Global Wave-Front Properties for Fourier Integral Operators and Hyperbolic Problems

Abstract

We illustrate the composition properties for an extended family of ${\text {SG}}$ Fourier integral operators. We prove continuity results on modulation spaces, and study mapping properties of global wave-front sets for such operators. These extend classical results to more general situations. For example, there are no requirements on homogeneity for the phase functions. Finally, we apply our results to the study of the propagation of singularities, in the context of modulation spaces, for the solutions to the Cauchy problems for the corresponding linear hyperbolic operators.

1 Introduction

In [26], global wave-front sets with respect to convenient Banach or Fréchet spaces were introduced, and global mapping properties of pseudo-differential operators of ${\text {SG}}$-type were established in terms of these wave-front sets (see, e.g. [14, 16, 18, 19, 25–27, 38, 40]). For any such Banach or Fréchet space ${\mathcal B}$ and tempered distribution f, the global wave-front set ${\text {WF}}_{{\mathcal B}}(f)$ is the union of three components ${\text {WF}}_{{\mathcal B}}^m(f)$, $m=1,2,3$. The first component (for $m=1$) describes the local wave-front set which informs where f locally fails to belong to ${\mathcal B}$, as well as the directions where the singularities (with respect to ${\mathcal B}$) propagates. The second and third components (for $m=2$ or $m=3$) inform where at infinity the growth and oscillations of f are strong enough such that f fails to belong to ${\mathcal B}$. We remark that ${\text {WF}}_{{{\mathscr {S}}}}^1(f)$, ${\text {WF}}_{{{\mathscr {S}}}}^2(f)$ and ${\text {WF}}_{{{\mathscr {S}}}}^3(f)$ agree with ${\text {WF}}_{{{\mathscr {S}}}}^\psi (f)$, ${\text {WF}}_{{{\mathscr {S}}}}^e(f)$ and ${\text {WF}}_{{{\mathscr {S}}}}^{\psi e}(f)$, respectively, in [19]. Note also that for admissible ${\mathcal B}$, these wave-front sets give suitable information for local and global behavior, since f belongs to ${\mathcal B}$ globally (locally), if and only if ${\text {WF}}_{{\mathcal B}}(f)=\emptyset$ (${\text {WF}}_{{\mathcal B}}^1(f)=\emptyset$).

It is convenient to formulate mapping properties for pseudo-differential operators of ${\text {SG}}$-type in terms of ${\text {SG}}$-ordered pairs $({\mathcal B},{\mathcal C})$, where ${\mathcal B}$ and ${\mathcal C}$ should be appropriate target and image spaces of the involved pseudo-differential operators. (Cf. [26].) More precisely, the pair $({\mathcal B},{\mathcal C})$ of spaces ${\mathcal B}$ and ${\mathcal C}$ containing ${{\mathscr {S}}}$ and contained in ${{\mathscr {S}}}'$, is called ${\text {SG}}$-ordered with respect to the weight $\omega _0$ if the mappings

$$\begin{aligned} {\text {Op}}(a)&: {\mathcal B}\rightarrow {\mathcal C},\quad {\text {Op}}(b)^{*} : {\mathcal C}\rightarrow {\mathcal B}, \nonumber \\ {\text {Op}}(c)&: {\mathcal B}\rightarrow {\mathcal B}\quad \text {and}\quad {\text {Op}}(c) : {\mathcal C}\rightarrow {\mathcal C}\end{aligned}$$

(1.1)

are continuous for every $a\in {\text {SG}}^{(\omega _0)}$, $b\in {\text {SG}}^{(1/\omega _0)}$ and $c\in {\text {SG}}^{0,0}$. If it is only required that the first mapping property in (1.1) holds, then the pair $({\mathcal B},{\mathcal C})$ is called weakly ${\text {SG}}$ -ordered. Here ${\text {SG}}^{(\omega )}$, the set of all ${\text {SG}}$-symbols with respect to $\omega$, belongs to an extended family of symbol classes of ${\text {SG}}$-type. We refer to [19] for the definition of (also classical) ${\text {SG}}$-symbols. We notice that (1.1) is true also after ${\text {Op}}(b)$ is replaced by its adjoint ${\text {Op}}(b)^{*}$, because ${\text {Op}}({\text {SG}}^{(\omega )})^{*} = {\text {Op}}({\text {SG}}^{(\omega )})$.

Important examples on function and distribution spaces which give rise to ${\text {SG}}$-ordered pair are the Schwartz space, or the set of tempered distributions. An other important example appears when these spaces are suitable modulation spaces, a family of function and distribution spaces, introduced by Feichtinger in [29] and further developed in [30, 31] by Feichtinger and Gröchenig. More precisely, in [26] it is noticed that $({\mathscr {S}},{\mathscr {S}})$ and $({\mathscr {S}}',{\mathscr {S}}')$ are ${\text {SG}}$-ordered pairs, and for any weight $\omega$ and any modulation space ${\mathcal B}$, there is a (unique) modulation space ${\mathcal C}$ such that $({\mathcal B},{\mathcal C})$ is an ${\text {SG}}$-ordered pair with respect to $\omega$. In particular, the family of ${\text {SG}}$-ordered pairs is broad in the sense that ${\mathcal B}$ can be chosen as a Sobolev space, or, more general, as a Sobolev–Kato space, since such spaces are special cases of modulation spaces. Moreover, if ${\text {SG}}^{(\omega )}$ is a classical symbol class of ${\text {SG}}$-type and ${\mathcal B}$ is a Sobolev–Kato space, then ${\mathcal C}$ is also a Sobolev–Kato space.

For any ${\text {SG}}$-ordered pairs $({\mathcal B},{\mathcal C})$ with respect to $\omega$, it is proved in [25, 26] that the wave-front sets with respect to ${\mathcal B}$ and ${\mathcal C}$ posses convenient mapping properties. For example, if $f\in {{\mathscr {S}}}'$ and $a\in {\text {SG}}^{(\omega )}$, then (1.1) is refined as

$$\begin{aligned} {\text {WF}}_{{\mathcal C}}({\text {Op}}(a)f)&\subseteq {\text {WF}}_{{\mathcal B}}(f),\nonumber \\ \text {i.e.,}\quad {\text {WF}}_{{\mathcal C}}^m({\text {Op}}(a)f)&\subseteq {\text {WF}}_{{\mathcal B}}^m(f),\quad m=1,2,3 , \end{aligned}$$

(1.2)

and that reversed inclusions are obtained by adding the set of characteristic points to the left-hand sides in (1.2). In particular, since the set of characteristic points is empty for elliptic operators, it follows that equalities are attained in (1.2) for such operators.

In this paper we establish similar properties for Fourier integral operators. More precisely, for any symbol a in ${\text {SG}}^{(\omega )}$ for some weight $\omega$, the Fourier integral operator (or FIO) ${\text {Op}}_{\varphi }(a)$ is given by

$$\begin{aligned} f\mapsto ({\text {Op}}_\varphi (a)f)(x)&\equiv (2\pi )^{-d}\int _{{\mathbf {R}}^{d}}e^{i\varphi (x,\xi )} a(x,\xi ) \widehat{f}(\xi )\, d\xi , \end{aligned}$$

and its formal $L^2$-adjoint by

$$\begin{aligned} f\mapsto ({\text {Op}}_\varphi (a)^{*}f)(x)&\equiv (2\pi )^{-d}\iint _{{\mathbf {R}}^{2d}} e^{i(\langle x,\xi \rangle -\varphi (y,\xi ))} \overline{a(y,\xi )} f(y)\, dyd\xi . \end{aligned}$$

The operator ${\text {Op}}_\varphi ^{*}(a)={\text {Op}}_\varphi (a)^{*}$ is here called Fourier integral operator of type II, while ${\text {Op}}_\varphi (a)$ is called a Fourier integral operator of type I, with phase function $\varphi$ and amplitude (or symbol) a. The phase function $\varphi$ should be in ${\text {SG}}^{1,1}_{1,1}$ and satisfy

$$\begin{aligned} \langle \varphi ' _x(x,\xi )\rangle \asymp \langle \xi \rangle \quad \text {and}\quad \langle \varphi ' _\xi (x,\xi )\rangle \asymp \langle x\rangle . \end{aligned}$$

(1.3)

Here and in what follows, $A\asymp B$ means that $A\lesssim B$ and $B\lesssim A$, where $A\lesssim B$ means that $A\le c\cdot B$, for a suitable constant $c>0$. Furthermore, $\varphi$ should also fulfill the usual (global) non-degeneracy condition

$$\begin{aligned} |\det (\varphi '' _{x\xi }(x,\xi ))|\ge c ,\quad x,\xi \in {\mathbf {R}}^{d}, \end{aligned}$$

for some constant $c>0$.

In Sect. 4, the notion on ${\text {SG}}$-ordered pair from [26] is reformulated to include such Fourier integral operators, where the operators ${\text {Op}}(a)$ and ${\text {Op}}(b)^{*}$ in (1.1) are replaced by ${\text {Op}}_{\varphi }(a)$ and ${\text {Op}}_{\varphi }(b)^{*}$, respectively, and takes into account the phase-function $\varphi$.

In order to establish wave-front results, similar to (1.2), it is also required that the phase functions fulfill some further natural conditions, namely, that they preserve shapes in certain ways near the points in the phase space $T^{*}{\mathbf {R}}^{d}\simeq {\mathbf {R}}^{2d}$ (see Sect. 5). In fact, the definitions of wave-front sets of appropriate distributions are based on the behavior in cones of corresponding Fourier transformations, after localizing the involved distributions near points or along certain directions.

