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

and Applied Analysis 3 In the case that3(8) := 8(0 < = < 1), the corresponding Roumieu class is the Gevrey class with exponent 1/=. In the limit case = = 1, not included in our setting, the corresponding Roumieu class E{*}(Ω) is the space of real analytic functions onΩ, whereas the Beurling classE(*)(R)) gives the entire functions. If a statement holds in the Beurling and theRoumieu case, thenwewill use the notationE∗(Ω). It means that in all cases, ∗ can be replaced either by (3) or {3}. For a compact set ( in R), define D∗ (() := {$ ∈ E∗ (R)) : supp$ ⊂ (} , (11) endowed with the induced topology. For an open setΩ inR), define D∗ (Ω) := ind 55555→ '⊂⊂ΩD∗ (() . (12) Following [14], we consider smooth functions in an open setΩ such that there exists ' > 0 verifying for each ∈ N0 := N ∪ {0}, *****&%(%)$*****2,' ≤ ' exp(JA∗ (-/ J )) , (13) where ( is a compact subset in Ω, ‖ ⋅ ‖2,' denotes the 22norm on (, and &%(%) is the -th iterate of the partial differential operator &(%) of order/; that is, &% (%) = & (%) ∘ ⋅ ⋅ ⋅ ∘ ⏟⏟⏟⏟⏟⏟⏟⏟⏟ % & (%) . (14) If = 0, then &0(%)$ = $. Given a polynomial & ∈ C[D1, . . . , D)] with degree /, &(D) = ∑|$|≤( X$D$, the partial differential operator &(%) is the following: &(%) = ∑|$|≤( X$%$, where% = (1/Y)Z. The spaces of ultradifferentiable functions with respect to the successive iterates of & are defined as follows. Let3 be aweight function.Given a polynomial&, an open set Ω of R), a compact subset ( ⊂⊂ Ω, and J > 0, we define the seminorm ****$****',1 := sup %∈N0*****&%(%)$*****2,' exp(−JA∗ (-/ J )) (15) and set E1!,* (() = {$ ∈ '∞ (() : ****$****',1 < +∞} . (16) It is a Banach space endowed with the ‖ ⋅ ‖',1-norm (it can be proved by the same arguments used for the standard class E1*(() in the sense of Braun et al.; see [15]). The space of ultradifferentiable functions of Beurling type with respect to the iterates of & is E!(*) (Ω) = {$ ∈ '∞ (Ω) : ****$****',1 < +∞ for each ( ⊂⊂ Ω, J > 0} , (17) endowed with the topology given by E!(*) (Ω) := proj ←55555 '⊂⊂Ω ←55 1>0 E1!,* (() . (18) If {()})∈N is a compact exhaustion ofΩ, we have E!(*) (Ω) = proj ←55 )∈N proj ←55 :∈N E:!,* (()) = proj ←55 )∈N E)!,* (()) . (19) This metrizable locally convex topology is defined by the fundamental system of seminorms {‖ ⋅ ‖'! ,)})∈N. The space of ultradifferentiable functions of Roumieu type with respect to the iterates of & is defined by E!{*} (Ω) = {$ ∈ '∞ (Ω) : ∀( ⊂⊂ Ω ∃J > 0 such that ****$****',1 < +∞} . (20) Its topology is defined by E!{*} (Ω) := proj ←55555 '⊂⊂Ω ind "→ 1>0E1!,* (() . (21) 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 A∗ 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 E∗(Ω) ]→ E!∗(Ω) is continuous (see [14, Theorem 4.1]). The space E!∗(Ω) is complete if and only if & is hypoelliptic (see [14, Theorem 3.3]). Moreover, under a mild condition on 3 introduced by Bonet et al. [26], E!