Wave Front Sets with respect to the Iterates of an Operator with Constant Coefficients

Boiti, C.; Jornet, D.; Juan-Huguet, J.

doi:https://doi.org/10.1155/2014/438716

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 438716 | https://doi.org/10.1155/2014/438716

Wave Front Sets with respect to the Iterates of an Operator with Constant Coefficients

C. Boiti,¹D. Jornet,²and J. Juan-Huguet³

Academic Editor: Luigi Rodino

Received28 Nov 2013

Accepted05 Feb 2014

Published08 May 2014

Abstract

We introduce the wave front set with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distribution in an open set Ω in the setting of ultradifferentiable classes of Braun, Meise, and Taylor. We state a version of the microlocal regularity theorem of Hörmander for this new type of wave front set and give some examples and applications of the former result.

1. Introduction

In the 1960s Komatsu characterized in [1] analytic functions in terms of the behaviour not of the derivatives , but of successive iterates of a partial differential elliptic operator with constant coefficients, proving that a function is real analytic in if and only if for every compact set there is a constant such that where is the order of the operator and is the norm on .

This result was generalized for elliptic operators with variable analytic coefficients by Kotake and Narasimhan [2, Theorem 1]. Later, this result was extended to the setting of Gevrey functions by Newberger and Zielezny [3] and completely characterized by Métivier [4] (see also [5]). Spaces of Gevrey type given by the iterates of a differential operator are called generalized Gevrey classes and were used by Langenbruch [6–9] for different purposes. We mention modern contributions like [10–13] also. More recently, Juan-Huguet [14] extended the results of Komatsu [1], Newberger and Zielezny [3], and Métivier [4] to the setting of nonquasianalytic classes in the sense of Braun et al. [15]. In [14], Juan-Huguet introduced the generalized spaces of ultradifferentiable functions on an open subset of for a fixed linear partial differential operator with constant coefficients and proved that these spaces are complete if and only if is hypoelliptic. Moreover, Juan-Huguet showed that, in this case, the spaces are nuclear. Later, the same author in [16] established a Paley-Wiener theorem for the classes again under the hypothesis of the hypoellipticity of .

The microlocal version of the problem of iterates was considered by Bolley et al. [17] to extend the microlocal regularity theorem of Hörmander [18, Theorem 5.4]. Bolley and Camus [19] generalized the microlocal version of the problem of iterates in [17] for some classes of hypoelliptic operators with analytic coefficients. We mention [20, 21] for investigations of the same problem for anisotropic and multianisotropic Gevrey classes. On the other hand, a version of the microlocal regularity theorem of Hörmander in the setting of [15] can be found in [22, 23] by one of the authors, which continues the study begun in [24].

Here, we continue in a natural way the previous work in [14] and study the microlocal version of the problem of iterates for generalized ultradifferentiable classes in the sense of Braun et al. [15]. We begin in Section 2 with some notation and preliminaries. In Section 3, we fix a hypoelliptic linear partial differential operator with constant coefficients and introduce the wave front set with respect to the iterates of of a distribution (Definition 7). To do this, we describe carefully the singular support in this setting (Proposition 6). We also prove that the new wave font set gives a more precise information for the study of the propagation of singularities than previous ones in Proposition 9, Theorem 13, and Example 15 (improving the previous works [22, 23] by one of the authors for operators with constant coefficients). More precisely, we clarify in Theorem 13 the necessity of the hypoellipticity of with a new version of the microlocal regularity theorem of Hörmander for an operator with constant coefficients. In Section 4, we prove that the product of a function in a suitable Gevrey class and a function in is still in (Proposition 17). This fact is used to give a more involved example, inspired in [25, Theorem 8.1.4], in which we construct a classical distribution with prescribed wave front set (Theorem 18). Finally, we mention that, as far as we know, this is the first time that a result like Proposition 17 is discussed.

2. Notation and Preliminaries

Let us recall from [15] the definitions of weight functions and of the spaces of ultradifferentiable functions of Beurling and Roumieu type.

Definition 1. A nonquasianalytic weight function is a continuous increasing function with the following properties:(α) s.t. ,(β), (γ) as ,(δ) is convex.

Normally, we will denote simply by .

For a weight function , we define by and again we denote this function by .

The Young conjugate is defined by There is no loss of generality to assume that vanishes on . Then has only nonnegative values, it is convex, is increasing and tends to as , and .

