The theory of ultradistributions was founded by Buerling and Roumieu in the sixties as a generalization of the theory of Schwartz distributions.

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POLONICI MATHEMATICI LXIII.1 (1996)

Laplace ultradistributions on a half line and a strong quasi-analyticity principle

by Grzegorz Lysik (Warszawa)

Abstract. Several representations of the space of Laplace ultradistributions sup- ported by a half line are given. A strong version of the quasi-analyticity principle of Phragm´ en–Lindel¨ of type is derived.

The theory of ultradistributions was founded by Buerling and Roumieu in the sixties as a generalization of the theory of Schwartz distributions.

Since then it was extensively studied by many authors: Bj¨ ork, Braun, Ko- matsu, Meise, Pilipovi´ c, Taylor , . . . , to mention but a few. The most system- atic treatment was presented by Komatsu [2], [3]. He derived, in particular, the boundary value representation of the space D

^(M^p⁾⁰

(Ω) of ultradistri- butions on an open set Ω ⊂ R

ⁿ

, structure theorems for D

^(M^p⁾⁰

(Ω) and described the image of the space D

^(M_K ^p⁾⁰

of ultradistributions with com- pact support in K under the Fourier–Laplace transformation. Following his approach Pilipovi´ c [9] recently introduced and investigated the space S

^(M^p⁾⁰

(R) of tempered ultradistributions. On the other hand, in the study of the Laplace transformation it is convenient to consider the space L

⁰_(ω)

(Γ ) of Laplace distributions of type ω ∈ R supported by a half line Γ . Since in the logarithmic variables the Laplace transformation is the Mellin transfor- mation we refer here to the book of Szmydt and Ziemian [11], where the latter transformation was systematically studied following the approach of Zemanian [12].

The aim of the present paper is to unify the theory of ultradistributions with that of Laplace distributions. We present it in the case of the space L

^(M_(ω)^p⁾⁰

(Γ ) of Laplace ultradistributions of Buerling type. Our theory is based on the Seeley type extension theorems for ultradifferentiable functions re-

1991 Mathematics Subject Classification: Primary 46F12, 44A10, 30D15.

Key words and phrases: ultradistributions, boundary values, quasi-analyticity.

Supported by KBN grant No. 2104591091.

[13]

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cently proved by Langenbruch [4] and Meise and Taylor [7]. We describe the image of the space L

^(M_(ω)^p⁾⁰

(Γ ) under the Laplace, Taylor and (modified) Cauchy transformations. In the latter case we follow the method of Mo- rimoto [8]. As an application of our theory we give, in the final section, a version of the quasi-analyticity principle of Phragm´ en–Lindel¨ of type. It says that a function holomorphic and of exponential type in the half plane {Re z > 0} vanishes if it satisfies some growth conditions along vertical lines and decreases superexponentially along a ray in {Re z > 0}.

0. Notation. Let t > 0. We denote by e B(t) the universal covering of the punctured disc B(t) \ {0} and by e C that of C \ {0}. We treat e B(t) and C as Riemann manifolds. Recall that any point x ∈ e e B(t) can be written in the form x = |x| exp i arg x with |x| < t.

We denote by µ : C → e C the biholomorphism µ(z) = e

^−z

for z ∈ C,

i.e. µ(z) = x ∈ e C with |x| = e

^{− Re z}

, arg x = − Im z. Then the inverse mapping µ

⁻¹

: e C → C is given by

µ

⁻¹

(x) = − ln x for x ∈ e C.

Let v ∈ R. We set

Γ

v

= [v, ∞) and I

v

= (0, e

^−v

].

Observe that I

v

= µ(Γ

v

). In the following we omit the subscript v as long as it is fixed. For z ∈ C we define the function exp

z

: R → C by

exp

_z

y = e

^yz

, y ∈ R.

For A ⊂ C we set

A

ε

= {z ∈ C : dist(z, A) < ε}, ε > 0.

We write D for the differential operator d/dx.

Let {P

τ

}

_{τ ∈T}

be a family of multivalued vector spaces. Then lim −→

^{τ ∈T}

T

τ

(resp. lim ←−

^{τ ∈T}

P

τ

) denotes the inductive limit (resp. projective limit) of P

τ

, τ ∈ T .

O(W ) denotes the set of holomorphic functions on an open subset W of some Riemann manifold. The value of a functional S on a test function ϕ is denoted by S[ϕ].

1. Laplace ultradistributions on a half line. Let (M

p

)

_p∈N0

be a

sequence of positive numbers. Throughout the paper we assume that (M

p

)

satisfies the following conditions:

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(M.1) (Logarithmic convexity )

M

_p²

≤ M

_p−1

M

p+1

for p ∈ N;

(M.2) (Stability under ultradifferential operators) There are constants A, H such that

M

p

≤ AH

^p

min

0≤q≤p

M

q

M

p−q

for p ∈ N

0

;

(M.3) (Strong non-quasi-analyticity) There is a constant A such that

∞

X

q=p+1

M

q−1

M

q

≤ Ap M

p

M

p+1

for p ∈ N.

Some results remain valid when (M.2), (M.3) are replaced by the follow- ing weaker conditions:

(M.2

⁰

) (Stability under differential operators) There are constants A, H such that

M

p+1

≤ AH

^p

M

p

for p ∈ N

0

; (M.3

⁰

) (Non-quasi-analyticity)

∞

X

p=1

M

p−1

M

p

< ∞.

Define

m

p

= M

p

/M

p−1

for p ∈ N.

Then (M.1) is equivalent to saying that the sequence m

p

is non-decreasing, and by (M.3

⁰

) it follows that m

p

→ ∞.

Note that the condition (M.3

⁰

) implies the following: for every h > 0 there exists δ > 0 such that

(1) M

p

h

^p

≥ δ for p ∈ N

0

,

which is equivalent to the finiteness of the associated function M defined by

(2) M (%) = sup

p∈N0

ln %

^p

M

0

M

p

for % > 0.

If M

p

/p! satisfies (1) the growth function M

^∗

is defined by

(3) M

^∗

(%) = sup

p∈N0

ln %

^p

p!M

0

M

p

for % > 0.

Example 1. The Gevrey sequence of order s > 1 is defined by M

p

=

(p!)

^s

, p ∈ N

0

. It satisfies all conditions (M.1)–(M.3) and M (%) ∼ %

^1/s

,

M

^∗

(%) ∼ %

^1/s−1

as % → ∞.

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R e m a r k 1. It follows from Lemma 4.1 of [2] that if M

p

satisfies (M.1) and (M.3

⁰

) then the associated function M is sublinear, i.e. M (%)/% → 0 as

% → ∞.

R e m a r k 2. If M

p

satisfies (M.1) and (M.3

⁰

) then

(4) lim

p→∞

(M

p

/p!)

^1/p

= ∞.

P r o o f. Take any l < ∞. Then by (M.1) and (M.3

⁰

) there exists p

l

∈ N such that M

p

≥ lpM

_p−1

for p ≥ p

l

. Hence

M

p

≥ C

_l

· l

^p

· p! for p ≥ p

l

, where C

l

= M

pl−1

M

0

(p

l

− 1)! l

^1−p^l

, and we easily get (4).