∗(Ω) coincides with the class of ultradifferentiable functions E∗(Ω) if and only if & is elliptic (see [14,Theorem 4.12]). Denoting by $̂ (_) := ∫ C−;⟨2,=⟩$ (M)=M (22) the classical Fourier transform of $ ∈ E#(Ω), 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 Ω ⊂ R) be an open set, ! ∈ D#(Ω), and M0 ∈ Ω. (a) Then ! is aE{*}-function in some neighborhood of M0 if and only if for some neighborhooda of M0 there exists a bounded sequence !? ∈ E#(Ω)which is equal to ! ina and satisfies, for some ' > 0 and b ∈ N, the estimates LLLL_LLLL? LLLL!̂? (_)LLLL ≤ 'C(1/:)@∗(?:), ∀c ∈ N, _ ∈ R). (23) (b) Then ! is aE(*)-function in some neighborhood of M0 if and only if for some neighborhood a of M0 there exists a bounded sequence !? ∈ E#(Ω) which is equal to ! in 4 Abstract and Applied Analysis a and such that for every b ∈ N there exists a constant ': > 0 satisfying LLLL_LLLL? LLLL!̂? (_)LLLL ≤ ':C:@∗(?/:), ∀c ∈ N, _ ∈ R). (24) This led, in [22, Definition 3.4], to the following definition of wave front set WF∗(!) in the sense of Braun et al. [15]. Definition 5. Let Ω be an open subset of R) and ! ∈ D#(Ω). The {3}-wave front set WF{*}(!), resp., (3)-wave front set WF(*)(!), of ! is the complement in Ω × (R) \ 0) of the set of points (M0, _0) such that there exist an open neighborhood a of M0 in Ω, a conic neighborhood Γ of _0, and a bounded sequence !? ∈ E#(Ω) (the set of classical distributions with compact support in Ω) equal to ! in a such that there are b ∈ N and ' > 0 with the property LLLL_LLLL? LLLL!̂? (_)LLLL ≤ 'C(1/:)@∗(:?), c = 1, 2, . . . , _ ∈ Γ (25) Resp., which satisfies that for every b ∈ N there is': > 0with the property LLLL_LLLL? LLLL!̂? (_)LLLL ≤ ':C:@∗(?/:), c = 1, 2, . . . , _ ∈ Γ. (26) 3. Wave Front Sets with respect to the Iterates of an Operator Now, we assume that A is a bounded open set in R) and we use the following notation: f . := {M ∈ f : = (M, Zf) > E} , (27) where =(M, Zf) is the distance of M to the boundary of f. 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 ' > 0 and > > 0 such that for every E ≥ 0 and 8 > 0 we have *****&($)(%)$*****2,A%+& ≤ ' (8|$|****&(%)$****2,A% + 8|$|−B****$****2,A%) , $ ∈ '∞ (f) . (28) 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 &(%)l or &(−%)l, with l ∈ N, 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. LetΩ be an open subset ofR), ! ∈ D#(Ω), M0 ∈ Ω and consider the following three conditions: (i) $? = &(%)?!, (ii) (Roumieu) ∃b ∈ N, ∀I ∈ R, ∃'D > 0, ∀c ∈ N, and _ ∈ R), we have LLLLL$̂? (_)LLLLL ≤ 'DC(1/:)@∗(:?()(1 + LLLL_LLLL)D, (29) (iii) (Beurling) ∀b ∈ N and I ∈ R,∃':,D > 0, ∀c ∈ N, and _ ∈ R), we have LLLLL$̂? (_)LLLLL ≤ ':,DC:@∗(?(/:)(1 + LLLL_LLLL)D. (30) Then, the distribution ! ∈ E!{*}(a) (! ∈ E!(*)(a)), where a is some neighborhood of M0, if and only if there exist a neighborhood j of M0 and a sequence {$?} in E#(Ω) that satisfies (i) and (ii) in j (that satisfies (i) and (iii) in j). Proof. Sufficiency (Roumieu Case). Let ! ∈ E!