On the Range of Two Convolution Operators

Math. Nachr. 216 (2000), 61 – 71

On the Range of Two Convolution Operators

By Carmen Fernandez of Valencia

(Received December 29, 1997)

(Revised Version January 14, 2000)

Abstract. Let E(ω)(IR) denote the non–quasianalytic class of Beurling type on IR. For

µ, ν ∈ E ′(ω)

(IR) we give necessary conditions for the inclusion Tν(E(ω)(IR)) ⊂ Tµ(E(ω)(IR)), thus

extending previous work of Malgrange and Ehrenpreis.

1. Introduction

The problem of characterizing surjective convolution operators in various spacesof infinitely differentiable functions and their duals has been investigated by manyauthors (see e. g. Malgrange [M], Ehrenpreis [E],Hormander [H1], Cioranescu

[C], Meise, Taylor and Vogt [MTV], Braun, Meise and Vogt [BRMV], Bonet,

Galbis and Meise [BGM]).In a recent paper, Bonet and Galbis [BG] studied the range of non – surjective

convolution operators in the non – quasianalytic classes of Beurling type E(ω)

(IRn

).

They analyzed which classes of (real analytic or) non –quasianalytic functions are stillcontained in the range of a non – surjective convolution operator.In the present note we investigate the range of the non – surjective convolution op-

erators on the non– quasianalytic classes of Beurling type from another poin of view.The problem we are interested in is the following. Fix a weight function ω and two ul-tradistributions µ, ν ∈ E ′

(ω)(IR). Under which conditions on µ and ν does Tµ(E(ω)(IR)

)contain Tν

(E(ω)(IR))?. Malgrange [M] and Ehrenpreis [E] considered the same

problem in the space of all infinitely differentiable functions.Our approach to the non – quasianalytic classes of Beurling type is as in Braun,

Meise and Taylor [BRMT].Although the paper only considers the one variable case, some results also hold for

several variables. This will be indicated in the proper places.

1991 Mathematics Subject Classification. 46F05, 46E10, 46F10, 35R50.Keywords and phrases. Ultradifferentiable functions, convolution operators.

62 Math. Nachr. 216 (2000)

Definition 1.1. Let ω : IR→ [0,∞[ be a continuous even function which is increas-ing on [0,∞[ and satisfies ω(0) = 0 and ω(1) > 0. ω is called a weight function if itsatisfies the following conditions:(α) ω(2t) ≤ K(1 + ω(t)) for all t ∈ IR;(β)

∫ ∞−∞

ω(t)1+t2

<∞;(γ) log

(1 + t2

)= o(ω(t)) as t tends to ∞;

(δ) ϕ(t) = ω(et

)is convex in IR.

For a weight function ω we define ω : C → [0,∞[ by ω(z) = ω(|z|) and againcall this function ω, by abuse of notation. The Young conjugate of ϕ is defined byϕ∗(x) = supy>0xy− ϕ(y).

Remark 1.2. (a) Each weight function ω satisfies limt→∞ω(t)t = 0 by the remark

following 1.3 of [MTV].(b) For each weight function ω there exists a weight function σ satisfying σ(t) = ω(t)

for all large t > 0 and σ∣∣[0,1[

= 0. This implies ϕσ(y) = ϕω(y) for all large y, ϕ∗∗σ = ϕσ.

From this it follows that all subsequent definitions do not change if ω is replaced by σ.On the other hand they also do not change if ω is replaced by ω + c, c some positivenumber.

Definition 1.3. Let ω be a weight funcion. We define

E(ω)(IR) := f ∈ C∞(IR) : ‖f‖K, <∞ ∀K ⊂⊂ IR, ∀ ∈ IN

where

‖f‖K, := supx∈K

supα

∣∣f(α)(x)∣∣ exp(

−ϕ∗( |α|

)).

The elements of E(ω)(IR) are called ω – ultradifferentiable functions of Beurling typeon IR. For a compact set K in IR we set

D(ω)(K) =f ∈ E(ω)(IR) : supp(f) ⊂ K

endowed with the induced topology. We put D(ω)(IR) := indj→D(ω)

(Kj

), where(

Kj)j∈IN

is any fundamental sequence of compact sets in IR. The dual D′(ω)(IR) of

D(ω)(IR) is endowed with its strong topology. The elements of D′(ω)(IR) are called

ω –ultradistributions of Beurling type on IR.