In order to explain our main results, let $\phi$ be the canonical transformation of $T^{*}{\mathbf {R}}^{d}$ generated by $\varphi$, and consider an elliptic Fourier integral operator ${\text {Op}}_\varphi (a)$ with amplitude $a\in {\text {SG}}^{(\omega _0)}$. If $({\mathcal B},{\mathcal C})$ are (weakly) ${\text {SG}}$-ordered with respect to $\omega _0$ and $\varphi$ (see Sect. 4 for precise definitions), then, under some natural invariance conditions on the weight $\omega _0$,

$$\begin{aligned} {\text {WF}}_{{\mathcal C}}({\text {Op}}_\varphi (a)f) = \phi ({\text {WF}}_{{\mathcal B}}(f)). \end{aligned}$$

(1.4)

A similar result holds for ${\text {Op}}_\varphi ^{*}(a)f$, namely

$$\begin{aligned} {\text {WF}}_{\widetilde{{\mathcal B}}}({\text {Op}}_\varphi ^{*} (a)f) = \phi ^{-1} ({\text {WF}}_{\widetilde{{\mathcal C}}}(f)), \end{aligned}$$

(1.5)

when ${\text {Op}}_\varphi ^{*}(a):\widetilde{{\mathcal C}}\rightarrow \widetilde{{\mathcal B}}$, with a (in general, different) couple of admissible spaces $\widetilde{{\mathcal C}}, \widetilde{{\mathcal B}}$, and the inverse $\phi ^{-1}$ of the canonical transformation in (1.4). More generally, by dropping the ellipticity of the amplitude functions, with $a\in {\text {SG}}^{(\omega _1)}$, $b\in {\text {SG}}^{(\omega _2)}$, $({\mathcal B}_1 ,{\mathcal C}_1, {\mathcal B}_2, {\mathcal C}_2 )$ being ${\text {SG}}$-ordered with respect to $\omega _1$, $\omega _2$ and $\varphi$, we show that

$$\begin{aligned} {\text {WF}}_{{\mathcal C}_1}({\text {Op}}_\varphi (a)f) \subseteq \phi ({\text {WF}}_{{\mathcal B}_1}(f))^\mathrm {con} \end{aligned}$$

(1.6)

and

$$\begin{aligned} {\text {WF}}_{{{\mathcal B}_2}}({\text {Op}}_\varphi ^{*} (b)f) \subseteq \phi ^{-1} ({\text {WF}}_{{{\mathcal C}_2}}(f))^\mathrm {con}, \end{aligned}$$

(1.7)

provided the phase function $\varphi$ additionally fulfills conditions similar to those in Kumano-Go [36]. In (1.6) and (1.7), we denoted by $W^{\mathrm {con}}$ the union of the smallest m-conical subsets which include the three components $W_m$, $m\in \{1,2,3\}$, of W (see [36] and Sect. 5 below).

Notice that the required conditions on the phase function are automatically satisfied by all the phase functions arising from the short-time solutions to hyperbolic Cauchy problems in the ${\text {SG}}$-classical context, see [14, 15, 17, 18]. We then apply our results to describe the propagation of singularities from the initial data to the solutions to such ${\text {SG}}$-hyperbolic Cauchy problems.

The results above are based on comprehensive investigations of algebraic and continuity properties of the involved Fourier integral operators. A significant part of these investigations concern compositions between Fourier integral operators of type I or II, with pseudo-differential operators. This is performed in [24], where it is proved that for any Fourier integral operators ${\text {Op}}_\varphi (a)$ and ${\text {Op}}_\varphi ^{*}(b)$ with $a,b\in {\text {SG}}^{(\omega _1)}$, and some $p\in {\text {SG}}^{(\omega _2 )}$, then, under suitable invariance conditions on the weights,

$$\begin{aligned} {\text {Op}}(p)\circ {\text {Op}}_\varphi (a)&= {\text {Op}}_\varphi (c_1) \mod {\text {Op}}({\mathcal B}_0), \\ {\text {Op}}(p)\circ {\text {Op}}_\varphi ^{*}(b)&= {\text {Op}}_\varphi ^{*}(c_2) \mod {\text {Op}}({\mathcal B}_0), \\ {\text {Op}}_\varphi (a) \circ {\text {Op}}(p)&= {\text {Op}}_\varphi (c_3) \mod {\text {Op}}({\mathcal B}_0) \\ {\text {Op}}_\varphi ^{*}(b) \circ {\text {Op}}(p)&= {\text {Op}}_\varphi ^{*}(c_4) \mod {\text {Op}}({\mathcal B}_0), \end{aligned}$$

for some $c_j\in {\text {SG}}^{(\omega _{0,j})}$, $j=1,\ldots ,4$, and suitable weights $\omega _{0,j}$. Here ${\text {Op}}({\mathcal B}_0)$ is a set of appropriate smoothing operators, depending on the symbols and the phase function. Furthermore, if $a\in {\text {SG}}^{(\omega _1)}$ and $b\in {\text {SG}}^{(\omega _2)}$, then it is also proved that ${\text {Op}}_\varphi ^{*}(b)\circ {\text {Op}}_\varphi (a)$ and ${\text {Op}}_\varphi (a)\circ {\text {Op}}_\varphi ^{*} (b)$ are equal to pseudo-differential operators ${\text {Op}}(c_5)$ and ${\text {Op}}(c_6)$, respectively, for some $c_5,c_6\in {\text {SG}}^{(\omega _{0,j})}$, $j=5,6$. We also present asymptotic formulae for $c_j$, $j=1,\ldots ,6$, in terms of a and b, or of a, b and p, modulo smoothing terms. The extensions of the calculus of ${\text {SG}}$ Fourier integral operators developed in [16] to the classes ${\text {SG}}^{(\omega _0)}_{r,\rho }$, introduced and systematically used in [25–27], is recalled in Sect. 3.

The formulae (1.4)–(1.7), given by the calculus recalled in Sect. 3, also rely on certain asymptotic expansions in the framework of symbolic calculus of ${\text {SG}}$ pseudo-differential operators, as well as on continuity properties for ${\text {SG}}$-ordered pairs.

The first of the above two points concerns making sense of expansions of the form

$$\begin{aligned} a\sim \sum a_j , \end{aligned}$$

in the framework of the generalised ${\text {SG}}$-classes ${\text {SG}}^{(\omega _0)}_{r,\rho }$. The ideas are similar to the corresponding properties in the usual Hörmander calculus in Sect. 18.1 in [35]. For this reason, in [22] we have established properties of asymptotic expansions for symbols classes of the form S(m, g), parameterized by the weight function m and Riemannian metric g on the phase space (cf. Sect. 18.4 in [35]). Note here that any ${\text {SG}}$-class is equal to S(m, g) for some choice of m and g, and that similar facts hold for the Hörmander classes $S^r _{\rho ,\delta }$. The results therefore cover several situations on asymptotic expansions for pseudo-differential operators.

With respect to the second point above, we study in Sect. 4 some specific spaces which are ${\text {SG}}$-ordered or weakly ${\text {SG}}$-ordered. For example, we present necessary and sufficient conditions for the involved weight functions and parameters, in order for Sobolev–Kato spaces, Sobolev spaces and modulation spaces should be ${\text {SG}}$-ordered or weakly ${\text {SG}}$-ordered. A direct proof of the continuity from $L^2({\mathbf {R}}^{d})$ to itself of ${\text {SG}}$ Fourier integral operators with a uniformly bounded amplitude (that is, the amplitude is of order 0, 0, or, equivalently, the weight $\omega$ is bounded), similar to the one given in [16], can be found in [24]. Moreover, taking advantage of the calculus developed in [24], recalled in Sect. 3 for the convenience of the reader, and relying on results in [13, 34], we prove that our classes of ${\text {SG}}$ Fourier integral operators are continuous between suitable couples of weighted modulation spaces $(M^p_{(\omega _1)}({\mathbf {R}}^{d}), M^p_{(\omega _2)}({\mathbf {R}}^{d}))$.

Finally, in Sect. 5 we prove our main propagation results and illustrate their application to Cauchy problems, for ${\text {SG}}$-hyperbolic linear operators and first order systems with constant multiplicities. In view of the mapping properties proved in Sect. 4, we observe that such problems are “well-posed with a loss of regularity” when considered in the environment of Lebesgue and modulation spaces, differently from other known situations, see, e.g, Bényi et al. [2], Cordero and Nicola [12], Wang and Hudzik [45] and the references quoted therein.

2 Preliminaries

We begin by fixing the notation and recalling some basic concepts which will be needed below. In Sects. 2.1–2.4 we mainly summarize parts of the contents of Sect. 2 in [24, 26, 27]. Some of the results that we recall, compared with their original formulation in the ${\text {SG}}$ context appeared in [16], are here given in a slightly more general form, adapted to the definitions given in Sect. 2.3.

2.1 Weight Functions

Let $\omega$ and v be positive measurable functions on ${\mathbf {R}}^{d}$. Then $\omega$ is called v-moderate if

$$\begin{aligned} \omega (x+y) \lesssim \omega (x)v(y) \end{aligned}$$

(2.1)

If v in (2.1) can be chosen as a polynomial, then $\omega$ is called a function or weight of polynomial type. We let ${\mathscr {P}}({\mathbf {R}}^{d})$ be the set of all polynomial type functions on ${\mathbf {R}}^{d}$. If $\omega (x,\xi )\in {\mathscr {P}}({\mathbf {R}}^{2d})$ is constant with respect to the x-variable or the $\xi$-variable, then we sometimes write $\omega (\xi )$, respectively $\omega (x)$, instead of $\omega (x,\xi )$, and consider $\omega$ as an element in ${\mathscr {P}}({\mathbf {R}}^{2d})$ or in ${\mathscr {P}}({\mathbf {R}}^{d})$ depending on the situation. We say that v is submultiplicative if (2.1) holds for $\omega =v$. For convenience we assume that all submultiplicative weights are even, and v and $v_j$ always stand for submultiplicative weights, if nothing else is stated.

