Besov wavefront set

Abstract

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.

1 Introduction

Microlocal analysis and the associated Hörmander’s wavefront set [18] 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. [6,7,8, 17, 23].

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 [19]. Still having in mind the realm of quantum field theory, one of the first remarkable uses has been discussed in [21], 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. [28, 29].

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 [14, 15] and in that of paracontrolled distributions [13]. 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 $C^\alpha (\mathbb {R}^d)\subset \mathcal {S}^\prime (\mathbb {R}^d)$, $\alpha \in \mathbb {R}$. The latter can be read as a specific instance of the so-called Besov spaces $B^\alpha _{p,q}(\mathbb {R}^d)$, $\alpha \in \mathbb {R}$, $p,q\in [1,\infty ]$, [27]. When working in this framework, one relies often in Bony paradifferential calculus [4] 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 $B^\alpha _{p,q}(\mathbb {R}^d)$.

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, [3, 9]. One of the weak points 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 $B^\alpha _{p,q}(\mathbb {R}^d)$ 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 $B^\alpha _{\infty ,\infty }(\mathbb {R}^d)\equiv C^\alpha (\mathbb {R}^d)$, which are, moreover, the most relevant ones in concrete applications. We highlight that an investigation in this direction, complementing our own, has appeared in [11]. 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.

Definition 1

Let $u \in \mathcal {D}^\prime (\mathbb {R}^d)$ and $\alpha \in \mathbb {R}$. We say that $(x,\xi )\in \mathbb {R}^d \times (\mathbb {R}^d{\setminus } \{0\})$ does not lie in the $B_{\infty ,\infty }^{\alpha }$-wavefront set, denoting $(x,\xi ) \not \in \textrm{WF}^\alpha (u)$, if there exist $\phi \in \mathcal {D}(\mathbb {R}^d)$ with $\phi (x)\ne 0$ as well as an open conic neighborhood $\Gamma$ of $\xi$ in $\mathbb {R}^d\setminus \{0\}$ such that

$$\begin{aligned} \bigg |\int _{\Gamma } \widehat{\phi u}(\eta ) \check{\underline{\kappa }}(\eta ) e^{i y \cdot \eta } d\eta \bigg |\lesssim&1\,, \end{aligned}$$

(1.1)

$$\begin{aligned} \bigg |\int _{\Gamma } \widehat{\phi u}(\eta ) \check{\kappa }(\lambda \eta ) e^{i y \cdot \eta } d\eta \bigg |\lesssim&\lambda ^\alpha \,, \end{aligned}$$

(1.2)

for any $\underline{\kappa }\in \mathcal {D}(B(0,1))$ with $\check{\underline{\kappa }}(0)\ne 0$, $\lambda \in (0,1)$, $y \in {\text {supp}}(\phi )$ and $\kappa \in \mathscr {B}_{\lfloor \alpha \rfloor }$, 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 $\textrm{WF}^\alpha (u)$, $u\in \mathcal {D}^\prime (\mathbb {R}^d)$, in terms of the intersection of the characteristic set of a suitable class of order zero, properly supported pseudodifferential operators, see Proposition 34. 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:

• We prove that, given an embedding $f\in C^\infty (\Omega ,\Omega ^\prime )$ between two open subsets $\Omega \subseteq \mathbb {R}^d$ and $\Omega ^\prime \subseteq \mathbb {R}^m$, one can establish a criterion, see Theorem 39, for the existence of the pull-back $f^*u$, $u\in \mathcal {D}^\prime (\Omega ^\prime )$ which generalizes the one devised by Hörmander in the smooth setting, [18, Thm. 8.2.4]. A noteworthy byproduct of this analysis is that, whenever f is a diffeomorphism, then, for any $\alpha \in \mathbb {R}$, $f^*WF^\alpha (u)=WF^\alpha (f^*u)$, see Theorem 46. 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, going along the path started with [24].

• 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 46. 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 56. This result is strongly tied to a preliminary analysis on the wavefront set $\textrm{WF}^\alpha (\mathcal {K}(u))$ where $\mathcal {K}$ is a linear map from $C^\infty _0(\Omega ^\prime )\rightarrow \mathcal {D}^\prime (\Omega )$ where $\Omega \subseteq \mathbb {R}^d$ while $\Omega ^\prime \subseteq \mathbb {R}^m$.

The paper is organized as a follows: In Sect. 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 Sect. 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 Sect. 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 $\mathcal {E}(\mathbb {R}^d)$ (resp. $\mathcal {D}(\mathbb {R}^d))$, we denote the space of smooth (resp. smooth and compactly supported) functions on $\mathbb {R}^d$, $d\ge 1$, while $\mathcal {S}(\mathbb {R}^d)$ stands for the space of rapidly decreasing smooth functions. Their topological dual spaces are denoted respectively $\mathcal {E}^\prime (\mathbb {R}^d)$, $\mathcal {D}^\prime (\mathbb {R}^d)$ and $\mathcal {S}^\prime (\mathbb {R}^d)$. In addition, given $u\in \mathcal {S}(\mathbb {R}^d)$, we adopt the following convention to define its Fourier transform

$$\begin{aligned} \mathcal {F}(u)(k)=\hat{u}(k):=\int _{\mathbb {R}^d}e^{-ik\cdot x}u(x)\,dx\,. \end{aligned}$$

At the same time, we indicate with the symbol $\check{\cdot }$ the inverse Fourier transform $\mathcal {F}^{-1}$, namely, for any $f\in \mathcal {S}(\mathbb {R}^d)$, $f=\check{\hat{f}}=\hat{\check{f}}$. Similarly, for any $v\in \mathcal {S}^\prime (\mathbb {R}^d)$, we indicate with $\widehat{v}\in \mathcal {S}^\prime (\mathbb {R}^d)$ its Fourier transform, defining it per duality as $\hat{v}(u)\doteq v(\hat{u})$ for all $u\in \mathcal {S}(\mathbb {R}^d)$. In general, given a function $f\in \mathcal {E}(\mathbb {R}^d)$, $x\in \mathbb {R}^d$ and $\lambda \in (0,1]$, we shall denote $f^\lambda _x(y):=\lambda ^{-d}f(\lambda ^{-1}(y-x))$. At last with $\langle x\rangle :=(1+|x|^2)^{\frac{1}{2}}$ we denote the Japanese bracket, while the symbol $\lesssim$ 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.

2 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 [2, 27]. The starting point lies in the notion of a Littlewood-Paley partition of unity.

Definition 2

Let $N\in \mathbb {N}$ and let $\psi \in \mathcal {D}(\mathbb {R}^d)$ be a positive function supported in $\{2^{-N}\le |\xi |\le 2^N\}$. We call Littlewood-Paley partition of unity a sequence $\{\psi _j\}_{j\in \mathbb {N}_0}$, $\mathbb {N}_0\doteq \mathbb {N}\cup \{0\}$ such that

• $\psi _0\in \mathcal {D}(\mathbb {R}^d)$ and $\text {supp}(\psi _0)\subseteq \{|\xi |\le 2^N\}$;

• $\psi _j(x):=\psi (2^{-j}x)$ for $j\ge 1$;

• $\sum _{j\in \mathbb {N}_0}\psi _j(\xi )=1$ for all $\xi \in \mathbb {R}^d$;

• for any multi-index $\alpha$, $\exists C_\alpha >0$ such that

  $$\begin{aligned} |D^\alpha \psi _j(\xi )|\le C_\alpha \langle \xi \rangle ^{-|\alpha |}\,,\qquad j\ge 1\,; \end{aligned}$$

• $\psi _j(-\xi )=\psi _j(\xi )$ for all $j\ge 0$.

In the following we shall always assume for definiteness $N=1$.

Definition 3

Let $\alpha \in \mathbb {R}$. We call Besov space $B^\alpha _{p,q}(\mathbb {R}^d)$, $p,q\in [1,\infty )$, the Banach space whose elements u are such that

$$\begin{aligned} \Vert u\Vert ^q_{B_{pq}^\alpha (\mathbb {R}^d)}:=\sum \limits _{j\ge 0}2^{j\alpha q} \Vert \psi _j(D)u\Vert ^q_{L^p(\mathbb {R}^d)}<\infty , \end{aligned}$$

(2.1)

At the same time if $q=\infty$, while $p\in [1,\infty ]$, we set

$$\begin{aligned} \Vert u\Vert _{B_{p,\infty }^\alpha (\mathbb {R}^d)}:=\sup _{j\ge 0}2^{j\alpha } \Vert \psi _j(D)u\Vert _{L^p(\mathbb {R}^d)}<\infty , \end{aligned}$$

(2.2)

where we used the Fourier multiplier notation $\psi _j(D)u(x):=\mathcal {F}^{-1}\{\psi _j(\xi )\hat{u}(\xi )\}(x)$. At the same time, we say that $u \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$ if $\varphi u \in B_{\infty ,\infty }^{\alpha }(\mathbb {R}^d)$ for any $\varphi \in \mathcal {D}(\mathbb {R}^d)$.

Remark 4

By definition of Fourier multiplier, it descends that

$$\begin{aligned} \psi _j(D) u (x) = \mathcal {F}^{-1}\{\psi _j(\xi ) \hat{u}(\xi )\}(x) = (\check{\psi }_j *u)(x)= u(2^{jd}\check{\psi }(2^j(\cdot -x)))\,, \end{aligned}$$

where we exploited $\mathcal {F}^{-1}\{uv\}= \check{u} *\check{v}$ and $\check{\psi }_j(x)=2^{jd}\check{\psi }(2^j x)$. As a consequence, if $u \in B_{\infty ,\infty }^\alpha (\mathbb {R}^d)$,

$$\begin{aligned} |u(\check{\psi }^{2^{-j}}_x) |\lesssim 2^{-j\alpha },\quad \forall j\ge 0, \quad \forall x \in \mathbb {R}^d. \end{aligned}$$

(2.3)

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 [27, Sec. 1.4 & Thm.1.10]. This is based on the following tool.

Definition 5

Let $B(0,1)=\{y \in \mathbb {R}^d: |y |< 1\}$. For $s\in \mathbb {N}_0$, we call $\mathscr {B}_s$ the subset of $\mathcal {D}(B(0,1))$ whose elements $\kappa$ are such that there exists $\epsilon >0$

$$\begin{aligned} \check{\kappa }(\xi )\ne 0\;\text { if }\;\frac{\varepsilon }{2}<|\xi |\le 2\varepsilon ,\quad \text {and}\quad (\partial ^\beta \check{\kappa })(0)=0 \,\;\text {if}\;|\beta |\le s. \end{aligned}$$

(2.4)

Observe that the second condition in Eq. (2.4) is empty if $s<0$.

Definition 6

Let $\alpha \in \mathbb {R}$, $\kappa \in \mathscr {B}_{\lfloor \alpha \rfloor }$, with $\lfloor \alpha \rfloor$ the biggest integer N such that $N\le \alpha$. Let $\underline{\kappa }\in \mathcal {D}(B(0,1))$ be such that $\check{\underline{\kappa }}(0)\ne 0$. We call $B_{p,\infty }^\alpha (\mathbb {R}^d)$, $p\in [1,\infty ]$, the space of distributions $u\in \mathcal {S}^\prime (\mathbb {R}^d)$ such that

$$\begin{aligned} \Vert u\Vert ^{\kappa , \underline{\kappa }}_{B_{p,\infty }^\alpha (\mathbb {R}^d)}:= \Vert u(\underline{\kappa }_{x})\Vert _{L^p(\mathbb {R}^d)} + \sup _{\lambda \in (0,1)} \frac{\Vert u(\kappa ^\lambda _{x})\Vert _{L^p(\mathbb {R}^d)}}{\lambda ^\alpha } < \infty , \end{aligned}$$

(2.5)

where the $L^\infty$-norm is taken with respect to the variable x.

Remark 7

We observe that different choices for $\kappa$ and $\underline{\kappa }$ yield in Eq. (2.5) equivalent norms. Therefore, henceforth we shall omit to indicate the superscripts $\kappa$ and $\underline{\kappa }$.

If $\alpha < 0$, there exists a further equivalent characterization for Besov spaces – see [5, Prop. A.5], [27, Cor. 1.12]. We focus on the case $p=\infty$.

Proposition 8

Let $\alpha < 0$ and $\kappa \in \mathcal {D}(B(0,1))$ be such that $\check{\kappa }(0)\ne 0$. Then $u \in B_{\infty ,\infty }^\alpha (\mathbb {R}^d)$ if and only if

$$\begin{aligned} \sup _{\lambda \in (0,1)} \frac{\Vert u(\kappa ^\lambda _{x})\Vert _{L^\infty (\mathbb {R}^d)}}{\lambda ^\alpha } < \infty , \end{aligned}$$

(2.6)

where the $L^\infty$-norm is taken with respect to the variable x.