Example 2. The following functions are, after a change in some interval , examples of weight functions:(i) for .(ii), .(iii), .(iv), .In what follows, denotes an arbitrary subset of and means that is a compact subset in .

Definition 3. Let be a weight function.
(a) For a compact subset in which coincides with the closure of its interior and , we define the seminorm where and set which is a Banach space endowed with the -topology.
(b) For an open subset in , the class of -ultradifferentiable functions of Beurling type is defined by The topology of this space is and one can show that is a Fréchet space.
(c) For a compact subset in which coincides with the closure of its interior and , set This space is the strong dual of a nuclear Fréchet space (i.e., a (DFN) space) if it is endowed with its natural inductive limit topology; that is,
(d) For an open subset in , the class of -ultradifferentiable functions of Roumieu type is defined by Its topology is the following: that is, it is endowed with the topology of the projective limit of the spaces when runs the compact subsets of . This is a complete PLS-space, that is, a complete space which is a projective limit of LB-spaces (i.e., a countable inductive limit of Banach spaces) with compact linking maps in the (LB) steps. Moreover, is also a nuclear and reflexive locally convex space. In particular, is an ultrabornological (hence barrelled and bornological) space.
The elements of (resp., ) are called ultradifferentiable functions of Beurling type (resp., Roumieu type) in .
In the case that (), the corresponding Roumieu class is the Gevrey class with exponent . In the limit case , not included in our setting, the corresponding Roumieu class is the space of real analytic functions on , whereas the Beurling class gives the entire functions.
If a statement holds in the Beurling and the Roumieu case, then we will use the notation . It means that in all cases, can be replaced either by or .
For a compact set in , define endowed with the induced topology. For an open set in , define
Following [14], we consider smooth functions in an open set such that there exists verifying for each , where is a compact subset in , denotes the -norm on , and is the th iterate of the partial differential operator of order ; that is, If , then .
Given a polynomial with degree , , the partial differential operator is the following: , where .
The spaces of ultradifferentiable functions with respect to the successive iterates of are defined as follows.
Let be a weight function. Given a polynomial , an open set of , a compact subset , and , we define the seminorm and set
It is a Banach space endowed with the -norm (it can be proved by the same arguments used for the standard class in the sense of Braun et al.; see [15]).
The space of ultradifferentiable functions of Beurling type with respect to the iterates of is

endowed with the topology given by

If is a compact exhaustion of , we have

This metrizable locally convex topology is defined by the fundamental system of seminorms .

The space of ultradifferentiable functions of Roumieu type with respect to the iterates of is defined by Its topology is defined by

As in the Gevrey case, we call these classes generalized nonquasianalytic classes. We observe that in comparison with the spaces of generalized nonquasianalytic classes as defined in [14] we add here as a factor inside in (15), where is the order of the operator , which does not change the properties of the classes and will simplify the notation in the following.

The inclusion map is continuous (see [14, Theorem 4.1]). The space is complete if and only if is hypoelliptic (see [14, Theorem 3.3]). Moreover, under a mild condition on introduced by Bonet et al. [26], coincides with the class of ultradifferentiable functions if and only if is elliptic (see [14, Theorem 4.12]).

Denoting by the classical Fourier transform of , we recall from [22, Proposition 3.3] the following characterization of the -singular support in the sense of Braun et al. [15].

Proposition 4. Let be an open set, , and .(a)Then is a -function in some neighborhood of if and only if for some neighborhood of there exists a bounded sequence which is equal to in and satisfies, for some and , the estimates(b)Then is a -function in some neighborhood of if and only if for some neighborhood of there exists a bounded sequence which is equal to in and such that for every there exists a constant satisfying

This led, in [22, Definition 3.4], to the following definition of wave front set in the sense of Braun et al. [15].

Definition 5. Let be an open subset of and . The wave front set , resp., wave front set , of is the complement in of the set of points such that there exist an open neighborhood of in , a conic neighborhood of , and a bounded sequence (the set of classical distributions with compact support in ) equal to in such that there are and with the property Resp., which satisfies that for every there is with the property

3. Wave Front Sets with respect to the Iterates of an Operator

Now, we assume that is a bounded open set in and we use the following notation: where is the distance of to the boundary of . Given a linear partial differential operator , we denote by the operator corresponding to the polynomial . If is hypoelliptic, by [27, Theorem 4.1] and the argument used in the proof of [3, Theorem 1], there are constants and such that for every and we have

We observe also that if has constant coefficients, its formal adjoint is and, if is hypoelliptic, is also hypoelliptic (because of the behavior of the associated polynomial ). Moreover, any power or , with , of or , is also hypoelliptic.