{*}(a) with a = k3E(M0), the ball in R) of center M0 and radius 3l, l > 0. We choose m ∈ D(Ω) such that m = 1 in kE(M0) and m = 0 in (k2E(M0))F. We set $? = m&(%)?!. Then, $? ∈ E#(Ω) and $? = &(%)?! in kE(M0). Now, fix l ∈ N. From the hypoellipticity of &(%), there are constants%, = > 0 such that, for |_| large enough, |&(_)| ≥ %|_|-.Then, from the definition of $? we obtain, for |_| large enough, %lLLLL_LLLL-l LLLLL$̂? (_)LLLLL ≤ LLLL& (_)LLLLl ⋅ LLLLL$̂? (_)LLLLL = LLLL& (_)LLLLl LLLLLLL∫R! m (M)&(%)?! (M) C−;⟨2,=⟩=MLLLLLLL = LLLLLLL∫R! m (M)&(%)?! (M)&(−%)l (C−;⟨2,=⟩) =M LLLLLLL . (31) We integrate by parts in the integral above, which will be equal to LLLLLLL∫R! &(%)l (m (M) ⋅ &(%)?! (M)) C−;⟨2,=⟩=MLLLLLLL . (32) From the generalized Leibniz rule, we can write (here/ is the order of &(%)) &(%)l (m (M) ⋅ &(%)?! (M)) = ∑ |$|≤(l 1 6!%$m (M) ⋅ (&l)($) (%) (&(%)?! (M)) . (33) Since &(%)l is hypoelliptic and &(%)?! is a '∞-function in the bounded set k3E(M0), we can apply formula (28) to the Abstract and Applied Analysis 5 operator &(%)l with 8 = o, for 0 < o < l, f .++ = k2E(M0), and $ = &(%)?! (and f . = k2E+G(M0)) to obtain constants 'l, > > 0 (which do not depend onc) such that *******(&l)($) (%) (&(%)?!)*******2,H2'(20) ≤ 'l (o|$|*****&(%)?+l!*****2,H2'+((20) + o|$|−B*****&(%)?!*****2,H2'+((20)) . (34) Now, as ! ∈ E!{*}(a), there are constants b ∈ N and ' > 0 such that (we use the convexity of A∗) *****&(%)?+l!*****2,H2'+( ≤ 'C(1/:)@∗(:((?+l)) ≤ 'C(1/2:)@∗(2:(?)+(1/2:)@∗(2:(l), l,c ∈ N. (35)and Applied Analysis 5 operator &(%)l with 8 = o, for 0 < o < l, f .++ = k2E(M0), and $ = &(%)?! (and f . = k2E+G(M0)) to obtain constants 'l, > > 0 (which do not depend onc) such that *******(&l)($) (%) (&(%)?!)*******2,H2'(20) ≤ 'l (o|$|*****&(%)?+l!*****2,H2'+((20) + o|$|−B*****&(%)?!*****2,H2'+((20)) . (34) Now, as ! ∈ E!{*}(a), there are constants b ∈ N and ' > 0 such that (we use the convexity of A∗) *****&(%)?+l!*****2,H2'+( ≤ 'C(1/:)@∗(:((?+l)) ≤ 'C(1/2:)@∗(2:(?)+(1/2:)@∗(2:(l), l,c ∈ N. (35) Therefore, we can estimate, by Hölder’s inequality, the Fourier transform $̂?(_) for |_| big enough in the following way (at the end, we use the fact that A∗(M)/M is an increasing function): %lLLLL_LLLL-l LLLLL$̂? (_)LLLLL ≤ 'l ∑ |$|≤(l 1 6!****%$m****2,H2'(20) ⋅ (o|$|*****&(%)?+l!*****2,H2'+((20) + o|$|−B*****&(%)?!*****2,H2'+((20)) ≤ %(,l (C(1/:)@∗(:((?+l)) + C(1/:)@∗(:(?)) ≤ r(,lC(1/2:)@∗(2:(?). (36) On the other hand, if |_| is bounded, we put %E = ‖m‖2,H2'(20) and, by Hölder’s inequality, we have LLLLL$̂? (_)LLLLL ≤ LLLLLLL∫R! m (M)&(%)?! (M) C−;⟨2,=⟩=MLLLLLLL ≤ %E*****&(%)?!*****2,H2' ≤ '%EC(1/2:)@∗(2:?(). (37) From the last estimates, we can conclude that ∃b ∈ N, ∀I ∈ R, ∃'D > 0, ∀c ∈ N and _ ∈ R), LLLLL$̂? (_)LLLLL ≤ 'DC(1/:)@∗(:?()(1 + LLLL_LLLL)D, (38) which finishes this implication. The Beurling case is similar. Necessity (Roumieu Case). Let {$?}?∈N ⊂ E#(Ω) satisfying (i) in some n


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 Ω if and only if for every compact set  ⊂⊂ Ω there is a constant  > 0 such that       ()       2, ≤  +1 (!)  , where  is the order of the operator and ‖ ⋅ ‖ 2, is the  2 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][7][8][9] for different purposes.We mention modern contributions like [10][11][12][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 E  * (Ω) on an open subset Ω of R  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 E  * (Ω) 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 ()   :   → (  ) is convex.