We conclude this subsection proving a last, useful characterization of the element lying in $B_{\infty ,\infty }^\alpha$.

Proposition 9

Let $u \in \mathcal {S}^\prime (\mathbb {R}^d)$ and let $\alpha \in \mathbb {R}$. Then $u \in B_{\infty ,\infty }^{\alpha }(\mathbb {R}^d)$ if and only if, given $\kappa \in \mathscr {B}_{\lfloor \alpha \rfloor }$ and $\underline{\kappa }\in \mathcal {D}(B(0,1))$ such that $\check{\underline{\kappa }}(0)\ne 0$, it holds that

$$\begin{aligned} |\langle \hat{u}(\xi ), e^{ix\cdot \xi } \check{\underline{\kappa }}(\xi ) \rangle |\lesssim 1, \qquad |\langle \hat{u}(\xi ), e^{ix\cdot \xi } \check{\kappa }(\lambda \xi ) \rangle |\lesssim \lambda ^\alpha , \end{aligned}$$

(2.7)

for any $\lambda \in (0,1)$ and $x \in \mathbb {R}^d$.

Proof

The statement is a direct consequence of Definition 6 combined with the following identities

$$\begin{aligned} u(\varphi _x) = \langle \hat{u}(\xi ), e^{ix\cdot \xi } \check{\varphi }(\xi ) \rangle \,, \qquad u(\varphi ^\lambda _x) = \langle \hat{u}(\xi ), e^{ix\cdot \xi } \check{\varphi }(\lambda \xi ) \rangle \,, \end{aligned}$$

(2.8)

where $\varphi \in \mathcal {S}(\mathbb {R}^d)$, $u \in \mathcal {S}^\prime (\mathbb {R}^d)$, $x\in \mathbb {R}^d$ and $\lambda \in (0,1]$. In turn these are a by-product of the identities $u(\varphi ) = \hat{u}(\check{\varphi })\,$, and $\check{(\varphi ^\lambda _x)}(\xi ) = e^{ix\cdot \xi } \check{\varphi }(\lambda \xi )$. $\square$

Remark 10

Observe that, if $\alpha < 0$, then it is sufficient to verify the second of the two conditions in Eq. (2.7).

2.1 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. An expert reader might skip this section, which is here to make the content of this work accessible to a less specialized audience. For later convenience, this section is mainly inspired by [16], though further details can be found in [12, 18]. We start by recalling the definition both of a symbol and of its quantization.

Definition 11

Let $m \in \mathbb {R}$ and $n,N \in \mathbb {N}$. A function $a \in C^\infty (\mathbb {R}^d\times \mathbb {R}^N)$ is called a symbol of order m if, for all $\alpha \in \mathbb {N}^n_0$, $\beta \in \mathbb {N}^N_0$, it satisfies

$$\begin{aligned} |\partial ^\alpha _x \partial _\xi ^\beta a(x,\xi ) |\le C_{\alpha \beta } \langle \xi \rangle ^{m-|\beta |} \end{aligned}$$

(2.9)

for some constant $C_{\alpha \beta } > 0$ and for any x in a compact set of $\mathbb {R}^d$. We denote the space of symbols of order m with $S^m(\mathbb {R}^d;\mathbb {R}^N)$. In addition, we define the space of residual symbols by

$$\begin{aligned} S^{-\infty }(\mathbb {R}^d;\mathbb {R}^N):= \bigcap _{m \in \mathbb {R}} S^m(\mathbb {R}^d;\mathbb {R}^N). \end{aligned}$$

(2.10)

At last we call $S^m_{\text {hom}}(\mathbb {R}^d;\mathbb {R}^N)\subset S^m(\mathbb {R}^d;\mathbb {R}^N)$ the collection of homogeneous symbols of order m, namely, when $|\xi |>1$, $a(x,\lambda \xi )=\lambda ^m a(x,\xi )$ for all $\lambda >0$ and, for all $\alpha \in \mathbb {N}^n_0$, $\beta \in \mathbb {N}^N_0$

$$\begin{aligned} |\partial ^\alpha _x \partial _\xi ^\beta a(x,\xi ) |\le C_{\alpha \beta } |\xi |^{m-|\beta |}. \end{aligned}$$

Definition 12

Let $m \in \mathbb {R}$, $n\in \mathbb {N}$ and let $a \in S^m(\mathbb {R}^d \times \mathbb {R}^d;\mathbb {R}^n)$. We define its quantization $\textrm{Op}(a):\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}^\prime (\mathbb {R}^d)$ as

$$\begin{aligned} (\textrm{Op}(a)u)(x):= (2\pi )^{-n} \int _{\mathbb {R}^d} \int _{\mathbb {R}^d} e^{i(x-y) \cdot \xi } a(x,y,\xi ) u(y) dy d\xi , \quad u \in \mathcal {S}(\mathbb {R}^d).\nonumber \\ \end{aligned}$$

(2.11)

$\textrm{Op}(a)$ is called a pseudodifferential operator ${\varvec{\Psi }}$DO of order ${\varvec{m}}$ and the whole set of these operators is denoted by $\Psi ^m(\mathbb {R}^d)$. Moreover, we set

$$\begin{aligned} \Psi ^{-\infty }(\mathbb {R}^d):= \bigcap _{m \in \mathbb {R}} \Psi ^m(\mathbb {R}^d). \end{aligned}$$

Since it plays a rôle in our analysis, we remark that Eq. (2.11) can be replaced either by the right quantization $\textrm{Op}_R(a)$ or by the left quantization $\textrm{Op}_L(a)$

$$\begin{aligned} (\textrm{Op}_R(a^\prime )u)(x):= (2\pi )^{-n} \int _{\mathbb {R}^d} \int _{\mathbb {R}^d} e^{i(x-y) \cdot \xi } a^\prime (y,\xi ) u(y)\, dy d\xi .\quad \forall a^\prime \in S^m(\mathbb {R}^d;\mathbb {R}^d)\nonumber \\ \end{aligned}$$

(2.12a)

$$\begin{aligned} (\textrm{Op}_L(\tilde{a})u)(x):= (2\pi )^{-n} \int _{\mathbb {R}^d} \int _{\mathbb {R}^d} e^{i(x-y) \cdot \xi } \tilde{a}(x,\xi ) u(y) dy d\xi \quad \quad \forall \tilde{a}\in S^m(\mathbb {R}^d;\mathbb {R}^d)\nonumber \\ \end{aligned}$$

(2.12b)

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 $a\in S^m(\mathbb {R}^d \times \mathbb {R}^d;\mathbb {R}^d)$, there always exist $a_L,a_R\in S^m(\mathbb {R}^d;\mathbb {R}^d)$ such that—see [16, Thm. 4.8]

$$\begin{aligned} \textrm{Op}(a)=\textrm{Op}_L(a_L)=\textrm{Op}_R(a_R). \end{aligned}$$

Remark 13

By means of a standard duality argument one can extend continuously, via a density argument of $\mathcal {S}$ in $\mathcal {S}^\prime$, the action of a pseudodifferential operator of order m, $m\in \mathbb {R}$, to tempered distributions. In order not to burdening the reader with an unnecessarily baroque notation, we still indicate any such extension as $\mathrm {Op(a)}:\mathcal {S}^\prime (\mathbb {R}^d)\rightarrow \mathcal {S}^\prime (\mathbb {R}^d)$ for all $a\in S^m(\mathbb {R}^d \times \mathbb {R}^d;\mathbb {R}^d)$.

As a last step we give a characterization of a notable subclass of pseudodifferential operators, based on their support properties.

Definition 14

Let $A \in \Psi ^m(\mathbb {R}^d)$ and let $K_A \in \mathcal {S}^\prime (\mathbb {R}^d \times \mathbb {R}^d)$ be the associated Schwartz kernel. We say that A is properly supported if the canonical projections $\pi _1 :\textrm{supp}(K)\subseteq \mathbb {R}^d \times \mathbb {R}^d \rightarrow \mathbb {R}^d$ and $\pi _2 :\textrm{supp}(K)\subseteq \mathbb {R}^d \times \mathbb {R}^d \rightarrow \mathbb {R}^d$ are proper maps.

Associated to a pseudodifferential operator, one can introduce the notion of operator wavefront set, which is a key ingredient in our construction outlined in Sect. 3. From now on, we shall consider only properly supported pseudodifferential operators unless stated otherwise.

Definition 15

Let $a \in S^m(\mathbb {R}^d;\mathbb {R}^N)$. We say that a point $(x_0,\xi _0) \in \mathbb {R}^d \times (\mathbb {R}^N \setminus \{0\})$ does not lie in the essential support of a

$$\begin{aligned} \text {ess supp}(a) \subset \mathbb {R}^d \times (\mathbb {R}^N\setminus \{0\}), \end{aligned}$$

if there exists $\varepsilon > 0$ such that for all $\ell \in \mathbb {N}_0^n$, $\beta \in \mathbb {N}^N_0$, $k \in \mathbb {R}$, it holds

$$\begin{aligned} |\partial ^\ell _x \partial ^\beta _\xi a(x,\xi ) |\le C \langle \xi \rangle ^{-k}, \quad \forall (x,\xi ),\;\text {such that}\; |\xi |\ge 1,\;\text {and}\; |x-x_0|+ \bigg |\frac{\xi }{|\xi |} - \frac{\xi _0}{|\xi _0 |} \bigg |< \varepsilon .\nonumber \\ \end{aligned}$$

(2.13)

Observe that $\text {ess supp}(a)$ is a closed subset of $\mathbb {R}^d \times (\mathbb {R}^N \setminus \{0\})$ whereas, for each $x\in \mathbb {R}^d$, $\pi _\xi [\text {ess supp}(a)]\subseteq \mathbb {R}^N{\setminus }\{0\}$ is a conical subset. At last we can state the main definition of this whole section:

Definition 16

Let $A = \textrm{Op}_L(a)\in \Psi ^m(\mathbb {R}^d)$. The operator wavefront set of A is

$$\begin{aligned} WF^\prime (A):= \text {ess supp}(a) \subset \mathbb {R}^d \times (\mathbb {R}^N \setminus \{0\}). \end{aligned}$$

(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.

Proposition 17

Let $A, B \in \Psi ^m(\mathbb {R}^d)$. The following properties hold:

• (1) If A has compactly supported Schwartz kernel, then $WF^\prime (A) = \emptyset$ if and only if $A \in \Psi ^{-\infty }(\mathbb {R}^d)$.

• (2) $WF^\prime (A + B) \subset WF^\prime (A) \cup WF^\prime (B)$.

• (3) $WF^\prime (AB) \subset WF^\prime (A) \cap WF^\prime (B)$.

• (4) $WF^\prime (A^*) = WF^\prime (A)$, where $A^*$ is the adjoint of A defined so that for all $u,v\in \mathcal {S}(\mathbb {R}^d)$

  $$\begin{aligned} \int \limits _{\mathbb {R}^d}dx\,(A^*u)(x)\overline{v}(x)=\int \limits _{\mathbb {R}^d}dx\,u(x)\overline{(Av)(x)}. \end{aligned}$$

A further concept, related to $\Psi$DOs 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 [12, Chap. 3], we underline that, given any $A\in \Psi ^m(\mathbb {R}^d)$, $m\in \mathbb {R}$, one can always associate to it a principal symbol $[\sigma _m(A)]\in {S^m(\mathbb {R}^d;\mathbb {R}^d)}/{S^{m-1}(\mathbb {R}^d;\mathbb {R}^d)}$. 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.

Definition 18

Given $A \in \Psi ^m(\mathbb {R}^d)$, a point $(x_0,\xi _0) \in \mathbb {R}^d \times (\mathbb {R}^d {\setminus } \{0\})$ lies in the elliptic set of A, $\textrm{Ell}(A)$, if there exists $\varepsilon > 0$ and a constant $C>0$ such that

$$\begin{aligned} |\sigma _m(A)(x,\xi ) |\ge C | \xi |^{m}, \quad \forall (x,\xi )\;\text {such that}\; |\xi |\ge 1,\;\text {and}\; |x-x_0|+ \bigg |\frac{\xi }{|\xi |} - \frac{\xi _0}{|\xi _0 |} \bigg |< \varepsilon ,\nonumber \\ \end{aligned}$$

(2.15)

where $[\sigma _m(A)]$ is the principal symbol of A. We call characteristic set of A, $\textrm{Char}(A)$, the complement of $\textrm{Ell}(A)$.

Remark 19

Definition 18 can be reformulated as follows: a point $(x_0,\xi _0)\in \textrm{Ell}(A)$ if there exist $b \in S^{-m}(\mathbb {R}^d;\mathbb {R}^d)$ and a conic neighbourhood of $(x_0,\xi _0)$ such that therein $P_m(A)b-1 \in S^{-1}(\mathbb {R}^d;\mathbb {R}^d)$.