Example 1.4. The following functions ω : [0,∞[→ [0,∞[ (after a suitable changeon [0, A] for some A > 0) are weight functions:(1) ω(t) = tβ, 0 < β < 1. In this case E(ω)(IR) is the Gevrey space of order 1/β;(2) ω(t) = (log(1 + t))β , β > 1;(3) ω(t) = t(log(e+ t))−β, β > 1.

Definition 1.5. For µ ∈ E ′(ω)(IR) the convolution operator Tµ : E(ω)(IR)→ E(ω)(IR)

is defined by Tµ(ϕ)(x) := (µ ∗ ϕ)(x) = 〈µy, ϕ(x − y)〉, which is a continuous linear

Fernandez, The Range of Two Convolutions 63

operator with transpose map T tµ : E ′(ω)(IR)→ E ′

(ω)(IR), given by Ttµ(η) := µ ∗ η, where

µ(ϕ) := µ(ϕ) and ϕ(x) := ϕ(−x). Moreover µ induces a continuous linear operatorSµ : D′

(ω)(IR) → D′(ω)(IR) where Sµ(ν)(ϕ) = 〈µ ∗ ν, ϕ〉 = 〈ν, µ ∗ ϕ〉 for ν ∈ D′

(ω)(IR),and ϕ ∈ D(ω)(IR).

If ω is a weight we put p(z) := ω(z) + |Imz|, z ∈ C. We let Ap denote the (DFS) –space of entire functions

Ap =f ∈ A(C ) : there exists k ∈ IN such that sup

z∈C|f(z)| exp(−kp(z)) <∞

.

It is well known that the Fourier – Laplace transform F : µ → µ, µ(z) :=⟨µx, e

−ixz⟩,is an algebra isomorphism between the convolution algebra

(E ′(ω)(IR), ∗

)andAp, which

is even a topological isomorphism when we endow E ′(ω)(IR) with the strong topology.

We will identify, via Fourier – Laplace transform, the transposed map T tµ with themultiplication operator FT tµF−1 : Ap → Ap, f → ˆµ f .

2. Results

For a given weight function ω and for two ultradistributions µ, ν ∈ E ′(ω)(IR) we give

necessary conditions for the inclusion Tµ(E(ω)(IR)

) ⊃ Tν(E(ω)(IR)

). The functional

analytic tool in our study is the following result due to Bonet, Meise and Taylor

([BMT, Proposition 2.1].

Theorem 2.1. Let E be a Frechet – Schwartz space and assume that T, S ∈ L(E)have dense range. Then, the following conditions are equivalent:(1) S(E) ⊂ T (E).(2) The map T t(E′) ⊂ E′

b → E′b, T

t(x)→ St(x) is continuous.Moreover, if T t(E′) is bornological for the topology induced by E′

b, then (1) and (2)are equivalent to(3) The map T t(E′) ⊂ E′

b → E′b, T

t(x) → St(x) maps bounded sets into boundedsets.

As a consequence of this Theorem we obtain the following extension of a resultof Malgrange ([M, 3.5]), which reduces our original problem to the continuity ofa division–multiplication operator. Observe that the proof also works for severalvariables.

Proposition 2.2. The following conditions are equivalent:(1) Tν

(E(ω)(IR)) ⊂ Tµ

(E(ω)(IR));

(2) For every f ∈ Ap such that f/ ˆµ is an entire function, we have that ˆν f/ ˆµ ∈ Ap.

Proof . (1) ⇒ (2). Applying Theorem 1 to E = E(ω)(IR), T = Tµ and S = Tν ,and taking the Fourier –Laplace transform, condition (1) holds if and only if the mapˆµAp ⊂ Ap → Ap, ˆµ f → ˆν f is continuous. Hence, we may find an extension to the

64 Math. Nachr. 216 (2000)

closure ˆµAp → Ap. In fact, given f ∈ ˆµAp, there is a Cauchy net( ˆµ fα) in ˆµAp

converging to f . Therefore(ˆν fα

)is a Cauchy net in Ap, which converges to some

g ∈ Ap. Now

ˆν f = limαˆν ˆµ fα = lim

αˆµ ˆν fα = ˆµ lim

αˆν fα = ˆµ g .

