The Gabor wave front set in spaces of ultradifferentiable functions
aa r X i v : . [ m a t h . F A ] J un THE GABOR WAVE FRONT SET IN SPACES OFULTRADIFFERENTIABLE FUNCTIONS
CHIARA BOITI, DAVID JORNET, AND ALESSANDRO OLIARO
Abstract.
Given a non-quasianalytic subadditive weight function ω we consider the weightedSchwartz space S ω and the short-time Fourier transform on S ω , S ′ ω and on the related modu-lation spaces with exponential weights. In this setting we define the ω -wave front set WF ′ ω ( u )and the Gabor ω -wave front set WF Gω ( u ) of u ∈ S ′ ω , and we prove that they coincide. Finallywe look at applications of this wave front set for operators of differential and pseudo-differentialtype. Introduction
The wave front set is a basic concept in the theory of linear partial differential operators.It deals with the analysis of singularities of a function (or distribution), and in the classicalSchwartz distributions theory it was originally defined in [H-1]. The idea is that, if a distribution u ∈ D ′ ( R d ) coincides with a C ∞ function in a neighborhood of a certain point x ∈ R d , thenthere exists a cut-off function ϕ ∈ D ( R d ) (i.e., ϕ ∈ C ∞ ( R d ) with compact support) such that ϕu ∈ D ( R d ), and consequently c ϕu is a rapidly decreasing function. If u is not C ∞ at x , then c ϕu does not decrease rapidly at least in some directions, and these directions are responsiblefor the absence of regularity of u at x . The wave front set collects all the points ( x , ξ ),with ξ = 0, where, roughly speaking, the distribution u is not C ∞ at x due (on the Fouriertransform side) to the absence of rapid decreasing in the direction ξ . The wave front set is thena subset of R d × ( R d \ { } ), is computed for a distribution u and has to do with the analysis ofthe points where u is not smooth in connection with the directions where absence of smoothnessis shown by the Fourier transform of u . In the case described above, the distribution space is D ′ , and ‘smooth’ means C ∞ .Some very natural questions arise. First, the spaces D ′ and C ∞ could be replaced by otherspaces of distributions connected to other concepts of ‘smoothness’; in this frame we referfor example to [R], [BJJ], [AJO-1], [AJO-2], where wave front sets connected with Gevreyand ultradifferentiable type regularity are considered. Moreover, in the case of D ′ and C ∞ the smoothness is intended in a local sense, but also some kind of global regularity could beconsidered, see for example [H-2], [CS], [RW], [N], [SW-2].The wave front set, moreover, has very important applications in the study of propagationof singularities for partial differential (or more generally for pseudodifferential) operators. Inthis frame, different classes of pseudodifferential operators lead to corresponding variants of the Mathematics Subject Classification.
Primary 35A18; Secondary 46F05, 42C15, 35S05.
Key words and phrases.
Gabor wave front set, weighted Schwartz classes, short-time Fourier transform,Gabor frames. The Gabor wave front set in spaces of ultradifferentiable functions wave front set, adapted to the class under consideration. Among the vast literature in this fieldwe refer for example to [H-3] for the C ∞ case, to [R] for the Gevrey case, and to [SW-1], [CW]for the case of global wave front set defined in the spirit of the present paper.Time-frequency analysis is a field of research that in the last decades has had a very biggrowth, with the development of many new techniques. One of the basic ideas of time-frequencyanalysis is the simultaneous analysis of a function (or distribution) with respect to variablesand covariables, in order to quantify the energy of a signal at some time x and some frequency ξ . Since the wave front set has to do with a simultaneous analysis of points (variables) anddirections (covariables), it is very natural to try to apply methods of time-frequency analysisin connection with the wave front set. The work [RW] is a very interesting contribution in thisdirection.The present paper deals with a global wave front set, in the spirit of [H-2], treated withtechniques from time-frequency analysis, following ideas from [RW]. In particular, we studythe case of ultradifferentiable functions in the sense of [B], [BMT], focusing on the space S ω ( R d ),defined as the space of functions f ∈ S ( R d ) such that for every λ > α ∈ N d we havesup x ∈ R d e λω ( x ) | ∂ α f ( x ) | < + ∞ , sup ξ ∈ R d e λω ( ξ ) | ∂ α ˆ f ( ξ ) | < + ∞ , where ω is a non-quasianalytic subadditive weight, cf. Definition 2.1. The spaces S ω ( R d ),together with the corresponding (ultra)distribution space S ′ ω ( R d ), play a role similar to S ( R d )and S ′ ( R d ) in the classical Schwartz frame. We study in this paper a global wave front setadapted to S ω regularity, giving two different definitions; one is based on Gabor transformand the other is related to Gabor frames. We show that these definitions are equivalent.Moreover, we give applications of this global wave front set to pseudodifferential operators,to partial differential operators with polynomial coefficients, to localization operators, and weanalyze some examples. The techniques are related with time-frequency analysis, in particularto Gabor transform, Gabor frames and modulation spaces.The paper is organized as follows. In Section 2 we present basic definitions and considertime-frequency analysis on ultradifferentiable spaces; we revise some known properties andprove other results that are needed in the paper, but that we could not find in the literature.In Section 3 we give the two definitions of wave front set and prove that they are equivalent.Moreover we show that the global wave front set of a distribution u ∈ S ′ ω ( R d ) is empty ifand only if u ∈ S ω ( R d ), and that the global wave front set is not affected by translations andmodulations, that are the basic operators of time-frequency analysis. In Section 4 we presentsome applications about the action of operators of differential and pseudodifferential type onthe wave front set, and finally in Section 5 we analyze some examples. Preliminaries and the short-time Fourier transform in S ω ( R d ) Given a function f ∈ L ( R d ), the Fourier transform of f is defined as F ( f ) = ˆ f ( ξ ) = Z R d e − i h x,ξ i f ( x ) dx, . Boiti, D. Jornet and A. Oliaro 3 with standard extensions to more general spaces of functions and distributions. Definition 2.1. A non-quasianalytic subadditive weight function is a continuous increasingfunction ω : [0 , + ∞ ) → [0 , + ∞ ) satisfying the following properties: ( α ) ω ( t + t ) ≤ ω ( t ) + ω ( t ) ∀ t , t ≥ ; ( β ) Z + ∞ ω ( t ) t dt < + ∞ ; ( γ ) log t = o ( ω ( t )) when t → + ∞ ; ( δ ) ϕ ω ( t ) := ω ( e t ) is convex.We then define ω ( ζ ) := ω ( | ζ | ) for ζ ∈ C d . We denote by ϕ ∗ ω the Young conjugate of ϕ ω , defined by ϕ ∗ ω ( s ) := sup t ≥ { ts − ϕ ω ( t ) } and recall that ϕ ∗ ω is convex and increasing, ϕ ∗∗ ω = ϕ ω and ϕ ∗ ω ( s ) /s is increasing (up to assume,without any loss of generality, that ω | [0 , ≡ Definition 2.2.
We define S ω ( R d ) as the set of all u ∈ S ( R d ) such that (i) ∀ λ > , α ∈ N d : sup R d e λω ( x ) | D α u ( x ) | < + ∞ , (ii) ∀ λ > , α ∈ N d : sup R d e λω ( ξ ) | D α b u ( ξ ) | < + ∞ ,where N := N ∪ { } and D α = ( − i ) | α | ∂ α .As usual, the corresponding dual space is denoted by S ′ ω ( R d ) and is the set of all linearand continuous functionals u : S ω ( R d ) → C . An element of S ′ ω ( R d ) is called an ω -tempereddistribution . In [BJO, Thm. 4.8] we provided the space S ω ( R d ) with different equivalent systems ofseminorms. For example, for u ∈ S ω ( R d ), the family of seminorms p λ,µ ( u ) := sup α,β ∈ N d sup x ∈ R d | x β D α u ( x ) | e − λϕ ∗ ω ( | α | λ ) − µϕ ∗ ω ( | β | µ ) , (2.1)for λ, µ >
0. On the other hand, it is not difficult to see (using, for instance, [BJO, Lemma4.7(ii)]) that the family of seminorms(2.2) q λ,µ ( u ) := sup α ∈ N d sup x ∈ R d | D α u ( x ) | e − λϕ ∗ ω ( | α | λ ) + µω ( x ) , λ, µ > , defines another equivalent system of seminorms for S ω ( R d ).We recall that S ω ( R d ) ⊆ S ( R d ) and for their correspondent dual spaces we have the inclusion S ′ ( R d ) ⊆ S ′ ω ( R d ).Let us denote by T x , M ξ and Π( z ), respectively, the translation , the modulation and the phase-space shift operators, defined by T x f ( y ) = f ( y − x ) , M ξ f ( y ) = e i h y,ξ i f ( y ) , Π( z ) f ( y ) = M ξ T x f ( y ) = e i h y,ξ i f ( y − x ) , for x, y, ξ ∈ R d and z = ( x, ξ ). The Gabor wave front set in spaces of ultradifferentiable functions
Definition 2.3.
For a window function ϕ ∈ S ω ( R d ) \ { } , the short-time Fourier transform (briefly STFT) of f ∈ S ′ ω ( R d ) is defined, for z = ( x, ξ ) ∈ R d , by: V ϕ f ( z ) := h f, Π( z ) ϕ i (2.3) = Z R d f ( y ) ϕ ( y − x ) e − i h y,ξ i dy, (2.4) where the bracket h· , ·i in (2.3) and the integral in (2.4) denote the conjugate linear action of S ′ ω on S ω , consistent with the inner product h· , ·i L . By [GZ, Lemma 1.1], for f, ϕ, ψ ∈ S ω ( R d ) we have the following inversion formula : h ψ, ϕ i f ( y ) = 1(2 π ) d Z R d V ϕ f ( z )(Π( z ) ψ )( y ) dz. (2.5)In particular, for ψ = ϕ ∈ S ω ( R d ) \ { } : f ( y ) = 1(2 π ) d k ϕ k L Z R d V ϕ f ( z )(Π( z ) ϕ )( y ) dz. (2.6)We recall, from [GZ], the following results: Theorem 2.4.
Let ϕ ∈ S ω ( R d ) \ { } and f ∈ S ′ ω ( R d ) . Then V ϕ f is continuous and there areconstants c, λ > such that | V ϕ f ( z ) | ≤ ce λω ( z ) ∀ z ∈ R d . (2.7) Proposition 2.5.
Let ϕ ∈ S ω ( R d ) \{ } and assume that F : R d → C is a measurable functionthat satisfies that for all λ > there is a constant C λ > such that | F ( z ) | ≤ C λ e − λω ( z ) ∀ z ∈ R d . Then f ( y ) := Z R d F ( z )(Π( z ) ϕ )( y ) dz defines a function f ∈ S ω ( R d ) . Theorem 2.6.
Let ϕ ∈ S ω ( R d ) \ { } . Then, for f ∈ S ′ ω ( R d ) , the following are equivalent: (i) f ∈ S ω ( R d ) ; (ii) for all λ > there exists C λ > such that | V ϕ f ( z ) | ≤ C λ e − λω ( z ) ∀ z ∈ R d ;(iii) V ϕ f ∈ S ω ( R d ) . The following lemma is well known for functions in S ( R d ), and hence in S ω ( R d ) . So we omitits proof.
Lemma 2.7.
For f, ϕ ∈ S ω ( R d ) we have that d V ϕ f ( η, y ) = (2 π ) d e i h η,y i f ( − y ) b ϕ ( η ) ∀ ( η, y ) ∈ R d . As a consequence, we can deduce the following result. . Boiti, D. Jornet and A. Oliaro 5
Proposition 2.8.