Without loss of generality we may assume that every $\omega \in {\mathscr {P}}({\mathbf {R}}^{d})$ is smooth and satisfies the ellipticity condition $\partial ^\alpha \omega / \omega \in L^\infty$. In fact, by Lemma 1.2 in [41] it follows that for each $\omega \in {\mathscr {P}}({\mathbf {R}}^{d})$, there is a smooth and elliptic $\omega _0\in {\mathscr {P}}({\mathbf {R}}^{d})$ which is equivalent to $\omega$ in the sense

$$\begin{aligned} \omega \asymp \omega _0. \end{aligned}$$

(2.2)

The weights involved in the sequel often have to satisfy additional conditions. More precisely let $r,\rho \ge 0$. Then ${\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ is the set of all $\omega (x,\xi )$ in ${\mathscr {P}}({\mathbf {R}}^{2d})\bigcap$ $C^\infty ({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} \langle x\rangle ^{r|\alpha |}\langle \xi \rangle ^{\rho |\beta |}\frac{\partial ^\alpha _x\partial ^\beta _\xi \omega (x,\xi )}{\omega (x,\xi )}\in L^\infty ({\mathbf {R}}^{2d}), \end{aligned}$$

(2.3)

for every multi-indices $\alpha$ and $\beta$. Any weight $\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ is called ${\text {SG}}$-moderate on ${\mathbf {R}}^{2d}$, of order r and $\rho$. Notice that ${\mathscr {P}}_{r,\rho }$ is different here compared to [25], and there are elements in ${\mathscr {P}}({\mathbf {R}}^{2d})$ which have no equivalent elements in ${\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$. On the other hand, if $s, t\in \mathbf R$ and $r, \rho \in [0,1]$, then ${\mathscr {P}}_{r,\rho } ({\mathbf {R}}^{2d})$ contains all weights of the form

$$\begin{aligned} \vartheta _{m,\mu } (x,\xi )\equiv \langle x\rangle ^m\langle \xi \rangle ^\mu , \end{aligned}$$

(2.4)

which are one of the most common type of weights.

It will also be useful to consider ${\text {SG}}$-moderate weights in one or three sets of variables. Let $\omega \in {\mathscr {P}}({\mathbf {R}}^{3d})\bigcap C^\infty ({\mathbf {R}}^{3d})$, and let $r_1,r_2,\rho \ge 0$. Then $\omega$ is called ${\text {SG}}$ moderate on ${\mathbf {R}}^{3d}$, of order $r_1$, $r_2$ and $\rho$, if it fulfills

$$\begin{aligned} \langle x_1\rangle ^{r_1|\alpha _1|} \langle x_2\rangle ^{r_2|\alpha _2|} \langle \xi \rangle ^{\rho |\beta |} \frac{\partial _{x_1}^{\alpha _1} \partial _{x_2}^{\alpha _2} \partial _\xi ^\beta \omega (x_1,x_2,\xi )}{\omega (x_1,x_2,\xi )} \in L^\infty ({\mathbf {R}}^{3d}). \end{aligned}$$

The set of all ${\text {SG}}$-moderate weights on ${\mathbf {R}}^{3d}$ of order $r_1$, $r_2$ and $\rho$ is denoted by ${\mathscr {P}}_{r_1,r_2,\rho }({\mathbf {R}}^{3d})$. Finally, we denote by ${\mathscr {P}}_{r}({\mathbf {R}}^{d})$ the set of all ${\text {SG}}$-moderate weights of order $r\ge 0$ on ${\mathbf {R}}^{d}$, which are defined in a similar fashion.

2.2 Modulation Spaces

Let $\phi \in {\mathscr {S}}({\mathbf {R}}^{d})$. Then the short-time Fourier transform of $f\in {\mathscr {S}}({\mathbf {R}}^{d})$ with respect to (the window function) $\phi$ is defined by

$$\begin{aligned} V_\phi f(x,\xi ) = (2\pi )^{-d/2}\int _{{\mathbf {R}}^{d}} f(y)\overline{\phi (y-x)}e^{-i\langle y,\xi \rangle }\, dy. \end{aligned}$$

(2.5)

More generally, the short-time Fourier transform of $f \in {\mathscr {S}}'({\mathbf {R}}^{d})$ with respect to $\phi \in {\mathscr {S}}'({\mathbf {R}}^{d})$ is defined by

Here ${\mathscr {F}}_2F$ is the partial Fourier transform of $F(x,y)\in {\mathscr {S}}'({\mathbf {R}}^{2d})$ with respect to the y-variable, and the Fourier transform $\mathscr {F}$ is the linear and continuous map on $\mathscr {S}'(\mathbf{R}^d)$ which takes the form

$$\begin{aligned} (\mathscr {F}f)(\xi )=\hat{f}(\xi )\equiv \int _{\mathbf{R}^d}f(x)e^{-i\langle x,\xi \rangle }dx \end{aligned}$$

when $f\in L^{1}({\mathbf {R}}^d$). We refer to [32, 33] for more facts about the short-time Fourier transform. To introduce the modulation spaces, we first recall that a Banach space ${\mathscr {B}}$, continuously embedded in $L^1_{\mathrm {loc}}({\mathbf {R}}^{d})$, is called a (translation) invariant BF-space on ${\mathbf {R}}^{d}$, with respect to a submultiplicative weight $v \in {\mathscr {P}}({\mathbf {R}}^{d})$, if there is a constant C such that the following conditions are fulfilled:

• (1) ${\mathscr {S}}({\mathbf {R}}^{d})\subseteq {\mathscr {B}}\subseteq {\mathscr {S}}'({\mathbf {R}}^{d})$ (continuous embeddings);

• (2) if $x\in {\mathbf {R}}^{d}$ and $f\in {\mathscr {B}}$, then $f(\cdot -x)\in {\mathscr {B}}$, and

  $$\begin{aligned} \Vert f(\cdot -x)\Vert _{{\mathscr {B}}}\le Cv(x)\Vert f\Vert _{{\mathscr {B}}}\text{; } \end{aligned}$$

  (2.6)

• (3) if $f,g\in L^1_{\mathrm {loc}}({\mathbf {R}}^{d})$ satisfy $g\in {\mathscr {B}}$ and $|f| \le |g|$ almost everywhere, then $f\in {\mathscr {B}}$ and

  $$\begin{aligned} \Vert f\Vert _{{\mathscr {B}}}\le C\Vert g\Vert _{{\mathscr {B}}}\text{; } \end{aligned}$$

• (4) if $f\in {\mathscr {B}}$ and $\varphi \in C^\infty _0({\mathbf {R}}^{d})$, then $f*\varphi \in {\mathscr {B}}$, and

  $$\begin{aligned} \Vert f*\varphi \Vert _{{\mathscr {B}}}\le \Vert \varphi \Vert _{L^1_{(v)}} \Vert f\Vert _{{\mathscr {B}}}. \end{aligned}$$

  (2.7)

The following definition of modulation spaces is due to Feichtinger [30]. Let $\mathscr {B}$ be a translation invariant BF-space on ${\mathbf {R}}^{2d}$ with respect to $v\in {\mathscr {P}}({\mathbf {R}}^{2d})$, $\phi \in \mathscr {S}({\mathbf {R}}^{d})\backslash {0}$ and let $\omega \in \mathscr {P}({\mathbf {R}}^{2d})$ be such that $\omega$ is v-moderate. The modulation space $M(\omega ,{\mathscr {B}})$ consists of all $f\in \mathscr {S}'({\mathbf {R}}^{d})$ such that $V_{\phi }f\cdot \omega \in \mathscr {B}$. We notice that $M(\omega ,{\mathscr {B}})$ is a Banach space with the norm

$$\begin{aligned} \Vert f\Vert _{M(\omega ,{\mathscr {B}})} \equiv \Vert V_{\phi } f \omega \Vert _{\mathscr {B}} \end{aligned}$$

(2.8)

(cf. [31]).

Remark 2.1

Assume that $p,q\in [1,\infty ]$, and let $L^{p,q}_{1}({\mathbf {R}}^{2d})$ and $L^{p,q}_{2}({\mathbf {R}}^{2d})$ be the sets of all $F\in L^1_{\mathrm {loc}} ({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} \Vert F\Vert _{L^{p,q}_1} \equiv \left( \int \left( \int |F(x,\xi )|^p\, dx\right) ^{q/p}\,d\xi \right) ^{1/q} <\infty \end{aligned}$$

and

$$\begin{aligned} \Vert F\Vert _{L^{p,q}_2} \equiv \left( \int \left( \int |F(x,\xi )|^q\, d\xi \right) ^{p/q}\, dx\right) ^{1/p}<\infty \end{aligned}$$

(with obvious modifications when $p = \infty$ or $q = \infty$). Then $M(\omega ,L^{p,q}_1({\mathbf {R}}^{2d}))$ is equal to the classical modulation space $M^{p,q}_{(\omega )}({\mathbf {R}}^{d})$, and $M(\omega ,L^{p,q}_2({\mathbf {R}}^{2d}))$ is equal to the space $W^{p,q}_{(\omega )}({\mathbf {R}}^{d})$, related to Wiener-amalgam spaces (cf. [29–31, 33]). We set $M^p_{(\omega )} = M^{p,p}_{(\omega )}= W^{p,p}_{(\omega )}$. Furthermore, if $\omega =1$, then we write $M^{p,q}$, $M^p$ and $W^{p,q}$ instead of $M^{p,q}_{(\omega )}$, $M^p_{(\omega )}$ and $W^{p,q}_{(\omega )}$ respectively.

Remark 2.2

Several important spaces agree with certain modulation spaces. In fact, let $s,\sigma \in \mathbf R$. If $\omega =\vartheta _{s,\sigma }$ (cf. (2.4)), then $M^2_{(\omega )}({\mathbf {R}}^{d})$ is equal to the weighted Sobolev space (or Sobolev–Kato space) $H^2_{\sigma ,s}({\mathbf {R}}^{d})$ in [19, 38], the set of all $f\in {\mathscr {S}}'({\mathbf {R}}^{d})$ such that $\langle x\rangle ^s\langle D\rangle ^\sigma f\in L^2({\mathbf {R}}^{d})$. In particular, if $s=0$ ($\sigma =0$), then $M^2_{(\omega )}({\mathbf {R}}^{d})$ equals to $H^2_\sigma ({\mathbf {R}}^{d})$ ($L^2_s({\mathbf {R}}^{d})$). Furthermore, if instead $\omega (x,\xi )=\langle x,\xi \rangle ^s\equiv \langle (x,\xi ) \rangle ^s$, then $M^2_{(\omega )}({\mathbf {R}}^{d})$ is equal to the Sobolev–Shubin space of order s. (Cf. e. g. [37]).