Proposition 20

Let $A \in \Psi ^m(\mathbb {R}^d)$ and let $\mathscr {C} \subset \textrm{Ell}(A)$ be a closed subset. Then there exists $B \in \Psi ^{-m}(\mathbb {R}^d)$ such that

$$\begin{aligned} \mathscr {C} \cap WF^\prime (AB-I) = \emptyset , \quad \mathscr {C} \cap WF^\prime (BA-I) = \emptyset . \end{aligned}$$

(2.16)

B is called microlocal parametrix for A on $\mathscr {C}$.

The proof of this proposition can be found in [16, Prop.6.15]. For later convenience we conclude the section stating a result on the properties of pseudodifferential operators acting on Besov spaces, see [1, Sect 6.6].

Theorem 21

Let $m \in \mathbb {R}$, $\alpha \in \mathbb {R}$ and let $a\in S^m(\mathbb {R}^d;\mathbb {R}^d)$. Let $A:\mathcal {S}^\prime (\mathbb {R}^d)\rightarrow \mathcal {S}^\prime (\mathbb {R}^d)$ be the associated element of $\Psi ^m(\mathbb {R}^d)$ as per Definition 12, Eq. (2.12b) and Remark 13. Then the restriction of A to a Besov space as per Definition 3 setting $p=q=\infty$ is a bounded linear operator $A:B^\alpha _{\infty ,\infty }(\mathbb {R}^d) \rightarrow B^{\alpha - m}_{\infty ,\infty }(\mathbb {R}^d)$.

Remark 22

We observe that, since we are working with properly supported operator, then the restriction of A to $B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$ is a continuous linear operator $A:B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d) \rightarrow B^{\alpha -m,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$.

2.1.1 Localization of a $\Psi$DO

In the next sections, we will be interested in the behaviour of $\Psi$DOs under the action of a local diffeomorphism. To this end we adapt to our framework and to our notations the analysis in [18, Chap. 18.1].

Hence, let $\Omega \subset \mathbb {R}^d$ be an open subset, we say that a function $a\in C^\infty (\Omega \times \mathbb {R}^d)$ identifies a local symbol on $\Omega \times \mathbb {R}^d$, i.e. $a\in S^m(\Omega ;\mathbb {R}^d)$ if $\phi a\in S^m(\mathbb {R}^d;\mathbb {R}^d)$ for all $\phi \in C^\infty _0(\Omega )$. Using Eq. (2.12b) one identifies an operator

$$\begin{aligned} \textrm{Op}_L(a):\mathcal {S}^\prime (\mathbb {R}^d)\rightarrow \mathcal {D}^\prime (\Omega ). \end{aligned}$$

(2.17)

Observing that $C^\infty _0(\Omega )\hookrightarrow \mathcal {E}^\prime (\Omega )\hookrightarrow \mathcal {S}^\prime (\mathbb {R}^d)$, one can restrict the domain in Eq. (2.17) to an operator $\textrm{Op}_L(a):\mathcal {E}^\prime (\Omega )\rightarrow \mathcal {D}^\prime (\Omega )$ or $\textrm{Op}_L(a):C^\infty _0(\Omega )\rightarrow C^\infty (\Omega )$, where with a slight abuse of notation we keep on using the same symbol $\textrm{Op}_L(a)$. In full analogy with Definition 12, we indicate the ensuing collection of pseudodifferential operators by $\Psi ^m(\Omega )$. The following theorem is the direct adaptation to our setting and notations of [18, Thm. 18.1.17].

Theorem 23

Let $\Omega ,\Omega ^\prime \subset \mathbb {R}^d$ be open subsets, $f\in \textrm{Diff}(\Omega ;\Omega ^\prime )$ and let $A \in \Psi ^m(\Omega ^\prime )$. Then

$$\begin{aligned} A_f :C_0^\infty (\Omega ) \rightarrow C^\infty (\Omega ), \quad u\mapsto A_f u:= A((f^{-1})^*u) \circ f \end{aligned}$$

(2.18)

is a pseudodifferential operator of order m. Moreover,

$$\begin{aligned} \sigma _m(A_f)(x,\xi ) = \sigma _m(A)(f(x), ({}^t df(x))^{-1} \xi ), \end{aligned}$$

(2.19)

where $\sigma _m(A_f)$ and $\sigma _m(A)$ are the principal symbols of $A_f$ and A respectively while df stands for the differential map associated with f.

3 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—[20, Ch. 8], while the second, outlined in Sect. 3.1, relies on pseudodifferential operators as introduced in Sect. 3. Observe that, in the following, we rely heavily on Proposition 9 as well as on Definition 5.

Definition 24

Let $u \in \mathcal {D}^\prime (\mathbb {R}^d)$ and $\alpha \in \mathbb {R}$. We say that $(x_0,\xi _0)\in \mathbb {R}^d \times (\mathbb {R}^d{\setminus } \{0\})$ does not lie in the $B_{\infty ,\infty }^{\alpha }$-wavefront set, denoting $(x_0,\xi _0) \not \in \textrm{WF}^\alpha (u)$, if there exist $\phi \in \mathcal {D}(\mathbb {R}^d)$ with $\phi (x)\ne 0$ as well as an open conic neighborhood $\Gamma$ of $\xi$ in $\mathbb {R}^d\setminus \{0\}$ such that for any compact set $K \subset \mathbb {R}^d$

$$\begin{aligned} \bigg |\int _{\Gamma } \widehat{\phi u}(\xi ) \check{\underline{\kappa }}(\xi ) e^{i x \cdot \xi } d\xi \bigg |\lesssim&1\,, \end{aligned}$$

(3.1)

$$\begin{aligned} \bigg |\int _{\Gamma } \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi } d\xi \bigg |\lesssim&\lambda ^\alpha \,, \end{aligned}$$

(3.2)

for any $\kappa \in \mathscr {B}_{\lfloor \alpha \rfloor }$, $\underline{\kappa }\in \mathcal {D}(B(0,1))$ with $\check{\underline{\kappa }}(0)\ne 0$, $\lambda \in (0,1]$ and $x \in K$.

Remark 25

Observe that, on account of Proposition 9 and of Remark 10, whenever $\alpha <0$ in Definition 24 it suffices to check that Eq. (3.2) holds true.

We are now in a position to prove some basic properties of the Besov wavefront set which are a direct consequence of its definition.

Proposition 26

Let $u\in \mathcal {D}^\prime (\mathbb {R}^d)$. Then

$$\begin{aligned} u\in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d) \iff \textrm{WF}^\alpha (u) = \emptyset \,. \end{aligned}$$

Proof

The implication

$$\begin{aligned} u \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d) \Longrightarrow \textrm{WF}^\alpha (u) = \emptyset \,, \end{aligned}$$

follows immediately combining Definition 3 and Proposition 9 with Definition 24. Conversely, if $\textrm{WF}^\alpha (u) = \emptyset$, then once more Definition 24 entails that, for any $\phi \in \mathcal {D}(\mathbb {R}^d)$, it holds

$$\begin{aligned} \bigg |\int _{\mathbb {R}^d} \widehat{\phi u}(\eta ) e^{iy\cdot \eta } \check{\kappa }(\lambda \eta ) d\eta \bigg |\lesssim \lambda ^\alpha \,, \qquad \bigg |\int _{\mathbb {R}^d} \widehat{\phi u}(\eta ) e^{iy\cdot \eta } \check{\underline{\kappa }}(\eta ) d\eta \bigg |\lesssim 1\,. \end{aligned}$$

From Proposition 9 it descends that $\phi u\in B_{\infty ,\infty }^\alpha (\mathbb {R}^d)$ for any $\phi \in \mathcal {D}(\mathbb {R}^d)$. This proves the sought statement. $\square$

Proposition 27

Let $u,v \in \mathcal {D}^\prime (\mathbb {R}^d)$. Then

$$\begin{aligned} \textrm{WF}^\alpha (u+v) \subset \textrm{WF}^\alpha (u) \cup \textrm{WF}^\alpha (v). \end{aligned}$$

Proof

Assume $(x_0,\xi _0) \in \textrm{WF}^\alpha (u+v)$. Then, for any test function $\phi \in \mathcal {D}(\mathbb {R}^d)$, open conic neighborhood $\Gamma$ of $\xi _0$, there exists a compact set $K\subset \mathbb {R}^d$ such that, for any $N \in \mathbb {N}$, it holds true

$$\begin{aligned} \bigg |\int _{\Gamma } \widehat{\phi (u+v)}(\xi ) \check{\kappa }(\overline{\lambda }\xi ) e^{i \overline{x} \cdot \xi } d\xi \bigg |> N {\overline{\lambda }}^\alpha , \end{aligned}$$

for some $\overline{x} \in K$ and $\overline{\lambda } \in (0,1]$. Applying the triangle inequality, it descends

$$\begin{aligned} N {\overline{\lambda }}^\alpha < \bigg |\int _{\Gamma } \widehat{\phi u}(\xi ) \check{\kappa }(\overline{\lambda }\xi ) e^{i \overline{x} \cdot \xi } d\xi \bigg |+ \bigg |\int _{\Gamma } \widehat{\phi v}(\xi ) \check{\kappa }(\overline{\lambda }\xi ) e^{i \overline{x} \cdot \xi } d\xi \bigg |, \end{aligned}$$

which entails that $(x_0,\xi _0) \in \textrm{WF}^\alpha (u)\cup \textrm{WF}^\alpha (v)$. $\square$

Corollary 28

Let $u \in \mathcal {D}^\prime (\mathbb {R}^d)$. If $\alpha _1 \le \alpha _2$, then

$$\begin{aligned} \textrm{WF}^{\alpha _1}(u) \subseteq \textrm{WF}^{\alpha _2}(u). \end{aligned}$$

(3.3)

Proof

The inclusion in Eq. (3.3) follows immediately from Definition 24, particularly Eq. (3.2). $\square$

Remark 29

Observe that, on account of the inclusion $C^\infty (\mathbb {R}^d)\subset B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$ for all $\alpha \in \mathbb {R}$, Proposition 26 entail that, for every $f\in C^\infty (\mathbb {R}^d)$

$$\begin{aligned} \textrm{WF}^\alpha (f)=\emptyset .\quad \forall \alpha \in \mathbb {R} \end{aligned}$$

In particular, this result entails that, given any $u\in \mathcal {D}^\prime (\mathbb {R}^d)$, if $x\notin \text {singsupp}(u)$, then $(x,\xi )\notin \textrm{WF}^\alpha (u)$ for all $\alpha \in \mathbb {R}$. Here $\textrm{singsupp}(u)$ refers to the singular support of u, see [20, Def. 2.2.3] for the definition.

In the following, we give some explicit examples of Besov wavefront sets. Observe that the results of Remark 29 are always implicitly taken into account.

Example 30

Let $u = \delta \in \mathcal {D}^\prime (\mathbb {R}^d)$ be the Dirac delta centered at the origin. Recalling that for any $\phi \in \mathcal {D}(\mathbb {R}^d)$ $\phi \delta =\phi (0)\delta$, Eq. (3.1) translates to

$$\begin{aligned} \left| \int _\Gamma \underline{\check{\kappa }}(\eta )e^{iy\cdot \eta }d\eta \right| \le \int _\Gamma \left| \underline{\check{\kappa }}(\eta )\right| d\eta \lesssim 1, \end{aligned}$$

since $\underline{\check{\kappa }}\in \mathcal {S}(\mathbb {R}^d)$. Here we have neglected $\phi (0)$ since it plays no rôle. Focusing instead on Eq. (3.2), for any choice of $\phi \in \mathcal {D}(\mathbb {R}^d)$ with $\phi (0)\ne 0$, it descends, neglecting once more $\phi (0)$, that

$$\begin{aligned} \left| \int _\Gamma \check{\kappa }(\eta )e^{iy\cdot \eta }d\eta \right| \le \int _\Gamma \left| \check{\kappa }(\lambda \eta )\right| d\eta \lesssim \lambda ^{-d}, \end{aligned}$$

where the last inequality descends from the change of variable $\eta \mapsto \eta ^\prime :=\lambda \eta$. While this estimate entails that $\textrm{WF}^\alpha (\delta )=\emptyset$ if $\alpha \le -d$, in order to obtain a sharp estimate observe that we can set $y=0$ in Eq. (3.2) since it lies in $\textrm{supp}(\phi )$ for any admissible $\phi$, being $\phi (0)\ne 0$. Hence it descends

$$\begin{aligned} \left| \int _\Gamma \check{\kappa }(\lambda \eta )d\eta \right| =\lambda ^{-d}\left| \int _\Gamma \check{\kappa }(\eta ^\prime )d\eta ^\prime \right| =C_{\check{\kappa }}\lambda ^{-d}, \end{aligned}$$

where $\eta ^\prime :=\lambda \eta$ and where we used implicitly both that $\Gamma$ is a cone and that $\check{\kappa }\in \mathcal {S}(\mathbb {R}^d)$. At this stage, comparing with Definition 24, we can conclude that

$$\begin{aligned} \textrm{WF}^\alpha (\delta )= \left\{ \begin{array}{ll} \emptyset &{} \alpha \le -d, \\ (0,\xi ):\xi \in \mathbb {R}^d \setminus \{0\} &{} \alpha > -d. \end{array}\right. \end{aligned}$$