Since f ∈ ˆµAp if and only if f/ ˆµ is an entire function by [BG, Prop. 1], condition(2) follows.(2) ⇒ (1). Condition (2) implies ˆν ˆµAp ⊂ ˆµAp. Given f ∈ ˆµAp, there is g ∈ Ap

such that ˆν f = ˆµg. Since multiplication is an injection on Ap, g is unique. Therefore,

T : ˆµAp → Ap , f → g

is well – defined, linear and continuous, since it has closed – graph. We can applyTheorem 2.1 to conclude (1).

It is clear now that in order to decide whether the inclusion Tν(E(ω)(IR)

)⊂Tµ(E(ω)(IR))

holds it would be convenient to have some condition in terms of the Fourier – Laplacetransforms of µ and ν, which in the sequel will be denote by F and G respectively.The following definition for distributions is due to Ehrenpreis [E].

Definition 2.3. (a) For f ∈ Ap and l ∈ IR, we writeMl(z, f) = max

w∈C , |Imw|≤|Imz|exp(−l |z − w|) |f(w)| .

(b) Given F, G ∈ Ap we say that F/G is ω – slowly– decreasing if, for each l andevery ε > 0 there is j such that for all z ∈ C

Ml(z, G)1+ε

Ml(z, F )≤ j exp jp(z) .

Remark 2.4. (a)Ml(z, f

)is well defined, that is, the maximum is attained. In fact,

as a function of w, exp(−l |w − z|) |f(w)| is continuous on w ∈ C : |Imw| ≤ |Imz|.Since ω(t) = o(t) we can apply the Paley –Wiener – Schwartz theorem to conclude thatlim|w|→∞, |Imw|≤|Imz| exp(−l |z −w|) |f(w)| = 0.(b) For ν = δ, i. e., for G ≡ 1, F/G is ω slowly decreasing if and only if F is

ω slowly decreasing in the usual sense (see for instance [BGM]). Indeed, let w besuch that |Imw| ≤ |Imz| and exp(−l |z − w|) |F (w)| = Ml(z, F ) and B > 0 with|F (t)| ≤ B expBp(t) for all t ∈ C. Then

Ml(z, G)1+ε

j exp jp(z)≤ exp(−l |z − w|) |F (w)|≤ B exp(Bp(w) − l |z −w|)≤ C exp(−(l − C) |z −w|+Cp(z))

for some constant C > 0. From where it follows

|z − w| ≤ 1l −C

((j + C

)ω(z) + logCj +

(C + j

) |Imz| − (1 + ε) logMl(z, G)).


Since G ≡ 1, logMl(z, G) = 0, and F is ω slowly decreasing. The converse is clear.

Theorem 2.5. Let µ, ν ∈ E ′(ω)(IR) be given. We consider the following assertions:

(1) Tµ(E(ω)(IR)

) ⊃ Tν(E(ω)(IR)

).

(2) Φ : T tµ(D(ω)(IR)

) → D(ω)(IR), given by Φ(T tµ(ϕ)

)= T tν(ϕ) is sequentially con-

tinuous.(3) For every B ⊂ E ′

(ω)(IR) such that T tµ(B) is bounded, T tν(B) is also bounded.(4) F/G is ω – slowly decreasing.(5) Tµ

(E(ω)(IR)) ⊃ Tν Tν Tν

(E(ω)(IR)).

Then (1), (2) and (3) are equivalent, (3) implies (4) and (4) implies (5).

Proof . To show (1) ⇒ (2), we follow the steps of [BGM, Proposition 2.2]. Assumethat Tν

(E(ω)

) ⊂ Tµ(E(ω)

). For a fixed compact set K we define

H :=v ∈ D(ω)(IR) : supp v ⊂ K andv ∈ T tµ

(D(ω)

)and

B : E(ω)(IR)×H −→ C , B(u, v) =⟨u, T tν

((T tµ

)−1v)⟩

.

The map B is bilinear and separately continuous. In fact, given v ∈ H we haveT tν

((T tµ

)−1v)∈ D(ω)(IR) ⊂ E ′

(ω)(IR). This gives the continuity in the first variable.And for a fixed u ∈ E(ω) take w ∈ E(ω) with Tνu = Tµw, then,

B(u, v) =⟨u, T tν

((T tµ

)−1v)⟩

=⟨Tνu,

(T tµ

)−1v⟩=

⟨Tµw,

(T tµ

)−1v⟩= 〈w, v〉 .