We now want to generalize the notion of -singular support of Proposition 4, using the iterates of a hypoelliptic linear partial differential operator with constant coefficients. The idea is to substitute the sequence which satisfies an estimate of the form (23) or (24) by the sequence whose Fourier transform satisfies the following estimates (29) or (30).

Proposition 6. Let be a linear partial differential operator of order with constant coefficients which is hypoelliptic. Let be an open subset of , , and consider the following three conditions:(i),(ii)(Roumieu) , , , , and , we have(iii)(Beurling) and , , , and , we have
Then, the distribution (), where is some neighborhood of , if and only if there exist a neighborhood of and a sequence in that satisfies (i) and (ii) in (that satisfies (i) and (iii) in ).

Proof.
Sufficiency (Roumieu case). Let with , the ball in of center and radius , . We choose such that in and in . We set . Then, and in .
Now, fix . From the hypoellipticity of , there are constants such that, for large enough, . Then, from the definition of we obtain, for large enough, We integrate by parts in the integral above, which will be equal to From the generalized Leibniz rule, we can write (here is the order of ) Since is hypoelliptic and is a -function in the bounded set , we can apply formula (28) to the operator with , for , , and (and ) to obtain constants (which do not depend on ) such that Now, as , there are constants and such that (we use the convexity of )
Therefore, we can estimate, by Hölder’s inequality, the Fourier transform for big enough in the following way (at the end, we use the fact that is an increasing function):
On the other hand, if is bounded, we put and, by Hölder’s inequality, we have
From the last estimates, we can conclude that , , , and , which finishes this implication.
The Beurling case is similar.
Necessity (Roumieu case). Let satisfying (i) in some neighborhood of and (ii). We fix a compact set and take . Now, by (ii), there is and a constant that depends on and such that, by Parseval’s formula, In a similar way, using the Fourier transform, we can see that the distributions satisfy analogous estimates for each multi-index on . By the hypoellipticity of we conclude that , and this finishes the proof in the Roumieu case.
As above, in the Beurling case we can argue in a similar way.

In the rest of the paper, it is assumed that the operator is hypoelliptic, but not elliptic. Hypoellipticity is not only useful for Proposition 6, but also because it gives some good properties of the space , such as completeness (cf. [14]). On the contrary, the elliptic case is not really interesting here since if and only if is elliptic, as we have already mentioned at the end of Section 2.

Proposition 6 leads us to define the wave front set with respect to the iterates of an operator.

Definition 7. Let be an open subset of , , and a linear partial differential hypoelliptic operator of order with constant coefficients. We say that a point is not in the -wave front set with respect to the iterates of , (-wave front set with respect to the iterates of , ), if there are a neighborhood of , an open conic neighborhood of , and a sequence such that (i) and (ii) of the following conditions hold ((i) and (iii) of the following conditions hold):(i)For every , in .(ii) Roumieu:(a)there are constants , , and , such that (b)there is a constant such that for all , there is with the property (iii) Beurling:(a)there are such that for all , there is such that (b)for all and there is such that

If we compare the last definition with Definition 5 we can deduce, as Proposition 9 will show, that the new wave front set gives more precise information about the propagation of singularities of a distribution than the -wave front set of a classical distribution ( or ). We first recall the following result that we state as a lemma (see [19, Proposition 1.8]).

Lemma 8. Let be an open subset of , , and a linear partial differential operator with analytic coefficients in of order . Let such that where does not depend on . Then the sequence , for large enough independent of satisfies for some constants and .

Proposition 9. Let be an open subset of , , a weight function, and a hypoelliptic linear partial differential operator of order with constant coefficients. Then, the following inclusions hold:

