Besov wavefront set

We develop a notion of wavefront set aimed at characterizing in Fourier space the directions along which a distribution behaves or not as an element of a specific Besov space. Subsequently we prove an alternative, albeit equivalent characterization of such wavefront set using the language of pseudodifferential operators. Both formulations are used to prove the main underlying structural properties. Among these we highlight the individuation of a sufficient criterion to multiply distributions with a prescribed Besov wavefront set which encompasses and generalizes the classical Young’s theorem. At last, as an application of this new framework we prove a theorem of propagation of singularities for a large class of hyperbolic operators.


Introduction
Microlocal analysis and the associated Hörmander's wavefront set [Hör94] have been an unmitigated success in analysis which has found in addition manifold applications ranging from engineering to mathematical physics.One of the most recent interplay with modern theoretical physics is related to the rôle played by microlocal techniques in the construction of a full-fledged theory of quantum fields on generic Lorentzian and Riemannian backgrounds as well as in the development of a mathematical formulation of renormalization with the language of distributions, see e.g.[BF00, Rej16, DDR20, CDDR20].
In the early developments of the interplay between microlocal analysis and renormalization, it has become clear that the original framework developed by Hörmander aimed at disentangling the directions of rapid decrease in Fourier space of a given distribution from the singular ones suffered from a substantial limitation.As a matter of fact, in many concrete scenarios one is interested in having a more refined estimate of the singular behavior of a distribution, for instance comparing it with that of an element lying in a suitable Sobolev space.This has lead to considering more specific forms of wavefront set, among which a notable rôle in application has been played by the so-called Sobolev wavefront set, see [Hör97].
Still having in mind the realm of quantum field theory, one of the first remarkable uses has been discussed in [JS02], while nowadays it has become an essential ingredient in many modern results among which noteworthy are those concerning the analysis of the wave equations on manifolds with boundaries or with corners, see e.g.[Vas08,Vas12].
An apparently completely detached branch of analysis in which distributions and their specific singular behavior plays a distinguished rôle is that of stochastic partial differential equations.Without entering in too many technical details, far from the scope of this work, remarkable leaps forward have been obtained in the past few years both within the framework of the theory of regularity structures [Hai14,Hai15] and in that of paracontrolled distributions [GIP12].In both approaches, despite the necessity of dealing with specific problems, such as renormalization, calling for the analysis of products or of extensions of a priori ill-defined distributions, microlocal techniques never enter the game.
The reasons are manifold but the main one lies in the fact that, in the realm of stochastic partial differential equations, often one considers Hölder distributions, i.e. elements of C α (R d ) ⊂ S ′ (R d ), α ∈ R. The latter can be read as a specific instance of the so-called Besov spaces B α p,q (R d ), α ∈ R, p, q ∈ [1, ∞], [Tri06].When working in this framework, one relies often in Bony paradifferential calculus [Bo81] as it is devised to better catch the specific features of elements lying in a Besov space.To this end microlocal techniques and the wavefront set in particular appear at first glance to be far from the optimal tool to be used, since it appears to be unable to grasp the peculiar singular behaviour of a distribution in comparison to an element of B α p,q (R d ).Nonetheless it has recently emerged that, in the analysis of a large class of nonlinear stochastic partial differential equations, microlocal analysis can be used efficiently to devise a recursive scheme to construct both solutions and correlation functions, while taking into account intrinsically the underlying renormalization freedoms, [DDRZ20,BDR21].One of the weak point of this novel approach lies in the lack of any control of the convergence of the underlying recursive scheme.This can be ascribed mainly to the fact that employing microlocal techniques appears to wash out all information concerning the behaviour of the underlying distributions as elements of a Besov space.Observe that each B α p,q (R d ) is endowed with the structure of a Banach space which is pivotal in setting up a fixed point argument to prove the existence of solutions for the considered class of nonlinear stochastic partial differential equations.