Definition. Let Γ = [v, ∞) with v ∈ R. The space D

^(M^p⁾⁰

(Γ ) of ultra- distributions on Γ of class (M

p

) is defined as the dual space of

D

^(M^p⁾

(Γ ) = lim −→

K⊂Γ

lim ←−

h>0

D

^(M_K,h^p⁾

(Γ ), where for any compact set K ⊂ Γ and h > 0,

D

^(M_K,h^p⁾

(Γ )

=

ϕ ∈ C

^∞

(Γ ) : supp ϕ ⊂ K and kϕk

^(M_K,h^p⁾

= sup

y∈K

sup

α∈N0

|D

^α

ϕ(y)|

h

^α

M

α

< ∞

. By ϕ ∈ C

^∞

(Γ ) we mean a restriction to Γ of some function ϕ ∈ C e

^∞

(R).

Definition. Let ω ∈ R∪{∞}. We define the space L

^(M_(ω)^p⁾⁰

(Γ ) of Laplace ultradistributions as the dual space of

L

^(M_(ω)^p⁾

(Γ ) = lim −→

a<ω

L

^(M_a ^p⁾

(Γ ), where for any a ∈ R,

L

^(M_a ^p⁾

(Γ ) = lim ←−

h>0

L

^(M_a,h^p⁾

(Γ ),

L

^(M_a,h^p⁾

(Γ ) =

ϕ ∈ C

^∞

(Γ ) : kϕk

^(M_a,h^p⁾

= sup

y∈Γ

sup

α∈N0

|e

^−ay

D

^α

ϕ(y)|

h

^α

M

α

< ∞

. Lemma 1. Assume that (M

p

) satisfies (M.1) and (M.3

⁰

). Then D

^(M^p⁾

(Γ ) is a dense subspace of L

^(M_(ω)^p⁾

(Γ ). Thus, L

^(M_(ω)^p⁾⁰

(Γ ) is a subspace of the space of ultradistributions D

^(M^p⁾⁰

(Γ ).

P r o o f. Making a translation if necessary we can assume that Γ = R

⁺

.

Let ϕ ∈ L

^(M_(ω)^p⁾

(Γ ). Then there exist a < b < ω such that ϕ ∈ L

^(Ma ^p⁾

(Γ ) ⊂

L

^(M_b ^p⁾

(Γ ). By the Denjoy–Carleman–Mandelbrojt theorem (cf. [2], [6]) there

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exists a function ψ ∈ D

^(M^p⁾

(Γ ) such that 0 ≤ ψ(y) ≤ 1 for y ∈ Γ , ψ(y) = 1 for 0 ≤ y ≤ 1 and ψ(y) = 0 for y ≥ 2. Put ϕ

ν

(y) = ψ(y/ν)ϕ(y) for y ∈ Γ , ν ∈ N. Then ϕ

^ν

∈ D

^(M^p⁾

(Γ ) and we shall show that ϕ

ν

→ ϕ in L

^(M_b ^p⁾

(Γ ) as ν → ∞. To this end take any h > 0. Noting that (M.1) implies M

q

M

p−q

≤ M

₀

M

p

for 0 ≤ q ≤ p, by the Leibniz formula we get

kϕ

ν

− ϕk

^(M_b,h^p⁾

= sup

y∈Γ

sup

α∈N0

|e

^−by

D

^α

(ϕ(y)(ψ(y/ν) − 1))|

h

^α

M

α

≤ sup

y∈Γ

sup

α∈N0

e

^−ay

|D

^α

ϕ(y)|

h

^α

M

α

e

^(a−b)y

|ψ(y/ν) − 1|

+ sup

y∈Γ

sup

α∈N

X

0≤β<α

α β

e

^−ay

|D

^β

ϕ(y)|

h

^β

M

β

· e

^(a−b)y

|D

^α−β

(ψ(y/ν) − 1)|M

0

h

^α−β

M

α−β

. Since ψ(y/ν) = 1 for 0 ≤ y ≤ ν the first summand tends to zero as ν → ∞.

Put K = [1, 2]. Then for β < α and any h

1

> 0 we have

|D

^α−β

(ψ(y/ν) − 1)| = |ν

^−(α−β)

ψ

^(α−β)

(y/ν)| ≤ ν

⁻¹

kψk

^(M_K,h^p⁾

1

h

^α−β₁

M

α−β

. We also have for any h

2

> 0 and β ≥ 0, e

^−ay

|D

^β

ϕ(y)| ≤ kϕk

^(M_a,h^p⁾

2

h

^β₂

M

β

. So the second summand is bounded by

M

0

e

^(a−b)ν

ν sup

α∈N

X

β<α

α β

kϕk

^(M_a,h^p⁾

2

h

2

h

β

kψk

^(M_K,h^p⁾

1

h

1

h

α−β

≤ M

0

ν kϕk

^(M_a,h^p⁾

2

kψk

^(M_K,h^p⁾

1

if h

2

+ h

1

≤ h and thus tends to zero as ν → ∞, proving the lemma.

Example 2. Let (M

p

) satisfy (1). Then the function Γ 3 y → exp

_z

y = e

^yz

belongs to L

^(M_(ω)^p⁾

(Γ ) if and only if Re z < ω. Furthermore, in this case for any a < ω and h > 0 we have

k exp

_z

k

^(M_a,h^p⁾

= M

₀⁻¹

exp{(Re z − a)v + M (|z|/h)}.

Let (M

p

) satisfy (M.1) and (1), and let z ∈ C. Then the operation of multiplication

exp

_z

: L

^(M_(ω)^p⁾

(Γ ) → L

^(M_{(ω+Re z)}^p⁾

(Γ ) is continuous. Thus the formula

exp

_z

S[ϕ] = S[exp

_z

ϕ] for ϕ ∈ L

^(M_{(ω+Re z)}^p⁾

(Γ ), S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) defines a continuous operation

exp

_z

: L

^(M_(ω)^p⁾⁰

(Γ ) → L

^(M_{(ω−Re z)}^p⁾⁰

(Γ ).

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Let (M

p

) satisfy (M.2). An ultradifferential operator P (D) of class (M

p

) is defined by

P (D) =

∞

X

α=0

a

α

D

^α

,

where a

α

∈ C satisfy the following condition: there are constants K < ∞ and C < ∞ such that

(5) |a

_α

| ≤ C K

^α

M

α

for α ∈ N

0

.

The entire function C 3 z → P (z) is called a symbol of class (M

p

). As in the proof of Theorem 2.12 of [2] one can show that an ultradifferential operator of class (M

p

) defines linear continuous mappings

P (D) : L

^(M_(ω)^p⁾

(Γ ) → L

^(M_(ω)^p⁾

(Γ ), P (D) : L

^(M_(ω)^p⁾⁰

(Γ ) → L

^(M_(ω)^p⁾⁰

(Γ ), where for S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) and ϕ ∈ L

^(M_(ω)^p⁾

(Γ ),

P (D)S[ϕ] = S[P

^∗

(D)ϕ] with P

^∗

(D) =

∞

X

α=0

(−1)

^α

a

α

D

^α

. For a ∈ R and ω ∈ R ∪ {∞} we define

(6) Y

a

= span{exp

_c

}

_c≤a

, Y

_(ω)

= [

a<ω

Y

a

.