Proof.
Roumieu Case. Let . From Definition 5, there exist a neighborhood of , an open conic neighborhood of , and a bounded sequence such that in for every and for some constants ,
By [18, Lemma 2.2], we can find a sequence such that in a neighborhood of and We select as in Lemma 8 (or bigger if necessary) and set . We first observe that, as in for all and , we have for all . We want to prove (i), (ii)(a), and (ii)(b) in Definition 7. By the choice of , condition (i) is fulfilled in the neighborhood . To see (ii)(a), we observe that from Lemma 8 there is such that for some constant . Since the weight function satisfies as tends to infinity, from [22, Remark 2.4(b)], for every there is such that In particular, for , we obtain which proves (ii)(a).
We prove now (ii)(b). We fix and set, for , where is the integral when , for to be chosen, and is the integral when , both considered with the factor . In , we have
Since is a bounded sequence in , there is such that for all and .
From (48), we can differentiate up to the order to obtain constants that depend on , , and such that (see [22, Lemma 3.5])
As has order , we also have for some constant and each and .
Moreover, in , and Therefore, from (54), we obtain for some .
On the other hand, if we consider the estimate , we obtain We observe that the integral is less than or equal to for some constant that depends on and the support of and some constant . Now, we write . If is a conic neighborhood of such that , we can select such that for and , we have . Consequently, we obtain, by assumption on (see (47)), and by the estimate for some positive constant , for , for some . We conclude, using the convexity of , that there are constants and such that
Beurling Case. Let us assume now that . From Definition 5, there exist a neighborhood of , an open conic neighborhood of , and a bounded sequence such that in for every and for every there is , such that We take and as in the Roumieu case. From (50), for any , there is satisfying which proves (iii)(a).
To prove (iii)(b), fix and consider now the estimate (use (48) and (50)) Here, where is the integral when , for to be chosen, and is the integral when . In this case, we use (60) and obtain a constant which depends on (and ) and a constant with the property that for every there is a constant such that for any and , This concludes the Beurling case.

Corollary 10. Let , and let be a compact subset of and a closed cone in . Let be a weight function. Suppose that is like in (48). Then, we have the following:(a)If , then the sequence , for large enough independent of , satisfies that there is such that for every , there is with (b)If , then the sequence , for large enough independent of , satisfies that for every there is with

Proof. We make a sketch of proof of (a) only. Let , and choose and , with a conic subset of and according to Definition 7. If the support of is in , we have . Now, the proof is like (ii)(b) of Proposition 9 for the set and instead of . To obtain a uniform estimate in , we can proceed as in [22, Lemma 3.5] at the end of the proof of (a). See also the proof of [25, Lemma 8.4.4].

The singular support of a classical distribution with respect to the class is the complement in of the biggest open set , where . As a consequence of Propositions 6 and 9 and Corollary 10, we obtain the following result.

Corollary 11. The projection in of is the singular support with respect to the class if .

Proof. Follow the lines of the proofs of [22, Theorem 3.6] and [25, Theorem 8.4.5].

Remark 12. We observe that from the definition it is obvious that if is hypoelliptic, then for or Then, by Proposition 9, the following inclusions hold:

Now, we can state an improvement of [22, Theorem 4.8] for operators with constant coefficients.

Theorem 13. Let , be a hypoelliptic linear partial differential operator with constant coefficients and order and let be an open subset of . Let be the principal part of and the characteristic set of . Then, for any distribution