Example 31

Let $u = \partial _j \delta \in \mathcal {D}^\prime (\mathbb {R}^d)$ be a derivative of the Dirac delta centered at the origin, i.e. $\partial _j=\frac{\partial }{\partial x_j}$, $x_j$ being an Euclidean coordinate on $\mathbb {R}^d$. Following Definition 24 and using the identity $\phi \partial _j\delta =\phi (0)\partial _j\delta -(\partial _j\phi )(0)\delta$ for any $\phi \in \mathcal {D}(\mathbb {R}^d)$, Eq. (3.1) translates to

$$\begin{aligned}{} & {} \left| (\partial _j\phi )(0)\int _\Gamma \eta _j\underline{\check{\kappa }}(\eta )e^{iy\cdot \eta }d\eta -\phi (0)\int _\Gamma \underline{\check{\kappa }}(\eta )e^{iy\cdot \eta }d\eta \right| \\{} & {} \qquad \le \int \limits _\Gamma \left| (\partial _j\phi )(0)\eta _j-\phi (0))\underline{\check{\kappa }}(\eta )\right| d\eta \lesssim 1, \end{aligned}$$

where, similarly to Example 30, we exploited that $\underline{\check{\kappa }}\in \mathcal {S}(\mathbb {R}^d)$. Focusing on Eq. (3.2), we can repeat the same procedure as in Example 30. For the sake of conciseness we focus directly only on $y=0$ since it lies in $\text {supp}(\phi )$, for any $\phi \in \mathcal {D}(\mathbb {R}^d)$ with $\phi (0)\ne 0$. In addition we can consider only the contribution due to $\phi (0)\partial _j\delta$ which yields, omitting $\phi (0)$ for simplicity of the notation,

$$\begin{aligned} \left| \int \limits _\Gamma \eta _j\check{\kappa }(\lambda \eta )d\eta \right| =\lambda ^{-d-1}\left| \int \limits _\Gamma \eta ^\prime _j\check{\kappa }(\eta ^\prime )d\eta ^\prime \right| =\widetilde{C}_{\check{\kappa }}\lambda ^{-d-1}, \end{aligned}$$

where $\eta ^\prime :=\lambda \eta$ and where we used implicitly both that $\Gamma$ is a cone and that $\check{k}\in \mathcal {S}(\mathbb {R}^d)$. Adding to this equality the outcome of Example 30, it descends

$$\begin{aligned} \textrm{WF}^\alpha (\partial _j\delta )= \left\{ \begin{array}{ll} \emptyset &{} \alpha \le -d-1, \\ (0,\xi ):\xi \in \mathbb {R}^d \setminus \{0\} &{} \alpha > -d-1. \end{array}\right. \end{aligned}$$

Example 32

Let $u \in \mathcal {E}^\prime (\mathbb {R}^d)$. Observe that there exists $C > 0$ such that

$$\begin{aligned} |\hat{u}(\xi ) |\le C \langle \xi \rangle ^M \end{aligned}$$

(3.4)

where M is the order of u and $\langle \xi \rangle :=(1+|\xi |^2)^{\frac{1}{2}}$, see [20, Lem. 8.1.1]. Fix $\Gamma$ an open conic neighborhood of $\xi \in \mathbb {R}^d {\setminus }\{0\}$. Given $\kappa$ as per Definition 5, $\lambda \in (0,1)$ and $y \in {\text {supp}}(u)$, it holds

$$\begin{aligned} \bigg |\int _\Gamma \hat{u}(\eta ) e^{iy\cdot \eta } \check{\kappa }(\lambda \eta ) d\eta \bigg |{} & {} \le \int _\Gamma |\hat{u}(\eta ) ||\check{\kappa }(\lambda \eta ) |d\eta \\{} & {} \le C \int _\Gamma \langle \eta \rangle ^M |\check{\kappa }(\lambda \eta ) |d\eta \approx \lambda ^{-M-d} \int _\Gamma |\eta |^{M} |\check{\kappa }(\eta ) |d\eta \lesssim \lambda ^{-M-d}, \end{aligned}$$

where, with reference to Eq. (3.1) and (3.2), we have implicitly chosen $\phi \in \mathcal {D}(\mathbb {R}^d)$ such that $\phi =1$ on $\text {supp}(u)$. As a result, we get $\textrm{WF}^\alpha (u)=\emptyset$ if $\alpha \le -d-M$.

Example 33

Let $u :\mathbb {R}^2 \rightarrow \mathbb {R}$ such that $u(x_1,x_2)=(x_1^2+x_2^2)^{\frac{1}{4}}$. We recall that $\hat{u}(\xi _1,\xi _2) = (\xi _1^2+\xi _2^2)^{-\frac{5}{4}}$, which should be interpreted as the integral kernel of an element lying in $\mathcal {S}^\prime (\mathbb {R}^2$). Since $\textrm{singsupp}(u)=\{(0,0)\}$, we consider $(0,0,\xi _1,\xi _2)$ such that $(\xi _1,\xi _2)\ne (0,0)$. Given $\phi \in \mathcal {D}(\mathbb {R}^2)$ with $\phi (0,0) = 1$ and an open conic neighborhood $\Gamma$ of $(\xi _1,\xi _2)$, we can still use the rationale followed in Example (30) studying Eq. (3.2) with $y=(0,0)$. It reads

$$\begin{aligned}&\bigg |\int _{\Gamma } \widehat{\phi u}(\eta _1,\eta _2) \check{\kappa }(\lambda \eta _1,\lambda \eta _2) d\eta _1 d\eta _2 \bigg |= \bigg |\int _{\Gamma } (\eta _1^2+\eta _2^2)^{-\frac{5}{4}} \check{\kappa }(\lambda \eta _1,\lambda \eta _2) d\eta _1 d\eta _2 \bigg |\\&\overset{(\lambda \eta _1,\lambda \eta _2) \mapsto (\eta _1,\eta _2)}{=}\ \int _\Gamma \lambda ^{\frac{1}{2}}(\eta _1^2+\eta _2^2)^{-\frac{5}{4}} |\check{\kappa }(\eta _1,\eta _2) |d\eta _1d\eta _2 = C_{\check{k}} \lambda ^{\frac{1}{2}}, \end{aligned}$$

where no singularity at the origin occurs since $\kappa$ is chosen in agreement with Definition 5. This entails that

$$\begin{aligned} \left\{ \begin{array}{ll} \textrm{WF}^\alpha (u)=\emptyset &{} \alpha \le \frac{1}{2}\\ \textrm{WF}^\alpha (u)=\{(0,0,\xi _1,\xi _2)\;|\;(\xi _1,\xi _2)\ne (0,0)\} &{} \alpha >\frac{1}{2} \end{array} \right. \end{aligned}$$

(3.5)

3.1 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 [12] for the smooth wavefront set and it is especially useful in discussing operations between distributions with a prescribed Besov wavefront set, see Sect. 4. In the following, we shall make use of the notions introduced in Definitions 12 and 14.

Proposition 34

Let $\alpha \in \mathbb {R}$ and $u \in \mathcal {D}^\prime (\mathbb {R}^d)$. Then

$$\begin{aligned} \textrm{WF}^\alpha (u) = \bigcap _{\begin{array}{c} A \in \Psi ^0(\mathbb {R}^d),\\ Au \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d) \end{array}} \textrm{Char}(A), \end{aligned}$$

(3.6)

where the intersection is taken only over properly supported pseudodifferential operators.

Proof

Suppose that $(x_0,\xi _0) \not \in \textrm{WF}^\alpha (u)$. By Definition 24, there exist $\phi \in \mathcal {D}(\mathbb {R}^d)$ with $\phi (x_0)\ne 0$ and $\Gamma$, a conic neighbourhood of $\xi _0$, such that for any compact set $K \subset \mathbb {R}^d$

$$\begin{aligned} \bigg |\int _{\Gamma } \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{ix\cdot \xi } d\xi \bigg |\lesssim \lambda ^\alpha \quad \forall x \in K, \forall \lambda \in (0,1], \end{aligned}$$

where $\kappa \in \mathscr {B}_{\alpha }$. Calling $\mathbb {I}_\Gamma (\xi )$ the characteristic function on $\Gamma$, it descends that

$$\begin{aligned} \mathcal {F}^{-1}\bigg [ \mathbb {I}_\Gamma (\xi ) \widehat{\phi u} \bigg ] \in B^{\alpha , \textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d). \end{aligned}$$

(3.7)

Set $\chi \in C^\infty (\mathbb {R}^n)$ to be such that $\chi (\xi )=0$ if $|\xi |\le a$ and $\chi (\xi ) = 1$ if $|\xi |\ge 2a$ where a is a non vanishing constant chosen so that $\chi (\xi _0)\ne 0$. In addition choose $\psi \in C^{\infty }(\mathbb {S}^{n-1})$ such that $\textrm{supp}(\psi ) \subset B_{\varepsilon } (\xi _0/|\xi _0 |)\subset \Gamma$, $\varepsilon >0$ and $\psi (\xi _0/|\xi _0|)\ne 0$. Consequently we can introduce $A:= \textrm{Op}(a)\in \Psi ^0(\mathbb {R}^d)$, where

$$\begin{aligned} a(x,y,\xi ) = \phi (x) \psi \bigg (\frac{\xi }{|\xi |}\bigg ) \chi (\xi ) \phi (y) \in S^0(\mathbb {R}^d \times \mathbb {R}^d;\mathbb {R}^d). \end{aligned}$$

(3.8)

Observe that, following standard arguments, A is by construction properly supported and elliptic at $(x_0,\xi _0)$. To conclude it suffices to notice that, combining Eq. (3.7) and Theorem 21, we can conclude that $Au\in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$.

Conversely, let $(x_0,\xi _0) \not \in \bigcap _{\begin{array}{c} A \in \Psi ^0(\mathbb {R}^d)\\ Au \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d) \end{array}} \textrm{Char}(A)$. Hence, taking into account Definition 18, there exists $B \in \Psi ^0$, elliptic at $(x_0,\xi _0)$, such that $Bu \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$. Consider once more $\phi$, $\psi$ and $\chi$ as in the previous part of the proof, so that

$$\begin{aligned} WF^\prime (A) \subset \textrm{Ell}(B) \end{aligned}$$

where $A:=Op_R(\psi (\xi /|\xi |) \chi (\xi ) \phi (y))$ and where $WF^\prime$ is as per Definition 16. We claim that $Au \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$. In view of Proposition 20, there exists a microlocal parametrix $Q \in \Psi ^0(\mathbb {R}^d)$ of B such that $QB = I-R$ with $R \in \Psi ^{-1}(\mathbb {R}^d)$ and $WF^\prime (R) \cap WF^\prime (A) = \emptyset$. Thus,

$$\begin{aligned} Au = A(QB+R) u = (AQ)(Bu) + ARu, \end{aligned}$$

where $ARu \in C^\infty (\mathbb {R}^d)$. Given $\rho \in \mathcal {D}(\mathbb {R}^d)$ such that $\rho = 1$ on $\textrm{supp}(\phi )$, it descends

$$\begin{aligned} (AQ)(Bu) = (AQ)(\rho Bu) + (AQ)((1-\rho )Bu). \end{aligned}$$

Since $1-\rho = 0$ on $\textrm{supp}(\phi )$, then $(AQ)((1-\rho )Bu) = 0$. At the same time $(AQ)(\rho Bu) \in B^\alpha _{\infty ,\infty }(\mathbb {R}^d)$ on account of Theorem 21. This entails that $Au \in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$. Hence, given $\kappa \in \mathscr {B}_\alpha$, see Definition 5, it holds

$$\begin{aligned} \bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} \psi \bigg (\frac{\xi }{|\xi |} \bigg ) \chi (\xi ) \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |\lesssim \lambda ^\alpha , \quad \forall \lambda \in (0,1], \quad \forall x \in K.\nonumber \\ \end{aligned}$$

(3.9)