2.3 Pseudo-differential Operators and SG Symbol Classes

Let $a\in {\mathscr {S}}({\mathbf {R}}^{2d})$, and $t\in \mathbf R$ be fixed. Then the pseudo-differential operator ${\text {Op}}_t(a)$ is the linear and continuous operator on ${\mathscr {S}}({\mathbf {R}}^{d})$ defined by the formula

$$\begin{aligned} ({\text {Op}}_t(a)f)(x) = (2\pi ) ^{-d}\iint e^{i\langle x-y,\xi \rangle } a((1-t)x+ty,\xi )f(y)\, dyd\xi \end{aligned}$$

(2.9)

(cf. Chap. XVIII in [35]). For general $a\in {\mathscr {S}}'({\mathbf {R}}^{2d})$, the pseudo-differential operator ${\text {Op}}_t(a)$ is defined as the continuous operator from ${\mathscr {S}}({\mathbf {R}}^{d})$ to ${\mathscr {S}}'({\mathbf {R}}^{d})$ with distribution kernel

$$\begin{aligned} K_{t,a}(x,y)=(2\pi )^{-d/2}({\mathscr {F}}_2^{-1}a)((1-t)x+ty,x-y). \end{aligned}$$

(2.10)

If $t=0$, then ${\text {Op}}_t(a)$ is the Kohn–Nirenberg representation ${\text {Op}}(a)=a(x,D)$, and if $t=1/2$, then ${\text {Op}}_t(a)$ is the Weyl quantization.

In most of our situations, a belongs to a generalized ${\text {SG}}$-symbol class, which we shall consider now. Let $m,\mu ,r, \rho \in \mathbf R$ be fixed. Then the ${\text {SG}}$-class ${\text {SG}}^{m,\mu }_{r,\rho }({\mathbf {R}}^{2d})$ is the set of all $a\in C^\infty ({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} |D _x^\alpha D _\xi ^\beta a(x,\xi )|\lesssim \langle x\rangle ^{m-r|\alpha |}\langle \xi \rangle ^{\mu -\rho |\beta |}, \end{aligned}$$

for all multi-indices $\alpha$ and $\beta$. Usually we assume that $r,\rho \ge 0$ and $\rho +r >0$.

More generally, assume that $\omega \in {\mathscr {P}}_{r ,\rho } ({\mathbf {R}}^{2d})$. Then ${\text {SG}}_{r,\rho }^{(\omega )}({\mathbf {R}}^{2d})$ consists of all $a\in C^\infty ({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} |D_x^\alpha D_\xi ^\beta a(x,\xi )|\lesssim \omega (x,\xi )\langle x\rangle ^{-r|\alpha |}\langle \xi \rangle ^{-\rho |\beta |}, \quad x,\xi \in {\mathbf {R}}^{d}, \end{aligned}$$

(2.11)

for all multi-indices $\alpha$ and $\beta$. We notice that

$$\begin{aligned} {\text {SG}}_{r,\rho }^{(\omega )}({\mathbf {R}}^{2d})=S(\omega ,g_{r,\rho } ), \end{aligned}$$

(2.12)

when $g=g_{r,\rho }$ is the Riemannian metric on ${\mathbf {R}}^{2d}$, defined by the formula

$$\begin{aligned} \big (g_{r,\rho }\big )_{(y,\eta )}(x,\xi ) =\langle y\rangle ^{-2r}|x|^2 +\langle \eta \rangle ^{-2\rho }|\xi |^2 \end{aligned}$$

(2.13)

(cf. Sect. 18.4–18.6 in [35]). Furthermore, ${\text {SG}}^{(\omega )}_{r,\rho } ={\text {SG}}^{m,\mu }_{r,\rho }$ when $\omega =\vartheta _{m,\mu }$ (see (2.4)).

For conveniency we set

$$\begin{aligned}&{\text {SG}}^{(\omega \vartheta _{-\infty ,0} )} _\rho ({\mathbf {R}}^{2d}) = {\text {SG}}^{(\omega \vartheta _{-\infty ,0} )} _{r,\rho } ({\mathbf {R}}^{2d}) \equiv \bigcap _{N\ge 0} {\text {SG}}^{(\omega \vartheta _{-N ,0} )} _{r,\rho } ({\mathbf {R}}^{2d}), \\&{\text {SG}}^{(\omega \vartheta _{0,-\infty } )} _r ({\mathbf {R}}^{2d}) = {\text {SG}}^{(\omega \vartheta _{0,-\infty } )} _{r,\rho } ({\mathbf {R}}^{2d}) \equiv \bigcap _{N\ge 0} {\text {SG}}^{(\omega \vartheta _{0,-N } )} _{r,\rho } ({\mathbf {R}}^{2d}), \end{aligned}$$

and

$$\begin{aligned} {\text {SG}}^{(\omega \vartheta _{-\infty ,-\infty } )} ({\mathbf {R}}^{2d}) = {\text {SG}}^{(\omega \vartheta _{-\infty ,-\infty } )} _{r,\rho } ({\mathbf {R}}^{2d}) \equiv \bigcap _{N\ge 0} {\text {SG}}^{(\omega \vartheta _{-N ,-N} )} _{r,\rho } ({\mathbf {R}}^{2d}). \end{aligned}$$

We observe that ${\text {SG}}^{(\omega \vartheta _{-\infty ,0} )} _{r,\rho } ({\mathbf {R}}^{2d})$ is independent of r, ${\text {SG}}^{(\omega \vartheta _{0,-\infty } )} _{r,\rho } ({\mathbf {R}}^{2d})$ is independent of $\rho$, and that ${\text {SG}}^{(\omega \vartheta _{-\infty ,-\infty } )} _{r,\rho } ({\mathbf {R}}^{2d})$ is independent of both r and $\rho$. Furthermore, for any $x_0,\xi _0\in {\mathbf {R}}^{d}$ we have

$$\begin{aligned} {\text {SG}}^{(\omega \vartheta _{-\infty ,0} )} _\rho ({\mathbf {R}}^{2d})&= {\text {SG}}^{(\omega _0\vartheta _{-\infty ,0} )} _\rho ({\mathbf {R}}^{2d}), \quad \text {when}\quad \omega _0(\xi ) = \omega (x_0,\xi ), \\ {\text {SG}}^{(\omega \vartheta _{0,-\infty } )} _r ({\mathbf {R}}^{2d})&= {\text {SG}}^{(\omega _0\vartheta _{0,-\infty } )} _r ({\mathbf {R}}^{2d}),\quad \text {when} \quad \omega _0(x ) = \omega (x,\xi _0), \end{aligned}$$

and

$$\begin{aligned} {\text {SG}}^{(\omega \vartheta _{-\infty ,-\infty } )} ({\mathbf {R}}^{2d})&= {\mathscr {S}}({\mathbf {R}}^{2d}). \end{aligned}$$

The following result shows that the concept of asymptotic expansion extends to the classes ${\text {SG}}^{(\omega )}_{r,\rho }({\mathbf {R}}^{2d})$. We refer to [22, Theorem 8] for the proof.

Proposition 2.3

Let $r,\rho \ge 0$ satisfy $r+\rho >0$, and let $\{ s_j \} _{j\ge 0}$ and $\{ \sigma _j \} _{j\ge 0}$ be sequences of non-positive numbers such that $\lim _{j\rightarrow \infty } s_j =-\infty$ when $r>0$ and $s_j=0$ otherwise, and $\lim _{j\rightarrow \infty } \sigma _j =-\infty$ when $\rho >0$ and $\sigma _j=0$ otherwise. Also let $a_j\in {\text {SG}}^{(\omega _j)}_{r,\rho }({\mathbf {R}}^{2d})$, $j=0,1,\ldots$, where $\omega _j=\omega \cdot \vartheta _{s_j,\sigma _j}$. Then there is a symbol $a\in {\text {SG}}^{(\omega )}_{r,\rho }({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} a-\sum _{j=0}^Na_j\in {\text {SG}}^{(\omega _{N+1})}_{r,\rho }({\mathbf {R}}^{2d}). \end{aligned}$$

(2.14)

The symbol a is uniquely determined modulo a remainder h, where

$$\begin{aligned} h&\in {\text {SG}}^{\omega \vartheta _{-\infty ,0})}_{\rho }({\mathbf {R}}^{2d}) \quad \text {when} \quad r>0, \nonumber \\ h&\in {\text {SG}}^{(\omega \vartheta _{0,-\infty } )}_{r}({\mathbf {R}}^{2d}) \quad \text {when} \quad \rho > 0, \nonumber \\ h&\in {\mathscr {S}}({\mathbf {R}}^{2d}) \quad \text {when} \quad r > 0, \rho >0. \end{aligned}$$

(2.15)

Definition 2.4

The notation $a\sim \sum a_j$ is used when a and $a_j$ fulfill the hypothesis in Proposition 2.3. Furthermore, the formal sum

$$\begin{aligned} \sum _{j\ge 0}a_j \end{aligned}$$

is called an asymptotic expansion.