Thus, B(u, · ) is continuous on H.Since E(ω) is a Frechet space and H is metrizable, the map B is continuous. Con-

sequently, given a compact set K, we find another compact set K, and C > 0 andm, l ∈ IN such that |B(u, v)| ≤ C ‖u‖

K,m‖v‖K,l for all u ∈ E(ω) and v ∈ H , i. e.,∣∣∫ fT tνϕ∣∣ ≤ ‖f‖

K,m

∥∥T tµϕ∥∥lfor every f ∈ E(ω) and every ϕ ∈ D(ω) with supp T tµϕ ⊂ K.

Now proceeding as in [BGM, 2.2] we reach the conclusion.(2) ⇒ (3). Fix B ⊂ E ′

(ω)(IR) such that Ttµ(B) is bounded. Then, there is a compact

set K such that suppT tµg ⊂ K for all g ∈ B. By the theorem of supports there isa compact set K with supp g ⊂ K for every g ∈ B. Without loss of generality wemay assume that K is absolutely convex. Let V be a relatively compact open setcontaining K and U an open relatively compact set containing the origin and suchthat K + (IR\U) does not intersect V . We take f ∈ D′

(ω)(IR) such that µ ∗ f = ν in U(the existence of f follows from (2)). We have µ ∗ f ∗ g = ν ∗ g on V for every g ∈ B.This implies that T tν(B) is bounded.(3) ⇒ (1). The space Tµ

(E(ω)(IR))is bornological by [BG, Remark 1]. The conclu-

sion follows from Theorem 1.(3) ⇒ (4). We will show that (3) implies the following inequality: Let D > 0 be

such that max(|F (z)|, |G(z)|) = O(e

D2 p(z)

), then for each l ∈ IN and every ε > 0 there

66 Math. Nachr. 216 (2000)

is j ∈ IN such that for each z ∈ C

(Ml(z, G))1+ε

Ml(z, F )< j exp

(jω(z) +D |Imz|) .(2.1)

Assume that the above inequality does not hold. Then, we find ε > 0 and a sequencezj in C , with limj→∞

∣∣zj∣∣ =∞ and

(Ml

(zj , G

))1+ε

Ml(zj , F

) > j exp(jω

(zj

)+D

∣∣Imzj ∣∣) .(2.2)

We first show that the sequence(zj

)can be chosen such that in (2.2) we may replace

Ml(zj , G

)by

∣∣G(zj

)∣∣. Let wj be such that Ml(zj , G)= exp

( − ∣∣zj − wj∣∣)∣∣G(

wj)∣∣.

Then, Ml(wj, G

) ≥ ∣∣G(wj

)∣∣ = exp(l∣∣zj − wj

∣∣)Ml(zj , G). Next, we prove that

Ml(wj , F

) ≤ exp(l∣∣zj − wj

∣∣)Ml(zj, F ). Indeed, if this is not the case, given v ∈ C

with |Imv| ≤ ∣∣Imwj∣∣ we haveMl

(wj , F

)> exp

(l∣∣zj −wj

∣∣) exp (− l∣∣zj − v

∣∣) ∣∣F (v)∣∣ ≥ exp(− l

∣∣wj − v∣∣) ∣∣F (v)∣∣ .

Since the maximum is attained, this implies that Ml(wj, F

)> Ml

(wj, F

), a contra-

diction.Then, Ml

(wj, F

) ≤ exp(l∣∣zj −wj

∣∣)Ml(zj , F ). Consequently,

j exp(jω

(zj

)+D

∣∣Imzj∣∣) <

(Ml

(zj , G

))1+ε

Ml(zj , F

)=

(exp

(l∣∣zj − wj

∣∣)Ml(zj , G))1+ε

exp(l∣∣zj −wj

∣∣)Ml(zj , F ) exp( − lε

∣∣zj −wj∣∣)

≤∣∣G(

wj)∣∣1+ε

Ml(wj, F

) exp( − lε

∣∣zj − wj∣∣) ,

hence

j exp(jω

(zj

)+D

∣∣Imzj ∣∣+ lε∣∣zj − wj

∣∣) ≤∣∣G(

wj)∣∣1+ε

Ml(wj, F

) .(2.3)