Proposition 1. Let b < a. Then L

^(M_b ^p⁾

(Γ ) is contained in the closure of Y

a

in L

^(Ma ^p⁾

(Γ ). Thus Y

_(ω)

is dense in L

^(M_(ω)^p⁾

(Γ ).

P r o o f. Since the multiplication by exp

_−a

is a continuous isomorphism of L

^(Mc ^p⁾

(Γ ) onto L

^(M_c−a^p⁾

(Γ ) and Y

c

onto Y

c−a

, where c ∈ R, it is sufficient to assume that a = 0. Let ϕ ∈ L

^(M_b ^p⁾

(Γ ). It is enough to show that for every ε > 0 and h > 0 there exists ψ ∈ Y

0

such that kϕ − ψk

^(M_0,h^p⁾

< ε. To this end fix ε > 0 and h > 0. By the proof of Lemma 1 there exists e ψ ∈ D

^(M^p⁾

(Γ ) such that kϕ − e ψk

^(M_0,h^p⁾

< ε/2. Put η(x) = ψ ◦ µ

⁻¹

(x) for x ∈ I. Then η has compact support in I = µ(Γ ) and by the Roumieu theorem ([10], Th´ eor` eme 13), η ∈ D

^(M^p⁾

(I). By the Weierstrass type theorem ([2], Theorem 7.3) for any δ > 0 and h

1

> 0 there exists a polynomial p = P

N

ν=0

c

ν

x

^ν

such that

(7) kη − pk

^(M_¯ ^p⁾

I,h1

< δ.

Put ψ(y) = p ◦ µ(y) = P

N

ν=0

c

ν

e

^−νy

for y ∈ Γ . Then ψ ∈ Y

0

and we shall

show that for a suitable choice of δ and h

1

, k e ψ − ψk

^(M_0,h^p⁾

< ε/2. To this end

(7)

put f = η − p. Following the proof of Th´ eor` eme 13 of [10] one can show that the derivatives D

^α

f ◦ µ are estimated by

(8) kf k

^(M_¯ ^p⁾

I,h1

·

α

X

β=1

e

^−vβ

M

β

β! h

^β₁

(α − 1)!α!

(α − β)!(β − 1)! . For γ ∈ N

0

put

(9) H

γ

= sup

β≥γ

(β!/M

β

)

^1/β

.

Then by Remark 2, H

γ

→ 0 as γ → ∞. Hence we can find γ ∈ N

0

and h

1

> 0 such that

(10) ( p

e

^−v

h

1

+ pH

γ

)

²

· H < h,

where H is the constant in (M.2). Since by (M.2), M

p+q

≤ AH

^p+q

M

p

M

q

for p, q ∈ N

0

and by (M.1), M

β

M

α−β

≤ M

₁

M

α

for 0 ≤ β ≤ α, we get for α ∈ N

0

, 0 ≤ β ≤ α,

(11) M

β

M

α−β+γ

≤ M

_β

AH

^α−β+γ

M

α−β

M

γ

≤ C

_γ

H

^α

M

α

,

where C

γ

= AM

1

M

γ

max(H

^γ

, 1).

Observe also that

α < β(α − β + γ) for 1 ≤ β ≤ α, α ∈ N, and

α

X

β=1

α − 1 β − 1

2

x

^β−1

y

^α−β

≤ ( √ x + √

y)

^2α−2

for α ∈ N, x ≥ 0, y ≥ 0.

Hence using (8), (9) and (11) we derive for α ∈ N, y ∈ Γ ,

|D

^α

f ◦ µ(y)|

≤ kf k

^(M_¯ ^p⁾

I,h1

·

α

X

β=1

e

^−vβ

h

^β₁

H

_γ^α−β+γ

(α − 1)!α!

(α − β)!(α − β + γ)!β!(β − 1)! M

β

M

α−β+γ

≤ kf k

^(M_¯ ^p⁾

I,h1

· C

_γ

H

_γ^γ

e

^−v

h

1 α

X

β=1

(e

^−v

h

1

)

^β−1

H

_γ^α−β

(α − 1)!

(α − β)!(β − 1)!

2

H

^α

M

α

≤ kf k

^(M_¯ ^p⁾

I,h1

· C

_γ

H

_γ^γ

e

^−v

h

1

( p

e

^−v

h

1

+ pH

γ

)

⁻²

(( p

e

^−v

h

1

+ pH

γ

)

²

H)

^α

M

α

≤ e C

γ

M

α

L

^α

kf k

^(M_¯ ^p⁾

I,h1

, where L = ( p

e

^−v

h

1

+ pH

γ

)

²

H. Finally, choosing δ < ε/(2 e C

γ

) in (7) we get by (10),

(8)

k e ψ − ψk

^(M_0,h^p⁾

= sup

y∈Γ

sup

α∈N0

|D

^α

f ◦ µ(y)|

h

^α

M

α

≤ sup

α∈N0

C e

γ

L

^α

h

^α

kf k

^(M_¯ ^p⁾

I,h1

< ε 2 . To end this section we quote the following version of

Seeley extension theorem ([4], [7]). Let Γ = [v, ∞), Γ

1

= [v

1

, ∞) with v

1

< v and a ∈ R. Then there exists a linear continuous extension operator

(12) E : L

^(M_a ^p⁾

(Γ ) → L

^(M_a ^p⁾

(Γ

1

) such that for every ϕ ∈ L

^(Ma ^p⁾

(Γ ), supp E ϕ ⊂ (v

1

, ∞).

Corollary 1. Let S be a linear functional on L

^(M_(ω)^p⁾

(Γ ). Then S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) if and only if for every a < ω there exist h > 0 and C < ∞ such that

(13) |S[ϕ]| ≤ Ckϕk

^(M_a,h^p⁾

for ϕ ∈ L

^(M_a ^p⁾

(Γ ).

2. The Paley–Wiener type theorem for Laplace ultradistribu- tions. We assume the conditions (M.1), (M.2) and (M.3). Let Γ = [v, ∞) with v ∈ R. By Example 2 the function exp

z

belongs to L

^(M_(ω)^p⁾

(Γ ) if and only if Re z < ω. Hence we can define the Laplace transform of S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) by

LS(z) = S[exp

_z

] for Re z < ω.

Since the mapping

{Re z < ω} 3 z → exp

_z

∈ L

^(M_(ω)^p⁾

(Γ ) is holomorphic, LS is a holomorphic function on {Re z < ω}.

Define

(14) O

_v^(M^p⁾

(Re z < ω)

= {F ∈ O(Re z < ω) :

for every a < ω there exist h > 0 and C < ∞ such that

|F (z)| ≤ C exp{v Re z + M (|z|/h)} for Re z ≤ a}.