It is a well-known fact that ${\text {SG}}$-operators give rise to linear continuous mappings from ${\mathscr {S}}({\mathbf {R}}^{d})$ to itself, extendable as linear continuous mappings from ${\mathscr {S}}'({\mathbf {R}}^{d})$ to itself. They also act continuously between modulation spaces, and in some situations between suitable Sobolev spaces $H^p_s({\mathbf {R}}^{d})$ and Lebesgue spaces $L^p_t({\mathbf {R}}^{d})$. Here $H^p_\sigma ({\mathbf {R}}^{d})$ consists of all $f\in {\mathscr {S}}'({\mathbf {R}}^{d})$ such that $\langle D\rangle ^\sigma f\in L^p({\mathbf {R}}^{d})$, and $L^p_s({\mathbf {R}}^{d})$ consists of all $f\in {\mathscr {S}}'({\mathbf {R}}^{d})$ such that $\langle \, \cdot \, \rangle ^sf\in L^p({\mathbf {R}}^{d})$. We also define $H^p_{s,\sigma }({\mathbf {R}}^{d})$ as the set of all $f\in {\mathscr {S}}'({\mathbf {R}}^{d})$ such that $\langle \, \cdot \, \rangle ^s\langle D\rangle ^\sigma f\in L^p({\mathbf {R}}^{d})$. Indeed, in the first one of the following propositions, the first part is a special case of [44, Theorem 3.2], and the second part follows from [8, Corollary 6]. (See also [26] for the first part and the proof of [34, Theorem 3.1] for the second part.) The second proposition follows from [46, Theorem 10.7] or [28, Theorem 1.1]. The proofs are therefore omitted.

Proposition 2.5

Let $r,\rho \ge 0$, $t\in \mathbf R$ and $\omega _0\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$. Then the following is true:

• (1) if $a\in {\text {SG}}^{(\omega _0)}_{r,\rho } ({\mathbf {R}}^{2d})$, then ${\text {Op}}_t(a)$ is continuous from $M(\omega ,{\mathscr {B}})$ to $M(\omega /\omega _0,{\mathscr {B}})$, for every choice of $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$ and every translation invariant BF-space ${\mathscr {B}}$ on ${\mathbf {R}}^{2d}$;

• (2) there exist $a\in {\text {SG}}^{(\omega _0)}_{r,\rho } ({\mathbf {R}}^{2d})$ and $b\in {\text {SG}}^{(1/\omega _0)}_{r,\rho } ({\mathbf {R}}^{2d})$ such that for every choice of $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$ and every translation invariant BF-space ${\mathscr {B}}$ on ${\mathbf {R}}^{2d}$, the mappings

  $$\begin{aligned} {\text {Op}}_t(a): {\mathscr {S}}({\mathbf {R}}^{d})\rightarrow {\mathscr {S}}({\mathbf {R}}^{d}),\quad {\text {Op}}_t(a): {\mathscr {S}}'({\mathbf {R}}^{d})\rightarrow {\mathscr {S}}'({\mathbf {R}}^{d}) \end{aligned}$$

  and

  $$\begin{aligned} {\text {Op}}_t(a): M(\omega ,{\mathscr {B}})\rightarrow M(\omega /\omega _0,{\mathscr {B}}). \end{aligned}$$

  are continuous bijections with inverses ${\text {Op}}_t(b)$.

Proposition 2.6

Let $r,\rho > 0$, $t\in \mathbf R$, $p\in (1,\infty )$ and $s,\sigma \in \mathbf R$. Then the following is true:

• (1) if $\mu \in \mathbf R$ and $a\in {\text {SG}}^{0,\mu }_{0,\rho }({\mathbf {R}}^{2d})$, then ${\text {Op}}_t(a)$ is continuous from $H^p_\sigma ({\mathbf {R}}^{d})$ to $H^p_{\sigma -\mu }({\mathbf {R}}^{d})$;

• (2) if $m \in \mathbf R$ and $a\in {\text {SG}}^{m,0}_{r,0}({\mathbf {R}}^{2d})$, then ${\text {Op}}_t(a)$ is continuous from $L^p_s({\mathbf {R}}^{d})$ to $L^p_{s-m}({\mathbf {R}}^{d})$;

• (3) if $m,\mu \in \mathbf R$ and $a\in {\text {SG}}^{m,\mu }_{r,\rho }({\mathbf {R}}^{2d})$, then ${\text {Op}}_t(a)$ is continuous from $H^p_{s,\sigma }({\mathbf {R}}^{d})$ to $H^p_{s-m,\sigma -\mu }({\mathbf {R}}^{d})$.

2.4 Composition and Further Properties of SG Classes of Symbols, Amplitudes, and Functions

We define families of smooth functions with ${\text {SG}}$ behaviour, depending on one, two or three sets of real variables (cfr. also [21]). We then introduce pseudo-differential operators defined by means of ${\text {SG}}$ amplitudes. Subsequently, we recall sufficient conditions for maps of ${\mathbf {R}}^{d}$ into itself to keep the invariance of the ${\text {SG}}$ classes.

In analogy of ${\text {SG}}$ amplitudes defined on ${\mathbf {R}}^{2d}$, we consider corresponding classes of amplitudes defined on ${\mathbf {R}}^{3d}$. More precisely, for any $m_1, m_2, \mu , r_1,r_2,\rho \in \mathbf R$, let ${\text {SG}}^{m_1,m_2,\mu }_{r_1,r_2,\rho } ({\mathbf {R}}^{3n})$ be the set of all $a \in C^\infty \left( {\mathbf {R}}^{3d} \right)$ such that

$$\begin{aligned} |\partial ^{\alpha _1}_{x_1} \partial ^{\alpha _2}_{x_2} \partial _\xi ^\beta a(x_1, x_2, \xi )| \lesssim \langle x_1\rangle ^{m_1 - r_1|\alpha _1|} \langle x_2\rangle ^{m_2 - r_2|\alpha _2|} \langle \xi \rangle ^{\mu - \rho |\beta |}, \end{aligned}$$

(2.16)

for every multi-indices $\alpha _1, \alpha _2, \beta$. We usually assume $r_1,r_2,\rho \ge 0$ and $r_1+r_2+\rho >0$. More generally, let $\omega \in {\mathscr {P}}_{r_1,r_2,\rho }({\mathbf {R}}^{3d})$. Then ${\text {SG}}^{(\omega )}_{r_1,r_2,\rho } ({\mathbf {R}}^{3d})$ is the set of all $a \in C^\infty \left( {\mathbf {R}}^{3d} \right)$ which satisfy

for every multi-indices $\alpha _1, \alpha _2, \beta$. The set ${\text {SG}}^{ (\omega ) }_{r_1,r_2,\rho } ({\mathbf {R}}^{3n})$ is equipped with the usual Fréchet topology based upon the seminorms implicitly given in (2.16)$'$.

As above,

$$\begin{aligned} {\text {SG}}^{(\omega )}_{r_1,r_2,\rho } ={\text {SG}}^{m_1,m_2,\mu }_{r_1,r_2,\rho } \quad \text {when}\quad \omega (x_1,x_2,\xi )=\langle x_1\rangle ^{m_1}\langle x_2\rangle ^{m_2}\langle \xi \rangle ^\mu . \end{aligned}$$

Definition 2.7

Let $r_1,r_2,\rho \ge 0$, $r_1+r_2+\rho >0$, and let $a\in {\text {SG}}^{(\omega )}_{r_1,r_2,\rho }({\mathbf {R}}^{3d})$, where $\omega \in {\mathscr {P}}_{r_1,r_2,\rho }({\mathbf {R}}^{3d})$. Then the pseudo-differential operator ${\text {Op}}(a)$ is the linear and continuous operator from ${{\mathscr {S}}}({\mathbf {R}}^{d})$ to ${{\mathscr {S}}}^\prime ({\mathbf {R}}^{d})$ with distribution kernel

$$\begin{aligned} K_{a}(x,y)=(2\pi )^{-d/2}({\mathscr {F}}_3^{-1}a)(x,y,x-y). \end{aligned}$$

For $f\in {{\mathscr {S}}}({\mathbf {R}}^{d})$, we have

$$\begin{aligned} ({\text {Op}}(a)f)(x)=(2\pi )^{-d}\iint e^{i \langle x-y,\xi \rangle } a(x,y,\xi )f(y)\,dyd\xi . \end{aligned}$$

The operators introduced in Definition 2.7 have properties analogous to the usual ${\text {SG}}$ operator families described in [14]. They coincide with the operators defined in the previous subsection, where corresponding symbols are obtained by means of asymptotic expansions, modulo remainders of the type given in (2.4). For the sake of brevity, we omit the details. Evidently, when neither the amplitude functions a, nor the corresponding weight $\omega$, depend on $x_2$, we obtain the definition of ${\text {SG}}$ symbols and pseudo-differential operators, given in the previous subsection.

Next we consider ${\text {SG}}$ functions, also called functions with ${\text {SG}}$ behavior. That is, amplitudes which depend only on one set of variables in ${\mathbf {R}}^{d}$. We denote them by ${\text {SG}}^{(\omega )}_r({\mathbf {R}}^{d})$ and ${\text {SG}}^{m}_r({\mathbf {R}}^{d})$, $r>0$, respectively, for a general weight $\omega \in {\mathscr {P}}_{r}({\mathbf {R}}^{d})$ and for $\omega (x)=\langle x\rangle ^m$. Furthermore, if $\phi :{\mathbf {R}}^{d_1} \rightarrow {\mathbf {R}}^{d_2}$, and each component $\phi _j$, $j=1, \ldots , d_2$, of $\phi$ belongs to ${\text {SG}}^{(\omega )}_r({\mathbf {R}}^{d_1})$, we will occasionally write $\phi \in {\text {SG}}^{(\omega )} _r({\mathbf {R}}^{d_1};{\mathbf {R}}^{d_2})$. We use similar notation also for other vector-valued ${\text {SG}}$ symbols and amplitudes.

In the sequel we need to consider compositions of ${\text {SG}}$ amplitudes with functions with ${\text {SG}}$ behavior. In particular, the latter will often be ${\text {SG}}$ maps (or diffeomorphisms) with ${\text {SG}}^0$-parameter dependence, generated by phase functions (introduced in [16]), see Definitions 2.8 and 2.9, and Sect. 3.1 below. For the convenience of the reader, we first recall, in a form slightly more general than the one adopted in [16], the definition of ${\text {SG}}$ diffeomorphisms with ${\text {SG}}^0$-parameter dependence.