Let ϕ ∈ S ω ( R d ) \ { } . Then V ϕ : S ω ( R d ) −→ S ω ( R d ) is continuous.Proof. Let us first remark that if f ∈ S ω ( R d ) then V ϕ f ∈ S ω ( R d ) by Theorem 2.6.Since S ω is a Fr´echet space, to prove the continuity of V ϕ we consider a sequence { f n } n ∈ N ⊂S ω ( R d ) such that f n −→ f ∈ S ω ( R d ) in S ω ( R d )(2.8)and prove that V ϕ f n → V ϕ f in S ω ( R d ).Indeed, (2.8) implies that e i h η,y i f n ( − y ) b ϕ ( η ) −→ e i h η,y i f ( − y ) b ϕ ( η ) in S ω ( R d )and hence, by Lemma 2.7, [ V ϕ f n → d V ϕ f in S ω ( R d ) . Applying the inverse Fourier transform, which is continuous on S ω , we have that V ϕ f n → V ϕ f in S ω ( R d ) . and the proof is complete. (cid:3) The short-time Fourier transform also provides a new equivalent system of seminorms for S ω ( R d ) Proposition 2.9. If ϕ ∈ S ω ( R d ) \ { } , then the collection of seminorms k V ϕ f k ω,λ := sup z ∈ R d | V ϕ f ( z ) | e λω ( z ) , for λ > , forms an equivalent system of seminorms for S ω ( R d ) .Proof. Set ˜ S ω ( R d ) := { f ∈ S ( R d ) : k V ϕ f k ω,λ < + ∞ ∀ λ > } . By Theorem 2.6 the sets ˜ S ω ( R d ) and S ω ( R d ) are equal. We have to prove that they have thesame topology.By the inversion formula (2.6) we have that, for z = ( x, ξ ) ∈ R d and λ, µ > e − λϕ ∗ ω ( | α | λ ) e − µϕ ∗ ω ( | β | µ ) | y β D αy f ( y ) |≤ Ce − λϕ ∗ ω ( | α | λ ) e − µϕ ∗ ω ( | β | µ ) Z R d | V ϕ f ( z ) | · | y β D αy (Π( z ) ϕ )( y ) | dz = Ce − λϕ ∗ ω ( | α | λ ) e − µϕ ∗ ω ( | β | µ ) Z R d | V ϕ f ( x, ξ ) | · | y β D αy e i h y,ξ i ϕ ( y − x ) | dxdξ ≤ C X γ ≤ α (cid:18) αγ (cid:19) −| α | Z R d | V ϕ f ( x, ξ ) | · | y | | β | e − µϕ ∗ ω ( | β | µ )(2.9) ·| ξ | | α − γ | | D γy ϕ ( y − x ) | e − λϕ ∗ ω ( | α | λ )2 | α | dxdξ for some C > The Gabor wave front set in spaces of ultradifferentiable functions
The following property is known and can be found, for instance, in Lemma 4.7(i) of [BJO](see also [FGJ]): | y | | β | e − µϕ ∗ ω ( | β | µ ) ≤ e µω ( y ) , for all y ∈ R d . (2.10)Moreover, since the weight function ω is subadittive and ϕ ∗ is convex, we can use, for instance,Proposition 2.1(e) of [BJ] to obtain2 | α | e − λϕ ∗ ω ( | α | λ ) ≤ e λ e − λϕ ∗ ω ( | α | λ ) . (2.11)Substituting (2.10) and (2.11) into (2.9), by the subadditivity of ω we have e − λϕ ∗ ω ( | α | λ ) − µϕ ∗ ω ( | β | µ ) | y β D αy f ( y ) | ≤ C λ X γ ≤ α (cid:18) αγ (cid:19) −| α | Z R d | V ϕ f ( x, ξ ) | e µω ( x ) e µω ( y − x ) ·| ξ | | α − γ | | D γy ϕ ( y − x ) | e − λϕ ∗ ω ( | α | λ ) dxdξ (2.12)for some C λ > ϕ ∈ S ω ( R d ), by (2.2), for every λ, µ > C λ,µ > γ ∈ N d and y ∈ R d , | D γy ϕ ( y ) | e µω ( y ) ≤ C λ,µ e λϕ ∗ ω ( | γ | λ ) . (2.13)From (2.13) with 3 λ instead of λ and y − x instead of y , we have that for every µ, λ > C µ,λ > e − λϕ ∗ ω ( | α | λ ) − µϕ ∗ ω ( | β | µ ) | y β D αy f ( y ) | ≤ C µ,λ X γ ≤ α (cid:18) αγ (cid:19) −| α | · Z R d | V ϕ f ( x, ξ ) | e µω ( x ) | ξ | | α − γ | e λϕ ∗ ω ( | γ | λ ) − λϕ ∗ ω ( | α | λ ) dxdξ. By (2.10) we have | ξ | | α − γ | ≤ e λω ( ξ )+3 λϕ ∗ ( | α − γ | λ ) . So, from the convexity of ϕ ∗ , we obtain e − λϕ ∗ ω ( | α | λ ) − µϕ ∗ ω ( | β | µ ) | y β D αy f ( y ) |≤ C µ,λ X γ ≤ α (cid:18) αγ (cid:19) −| α | Z R d | V ϕ f ( x, ξ ) | e µω ( x ) e λω ( ξ ) dxdξ (2.14) ≤ C µ,λ Z R d | V ϕ f ( z ) | e ( µ +3 λ +1) ω ( z ) e − ω ( z ) dz ≤ C ′ µ,λ k V ϕ f k ω,µ +3 λ +1 , (2.15)for C ′ µ,λ := C µ,λ R R d e − ω ( z ) dz , which is finite by condition ( γ ) of Definition 2.1.It is easy to see that ˜ S ω ( R d ) is a Fr´echet space. Indeed, the estimate (2.15) implies that theidentity operator I : ˜ S ω ( R d ) → S ω ( R d ) is continuous. Hence, any Cauchy sequence { f n } n ∈ N in ˜ S ω ( R d ) is a Cauchy sequence in S ω ( R d ). So, it converges in S ω ( R d ) to some f (because S ω ( R d ) is complete). From Proposition 2.8, { V ϕ f n } n ∈ N converges to V ϕ f in S ω ( R d ). Therefore, { f n } n ∈ N converges to f in ˜ S ω ( R d ) . We can apply the open mapping theorem to conclude that I is an isomorphism and hencethe two topologies on S ω ( R d ) coincide. (cid:3) . Boiti, D. Jornet and A. Oliaro 7 Now, we can prove the following
Proposition 2.10.
Assume that ψ, γ ∈ S ω ( R d ) \ { } with h ψ, γ i 6 = 0 . Then the followingassertions hold: (a) If F : R d → C is a measurable function that satisfies, for some c, λ > , | F ( z ) | ≤ ce λω ( z ) ∀ z ∈ R d , (2.16) then S ω ( R d ) ∋ ϕ
7→ h f, ϕ i := Z R d F ( z ) h Π( z ) γ, ϕ i dz define an ω -tempered distribution f ∈ S ′ ω ( R d ) . (b) In particular, if F = V ψ f for some f ∈ S ′ ω ( R d ) , then the following inversion formula holds: f = 1(2 π ) d h γ, ψ i Z R d V ψ f ( z )Π( z ) γdz. (2.17) Proof.
From (2.16) we have, for all ϕ ∈ S ω ( R d ), |h f, ϕ i| ≤ Z R d | F ( z ) | · | V γ ϕ ( z ) | dz ≤ c Z R d e λω ( z )+ ω ( z ) | V γ ϕ ( z ) | e − ω ( z ) dz ≤ c ′ k V γ ϕ k ω,λ +1 (2.18)for some c ′ > f defines a continuous linear func-tional on S ω ( R d ), i.e. f ∈ S ′ ω ( R d ). This proves ( a ).In particular, if F = V ψ f for some f ∈ S ′ ω ( R d ) then F satisfies (2.16) for Theorem 2.4 andhence (2.17) defines an ω -tempered distribution ˜ f ∈ S ′ ω ( R d ) given by h ˜ f , ϕ i = 1(2 π ) d h γ, ψ i Z R d V ψ f ( z ) h Π( z ) γ, ϕ i dz ∀ ϕ ∈ S ω ( R d ) . However, from (2.5) we have that ϕ = 1(2 π ) d h ψ, γ i Z R d V γ ϕ ( z )Π( z ) ψdz and then (see also [G, pg 43] for vector valued integrals) h f, ϕ i = 1(2 π ) d h ψ, γ i Z R d V γ ϕ ( z ) h f, Π( z ) ψ i dz = 1(2 π ) d h γ, ψ i Z R d h Π( z ) γ, ϕ i V ψ f ( z ) dz = h ˜ f , ϕ i , ϕ ∈ S ω ( R d ) . Therefore f = ˜ f and ( b ) is proved. (cid:3) The Gabor wave front set in spaces of ultradifferentiable functions
Let us now recall the definition of the adjoint operator of V ϕ . We consider, for ϕ ∈ L ( R d ),the operator A ϕ : L ( R d ) −→ L ( R d )defined by A ϕ F = Z R d F ( z )Π( z ) ϕ dz. This is the adjoint operator of V ϕ : L ( R d ) → L ( R d ) since, for all F ∈ L ( R d ) and h ∈ L ( R d ), h A ϕ F, h i = Z R d F ( z ) h Π( z ) ϕ, h i dz = h F, V ϕ h i = h V ∗ ϕ F, h i . In particular, for ϕ ∈ S ω ( R d ) and F ∈ S ω ( R d ) we can define the adjoint operator V ∗ ϕ F = A ϕ F . We observe that V ∗ ϕ F ∈ S ω ( R d ). In fact, if G ( x, ξ, t ) := F ( x, ξ ) ϕ ( t − x ) ∈ S ω ( R d ), wecan write A ϕ F as a partial Fourier transform: A ϕ F ( t ) = Z R d F ( x, ξ ) ϕ ( t − x ) e i h t,ξ i dxdξ = (cid:0) F ( x,ξ ) G (cid:1) ( x ′ , ξ ′ , t ) (cid:12)(cid:12) ( x ′ ,ξ ′ ,t )=(0 , − t,t ) . (2.19)Then V ∗ ϕ : S ω ( R d ) −→ S ω ( R d )(2.20)continuously.Moreover, the inversion formula (2.5) gives, for ϕ, ψ, f ∈ S ω ( R d ) with h ϕ, ψ i 6 = 0,1 h ϕ, ψ i V ∗ ϕ V ψ f = 1 h ϕ, ψ i Z R d V ψ f ( z )Π( z ) ϕdz = (2 π ) d f, i.e. 1(2 π ) d h ϕ, ψ i V ∗ ϕ V ψ = I S ω ( R d ) . (2.21)More in general, if ϕ ∈ S ω ( R d ) \ { } and F is a measurable function on R d , we define theadjoint operator V ∗ ϕ F = Z R d F ( z )Π( z ) ϕdz, (2.22)where the integral is interpreted, if necessary, in a weak sense, i.e. h V ∗ ϕ F, g i = Z R d F ( z ) h Π( z ) ϕ, g i dz = Z R d F ( z ) V ϕ g ( z ) dz = h F, V ϕ g i for g ∈ S ω ( R d ).In particular, if ϕ, ψ ∈ S ω ( R d ) \ { } with h ϕ, ψ i 6 = 0, by Theorem 2.4 and Proposition 2.10we can define the adjoint operator (2.22) for F = V ψ f with f ∈ S ′ ω ( R d ) and obtain that, for all g ∈ S ω ( R d ), h V ∗ ϕ V ψ f, g i = Z R d V ψ f ( z ) h Π( z ) ϕ, g i dz = (2 π ) d h ϕ, ψ ih f, g i , (2.23) . Boiti, D. Jornet and A. Oliaro 9 i.e. 1(2 π ) d h ϕ, ψ i V ∗ ϕ V ψ = I S ′ ω ( R d ) . (2.24)We can now prove the following proposition in a standard way. Proposition 2.11.
Let ϕ, ψ, γ ∈ S ω ( R d ) with h γ, ψ i 6 = 0 and let f ∈ S ′ ω ( R d ) . Then | V ϕ f ( z ) | ≤ π ) d |h γ, ψ i| ( | V ψ f | ∗ | V ϕ γ | )( z ) , z = ( x, ξ ) ∈ R d . The ω -Gabor wave front set In this section, we consider a global wave front set defined in terms of rapid decay of the STFTin conical sets. After that, for a Gabor frame we define the
Gabor wave front set , where conicalsets are intersected with a lattice. We prove that these wave front sets coincide.
Definition 3.1.
Let u ∈ S ′ ω ( R d ) and ϕ ∈ S ω ( R d ) \ { } . We say that z = ( x , ξ ) ∈ R d \ { } is not in the ω -wave front set WF ′ ω ( u ) of u if there exists an open conic set Γ ⊆ R d \ { } containing z and such that sup z ∈ Γ e λω ( z ) | V ϕ u ( z ) | < + ∞ , λ > . (3.1)We observe that WF ′ ω ( u ) is a closed conic subset of R d \ { } . Moreover, it does not dependon the choice of the window function ϕ , as the following proposition shows. Proposition 3.2.
Let u ∈ S ′ ω ( R d ) , ϕ ∈ S ω ( R d ) \ { } and z ∈ R d \ { } . Assume that thereexists an open conic set Γ ⊆ R d \ { } containing z such that (3.1) is satisfied. Then, forany ψ ∈ S ω ( R d ) \ { } and for any open conic set Γ ′ ⊆ R d \ { } containing z and such that Γ ′ ∩ S d − ⊆ Γ , where S d − is the unit sphere in R d , we have sup z ∈ Γ ′ e λω ( z ) | V ψ u ( z ) | < + ∞ , λ > . (3.2) Proof.