Proof. Let such that . Then, there are a neighborhood of , a conic neighborhood of , and a sequence that verify (i), (ii)(a)-(ii)(b) in the Roumieu case, and (iii)(a)-(iii)(b) in the Beurling case of Definition 7. We take such that for . We take a compact neighborhood of and consider a sequence satisfying (48) such that on .
We set now . We want to estimate To estimate in , we will solve in an approximate way the following equation: As in [17], we put . For , we have where , with a differential operator of order which depends on the parameter such that is homogeneous of order . Recursively, it is easy to compute then Therefore, we want to give an approximate solution of A formal solution of (74) is given by the series: For we can write We observe that the coefficient of with is given by by the Chu-Vandermonde identity. For , the term does not appear anymore for . So, we do not have all the summands needed in the identity above and hence the coefficients of are not zero. Therefore, (we write for for simplicity) for Then,
If we apply these identities to , we obtain where the integrals denote action of distributions.
We suppose now that has order in a neigborhood of . Since , we have In order to estimate this expression, first we estimate The number of terms in depends on Now, since and in the sum of the expression of , , we obtain (we recall that ) In the last expression, we obtain a sum of terms, for some constant , of the form which contain derivatives of order and are homogeneous of degree , where . Then, if we take , we get a new constant , such that Therefore, we obtain a new constant such that
We study now where we have splitted in the sum of and , the first when and the second when , for a constant to be chosen.
First, we estimate defined in formula (76). Proceeding in a similar way as before with the expression of , if we take and and estimate the binomials as in (85), we find a constant such that At this point, we have to separate Beurling and Roumieu cases.
Roumieu Case. From Definition 7(ii)(a), we have for some constants , , and . Now, as , by (90), we have, as in [22, Lemma 3.5],
We proceed now as in the proof of (ii)(b) of Proposition 9 in order to estimate . In , we have and, by (92), we deduce for some constants .
For we have As in the proof of Proposition 9, we can estimate , in the Roumieu case, with the use of (ii)(b) of Definition 7 in the following way: we select for which there are and such that for in some neighborhood of (see the argument before inequality (58)), Consequently, since for some constant ,
Therefore, if we combine (96) and (93), we obtain two constants and such that for in some conic neighborhood of and , by (89), As in (50), we have for some constant and every . Then, from (88), we deduce a similar estimate to the one of for . Now, from the bounds for and , there are constants such that, for in some conic neighborhood of and , We have a similar estimate when . In fact, since the sequence is bounded in , there are constants and which satisfy Then, we have
Beurling Case. In this setting we will proceed in a similar way. We can select and apply now (iii)(b) of Definition 7 to obtain, for every , a constant such that, for all in some neighborhood of , In a similar way to (92), we can obtain here where the constant comes from Definition 7(iii)(a).
Now, as in (93), we have a constant and for every a constant such that Therefore, from (101) and (103), we have and for a fixed a constant such that for in some conic neighborhood of and , As in the Roumieu case, we deduce a similar estimate for . Then, the bounds for and give a constant and, for every , a constant such that for in some conic neighborhood of and () (if is large enough),
Finally, we also have a similar estimate when , which concludes the proof of the theorem.

Remark 14. If is elliptic, then and Theorem 13 and Remark 12 imply that

Example 15. We show that the inclusions of Remark 12 are strict. As in [14] (see [26]), we consider a nonquasianalytic weight function satisfying the following condition: there exists a constant such that for all , For example, if is a Gevrey weight, then it satisfies such a property. We consider now a polynomial with constant complex coefficients such that it is hypoelliptic but not elliptic (for instance, the heat operator). Then by [14, Theorem 4.12], there is (for some open subset of ). Then, but , which implies that the inclusion is strict.
On the other hand, if we consider now a -hypoelliptic polynomial which is not elliptic (e.g., the heat operator in for ), then as before there will be . In particular, . Now, if , we will have and since is -hypoelliptic, , which is a contradiction. Therefore, and we conclude that the inclusion is strict.
Let us also remark that for the heat operator , we can explicitly write its characteristic set , so that the previous considerations give, for , the following information on , because of Theorem 13:
In the Beurling setting, we can proceed in a similar way.
Let us finally notice that the inclusion of Remark 12 is strict in general.

4. Distributions with Prescribed Wave Front Set

The proof of the following lemma is straightforward.

Lemma 16. Let be a weight function. Then, for every and (i);(ii).
Now, we show that the product of a Gevrey function with a function in belongs to the last space.

Proposition 17. Let be a nonquasianalytic weight function such that as , where is the constant in (28) and is a Gevrey weight, with . If and , where or , then the multiplication .

Proof. We will analyse the -norms of on a compact set in . First, we observe that, by the generalized Leibniz rule over applied times, We fix now a compact set in such that . We apply -norms in the compact set Since , there is a constant such that, for each and we have Consequently, Therefore, Now, we apply (28) times to the factor . We will use the notation , for . In the first step, In the second step, is replaced by and so on in the next steps. Therefore, to avoid that, after steps, the set leaves and to keep it bounded for all , we may take depending on for all . We take with a constant such that for all . It is obvious that for all . Moreover, we can assume that for all .
After steps we get With our selection of for , we have for all . Moreover, for all , , which is compact and a subset of . Consequently, since for all , we have (we can assume that the constant and then for all ) Summing up, we obtain If we use the multinomial theorem, where is the dimension of the multi-index or . Then, it is clear that for some constant that depends on , , and the compact set .
Now, we control the sequence for , which is the factor of and less than or equal to
For , since as , there is a constant such that Since as , for any constant , On the other hand, since , there are constants and that depend on such that Then, from the convexity of ,
If , since as for every , there is such that Moreover, if for each , there is such that Now, we can proceed as in the Roumieu case to obtain which concludes the proof.

Let us recall that, by Proposition 9 and Theorem 13 if is a nonquasianalytic weight and is elliptic, then for being equal to or . Let us then assume is not elliptic and prove the following result, which generalizes Theorems 8.1.4 and 8.4.14 of [25].