Definition 2.8

Let $\Omega _j \subseteq {\mathbf {R}}^{d_j}$ be open, $\Omega = \Omega _1\times \cdots \times \Omega _k$ and let $\phi \in C^\infty ({\mathbf {R}}^{d}\times \Omega ;{\mathbf {R}}^{d})$. Then $\phi$ is called an ${\text {SG}}$ map (with ${\text {SG}}^0$-parameter dependence) when the following conditions hold:

• (1) $\langle \phi (x,\eta )\rangle \asymp \langle x\rangle$, uniformly with respect to $\eta \in \Omega$;

• (2) for all $\alpha \in \mathbf {Z}^{d}_+$, $\beta = (\beta _1,\ldots ,\beta _k)$, $\beta _j \in \mathbf {Z}^{d_j}_+$, $j=1,\ldots , k$, and any $(x,\eta )\in {\mathbf {R}}^{d} \times \Omega$,

  $$\begin{aligned} |\partial ^\alpha _x\partial ^{\beta _1}_{\eta _1}\cdots \partial ^{\beta _k}_{\eta _k}\phi (x,\eta )| \lesssim \langle x\rangle ^{1-|\alpha |}\langle \eta _1\rangle ^{-|\beta _1|}\cdots \langle \eta _k\rangle ^{-|\beta _k|}, \end{aligned}$$

  where $\eta =(\eta _1,\ldots ,\eta _k)$ and $\eta _j\in \Omega _j$ for every j.

Definition 2.9

Let $\phi \in C^\infty ({\mathbf {R}}^{d}\times \Omega ;{\mathbf {R}}^{d} )$ be an ${\text {SG}}$ map. Then $\phi$ is called an ${\text {SG}}$ diffeomorphism (with ${\text {SG}}^0$-parameter dependence) when there is a constant $\varepsilon >0$ such that

$$\begin{aligned} |\det \phi ^\prime _x(x,\eta )|\ge \varepsilon , \end{aligned}$$

(2.17)

uniformly with respect to $\eta \in \Omega$.

Remark 2.10

The condition (1) in Definition 2.8 and (2.17), together with abstract results (see, e.g., [3], page 221) and the inverse function theorem, imply that, for any $\eta \in \Omega$, an ${\text {SG}}$ diffeomorphism $\phi (\, \cdot \, ,\eta )$ is a smooth, global bijection from ${\mathbf {R}}^{d}$ to itself with smooth inverse $\psi (\, \cdot \, ,\eta ) =\phi ^{-1}(\, \cdot \, ,\eta )$. It can be proved that also the inverse mapping $\psi (y,\eta )=\phi ^{-1}(y,\eta )$ fulfills Conditions (1) and (2) in Definition 2.8, as well as (2.17), see [16].

Definition 2.11

Let $r,\rho \ge 0$, $r+\rho >0$, $\omega \in {\mathscr {P}}_{r,\rho } ({\mathbf {R}}^{2d})$, and let $\phi ,\phi _1,\phi _2\in C^\infty ({\mathbf {R}}^{d}\times {\mathbf {R}}^{d_0};{\mathbf {R}}^{d})$ be ${\text {SG}}$ mappings.

• (1) $\omega$ is called $(\phi ,1)$ -invariant when

  $$\begin{aligned} \omega (\phi (x,\eta _1+\eta _2),\xi )\lesssim \omega (\phi (x,\eta _1),\xi ), \end{aligned}$$

  for any $x, \xi \in {\mathbf {R}}^{d}$, $\eta _1,\eta _2\in {\mathbf {R}}^{d_0}$, uniformly with respect to $\eta _2\in {\mathbf {R}}^{d_0}$. The set of all $(\phi ,1)$-invariant weights in ${\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ is denoted by ${\mathscr {P}}_{r,\rho }^{\phi ,1}({\mathbf {R}}^{2d})$;

• (2) $\omega$ is called $(\phi ,2)$ -invariant when

  $$\begin{aligned} \omega (x,\phi (\xi ,\eta _1+\eta _2))\lesssim \omega (x,\phi (\xi ,\eta _1)), \end{aligned}$$

  for any $x, \xi \in {\mathbf {R}}^{d}$, $\eta _1,\eta _2\in {\mathbf {R}}^{d_0}$, uniformly with respect to $\eta _2\in {\mathbf {R}}^{d_0}$. The set of all $(\phi ,2)$-invariant weights in ${\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ is denoted by ${\mathscr {P}}_{r,\rho }^{\phi ,2}({\mathbf {R}}^{2d})$;

• (3) $\omega$ is called $(\phi _1,\phi _2)$ -invariant if $\omega$ is both $(\phi _1,1)$-invariant and $(\phi _2,2)$-invariant. The set of all $(\phi _1,\phi _2)$-invariant weights in ${\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ is denoted by ${\mathscr {P}}_{r,\rho }^{(\phi _1,\phi _2)}({\mathbf {R}}^{2d})$

The next Lemma 2.12, proved in [24], shows that, under mild additional conditions, the families of weights introduced in Sect. 2.1 are indeed “invariant” under composition with ${\text {SG}}$ maps with ${\text {SG}}^0$-parameter dependence. That is, the compositions introduced in Definition 2.11 are still weight functions in the sense of Sect. 2.1, belonging to suitable sets $\mathscr {P}_{r,\rho }({\mathbf {R}}^{2d})$.

Lemma 2.12

Let $r,\rho \in [0,1]$, $r+\rho >0$, $\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, and let $\phi :{\mathbf {R}}^{d}\times {\mathbf {R}}^{d}\rightarrow {\mathbf {R}}^{d}$ be an ${\text {SG}}$ map as in Definition 2.8. The following statements hold true.

• (1) Assume $\omega \in {\mathscr {P}}_{1,\rho }^{\phi ,1}({\mathbf {R}}^{2d})$, and set $\omega _1(x,\xi ):=\omega (\phi (x,\xi ),\xi )$. Then $\omega _1\in {\mathscr {P}}_{1,\rho }({\mathbf {R}}^{2d})$.

• (2) Assume $\omega \in {\mathscr {P}}_{r,1}^{\phi ,2}({\mathbf {R}}^{2d})$, and set $\omega _2(x,\xi ):=\omega (x,\phi (\xi ,x))$. Then $\omega _2\in {\mathscr {P}}_{r,1}({\mathbf {R}}^{2d})$.

Remark 2.13

It is obvious that, when dealing with Fourier integral operators, the requirements for $\phi$ and $\omega$ in Lemma 2.12 need to be satisfied only on the support of the involved amplitude. By Lemma 2.12, it also follows that if $a \in {\text {SG}}^{(\omega )}_{1,1}({\mathbf {R}}^{2d})$ and $\phi =(\phi _1,\phi _2)$, where $\phi _1\in {\text {SG}}^{1,0}_{1,1}({\mathbf {R}}^{2d})$ and $\phi _2\in {\text {SG}}^{0,1}_{1,1}({\mathbf {R}}^{2d})$ are ${\text {SG}}$ maps with ${\text {SG}}^0$ parameter dependence, then $a\circ \phi \in {\text {SG}}^{({\omega _0})}_{1,1}({\mathbf {R}}^{2d})$ when ${\omega _0}:=\omega \circ \phi$, provided $\omega$ is $(\phi _1,\phi _2)$-invariant. Similar results hold for ${\text {SG}}$ amplitudes and weights defined on ${\mathbf {R}}^{3d}$.

Remark 2.14

By the definitions it follows that any weight $\omega =\vartheta _{s,\sigma }$, $s,\sigma \in \mathbf R$, is $(\phi ,1)$-, $(\phi ,2)$-, and $(\phi _1,\phi _2)$-invariant with respect to any ${\text {SG}}$ diffeomorphism with ${\text {SG}}^0$ parameter dependence $\phi$, $(\phi _1,\phi _2)$.

3 Symbolic Calculus for Generalised FIOs of ${\text {SG}}$ Type

We here recall the class of Fourier integral operators we are interested in, generalizing those studied in [16]. The corresponding symbolic calculus has been obtained in [24], from which we recall the results listed below, and to which we refer the reader for the details. A key tool in the proofs of the composition theorems below are the results on asymptotic expansions in the Weyl–Hörmander calculus obtained in [22].

3.1 Phase Functions of SG Type

We recall the definition of the class of admissible phase functions in the ${\text {SG}}$ context, as it was given in [16]. We then observe that the subclass of regular phase functions generates (parameter-dependent) mappings of ${\mathbf {R}}^{d}$ onto itself, which turn out to be ${\text {SG}}$ maps with ${\text {SG}}^0$ parameter-dependence. Finally, we define some regularizing operators, which are used to prove the properties of the ${\text {SG}}$ Fourier integral operators introduced in the next subsection.

Definition 3.1

A real-valued function $\varphi \in {\text {SG}}^{1,1}_{1,1}({\mathbf {R}}^{2d})$ is called a simple phase function (or simple phase), if

$$\begin{aligned} \langle \varphi _{\xi }^\prime (x,\xi )\rangle \asymp \langle x\rangle \quad \text{ and } \quad \langle \varphi _{x}^\prime (x,\xi )\rangle \asymp \langle \xi \rangle , \end{aligned}$$

(3.1)

are fulfilled, uniformly with respect to $\xi$ and x, respectively. The set of all simple phase functions is denoted by $\mathfrak {F}$. Moreover, the simple phase function $\varphi$ is called regular, if $\left| \det (\varphi ^{\prime \prime }_{x \xi } (x,\xi ) ) \right| \ge c$ for some $c>0$ and all $x,\xi \in {\mathbf {R}}^{d}$. The set of all regular phases is denoted by $\mathfrak {F}^r$.

We observe that a regular phase function $\varphi$ defines two globally invertible mappings, namely $\xi \mapsto \varphi ^\prime _x(x,\xi )$ and $x \mapsto \varphi ^\prime _\xi (x,\xi )$, see the analysis in [16]. Then the following result holds true for the mappings $\phi _1$ and $\phi _2$ generated by the first derivatives of the admissible regular phase functions.

