Global Wave Front Sets in Ultradifferentiable Classes

We introduce a global wave front set using Weyl quantizations of pseudodifferential operators of infinite order in the ultradifferentiable setting. We see that in many cases it coincides with the Gabor wave front set already studied by the last three authors of the present work. In this sense, we also extend, to the ultradifferentiable setting, previous work by Rodino and Wahlberg. Finally, we give applications to the study of propagation of singularities of pseudodifferential operators.


Introduction
In the theory of partial differential equations, the wave front set locates the singularities of a distribution and, at the same time, describes the directions of the high frequencies (in terms of the Fourier transform) responsible for those singularities. In the classical context of Schwartz distributions theory, it was originally defined by Hörmander [27]. There is a huge literature on wave front sets for the study of the regularity of linear partial differential operators in spaces of distributions or ultradistributions in a local sense; see, for instance, [1,2,9,10,23,36,37] and the references therein.
In global classes of functions and distributions (like the Schwartz class S(R d ) and its dual) the concept of singular support does not make sense, since we require the information on the whole R d . However, we can still define a

Preliminaries
We denote the Fourier transform of f ∈ L 1 (R d ) by with standard extensions to more general spaces of functions and distributions. We work with weight functions as in Braun, Meise, and Taylor [16]. for all x, y ∈ R d . We also recall that (ii) For all σ > 0 there exists C σ > 0 such that for each λ > 0, inf j∈N0 t −σj e λϕ * ( σj λ ) ≤ C σ e −(λ−1)ω(t) , t≥ 1.
(v) For all λ, B > 0 there exists C > 0 such that From now on L ≥ 1 stands for the constant in Lemma 2.2(iii). When considering a suitable change of weights, the following estimates included in [6, Lemma 2.9] will appear on the stage.

The Weyl Quantization
In this section we study properties of the kernel of an operator given by a Weyl quantization and the short-time Fourier transform. First, we recall the definition of the global symbols defined in [4,6]. From now on, m ∈ R and 0 < ρ ≤ 1.
We observe that the only difference with the global symbols in [6, Definition 3.1] is the factor e mω(x,ξ) instead of e mω(x) e mω(ξ) , which is more convenient for our purposes here, but the corresponding theory of pseudodifferential operators remains the same. The constant m is called the order of the symbol.
For b ∈ GS m,ω ρ , we consider the Weyl quantization for u ∈ S ω (R d ) (see for example [39,Definition 23.5] or [37, page 631] By [4,Lemma 3.3] and [6, Theorem 3.7], given a global symbol in GS m,ω ρ , the corresponding Weyl quantization is well defined and continuous from S ω (R d ) into itself. Given two symbols a(x, ξ) and b(x, ξ), we write a#b(x, ξ) to denote the Weyl product of the two symbols, i.e. the symbol corresponding to the composition of the Weyl quantizations of a(x, ξ) and b(x, ξ): By [39,Theorem 23.6 and Problem 23.2] (cf. [4,Corollary 4.5]), the Weyl product of a and b has the following asymptotic expansion: When dealing with asymptotic expansions, a special type of operator appears, usually denoted by R, and called globally ω-regularizing operator, which acts [6,Proposition 3.11]. We also have that given a global symbol r(x, ξ) ∈ GS m,ω ρ , the pseudodifferential operator associated R = r w (x, D) is globally ω-regularizing if and only if r ∈ S ω (R 2d ) (see [6,Proposition 3.11], [31, Proposition 1.2.1]).
The following result provides sufficient conditions for a symbol to admit a left parametrix and is an extension of [22,Theorem 3.4] for global pseudodifferential operators. These conditions are the basis to define the Weyl wave front set. By [4,Theorems 3.11 and 5.4], it is easy to see that the same conditions are valid for any global quantization. In particular, for Weyl quantizations.