Normally, we will denote   simply by . ( There is no loss of generality to assume that  vanishes on [0, 1].Then  * has only nonnegative values, it is convex,  * ()/ is increasing and tends to ∞ as  → ∞, and  * * = .
In what follows, Ω denotes an arbitrary subset of R  and  ⊂⊂ Ω means that  is a compact subset in Ω.
(b) For an open subset Ω in R  , the class of -ultradifferentiable functions of Beurling type is defined by for every  ⊂⊂ Ω and every  > 0} . (5) The topology of this space is and one can show that E () (Ω) is a Fréchet space.
(c) For a compact subset  in R  which coincides with the closure of its interior and  > 0, set E {} () = { ∈  ∞ () : there exists  ∈ N such that  ,1/ () < ∞} .(7) 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 Ω in R  , the class of ultradifferentiable functions of Roumieu type is defined by E {} (Ω) := { ∈  ∞ (Ω) : ∀ ⊂⊂ Ω ∃ > 0 such that  , () < ∞} .(9) Its topology is the following: that is, it is endowed with the topology of the projective limit of the spaces E {} () when  runs the compact subsets of Ω.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, E {} (Ω) is also a nuclear and reflexive locally convex space.In particular, E {} (Ω) is an ultrabornological (hence barrelled and bornological) space.The elements of E () (Ω) (resp., E {} (Ω)) are called ultradifferentiable functions of Beurling type (resp., Roumieu type) in Ω.
In the case that () :=   (0 <  < 1), the corresponding Roumieu class is the Gevrey class with exponent 1/.In the limit case  = 1, not included in our setting, the corresponding Roumieu class E {} (Ω) is the space of real analytic functions on Ω, whereas the Beurling class E () (R  ) gives the entire functions.
If a statement holds in the Beurling and the Roumieu case, then we will use the notation E * (Ω).It means that in all cases, * can be replaced either by () or {}.
For a compact set  in R  , define endowed with the induced topology.For an open set Ω in R  , define Following [14], we consider smooth functions in an open set Ω such that there exists  > 0 verifying for each  ∈ N where  is a compact subset in Ω, ‖ ⋅ ‖ 2, denotes the  2norm on , and   () is the th iterate of the partial differential operator () of order ; that is, If  = 0, then  0 () = .Given a polynomial  ∈ C[ 1 , . . .,   ] with degree , () = ∑ ||≤     , the partial differential operator () is the following: () = ∑ ||≤     , where  = (1/).
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 Ω of R  , a compact subset  ⊂⊂ Ω, and  > 0, 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 E   () 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 {  } ∈N is a compact exhaustion of Ω, we have This metrizable locally convex topology is defined by the fundamental system of seminorms {‖ ⋅ ‖   , } ∈N .
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.
Denoting by the classical Fourier transform of  ∈ E  (Ω), we recall from [22,Proposition 3.3] the following characterization of the * -singular support in the sense of Braun et al. [15].
This led, in [22,Definition 3.4], to the following definition of wave front set WF * () in the sense of Braun et al. [15].Definition 5. Let Ω be an open subset of R  and  ∈ D  (Ω).The {}-wave front set WF {} (), resp., ()-wave front set WF () (), of  is the complement in Ω × (R  \ 0) of the set of points ( 0 ,  0 ) such that there exist an open neighborhood  of  0 in Ω, a conic neighborhood Γ of  0 , and a bounded sequence   ∈ E  (Ω) (the set of classical distributions with compact support in Ω) equal to  in  such that there are  ∈ N and  > 0 with the property

Wave Front Sets with respect to the Iterates of an Operator
Now, we assume that  is a bounded open set in R  and we use the following notation: where (, ) is the distance of  to the boundary of .