Hence, it appears natural to seek a way to combine the best of both worlds, trying to use the language of microlocal analysis on the one hand, while keeping track of the underlying Besov space structure on the other hand.In this paper we plan to make the first step in this direction, developing a modified notion of wavefront set, specifically devised to keep track of the behaviour of a distribution in comparison to that of an element of a Besov space.For definiteness and in order to avoid unnecessary technical difficulties, focusing instead on the main ideas and constructions, we shall focus on the Besov spaces ), which are, moreover, the most relevant ones in concrete applications.We highlight that an investigation in this direction, complementing our own, has appeared in [GM15].
Specifically our proposal hinges on the following starting point, a definition of Besov wavefront set which focuses on the behaviour of a distribution in Fourier space.
for any κ ∈ D(B(0, 1)) with κ(0) = 0, λ ∈ (0, 1), y ∈ supp(φ) and κ ∈ B ⌊α⌋ , see Definition 5.While conceptually the above definition enjoys all desired structural properties, from an operational viewpoint, it is rather difficult to use it concretely both in examples and in the proof of various results.For this reason we give an alternative, albeit equivalent, characterization of WF α (u), u ∈ D ′ (R d ), in terms of the intersection of the characteristic set of a suitable class of order zero, properly supported pseudodifferential operators, see Proposition 33.Using this tool we are able to prove a large set of structural properties of the Besov wavefront set.The three main results that we obtain are the following: one can establish a criterion, see Theorem 38, for the existence of the pull-back f * u, u ∈ D ′ (Ω ′ ) which generalizes the one devised by Hörmander in the smooth setting, [Hör94, Thm.8.2.4].
A noteworthy byproduct of this analysis is that, whenever f is a diffeomorphism, then, for any This result is noteworthy since it entails that the notion of Besov wavefront set can be applied also to distributions supported on an arbitrary smooth manifold [RS21].
• We establish a sufficient criterion for the existence of the product of two distributions with prescribed Besov wavefront set and we provide an estimate for the wavefront set of the product, see Theorem 45.This result contains and actually extends the renown Young's theorem on the product of two Hölder distributions, which is often used in the applications to stochastic partial differential equations.
• We apply the whole construction of the Besov wavefront set to prove a propagation of singularities theorem for a large class of hyperbolic partial differential equations, see Theorem 55.This result is strongly tied to a preliminary analysis on the wavefront set WF α (K(u)) where K is a linear map from where Ω ⊆ R d while Ω ′ ⊆ R m .The paper is organized as a follows: In Section 2, we present the definition of Besov spaces outlining some of its main properties and alternative, equivalent characterizations.Subsequently we review succinctly the basic notions of pseudodifferential operators and of the associated operator wavefront set.In Section 3 we present the main object of our investigation, giving the definition of Besov wavefront set in terms of the behaviour of a distribution in Fourier space, outlining subsequently some of the basic structural properties and discussing a few notable examples.In Section 3.1 we prove that the Besov wavefront set can be equivalently characterized in terms of the characteristic set of a suitable class of properly supported pseudodifferential operators.Section 4 contains the main results concerning the structural properties of the Besov wavefront set.In particular we discuss its interplay with pullbacks, we devise a sufficient criterion for the product of two distributions with prescribed Besov wavefront set and we prove a theorem of propagation of singularities for a class of hyperbolic partial differential operators.
Notations In this short paragraph we fix a few recurring notations used in this manuscript.With E(R d ) (resp.D(R d )), we denote the space of smooth (resp.smooth and compactly supported) functions on R d , d ≥ 1, while S(R d ) stands for the space of rapidly decreasing smooth functions.Their topological dual spaces are denoted respectively E ′ (R d ), D ′ (R d ) and S ′ (R d ).In addition, given u ∈ S(R d ), we adopt the following convention to define its Fourier transform At the same time, we indicate with the symbol • the inverse Fourier transform F −1 , namely, for any f ∈ S(R d ), f = f = f .Similarly, for any v ∈ S ′ (R d ), we indicate with v ∈ S ′ (R d ) its Fourier transform, defining it per duality as v(u) .= v(û) for all u ∈ S(R d ).In general, given a function f ∈ E(R d ), x ∈ R d and λ ∈ (0, 1], we shall denote f λ x (y) := λ −d f (λ −1 (y − x)).At last with x := (1 + |x| 2 ) 1 2 we denote the Japanese bracket, while the symbol refers to an inequality holding true up to a multiplicative finite constant.Observe that, depending on the case in hand, such constant might depend on other data, such as for example the choice of an underlying compact set.For the ease of notation we shall omit making such dependencies explicit, since they shall become clear from the context.