From Proposition 2.11 we have that | V ψ u ( z ) | ≤ (2 π ) − d k ϕ k − L ( | V ϕ u | ∗ | V ψ ϕ | )( z ) ∀ z ∈ R d . (3.3)Moreover, since ϕ ∈ S ω ( R d ), from Theorem 2.6 we have that for every µ > C µ > e µω ( z ) | V ψ ϕ ( z ) | ≤ C µ ∀ z ∈ R d . (3.4)Then( | V ϕ u | ∗ | V ψ ϕ | )( z ) = Z R d | V ϕ u ( z − z ′ ) | · | V ψ ϕ ( z ′ ) | dz ′ ≤ Z h z ′ i≤ ε h z i | V ϕ u ( z − z ′ ) | · | V ψ ϕ ( z ′ ) | dz ′ + Z h z ′ i >ε h z i | V ϕ u ( z − z ′ ) | · | V ψ ϕ ( z ′ ) | dz ′ =: I + I . (3.5)Let us choose ε > z ∈ Γ ′ , | z | ≥ , h z ′ i ≤ ε h z i ⇒ z − z ′ ∈ Γ , The Gabor wave front set in spaces of ultradifferentiable functions and hence, from (3.1), the subadditivity of ω and (3.4): I ≤ C λ Z h z ′ i≤ ε h z i e − λω ( z − z ′ ) | V ψ ϕ ( z ′ ) | dz ′ ≤ C λ e − λω ( z ) Z R d e ( λ +1) ω ( z ′ ) | V ψ ϕ ( z ′ ) | e − ω ( z ′ ) dz ′ ≤ C ′ λ e − λω ( z ) , λ > , z ∈ Γ ′ , | z | ≥ . (3.6)On the other hand, from Theorem 2.4 and (3.4): I ≤ c Z h z ′ i >ε h z i e λω ( z − z ′ ) | V ψ ϕ ( z ′ ) | dz ′ ≤ ce λω ( z ) Z h z ′ i >ε h z i e ( λ +1 − µ ) ω ( z ′ ) | V ψ ϕ ( z ′ ) | e µω ( z ′ ) e − ω ( z ′ ) dz ′ ≤ c ′ e λω ( z ) e A ε ( λ +1 − µ ) ω ( z ) C µ (3.7)for some c ′ >
0, if µ > λ + 1, since − ω ( z ′ ) ≤ − A ε (1 + ω ( z )) , if h z ′ i > ε h z i , for some constant A ε > ε > µ > λ + 1 in (3.7) implies that for every λ ′ > C λ ′ > I ≤ C λ ′ e − λ ′ ω ( z ) , z ∈ R d . (3.8)This gives the conclusion. (cid:3) Given α, β >
0, consider the lattice
Λ = α Z d × β Z d ⊂ R d . For a window ϕ ∈ L ( R d ) \ { } the collection { Π( σ ) ϕ } σ ∈ Λ is called a Gabor frame for L ( R d ) provided there exists constants A, B > A k f k L ≤ X σ ∈ Λ |h f, Π( σ ) ϕ i| ≤ B k f k L , f ∈ L ( R d )(see [G] for the analysis of the conditions on α and β for which { Π( σ ) ϕ } σ ∈ Λ is a Gabor frame).Now, we define the Gabor ω -wave front set. Definition 3.3.
Let ϕ ∈ S ω ( R d ) \ { } and Λ = α Z d × β Z d ⊆ R d a lattice with α , β > sufficiently small so that { Π( σ ) ϕ } σ ∈ Λ is a Gabor frame for L ( R d ) . If u ∈ S ′ ω ( R d ) , we say that z ∈ R d \ { } is not in the Gabor ω -wave front set WF Gω ( u ) of u if there exists an open conicset Γ ⊂ R d \ { } containing z such that sup σ ∈ Λ ∩ Γ e λω ( σ ) | V ϕ u ( σ ) | < + ∞ ∀ λ > . (3.9)Our aim is to prove that WF ′ ω ( u ) = WF Gω ( u ). We follow the lines of [RW] and we needsome properties of modulation spaces that are already true in S ( R d ) (see [G]), adapted to oursetting.We consider, for λ ∈ R \ { } , m λ ( z ) = e λω ( z ) , v λ ( z ) = e | λ | ω ( z ) , z ∈ R n . (3.10) . Boiti, D. Jornet and A. Oliaro 11 The weights m λ ( z ) are v λ -moderate, in the sense that m λ ( z + z ) ≤ v λ ( z ) m λ ( z ) , for every λ = 0 and z , z ∈ R n . This is immediate from the subadditivity of ω .We denote, following [G], the weighted L p,q spaces by L p,qm λ ( R d ) := n F measurable on R d such that k F k L p,qmλ := (cid:16) Z R d (cid:16) Z R d | F ( x, ξ ) | p m λ ( x, ξ ) p dx (cid:17) q/p dξ (cid:17) /q < + ∞ o , for 1 ≤ p, q < + ∞ , and L ∞ ,qm λ ( R d ) := n F measurable on R d such that k F k L ∞ ,qmλ := (cid:16) Z R d (cid:0) ess sup x ∈ R d | F ( x, ξ ) | m λ ( x, ξ ) (cid:1) q dξ (cid:17) /q < + ∞ o ,L p, ∞ m λ ( R d ) := n F measurable on R d such that k F k L p, ∞ mλ := ess sup ξ ∈ R d (cid:16) Z R d | F ( x, ξ ) | p m λ ( x, ξ ) p dx (cid:17) /p < + ∞ o , for 1 ≤ p, q ≤ + ∞ with p = + ∞ or q = + ∞ respectively.By [G, Lemma 11.1.2] these are Banach spaces for all 1 ≤ p, q ≤ + ∞ . Moreover, for F ∈ L p,qm λ ( R d ) and H ∈ L p ′ ,q ′ /m λ ( R d ), where p ′ and q ′ are the conjugate exponents of p and q respectively (i.e. p + p ′ = 1 if 1 < p < + ∞ , p ′ = + ∞ if p = 1, p ′ = 1 if p = + ∞ , and the samefor q ), then F · H ∈ L ( R d ) and (cid:12)(cid:12)(cid:12)(cid:12)Z R d F ( z ) H ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ k F k L p,qmλ k H k L p ′ ,q ′ /mλ . (3.11)If 1 ≤ p, q < + ∞ , the dual of L p,qm λ ( R d ) is given by L p ′ ,q ′ /m λ ( R d ).From [G, Proposition 11.1.3] we have the following Young inequality for weighted L p,q spaces.For F ∈ L p,qm λ and G ∈ L v λ , k F ∗ G k L p,qmλ ≤ C k F k L p,qmλ k G k L vλ , (3.12)for some C > Remark 3.4.
It is easy to see that for every λ ∈ R \ { } and 1 ≤ p, q ≤ + ∞ we have S ω ( R d ) ⊂ L p,qm λ ( R d ) . Definition 3.5.
Let ϕ ∈ S ω ( R d ) \ { } , and m λ ( z ) as in (3.10) for some λ = 0 . For ≤ p, q ≤ + ∞ , the modulation space M p,qm λ ( R d ) is defined by M p,qm λ ( R d ) := { f ∈ S ′ ω ( R d ) : V ϕ f ∈ L p,qm λ ( R d ) } , with norm k f k M p,qmλ = k V ϕ f k L p,qmλ . We denote then M pm λ ( R d ) := M p,pm λ ( R d ) . The Gabor wave front set in spaces of ultradifferentiable functions
Observe that Definition 3.5 is similar to the definition of modulation spaces in [G]; thedifference is that here M p,qm λ ( R d ) is a subset of S ′ ω ( R d ), and we take a window ϕ ∈ S ω ( R d ),while in [G] the modulation space M p,qm ( R d ) is a subset of S ′ ( R d ) and the window belongs to S ( R d ) (or a subset of ( M v ) ∗ for a suitable weight v , in a suitable space of ‘special’ windows S C ( R d )). Moreover, here we always need weights of exponential type. We refer to [T-1, T-2]for modulation spaces in the setting of Gelfand-Shilov spaces, among other type of spaces ofultradifferentiable functions and ultradistributions.The definition of M p,qm λ is independent of the window ϕ , in the sense that different (non-zero)windows in S ω ( R d ) give equivalent norms. Indeed for ϕ, ψ ∈ S ω ( R d ), ϕ, ψ = 0, we have fromProposition 2.11, applied with γ = ψ , that k V ϕ f k L p,qmλ ≤ π ) d k ψ k L k| V ψ f | ∗ | V ϕ ψ |k L p,qmλ ≤ C k V ψ f k L p,qmλ , (3.13)where C = k V ϕ ψ k L vλ (2 π ) d k ψ k L , as we can deduce from Young inequality (3.12) (observe that C is finiteby Proposition 2.8 and Remark 3.4). Then, by interchanging the roles of ϕ and ψ we have that V ϕ f ∈ L p,qm λ if and only if V ψ f ∈ L p,qm λ , and the corresponding modulation space norms of f withrespect to the two windows are equivalent. Remark 3.6.
From Theorems 2.6 and 2.4 and Proposition 2.10 we have that S ω ( R d ) = \ λ> M ∞ m λ ( R d ); S ′ ω ( R d ) = [ λ< M ∞ m λ ( R d ) . The inversion formula of Proposition 2.10 holds also in modulation spaces, as stated in thefollowing result.
Proposition 3.7.
Let γ ∈ S ω ( R d ) be a not identically zero window, and consider, for a mea-surable function F on R d , the adjoint V ∗ γ F defined as in (2.22) . Then: (i) The operator V ∗ γ acts continuously as V ∗ γ : L p,qm λ ( R d ) → M p,qm λ ( R d ) , and there exists C > such that k V ∗ γ F k M p,qmλ ≤ C k V ϕ γ k L vλ k F k L p,qmλ , where ϕ is the window in the corresponding M p,qm λ norm. (ii) In the particular case when F = V g f , for g ∈ S ω ( R d ) , and f ∈ M p,qm λ , if h γ, g i 6 = 0 thefollowing inversion formula holds: f = 1(2 π ) d h γ, g i Z R d V g f ( z )Π( z ) γ dz. Proof. (i) We start by proving that V ∗ γ F is an element of S ′ ω ( R d ). For ψ ∈ S ω ( R d ) we havefrom (3.11), |h V ∗ γ F, ψ i| = |h F, V γ ψ i| ≤ k F k L p,qmλ k V γ ψ k L p ′ ,q ′ /mλ ≤ k F k L p,qmλ k e µω ( z ) V γ ψ k ∞ k e − µω ( z ) k L p ′ ,q ′ /mλ ; . Boiti, D. Jornet and A. Oliaro 13 this expression is finite for µ > V ∗ γ F is a well defined element of S ′ ω ( R d ). From Theorem 2.4 we have that V ϕ V ∗ γ F is a continuous function; it is explicitly givenby V ϕ V ∗ γ F ( z ) = h V ∗ γ F, Π( z ) ϕ i = Z R d F ( y, η ) V γ (Π( z ) ϕ )( y, η ) dy dη. Writing z = ( x, ξ ) we have | V ϕ V ∗ γ F ( x, ξ ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R d F ( y, η ) V ϕ γ ( x − y, ξ − η ) e − i h y,ξ − η i dy dη (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( | F | ∗ | V ϕ γ | )( x, ξ ) . Then, from Young inequality (3.12) we obtain k V ∗ γ F k M p,qmλ = k V ϕ V ∗ γ F k L p,qmλ ≤ C k F k L p,qmλ k V ϕ γ k L vλ , (3.14)and this expression is finite since V ϕ γ ∈ S ω ( R d ) ⊂ L v λ ( R d ) for every λ ∈ R from Remark 3.4.(ii) We first observe that, by (3.13), V g f ∈ L p,qm λ . Then, from point (i), ˜ f = π ) d h γ,g i V ∗ γ V g f ∈ M p,qm λ . Since M p,qm λ ⊂ S ′ ω , we have that ˜ f = f by (2.24). (cid:3) Theorem 3.8.
Let ≤ p, q < ∞ . We have ( M p,qm λ ) ∗ = M p ′ ,q ′ /m λ , and the duality is given by h f, h i = Z R d V ϕ f ( z ) V ϕ h ( z ) dz for f ∈ M p,qm λ and h ∈ M p ′ ,q ′ /m λ .Proof. The proof of this result relies on the duality of weighted L p,q spaces, and it is the sameas in Theorem 11.3.6 of [G]. (cid:3) Proposition 3.9.
For ≤ p, q < ∞ we have that S ω ( R d ) is a dense subspace of M p,qm λ .Proof. We first observe that, from property ( γ ) of the weight function ω (see Definition 2.1) wehave that, for µ > e − µω ( z ) ∈ L p,qm λ . Hence, for every f ∈ S ω ( R d ) we obtain k f k M p,qmλ = k V ϕ f k L p,qmλ ≤ k V ϕ f ( z ) e µω ( z ) k ∞ k e − µω ( z ) k L p,qmλ . From Proposition 2.9 we have S ω ( R d ) ⊂ M p,qm λ , with continuous inclusion. It remains to prove the density. We denote by K n := { z ∈ R d : | z | ≤ n } , and we fix ϕ ∈ S ω with k ϕ k L = (2 π ) − d . Consider f ∈ M p,qm λ and define F n = V ϕ f · χ K n and f n = V ∗ ϕ F n . The Gabor wave front set in spaces of ultradifferentiable functions
From Proposition 2.5 we have that f n ∈ S ω ( R d ). Moreover, using (2.24) and Proposition 3.7we obtain k f n − f k M p,qmλ = k V ∗ ϕ F n − V ∗ ϕ V ϕ f k M p,qmλ ≤ C k F n − V ϕ f k L p,qmλ = C k V ϕ f k L p,qmλ ( R d \ K n ) . So, k f n − f k M p,qmλ tends to 0 for n → ∞ , which finishes the proof. (cid:3) We recall now from [G] some basic facts about amalgam spaces . Definition 3.10.