Applying Corollary 1 with ϕ = exp

_z

and Re z ≤ a, by Example 2 we get Theorem 1. Let S ∈ L

^(M(ω)^p⁾⁰

(Γ ) and F (z) = LS(z) for Re z < ω. Then F ∈ O

v^(M^p⁾

(Re z < ω).

Theorem 2. Let ω

¹

≤ ω

2

, S

1

∈ L

^(M_(ω^p⁾⁰

1)

(Γ ) and S

2

∈ L

^(M_(ω^p⁾⁰

2)

(Γ ). If (15) LS

₁

(z) = LS

2

(z) for Re z < ω

1

then S

1

= S

2

in L

^(M_(ω^p⁾⁰

1)

(Γ ).

(9)

P r o o f. We have to prove that for arbitrary ϕ ∈ L

^(M_(ω^p⁾

1)

(Γ ), S

1

[ϕ] = S

2

[ϕ].

To this end fix ϕ ∈ L

^(M_(ω^p⁾

1)

(Γ ), choose b < ω

1

such that ϕ ∈ L

^(M_b ^p⁾

(Γ ) and take b < a < ω

1

. Since by (15), S

1

[exp

_c

] = S

2

[exp

_c

] for c ≤ a the proof follows from Proposition 1.

To prove the converse of Theorem 1 we need two lemmas. The first one is a restatement of Lemma 9.1 of [11] (cf. also [12]).

Lemma 2. Let a ∈ R. Suppose that G is holomorphic on the set {Re z ≤ b} and satisfies there the estimate

|G(z)| ≤ C

hzi

²

e

^{v Re z}

with some C < ∞, v ∈ R. Put

g(y) = 1 2πi

R

c+iR

G(z)e

^−zy

dz for y ∈ R.

Then g does not depend on the choice of c ≤ b; it is a continuous function on R with support in Γ = [v, ∞); the function Γ 3 y → e

^by

g(y) is bounded ; g ∈ L

^(M_(b)^p⁾⁰

(Γ ) and G(z) = Lg(z) for Re z < b.

Lemma 3. Let ω ∈ R and k > 0. Then there exists a symbol P of class e (M

p

) not vanishing on {Re z < ω + 1} such that e

(16) exp M (k|z|)

P (z) ≤ 1

hzi

²

for Re z ≤ ω. e

P r o o f. Since m

p

→ ∞ as p → ∞ (by (M.3

⁰

)) we can find p

0

∈ N such that m

p

> 2k| ω| + k and |m e

p

− kz| ≥ k|z| for p ≥ p

₀

and Re z ≤ ω. Put e

P (z) = (z − e ω − 1)

^p⁰⁺¹

∞

Y

p=p0

1 − kz

m

p

for z ∈ C.

Then P does not vanish on {Re z < e ω} and by the Hadamard factorization theorem (cf. [2], Propositions 4.5 and 4.6) it is a symbol of class (M

p

). On the other hand, if Re z ≤ ω we estimate from below: e

∞

Y

p=p0

1 − kz

m

p

≥

∞

Y

p=p0

1 − k| ω| e m

p

sup

q≥p0

q

Y

p=p0

1 − kz m

p

≥

∞

Y

p=p0

1 − k| ω| e m

p

sup

q≥p0

q

Y

p=p0

k|z|

m

p

= C|z|

^−p⁰⁺¹

exp M (k|z|),

(10)

where

C = M

p0−1

M

0

∞

Y

p=p0

1 − k| ω| e m

p

> 0.

Hence, possibly multiplying P by a suitable constant, we get (16).

Theorem 3. Let ω ∈ R∪{∞} and let F ∈ O

v^(M^p⁾

(Re z < ω). Then there exists a Laplace ultradistribution S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) such that

(17) F (z) = LS(z) for Re z < ω.

P r o o f. Fix a < ω. Choose ω ∈ R such that a < e ω < ω and assume e (14). By Lemma 3 we can find a symbol P of class (M

p

), not vanishing on {Re z < e ω + 1} and satisfying (16). Next we apply Lemma 2 to the function

G(z) = F (z)

P (z) , Re z ≤ e ω.

We get a continuous function g which belongs to L

^(M_(a)^p⁾⁰

(Γ ) and satisfies Lg(z) = G(z) for Re z < a. Put S = P (−D)g. Then S ∈ L

^(M_(a)^p⁾⁰

(Γ ) and LS(z) = P (z)Lg(z) = F (z) for Re z < a.

Thus for every a < ω we can find S

a

∈ L

^(M_(a)^p⁾⁰

(Γ ) such that LS

a

(z) = F (z) for Re z < a. By Theorem 2 the definition S = S

a

on L

^(M_(a)^p⁾

(Γ ), a < ω, defines correctly a functional S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) which satisfies (17).

It follows from the proof of Theorem 3 that Laplace ultradistributions can be characterized as follows.

Theorem 4 (Structure theorem). An ultradistribution S ∈ D

^(M^p⁾⁰

(R) is in L

^(M_(ω)^p⁾⁰

(Γ ) if and only if for every a < ω there exist an ultradifferential operator P

a

of class (M

p

) and a function g

a

continuous on R with support in Γ such that

|g

_a

(y)| ≤ Ce

^−ay

for y ∈ Γ,

|Lg

_a

(z)| ≤ C

hzi

²

e

^{v Re z}

for Re z < a and

(18) S = P

a

(D)g

a

in L

^(M_(a)^p⁾⁰

(Γ ).

3. Boundary value representation. In this section we use the follow- ing version of the Phragm´ en–Lindel¨ of theorem.

3-line theorem ([1]). Let R > 0 and F ∈ O(Γ

R

) ∩ C

⁰

(Γ

R

). Suppose that for some k > 0 the function

Γ

R

3 z → exp{−k|z|}F (z)

(11)

is bounded. If F is bounded on the boundary of Γ

R

then it is also bounded on Γ

R

.

Definition. Let R > 0, k > 0 and a ∈ R. We define the spaces L e

a

(Γ

R

) = {ϕ ∈ O(Γ

R

) ∩ C

⁰

(Γ

R

) : kϕk

a,ΓR

= sup

z∈ ¯ΓR

|ϕ(z)e

^az

| < ∞},

L e

^(M_a,k^p⁾

(Γ

R

\ Γ ) = {ϕ ∈ O(Γ

R

\ Γ ) : ϕ · exp{−M

^∗

(k/|Im z|)} ∈ C

⁰

(Γ

R

), kϕk

^(M_a,k,Γ^p⁾

R

= sup

z∈ ¯ΓR

|ϕ(z) exp{az − M

^∗

(k/|Im z|)}| < ∞}.

By the 3-line theorem e L

a

(Γ

R

) is a closed subspace of the Banach space L e

^(M_a,k^p⁾

(Γ

R

\ Γ ) and we can define

H

_a,k^(M^p⁾

(Γ

R

, Γ ) = e L

^(M_a,k^p⁾

(Γ

R

\ Γ )/e L

a

(Γ

R

).