Preliminaries
The aim of this section is to introduce the key function spaces and some of their notable properties.The content of this specific subsection is mainly inspired by [BCD11,Tri06].The starting point lies in the notion of a Littlewood-Paley partition of unity.
In the following we shall always assume for definiteness N = 1.
Definition 3: Let α ∈ R. We call Besov space B α p,q (R d ), p, q ∈ [1, ∞), the Banach space whose elements u are such that where we used the Fourier multiplier notation ψ j (D)u(x) := F −1 {ψ j (ξ)û(ξ)}(x).At the same time, we say that Remark 4: By definition of Fourier multiplier, it descends that where we exploited In our analysis it will be often convenient not to consider directly Definition 3, rather to work with an equivalent characterization, dubbed the local means formulation -see [Tri06,Sec. 1.4 & Thm.1.10].This is based on the following tool.
where the L ∞ -norm is taken with respect to the variable x.
Remark 7: We observe that different choices for κ and κ yield in Equation (2.5) equivalent norms.Therefore, henceforth we shall omit to indicate the superscripts κ and κ.
Proof.The statement is a direct consequence of Definition 6 combined with the following identities In turn these are a by-product of the identities u(ϕ) = û( φ) , and (ϕ λ x )(ξ) = e ix•ξ φ(λξ).
Remark 10: Observe that, if α < 0, then it is sufficient to verify the second of the two conditions in Equation (2.7).

Pseudodifferential Operators
In this section we shall focus on the second functional tool which plays a distinguished rôle in our analysis.
Hence we recall succinctly the definition and some notable properties of pseudodifferential operators.For later convenience, this section is mainly inspired by [Hin21], though further details can be found in [GS94,Hör94].We start by recalling the definition both of a symbol and of its quantization.
for some constant C αβ > 0 and for any x in a compact set of R d .We denote the space of symbols of order m with S m (R d ; R N ).In addition, we define the space of residual symbols by (2.10) We define its quantization Op(a) : Op(a) is called a pseudodifferential operator ΨDO of order m and the whole set of these operators is denoted by Ψ m (R d ).Moreover, we set Since it plays a rôle in our analysis, we remark that Equation (2.11) can be replaced either by the right quantization Op R (a) or by the left quantization Op L (a) It is important to stress that, at the level of pseudodifferential operators, the choices of quantization procedure is to a certain extent immaterial, since, for any Remark 13: By means of a standard duality argument one can extend continuously the action of a pseudodifferential operator of order m, m ∈ R, to tempered distributions.In order not to burdening the reader with an unnecessarily baroque notation, we still indicate any such extension as Op(a) : As a last step we give a characterization of a notable subclass of pseudodifferential operators, based on their support properties.
Associated to a pseudodifferential operator, one can introduce the notion of operator wavefront set, which is a key ingredient in our construction outlined in Section 3.
At last we can state the main definition of this whole section: (2.14) In the following proposition we summarize a few notable properties of the operator wave set.Since the proof is a direct application of Definition 15 and 16, we omit it.
The following properties hold: (1) If A has compactly supported Schwartz kernel, then ( ( , where A * is the adjoint of A defined so that for all u, v ∈ S(R d ) A further concept, related to ΨDOs and of great relevance in the following sections is that of microlocal parametrix.Here we recall its construction.Without entering into many details, for which we refer in particular to [GS94, Chap.3], we underline that, given any In the following, when we do not write explicitly the square brackets, we are considering a representative within the equivalence class identifying the principal symbol.
where [σ m (A)] is the principal symbol of A. We call characteristic set of A, Char(A), the complement of Ell(A).