We indicate with ℓ p,qm λ ( Z d ) the space of all sequences ( a kn ) k,n ∈ Z d , with a kn ∈ C for every k, n ∈ Z d , such that the following norm is finite k a k ℓ p,qmλ = (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a kn | p m λ ( k, n ) p (cid:19) q/p (cid:19) /q . Definition 3.11.
Let F be a measurable function on R d , and define a kn = ess sup ( x,ξ ) ∈ [0 , d | F ( k + x, n + ξ ) | . We say that F ∈ W ( L p,qm λ ) if the sequence a = ( a kn ) k,n ∈ Z d belongs to ℓ p,qm λ ( Z d ) . The space W ( L p,qm λ ) is called amalgam space , and has the norm defined by k F k W ( L p,qmλ ) = k a k ℓ p,qmλ . Let ϕ ∈ S ω ( R d ) and Λ = α Z d × β Z d a lattice with α , β > { Π( σ ) ϕ } σ ∈ Λ is a Gabor frame for L ( R d ). We indicate with e m λ the restriction of the weight(3.10) to the lattice Λ, in the sense that e m λ ( k, n ) := m λ ( α k, β n ) . We recall the following result (see Proposition 11.1.4 of [G]).
Proposition 3.12.
Let F ∈ W ( L p,qm λ ) be a continuous function, and α , β > . Then F | Λ ∈ ℓ p,q e m λ , and there exists a constant C = C ( α , β , λ ) such that k F | Λ k ℓ p,q e mλ ≤ C k F k W ( L p,qmλ ) . Now, we study the
Gabor frame operator associated to the lattice Λ, given by S ϕ,ψ f = X σ ∈ Λ h f, Π( σ ) ϕ i Π( σ ) ψ, (3.15)for ϕ, ψ, f ∈ L ( R d ).We write as usual S ϕ,ψ = D ψ C ϕ , where C ϕ is the ‘analysis’ operator, acting on a function f as C ϕ f = h f, Π( σ ) ϕ i , σ ∈ Λ , (3.16)and D ψ is the ‘synthesis’ operator, acting on a sequence c = ( c kn ) k,n ∈ Z d as D ψ c = X k,n ∈ Z d c kn Π( α k, β n ) ψ. (3.17) . Boiti, D. Jornet and A. Oliaro 15 We analyse the action of the previous operators on the modulation spaces M p,qm λ . The proofsof the next two results are very similar to [G, Thms. 12.2.3, 12.2.4], so we omit them. We justremark that, since ϕ ∈ S ω ⊂ S , we have that V ϕ ϕ ∈ S ; then by Proposition 12.1.11 of [G] wehave V ϕ ϕ ∈ W ( L v λ ), and so we can apply Theorem 11.1.5 of [G]. Theorem 3.13.
Let ϕ ∈ S ω ( R d ) and Λ a lattice as before. Then the operator C ϕ : M p,qm λ ( R d ) −→ ℓ p,q e m λ ( Z d ) is bounded for every λ ∈ R \ { } , α , β > , and ≤ p, q ≤ ∞ . Theorem 3.14.
Let ψ ∈ S ω ( R d ) . Then we have: (i) The operator D ψ : ℓ p,q e m λ ( Z d ) −→ M p,qm λ ( R d ) is bounded, for every ≤ p, q ≤ ∞ , α , β > , and λ ∈ R \ { } . (ii) For every c ∈ ℓ p ′ ,q ′ e m − λ and f ∈ M p,qm λ we have that h D ψ c, f i = h c, C ψ f i , for ≤ p, q < ∞ (3.18) and h C ψ f, c i = h f, D ψ c i , for < p, q ≤ ∞ . (3.19)(iii) For p, q < ∞ , we have that D ψ c converges unconditionally in M p,qm λ ; if p = q = ∞ , then D ψ c converges unconditionally weak ∗ in M ∞ /v λ . Now, we study the Gabor frame operator (3.15). We recall (see [G, Prop. 5.1.1 and 5.2.1])that if we take a window ϕ ∈ L ( R d ) and a lattice Λ such that { Π( σ ) ϕ } σ ∈ Λ is a Gabor framefor L ( R d ), the operator (3.15) is invertible on L ( R d ). Moreover, if we define the dual window ψ of ϕ by ψ := S − ϕ,ϕ ϕ , we have that for every f ∈ L ( R d ), f = X σ ∈ Λ h f, Π( σ ) ϕ i Π( σ ) ψ with unconditional convergence in L ( R d ). We observe also that if ϕ ∈ S ω ( R d ) then the dualwindow ψ ∈ S ω ( R d ) by [GZ, Thm. 4.2]. Lemma 3.15.
Fix ϕ ∈ S ω ( R d ) \ { } , and let ψ ∈ S ω ( R d ) \ { } be the dual window of ϕ . For f ∈ M p,qm λ ( R d ) , λ ∈ R \ { } , we have f = D ψ C ϕ f = X σ ∈ Λ h f, Π( σ ) ϕ i Π( σ ) ψ and f = D ϕ C ψ f = X σ ∈ Λ h f, Π( σ ) ψ i Π( σ ) ϕ, with convergence in M p,qm λ for p, q < ∞ , and weak ∗ convergence in M ∞ /v λ in the case p = q = ∞ . The Gabor wave front set in spaces of ultradifferentiable functions
Proof.
We first consider the case p, q < ∞ . From Proposition 3.9 we have that there exists asequence f n ∈ S ω ( R d ) such that f n → f in M p,qm λ as n → ∞ . Since S ω ( R d ) ⊂ L ( R d ), we havethat f n = D ψ C ϕ f n = D ϕ C ψ f n . (3.20)From Theorems 3.13 and 3.14 we obtain D ψ C ϕ f n → D ψ C ϕ f and D ϕ C ψ f n → D ϕ C ψ f in M p,qm λ ,and so from (3.20) the result is proved.We now pass to the case p = q = ∞ . Let f ∈ M ∞ /v λ and g ∈ M v λ . We have to prove that h f, g i = h D ψ C ϕ f, g i = h D ϕ C ψ f, g i . (3.21)From (3.18) and (3.19) we have that h D ψ C ϕ f, g i = h f, D ϕ C ψ g i ;from the previous point we have that D ϕ C ψ g = g in M v λ , so the first equality in (3.21) isproved. The other is similar. (cid:3) Remark 3.16.
Let u ∈ S ′ ω ( R d ), and ϕ, ψ ∈ S ω ( R d ) as in Lemma 3.15. Then for every θ ∈ S ω ( R d ) we have h u, θ i = X σ ∈ Λ h u, Π( σ ) ϕ ih Π( σ ) ψ, θ i . (3.22)We have indeed that from Remark 3.6 there exists λ < u ∈ M ∞ m λ = M ∞ /v λ . Then,from Lemma 3.15, for every g ∈ M v λ , h u, g i = X σ ∈ Λ h u, Π( σ ) ϕ ih Π( σ ) ψ, g i . From Proposition 3.9, the previous formula then holds for g = θ ∈ S ω ( R d ), so we have (3.22).We can now prove the main result of this section. Theorem 3.17. If u ∈ S ′ ω ( R d ) then WF ′ ω ( u ) = WF Gω ( u ) . Proof.
The inclusion WF Gω ( u ) ⊆ WF ′ ω ( u ) is trivial, so that we only have to prove thatWF ′ ω ( u ) ⊆ WF Gω ( u ) . Let 0 = z / ∈ WF Gω ( u ). So, there exists an open conic set Γ ⊂ R d \{ } containing z such that(3.9) is satisfied. By Remark 3.16 we have that, for ϕ ∈ S ω ( R d ) \ { } and ˜ ϕ = S − ϕϕ ϕ ∈ S ω ( R d )its dual window, h u, ψ i = X σ ∈ Λ V ϕ u ( σ ) h Π( σ ) ˜ ϕ, ψ i ∀ ψ ∈ S ω ( R d ) . We denote u = X σ ∈ Λ ∩ Γ V ϕ u ( σ )Π( σ ) ˜ ϕ,u = X σ ∈ Λ \ Γ V ϕ u ( σ )Π( σ ) ˜ ϕ. . Boiti, D. Jornet and A. Oliaro 17 Clearly V ϕ u ( z ) = V ϕ u ( z ) + V ϕ u ( z ). Denoting σ = ( σ , σ ) ∈ R d × R d , by (2.10), (2.11), thesubadditivity of ω and (2.13), we can estimate, for every α, β ∈ N d , λ, µ > e − λϕ ∗ ω ( | α | λ ) e − µϕ ∗ ω ( | β | µ ) | x β ∂ α u ( x ) |≤ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | · (cid:12)(cid:12) x β ∂ α (cid:0) e i h σ ,x i ˜ ϕ ( x − σ ) (cid:1)(cid:12)(cid:12) e − λϕ ∗ ω ( | α | λ ) e − µϕ ∗ ω ( | β | µ ) ≤ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | X γ ≤ α (cid:18) αγ (cid:19) −| α | | x | | β | e − µϕ ∗ ω ( | β | µ ) h σ i | α − γ | | ∂ γ ˜ ϕ ( x − σ ) | e − λϕ ∗ ω ( | α | λ )2 | α | ≤ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | X γ ≤ α (cid:18) αγ (cid:19) −| α | e µω ( x ) | ∂ γ ˜ ϕ ( x − σ ) |h σ i | α − γ | e λ e − λϕ ∗ ω ( | α | λ ) ≤ C λ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | X γ ≤ α (cid:18) αγ (cid:19) −| α | e µω ( σ ) e µω ( x − σ ) | ∂ γ ˜ ϕ ( x − σ ) |h σ i | α − γ | e − λϕ ∗ ω ( | α | λ ) ≤ C λ,λ ′ ,µ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | X γ ≤ α (cid:18) αγ (cid:19) −| α | e µω ( σ ) e λ ′ ϕ ∗ ω ( | γ | λ ′ ) − λϕ ∗ ω ( | α | λ ) h σ i | α − γ | for some C λ , C λ,λ ′ ,µ > λ ′ = 6 λ we apply [BJ, Prop. 2.1(g)], then (2.10) and (3.9), and finally obtain, for someconstants depending on λ and µ : e − λϕ ∗ ω ( | α | λ ) e − µϕ ∗ ω ( | β | µ ) | x β ∂ α u ( x ) |≤ C λ, λ,µ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | X γ ≤ α (cid:18) αγ (cid:19) −| α | e µω ( σ ) e − λϕ ∗ ω ( | α − γ | λ ) h σ i | α − γ | ≤ C λ, λ,µ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | X γ ≤ α (cid:18) αγ (cid:19) −| α | e µω ( σ ) e λω ( h σ i ) ≤ C λ,µ X σ ∈ Λ ∩ Γ | V ϕ u ( σ ) | e ( µ +6 λ ) ω ( h σ i )+ ω ( h σ i ) e − ω ( h σ i ) ≤ C ′ λ,µ X σ ∈ Λ ∩ Γ e − ω ( h σ i ) ≤ C ′′ λ,µ , x ∈ R d . (3.23)This proves that u ∈ S ω ( R d ) (here, we consider the seminorms given in (2.1)). Therefore, fromTheorem 2.6, V ϕ u ∈ S ω ( R d ) and for every λ > C λ > e λω ( z ) | V ϕ u ( z ) | ≤ C λ ∀ z ∈ R d . (3.24)Let us now fix an open conic set Γ ′ ⊂ R d \ { } containing z and such that Γ ′ ∩ S d − ⊆ Γ.Then inf = σ ∈ Λ \ Γ z ∈ Γ ′ (cid:12)(cid:12)(cid:12)(cid:12) σ | σ | − z (cid:12)(cid:12)(cid:12)(cid:12) = ε > | σ − z | ≥ ε | σ | for 0 = σ ∈ Λ \ Γ and z ∈ Γ ′ . The Gabor wave front set in spaces of ultradifferentiable functions
From the subadditivity of ω we have e λω ( z ) | V ϕ u ( z ) | ≤ X σ ∈ Λ \ Γ e λω ( σ )+ λω ( z − σ ) | V ϕ u ( σ ) | · |h Π( σ ) ˜ ϕ, Π( z ) ϕ i|≤ C X σ ∈ Λ \ Γ e ( λ +¯ λ ) ω ( σ ) e λω ( z − σ ) | V ϕ ˜ ϕ ( z − σ ) | , (3.26)for some C, ¯ λ >
0, because of Theorem 2.4 and since ([G, pg 41]) |h Π( σ ) ˜ ϕ, Π( z ) ϕ i| = | e − i h σ ,z − σ i V ϕ ˜ ϕ ( z − σ ) | = | V ϕ ˜ ϕ ( z − σ ) | . (3.27)Since ˜ ϕ ∈ S ω ( R d ), from Theorem 2.6 we have that for every µ > C µ > | V ϕ ˜ ϕ ( z − σ ) | ≤ C µ e − µω ( z − σ ) and hence, substituting in (3.26): e λω ( z ) | V ϕ u ( z ) | ≤ CC µ X σ ∈ Λ \ Γ e ( λ +¯ λ ) ω ( σ ) e ( λ − µ ) ω ( z − σ ) . (3.28)However, for z ∈ Γ ′ and σ ∈ Λ \ Γ we have | σ − z | ≥ ε | σ | and therefore, by the subadditivityof ω , − ω ( z − σ ) ≤ − ω ( εσ ) ≤ − M ω ( σ )for some M > ε defined in (3.25). By formula (3.28) we obtain e λω ( z ) | V ϕ u ( z ) | ≤ CC µ X σ ∈ Λ \ Γ e ( λ +¯ λ + λM − µM ) ω ( σ ) ≤ C λ , z ∈ Γ ′ , (3.29)for some C λ >
0, if µ is chosen large enough.From (3.24) and (3.29) we finally deducesup z ∈ Γ ′ e λω ( z ) | V ϕ u ( z ) | < + ∞ , λ > , and hence z / ∈ WF ′ ω ( u ). (cid:3) From Theorem 3.17, in what follows we use WF ′ ω ( u ) for WF Gω ( u ) and any u ∈ S ′ ω ( R d ). Proposition 3.18.