Next, we prove

j exp(jω

(wj

)+D

∣∣Imwj∣∣) ≤ j exp(jω

(zj

)+D

∣∣Imzj ∣∣+ lε∣∣zj − wj

∣∣) .This follows immediately from the fact that

∣∣Imzj ∣∣ ≥ ∣∣Imwj∣∣ and ω(t) = o(t). Conse-quently, (2.2) holds with wj instead of zj .It remains to show that

∣∣wj ∣∣ tends to ∞. If this is not the case, without loss ofgenerality, we may find M > 0 with

∣∣wj∣∣ ≤M , for all j ∈ IN. We set

H :=(z, w) ∈ C 2 : |w| ≤M, |Imz| ≤ |Imw| , K(z, w) := |F (z)| exp(−l |z−w|) .


Since ω(t) = o(|t|), we deduce from the Paley –Wiener – Schwartz theorem that K isuniformly continuous on H . But then, it is easy to see that

Ml(w, F ) = max|Imz|≤|Imw|

K(z, w)

is a continuous positive function on |w| ≤M . It is clear that the argument above alsoworks with G instead of F . This implies(

G(wj

))1+ε

Ml(wj , F

)is bounded and, by (2.2), it is greater than j for every j, which is a contradiction.Next, we give an estimate for a cube containing wj such that for all z ∈ C with

Imz = Imwj in this cube we have

|F (z)| ≤ ∣∣G(wj

)∣∣ j −12 exp

12

( −D∣∣wj∣∣ − jω

(wj

)).

Let z be a point with Imz = Imwj and

|F (z)| > ∣∣G(wj

)∣∣ j −12 exp

12

( −D∣∣Imwj∣∣ − jω

(wj

)).

We have

Ml(wj, F

) ≥ exp( − l

∣∣wj − z∣∣) ∣∣F (z)∣∣

> exp( − l

∣∣wj − z∣∣) ∣∣G(

wj)∣∣ j −1

2 exp12

( −D∣∣Imwj∣∣ − jω

(wj

)).

But

Ml(wj , F

)<

∣∣G(wj

)∣∣1+ε

j exp(jω

(wj

)+D

∣∣Imwj∣∣) .Thus, ∣∣G(

wj)∣∣ exp ( − l

∣∣wj − z∣∣) <

∣∣G(wj

)∣∣1+ε

exp 12

(D

∣∣Imwj ∣∣+ jω(wj

)) ,therefore

exp( − l

∣∣wj − z∣∣) <

∣∣G(wj

)∣∣εexp 1

2

(D

∣∣Imwj∣∣+ jω(wj

)) ,from where it follows

∣∣wj − z∣∣ >

−εllog

∣∣G(wj

)∣∣ + 12llog j +

j

2lω(wj

)+D

2l∣∣Imwj∣∣ .

Since log∣∣G(

wj)∣∣ = O

(D2p(wj

))the right –hand side tends to ∞. We assume that

Imwj ≥ 0 and observe that ε is arbitrary small.To finish we constuct a sequence

(Fj

)in E ′

(ω)(IR) such that(FFj

)is bounded in

E ′(ω)(IR) but

(GFj

)is not bounded in this space. To do this we follow an idea of [E].

68 Math. Nachr. 216 (2000)

We define, for all j ∈ IN

Hj(z) =(j

πzsin

πz

j

)jand Gj(z) = exp

(kj

)Hkj

(z − wjεl

),

where kj = E( − log

∣∣G(wj

)∣∣ + D|Imzj |

2 + log j2 + j

2 ω(wj

)). It is easy to see that kj

tends to ∞. We have(1) Gj is of exponential type π,

(2)∣∣Gj(wj)∣∣ = exp

(kj

)Hkj(0) hence,

∣∣Gj(wj)∣∣ ≥ j12 exp( 1

2 (D |Imwj|+jω(wj)))

e |G(wj)| ,

(3) |Gj(z)| ≤ j12 exp( 1

2 (D |Imwj|+jω(wj)))

|G(wj)| provided that Imz = Imwj ,

(4) |Gj(z)| ≤ 1 if Imz = Imwj and |z −wj| ≥ kjεl .

From (2) we deduce that (GGj) is not bounded in E ′(ω)(IR).