On account of Remark 19, there exists a symbol $p \in S^0(\mathbb {R}^n;\mathbb {R}^n)$ such that

$$\begin{aligned}{} & {} r(\xi ): = 1-\psi \bigg (\frac{\xi }{|\xi |} \bigg ) \chi (\xi ) p(\xi ) \in S^{-1} \end{aligned}$$

for any $\xi \in \textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )$. It descends

$$\begin{aligned}&\bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |\nonumber \\&= \bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} \bigg (\psi \bigg (\frac{\xi }{|\xi |} \bigg ) \chi (\xi ) p(\xi ) + r(\xi ) \bigg ) \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |\nonumber \\&\le \bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} \psi \bigg (\frac{\xi }{|\xi |} \bigg ) \chi (\xi ) p(\xi ) \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |\nonumber \\&\qquad + \bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} r(\xi ) \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |\nonumber \\&= \underbrace{\bigg |\bigg \langle p(D)\psi \bigg (\frac{D}{|D |}\bigg ) \chi (D)(\phi u), \kappa ^\lambda _x \bigg \rangle \bigg |}_{|A |} + \bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} r(\xi ) \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |, \end{aligned}$$

(3.10)

for any $x \in K$ and $\lambda \in (0,1]$. On the one hand, as a result of Theorem 21 and Eq. (3.9), it holds that

$$\begin{aligned} |A |\lesssim \lambda ^\alpha . \end{aligned}$$

On the other hand,

$$\begin{aligned} |B |&\le \bigg |\bigg \langle r(D)p(D)\psi \bigg (\frac{D}{|D |}\bigg ) \chi (D)(\phi u), \kappa ^\lambda _x \bigg \rangle \bigg |\nonumber \\&\quad + \bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} r^2(\xi ) \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |\nonumber \\&\lesssim \lambda ^{\alpha + 1} + \bigg |\int _{\textrm{Ell}\big ( \psi (D/|D|) \chi (D)\big )} r^2(\xi ) \widehat{\phi u}(\xi ) \check{\kappa }(\lambda \xi ) e^{i x \cdot \xi }d\xi \bigg |, \end{aligned}$$

(3.11)

where we applied once more Theorem 21 with $r(D)\in \Psi ^{-1}(\mathbb {R}^d)$ and $p(D)\psi \bigg (\frac{D}{|D |}\bigg ) \chi (D)(\phi u) \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$. This concludes the proof. $\square$

Remark 35

The content of Proposition 34 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 [16, Cor. 6.18]. For later convenience and to fix the notation, we recall it. Let $v\in \mathcal {D}^\prime (\mathbb {R}^d)$. It holds

$$\begin{aligned} WF(v)=\bigcap _{\begin{array}{c} A \in \Psi ^0(\mathbb {R}^d)\\ Av \in C^\infty (\mathbb {R}^d) \end{array}}\textrm{Char}(A), \end{aligned}$$

where $\textrm{Char}(A)$ 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.

Proposition 36

Let $u \in \mathcal {D}^\prime (\mathbb {R}^d)$. It holds that

$$\begin{aligned} (x,\xi ) \in \textrm{WF}^\alpha (u) \iff (x,\xi ) \in WF(u-v)\quad \forall v \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d), \end{aligned}$$

(3.12)

where WF stands for the (smooth) wavefront set.

Proof

Suppose $(x,\xi ) \in \textrm{WF}^\alpha (u)$. On account of Remark 35, given $v\in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$ we consider $A \in \Psi ^0(\mathbb {R}^d)$ such that $A(u-v) \in C^\infty (\mathbb {R}^d)$. This entails that $Au \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$. Yet, since $(x,\xi ) \in \textrm{WF}^\alpha (u)$, Proposition 34 entails that $(x,\xi )\in \textrm{Char}(A)$.

Conversely, let $(x,\xi ) \not \in \textrm{WF}^\alpha (u)$. By Definition 24, there exist $\phi \in \mathcal {D}(\mathbb {R}^d)$ normalized so that $\phi (x)=1$ and an open conic neighborhood $\Gamma$ of $\xi$ such that Eq. (3.1) is satisfied. Let $v \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$ be such that

$$\begin{aligned} \hat{v}(\eta ) = {\left\{ \begin{array}{ll} \widehat{\phi u}(\eta ) \quad \text {if } \eta \in \Gamma ,\\ 0 \quad \text {otherwise} \end{array}\right. } \end{aligned}$$

(3.13)

Then ${\hat{\theta }} = \widehat{\phi u} - \hat{v}$ vanishes on $\Gamma$ and therefore $(x,\xi ) \not \in WF(\theta )$. Consider $\chi \in \mathcal {D}(\mathbb {R}^d)$ such that $\chi \phi = 1$ in a neighbourhood of $x\in \mathbb {R}^d$. Then $\chi v \in B_{\infty ,\infty }^\alpha (\mathbb {R}^d)$ and $(x,\xi ) \not \in WF(\chi \theta )$. After observing that $u-\chi v = (1-\chi \phi ) u + \chi \theta$, we conclude $(x,\xi ) \not \in WF(u-\chi v)$ exploiting that $(1-\chi \phi )u$ vanishes in a neighbourhood of x. Since $\chi =1$ at x, we can conclude that $(x,\xi )\notin WF(u-v)$. $\square$

We can now establish a relation between the Besov wavefront set and the smooth counterpart, see Remark 35. 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 [16, Prop. 6.32].

Corollary 37

Let $u\in \mathcal {D}^\prime (\mathbb {R}^d)$. It holds that

$$\begin{aligned} WF(u)=\overline{\bigcup _{\alpha \in \mathbb {R}}\textrm{WF}^\alpha (u)}. \end{aligned}$$

(3.14)

Proof

Assume $(x,\xi )\in \textrm{WF}^\alpha (u)$ for any $\alpha \in \mathbb {R}$. Using Proposition 36, we can choose $v\in C^\infty _0(\mathbb {R}^d)\subset B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$ concluding that $(x,\xi )\in WF(u-v)=WF(u)$. Hence $\bigcup _{\alpha \in \mathbb {R}}\textrm{WF}^\alpha (u)\subseteq WF(u)$. Taking the closure and recalling that WF(u) is per construction a closed set, it descends $\overline{\bigcup _{\alpha \in \mathbb {R}}\textrm{WF}^\alpha (u)}\subseteq WF(u)$.

To prove the other inclusion, assume $(x,\xi )\notin WF^\alpha (u)$ for all $\alpha \in \mathbb {R}$. Hence there must exist a conic, open set $\Gamma \subseteq \mathbb {R}^d\times \mathbb {R}^d\setminus \{0\}$ such that $(x,\xi )\in \Gamma$ and $\Gamma \cap \textrm{WF}^\alpha (u)=\emptyset$ for all $\alpha \in \mathbb {R}$. We can thus choose $A\in \Psi ^0(\mathbb {R}^d)$ to be properly supported, elliptic at $(x,\xi )$ and such that $WF^\prime (A)\subset \Gamma$ and $Au\in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$ for all $\alpha \in \mathbb {R}$. This entails that $Au\in C^\infty (\mathbb {R}^d)$. It descends that, since $(x,\xi )\notin \textrm{Char}(A)$, then $(x,\xi )\notin WF(u)$. $\square$

4 Structural properties

In this section we discuss the main structural properties of distributions with a prescribed Besov wavefront set as per Definition 24 and Proposition 34, including notable operations.

Transformation Properties under Pullback—We start by investigating the interplay between Definition 24 and the pull-back of a distribution. In the following we enjoy the analysis outlined in Sect. 2.1.1.

Remark 38

In Definition 24 as well as in Proposition 34 we have always assumed implicitly that the underlying distribution is globally defined, i.e. $u\in \mathcal {D}^\prime (\mathbb {R}^d)$. Yet, mutatis mutandis, the whole construction and the results obtained so far can be slavishly adapted to distributions $v\in \mathcal {D}^\prime (\Omega )$, $\Omega \subseteq \mathbb {R}^d$.

Theorem 39

(Pull-back - I) Let $\Omega \subseteq \mathbb {R}^d$, $\Omega ^\prime \subseteq \mathbb {R}^m$ be open sets and let $f\in C^\infty (\Omega ;\Omega ^\prime )$ be an embedding. Moreover let

$$\begin{aligned} N_f:=\{(f(x),\xi ) \in \Omega ^\prime \times \mathbb {R}^m: {}^t df(x) \xi = 0\}, \end{aligned}$$

(4.1)

be the set of normals of f. For any $u \in \mathcal {D}^\prime (\Omega ^\prime )$ such that there exists $\alpha > 0$ so that

$$\begin{aligned} N_f \cap \textrm{WF}^\alpha (u) = \emptyset , \end{aligned}$$

(4.2)

there exists $f^*u \in \mathcal {D}^\prime (\Omega )$. In addition, provided $f^*u$ exists as a distribution, for any $\alpha '\in \mathbb {R}$

$$\begin{aligned} \textrm{WF}^{\alpha '}(f^*u) \subseteq f^*\textrm{WF}^{\alpha '}(u), \end{aligned}$$

(4.3)

for every $u \in \mathcal {D}^\prime (\Omega ^\prime )$ abiding to Eq. (4.2), where

$$\begin{aligned} f^*\textrm{WF}^{\alpha '}(u):= \{(x,{}^t df(x)\eta ):(f(x),\eta ) \in \textrm{WF}^{\alpha '}(u)\}. \end{aligned}$$

(4.4)

Proof

As a consequence of Proposition 36, Eq. (4.2) is equivalent to

$$\begin{aligned} N_f \cap WF(u-v) = \emptyset , \quad \forall v \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\Omega ^\prime ). \end{aligned}$$

Then, there exists the pullback $f^*(u-v) \in \mathcal {D}^\prime (\Omega )$. Taking into account that $B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\Omega ^\prime ) \subset C^0(\Omega ^\prime )$ for $\alpha > 0$, we have that $f^*v = v \circ f$. Thus,

$$\begin{aligned} f^*u = f^*(u-v) + f^*v \end{aligned}$$

identifies an element lying in $\mathcal {D}^\prime (\Omega )$. Focusing on Eq. (4.3), let $(x,{}^t df(x)\eta ) \not \in f^*\textrm{WF}^\alpha (u)$. It implies $(f(x),\eta ) \not \in \textrm{WF}^\alpha (u)$. By Proposition 34, there exists $A \in \Psi ^0(\Omega ^\prime )$, elliptic in $(f(x),\eta )$, such that $Au \in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\Omega ^\prime )$. Bearing in mind that f is a diffeomorphism on $f[\Omega ]$,

$$\begin{aligned} A_f(f^*u) = (Au) \circ f, \end{aligned}$$

identifies a pseudodifferential operator of order 0 per Theorem 23. Since $(Au) \circ f \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$, Theorem 23 entails

$$\begin{aligned} \sigma _0(A_f)(x,{}^t df(x)\eta ) = \sigma _0(A)(f(x), \eta ) \ne 0. \end{aligned}$$

This proves $(x,{}^t df(x)\eta ) \not \in \textrm{WF}^\alpha (f^*u)$. $\square$

To conclude this first part of the section, we shall prove that Besov wavefront set is invariant under the action of diffeomorphisms.

Theorem 40

(Pull-back - II) Let $\Omega ,\Omega ^\prime \subseteq \mathbb {R}^d$ be two open subsets and let $f :\Omega \rightarrow \Omega ^\prime$ be a diffeomorphism. Then, given $u \in \mathcal {D}^\prime (\Omega ^\prime )$, for any $\alpha \in \mathbb {R}$, it holds

$$\begin{aligned} \textrm{WF}^\alpha (f^*u) = f^*\textrm{WF}^\alpha (u). \end{aligned}$$

Proof

We prove the inclusion $\textrm{WF}^\alpha (f^*u) \subseteq f^*\textrm{WF}^\alpha (u)$. Let $(x,\xi ) \not \in f^*\textrm{WF}^\alpha (u)$, i.e., $(f(x),({}^t df(x))^{-1} \xi )\not \in \textrm{WF}^\alpha (u)$. Thus, there exists $A \in \Psi ^0(\Omega ^\prime )$, elliptic at $(f(x),({}^t df(x))^{-1} \xi )$, such that $Au \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\Omega ^\prime )$. If one introduces $A_f \in \Psi ^0(\Omega )$ such that

$$\begin{aligned} A_f(f^*u) = f^*(Au), \end{aligned}$$

Equation (2.19) entails that

$$\begin{aligned} \sigma _0(A_f)(x,\xi ) = \sigma _0(A)(f(x), ({}^t df(x))^{-1} \xi ) \ne 0, \end{aligned}$$

that is $A_f$ is elliptic at $(f(x), ({}^t df(x))^{-1} \xi )$. To conclude, we need to prove that $A_f(f^*u) \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\Omega )$. For any but fixed $\phi \in C_0^\infty (\Omega )$, it holds

$$\begin{aligned} |\langle \phi A_f(f^*u), \kappa ^\lambda _z \rangle |&= |\langle f^*(Au), \phi \kappa ^\lambda _z \rangle |= |\langle Au, (f_*\phi ) (f_*\kappa )^\lambda _{f(z)} |\det (df^{-1}) |\rangle |\\&\lesssim |\langle Au, (f_*\phi ) (f_*\kappa )^\lambda _{f(z)} \rangle |\lesssim \lambda ^\alpha , \end{aligned}$$

for any $z \in \Omega$, $\lambda \in (0,1]$ and $\kappa \in \mathscr {B}_\alpha$. An analogous estimate yields

$$\begin{aligned} |\langle \phi A_f(f^*u), \underline{\kappa }_z \rangle |\lesssim 1, \end{aligned}$$

for any $z \in \Omega$ and $\underline{\kappa } \in \mathcal {D}(B(0,1))$ such that $\check{\underline{\kappa }}(0)\ne 0$. This proves that $(x,\xi ) \not \in \textrm{WF}^\alpha (f^*u)$.