For every u ∈ S ′ ω ( R d ) we have WF ′ ω ( u ) = ∅ if and only if u ∈ S ω ( R d ) .Proof. Suppose that u ∈ S ω ( R d ), and fix a window function ϕ ∈ S ω ( R d ) \ { } ; then fromTheorem 2.6 we have that for every λ > C λ > | V ϕ u ( z ) | ≤ C λ e − λω ( z ) , ∀ z ∈ R d . Then for every open conic set Γ ⊆ R d \ { } condition (3.1) holds, so WF ′ ω ( u ) = ∅ .Suppose now that WF ′ ω ( u ) = ∅ . From Definition 3.1 we have that for every z ∈ R d \ { } thereexists an open conic set Γ z ⊆ R d \ { } containing z such that for every λ > C λ,z > | V ϕ u ( z ) | ≤ C λ,z e − λω ( z ) ∀ z ∈ Γ z . . Boiti, D. Jornet and A. Oliaro 19 Let Υ z = Γ z ∩ S d − . We have that { Υ z , z ∈ R d \ { }} is an open covering of S d − ; since S d − is compact and Γ z is conic, there exist z , . . . , z k ∈ R d \ { } such thatΓ z ∪ · · · ∪ Γ z k = R d \ { } . We then have that for every λ > | V ϕ u ( z ) | ≤ C λ e − λω ( z ) ∀ z ∈ R d , where C λ = max { C λ,z , . . . , C λ,z k , | V ϕ u (0) | e λω (0) } . From Theorem 2.6 we finally have u ∈S ω ( R d ). (cid:3) We now prove that the wave front set WF ′ ω is not affected by the phase-space shift operator. Proposition 3.19.
For every w = ( y, η ) ∈ R d and for every u ∈ S ′ ω ( R d ) we have WF ′ ω (Π( w ) u ) = WF ′ ω ( u ) . Proof.
Since Π( w ) = M η T y , it is enough to prove that translation and modulation do not affectthe wave front set. Concerning translation, we have that for z = ( x, ξ ) ∈ R d , V ϕ ( T y u )( z ) = h T y u, Π( z ) ϕ i = h u, T − y Π( z ) ϕ i = e − i h y,ξ i V T − y ϕ u ;writing ψ = T − y ϕ ∈ S ω ( R d ) we have that | V ϕ ( T y u )( z ) | = | V ψ u ( z ) | , and since the wave front set does not depend on the window (Proposition 3.2) we haveWF ′ ω ( T y u ) = WF ′ ω ( u ). Concerning modulation, we have V ϕ ( M η u )( z ) = h M η u, Π( z ) ϕ i = h u, M − η Π( z ) ϕ i = e i h η,x i V M − η ϕ u ( z );then, writing θ = M − η ϕ ∈ S ω ( R d ), we get | V ϕ ( M η u )( z ) | = | V θ u ( z ) | , and as before we conclude that WF ′ ω ( M η u ) = WF ′ ω ( u ). (cid:3) The results obtained in Sections 2 and 3 are true with the weaker assumption (see Bj¨orck[B]): “there exist a ∈ R , b > ω ( t ) ≥ a + b log(1 + t ) for t ≥
0” instead of ( γ ) ofDefinition 2.1. A detailed and modern treatment of these type of weights can be found [BG].Moreover, the results above are true in the quasi-analytic case also, i.e. when we consider that ω ( t ) = o ( t ), as t → + ∞ , instead of condition ( β ) of Definition 2.1. Applications to (pseudo-)differential operators
In this section we analyze the action of several operators of pseudo-differential (or differential)type on the global wave front set WF ′ ω ( u ) of u ∈ S ′ ω ( R d ). We will use the kernel theorem in S ω . It is known that S ω is nuclear for many weight functions ω . For example, whenever theysatisfy the following condition:(4.1) ∃ H ≥ ∀ t ≥ , ω ( t ) ≤ ω ( Ht ) + H. Morever, Bonet, Meise and Melikhov [BMM] proved that under such a condition the classesof ultradifferentiable functions defined by sequences in the sense of Komatsu satisfying the The Gabor wave front set in spaces of ultradifferentiable functions standard conditions ( M M M
2) and ( M Definition 4.1.
For m ∈ R we define S mω := { a ∈ C ∞ ( R d ) : ∀ λ, µ > ∃ C λ,µ > such that | ∂ αx ∂ βξ a ( x, ξ ) | ≤ C λ,µ e λϕ ∗ ω ( | α | λ ) e µϕ ∗ ω ( | β | µ ) e mω ( ξ ) , ∀ ( x, ξ ) ∈ R d , α, β ∈ N d } . Then we consider the Kohn-Nirenberg quantization defined by a ( x, D ) f ( x ) := (2 π ) − d Z R d e i h x,ξ i a ( x, ξ ) b f ( ξ ) dξ, a ∈ S mω , f ∈ S ω ( R d ) . (4.2)The above Kohn-Nirenberg quantization is well defined since b f ∈ S ω ( R d ) and hence for every λ > C λ > | a ( x, ξ ) | · | b f ( ξ ) | ≤ e mω ( ξ ) C λ e − λω ( ξ ) which is integrable in R d if we choose λ > a ( x, D ) : S ω −→ S ′ ⊆ S ′ ω . If S ω is nuclear, we can use the kernel theorem and find a unique distribution K ∈ S ′ ω ( R d )of the linear operator V ϕ a ( x, D ) V ∗ ϕ : S ω ( R d ) −→ S ′ ω ( R d )such that V ϕ a ( x, D ) V ∗ ϕ F ( y ′ , η ′ ) = (2 π ) d Z R d K ( y ′ , η ′ ; y, η ) F ( y, η ) dydη ∀ F ∈ S ω ( R d ) , (4.3)in the sense that h V ϕ a ( x, D ) V ∗ ϕ F, G i = (2 π ) d h K ( y ′ , η ′ ; y, η ) , G ( y ′ , η ′ ) F ( y, η ) i ∀ G ∈ S ω ( R d ) . (4.4)If u ∈ S ω ( R d ) and F = V ϕ u ∈ S ω ( R d ) for ϕ ∈ S ω ( R d ) with k ϕ k L = 1, then, from (2.21), V ϕ a ( x, D ) u ( y ′ , η ′ ) = (2 π ) − d V ϕ a ( x, D ) V ∗ ϕ V ϕ u ( y ′ , η ′ )= Z R d K ( y ′ , η ′ ; y, η ) V ϕ u ( y, η ) dydη and we can compute the kernel directly: Lemma 4.2.
For a ∈ S mω , ϕ ∈ S ω ( R d ) with k ϕ k L = 1 and u ∈ S ω ( R d ) we have that V ϕ ( a ( x, D ) u )( z ′ ) = Z R d K ( z ′ , z ) V ϕ u ( z ) dz, (4.5) where, for all z = ( y, η ) , z ′ = ( y ′ , η ′ ) ∈ R d , K ( z ′ , z ) = (2 π ) − d e i h y,η i Z R d e i ( h x,ξ i−h y,ξ i−h x,η ′ i ) a ( x, ξ ) b ϕ ( ξ − η ) ϕ ( x − y ′ ) dxdξ. (4.6) . Boiti, D. Jornet and A. Oliaro 21 Proof.
Let F ∈ S ω ( R d ) and consider the Kohn-Nirenberg quantization (4.2) of V ∗ ϕ F ∈ S ω ( R d ): a ( x, D ) V ∗ ϕ F ( x ) = (2 π ) − d Z R d e i h x,ξ i a ( x, ξ ) d V ∗ ϕ F ( ξ ) dξ. Then, by (2.22), V ϕ a ( x, D ) V ∗ ϕ F ( y ′ , η ′ ) = Z R d ( a ( x, D ) V ∗ ϕ F )( x ) ϕ ( x − y ′ ) e − i h x,η ′ i dx = (2 π ) − d Z R d Z R d e i h x,ξ i a ( x, ξ ) d V ∗ ϕ F ( ξ ) ϕ ( x − y ′ ) e − i h x,η ′ i dξdx = (2 π ) − d Z R d Z R d Z R d e i h x,ξ i a ( x, ξ ) V ∗ ϕ F ( x ′ ) e − i h x ′ ,ξ i ϕ ( x − y ′ ) e − i h x,η ′ i dx ′ dξdx = (2 π ) − d Z R d Z R d Z R d Z R d e i h x,ξ i a ( x, ξ ) F ( y, η ) e i h x ′ ,η i ϕ ( x ′ − y ) · e − i h x ′ ,ξ i ϕ ( x − y ′ ) e − i h x,η ′ i dydηdx ′ dξdx. Since a ∈ S mω , F ∈ S ω ( R d ) and ϕ ∈ S ω ( R d ), we have that for every λ , λ , λ > C λ > | a ( x, ξ ) F ( y, η ) ϕ ( x ′ − y ) ϕ ( x − y ′ ) |≤ C λ e mω ( ξ ) e − λ ω ( y,η ) e − λ ω ( x ′ − y ) e − λ ω ( x − y ′ ) ≤ C λ e mω ( ξ ) e − λ ω ( y ) e − λ ω ( η ) e − λ ω ( x ′ )+ λ ω ( y ) e − λ ω ( x )+ λ ω ( y ′ ) . Choosing λ , λ > y, η and x ′ , obtaining: V ϕ a ( x, D ) V ∗ ϕ F ( y ′ , η ′ ) == (2 π ) − d Z R d Z R d Z R d e i h x,ξ i a ( x, ξ ) F ( y, η ) · (cid:18)Z R d e i h x ′ ,η i ϕ ( x ′ − y ) e − i h x ′ ,ξ i dx ′ (cid:19) ϕ ( x − y ′ ) e − i h x,η ′ i dydηdξdx = (2 π ) − d Z R d Z R d Z R d e i h x,ξ i a ( x, ξ ) F ( y, η ) · (cid:18)Z R d e i h y + s,η i e − i h y + s,ξ i ϕ ( s ) ds (cid:19) ϕ ( x − y ′ ) e − i h x,η ′ i dydηdξdx = (2 π ) − d Z R d Z R d Z R d e i h x,ξ i a ( x, ξ ) F ( y, η ) e i h y,η i e − i h y,ξ i · (cid:18)Z R d e − i h s,ξ − η i ϕ ( s ) ds (cid:19) ϕ ( x − y ′ ) e − i h x,η ′ i dydηdξdx = (2 π ) − d Z R d Z R d Z R d e i h x,ξ i a ( x, ξ ) F ( y, η ) e i h y,η i e − i h y,ξ i · b ϕ ( ξ − η ) ϕ ( x − y ′ ) e − i h x,η ′ i dydηdξdx. The Gabor wave front set in spaces of ultradifferentiable functions
Since a ∈ S mω , F ∈ S ω ( R d ) and ϕ ∈ S ω ( R d ), for every µ , µ , µ > C µ > | a ( x, ξ ) F ( y, η ) b ϕ ( ξ − η ) ϕ ( x − y ′ ) |≤ C µ e mω ( ξ ) e − µ ω ( y ) e − µ ω ( η ) e − µ ω ( ξ )+ µ ω ( η ) e − µ ω ( x )+ µ ω ( y ′ ) , so that, for µ , µ > µ sufficiently large, we can apply Fubini’s theorem and obtain V ϕ a ( x, D ) V ∗ ϕ F ( y ′ , η ′ ) == (2 π ) − d Z R d F ( y, η ) e i h y,η i · (cid:18)Z R d e i ( h x,ξ i−h y,ξ i−h x,η ′ i ) a ( x, ξ ) b ϕ ( ξ − η ) ϕ ( x − y ′ ) dxdξ (cid:19) dydη. Applying the above result to F = V ϕ u for some u ∈ S ω ( R d ), since k ϕ k L = 1 and hence V ∗ ϕ F = V ∗ ϕ V ϕ u = (2 π ) d u by (2.21), we have V ϕ ( a ( x, D ) u )( y ′ , η ′ ) = Z R d K ( y ′ , η ′ ; y, η ) V ϕ u ( y, η ) dydη, for K ( y ′ , η ′ ; y, η ) = (2 π ) − d e i h y,η i Z R d e i ( h x,ξ i−h y,ξ i−h x,η ′ i ) a ( x, ξ ) b ϕ ( ξ − η ) ϕ ( x − y ′ ) dxdξ, which concludes the proof of the lemma. (cid:3) In the next result the following property on the weight function ω will be useful: from [BJO,Lemma 4.7(ii)] (for instance), for every µ > t ≥ β ∈ N do t −| β | e µϕ ∗ ω ( | β | µ ) ≤ e − ( µ − b ) ω ( t ) − ab , (4.7)where a ∈ R and b > ω . Proposition 4.3. If a ∈ S mω , m ∈ R and K ∈ C ∞ ( R d ) is defined by (4.6) , then for every λ > there exists a constant C λ > such that | K ( z ′ , z ) | ≤ C λ e − λω ( y − y ′ ) e ( m − λ ) ω ( η − η ′ ) e mω ( η ′ ) , z = ( y, η ) , z ′ = ( y ′ , η ′ ) ∈ R d . (4.8) Moreover, if a ( z ) = 0 for z ∈ Γ \ B (0 , R ) for an open conic set Γ ⊆ R d \ { } and for some R > (here B (0 , R ) is the ball of center and radius R in R d ), then for every open conic set Γ ′ ⊆ R d \ { } such that Γ ′ ∩ S d − ⊆ Γ we have that for every λ > there exists a constant C λ > such that for all z ′ = ( y ′ , η ′ ) ∈ Γ ′ and z = ( y, η ) ∈ R d , | K ( z ′ , z ) | ≤ C λ e − λω ( y − y ′ ) e − λω ( η − η ′ ) e − λω ( y ′ ) e − λω ( η ′ ) . (4.9) Proof.