On the other hand, given z with Imz = Imwj and

∣∣wj − z∣∣ >

−εllog

∣∣G(wj

)∣∣+ 12llog j +

j

2lω(wj

)+D

2l∣∣Imwj∣∣

we obtain∣∣F (z)Gj(z)∣∣ ≤ |F (z)| whereas for Imz = Imwj and the opposite inequality

we get∣∣F (z)Gj(z)∣∣ ≤ 1. Hence, for z ∈ C with Imz = Imwj we have shown that∣∣F (z)Gj(z)∣∣ ≤ M exp(Mω(z) +M |Imz|) where the constant M does not depend on

j.We set Fj(z) = Gj(z) exp

(− iπ+1ε

(z−wj

)). Observe that for Imz = Imwj, we have∣∣Fj(z)∣∣ = ∣∣Gj(z)∣∣, thus (

GFj)is unbounded.

We show that(FFj

)is bounded. Given x ∈ IR we apply [B, 6.5.9] to get∣∣F (x)Gj(x)∣∣ =

∣∣F (x− i Imwj + i Imwj

)Gj

(x− i Imwj + i Imwj

)∣∣≤

∣∣Imwj∣∣π

∫ +∞

−∞

log(∣∣F (

t+ i Imwj)Gj

(t+ i Imwj

)∣∣)(t− x)2 +

(Imwj

)2 dt + C Imwj

≤ Imwjπ

∫ +∞

−∞

logM +Mω(t) +M Imwj(t− x)2 +

(Imwj

)2 dt +C Imwj

≤ logM +M Imwj +CImwj +MPω(x+ Imwj

)≤ Lω(x) + L

∣∣Imwj∣∣+ L ,

where for the last inequality we use [MTV, Remark 1.5], and C depends on theexponential type of FGj, hence it is independent of j. Thus for arbitrary x ∈ IR

∣∣F (x)Fj(x)∣∣ =∣∣∣∣F (x)Fj(x) exp

(−i π + 1

ε

(z − wj

))∣∣∣∣≤ exp

(L+ Lω(x) +

(L − π + 1

ε

)Imwj

).

Taking ε small enough,(L − π+1

ε

)< 0. Consequently,

∣∣F (x)Fj(x)∣∣≤exp(L + Lω(x)).Now, we apply again the Phragmen –Lindelof Principle to obtain that

(FFj

)is boun-

ded in E ′(ω)(IR).


(4) ⇒ (5). We use [H1, Lemma 3.2 ] to prove condition (2) in Proposition 1.1.Assume that F/G is ω slowly decreasing , and take

r =1

l−B

((j+B)p(z)+logBj−(1+ε) logMl(z, G)

) ≤ Ap(z)+A−δ logMl(z, G) ,

for some δ > 0 arbitrarily small. Then

|G(z)|3 |f(z)||F (z)| ≤ |G(z)|3

(sup|ζ−z|<4r |f(ζ)|

)(sup|ζ−z|<4r |F (ζ)|

)(sup|ζ−z|<r |F (ζ)|

)2 .

The choice of r guarantees the existence of w with |z−w| < r and |F (w)| ≥ Ml(z,G)1+ε

j exp jp(z) ,thus

|G(z)|3 |f(z)||F (z)| ≤ |G(z)|3

(sup|ζ−z|<4r |f(ζ)

)(sup|ζ−z|4r |F (ζ)|

)(Ml(z,G)1+ε

j exp jp(z)

)2 .

Since |f(ζ)| ≤ C exp(Cp(ζ)) ≤ C exp(Cp(z) + C4r) we can apply the estimate of rto get

|f(ζ)| ≤ C exp((C + 4AC)p(z) + 4AC − 4Cδ logMl(z, G)) .

We proceed in the same way for |F (ζ)| to get

|G(z)|3 |f(z)||F (z)| ≤ (Ml(z, G))3

D expDp(z)(Ml(z, G))2+α

for α arbitrarily small. In particular, if l is big enough, α is smaller than 1.

Remark 2.6. It is clear that all the conditions in Theorem 1 are equivalent incase ν = δ. In fact, in this case, we have even more, since, as has been shown byBonet, Galbis and Meise in [BGM], Φ is not only sequentially continuous but alsocontinuous.It is worth mentioning that in the previous theorem (1)⇒ (2)⇒ (3) and (4)⇒ (5)

also work for higher dimensions. We have not been able to show (3)⇒ (1) for it is anopen problem whether Tµ

(E ′(ω)

)is bornological for all µ and ω. The techniques used

to show (3)⇒ (4) work for the one variable case.