Conversely, let $(x,\xi ) \not \in \textrm{WF}^\alpha (f^*u)$. Then, there exists $\tilde{A} \in \Psi ^0(\Omega )$, elliptic in $(x,\xi )$, such that $\tilde{A}(f^*u) \in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\Omega )$. Let $A \in \Psi ^0(\Omega ^\prime )$ be such that

$$\begin{aligned} Au = (f^{-1})^*(\tilde{A}(f^*u)). \end{aligned}$$

Still on account of Theorem 23, it holds that

$$\begin{aligned} \sigma _0(A)(f(x),({}^t df(x))^{-1}\xi ) = \sigma _0(\tilde{A})(x,\xi ) \ne 0. \end{aligned}$$

As a consequence, A is elliptic at $(f(x),({}^t df(x))^{-1}\xi )$. Reasoning as in the first part of the proof, it turns out that $Au \in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\Omega ^\prime )$. This entails that $(x,\xi ) \not \in f^*\textrm{WF}^\alpha (u)$. $\square$

Remark 41

Theorem 40 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 $\textrm{WF}^\alpha (u)$, $u\in \mathcal {D}^\prime (\mathbb {R}^d)$, as an element of $\mathbb {R}^d\times \mathbb {R}^d\setminus \{0\}$. Yet, from a geometrical viewpoint each element of $\textrm{WF}^\alpha (u)$ should be better read as lying in the cotangent bundle $\textrm{T}^*\mathbb {R}^d\setminus \{0\}$. For the sake of conciseness, we shall not dwell into further details which are left to the reader.

Microlocal Properties of $\Psi$DOs—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 [16, Prop. 6.27], namely, if $A\in \Psi ^m(\mathbb {R}^d)$ and $u\in \mathcal {D}^\prime (\mathbb {R}^d)$, then A is microlocal:

$$\begin{aligned} WF(Au) \subseteq WF^\prime (A) \cap WF(u), \end{aligned}$$

(4.5)

where $WF^\prime$ 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.

Proposition 42

Let $A \in \Psi ^m(\mathbb {R}^d)$, $u \in \mathcal {D}^\prime (\mathbb {R}^d)$ and $\alpha \in \mathbb {R}$. Then

$$\begin{aligned} \textrm{WF}^{\alpha - m}(Au) \subseteq WF^\prime (A) \cap \textrm{WF}^\alpha (u). \end{aligned}$$

(4.6)

Proof

Suppose that $(x_0,\xi _0) \not \in WF^\prime (A)$. As a consequence of Proposition 20 there exists $B \in \Psi ^0(\mathbb {R}^d)$, elliptic at $(x_0,\xi _0)$. In addition, we find B such that $WF^\prime (A) \cap WF^\prime (B) = \emptyset$. Proposition 17 entails that $BA \in \Psi ^{-\infty }(\mathbb {R}^d)$, which implies in turn that $B(Au)\in C^\infty (\mathbb {R}^d)\subseteq B_{\infty ,\infty }^{\alpha -m,\textrm{loc}}(\mathbb {R}^d)$. Proposition 34 yields that $(x_0,\xi _0)\notin \textrm{WF}^{\alpha -m}(Au)$.

Conversely, suppose $(x_0,\xi _0) \not \in \textrm{WF}^\alpha (u)$. Then, still in view of Proposition 34, there exists $\tilde{A} \in \Psi ^0(\mathbb {R}^d)$, elliptic at $(x_0,\xi _0)$, such that $\tilde{A}u \in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$. Take $B \in \Psi ^0(\mathbb {R}^d)$ elliptic at $(x_0,\xi _0)$ with $WF^\prime (B) \subseteq \textrm{Ell}(\tilde{A})$. On account of Proposition 20, there exists a parametrix $Q \in \Psi ^0(\mathbb {R}^d)$ of $\tilde{A}$, that is, $Q\tilde{A} = I-R$ with $R \in \Psi ^0(\mathbb {R}^d)$ and $WF^\prime (R) \cap WF^\prime (B) = \emptyset$. Therefore,

$$\begin{aligned} B(Au) = BA(Q\tilde{A}+R)u = BAQ(\tilde{A}u) + (BAR)u. \end{aligned}$$

Since $BAR \in \Psi ^{-\infty }(\mathbb {R}^d)$, then $(BAR)u \in C^\infty (\mathbb {R}^d)$. At the same time $BAQ(\tilde{A}u) \in B^{\alpha - m,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$ because $\tilde{A}u \in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$ and $BAQ \in \Psi ^{m}(\mathbb {R}^d)$. Yet, since $(x_0,\xi _0)\notin \textrm{Char}(B)$, it descends $(x_0,\xi _0)\notin \textrm{WF}^{\alpha -m}(Au)$. This concludes the proof. $\square$

The second result we present in this section provides a sort of inverse result, with respect to the previous one, which is more relevant from a PDE viewpoint.

Proposition 43

Let $u \in \mathcal {D}^\prime (\mathbb {R}^d)$, $A \in \Psi ^m(\mathbb {R}^d)$ and $m,\alpha \in \mathbb {R}$. Then

$$\begin{aligned} \textrm{WF}^\alpha (u) \subseteq \textrm{Char}(A) \cup \textrm{WF}^{\alpha - m}(A u). \end{aligned}$$

(4.7)

Proof

Let $(x_0,\xi _0) \not \in \textrm{Char}(A) \cup \textrm{WF}^{\alpha - m}(A u)$. Thus there exists $B \in \Psi ^0(\mathbb {R}^d)$, elliptic at $(x_0,\xi _0)$, such that $B(Au) \in B^{\alpha -m,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$. Let K be any properly supported pseudodifferential operator lying in $\Psi ^{-m}(\mathbb {R}^n)$, which can be chosen without loss of generality to be elliptic at $(x_0,\xi _0)$. It descends $(KBA) u \in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\mathbb {R}^d)$. Since $(x_0,\xi _0)\notin \textrm{Char}(KBA)$, it descends that $(x_0,\xi _0)\notin \textrm{WF}^\alpha (u)$. $\square$

Corollary 44

(Elliptic Regularity) Let $u \in \mathcal {D}^\prime (\mathbb {R}^d)$, $m,\alpha \in \mathbb {R}$ and let $A\in \Psi ^m(\mathbb {R}^d)$ be elliptic. Then

$$\begin{aligned} \textrm{WF}^\alpha (u) = \textrm{WF}^{\alpha - m}(A u). \end{aligned}$$

Proof

Since A is an elliptic pseudodifferential operator $\textrm{Char}(A)=\emptyset$. The statement is thus a direct consequence of Propositions 42 and 43. $\square$

Product of Distributions—In the following we 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 [20], we rely on two ingredients. The first has already been discussed in Theorem 39, while the second one concerns the tensor product of two distributions. In particular we wish to establish an estimate on the singular behaviour of $u\otimes v$ for given $u,v\in \mathcal {D}^\prime (\mathbb {R}^d)$. This can be read as a direct adaptation to this context of [21, Prop. B.5] which is based in turn on [19, Lemma 11.6.3]. For this reason we shall omit the proof.

Proposition 45

(Tensor product) Let $\Omega \subseteq \mathbb {R}^d$ and $\Omega ^\prime \subseteq \mathbb {R}^m$ be two open sets. If $u\in \mathcal {D}^\prime (\Omega )$ and $v \in \mathcal {D}^\prime (\Omega ^\prime )$, then the following two inclusions hold true:

$$\begin{aligned} \textrm{WF}^{\alpha +\beta }(u\otimes v)\subseteq \textrm{WF}_0^\alpha (u)\times WF(v)\cup WF(u)\times \textrm{WF}^\beta _0(v), \end{aligned}$$

(4.8)

and, calling $\gamma :=\min \{\alpha ,\beta ,\alpha +\beta \}$,

$$\begin{aligned} \textrm{WF}^\gamma (u\otimes v)\subseteq \textrm{WF}^\alpha (u)\times WF_0(v)\cup WF_0(u)\times \textrm{WF}^\beta (v), \end{aligned}$$

(4.9)

where we adopted the notation $WF_0(u):=WF(u)\cup \left( \text {supp}(u)\times \{0\}\right)$ and similarly $\textrm{WF}^\alpha _0(u):=\textrm{WF}^\alpha (u)\cup \left( \text {supp}(u)\times \{0\}\right)$.

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.

Theorem 46

Let $u,v \in \mathcal {D}^\prime (\Omega )$ where $\Omega \subseteq \mathbb {R}^d$ is any open set. If $\forall (x,\xi )\in \Omega \times \mathbb {R}^d{\setminus }\{0\}$ there exist $\alpha ,\beta \in \mathbb {R}$ with $\alpha + \beta > 0$ such that

$$\begin{aligned} (x,\xi ) \not \in \textrm{WF}^\alpha (u)\cup (-WF^\beta (v)) \end{aligned}$$

then the product $uv\in \mathcal {D}^\prime (\Omega )$ can be defined by

$$\begin{aligned} u v = \Delta ^*(u \otimes v), \end{aligned}$$

where $\Delta :\Omega \rightarrow \Omega \times \Omega$ is the diagonal map. Moreover, provided $u v = \Delta ^*(u \otimes v)$ is well-defined, calling $\gamma :=\min \{\alpha ,\beta ,\alpha +\beta \}$,

$$\begin{aligned} \textrm{WF}^\gamma (uv){} & {} \subset \{(x,\xi +\eta ):(x,\xi ) \in \textrm{WF}^\alpha (u), (x,\eta )\nonumber \\{} & {} \in WF_0(v)\;\text {or}\;(x,\xi )\in WF_0(u), (x,\eta ) \in \textrm{WF}^\beta (v)\}. \end{aligned}$$

(4.10)

Proof

Observe that, per hypothesis, there exist $\alpha ,\beta \in \mathbb {R}$ with $\alpha +\beta > 0$ such that

$$\begin{aligned} \textrm{WF}^{\alpha +\beta }(u\otimes v) \cap N_{\Delta } = \emptyset , \end{aligned}$$

where $N_{\Delta }=\{(x,x,\xi ,-\xi )\}$ is the set of normal directions of the diagonal map as defined in Eq. (4.1). Hence, on account of Theorem 39 combined with Proposition 45, Eq. (4.9) in particular, there exists $\Delta ^*(u\otimes v) \in \mathcal {D}^\prime (\mathbb {R}^d)$ and

$$\begin{aligned} \textrm{WF}^\gamma (\Delta ^*(u\otimes v)){} & {} \subset \Delta ^*\textrm{WF}^\gamma (u\otimes v) \\{} & {} = \{(x,\xi +\eta ):(x,\xi ) \in \textrm{WF}^\alpha (u), (x,\eta ) \in WF_0(v)\;\text {or}\;(x,\xi )\\{} & {} \in WF_0(u), (x,\eta ) \in \textrm{WF}^\beta (v)\}. \end{aligned}$$

This concludes the proof. $\square$

Remark 47

Observe that, if we consider $u\in B^{\alpha ,\text {loc}}_{\infty \infty }(\mathbb {R}^d)$ and $v\in B^{\beta ,\text {loc}}_{\infty \infty }(\mathbb {R}^d)$ with $\alpha +\beta >0$, it descends that $\textrm{WF}^\alpha (u)=\textrm{WF}^\beta (v)=\emptyset$. Hence, on account of Theorem 46, there exists $uv\in \mathcal {D}^\prime (\mathbb {R}^d)$ and $\textrm{WF}^\gamma (uv)=\emptyset$ with $\gamma =\min \{\alpha ,\beta \}$. This is nothing but the statement of the renown Young’s theorem on the product of two Hölder distributions, see [2, 10].

To conclude this section, we discuss an application of Theorem 46 which is of relevance in many concrete scenarios. More precisely, we consider a continuous $\mathcal {K}:C_0^\infty (\Omega ^\prime ) \rightarrow \mathcal {D}^\prime (\Omega )$ with kernel $K \in \mathcal {D}^\prime (\Omega \times \Omega ^\prime )$. Given $u \in \mathcal {E}^\prime (\Omega ^\prime )$, we investigate the existence of $\mathcal {K}u$ and we seek to establish a bound on the associated Besov wavefront set. As a preliminary step, we need to prove two ancillary results.