Theorem 3.2.
Let ω be a weight function and let σ be a subadditive weight function with The following example is inspired by [3,Capitolo 4]. [39] Example 3.3. Let ω(t) = t a be a Gevrey weight, for some 0 < a < 1/2. For m ∈ R we consider We want to show that (i) and (ii) in Theorem 3.2 hold. It is clear that On the other hand, by using Faà di Bruno formula for several variables (see, for example [29, Page 234]), we obtain that there exists C > 0 such that, for ρ = 1 − a, Let σ be as in Theorem 3.2. By Lemma 2.2(v) there exists C ≥ 1 such that for all α ∈ N 2d 0 and z ∈ R 2d . We claim that p ∈ GS |m|,ω ρ . Indeed, from Lemmas 2.2(v) and 2.3(i), for all λ > 0 there exist C λ , C λ > 0 such that So, from (3.2) and the subadditivity of the weight ω, for all λ > 0 there exists C λ > 0 such that for all α ∈ N 2d 0 and z ∈ R 2d . We observe that in this example we have the restriction 0 < a < 1/2, since a = 1 − ρ, ω(t 1/ρ ) = t a 1−a , and ω is non-quasianalytic.
We can therefore use Fubini's theorem in (3.5), and we obtain Then, by (2.7) it follows by assumption for all (y , η ) ∈ R 2d , where the kernel K(y , η , y, η) is as in (3.4).
for some C > 0, satisfying for the same constant C > 0 as in (3.7). For this C > 0, by using Lemma 2.2(v) we have that for all λ > 0 there exists C λ > 0 such that for some K > 0. We integrate by parts using the ultradifferential operator of (ω)-class associated to G (see [6, p. 3483] for the definition) in (3.6) with the next formula, which follows from [6, (3.3)]: Under the assumptions in Theorem 3.4, we estimate the kernel (3.4) as done in [12,Proposition 4.4], with the corresponding modifications for the general symbols of [4,6].
Proof. Let m > 0. We use the following change of variables in the kernel (3.4) x − y = x , s− y = s .
By abuse of notation, we will denote x by x and s by s. We have, by Theorem 3.4, (3.14) Let G ∈ H(C d ) be as in (3.6). For , h ∈ N, k ∈ N 0 , we use (3.11) as follows: We want to apply this into (3.14) and then integrate by parts in order to write We can integrate by parts in ds and dx since ψ ∈ S ω (R d Then, for j = (j 1 , . . . , We take M ∈ N, to be determined later. By (3.10) there exist C M , C 1 , C 3 > 0 so that By Lemma 2.2(i), it holds that for M ∈ N, Analogously, We also have where a δ and b τ correspond to the coefficients of G h and of G in (3.8). Moreover, by (3.9) there exists C 4 > 0 so that they can be estimated by Since b ∈ GS m,ω ρ and ψ ∈ S ω (R d ), for the above M ∈ N and for all μ > 0 there exists C M,μ > 0 such that Similarly as in (2.2), we have e mω( x+y +s+y 2 ,ξ) ≤ e mLω( x+y +s+y 2 ) e mLω(ξ) e mL ≤ e mLω(2 max{|x|,|y |,|s|,|y|}) e mLω(ξ) e mL ≤ e mL 2 ω(x) e mL 2 ω(y ) e mL 2 ω(s) e mL 2 ω(y) e mLω(ξ) e mL 2 +mL .

The Weyl Wave Front Set
In the present section we introduce a new global wave front set given in terms of Weyl quantizations in the ultradifferentiable setting, similarly to the one introduced by Hörmander [28, Definition 2.1] in the classical setting. We have some restrictions on the weight functions since the definition is based on the construction of parametrices given in [4,22]. We also show that in the definition it is enough to use symbols of order zero, so we extend [37, Proposition 2.7], which is crucial for the next sections. for all α ∈ N 2d 0 , z ∈ Γ, |z| ≥ R. We recall that there are non-quasianalytic weight functions in the sense of [16] that cannot be dominated by any subadditive function that satisfies property (β) (see [24]). This motivates the following definition. such that for some Gevrey weight function σ with ω(t 1/ρ ) = o(σ(t)) as t → +∞, the inequalities (4.1) and (4.2) hold for all z ∈ R 2d with |z| ≥ R, for some R ≥ 1.

Definition 4.3.
Let ω be a weight function, 0 < ρ ≤ 1 and u ∈ S ω (R d ). We say that z ∈ R 2d \ {0} is not in the Weyl wave front set WF ω ρ (u) of u if there exist m ∈ R and a ∈ GS m,ω ρ such that a w (x, D)u ∈ S ω (R d ) and z is non-characteristic for a.
We need to introduce the notion of conic support [37, Definition 2.1] before the next result.

Definition 4.4.
Given u ∈ S ω (R 2d ), the conic support of u, denoted by conesupp (u), is defined as the set of all x ∈ R 2d \ {0} such that any conic open set Γ ⊆ R 2d \ {0} that contains x satisfies that supp (u) ∩ Γ is not a compact set in R 2d .