Theorem 18. Let be a nonquasianalytic weight function such that as tends to infinity, where is a Gevrey weight function, with and , with the constant in (28). Let be a linear partial differential operator with constant coefficients which is hypoelliptic but not elliptic. Given an open subset of and a closed conic subset of , then there is a distribution with . In particular, if for some and with , we have .

Proof. Let us first remark that it is sufficient to prove the statement when .
Moreover, since is hypoelliptic but not elliptic, we can find and such that for big enough. Choose a sequence with so that every with is the limit of a subsequence.
Let us now set and separate Beurling and Roumieu cases.
Roumieu Case. Take with .
Then, there exist and such that Since as , by definition of weight function, by Lemma 1.7 of [15], there exists a weight function such that and for .
Note that for every , there is such that and define then This is a continuous function in and we will prove that .
To prove first that , we take and prove that . To this aim, we choose an open neighborhood of and an open conic neighborhood of such that Write , where is the sum of terms in (138) with and is the sum of terms with .
Therefore, there is a neighborhood of with such that is in since all but a finite number of terms vanish in . Moreover, every weight function is increasing by definition, so that , and hence .
Consider then Note that it is a totally convergent series since for some and because of (137) with .
Let us then compute the Fourier transform with because of (139).
If is a conic neighborhood of with , then when and , for some , since this is true when . Thus, It follows from (136) that for some , since for some by definition of weight function. Therefore, by (142) and Lemma 16, if we fix , for , , for some . Now, from the convexity of , it follows easily that condition of Definition 7 is satisfied. But also condition of Definition 7 is satisfied for some . This, together with , proves that .
Let us now prove that .
Choose equal to 1 near , where is the Gevrey weight of the hypotheses. To prove that , we proceed by contradiction and assume that the wave front set is empty. Then, .
Set By hypothesis which implies in particular that . Then, the sequence is a bounded set in and, in fact, the supports for all . We can use [15, Proposition 3.4] to obtain constants such that for all and all .
The Fourier transform of is a sum of the form (142) with replaced by . We observe that Moreover, for close to and large enough, the equality is satisfied. Consequently, from (135), we have, for some , In fact, for large enough since, for , , is bounded ( in (135)), , and .
On the other hand, by Proposition 17, the product . We obtain and such that, for all , where is a constant that depends on the Lebesgue measure of . Consequently, from (150), we have for every and .
Now, (153) implies, by Lemma 16, But for every fixed , there is large enough so that since we can argue as in (151), which is a contradiction. Therefore, .
Beurling Case. Take with .
For every , there exists then a constant such that
Note that for every fixed , for large enough since as . Define then This is a continuous function in and we will prove that .
The proof of the inclusion is similar to that in the Roumieu case. We take , choose an open neighborhood of and an open conic neighborhood of such that , and write , where is the sum of terms in (158) with and is the sum of terms with .
We choose a neighborhood of with such that is in since all but a finite number of terms vanish in .
Then, we consider the totally convergent series (because of (157) with large enough) and compute its Fourier transform with .
For a conic neighborhood of with , we have that (143) is satisfied and hence, from (156), for some , since for some . Now, we fix . By Lemma 16, for some . From the convexity of , we conclude that condition of Definition 7 is satisfied. But also condition of Definition 7 is satisfied for some . This, together with , proves that and hence .
Let us prove now that .
Choose equal to 1 near . We proceed by contradiction and assume that . Then, .
Set as in the Roumieu case. Since , ([15, Proposition 4.7]). Then the sequence is a bounded set in and for all , as in the Roumieu case. By [15, Proposition 3.4], for each , there is such that for all and ,
If is close to and is large enough, then and by (149), we have
On the other hand, by Proposition 17, and proceeding as in the Roumieu case, we obtain that for every , there would exist such that
But (166) and (165) give a contradiction since they imply, by Lemma 16, that must hold for every and large enough.
However, since for some , there exists a constant such that contradicting (167) for large enough. Then .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of the first and the second authors was partially supported by Grants PRIN2008 (MIUR) and FAR2009 (University of Ferrara). The research of the second and third authors was partially supported by MEC and FEDER, Project MTM2010-15200. The research of the second author was partially supported by Programa de Apoyo a la Investigación y Desarrollo de la UPV PAID-06-12. The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Instituto Nazionale di Alta Matematica (INdAM).

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