Corollary 48

Let $\Omega \times \Omega ^\prime \subseteq \mathbb {R}^d\times \mathbb {R}^m$ and let $v\in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\Omega \times \Omega ^\prime )$, $\alpha \in \mathbb {R}$. Calling $\pi :\Omega \times \Omega ^\prime \rightarrow \Omega$ the projection map on the first factor, it holds that $\pi _*v\in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\Omega )$, $\pi _*$ being the push-forward map.

Proof

Without loss of generality, let us consider $v\in B^{\alpha ,\textrm{loc}}_{\infty ,\infty }(\Omega \times \Omega ^\prime )\cap \mathcal {E}^\prime (\Omega \times \Omega ^\prime )$. We recall that

$$\begin{aligned} (\pi _*v)(\phi ):= v(\phi \otimes 1), \end{aligned}$$

where $\phi \in \mathcal {E}(\Omega )$. Then, for any $\kappa \in \mathscr {B}_{\lfloor \alpha \rfloor }(\Omega )$ as per Definition 5, $x^\prime \in \Omega$, $\lambda \in (0,1]$,

$$\begin{aligned} |(\pi _*v)(\kappa ^\lambda _{x^\prime }) |= |v((\kappa \otimes 1)^\lambda _{(x^\prime ,y^\prime )}) |\lesssim \lambda ^\alpha . \end{aligned}$$

Observe that $\kappa \otimes 1 \in \mathscr {B}_{\lfloor \alpha \rfloor }(\Omega \times \Omega ^\prime )$. At the same time, for any $\underline{\kappa } \in \mathcal {D}(B(0,1))$ with $\check{\underline{\kappa }}(0)\ne 0$, $x^\prime \in \Omega$, $\lambda \in (0,1]$, it holds true

$$\begin{aligned} |(\pi _*v)(\underline{\kappa }_{x^\prime }) |= |v((\underline{\kappa } \otimes 1)_{(x^\prime ,y^\prime )}) |\lesssim \lambda ^\alpha , \end{aligned}$$

which concludes the proof. $\square$

Proposition 49

Let $v\in \mathcal {E}^\prime (\Omega \times \Omega ^\prime )$, where $\Omega \times \Omega ^\prime \subseteq \mathbb {R}^d\times \mathbb {R}^m$ is an open subset. Assume that the projection map on the first factor $\pi :\Omega \times \Omega ^\prime \rightarrow \Omega$ is proper on $\text {supp}(v)$. Then it holds that, for all $\alpha \in \mathbb {R}$

$$\begin{aligned} \textrm{WF}^\alpha (\pi _*v)\subset \{(x,\xi )\in (\Omega \times \mathbb {R}^d\setminus \{0\})\;|\;\exists y\in \text {supp}(v)\;\text {for which}\;(x,y,\xi ,0)\in \textrm{WF}^\alpha (v)\}, \end{aligned}$$

where $\pi _*$ is the push-forward map.

Proof

Since v is compactly supported and since the action of $\pi _*$ is tantamount to a partial evaluation against the constant function $1\in C^\infty (\Omega ^\prime )$, i.e., $\pi _*(v)(\phi )=v(\phi \otimes 1)$ for all $\phi \in \mathcal {E}(\Omega )$, then $\pi _*(v)\in \mathcal {E}^\prime (\Omega )$. On account of Proposition 36, a pair $(x,\xi )\in \textrm{WF}^\alpha (\pi _*v)$ if and only if $(x,\xi )\in WF(\pi _*v - u)$ for all $u\in B^\alpha _{\infty ,\infty }(\Omega )\cap \mathcal {E}^\prime (\Omega )$. Here we can restrict the attention to compactly supported elements lying in $B^\alpha _{\infty ,\infty }(\Omega )$ since $\pi _*v\in \mathcal {E}^\prime (\Omega )$.

In turn, on account of Corollary 48, we can replace u by $\pi _*(\tilde{u})$, where $\tilde{u}\in B^\alpha _{\infty ,\infty }(\Omega \times \Omega ^\prime )\cap \mathcal {E}^\prime (\Omega \times \Omega ^\prime )$. In other words it turns out that

$$\begin{aligned} (x,\xi )\in \textrm{WF}^\alpha (\pi _*v)\Longleftrightarrow (x,\xi )\in WF(\pi _*(v-\tilde{u}))\quad \forall \tilde{u}\in B^\alpha _{\infty ,\infty }(\Omega \times \Omega ^\prime )\cap \mathcal {E}^\prime (\Omega \times \Omega ^\prime ). \end{aligned}$$

Applying [20, Thm. 8.2.12], it descends that

$$\begin{aligned} WF(\pi _*(v-\tilde{u}))\subseteq \{(x,\xi )\;|\;\exists y\in \text {supp}(v-\tilde{u})\;\text {for which}\; (x,y,\xi ,0)\in WF(v-\tilde{u})\}. \end{aligned}$$

Yet, on account of the arbitrariness of $\tilde{u}$ and using Proposition 36, it descends that $y\in \textrm{supp}(v)$ and $(x,y,\xi ,0)\in \textrm{WF}^\alpha (v)$, which is nothing but the sought statement. $\square$

We can prove the main result of this part of our work and we divide it in two statements.

Theorem 50

Let $\Omega \subseteq \mathbb {R}^n, \Omega ^\prime \subseteq \mathbb {R}^m$ be open subsets, $K \in \mathcal {D}^\prime (\Omega \times \Omega ^\prime )$ be the kernel of $\mathcal {K}:C_0^\infty (\Omega ^\prime ) \rightarrow \mathcal {D}^\prime (\Omega )$. Then, for all $\alpha \in \mathbb {R}$ and for all $u\in C^\infty _0(\Omega ^\prime )$,

$$\begin{aligned} \textrm{WF}^\alpha (\mathcal {K}(u))\subset \{(x,\xi )\;|\;\exists y\in \text {supp}(u)\;\text {for which}\;(x,y,\xi ,0)\in \textrm{WF}^\alpha (K)\}. \end{aligned}$$

Proof

Let $\pi :\Omega \times \Omega ^\prime \rightarrow \Omega$ be the projection map on the second factor and assume for the time being that $K\in \mathcal {E}^\prime (\Omega \times \Omega ^\prime )$. It descends that $\mathcal {K}(u)=\pi _*(K\cdot (1\otimes u))$, where $\pi _*$ is the push-forward along $\pi$ while $\cdot$ stands for the product of distributions. Observe that, since $\textrm{WF}^\alpha (1\otimes u)=\emptyset$ for all $\alpha \in \mathbb {R}$ then, the pointwise product is well-defined on account of Theorem 46. The latter also entails that, for all $\alpha \in \mathbb {R}$,

$$\begin{aligned} \textrm{WF}^\alpha (K\cdot (1\otimes u))\subset \{(x,y,\xi ,\eta )\in \textrm{WF}^\alpha (K)\;|\;y\in \text {supp}(u)\}. \end{aligned}$$

At this stage, observing that by localizing the underlying distribution around each point of the wavefront set, we can apply Proposition 49. It descends

$$\begin{aligned} \textrm{WF}^\alpha (\pi _*(K\cdot (1\otimes u)))\subset \{(x,\xi )\;|\;\exists y\in \text {supp}(u)\;\text {for which}\;(x,y,\xi ,0)\in \textrm{WF}^\alpha (K)\}, \end{aligned}$$

which concludes the proof. $\square$

At last we generalize the preceding theorem so to investigate under which circumstances u can be taken to be an element lying $\mathcal {E}^\prime (\Omega ^\prime )$ and with a non empty wavefront set.

Theorem 51

Let $\Omega \subseteq \mathbb {R}^n, \Omega ^\prime \subseteq \mathbb {R}^m$ be open subsets, $K \in \mathcal {D}^\prime (\Omega \times \Omega ^\prime )$ be the kernel of $\mathcal {K}:C_0^\infty (\Omega ^\prime ) \rightarrow \mathcal {D}^\prime (\Omega )$ and $u \in \mathcal {E}^\prime (\Omega ^\prime )$. In addition, for any $\alpha \in \mathbb {R}$, we call

$$\begin{aligned} -\textrm{WF}^{\alpha }_{\Omega ^\prime }(K):=\{(y,\eta )\in \Omega ^\prime \times \mathbb {R}^m\setminus \{0\}:\;\exists x\in \Omega \;|\; (x,y,0,-\eta ) \in \textrm{WF}^\alpha (K)\}.\nonumber \\ \end{aligned}$$

(4.11)

If for any $(y,\eta ) \in \Omega ^\prime \times (\mathbb {R}^m{\setminus }\{0\})$ there exists $\alpha _1,\alpha _2 \in \mathbb {R}$ with $\alpha _1 + \alpha _2 > 0$ such that

$$\begin{aligned} (y,\eta ) \not \in -\textrm{WF}^{\alpha _1}_{\Omega ^\prime }(K) \cup \textrm{WF}^{\alpha _2}(u), \end{aligned}$$

(4.12)

then there exists $\mathcal {K}u \in \mathcal {D}^\prime (\Omega )$. Furthermore, for any $\alpha _1,\alpha _2\in \mathbb {R}$, setting $\alpha =\textrm{min}\{\alpha _1,\alpha _2,\alpha _1+\alpha _2\}$, then

$$\begin{aligned} WF^\alpha (\mathcal {K}(u))\subseteq \{(x,\xi )\in \Omega \times (\mathbb {R}^n\setminus \{0\}):\exists (y,\eta ) \in \Omega ^\prime \times (\mathbb {R}^m\setminus \{0\}) | (x,y,\xi ,\eta ) \in X\cup Y\}, \end{aligned}$$

where

$$\begin{aligned}{} & {} X:=\{(x,y,\xi ,\eta )\in \textrm{WF}^{\alpha _1}(K)\;|\;(y,-\eta )\in WF_0(u)\},\\{} & {} Y=\{(x,y,\xi ,\eta )\in WF(K)\;|\;(y,-\eta )\in \textrm{WF}^{\alpha _2}(u)\}. \end{aligned}$$

Proof

Following the same strategy as in the proof of Theorem 50, we aim at writing $\mathcal {K}(u):=\pi _*(K\cdot (1\otimes u))$ where $\pi _*$ is the push-forward built out of the projection map $\pi :\Omega \times \Omega ^\prime \rightarrow \Omega$. Given $\alpha _2\in \mathbb {R}$, Eq. (4.9) entails that $\textrm{WF}^{\alpha _2}(1\otimes u)\subseteq (\text {supp}(u)\times 0)\times \textrm{WF}^{\alpha _2}(u)$, which combined with Theorem 46 and Eq. (4.12), entails that there exists $K\cdot (1\otimes u)\in \mathcal {D}^\prime (\Omega \times \Omega ^\prime )$. Yet, being u compactly supported we can act with the push-forward along the map $\pi :\Omega \times \Omega ^\prime \rightarrow \Omega$, hence obtaining that $\pi _*(K\cdot (1\otimes u))\in \mathcal {D}^\prime (\Omega )$.

A straightforward adaptation to the case in hand of Proposition 49 entails that, for every $\alpha \in \mathbb {R}$, $\textrm{WF}^\alpha (\pi _*(K\cdot (1\otimes u)))$ is contained within the collection of points $(x,\xi )\in \Omega \times \mathbb {R}^d\setminus \{0\}$ for which there exists $y\in \Omega ^\prime$ such that $(x,y,\xi ,0)\in \textrm{WF}^\alpha (K\cdot (1\otimes u))$.

Set now $\alpha =\textrm{min}\{\alpha _1,\alpha _2,\alpha _1+\alpha _2\}$. Theorem 46, Eq. (4.10) in particular entails that the collection of points $(x,y,\xi ,0)\in \textrm{WF}^\alpha (K\cdot (1\otimes u))$ is contained in those of the form $(x,y,\xi ,0)$ such that one of the two following conditions is met:

• 1. there exists $\eta \in \mathbb {R}^m$ such that $(x,y,\xi ,\eta )\in \textrm{WF}^{\alpha _1}(K)$ and $(y,-\eta )\in WF_0(u)$,

• 2. there exists $\eta \in \mathbb {R}^m$ such that $(x,y,\xi ,\eta )\in WF(K)$ and $(y,-\eta )\in \textrm{WF}^{\alpha _2}(u)$.