By the linear change of variables ξ ′ = ξ − η and x ′ = x − y ′ in (4.6) we have K ( z ′ , z ) = (2 π ) − d e i h y,η i Z R d e i ( h x ′ + y ′ ,ξ ′ + η i−h y,ξ ′ + η i−h x ′ + y ′ ,η ′ i ) a ( x ′ + y ′ , ξ ′ + η ) b ϕ ( ξ ′ ) ϕ ( x ′ ) dx ′ dξ ′ = (2 π ) − d e i ( h y ′ ,η i−h y ′ ,η ′ i ) · Z R d e i ( h x ′ ,ξ ′ i + h x ′ ,η i + h y ′ ,ξ ′ i−h y,ξ ′ i−h x ′ ,η ′ i ) a ( x ′ + y ′ , ξ ′ + η ) b ϕ ( ξ ′ ) ϕ ( x ′ ) dx ′ dξ ′ , . Boiti, D. Jornet and A. Oliaro 23 and hence, setting x = x ′ and ξ = ξ ′ : | K ( z ′ , z ) | = (2 π ) − d (cid:12)(cid:12)(cid:12)(cid:12)Z R d e i ( h x,η − η ′ + ξ i + h ξ,y ′ − y i ) a ( x + y ′ , ξ + η ) b ϕ ( ξ ) ϕ ( x ) dxdξ (cid:12)(cid:12)(cid:12)(cid:12) . (4.10)Writing, for M, N ∈ N , e i ( h x,η − η ′ + ξ i + h ξ,y ′ − y i ) = h η − η ′ + ξ i − M (1 − ∆ x ) M e i ( h x,η − η ′ + ξ i + h ξ,y ′ − y i ) = h y − y ′ i − N h η − η ′ + ξ i − M (1 − ∆ x ) M e i h x,η − η ′ + ξ i (1 − ∆ ξ ) N e i h ξ,y ′ − y i and integrating by parts in (4.10), we have | K ( z ′ , z ) | = (2 π ) − d h y − y ′ i − N (cid:12)(cid:12)(cid:12)(cid:12)Z R d e i ( h x,η − η ′ i + h ξ,y ′ − y i ) λ N,M ( y ′ , η ′ , η, x, ξ ) dxdξ (cid:12)(cid:12)(cid:12)(cid:12) , (4.11)where λ N,M ( y ′ , η ′ , η, x, ξ )= (1 − ∆ ξ ) N h e i h x,ξ i h η − η ′ + ξ i − M (1 − ∆ x ) M (cid:16) a ( x + y ′ , ξ + η ) b ϕ ( ξ ) ϕ ( x ) (cid:17)i . For a ∈ S mω , since ϕ, b ϕ ∈ S ω ( R d ), by [BJO, Thm. 4.8(5)] we have that for every λ , µ , λ ′ , µ ′ , λ ′′ , µ ′′ > C λ,µ , C λ ′ ,λ ′′ , C µ ′ ,µ ′′ , depending only on the indexedconstants, such that for every M, N, k, ℓ ∈ N : | λ N,M ( y ′ , η ′ , η, x, ξ ) | ≤ X γ + γ + γ + γ =2 N (2 N )! γ ! γ ! γ ! γ ! X σ + σ =2 M (2 M )! σ ! σ ! h x i | γ | h η − η ′ + ξ i − M −| γ | · C λ,µ e λϕ ∗ ω (cid:16) | γ | λ (cid:17) e µϕ ∗ ω (cid:16) | σ | µ (cid:17) e mω ( ξ + η ) C λ ′ ,λ ′′ h ξ i − k e λ ′ ϕ ∗ ω ( kλ ′ ) e λ ′′ ϕ ∗ ω (cid:16) | γ | λ ′′ (cid:17) · C µ,µ ′ h x i − ℓ e µ ′ ϕ ∗ ω (cid:16) ℓµ ′ (cid:17) e µ ′′ ϕ ∗ ω (cid:16) | σ | µ ′′ (cid:17) . (4.12)Note that h η − η ′ + ξ i − ≤ √ h η − η ′ i − h ξ i . (4.13)Moreover, we can choose λ ′′ = λ , µ ′′ = µ and apply Proposition 2.1(g) of [BJ]. Taking alsointo account the subadditivity of ω , we have that for every λ , µ , λ ′ , µ ′ > C λ,µ,λ ′ ,µ ′ > M, N, k, ℓ ∈ N : | λ N,M ( y ′ , η ′ , η, x, ξ ) | ≤ C λ,µ,λ ′ ,µ ′ X γ + γ + γ + γ =2 N (2 N )! γ ! γ ! γ ! γ ! 2 − N X σ + σ =2 M (2 M )! σ ! σ ! 2 − M ·h x i | γ |− ℓ h η − η ′ i − M −| γ | h ξ i M + | γ | e mω ( ξ ) e mω ( η − η ′ ) e mω ( η ′ ) · e λ ϕ ∗ ω (cid:16) | γ γ | λ/ (cid:17) N e µ ϕ ∗ ω (cid:16) | σ σ | µ/ (cid:17) M h ξ i − k e λ ′ ϕ ∗ ω ( kλ ′ ) e µ ′ ϕ ∗ ω (cid:16) ℓµ ′ (cid:17) . Taking the infimum on k ∈ N and applying (4.7) and (2.11), we get: | λ N,M ( y ′ , η ′ , η, x, ξ ) | ≤ C λ,µ,λ ′ ,µ ′ X γ + γ + γ + γ =2 N (2 N )! γ ! γ ! γ ! γ ! 2 − N X σ + σ =2 M (2 M )! σ ! σ ! 2 − M · e µ ′ ϕ ∗ ω (cid:16) ℓµ ′ (cid:17) h x i N − ℓ h η − η ′ i − M e µ ϕ ∗ ω ( Mµ/ ) e µ/ ·h ξ i M +2 N e mω ( ξ ) e − ( λ ′ − b ) ω ( ξ ) − ab e mω ( η − η ′ ) e mω ( η ′ ) e λ ϕ ∗ ω ( Nλ/ ) e λ/ . The Gabor wave front set in spaces of ultradifferentiable functions
Substituting in (4.11) we have that for all λ, µ, λ ′ , µ ′ > C ′ λ,µ,λ ′ ,µ ′ > M, N, ℓ ∈ N : | K ( z ′ , z ) | ≤ C ′ λ,µ,λ ′ ,µ ′ h y − y ′ i − N e λ ϕ ∗ ω ( Nλ/ ) h η − η ′ i − M e µ ϕ ∗ ω ( Mµ/ ) e mω ( η − η ′ ) e mω ( η ′ ) · e µ ′ ϕ ∗ ω (cid:16) ℓµ ′ (cid:17) Z R d h x i N − ℓ dx Z R d h ξ i M +2 N e ( m − λ ′ + b ) ω ( ξ ) dξ. (4.14)Let us now fix µ ′ >
0, choose ℓ ∈ N and λ ′ > M and N , apply (4.7) and obtain: | K ( z ′ , z ) | ≤ C λ,µ e − ( λ − b ) ω ( y − y ′ ) e − ( µ − b ) ω ( η − η ′ ) e mω ( η − η ′ ) e mω ( η ′ ) . (4.15)In particular, by the arbitrariness of λ and µ in (4.15), we have that for every λ, µ > C λ,µ > | K ( z ′ , z ) | ≤ C λ,µ e − λω ( y − y ′ ) e ( m − µ ) ω ( η − η ′ ) e mω ( η ′ ) ∀ z = ( y, η ) , z ′ = ( y ′ , η ′ ) ∈ R d , (4.16)which proves (4.8) for µ = λ .Applying (4.13) only to h η − η ′ + ξ i − M in (4.12), by the same computations as to get (4.14)we obtain that if a ( z ) = 0 for z ∈ Γ \ B (0 , R ), then | K ( z ′ , z ) | ≤ C ′ λ,µ,λ ′ ,µ ′ h y − y ′ i − N e λ ϕ ∗ ω ( Nλ/ ) h η − η ′ i − M e µ ϕ ∗ ω ( Mµ/ ) e mω ( η − η ′ ) e mω ( η ′ ) · Z D y ′ ,η h η ′ − ( ξ + η ) i − M e µ ′ ϕ ∗ ω (cid:16) ℓµ ′ (cid:17) h x i N − ℓ h ξ i M +2 N e ( m − λ ′ + b ) ω ( ξ ) dxdξ, (4.17)where D y ′ ,η := { ( x, ξ ) ∈ R d : ( x + y ′ , ξ + η ) ∈ ( R d \ Γ) ∪ B (0 , R ) } . We now want to estimate (4.17) for z ′ = ( y ′ , η ′ ) ∈ Γ ′ and z = ( y, η ) ∈ R d . It has beenproved in [RW, pg 643] that h y ′ ih η ′ i ≤ C h x i h η ′ − ( ξ + η ) i ∀ z ′ ∈ Γ ′ \ B (0 , R ) , z ∈ R d , ( x, ξ ) ∈ D y ′ ,η (4.18)for some constant C > z ′ ∈ Γ ′ \ B (0 , R )and z ∈ R d : | K ( z ′ , z ) | ≤ C M/ C ′ λ,µ,λ ′ ,µ ′ h y − y ′ i − N e λ ϕ ∗ ω ( Nλ/ ) ·h η − η ′ i − M e µ ϕ ∗ ω ( Mµ/ ) e µ ϕ ∗ ω ( M/ µ/ ) h y ′ i − M/ e µ ϕ ∗ ω ( M/ µ/ ) h η ′ i − M/ · e mω ( η − η ′ ) e mω ( η ′ ) Z D y ′ ,η h x i M +2 N − ℓ e µ ′ ϕ ∗ ω (cid:16) ℓµ ′ (cid:17) h ξ i M +2 N e ( m − λ ′ + b ) ω ( ξ ) dxdξ. We now fix µ ′ > ℓ ∈ N and λ ′ > M and N and apply (4.7). We obtain that for every λ, µ > C λ,µ > | K ( z ′ , z ) | ≤ C λ,µ e − ( λ − b ) ω ( y − y ′ ) e − ( µ − b ) ω ( η − η ′ ) · e − ( µ − b ) ω ( y ′ ) e − ( µ − b ) ω ( η ′ ) e mω ( η − η ′ ) e mω ( η ′ ) ∀ z ′ ∈ Γ ′ \ B (0 , R ) , z ∈ R d . . Boiti, D. Jornet and A. Oliaro 25 In particular, for ¯ λ = λ − b and ¯ µ = µ − b we have that there is C ¯ λ, ¯ µ > z ′ ∈ Γ ′ \ B (0 , R ) and z ∈ R d : | K ( z ′ , z ) | ≤ C ¯ λ, ¯ µ e − ¯ λω ( y − y ′ ) e ( m − ¯ µ ) ω ( η − η ′ ) e ( m − ¯ µ ) ω ( η ′ ) e − ¯ µ ω ( y ′ ) . For ¯ µ ≥ λ + 2 m we have that m − ¯ µ ≤ − ¯ λ and − ¯ µ ≤ m − ¯ µ ≤ − λ which proves (4.9) for z ′ ∈ Γ ′ \ B (0 , R ) , z ∈ R d .Since the estimate (4.9) for | z ′ | ≤ R follows from (4.16), the proof is complete. (cid:3) Remark 4.4.