Further, we define L e

a

(C) = lim ←−

R→∞

L e

a

(Γ

R

), L e

^(M_a,k^p⁾

(C \ Γ ) = lim ←−

R→∞

L e

^(M_a,k^p⁾

(Γ

R

\ Γ ), L e

^(M_a ^p⁾

(C \ Γ ) = lim −→

k→∞

L e

^(M_a,k^p⁾

(C \ Γ ), H e

_a^(M^p⁾

(C, Γ ) = e L

^(M_a ^p⁾

(C \ Γ )/e L

a

(C),

e

H

_a^(M^p⁾

(Γ ) = lim −→

R→0 k→∞

H

_a,k^(M^p⁾

(Γ

R

, Γ ).

Let a < b. Then the natural mappings H e

_b^(M^p⁾

(C, Γ ) → e H

_a^(M^p⁾

(C, Γ ),

e

H

_b^(M^p⁾

(Γ ) → e

H

_a^(M^p⁾

(Γ )

are well defined and by the 3-line theorem they are injections. Thus, for ω ∈ R ∪ {∞}, we can define

H e

_(ω)^(M^p⁾

(C, Γ ) = lim ←−

a<ω

H e

_a^(M^p⁾

(C, Γ ), e

H

_(ω)^(M^p⁾

(Γ ) = lim ←−

a<ω

e

H

_a^(M^p⁾

(Γ ).

An element f ∈ e H

_(ω)^(M^p⁾

(C, Γ ) is given by a set {F

a

}

_a<ω

of functions such that for a < ω, F

a

∈ e L

^(Ma ^p⁾

(C \ Γ ) and for a < b < ω, F

a

− F

_b

∈ e L

a

(C).

On the other hand, an element g ∈ e

H

_(ω)^(M^p⁾

(Γ ) is given by a set {G

a

}

_a<ω

of functions such that for a < ω there exist R

a

> 0 and k

a

< ∞ such that G

a

∈ e L

^(M_a,k^p⁾

a

(Γ

Ra

\ Γ ) and for a < b < ω, G

_a

− G

_b

∈ e L

a

(Γ

Ra

∩ Γ

_R_b

, Γ ).

The natural mapping

(19) i : e H

_(ω)^(M^p⁾

(C, Γ ) → e

H

_(ω)^(M^p⁾

(Γ )

is defined by retaining the same set of defining functions. Obviously it is an

injection.

(12)

Lemma 4. Let S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) and a < ω. Put C

_a

S(z) = 1

2πi S e

^a(·−z)

z − ·

for z ∈ C \ Γ.

Then C

a

S ∈ e L

^(Ma ^p⁾

(C \ Γ ). Furthermore, if a < b < ω then C

^a

S − C

b

S ∈ L e

a

(C).

P r o o f. Take a < c < ω. By Theorem 4 we can find an ultradifferential operator P

c

of class (M

p

) and a continuous function g

c

with support in Γ satisfying |g

c

(y)| ≤ Ce

^−cy

for y ∈ Γ and S = P

c

(D)g

c

in L

^(M_(c)^p⁾⁰

(Γ ). Since for fixed z ∈ C \ Γ the function

Γ 3 y → e

^a(y−z)

z − y belongs to L

^(M_(c)^p⁾

(Γ ) we have

C

_a

(z) = 1

2πi e

^−az

R

Γ

g

c

(y)P

_c^∗

(D

y

)

e

^ay

z − y

dy.

Let P

_c^∗

(D) = P

∞

α=0

(−1)

^α

a

α

D

^α

with a

α

satisfying (5) and let R > 0. Then for z ∈ Γ

R

\ Γ we estimate

P

_c^∗

(D)

e

^ay

z − y

≤

∞

X

α=0

|a

α

|

D

^α

e

^ay

z − y

≤ e

^ay

∞

X

β=0

|a|

^β

β!

∞

X

α=β

|a

_α

|α!

|Im z|

^α−β+1

≤ e

^ay

∞

X

β=0

|a|

^β

R

^β

β!

∞

X

α=0

|a

α

|α!

|Im z|

^α+1

≤ e

^ay+|a|R

· AC

HK exp M

^∗

2HK

|Im z|

since by (5) and (M.2

⁰

) we have

∞

X

α=0

|a

α

|α!

|Im z|

^α+1

≤ C

∞

X

α=0

K

^α

α!

|Im z|

^α+1

M

α

≤ 2C sup

α∈N0

(2K)

^α

α!

|Im z|

^α+1

M

α

≤ 2AC 2HK sup

α∈N0

(2HK)

^α+1

α!

|Im z|

^α+1

M

α+1

≤ AC

HK exp M

^∗

2HK

|Im z|

. Put k = 2HK. Then for every R > 0 there exists C < ∞ such that

|C

_a

S(z)| ≤ C exp{− a Re z + M

^∗

(k/|Im z|)} for z ∈ Γ

R

\ Γ.

Thus, C

a

S ∈ e L

^(Ma ^p⁾

(C \ Γ ). If a < b < ω we take c < ω such that b < c and

(13)

note that for z ∈ C the function

Γ 3 y → e

^a(y−z)

− e

^b(y−z)

z − y

belongs to L

^(M_(c)^p⁾

(Γ ). Thus, the holomorphic extension of C

a

S − C

b

S is given by

(C

a

S − C

b

S)(z) = 1 2πi

R

Γ

g

c

(y)P

_c^∗

(D) e

^a(y−z)

− e

^b(y−z)

z − y

dy

and we easily find that C

a

− C

_b

∈ e L

a

(C).

Definition. Let S ∈ L

^(M_(ω)^p⁾⁰

(Γ ). Then by Lemma 4 the set {C

a

S}

a<ω

of functions defines an element f ∈ e H

_(ω)^(M^p⁾

(Γ ). We call f the Cauchy transform of S and write f = CS.

Proposition 2. Let F

a

∈ e L

^(M_a,k^p⁾

(Γ

R

\Γ ) with a ≤ 0. Then there exist an ultradifferential operator P

a

(D) of class (M

p

) and functions H

_a^±

∈ O(Γ

R

∩ {± Im z > 0 or Re z < v}) such that

1

^◦

P

a

(D)H

_a^±

= F

a

;

2

^◦

For every 0 < R

⁰

< R and a

⁰

< a there exists C < ∞ such that

|H

_a^±

(z)| ≤ C exp{−a

⁰

Re z} for z ∈ Γ

R⁰

∩ {± Im z > 0};

3

^◦

H

_a^±

(x + iy) converges uniformly as y → 0+ to a function h

^±_a

contin- uous on (v − R, ∞) and analytic on (v − R, v) satisfying

|h

^±_a

(x)| ≤ C exp{−a

⁰

x} for x ∈ (v − R

⁰

, ∞).

Furthermore, if we put

S

a

= F

_a⁺

(x + i0) − F

_a⁻

(x − i0), where F

_a^±

(x ± i0) = P

a

(D)h

^±_a

, then S

a

extends to a Laplace ultradistribution e S

a

∈ L

^(M_(a)^p⁾⁰

(Γ ) defined by (20) S e

a

[ϕ] =

∞

R

v−R

(h

⁺_a

(x) − h

⁻_a

(x))P

_a^∗

(D)E ϕ(x) dx for ϕ ∈ L

^(M_(a)^p⁾

(Γ ).

