, Poland 00-950 Warszawa, Poland

E-mail: smplis@cyf-kr.edu.pl E-mail: ziemian@impan.impan.gov.pl

Re¸ cu par la R´ edaction le 2.10.1995

R´ evis´ e le 10.10.1996