Localization of a ΨDO
In the next sections, we will be interested in the behaviour of ΨDOs under the action of a local diffeomorphism.To this end we adapt to our framework and to our notations the analysis in [Hör94, Chap. (2.17) , one can restrict the domain in Equation (2.17) to an operator Op L (a) : , where with a slight abuse of notation we keep on using the same symbol Op L (a).In full analogy with Definition 12, we indicate the ensuing collection of pseudodifferential operators by Ψ m (Ω).The following theorem is the direct adaptation to our setting and notations of [Hör94, Thm.18.1.17].
is a pseudodifferential operator of order m.Moreover, where σ m (A f ) and σ m (A) are the principal symbols of A f and A respectively while df stands for the differential map associated with f .

Besov Wavefront Set
The aim of this section is to introduce our main object of investigation.We shall therefore give a definition of Besov wavefront set, discussing subsequently its main structural properties.We proceed in two different, albeit ultimately equivalent ways.The first is based on the prototypical notion of wavefront set based on Fourier transforms -[Hör03, Ch. 8], while the second, outlined in Section 3.1, relies on pseudodifferential operators as introduced in Section 3. Observe that, in the following, we rely heavily on Proposition 9 as well as on Definition 5.
We are now in a position to prove some basic properties of the Besov wavefront set which are a direct consequence of its definition.
Proof.The implication From Proposition 9 it descends that φu ∈ B α ∞,∞ (R d ) for any φ ∈ D(R d ).This proves the sought statement.
Remark 28: Observe that, on account of the inclusion In particular, this result entails that, given any u where η ′ := λη and where we used implicitly both that Γ is a cone and that κ ∈ S(R d ).At this stage, comparing with Definition 23, we can conclude that ) be a derivative of the Dirac delta centered at the origin, i.e. ∂ j = ∂ ∂xj , x j being an Euclidean coordinate on R d .Following Definition 23 and using the identity where, similarly to Example 29, we exploited that κ ∈ S(R d ).Focusing on Equation (3.2), we can repeat the same procedure as in Example 29.For the sake of conciseness we focus directly only on y = 0 since it lies in supp(φ), for any φ ∈ D(R d ) with φ(0) = 0.In addition we can consider only the contribution due to φ(0)∂ j δ which yields, omitting φ(0) for simplicity of the notation, where η ′ := λη and where we used implicitly both that Γ is a cone and that ǩ ∈ S(R d ).Adding to this equality the outcome of Example 29, it descends where M is the order of u and ξ := (1 + |ξ| 2 ) 1 2 , see [FJ99].Fix Γ an open conic neighborhood of ξ ∈ R d \ {0}.Given κ as per Definition 5, λ ∈ (0, 1) and y ∈ supp(u), it holds where, with reference to Equation (3.1) and (3.2), we have implicitly chosen φ ∈ D(R d ) such that φ = 1 on supp(u).As a result, we get , which should be interpreted as the integral kernel of an element lying in S ′ (R 2 ).Since singsupp(u) = {(0, 0)}, we consider (0, 0, ξ 1 , ξ 2 ) such that (ξ 1 , ξ 2 ) = (0, 0).Given φ ∈ D(R 2 ) with φ(0, 0) = 1 and an open conic neighborhood Γ of (ξ 1 , ξ 2 ), we can still use the rationale followed in Example (29) studying Equation (3.2) with y = (0, 0).It reads where no singularity at the origin occurs since κ is chosen in agreement with Definition 5.This entails that

Pseudodifferential Characterization
The aim of this section is to give a second, albeit equivalent, characterization of the Besov wavefront set of a distribution by means of pseudodifferential operators.This is in spirit very much akin to the one outlined in [GS94] for the smooth wavefront set and it is especially useful in discussing operations between distributions with a prescribed Besov wavefront set, see Section 4. In the following, we shall make use of the notions introduced in Definition 12 and 14.
where the intersection is taken only over properly supported pseudodifferential operators.