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 Ω be an open subset of R  ,  ∈ D  (Ω),  0 ∈ Ω and consider the following three conditions: 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 () ℓ is hypoelliptic and ( ≤  ,ℓ ( (1/) * ((+ℓ)) +  (1/) * () ) ≤  ,ℓ  (1/2) * (2) . (36) On the other hand, if || is bounded, we put   = ‖‖ 2, 2 ( 0 ) and, by Hölder's inequality, we have From the last estimates, we can conclude that ∃ ∈ N, which finishes this implication.The Beurling case is similar.
in some neighborhood  of  0 and (ii).We fix a compact set  ⊂⊂  and take  > ( + 1)/2.Now, by (ii), there is  ∈ N and a constant  > 0 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 E  * (Ω), such as completeness (cf.[14]).On the contrary, the elliptic case is not really interesting here since E  * (Ω) = E * (Ω) 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 Ω be an open subset of R  ,  ∈ D  (Ω), and () a linear partial differential hypoelliptic operator of order  with constant coefficients.We say that a point ( 0 ,  0 ) ∈ Ω × (R  \ {0}) is not in the {}-wave front set with respect to the iterates of , WF  {} () (()-wave front set with respect to the iterates of , WF  () ()), if there are a neighborhood  of  0 , an open conic neighborhood Γ of  0 , and a sequence {  } ∈N ⊂ E  (Ω) such that (i) and (ii) of the following conditions hold ((i) and (iii) of the following conditions hold): (ii) Roumieu: (a) there are constants  ∈ N,  > 0, and  > 0, such that (b) there is a constant  ∈ N such that for all ℓ ∈ N 0 , there is  ℓ > 0 with the property (iii) Beurling: (a) there are ,  > 0 such that for all  ∈ N, there is   > 0 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]).
and () a linear partial differential operator with analytic coefficients in Ω of order .

Proposition 9.
Let Ω be an open subset of R  ,  ∈ D  (Ω),  a weight function, and () a hypoelliptic linear partial differential operator of order  with constant coefficients.Then, the following inclusions hold: Proof.
On the other hand, if we consider the estimate for some C > 0. We conclude, using the convexity of  * , that there are constants  ℓ > 0 and  > 0 such that Here, where  1 () is the integral when || ≤ ||, for  > 0 to be chosen, and  2 () is the integral when || ≥ ||.In this case, we use (60) and obtain a constant  ℓ > 0 which depends on ℓ (and , ) and a constant  > 0 with the property that for every  ∈ N there is a constant   > 0 such that for any  ∈ Γ and  ∈ N, Then, by Proposition 9, the following inclusions hold: Proof.Let ( 0 ,  0 ) ∉ WF  * () such that   ( 0 ) ̸ = 0.Then, there are a neighborhood  of  0 , a conic neighborhood Γ of  0 , and a sequence {  } ∈N ⊂ E  (Ω) 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  ⊂ Γ such that   () ̸ = 0 for  ∈ .We take a compact neighborhood  ⊂  of  0 and consider a sequence {  } ∈N ⊂ D() satisfying (48) such that   ≡ 1 on .
We set now   =  3 2  .We want to estimate To estimate |û  ()| in , we will solve in an approximate way the following equation: As in [17], we put V() =  −⟨,⟩ (, )/  ()  .For (, ) ∈  × , we have where  =  1 + ⋅ ⋅ ⋅ +   , with   =   (, ) a differential operator of order ≤ which depends on the parameter  such that   ||  is homogeneous of order 0. 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  ℎ+  3 2  =    3 2  with ℎ +  =  ≤  is given by by the Chu-Vandermonde identity.For  ≥  + 1, the term   does not appear anymore for ℎ = 0. 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  3 2  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  > 0 in a neigborhood of .Since  1 () = ⟨,    −⟨,⟩ ⟩, we have Now, since ∑  ℎ=0 (  ℎ ) = 2  and in the sum of the expression of   ,  <  = ℎ +  ≤  + , we obtain (we recall that In the last expression, we obtain a sum of   terms, for some constant  > 0, of the form   We study now where we have splitted  2 () in the sum of  1 () and  2 (), the first when || ≤ || and the second when || ≥ ||, for a constant  > 0 to be chosen.
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  ∈ E  {} (Ω)\E {} (Ω) (for some open subset Ω of R  ).Then, WF  {} () = 0 but WF {} () ̸ = 0, which implies that the inclusion is strict.
Let us also remark that for the heat operator () =   − Δ  , we can explicitly write its characteristic set Σ, so that the previous considerations give, for  ∈ E  {} (Ω) \ E {} (Ω), the following information on WF {} (), 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.

Distributions with Prescribed Wave Front Set
The proof of the following lemma is straightforward.

Lemma 16.