Proposition 3.2

Let $\varphi \in \mathfrak {F}$. Then $\phi _1:{\mathbf {R}}^{d}\rightarrow {\mathbf {R}}^{d}:x \mapsto \varphi ^\prime _\xi (x,\xi _0)$ and $\phi _2:{\mathbf {R}}^{d}\rightarrow {\mathbf {R}}^{d}:\xi \mapsto \varphi ^\prime _x(x_0,\xi )$ are ${\text {SG}}$ maps (with ${\text {SG}}^0$ parameter dependence) from ${\mathbf {R}}^{d}$ to itself, for any $x_0,\xi _0\in {\mathbf {R}}^{d}$. If $\varphi \in \mathfrak {F}^r$, $\phi _1$ and $\phi _2$ give rise to ${\text {SG}}$ diffeomorphism with ${\text {SG}}^{0}$ parameter dependence.

For any $\varphi \in \mathfrak {F}$, the operators $\Theta _{1,\varphi }$ and $\Theta _{2,\varphi }$ are defined by

$$\begin{aligned} (\Theta _{1,\varphi }f)(x,\xi ) \equiv f(\varphi '_\xi (x,\xi ), \xi ) \quad \text {and}\quad (\Theta _{2,\varphi }f)(x,\xi ) \equiv f(x, \varphi '_x(x,\xi )), \end{aligned}$$

when $f\in C^1({\mathbf {R}}^{2d})$, and remark that the modified weights

$$\begin{aligned} (\Theta _{1,\varphi }\omega ) (x,\xi ) = \omega (\varphi '_\xi (x,\xi ), \xi ) \quad \text {and}\quad (\Theta _{2,\varphi }\omega ) (x,\xi ) = \omega (x, \varphi '_x(x,\xi )), \end{aligned}$$

(3.2)

will appear frequently in the sequel. In the following lemma we show that these weights belong to the same classes of weights as $\omega$, provided they additionally fulfill

$$\begin{aligned} \Theta _{1,\varphi }\omega \asymp \Theta _{2,\varphi }\omega \end{aligned}$$

(3.3)

when $\varphi$ is the involved phase function. That is, (3.3) is a sufficient condition to obtain $(\phi _1,1)$- and/or $(\phi _2,2)$-invariance of $\omega$ in the sense of Definition 2.11, depending on the values of the parameters $r,\rho \ge 0$.

Lemma 3.3

Let $\varphi$ be a simple phase on ${\mathbf {R}}^{2d}$, $r,\rho \in [0,1]$ be such that $r=1$ or $\rho =1$, and let $\Theta _{j,\varphi }\omega$, $j=1,2$, be as in (3.2), where $\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ satisfies (3.3). Then

$$\begin{aligned} \Theta _{j,\varphi }\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d}),\quad j=1,2. \end{aligned}$$

In what follows we let

$$\begin{aligned} ^{t}a(x,\xi )=a(\xi ,x)\quad \text {and} \quad (a^{*})(x,\xi ) =\overline{a(\xi ,x)}, \end{aligned}$$

when $a(x,\xi )$ is a function.

3.2 Generalised Fourier Integral Operators of SG Type

In analogy with the definition of generalized ${\text {SG}}$ pseudo-differential operators, recalled in Sect. 2.1, we define the class of Fourier integral operators we are interested in terms of their distributional kernels. These belong to a class of tempered oscillatory integrals, studied in [21]. Thereafter we prove that they posses convenient mapping properties.

Definition 3.4

Let $\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ satisfy (3.3), $r,\rho \ge 0$, $r+\rho >0$, $\varphi \in \mathfrak {F}$, $a,b\in {\text {SG}}^{(\omega )}_{r,\rho }({\mathbf {R}}^{2d})$.

• (1) The generalized Fourier integral operator $A={\text {Op}}_\varphi (a)$ of ${\text {SG}}$ type I (${\text {SG}}$ FIOs of type I) with phase $\varphi$ and amplitude a is the linear continuous operator from ${{\mathscr {S}}}({\mathbf {R}}^{d})$ to ${{\mathscr {S}}}^\prime ({\mathbf {R}}^{d})$ with distribution kernel $K_A\in {{\mathscr {S}}}^\prime ({\mathbf {R}}^{2d})$ given by

  $$\begin{aligned} K_A(x,y)=(2\pi )^{-d/2}({{\mathscr {F}}}_2(e^{i\varphi }a))(x,y)\text{; } \end{aligned}$$

• (2) The generalized Fourier integral operator $B={\text {Op}}_\varphi ^{*}(b)$ of ${\text {SG}}$ type II (${\text {SG}}$ FIOs of type II) with phase $\varphi$ and amplitude b is the linear continuous operator from ${{\mathscr {S}}}({\mathbf {R}}^{d})$ to ${{\mathscr {S}}}^\prime ({\mathbf {R}}^{d})$ with distribution kernel $K_B\in {{\mathscr {S}}}^ \prime ({\mathbf {R}}^{2d})$ given by

  $$\begin{aligned} K_B(x,y)=(2\pi )^{-d/2}({{\mathscr {F}}}^{-1}_2(e^{-i\varphi }\overline{b}))(y,x). \end{aligned}$$

Evidently, if $f \in {{\mathscr {S}}}({\mathbf {R}}^{d})$, and A and B are the operators in Definition 3.4, then

$$\begin{aligned} Af(x)&= {\text {Op}}_\varphi (a)u(x) = (2\pi )^{-d/2}\int e^{i \varphi (x, \xi )} \, a(x, \xi ) \, ({{{\mathscr {F}}}{f}})(\xi )\,d \xi , \end{aligned}$$

(3.4)

and

$$\begin{aligned} Bf(x)&= {\text {Op}}^{*}_\varphi (b)u(x)\nonumber \\&=(2\pi )^{-d} \iint e^{i(\langle x, \xi )- \varphi (y, \xi ))} \, \overline{b(y, \xi )} \, f(y) \,dy d\xi . \end{aligned}$$

(3.5)

Remark 3.5

In the sequel the formal ($L^2$-)adjoint of an operator Q is denoted by $Q^{*}$. By straightforward computations it follows that the ${\text {SG}}$ type I and ${\text {SG}}$ type II operators are formal adjoints to each others, provided the amplitudes and phase functions are the same. That is, if b and $\varphi$ are the same as in Definition 3.4, then ${\text {Op}}^{*}_\varphi (b)={\text {Op}}_\varphi (b)^{*}$.

Obviously, for any $\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, $^{t}\omega =\omega ^{*}$ is also an admissible weight which belongs to ${\mathscr {P}}_{\rho ,r}({\mathbf {R}}^{2d})$. Similarly, for arbitrary $\varphi \in \mathfrak {F}$ and $a \in {\text {SG}}^{(\omega )}_{r,\rho }({\mathbf {R}}^{2d})$, we have $^{t}\varphi =\varphi ^{*}\in \mathfrak {F}$ and $^{t}a, a^{*}\in {\text {SG}}^{(\omega ^{*})} _{\rho ,r}({\mathbf {R}}^{2d})$. Furthermore, by Definition 3.4 we get

$$\begin{aligned} \begin{array}{ll} {\text {Op}}_\varphi ^{*}(b) &{}= {{\mathscr {F}}}^{-1} \circ {\text {Op}}_{- \varphi ^{*}}(b^{*}) \circ {{\mathscr {F}}}^{-1} \\ &{}\qquad \Longleftrightarrow \\ {\text {Op}}_{\varphi }(a) &{}= {{\mathscr {F}}}\circ {\text {Op}}_{- \varphi ^{*}}^{*}(a^{*}) \circ {{\mathscr {F}}}. \end{array} \end{aligned}$$

(3.6)

The following result shows that type I and type II operators are linear and continuous from ${{\mathscr {S}}}({\mathbf {R}}^{d})$ to itself, and extendable to linear and continuous operators from ${{\mathscr {S}}}^\prime ({\mathbf {R}}^{d})$ to itself.

Theorem 3.6

Let a, b and $\varphi$ be the same as in Definition 3.4. Then ${\text {Op}}_\varphi (a)$ and ${\text {Op}}_\varphi ^{*}(b)$ are linear and continuous operators on ${{\mathscr {S}}}({\mathbf {R}}^{d})$, and uniquely extendable to linear and continuous operators on ${{\mathscr {S}}}'({\mathbf {R}}^{d})$.

3.3 Composition with Pseudo-differential Operators of SG Type

The composition theorems presented in this and the subsequent subsections are variants of those originally appeared in [16]. The notation used in the statements of the composition theorems are those introduced in Sects. 2.3, 3.1 and 3.2. The proofs and more details can be found in [24].

