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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 438716, 17 pages
http://dx.doi.org/10.1155/2014/438716
Research Article

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

1Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy
2Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, C/Camino de Vera, S/N, E-46071 Valencia, Spain
3Departament de Didàctica de la Matemàtica, Universitat de València, Avenida dels Tarongers 4, 46022 Valencia, Spain

Received 28 November 2013; Accepted 5 February 2014; Published 8 May 2014

Academic Editor: Luigi Rodino

Copyright © 2014 C. Boiti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [69] for different purposes. We mention modern contributions like [1013] 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