For a ∈ S mω , m ∈ R , and K ∈ C ∞ ( R d ) defined by (4.6) the integral in (4.5) iswell defined also for u ∈ S ′ ω ( R d ). In fact, (2.7) and (4.8) imply that there exist ˜ C, ˜ λ > λ > C λ > | K ( z ′ , z ) V ϕ u ( z ) | ≤ ˜ CC λ e − λω ( y − y ′ )+( m − λ ) ω ( η − η ′ ) e mω ( η ′ ) e ˜ λω ( y )+˜ λω ( η ) ≤ ˜ CC λ e λω ( y ′ )+( m + λ ) ω ( η ′ ) e (˜ λ − λ ) ω ( y )+( m +˜ λ − λ ) ω ( η ) ∈ L ( R dz =( y,η ) )(4.19)if λ > max { ˜ λ, m + ˜ λ } .We now want to extend Lemma 4.2 for u ∈ S ′ ω ( R d ). To this aim we first need the next tworesults. Proposition 4.5.
The space S ω ( R d ) is dense in S ′ ω ( R d ) .Proof. Let us consider the inclusion i : S ω ( R d ) ֒ → S ′ ω ( R d ) f
7→ h i ( f ) , ϕ i := Z R d f ( x ) ϕ ( x ) dx ∀ ϕ ∈ S ω ( R d ) . To show that the image is dense we take T ∈ (cid:0) S ′ ω ( R d ) (cid:1) ′ such that T | S ω = 0 and prove that T ≡ S ω ( R d ) is reflexive, there exists a unique f ∈ S ω ( R d ) such that T ( ϕ ) = Z R d f ( x ) ϕ ( x ) dx = 0 , ∀ ϕ ∈ S ω ( R d ) , because of T | S ω = 0. Therefore f = 0, i.e. T ≡ (cid:3) Proposition 4.6.
Let ϕ ∈ S ω ( R d ) \ { } . Then V ϕ : S ′ ω ( R d ) −→ S ′ ω ( R d ) is continuous.Proof. We already know that V ∗ ϕ : S ω ( R d ) −→ S ω ( R d )is continuous by (2.20). It follows that( V ∗ ϕ ) ∗ : S ′ ω ( R d ) −→ S ′ ω ( R d )is continuous and moreover ( V ∗ ϕ ) ∗ (cid:12)(cid:12) S ω ( R d ) = V ϕ because, for f, g ∈ S ω ( R d ), h ( V ∗ ϕ ) ∗ f, g i = h f, V ∗ ϕ g i = h V ϕ f, g i . The Gabor wave front set in spaces of ultradifferentiable functions
Since S ω ( R d ) is dense in S ′ ω ( R d ) by Proposition 4.5, we have that ( V ∗ ϕ ) ∗ is the continuousextension of V ϕ to S ′ ω ( R d ) and, hence, V ϕ is continuous on S ′ ω ( R d ) also. (cid:3) Now, we need amplitudes a ( x, y, ξ ), instead of symbols a ( x, ξ ). Definition 4.7.
Given m ∈ R , we say that a ( x, y, ξ ) ∈ C ∞ ( R d ) is an amplitude in the space S mω if for every λ, µ > there is C λ,µ > such that | ∂ αx ∂ γy ∂ βξ a ( x, y, ξ ) | ≤ C λ,µ e λϕ ∗ ( | α + γ | λ )+ µϕ ∗ ( | β | µ ) e mω ( ξ ) , for all ( x, y, ξ ) ∈ R d and α, β, γ ∈ N d . Now, proceeding in a similar way to that of Proposition 1.9 and Theorem 2.2 of [FGJ], onecan prove that if a ( x, y, ξ ) ∈ S mω is an amplitude as in Definition 4.7, the operator acting on S ω , given by the iterated integral A ( f )( x ) := Z R d (cid:18)Z R d e i h x − y,ξ i a ( x, y, ξ ) f ( y ) dy (cid:19) dξ, f ∈ S ω , is well defined and continuous from S ω into itself. The operator A is called pseudo-differentialoperator of type ω with amplitude a ( x, y, ξ ). Moreover, A can be extended continuously tothe dual space ˜ A : S ′ ω → S ′ ω in a standard way (see [FGJ, Theorem 2.5]). In particular, theKohn-Nirenberg quantization defined in (4.2) is a pseudo-differential operator with amplitude a ( x, y, ξ ) := (2 π ) − d p ( x, ξ ) , where p ( x, ξ ) is a symbol as in Definition 4.1.As a consequence of the above considerations and of the estimates of the kernel in Proposi-tion 4.3, we obtain the following result: Corollary 4.8.
Let a ( x, ξ ) ∈ S mω a symbol as in Definition 4.1, ϕ ∈ S ω ( R d ) with k ϕ k L = 1 and u ∈ S ′ ω ( R d ) . Then, for K ( z ′ , z ) as in (4.6) , we have V ϕ a ( x, D ) u ( z ′ ) = Z R d K ( z ′ , z ) V ϕ u ( z ) dz, (4.20) for all z ′ ∈ R d .Proof. Since V ϕ operates on S ′ ω , from the previous comments it is clear that V ϕ a ( x, D ) canbe extended to S ′ ω ( R d ). We take u ∈ S ′ ω ( R d ). By Proposition 4.5, there exists a sequence { u n } n ∈ N ⊂ S ω ( R d ) which converges to u in S ′ ω and, hence, Z R d K ( z ′ , z ) V ϕ u n ( z ) dz = V ϕ a ( x, D ) u n ( z ′ ) −→ V ϕ a ( x, D ) u ( z ′ ) in S ′ ω ( R d ) . (4.21)We want to prove that Z R d K ( z ′ , z ) V ϕ u n ( z ) dz −→ Z R d K ( z ′ , z ) V ϕ u ( z ) dz (4.22)using Lebesgue’s dominated convergence theorem. First, it is easy to see that { V ϕ u n ( z ) } n ∈ N converges pointwise to V ϕ u ( z ) for every z ∈ R d from the definition of the short-time Fouriertransform. . Boiti, D. Jornet and A. Oliaro 27 Now, since { u n } n ∈ N is bounded in S ′ ω ( R d ), it is equicontinuous there. So, there exist aconstant C > q on S ω ( R d ) such that |h u n , ϕ i| ≤ Cq ( ϕ ) , ϕ ∈ S ω ( R d ) . This yields a uniform estimate of the inequality (2.7) (see the proof of [GZ, Theorem 2.4]) inthe sense: | V ϕ u n ( z ) | ≤ ˜ Ce ˜ λω ( z ) , z ∈ R d , n ∈ N , (4.23)for some ˜ C, ˜ λ > n and z . From (4.23) and (4.19) we have that K ( z ′ , z ) V ϕ u n ( z )is dominated by a function in L ( R dz ).Therefore (4.22) is satisfied and hence, from (4.21), V ϕ a ( x, D ) u ( z ′ ) = Z R d K ( z ′ , z ) V ϕ u ( z ) dz also for u ∈ S ′ ω ( R d ). (cid:3) We recall the notion of conic support from [RW]:
Definition 4.9.
For a ∈ D ′ ( R d ) the conic support of a , denoted by cone supp( a ) , is the set ofall z ∈ R d \ { } such that any open conic set Γ ⊂ R d \ { } containing z satisfies that supp( a ) ∩ Γ is not compact in R d . We have the following
Proposition 4.10. If m ∈ R , a ∈ S mω and u ∈ S ′ ω ( R d ) , then WF ′ ω ( a ( x, D ) u ) ⊆ cone supp( a ) . Proof.
Let 0 = z / ∈ cone supp( a ). This means that there exists an open conic set Γ ⊂ R d \ { } containing z and such that a ( z ) = 0 for z ∈ Γ \ B (0 , R ) for some R >
0. Then, fromProposition 4.3, for every open conic set Γ ′ ⊆ R d \ { } with Γ ′ ∩ S d − ⊆ Γ we have that thekernel K ( z ′ , z ) defined by (4.6) satisfies the estimate (4.9) for all z ′ ∈ Γ ′ and z ∈ R d .We argue as in Corollary 4.8 and use (4.9) to obtain that formula (4.20) holds for all z ′ ∈ Γ ′ and therefore there exist C, ¯ λ > λ, N > C λ,N > z ′ ∈ Γ ′ , | V ϕ ( a ( x, D ) u )( z ′ ) | ≤ Z R d | K ( z ′ , z ) | · | V ϕ u ( z ) | dz ≤ C λ,N e − λ + N ) ω ( y ′ ) e − λ + N ) ω ( η ′ ) · Z R d e − ( λ + N ) ω ( y − y ′ ) e − ( λ + N ) ω ( η − η ′ ) | V ϕ u ( y, η ) | dydη ≤ CC λ,N e − λ + N ) ω ( y ′ ) e − λ + N ) ω ( η ′ ) · Z R d e − ( λ + N ) ω ( y − y ′ ) e − ( λ + N ) ω ( η − η ′ ) e ¯ λω ( y,η ) dydη. The Gabor wave front set in spaces of ultradifferentiable functions
It follows, by the subadditivity of ω , that | V ϕ a ( x, D ) u ( z ′ ) | ≤ CC λ,N e − λ + N ) ω ( y ′ ) e − λ + N ) ω ( η ′ ) · Z R d e − ( λ + N ) ω ( y )+( λ + N ) ω ( y ′ ) e − ( λ + N ) ω ( η )+( λ + N ) ω ( η ′ ) e ¯ λω ( y )+¯ λω ( η ) dydη ≤ CC λ,N e − λω ( y ′ ) e − λω ( η ′ ) Z R d e (¯ λ − N ) ω ( y ) e (¯ λ − N ) ω ( η ) dydη (4.24) ≤ C λ e − λω ( y ′ ) e − λω ( η ′ ) ≤ C λ e − λω ( z ′ ) ∀ z ′ = ( y ′ , η ′ ) ∈ Γ ′ for some C λ > N sufficiently large so that the integral in (4.24) converges.This proves that z / ∈ WF ′ ω ( a ( x, D ) u ) by Definition 3.1, and the proof is complete. (cid:3) Since our weight functions are non-quasianalytic, we can obtain the following consequenceof Proposition 4.10.
Corollary 4.11.
Let a ∈ S ω ( R d ) with compact support, and consider the corresponding pseudo-differential operator a ( x, D ) , cf. (4.2) . Then a ( x, D ) is globally ω -regularizing, in the sense thatfor every u ∈ S ′ ω ( R d ) we have a ( x, D ) u ∈ S ω ( R d ) .Proof. It is easy to see that a ∈ S ω . Consequently, the corresponding pseudo-differential op-erator a ( x, D ) can be extended to S ′ ω ( R d ). Since the support of a is compact, we have thatcone supp( a ) = ∅ . From Proposition 4.10 we get WF ′ ω ( a ( x, D ) u ) = ∅ . We apply Proposition 3.18to conclude. (cid:3)
In the next part of the section we consider other kind of operators, proving that they do notenlarge the wave front set. We start from the operators with polynomial coefficients.
Theorem 4.12.
Let m > be an integer, and consider A ( x, D ) = X | α + β |≤ m c αβ x α D βx , where c αβ ∈ C . Then for every u ∈ S ′ ω ( R d ) we have WF ′ ω ( A ( x, D ) u ) ⊆ WF ′ ω ( u ) . Proof.