To conclude it suffices to recall that, on account of Eq. (3.3) $\textrm{WF}^\alpha (K\cdot (1\otimes u))\subseteq \textrm{WF}^{\alpha _1+\alpha _2}(K\cdot (1\otimes u))$ whenever $\alpha \le \alpha _1+\alpha _2$. $\square$

To conclude, we prove a statement which adapts to the current scenario an important result for the Sobolev wavefront set, see [21, Prop. B.9].

Corollary 52

Let $\Omega \subseteq \mathbb {R}^n, \Omega ^\prime \subseteq \mathbb {R}^m$ be open subsets, $K \in \mathcal {D}^\prime (\Omega \times \Omega ^\prime )$ be the kernel of $\mathcal {K}:C_0^\infty (\Omega ^\prime ) \rightarrow \mathcal {D}^\prime (\Omega )$ and $u \in \mathcal {E}^\prime (\Omega ^\prime )$. Assume in addition that for any $(y,\eta ) \in \Omega ^\prime \times (\mathbb {R}^m{\setminus }\{0\})$ there exists $\alpha _1,\alpha _2 \in \mathbb {R}$ with $\alpha _1 + \alpha _2 > 0$ such that

$$\begin{aligned} (y,\eta ) \not \in -WF^{\alpha _1}_{\Omega ^\prime }(K) \cup WF^{\alpha _2}(u). \end{aligned}$$

(4.13)

If $WF_{\Omega ^\prime }(K) =\emptyset$ and if there exists $\gamma \in \mathbb {R}$ such that $\mathcal {K}(B_{\infty ,\infty }^\alpha (\Omega ^\prime )\cap \mathcal {E}^\prime (\Omega ^\prime )) \subset B_{\infty ,\infty }^{\alpha -\gamma ,\textrm{loc}}(X)$, then for any $\alpha \in \mathbb {R}$,

$$\begin{aligned} \textrm{WF}^{\alpha -\gamma }(\mathcal {K}u) \subseteq WF^\prime (K) \circ \textrm{WF}^\alpha (u) \cup WF_\Omega (K), \end{aligned}$$

(4.14)

where $WF^\prime (K) \circ \textrm{WF}^\alpha (u):= \{(x,\xi )\;|\;\exists (y,\eta ) \in \textrm{WF}^\alpha (u)\;\text {for which}\; (x,y,\xi ,-\eta )\in WF(K) \}$ while $W_\Omega (K):=\{(x,\xi )\in \Omega \times \mathbb {R}^n\;:\;\exists y\in \Omega ^\prime \;|\;(x,y,\xi ,0)\in WF(K)\}$.

Proof

On account of Theorem 51, Eq. (4.13) entails that $\mathcal {K}(u)\in \mathcal {D}^\prime (\Omega )$. Bearing in mind Proposition 36, we can find an open conic neighborhood $\Gamma \subset \textrm{WF}^\alpha (u)$ such that $WF(u-v) \subset \Gamma$ for all $v \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\Omega ^\prime )$. Per assumption $\mathcal {K}(v)\in B_{\infty ,\infty }^{\alpha -\gamma ,\textrm{loc}}(\Omega )$, which entails in turn on account of [20, Theorem 8.2.13]

$$\begin{aligned} \textrm{WF}^{\alpha -\gamma }(\mathcal {K}u){} & {} \subseteq WF(\mathcal {K}(u-v)) \subseteq WF^\prime (K) \circ WF(u-v)\\{} & {} \cup WF_X(K) \subset WF^\prime (K) \circ \Gamma \cup WF_X(K). \end{aligned}$$

To conclude, in view of the arbitrariness of $\Gamma$, we infer

$$\begin{aligned} \textrm{WF}^{\alpha -\gamma }(\mathcal {K}u) \subseteq WF^\prime (K) \circ \textrm{WF}^\alpha (u) \cup WF_\Omega (K). \end{aligned}$$

$\square$

Example 53

Let us consider the heat kernel operator, namely the fundamental solution of the heat equation $G\in \mathcal {D}^\prime (\mathbb {R}^{d+1}\times \mathbb {R}^{d+1})$, whose integral kernel reads in standard Cartesian coordinates

$$\begin{aligned} G(t,x,t^\prime ,x^\prime )= \frac{\Theta (t-t^\prime )}{(4\pi (t-t^\prime ))^{d/2}} e^{-\frac{|x-x^\prime |^2}{4(t-t^\prime )}}, \end{aligned}$$

where $\Theta$ is the Heaviside function. By Schauder estimates, c.f. [25], G can also be read as the kernel of an operator $\mathcal {G} :B^\alpha _{\infty ,\infty }(\mathbb {R}^{1+d}) \rightarrow B^{\alpha + 2}_{\infty ,\infty }(\mathbb {R}^{1+d})$. Furthermore it holds that

$$\begin{aligned} WF(G)=\{(t,x,t,x,\tau ,\xi ,-\tau ,-\xi )\;|\; (t,x)\in \mathbb {R}^{d+1}\;\text {and}\; (\tau ,\xi )\in \mathbb {R}^{d+1}\setminus \{0\}\}.\nonumber \\ \end{aligned}$$

(4.15)

Therefore, we are in position to apply(4.14). Considering any $u\in \mathcal {E}^\prime (\mathbb {R})$, we can infer that the hypotheses of Corollary 52 are met since $\textrm{WF}^\alpha _{\mathbb {R}^{d+1}}(G)=\emptyset$ for all $\alpha \in \mathbb {R}$, where the subscript $\mathbb {R}^{d+1}$ should be read in the sense of Eq. (4.11). At the same time, on account of Remark 32, there must exist $\alpha <0$ such that $\textrm{WF}^\alpha (u)=\emptyset$. This entails that

$$\begin{aligned} \textrm{WF}^{\alpha +2}(\mathcal {G}(u)) \subseteq WF^\prime (G) \circ \textrm{WF}^\alpha (u), \end{aligned}$$

which, combined with Eq. (4.15), yields $WF^\prime (G) \circ \textrm{WF}^\alpha (u) = \textrm{WF}^\alpha (u)$. This leads to the inclusion

$$\begin{aligned} \textrm{WF}^{\alpha +2}(\mathcal {G}u) \subseteq \textrm{WF}^\alpha (u). \end{aligned}$$

4.1 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

$$\begin{aligned} \partial _t u = i a(D_x) u, \quad (t,x) \in \mathbb {R}\times \mathbb {R}^d, \end{aligned}$$

(4.16)

where we assume $a=a_1+a_0$ where $a_1\in S^1_{\text {hom}}(\mathbb {R}^d)$, while $a_0\in S^0(\mathbb {R}^d)$ see Definition 11. Using standard Fourier analysis, we can infer that the fundamental solution associated to the operator $\partial _t-ia(D_x)$ is the distribution $G\in \mathcal {D}^\prime (\mathbb {R}\times \mathbb {R}^d)$, whose integral kernel reads

$$\begin{aligned} G(t,x) = \Theta (t) [e^{it a(D)} \delta ](x), \end{aligned}$$

where $\Theta$ is once more the Heaviside function.

Proposition 54

Let $\alpha \in \mathbb {R}$. Then $B_{\infty ,\infty }^\alpha (\mathbb {R}^d) \cap \mathcal {E}^\prime (\mathbb {R}^d)\subset B_{2,\infty }^\alpha (\mathbb {R}^d)$.

Proof

Let $v \in B_{\infty ,\infty }^\alpha (\mathbb {R}^d) \cap \mathcal {E}^\prime (\mathbb {R}^d)$. For any $\kappa \in \mathscr {B}_{\lfloor \alpha \rfloor }$ as per Definition 5, it

$$\begin{aligned} \Vert v(\kappa ^\lambda _{x})\Vert _{L^2(\mathbb {R}^d)} \lesssim \Vert v(\kappa ^\lambda _{x})\Vert _{L^\infty (\mathbb {R}^d)} \lesssim \lambda ^\alpha , \end{aligned}$$

where the first estimate is a a byproduct of v being compactly supported. A similar reasoning applies when considering any $\underline{\kappa }\in \mathcal {D}(B(0,1))$ such that $\check{\kappa }(0) \ne 0$. As a consequence of Definition 6, we infer that $v \in B_{2,\infty }^\alpha (\mathbb {R}^d)$. $\square$

Proposition 55

Let $G \in \mathcal {D}^\prime (\mathbb {R}\times \mathbb {R}^d)$ be the fundamental solution of the hyperbolic operator $\partial _t - i a(D_x)$. Then, $G(t,\cdot ) \in B_{2,\infty }^{-\frac{d}{2}}(\mathbb {R}^d)$ for any $t \in \mathbb {R}$. Moreover, given $v \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$ with $\alpha \in \mathbb {R}$,

$$\begin{aligned} G(t,\cdot ) *v \in B_{\infty ,\infty }^{\alpha -\frac{d}{2},\textrm{loc}}(\mathbb {R}^d), \end{aligned}$$

where $*$ stands for the convolution.

Proof

Let $\{\psi _j\}_{j\ge 0}$ be a Littlewood-Paley partition of unity as per Definition 2. For any $j\ge 1$, it descends

$$\begin{aligned} \Vert \psi _j(D_x)e^{ita(D_x)}\delta \Vert _{L^2(\mathbb {R}^d)} = \Vert \psi _j\Vert _{L^2(\mathbb {R}^d)} = 2^{j \frac{d}{2}} \Vert \psi \Vert _{L^2(\mathbb {R}^d)}, \end{aligned}$$

(4.17)

where we applied Fourier-Plancherel theorem in the first equality. Hence we can conclude that

$$\begin{aligned} \sup _{j\ge 0}2^{-j \frac{d}{2}} \Vert \psi _j(D_x)e^{ita(D_x)}\delta \Vert _{L^2(\mathbb {R}^d)} < \infty , \end{aligned}$$

which entails that $G(t,\cdot ) \in B_{2,\infty }^{-\frac{d}{2}}(\mathbb {R}^d)$. Observe that, for every $\phi \in \mathcal {D}(\mathbb {R}^d)$, $\phi v \in B^\alpha _{2,\infty }(\mathbb {R}^d)$ on account of Proposition 54. Then, as a consequence of [22, Thm 2.2], we can infer that $G(t,\cdot ) *(\phi v) \in B_{\infty ,\infty }^{\alpha -\frac{d}{2}}(\mathbb {R}^d)$ for any $t \in \mathbb {R}$. $\square$

Proposition 55 can be read as a statement that the solution map associated to Eq. (4.16)

$$\begin{aligned} S(t,0): u(0) \mapsto u(t) \end{aligned}$$

is continuous from $B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$ to $B_{\infty ,\infty }^{\alpha -\frac{d}{2},\textrm{loc}}(\mathbb {R}^d)$. Moreover, S(t, 0) can be inverted and $S(t,0)^{-1}=S(0,t)$.

Theorem 56

Let a be as per Eq. (4.16) and let $u_0 \in \mathcal {S}^\prime (\mathbb {R}^d)$. Suppose that u is the solution of the initial value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u = i a(D_x) u,\\ u(0)= u_0. \end{array}\right. } \end{aligned}$$

(4.18)

Then, for every $\alpha \in \mathbb {R}$,

$$\begin{aligned} \textrm{WF}^{\alpha -\frac{d}{2}}(u(t)) = \mathcal {C}(t) \textrm{WF}^\alpha (u_0), \end{aligned}$$

(4.19)

where $\mathcal {C}(t)$ is the flow from t to 0 associated the Hamiltonian vector field $H_{a(\xi )}$.

Proof

We just prove the inclusion $\subset$, the other following suite. Let us consider $(x,\xi )\not \in \textrm{WF}^\alpha (u_0)$. Then there exists $A \in \Psi ^0(\mathbb {R}^d)$, elliptic at $(x,\xi )$, such that $A u_0 \in B_{\infty ,\infty }^{\alpha ,\textrm{loc}}(\mathbb {R}^d)$. Let us define $A(t):=S(t,0)\circ A\circ S(0,t)$ so that $A(t)u(t) = S(t,0)Au_0 \in B_{\infty ,\infty }^{\alpha -\frac{d}{2},\textrm{loc}}(\mathbb {R}^d)$. On account of Egorov’s theorem, see e.g. [16] and [26, Thm. 1.2 and Thm.5.5], we can conclude that A(t) still lies in $\Psi ^0(\mathbb {R}^d)$ and it is elliptic at $\mathcal {C}(t)^{-1}(x,\xi )$. This implies $\mathcal {C}(t)^{-1}(x,\xi ) \not \in \textrm{WF}^{\alpha -\frac{d}{2}}(u(t))$. $\square$

Remark 57

It is worth mentioning that the estimate on the Besov wavefront set as per Theorem 56 might be improved if working with a generic Besov space $B^\alpha _{pq}(\mathbb {R}^d)$ rather than with $B^\alpha _{\infty \infty }(\mathbb {R}^d)$. Yet this step requires first of all to establish an improved version of Proposition 55, which appears to be elusive at this stage.