Let  be a weight function.Then, for every  > 0 and  ∈ N

Now, we show that the product of a Gevrey function with a function in E 𝑃
* (Ω) belongs to the last space.
Proof.We will analyse the  2 -norms of ()  () on a compact set  in Ω.First, we observe that, by the generalized Leibniz rule over () applied  times, We fix now a compact set  in Ω such that dist(, Ω) ≥  > 0. We apply  2 -norms in the compact set Since  ∈ E {} (Ω), there is a constant  > 0 such that, for each  ∈ N  0 and  ∈  we have Consequently, sup Therefore, Now, we apply (28)  times to the factor ‖ ( 1 ) ⋅ ⋅ ⋅  (  ) ‖ 2, .We will use the notation () =  + (0, ), for  > 0. In the first step,       ( 1 ) ⋅ ⋅ ⋅  (  )      2, ≤  ( In the second step, ( 1 ) is replaced by ( 1 +  2 ) and so on in the next steps.Therefore, to avoid that, after  steps, the set ( 1 + ⋅ ⋅ ⋅ +   ) leaves Ω and to keep it bounded for all , we may take   depending on  for all 1 ≤  ≤ .We take   =  − with  > 0 a constant such that for all .It is obvious that +1 for all 1 ≤  ≤  − 1.Moreover, we can assume that   < 1 for all 1 ≤  ≤ .
(132)  (138) This is a continuous function in R  and we will prove that 0 ̸ = WF  {}  ⊂ .
To prove first that WF  {}  ⊂ , we take ( 0 ,  0 ) ∉  and prove that ( 0 ,  0 ) ∉ WF  {} .To this aim, we choose an open neighborhood  of  0 and an open conic neighborhood Γ of  0 such that Write  =  1 +  2 , where  1 is the sum of terms in (138) with   ∉  and  2 is the sum of terms with   ∈ .Therefore, there is a neighborhood  1 of  0 with  1 ⊂  such that  1 is in E {} ( 1 ) since all but a finite number of terms vanish in  1 .Moreover, every weight function  is increasing by definition, so that  ≤ , E {} ⊂ E {} and hence  1 ∈ E {} ( 1 ).
Consider then Note that it is a totally convergent series since sup for some   > 0 and because of (137) with ℓ ≥ 3 + 2.
Let us then compute the Fourier transform with   ∉ Γ because of (139).
Note that for every fixed ℓ ∈ N, This is a continuous function in R  and we will prove that 0 ̸ = WF  ()  ⊂ .The proof of the inclusion WF  ()  ⊂  is similar to that in the Roumieu case.We take ( 0 ,  0 ) ∉ , choose an open neighborhood  of  0 and an open conic neighborhood Γ of  0 such that ( × Γ) ∩  ̸ = 0, and write  =  1 +  2 , where  1 is the sum of terms in (158) with   ∉  and  2 is the sum of terms with   ∈ .
We choose a neighborhood  1 of  0 with  1 ⊂  such that  1 is in E () ( 1 ) ⊂ E () ( 1 ) since all but a finite number of terms vanish in  1 .

)
Beurling Case.Let us assume now that ( 0 ,  0 ) ∉ WF () .We take   and   as in the Roumieu case.From (50), for any  ∈ N, there is   > 0 satisfying      f ()      ≤   C ( (/) * (/) +          ) From Definition 5, there exist a neighborhood  of  0 , an open conic neighborhood  of  0 , and a bounded sequence {  } ∈N ⊂ E  (Ω) such that   =  in  for every  ∈ N and for every  ∈ N there is   > 0, such that               û ()     ≤     * (/) ,  ∈ ,  ∈ N. (60) * (/) [25, {} () ∩ ( × ) = 0, then the sequence   =   ()  , for  ∈ N large enough independent of , satisfies that there is  ∈ N such that for every ℓ ∈ N, there is  ℓ > 0 with Proof.We make a sketch of proof of (a) only.Let  0 ∈ ,  0 ∈ \{0} and choose  and Γ, with Γ 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 Γ and   instead of ()   +ℓ .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  ∈ D  (Ω) with respect to the class E  * is the complement in Ω of the biggest open set , where |  ∈ E  * ().As a consequence of Propositions 6 and 9 and Corollary 10, we obtain the following result.The projection in Ω of   * () is the singular support with respect to the class E  * (Ω) if  ∈ D  (Ω).