In (20), E is a linear continuous extension mapping E : L

^(M_(a)^p⁾

(Γ ) → L

^(M_(a)^p⁾

([v − R, ∞)), which exists by the Seeley extension theorem.

P r o o f. Put

P (ζ) = (1 + ζ)

²

∞

Y

p=1

1 + kζ

m

p

for ζ ∈ C.

(14)

Then P is a symbol of class (M

p

). Define the Green kernel for P by G(z) = 1

2πi

∞

R

0

e

^zζ

P (ζ) dζ for Re z < 0.

Then by Lemma 11.4 of [2], G can be holomorphically continued to the Riemann domain {z : −π/2 < arg z < 5π/2}, on which we have

P (D)G(z) = − 1 2πi

1 z . Furthermore, since for any 0 ≤ ψ < π/2,

|z|

|1 − z| ≤ 1

cos ψ and 1

|1 − z| ≤ 1 cos ψ

for z ∈ C with |arg z| ≤ ψ + π/2, following the proof of the above-mentioned lemma we conclude that on the domain {−ψ ≤ arg z ≤ 2π + ψ}, G is bounded by C/cos ψ with C < ∞ not depending on ψ. We also have

|g(x)| ≤ A √

x exp{−M

^∗

(k/x)} for x > 0, where

g(z) = G

+

(z) − G

₋

(e

^2πi

z) for Re z > 0,

G

+

being the branch of G on {−π/2 < arg z < π/2} and G

₋

that on {3π/2 < arg z < 5π/2}. Put

H

_a^±

(z) = ±i R

γ±

G(±i(z − w))F

a

(w) dw,

where γ

±

is a closed curve encircling z once, in the anticlockwise direction, such that −π/2 < arg(±i(z − w)) < 5π/2 for w ∈ γ

±

. We choose a starting point z

^◦±

of γ

±

in such a way that |arg{±i(z − z

^◦±

)}| < π/2. Then H

_a^±

is a holomorphic function on Γ

R

∩ {± Im z > 0 or Re z < v} and does not depend on the choice of γ

±

with a fixed starting point z

^◦±

. For a fixed γ and z changing in a compact set in the domain bounded by γ we have

P (D)H

_a^±

(z) = ±i R

γ

P (D

z

)G(±i(z − w))F

a

(w) dw

= −1 2πi

R

γ

F

a

(w)

z − w dw = F

a

(z).

Let 0 < R

⁰

< R and z ∈ Γ

R⁰

∩{Im z > 0}. Fix z

^◦

∈ Γ

R

∩{Im z > r

⁰

}∩{Re z <

v − R

⁰

} and take γ

₊

= γ

1

∪ γ

₂

∪ γ

₃

∪ γ

₄

, where γ

1

= [x

^◦

+ iy

^◦

, x + iy

^◦

],

γ

2

= [x + iy

^◦

, x + iy], γ

3

= [x + iy, x + iy

^◦

] and γ

4

= [x + iy

^◦

, x

^◦

+ iy

^◦

]. Since

0 ≤ arg(i(z − w)) ≤ ψ for w ∈ γ

1

and 2π ≤ arg(i(z − w)) ≤ 2π + ψ for

w ∈ γ

4

, where 0 ≤ ψ < π/2 is such that tan ψ = (x − x

^◦

)/(y

^◦

− y), by the

(15)

boundedness of G on {0 ≤ arg ψ ≤ 2π + ψ} we have |G(i(z − w))| ≤ Cx for w ∈ γ

1

∪ γ

₄

, where C does not depend on x. So

R

γ1∪γ4

G(i(z − w))F

a

(w) dw

≤ Cx

²

e

^−ax

for z ∈ Γ

R⁰

∩ {Im z > 0}.

On the other hand,

R

γ2∪γ3

G(i(z − w))F

a

(w) dw =

− i

y^◦−y

R

0

g(t)F (x + i(y + t)) dt

≤ AC

y^◦−y

R

0

√ t exp

M

^∗

k y + t

− M

^∗

k t

− ax

dt

≤ AC(y

^◦

− y)

^3/2

e

^−ax

for z ∈ Γ

R⁰

∩ {Im z > 0}.

Thus, for any a

⁰

< a one can find C < ∞ such that

|H

_a⁺

(z)| ≤ Ce

^−a⁰^x

for z ∈ Γ

R⁰

∩ {Im z > 0}.

The estimate of H

_a⁻

is obtained in an analogous way.

The assertion 3

^◦

is clear from the above estimates.

Let ψ ∈ D

^(M^p⁾

((v − R, ∞)). By 1

^◦

and 2

^◦

we derive F

a

(x ± i0)[ψ] = P

a

(D)h

^±_a

[ψ] =

∞

R

v−R

h

^±_a

(x)P

_a^∗

(D)ψ(x) dx

= lim

y→0+

∞

R

v−R

H

a

(x ± iy)P

_a^∗

(D)ψ(x) dx

= lim

y→0+

∞

R

v−R

F

a

(x ± iy)ψ(x) dx.

Since for ψ ∈ D

^(M^p⁾

((v − R, v)), S

a

[ψ] = lim

y→0+

v

R

v−R

P

a

(D)(H

_a⁺

(x + iy) − H

_a⁻

(x − iy))ψ(x) dx = 0, S

a

has support in Γ and we can define the extension of S

a

by (20).

Let f ∈ e

H

a^(M^p⁾

(Γ ). Then there exist R > 0, k < ∞ and a function

F

a

∈ e L

^(M_a,k^p⁾

(Γ

R

\ Γ ) such that f = [F

_a

]. If a ≤ 0 we can apply Proposition 2

to F

a

. If a > 0 then we apply Proposition 2 to F

_a^∗

= e

^az

F

a

instead of F

a

.

In this case denote by e S

_a^∗

the element of L

^(M₍₀₎^p⁾⁰

(Γ ) given by (20) and define

S e

a

= e

^−ax

S e

_a^∗

. In both cases e S

a

∈ L

^(M_(a)^p⁾⁰

(Γ ) does not depend on the choice

of a defining function F

a

for f . Thus, the assignment f → e S

a

defines a

(16)

mapping

(21) b :

e

H

_a^(M^p⁾

(Γ ) → L

^(M_(a)^p⁾⁰

(Γ ).

Since (21) holds for every a < ω we have b :

e

H

_(ω)^(M^p⁾

→ lim ←−

a<ω

L

^(M_(a)^p⁾⁰

(Γ ) ' ( lim −→

a<ω

L

^(M_(a)^p⁾

(Γ ))

⁰

= L

^(M_(ω)^p⁾⁰

(Γ ), where the isomorphism ' follows by the formula (1.2) of [2].

Theorem 5. The mapping

C : L

^(M_(ω)^p⁾⁰

(Γ ) → e H

_(ω)^(M^p⁾

(C, Γ ) is a topological isomorphism with inverse b ◦ i.