Lemma 4.5. Given a weight function σ and two cones
ρ . Now we show that in Definition 4.3, similarly as in [37,Proposition 2.7], the symbol can be taken of order zero for regular weight functions.
Without losing generality, we can assume that Γ is connected (if Γ is not connected, then we take the connected component in which z 0 lies). Then, since (4.1) is satisfied it is not restrictive to assume [40] a(z) ≥ 0, z ∈ Γ, |z| ≥ R.
Moreover, we have Since ω is a ρ-regular weight there is a symbol a 0 ∈ GS m,ω ρ and a Gevrey weight function that without loss of generality we can assume to be σ (if not, we take the minimum of the two Gevrey weights, which is also a Gevrey weight) such that for the same C 1 , C 2 > 0, n ∈ N, R ≥ 1, formulas (4.1) and (4.2) are satisfied for a 0 , for all z ∈ R 2d with |z| ≥ R. As R 2d \ B(0, R) is connected, a 0 (z) ≥ 0 for all z ∈ R 2d , |z| ≥ R and also For the weight function σ, let χ be as in Lemma 4.5 for Γ and Γ . By proceeding in a similar way for Γ and Γ , we can obtain b ∈ GS 0,ω It is clear that b 0 ∈ GS m,ω ρ since a, a 0 ∈ GS m,ω ρ , χ ∈ GS 0,ω ρ . For any z / ∈ Γ, we have that χ(z) = 0 and therefore (since a 0 satisfies (4.1) for all |z| ≥ R), On the other hand, as a(z), a 0 (z) ≥ 0 for all z ∈ Γ with |z| ≥ R, and 0 ≤ χ ≤ 1, it follows b 0 (z) ≥ 0. Furthermore, from (4.3) and (4.4), Hence, we obtain This obviously implies condition (i) of Theorem 3.2 for b 0 . Since χ is as in Lemma 4.5, there exists C > 0 such that, for the previous n ∈ N, for all α ∈ N 2d 0 , z ∈ R 2d . Therefore, as a, a 0 satisfy (4.2) for z ∈ Γ with |z| ≥ R, by Leibniz rule we have Since a, a 0 ∈ GS m,ω ρ there exists C > 0 such that (we observe that β≤α We consider D = 2C 2 max{1, 2CC C1 } > 0. Then from (4.5) we obtain On the other hand, if z / ∈ Γ, then by construction b 0 = a 0 , which satisfies (4.2). Hence b 0 satisfies condition (ii) of Theorem 3.2 for all z ∈ R 2d with |z| ≥ R.
vanishes for z ∈ Γ , |z| ≥ 1 (because χ(z) = 1) we deduce that is a compact set. This implies Indeed, let χ ∈ S ω (R 2d ) with compact support with χ = 1 on E. Then b#c#(b 0 − a) has the same asymptotic expansion of b#c# ( χ(b 0 − a)). By [6,Proposition 4.3] we deduce has compact support, we have The continuity of the Weyl operator yields Hence, by (4.7), Moreover, since s ∈ S ω (R 2d ), we have By assumption a w (x, Hence, from (4.6), we finally obtain that , and the proof is complete.

A Comparison Between Different Wave Front Sets
The following definition of wave front set has been introduced and studied in [12,Definition 3.1], which extends the Gabor wave front set given in [37] for the classical setting. In [12,Theorem 4.13] an inclusion like (5.2) for linear partial differential operators with polynomial coefficients is proven. Now, we present an extension of this result for any linear partial differential operator of order m with variable coefficients of the form where a γ ∈ S ω (R d ). We observe that, in general, a function in S ω (R 2d ) is not automatically a global symbol in GS m,ω ρ . Hence (5.1) is not necessarily an operator with symbol in these classes. It is proven in [6,Example 3.13(b)] that in general S σ (R 2d ) ⊆ m∈R GS m,ω ρ ⊆ S ω (R 2d ) for every pair of weights ω and σ satisfying ω(t (1+ρ)/ρ ) = O(σ(t)), t → ∞, for some 0 < ρ ≤ 1. Also it is given there a suitable example of a weight ω for which S ω (R 2d ) = m∈R GS m,ω ρ . We show that the action of the differential operator in (5.1) to an ultradistribution u ∈ S ω (R d ) shrinks the ω-wave front set WF ω (u).
Theorem 5.2. For the differential operator defined in (5.1), we have The ω-wave front set does not depend on the choice of the window function ψ. The following lemma is an improvement of [12, Proposition 3.2].
By Lemma 2.6, Then, For ε > 0, we denote for all z ∈ R 2d , We choose ε > 0 sufficiently small so that To estimate I 1 , we use the assumption made on Γ for V ψ u as follows: for all λ > 0 there exists C λ > 0 such that for some constant C λ > 0, for all z ∈ Γ , |z| ≥ 1, and all φ ∈ B.
On the other hand, by [26,Theorem 2.4] (see also [12,Theorem 2.5]), V ψ u is continuous and there are constants c, μ > 0 such that Let q ∈ N 0 be such that ε −1 < 2 q . Then, for z ≥ ε z , the properties of the weight ω yield Then, we have Therefore, for all λ > 0 and all φ ∈ B, we have Hence for all λ > 0 there exists C λ > 0 such that This finishes the proof.

First Inclusion
Now, we compare the Weyl wave front set defined in Sect. 4 with the ω-wave front set WF ω (u), for certain weight functions ω and any ultradistribution u ∈ S ω (R d ).

Proposition 5.4.
Let ω be a weight function, b ∈ GS m,ω ρ , and u ∈ S ω (R d ). Then, Proof. For a window function ψ ∈ S ω (R d ) \ {0}, by formula (3.13) the kernel K in (3.4) satisfies the same estimates as in [12,Proposition 4.4]. The proof is therefore analogous to that of [12,Proposition 4.11] (with the only difference that now ω is not necessarily subadditive).
The same result holds for the Kohn-Nirenberg quantization, and the proof is analogous. As a consequence of Proposition 5.4, we obtain as in [12,Corollary 4.12] the following