Proof.Suppose that (x 0 , ξ 0 ) ∈ WF α (u).By Definition 23, there exist φ ∈ D(R d ) with φ(x 0 ) = 0 and Γ , a conic neighbourhood of ξ 0 , such that for any compact set where κ ∈ B α .Calling I Γ (ξ) the characteristic function on Γ, it descends that where a is a non vanishing constant chosen so that χ(ξ 0 ) = 0.In addition choose Observe that, following standard arguments, A is by construction properly supported and elliptic at (x 0 , ξ 0 ).To conclude it suffices to notice that, combining Equation (3.7) and Theorem 21, we can conclude that Char(A).Hence, taking into account Definition 18, there exists B ∈ Ψ 0 , elliptic at (x 0 , ξ 0 ), such that Bu ∈ B α,loc ∞,∞ (R d ).Consider once more φ, ψ and χ as in the previous part of the proof, so that where A := Op R (ψ(ξ/|ξ|)χ(ξ)φ(y)) and where W F ′ is as per Definition 16.We claim that Au ∈ B α,loc ∞,∞ (R d ).In view of Proposition 20, there exists a microlocal parametrix Remark 34: The content of Proposition 33 is an adaptation to the case in hand of the characterization of the smooth wavefront set of a distribution in terms of pseudodifferential operators, see [Hin21,Cor. 6.18].For later convenience and to fix the notation, we recall it.Let v ∈ D ′ (R d ).It holds where Char(A) is the characteristic set of A introduced in Definition 18.
We prove a proposition aimed at stating another useful characterization of the Besov wavefront set of a distribution.
where W F stands for the (smooth) wavefront set.
Proof.Suppose (x, ξ) ∈ WF α (u).On account of Remark 34, given ).Yet, since (x, ξ) ∈ WF α (u), Proposition 33 entails that (x, ξ) ∈ Char(A).Conversely, let (x, ξ) ∈ WF α (u).By Definition 23, there exist φ ∈ D(R d ) normalized so that φ(x) = 1 and an open conic neighborhood Γ of ξ such that Equation We can now establish a relation between the Besov wavefront set and the smooth counterpart, see Remark 34.The second part of the proof of the following corollary is inspired by a similar one, valid in the context of the Sobolev wavefront set [Hin21, Prop.6.32].
Corollary 36: Taking the closure and recalling that W F (u) is per construction a closed set, it descends α∈R WF α (u) ⊆ W F (u).

Structural Properties
In this section we discuss the main structural properties of distributions with a prescribed Besov wavefront set as per Definition 23 and Proposition 33, including notable operations.
Transformation Properties under Pullback -We start by investigating the interplay between Definition 23 and the pull-back of a distribution.In the following we enjoy the analysis outlined in Section 2.1.1.
Remark 37: In Definition 23 as well as in Proposition 33 we have always assumed implicitly that the underlying distribution is globally defined, i.e. u ∈ D ′ (R d ).Yet, mutatis mutandis, the whole construction and the results obtained so far can be slavishly adapted to distributions v ∈ D ′ (Ω), Ω ⊆ R d .
Theorem 38 (Pull-back -I): be the set of normals of f .For any u ∈ D ′ (Ω ′ ) such that there exists α > 0 so that there exists f * u ∈ D ′ (Ω).In addition for every u ∈ D ′ (Ω ′ ) abiding to Equation (4.2), where Proof.As a consequence of Proposition 35, Equation (4.2) is equivalent to Then, there exists the pullback identifies a pseudodifferential operator of order 0 per Theorem 22. Since (Au) This proves (x, t df (x)η) ∈ WF α (f * u).
To conclude this first part of the section, we shall prove that Besov wavefront set is invariant under the action of diffeomorphisms.
Remark 40: Theorem 39 is especially noteworthy since it is the building block to extend the notion of Besov wavefront set to distributions supported on any arbitrary smooth manifold M , following the same rationale used when working with the smooth counterpart.On a similar note, we observe that for the sake of simplicity of the presentation, we decided to stick to individuating a point of Yet, from a geometrical viewpoint each element of WF α (u) should be better read as lying in the cotangent bundle T * R d \ {0}.For the sake of conciseness, we shall not dwell into further details which are left to the reader.
Microlocal Properties of ΨDOs -Our next task is the study of the interplay between pseudodifferential operators and distributions at the level of wavefront set.To this end we recall a notable result, valid in the smooth setting, see [Hin21, Prop.6.27], namely, if where W F ′ stands for the operator wavefront set as per Definition 16.At the level of Besov wavefront set the counterpart of this statement is the following proposition.