Corollary 2.7. If Tν(E(ω)(IR)

) ⊂ Tµ(E(ω)(IR)

), then for every relatively compact

open set U we find S in D′(ω)(IR) such that µ ∗ S = ν in U .

Proof .: Tν(E(ω)(IR)

) ⊂ Tµ(E(ω)(IR)

)implies (2) in Theorem 1. Given a relatively

compact open set U in IR we take a compact set K with U ⊂ K. The map

Φ : D(ω)(K)⋂T tµ

(D(ω)(IR)) −→ D(ω)(IR) , given by Φ

(T tµ(ϕ)

)= ν ∗ ϕ(0)

70 Math. Nachr. 216 (2000)

is continuous. By the Hahn –Banach’s theorem we find S ∈ D′(ω)(IR) with 〈S, µ ∗ϕ〉 =

〈ν, ϕ〉 for each ϕ ∈ D(ω)(K). Then S ∗ µ = ν on K, hence on U .

By Corollary 2.7 the inclusion Tν(E(ω)(IR)

) ⊂ Tµ(E(ω)(IR)

)provides local solutions

of the equation µ∗f = ν . The existence of a global solution is implied by the inclusionSν

(D′(ω)(IR)

) ⊂ Sµ(D′

(ω)(IR))which of course implies Tν

(E(ω)(IR)) ⊂ Tµ

(E(ω)(IR)).

Next, we will consider a condition on F and G which is stronger than being slowlydecreasing (but equivalent in case ν = δ) and which is sufficient to show that the rangeof Sµ contains the range of some iteration of Sν . The proof is a modification of [BGM,Proposition 2.5].

Proposition 2.8. Let µ, ν ∈ E ′(ω)(IR) be given and assume that for arbitrary l, n ∈ IN

and each ε > 0 we find k ∈ IN such that for every z ∈ C there is w ∈ C with|z−w| ≤ kω(z)+ |Imz|

n and |F (w)| >Ml(G, z)1+ε exp(−kp(z)). Then Sµ(D′

(ω)(IR)) ⊃

Sν Sν Sν(D′

(ω)(IR)).

In what follows, we will consider classical distributions.

Definition 2.9. Given S ∈ D′(IR) we denote by sing supp S the closure of thesmallest open set Ω of IR such that S is C∞ in IR\Ω.Proceeding as in the case of partial differential operators with constant coefficients

we obtain the following necessary condition for the inclusion Sµ(D′(IR)) ⊃ Sν Sν Sν(D′(IR)). The proof follows the steps of ([H3, Th. 3.6.3]) with minor modifications.

Proposition 2.10. If Sν(D(IR)′) ⊂ Sµ(D(IR)′), then for every compact set K wefind another compact set K such that for each T ∈ E(IR)′ with sing supp (µ ∗ T ) ⊂ K

it follows that sing supp (ν ∗ T ) ⊂ K.

Proposition 2.11. Let µ, ν ∈ E ′ be given and assume that for each l ∈ IN andeach ε > 0 we find k ∈ IN such that for every z ∈ C there is w ∈ C with |w − z| <kp(z) and |F (w)| > Ml(G, z)1+εexp(−kp(z)). Then, for each compact set K thereis another compact set K such that if T ∈ E ′ with sing supp (µ ∗ T ) ⊂ K we havesing supp (ν ∗ T ) ⊂ K.

Proof . Proceeding as in [H1, Theorem 4.3], we show that for every compact set K inIR we find another compact set K such that for every T ∈ E ′(IR) with sing suppT tµ(T ) ⊂K we have sing supp T tν T tν T tν(T ) ⊂ K. To conclude, we observe that from [H2,Definition 5.1, Theorem 5.1, Corollary 5.2] we may obtain another compact K in IRsuch that sing supp T tν(T ) ⊂ K.

Acknowledgements

This research has been supported by DGICYT under proyecto PB94 – 0541. The author isvery indebted with Jose Bonet for his interest in the present research.


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Universidad de ValenciaFacultad de MatemticasDr. Moliner, 50E – 46100, BurjassotSpainE–mail:[email protected]

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On the Range of Two Convolution Operators