P r o o f. Let S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) and CS = f ∈ e H

_(ω)^(M^p⁾

(Γ ). Then f = [{F

a

}

_a<ω

] with F

a

(z) = 1

2πi S e

^a(·−z)

z − ·

for z ∈ C \ Γ.

Treat f as an element of e

H

_(ω)^(M^p⁾

(Γ ) and put e S = b(f ) ∈ L

^(M_(ω)^p⁾⁰

(Γ ). Fix a < ω. Then for ϕ ∈ Y

_(a)

we have

S[ϕ] = − e R

∂Γε

F

a

(z)ϕ(z) dz = S

− 1 2πi

R

∂Γε

e

^a(·−z)

z − · ϕ(z) dz

= S[ϕ],

by the Cauchy integral formula. Since Y

_(a)

is dense in L

^(M_(a)^p⁾

(Γ ) and a < ω is arbitrary, we have e S[ϕ] = S[ϕ] for ϕ ∈ L

^(M_(ω)^p⁾

(Γ ). Thus b ◦ i ◦ C = id.

Let f ∈ e H

_(ω)^(M^p⁾

(C, Γ ), f = [{F

^a

}

a<ω

] with F

a

∈ e L

^(Ma ^p⁾

(C \ Γ ) and F

^a

− F

b

∈ e L

a

(C) for a < b < ω. Put

e

f = i(f ) and fix a < ω. Then e

f = [F

a

] in e

H

a^(M^p⁾

(Γ ) and by (21), e S

a

= b(

e

f ) ∈ L

^(M_(a)^p⁾⁰

(Γ ). Observe that for ε > 0 we have

S e

a

[ϕ] = − R

∂Γε

F

a

(z)ϕ(z) dz for ϕ ∈ Y

(a)

. On the other hand, by the part of the theorem just proved,

S e

a

[ϕ] = − R

∂Γε

1 2πi S e

a

e

^a(·−z)

z − ·

ϕ(z) dz for ϕ ∈ Y

(a)

.

So for ϕ ∈ Y

(a)

, (22) R

∂Γε

ψ

a

(z)ϕ(z) dz = 0, where ψ

a

(z) = 1 2πi S e

a

e

^a(·−z)

z − ·

− F

_a

(z).

(17)

Then ψ

a

∈ e L

^(M_a,k^p⁾

(C \ Γ ) and we shall show that ψ

a

extends holomorphically to a function e ψ

a

∈ e L

a

(C), which proves that C ◦b◦i = id. To this end observe that (22) holds also for ϕ ∈ L

^(M_(a)^p⁾

(Γ ) ∩ O(Γ

ε

) and put for any b < a, R > ε,

G

b

(z) = R

∂ΓR

ψ

a

(ζ) e

^b(ζ−z)

z − ζ dζ for z ∈ Γ

R

.

Then |G

b

(z)| ≤ C exp{−b Re z} for z ∈ Γ

R⁰

with R

⁰

< R. Using (22) with ϕ(ζ) = exp{b(ζ − z)}/(z − ζ), z ∈ Γ

R

\ Γ

_ε

, we get G

b

(z) = ψ

a

(z) for z ∈ Γ

R

\ Γ

_ε

. Put

ψ e

a

(z) = ψ

_a

(z) for z ∈ Γ

R

\ Γ , G

b

(z) for z ∈ Γ

R

.

Then e ψ

a

∈ O(Γ

_R

) and by the 3-line theorem e ψ

a

∈ e L

a

(Γ

R

). Since R > ε was arbitrary we have e ψ ∈ e L

a

(C).

4. Mellin ultradistributions

Definition. Let ω ∈ R ∪ {∞}, v ∈ R and I = (0, e

^−v

]. We define the space M

_(ω)^(M^p⁾⁰

(I) of Mellin ultradistributions as the dual space of

M

_(ω)^(M^p⁾

(I) = lim −→

a<ω

lim ←−

h>0

M

_a,h^(M^p⁾

(I), where for any a ∈ R and h > 0,

M

_a,h^(M^p⁾

(I) =

ψ ∈ C

^∞

(I) : %

^(M_a,h^p⁾

(ψ) = sup

x∈I

sup

α∈N0

|x

^a+1

(Dx)

^α

ψ(x)|

h

^α

M

α

< ∞

. Lemma 5. Let a ∈ R, h > 0, ψ ∈ M

a,h^(M^p⁾

(I) and ϕ = µ · ψ ◦ µ. Then ϕ ∈ L

^(M_a,h^p⁾

(Γ ) and kϕk

^(M_a,h^p⁾

= %

^(M_a,h^p⁾

(ψ). Thus, the mapping

M

_(ω)^(M^p⁾

(I) 3 ψ → µ · ψ ◦ µ ∈ L

^(M_(ω)^p⁾

(Γ ) is a continuous isomorphism with inverse

L

^(M_(ω)^p⁾

(Γ ) 3 ϕ → exp

₁

◦µ

⁻¹

· ϕ ◦ µ

⁻¹

∈ M

_(ω)^(M^p⁾

(I).

P r o o f. The proof follows easily from the formula

D

^α_y

(µ(y)ψ ◦ µ(y)) = (−1)

^α

x(D

x

x)

^α

ψ(x), for α ∈ N

0

, x = µ(y), which can be proved by induction.

Let S ∈ L

^(M_(ω)^p⁾⁰

(Γ ). Put

S ◦ µ

⁻¹

[ψ] = S[µ · ψ ◦ µ] for ψ ∈ M

_(ω)^(M^p⁾

(I).

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Then by Lemma 5, S ◦ µ

⁻¹

is a well defined element of M

_(ω)^(M^p⁾⁰

(I) and the mapping

L

^(M_(ω)^p⁾⁰

(Γ ) 3 S → S ◦ µ

⁻¹

∈ M

_(ω)^(M^p⁾⁰

(I) is continuous.

Observe that the function

I 3 x → x

^−z−1

= exp

_z+1

◦µ

⁻¹

(x)

belongs to M

_(ω)^(M^p⁾

(I) if and only if Re z < ω. Thus, we can define the Mellin transform of T ∈ M

_(ω)^(M^p⁾⁰

(I) by

MT (z) = T [exp

_z+1

◦µ

⁻¹

] for Re z < ω.

Let S ∈ L

^(M_(ω)^p⁾⁰

(Γ ) and T = S ◦ µ

⁻¹

. Then for Re z < ω we have MT (z) = S ◦ µ

⁻¹

[exp

_z+1

◦µ

⁻¹

] = S[exp

_z

] = LS(z).

5. Strong quasi-analyticity principle

Definition. Let S ∈ L

^(M_(ω)^p⁾⁰

(Γ ). We define the Taylor transform of S by

T S(x) = LS(ln x) for x ∈ e B(e

^ω

).

We also define O

^(M_v ^p⁾

( e B(e

^ω

))

= {u ∈ O( e B(e

^ω

)) :

for every t < e

^ω

there exist k < ∞ and C < ∞ such that

|u(x)| ≤ C exp{M (k(ω − ln |x| + |arg x|))} · |x|

^v

for |x| ≤ t}.