Corollary 43 (Elliptic Regularity): Proof.Since A is an elliptic pseudodifferential operator Char(A) = ∅ and, in view of Definition 16, W F ′ (A) = ∅.The statement is thus a direct consequence of Propositions 41 and 42.
Product of Distributions -In the following investigate the formulation of a version of Hörmander's criterion for the product of distributions, tied to the Besov wavefront set.In the spirit of [Hör03], we rely on two ingredients.The first has already been discussed in Theorem 38, while the second one concerns the tensor product of two distributions.In particular we wish to establish an estimate on the singular behaviour of u ⊗ v for given u, v ∈ D ′ (R d ).This can be read and, calling γ := min{α, β, α + β}, where we adopted the notation W F 0 (u) := W F (u)∪(supp(u) × {0}) and similarly WF α 0 (u) := WF α (u)∪ (supp(u) × {0}).
At last, we are in a position to prove a counterpart of Hörmander's criterion for the product of two distributions within the framework of the Besov wavefront set.
To conclude this section, we discuss an application of Theorem 45 which is of relevance in many concrete scenarios.More precisely, we consider a continuous K : ), we investigate the existence of Ku and we seek to establish a bound on the associated Besov wavefront set.As a preliminary step, we need to prove two ancillary results.
Assume that the projection map on the first factor π : Ω × Ω ′ → Ω is proper on supp(v).Then it holds that, for all α ∈ R where π * is the push-forward map.
Proof.Since v is compactly supported and since the action of π * is tantamount to a partial evaluation against the constant function 1 Here we can restrict the attention to compactly supported elements lying in In turn, on account of Corollary 47, we can replace u by π * (ũ), where ũ Applying [Hör03, Thm.8.2.12], it descends that Yet, on account of the arbitrariness of ũ and using Proposition 35, it descends that y ∈ supp(v) and (x, y, ξ, 0) ∈ WF α (v), which is nothing but the sought statement.
We can prove the main result of this part of our work and we divide it in two statements.
Theorem 49: Proof.Let π : Ω × Ω ′ → Ω be the projection map on the second factor and assume for the time being that , where π * is the push-forward along π while • stands for the product of distributions.Observe that, since WF α (1 ⊗ u) = ∅ for all α ∈ R then, the pointwise product is well-defined on account of Theorem 45.The latter also entails that, for all α ∈ R, At this stage, observing that by localizing the underlying distribution around each point of the wavefront set, we can apply Proposition 48.It descends which concludes the proof.
At last we generalize the preceding theorem so to investigate under which circumstances u can be taken to be an element lying E ′ (Ω ′ ) and with a non empty wavefront set.
To conclude, we prove a statement which adapts to the current scenario an important result for the Sobolev wavefront set, see [JS02, Prop.B.9].

Besov Wavefront Set and Hyperbolic Partial Differential Equations
As an application of the results of the previous sections, we study the interplay between the Besov wavefront set and a large class of hyperbolic partial differential equations of the form where we assume a = a 1 + a 0 where a 1 ∈ S 1 hom (R d ), while a 0 ∈ S 0 (R d ) see Definition 11.Using standard Fourier analysis, we can infer that the fundamental solution associated to the operator Proof.
For any κ ∈ B ⌊α⌋ as per Definition 5, it where the first estimate is a a byproduct of v being compactly supported.A similar reasoning applies when considering any κ ∈ D(B(0, 1)) such that κ(0) = 0.As a consequence of Definition 6, we infer that v ∈ B α 2,∞ (R d ).
Proposition 54: Let G ∈ D ′ (R×R d ) be the fundamental solution of the hyperbolic operator ∂ t −ia(D x ).
16) B is called microlocal parametrix for A on C .The proof of this proposition can be found in [Hin21, Prop.6.15].For later convenience we conclude the section stating a result on the properties of pseudodifferential operators acting on Besov spaces, see [Abe12, Sect 6.6].
as per Definition 12, Equation (2.12b) and Remark 13.Then the restriction of A to a Besov space as per Definition 3 setting