We fix a window function ϕ ∈ S ω ( R d ), and, for ν ∈ N d we write ϕ ν for the function ϕ ν ( x ) = x ν ϕ ( x ) . For every α ∈ N d and z = ( y, η ) ∈ R d we obtain by induction on | α | that x α Π( z ) ϕ = X ν ≤ α (cid:18) αν (cid:19) y α − ν Π( z ) ϕ ν . (4.25)We have indeed that for | α | = 1, writing j for the multi-index in N d having 1 in the j -thposition and 0 elsewhere, we have x j Π( z ) ϕ = y j Π( z ) ϕ + Π( z ) ϕ j ; . Boiti, D. Jornet and A. Oliaro 29 we suppose now that (4.25) is true for every | α | = n , and prove it for ˜ α with | ˜ α | = n + 1. Thereexists j ∈ { , . . . , d } such that ˜ α = α + j . Then by the inductive hypothesis we have x ˜ α Π( z ) ϕ = x j X ν ≤ α (cid:18) αν (cid:19) y α − ν Π( z ) ϕ ν = X ν ≤ α (cid:18) αν (cid:19) (cid:2) y α − ν + j Π( z ) ϕ ν + y α − ν Π( z ) ϕ ν + j (cid:3) = y ˜ α Π( z ) ϕ + Π( z ) ϕ ˜ α + X ν ≤ αν =0 (cid:20)(cid:18) αν (cid:19) + (cid:18) αν − j (cid:19)(cid:21) y ˜ α − ν Π( z ) ϕ ν = X ν ≤ ˜ α (cid:18) ˜ αν (cid:19) y ˜ α − ν Π( z ) ϕ ν , and so (4.25) is proved. From the definition of short-time Fourier transform we have V ϕ ( x α u )( z ) = h x α u, Π( z ) ϕ i = h u, x α Π( z ) ϕ i and so by (4.25) we get V ϕ ( x α u )( z ) = X ν ≤ α (cid:18) αν (cid:19) y α − ν V ϕ ν u ( z ) . (4.26)Concerning derivation, since V ϕ ( D β u )( z ) = h D β u, Π( z ) ϕ i = h u, D β (Π( z ) ϕ ) i a direct computation shows that V ϕ ( D β u )( z ) = X µ ≤ β (cid:18) βµ (cid:19) η β − µ V D µ ϕ u. (4.27)From (4.26) and (4.27) we finally obtain V ϕ ( A ( x, D ) u )( y, η ) = X | α + β |≤ m c αβ V ϕ ( x α D βx u )( y, η )= X | α + β |≤ m X ν ≤ αµ ≤ β c αβ (cid:18) αν (cid:19)(cid:18) βµ (cid:19) y α − ν η β − µ V D µ ϕ ν u ( y, η ) . (4.28)On the other hand, it is not difficult to see that for every µ, ν ∈ N d , D µ ϕ ν ∈ S ω ( R d ).Suppose now that z = ( y , η ) / ∈ WF ′ ω ( u ), z ∈ R d \ { } . Then, there exists an open conicset Γ ⊆ R d \ { } containing z and such thatsup z ∈ Γ e λω ( z ) | V ϕ u ( z ) | < + ∞ , λ > . From Proposition 3.2 we have that for every µ, ν ∈ N d and for every open conic set Γ ′ ⊆ R d \{ } containing z and such that Γ ′ ∩ S d − ⊆ Γ,sup z ∈ Γ ′ e λω ( z ) | V D µ ϕ ν u ( z ) | < + ∞ ∀ λ > . (4.29) The Gabor wave front set in spaces of ultradifferentiable functions
From (4.28), for every k > e λω ( z ) | V ϕ ( A ( x, D ) u )( z ) | ≤ X | α + β |≤ m X ν ≤ αµ ≤ β c αβ (cid:18) αν (cid:19)(cid:18) βµ (cid:19) e − kω ( z ) | y α − ν η β − µ | e ( λ + k ) ω ( z ) | V D µ ϕ ν u ( z ) | . Since | α − ν | + | β − µ | ≤ m , from the property ( γ ) of the weight function ω we obtainsup z ∈ R d e − kω ( z ) | y α − ν η β − µ | < + ∞ , for every ν ≤ α , µ ≤ β . Therefore, from (4.29) we obtainsup z ∈ Γ ′ e λω ( z ) | V ϕ ( A ( x, D ) u )( z ) | < + ∞ , λ > , which means that z / ∈ WF ′ ω ( A ( x, D ) u ), and the proof is complete. (cid:3) We now want to prove an analogue of Theorem 4.12 for the case of localization operators.We recall here the definition of such operators and prove some results that are needed for ourpurpose. Given two window functions ψ, γ ∈ S ω ( R d ) \ { } and a symbol a ∈ S ′ ω ( R d ), thecorresponding localization operator L aψ,γ is defined, for f ∈ S ω ( R d ), as L aψ,γ f = V ∗ γ ( a · V ψ f ) . (4.30)From Proposition 2.8 we have that L aψ,γ : S ω ( R d ) → S ′ ω ( R d ) . We want now to consider symbols in a smaller class than S ′ ω ( R d ), in order to apply thecorresponding localization operator to distributions. We have the following result. Lemma 4.13.
Let a ( z ) , z ∈ R d , be a measurable function such that there exist τ, C > suchthat | a ( z ) | ≤ Ce τω ( z ) ∀ z ∈ R d . (4.31) Then L aψ,γ : S ω ( R d ) → S ω ( R d )(4.32) and L aψ,γ : S ′ ω ( R d ) → S ′ ω ( R d )(4.33) are continuous.Proof. Let f ∈ S ω ( R d ). From Theorem 2.6 we have that for every λ, ρ > C λ > e ρω ( z ) | a ( z ) || V ψ f ( z ) | ≤ C λ e ( ρ + τ − λ ) ω ( z ) , and so, choosing λ ≥ ρ + τ , we have that a · V ψ f ∈ L ∞ m ρ ( R d ) for every ρ >
0, where m ρ isdefined by (3.10). From Proposition 3.7 and (4.30), we have that L aψ,γ f ∈ M ∞ m ρ ( R d ) for every ρ >
0, and then, from Remark 3.6, L aψ,γ f ∈ S ω ( R d ). To prove the continuity of L aψ,γ on S ω ( R d ) . Boiti, D. Jornet and A. Oliaro 31 we fix ϕ ∈ S ω ( R d ) \ { } , ρ >
0, and we observe that from (3.14) (with p = q = ∞ ) and (4.31)we get sup z ∈ R d | V ϕ ( L aψ,γ f )( z ) | e ρω ( z ) = sup z ∈ R d | V ϕ V ∗ γ ( a · V ψ f ) | e ρω ( z ) ≤ C k V ϕ γ k L vρ sup z ∈ R d | a ( z ) V ψ f ( z ) | e ρω ( z ) ≤ C ′ sup z ∈ R d | V ψ f ( z ) | e ( τ + ρ ) ω ( z ) . From Proposition 2.9 we have that (4.32) is continuous.Let now f ∈ S ′ ω ( R d ). From Remark 3.6 there exists λ < f ∈ M ∞ m λ ( R d ); then,choosing ρ = −| τ | − | λ | we have e ρω ( z ) | a ( z ) || V ψ f ( z ) | ≤ Ce ( ρ + τ − λ ) ω ( z ) < + ∞ for every z ∈ R d , so a · V ψ f ∈ L ∞ m ρ ( R d ). Then by Proposition 3.7 we have L aψ,γ f ∈ M ∞ m ρ ( R d ),and from Remark 3.6 we finally have L aψ,γ f ∈ S ′ ω ( R d ). Observe now that for every u ∈ S ′ ω ( R d )and v ∈ S ω ( R d ) we have h L aψ,γ u, v i = h V ∗ γ ( a · V ψ u ) , v i = h u, V ∗ ψ ( a · V γ v ) i = h u, L aγ,ψ v i . Then L aψ,γ = ( L aγ,ψ ) ∗ ; since a satisfies the same estimates as a , the continuity of (4.33) followsfrom that of (4.32). (cid:3) Theorem 4.14.
Let ψ, γ ∈ S ω ( R d ) \ { } , and let a be a symbol satisfying (4.31) . Then forevery u ∈ S ′ ω ( R d ) we have WF ′ ω ( L aψ,γ u ) ⊆ WF ′ ω ( u ) . Proof.
Let z / ∈ WF ′ ω ( u ), z ∈ R d \ { } . Then there exists an open conic set Γ ⊆ R d \ { } containing z such that sup z ∈ Γ e λω ( z ) | V ψ u ( z ) | < + ∞ ∀ λ > . From (4.31), since λ is arbitrary we havesup z ∈ Γ e λω ( z ) | a ( z ) V ψ u ( z ) | < + ∞ ∀ λ > . For windows functions ϕ, γ ∈ S ω ( R d ) we can then repeat the same procedure used in the proofof Proposition 3.2. First, we observe that from de definition of localization operator V ϕ ( L aψ,γ u ) = V ϕ V ∗ γ ( a · V ψ u ) . Now, it is not difficult to see that V ϕ ( L aψ,γ u )( x, ξ ) = Z R d ( a · V ψ u )( s, η ) V γ (Π( z ) ϕ )( s, η ) dsdη,V γ (Π( z ) ϕ )( s, η ) = V ϕ γ ( x − s, ξ − η ) e − i h s,ξ − η i , and hence | V ϕ ( L aψ,γ u ) | ≤ | a · V ψ u | ∗ | V ϕ γ | . The Gabor wave front set in spaces of ultradifferentiable functions
Consequently, for every open conic set Γ ′ ⊆ R d \{ } containing z and such that Γ ′ ∩ S d − ⊆ Γwe have (see the proof of Proposition 3.2)sup z ∈ Γ ′ e λω ( z ) | V ϕ ( L aψ,γ u )( z ) | < + ∞ , λ > . This implies that z / ∈ WF ′ ω ( L aψ,γ u ) and the proof is complete. (cid:3) Examples
In this section we compute the Gabor wave front set for some particular u ∈ S ′ ω ( R d ) (see alsothe examples in [RW]). Example 5.1.
Consider the Dirac distribution u = δ ∈ S ′ ω ( R d ) for every weight ω . We havethat V ϕ δ ( x, ξ ) = ϕ ( − x ) . Since V ϕ δ (0 , ξ ) = ϕ (0), choosing ϕ in such a way that ϕ (0) = 0 we have { } × ( R d \ { } ) ⊆ WF ′ ω ( δ ) . Let now ( x , ξ ) ∈ R d \ { } such that x = 0, and consider an open conic set containing ( x , ξ )of the form Γ = { ( x, ξ ) ∈ R d \ { } : | ξ | < C | x |} for C >
0. From the subadditivity of ω , there exists C > z = ( x, ξ ),sup z ∈ Γ e λω ( z ) | V ϕ δ ( z ) | ≤ sup x ∈ R d e λC ω ( x ) | ϕ ( − x ) | < + ∞ since ϕ ∈ S ω ( R d ). Then ( x , ξ ) / ∈ WF ′ ω ( δ ), and so WF ′ ω ( δ ) = { } × ( R d \ { } ). FromProposition 3.19 we have that for every x ∈ R d , writing δ x for the Dirac distribution centeredat x , WF ′ ω ( δ x ) = { } × ( R d \ { } ) . (5.1) Example 5.2.
Let u = be the function identically 1, that belong to S ′ ω ( R d ) for every weight ω . A direct computation shows that V ϕ ( ) = e − i h x,ξ i ˆ ϕ ( − ξ );since ˆ ϕ ∈ S ω ( R d ) we can proceed as in Example 5.1, obtaining that for every weight ω ,WF ′ ω ( ) = ( R d \ { } ) × { } . From Proposition 3.19 we then have that for every ξ ∈ R d and for every weight ω , WF ′ ω ( e i h· ,ξ i ) = ( R d \ { } ) × { } . (5.2) Example 5.3.
We consider now the function u ( x ) = e icx / , for x ∈ R and c ∈ R \{ } . Observethat u ∈ S ′ ω ( R ) for every ω . Choosing as window function the Gaussian ϕ ( t ) = e − t / , thatbelongs to S ω ( R ) for every ω , we have, as in Example 6.6 of [RW], that there exists C > | V ϕ u ( x, ξ ) | = C exp (cid:18) − ( ξ − cx ) c ) (cid:19) . . Boiti, D. Jornet and A. Oliaro 33 Then, proceeding in a similar way as in the previous cases we haveWF ′ ω ( u ) = { ( x, cx ) : x ∈ R \ { }} (5.3)for every weight ω .We observe that in the cases (5.1) and (5.2) the Gabor wave front set gives rougher informa-tion since it does not take into account translations and modulations, while for the case (5.3)it gives finer information, since it identifies the so-called instantaneous frequency , that is theonly direction along which the time-frequency content of u does not decay. For a comparisonof the Gabor wave front set of the element considered in the previous examples with other typeof global wave front set (at least in the frame of tempered distributions) we refer to [RW].We observe now that in the previous examples the considered distributions have the samewave front set for every weight ω . In general the Gabor wave front set may depend on ω , asshown in the next example. Example 5.4.
Let ω and σ be two weight functions, such that ω ( t ) ≤ σ ( t ) and S σ ( R d ) ∩D ( R d ) ( S ω ( R d ) ∩ D ( R d ). We then fix a function f ∈ S ω ( R d ) with compact support such that f / ∈ S σ ( R d ). From Proposition 3.18 we haveWF ′ ω ( f ) = ∅ . Fix now a window ϕ ∈ S σ ( R d ) with compact support such that ϕ ≡ f ). From thedefinition of short-time Fourier transform, we then have that the orthogonal projection on R dx of the support of V ϕ f ( x, ξ ) is compact. Let now z = ( x , ξ ) ∈ R d with x = 0, and fix anopen conic set containing z of the formΓ = { ( x, ξ ) ∈ R d \ { } : | ξ | < C | x |} , for C >
0. We then have that Γ ∩ supp( V ϕ f ) is compact, so the condition (3.1) is satisfied forevery λ >
0. Then ( x , ξ ) / ∈ WF ′ σ ( f ) for every x = 0. Consider now a point of the type (0 , ξ )with ξ = 0, ξ ∈ R d . From the fact that ϕ ≡ f ), we have V ϕ f (0 , ξ ) = Z e − i h t,ξ i f ( t ) ϕ ( t ) dt = ˆ f ( ξ ) . Since f / ∈ S σ ( R d ), we have that there exists λ > ξ ∈ R d e λσ ( ξ ) | V ϕ f (0 , ξ ) | = + ∞ , so (3.1) cannot be satisfied in an open conic set containing (0 , ξ ), and then (0 , ξ ) ∈ WF ′ ω ( f ).We then have that WF ′ σ ( f ) = { } × ( R d \ { } );in particular WF ′ σ ( f ) = WF ′ ω ( f ). Acknowledgments.
The authors were partially supported by the INdAM-Gnampa Project2016 “Nuove prospettive nell’analisi microlocale e tempo-frequenza”, by FAR 2013 (Universityof Ferrara) and by the project “Ricerca Locale - Analisi di Gabor, operatori pseudodifferen-ziali ed equazioni differenziali” (University of Torino). The research of the second author waspartially supported by the project MTM2016-76647-P The Gabor wave front set in spaces of ultradifferentiable functions
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