By Theorems 1 and 3 we get

Theorem 6. The Taylor transformation is an isomorphism of L

^(M_(ω)^p⁾⁰

(Γ ) onto O

^(Mv ^p⁾

( e B(e

^ω

)).

Let u ∈ O

^(Mv ^p⁾

( e B(e

^ω

)). Then for any t < e

^ω

, u

_|(0,t]

∈ M

_(v)^(M^p⁾⁰

((0, t]) and M

t

u(z) =

t

R

0

u(x)x

^−z−1

dx for Re z < v.

By Theorem 6, u(x) = S[x

^·

] for x ∈ e B(e

^ω

) with S = T

⁻¹

u ∈ L

^(M_(ω)^p⁾⁰

(Γ ), Γ = [v, ∞). For Re z < v we derive

M

_t

u(z) = S h R

^t

0

x

^·−z−1

dx i

= S t

^·−z

· − z

= −2πiC

ln t

S(z).

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Thus, by Lemma 4, M

t

u extends holomorphically to a function M

t

u ∈ L e

^(M_{ln t}^p⁾

(C \ Γ ) and the set of functions {M

^t

u}

t<e^ω

defines an element of H e

_(ω)^(M^p⁾

(C, Γ ), which will be denoted by Mu and called the Mellin transform of u.

We can summarize Theorems 1, 3, 5 and 6 as follows:

Corollary 2. We have the following diagram of linear topological iso- morphisms:

M

_(ω)^(M^p⁾⁰

(I) O

v^(M^p⁾

(Re z < ω)

L

^(M_(ω)^p⁾⁰

(Γ ) O

v^(M^p⁾

( e B(e

^ω

))

e

H

_(ω)^(M^p⁾

(Γ ) H e

_(ω)^(M^p⁾

(C, Γ ).

M

//

◦µ

^◦−µ

−1

T

//

L

ooo ooo ooo oo77

OOOOO

C

OOOOO ''

◦µ⁻¹

OO

M b

OO

oo

i

Following [13] we call the elements of O

v^(M^p⁾

(Re z < ω) generalized ana- lytic functions determined by L

^(M_(ω)^p⁾⁰

(Γ ). Generalized analytic functions have the following quasi-analyticity property:

Theorem 7. Let u ∈ O

v^(M^p⁾

( e B(e

^ω

)). Suppose that for some t < e

^ω

and every m ∈ N there exist C

m

such that

|u(x)| ≤ C

_m

x

^m

for 0 < x ≤ t.

Then u ≡ 0.

P r o o f. By Theorem 6, u(x) = T S(x) for x ∈ e B(e

^ω

) with some S ∈ L

^(M_(ω)^p⁾⁰

(Γ ). The assumption that u is flat of arbitrary order m ∈ N on (0, t) implies that M

t

u ∈ O(C). Since for every R > 0, L

^(M_a,k^p⁾

(Γ

R

)∩O(Γ

R

) = L

a

(Γ

R

), Mu defines the zero element in e H

_(ω)^(M^p⁾

(Γ ). Thus S = 0 and u ≡ 0.

Theorem 8 (Strong quasi-analyticity principle). Let −π/2 < θ < π/2, l

θ

= {z = re

^iθ

: r > 0} and F ∈ O(Re z > 0). Suppose that for some v ∈ R and every κ > 0 there exist k < ∞ and C < ∞ such that

(23) |F (z)| ≤ C exp{v Re z + M (k|z|)} for Re z ≥ κ.

If for some τ > 0 and every m ∈ N there exists C

m

< ∞ such that (24) |F (z)| ≤ C

m

e

^{−m Re z}

for z ∈ l

θ

, Re z ≥ τ,

then F ≡ 0.

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P r o o f. Put u(x) = F ◦ µ

⁻¹

(x) for x ∈ e B(1). Then u ∈ O

v^(M^p⁾

( e B(1)).

Set t = e

^−τ

, let γ

t,θ

be the set of x ∈ e B(1) that satisfy

x = t exp{−ir sin θ} for 0 ≤ r ≤ τ /cos θ, exp{−r(cos θ + i sin θ)} for r ≥ τ /cos θ, and observe that

(25) M

_t

u(z) = R

γt,θ

u(x)x

^−z−1

for z ∈ Ω

v,θ

,

where Ω

v,θ

= {z ∈ C : Re z < v and sin θ Im z > cos θ(Re z − v)}. Using (24) we infer that the right hand side of (25) is defined for z ∈ C. Thus, M

t

u ∈ O(C). As in the proof of Theorem 7 this implies that u ≡ 0 and hence F ≡ 0.

R e m a r k 3. The conclusion of Theorem 8 does not hold if instead of (23) we assume that for every ε > 0 and κ > 0 there exists C

ε,κ

such that

|F (z)| ≤ C

_ε,κ

exp{v Re z + ε|z|} for Re z ≥ κ.

In this case the function u = F ◦ µ

⁻¹

is the Taylor transform of an analytic functional with carrier at {∞} and need not be equal to zero.

R e m a r k 4. The results of the paper can be easily extended to the n-dimensional case if Γ is a cone of product type. The case of an arbi- trary convex, proper cone in R

ⁿ

is more difficult and will be studied in a subsequent paper.

References

[1] E. H i l l e, Analytic Function Theory , Vol. 2, Chelsea, New York, 1962.

[2] H. K o m a t s u, Ultradistributions, I. Structure theorems and a characterization, J.

Fac. Sci. Univ. Tokyo 20 (1973), 25–105.

[3] —, Ultradistributions, II. The kernel theorem and ultradistributions with support in a submanifold , ibid. 24 (1977), 607–628.

[4] M. L a n g e n b r u c h, Bases in spaces of ultradifferentiable functions with compact support , Math. Ann. 281 (1988), 31–42.

[5] G. L y s i k, Generalized analytic functions and a strong quasi-analyticity principle, Dissertationes Math. 340 (1995), 195–200.

[6] S. M a n d e l b r o j t, S´ eries adh´ erentes, r´ egularisation de suites, applications, Gau- thier-Villars, Paris, 1952.

[7] R. M e i s e and A. T a y l o r, Linear extension operators for ultradifferentiable func- tions of Beurling type on compact sets, Amer. J. Math. 111 (1989), 309–337.

[8] M. M o r i m o t o, Analytic functionals with non-compact carrier , Tokyo J. Math. 1 (1978), 77–103.

[9] S. P i l i p o v i ´ c, Tempered ultradistributions, Boll. Un. Mat. Ital. B (7) 2 (1988),

235–251.

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[10] C. R o u m i e u, Ultra-distributions d´ efinies sur R

ⁿ

et sur certaines classes de vari´ et´ es diff´ erentiables, J. Anal. Math. 10 (1962-63), 153–192.

[11] Z. S z m y d t and B. Z i e m i a n, The Mellin Transformation and Fuchsian Type Par- tial Differential Equations, Kluwer, Dordrecht, 1992.

[12] A. H. Z e m a n i a n, Generalized Integral Transformations, Interscience, 1969.

[13] B. Z i e m i a n, Generalized analytic functions, Dissertationes Math., to appear.

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