Theorem 3.7

Let $r_j,\rho _j\in [0,1]$, $\varphi \in \mathfrak {F}$ and let $\omega _j \in {\mathscr {P}}_{r_j,\rho _j}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that

$$\begin{aligned} \rho _2=1, \quad r_0=\min \{r_1,r_2,1\} ,\quad \rho _0=\min \{ \rho _1,1\}, \quad \omega _0 =\omega _1\cdot (\Theta _{2,\varphi }\omega _2), \end{aligned}$$

and $\omega _2\in \mathscr {P}_{r,1}({\mathbf {R}}^{2d})$ is $(\phi ,2)$-invariant with respect to $\phi :\xi \mapsto \varphi ^\prime _x(x,\xi )$. Also let $a \in {\text {SG}}^{(\omega _1)} _{r_1,\rho _1}({\mathbf {R}}^{2d})$, $p \in {\text {SG}}^{(\omega _2)}_{r_2,1}({\mathbf {R}}^{2d})$, and let

$$\begin{aligned} \psi (x,y,\xi ) = \varphi (y,\xi ) - \varphi (x,\xi ) - \langle y - x,\varphi '_x(x,\xi )\rangle . \end{aligned}$$

(3.7)

Then

$$\begin{aligned} {\text {Op}}(p) \circ {\text {Op}}_\varphi (a)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}_\varphi ({\text {SG}}^{(\omega \vartheta _{0,-\infty })}_0 ),&\quad r_1=0, \\ {\text {Op}}(p) \circ {\text {Op}}_\varphi (a)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}({\mathscr {S}} ),&\quad r_1>0, \end{aligned}$$

where $c \in {\text {SG}}^{(\omega _0)}_{r_0,\rho _0}({\mathbf {R}}^{2d})$ admits the asymptotic expansion

$$\begin{aligned} c(x,\xi ) \sim \sum _{\alpha } \frac{i^{|\alpha |}}{\alpha !} (D^\alpha _\xi p)(x, \varphi '_x(x,\xi )) \,D^\alpha _y \left[ e^{i \psi (x,y,\xi )} a(y,\xi ) \right] _{y=x}. \end{aligned}$$

(3.8)

Theorem 3.8

Let $r_j,\rho _j\in [0,1]$, $\varphi \in \mathfrak {F}$ and let $\omega _j \in {\mathscr {P}}_{r_j,\rho _j}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that

$$\begin{aligned} r_2=1, \quad r_0=\min \{r_1,1\} ,\quad \rho _0=\min \{ \rho _1,\rho _2,1\}, \quad \omega _0 =\omega _1\cdot (\Theta _{1,\varphi } \omega _2), \end{aligned}$$

and $\omega _2\in \mathscr {P}_{r,1}({\mathbf {R}}^{2d})$ is $(\phi ,1)$-invariant with respect to $\phi :x \mapsto \varphi ^\prime _\xi (x,\xi )$. Also let $a \in {\text {SG}}^{(\omega _1)} _{r_1,\rho _1}({\mathbf {R}}^{2d})$ and $p \in {\text {SG}}^{(\omega _2)}_{1,\rho _2}({\mathbf {R}}^{2d})$. Then

$$\begin{aligned} {\text {Op}}_\varphi (a) \circ {\text {Op}}(p)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}_\varphi ({\text {SG}}^{(\omega \vartheta _{-\infty ,0})}_0 ), \quad \rho _1=0, \\ {\text {Op}}_\varphi (a) \circ {\text {Op}}(p)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}({\mathscr {S}} ), \quad \rho _1>0, \end{aligned}$$

where the transpose ${^t}c$ of $c \in {\text {SG}}^{(\omega _0)} _{r_0,\rho _0}({\mathbf {R}}^{2d})$ admits the asymptotic expansion (3.8), after p and a have been replaced by ${^t}p$ and ${^t}a$, respectively.

Theorem 3.9

Let $r_j,\rho _j\in [0,1]$, $\varphi \in \mathfrak {F}$ and let $\omega _j \in {\mathscr {P}}_{r_j,\rho _j}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that

$$\begin{aligned} \rho _2=1, \quad r_0=\min \{r_1,r_2,1\} ,\quad \rho _0=\min \{ \rho _1,1\}, \quad \omega _0 =\omega _1\cdot (\Theta _{2,\varphi } \omega _2), \end{aligned}$$

and $\omega _2\in \mathscr {P}_{r,1}({\mathbf {R}}^{2d})$ is $(\phi ,2)$-invariant with respect to $\phi :\xi \mapsto \varphi ^\prime _x(x,\xi )$. Also let $b \in {\text {SG}}^{(\omega _1)} _{r_1,\rho _1}({\mathbf {R}}^{2d})$, $p \in {\text {SG}}^{(\omega _2)}_{r_2,1}({\mathbf {R}}^{2d})$, $\psi$ be the same as in (3.7), and let $q \in {\text {SG}}^{(\omega _2)} _{r_2,1}({\mathbf {R}}^{2d})$ be such that

$$\begin{aligned} q(x,\xi )\sim \sum _{\alpha }\frac{i^{|\alpha |}}{\alpha !}D^\alpha _x D^\alpha _\xi \overline{p(x,\xi )}. \end{aligned}$$

(3.9)

Then

$$\begin{aligned} {\text {Op}}_\varphi ^{*}(b) \circ {\text {Op}}(p)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}_\varphi ^{*}({\text {SG}}^{(\omega \vartheta _{0,-\infty })}_0 ), \quad r_1=0, \\ {\text {Op}}_\varphi ^{*}(b) \circ {\text {Op}}(p)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}({\mathscr {S}} ), \quad r_1>0, \end{aligned}$$

where $c \in {\text {SG}}^{(\omega _0)}_{r_0,\rho _0}({\mathbf {R}}^{2d})$ admits the asymptotic expansion

$$\begin{aligned} c(x,\xi ) \sim \sum _{\alpha } \frac{i^{|\alpha |}}{\alpha !} (D^\alpha _\xi q)(x, \varphi ^\prime _x(x,\xi )) D^\alpha _y \!\!\left[ e^{i \psi (x,y,\xi )} b(y,\xi ) \right] _{y=x}. \end{aligned}$$

(3.10)

Theorem 3.10

Let $r_j,\rho _j\in [0,1]$, $\varphi \in \mathfrak {F}$ and let $\omega _j \in {\mathscr {P}}_{r_j,\rho _j}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that

$$\begin{aligned} r_2=1, \quad r_0=\min \{r_1,1\} ,\quad \rho _0=\min \{ \rho _1,\rho _2,1\}, \quad \omega _0 =\omega _1\cdot (\Theta _{1,\varphi } \omega _2), \end{aligned}$$

and $\omega _2\in \mathscr {P}_{r,1}({\mathbf {R}}^{2d})$ is $(\phi ,1)$-invariant with respect to $\phi :x \mapsto \varphi ^\prime _\xi (x,\xi )$. Also let $a \in {\text {SG}}^{(\omega _1)} _{r_1,\rho _1}({\mathbf {R}}^{2d})$ and $p \in {\text {SG}}^{(\omega _2)}_{1,\rho _2}({\mathbf {R}}^{2d})$. Then

$$\begin{aligned} {\text {Op}}(p)\circ {\text {Op}}_\varphi ^{*}(b)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}_\varphi ^{*}({\text {SG}}^{(\omega \vartheta _{-\infty ,0})}_0 ), \quad \rho _1=0, \\ {\text {Op}}(p)\circ {\text {Op}}_\varphi ^{*}(b)&= {\text {Op}}_{\varphi }(c) {\text {Mod}} {\text {Op}}({\mathscr {S}} ), \quad \rho _1>0, \end{aligned}$$

where the transpose ${^t}c$ of $c \in {\text {SG}}^{(\omega _0)} _{r_0,\rho _0}({\mathbf {R}}^{2d})$ admits the asymptotic expansion (3.10), after q and b have been replaced by ${^t}q$ and ${^t}b$, respectively.

3.4 Composition Between SG FIOs of Type I and Type II

The subsequent Theorems 3.12 and 3.13 deal with the composition of a type I operator with a type II operator, and show that such compositions are pseudo-differential operators with symbols in natural classes.

The main difference, with respect to the arguments in [16] for the analogous composition results, is that we again make use, in both cases, of the generalized asymptotic expansions introduced in Definition 2.4. This allows to overcome the additional difficulty, not arising there, that the amplitudes appearing in the computations below involve weights which are still polynomially bounded, but which do not satisfy, in general, the moderateness condition (2.1). On the other hand, all the terms appearing in the associated asymptotic expansions belong to ${\text {SG}}$ classes with weights of the form $\widetilde{\omega }_{2,\varphi }\cdot \vartheta _{-k,-k}$, where $\widetilde{\omega }=\omega _1\cdot \omega _2$, which can be handled through the results in [22].

Let $S_\varphi$, $\varphi \in \mathfrak {F}$, be the operator defined by the formulae

$$\begin{aligned}&(S_\varphi f)(x,y,\xi ) = f(x,y,\Phi (x,y,\xi ))\cdot \left| \det \Phi '_\xi (x,y,\xi )\right| \nonumber \\&\quad \text {where}\quad \int _0^1 \varphi _x'(y+t(x-y),\Phi (x,y,\xi ))\, dt =\xi . \end{aligned}$$

(3.11)

That is, for every fixed $x,y\in {\mathbf {R}}^{d}$, $\xi \mapsto \Phi (x,y,\xi )$ is the inverse of the map

$$\begin{aligned} \xi \mapsto \int _0^1 \varphi _x'(y+t(x-y),\xi )\, dt . \end{aligned}$$

(3.12)

Notice that, as proved in [16], the map (3.12) is indeed invertible for (x, y) belonging to a suitable neighborhood of the diagonal $y=x$ of ${\mathbf {R}}^{d}\times {\mathbf {R}}^{d}$, and it turns out to be an ${\text {SG}}$ diffeomorphism with ${\text {SG}}^0$ parameter dependence. We also recall, from [16], the definition of the ${\text {SG}}$ compatible cut-off functions localizing to such neighborhoods.

Definition 3.11

The sets $\Xi ^\Delta (k)$, $k > 0$, of the ${\text {SG}}$ compatible cut-off functions along the diagonal of ${\mathbf {R}}^{d}\times {\mathbf {R}}^{d}$, consist of all $\chi = \chi (x,y) \in {\text {SG}}^{0,0}_{1,1}({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} \begin{array}{rcl} |y-x| \le k\langle x\rangle /2 &{} \Longrightarrow &{} \chi (x,y) = 1,\\ |y-x| > k \langle x\rangle &{} \Longrightarrow &{} \chi (x,y) = 0. \end{array} \end{aligned}$$

(3.13)

If not otherwise stated, we always assume $k \in (0,1)$.

Theorem 3.12

Let $r_j\in [0,1]$, $\varphi \in \mathfrak {F}$ and let $\omega _j \in {\mathscr {P}}_{r_j,1}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that $\omega _1$ and $\omega _2$ are $(\phi ,2)$-invariant with respect to $\phi :\xi \mapsto (\varphi ^\prime _x)^{-1}(x,\xi )$,

$$\begin{aligned} \quad r_0=\min \{r_1,r_2,1\} \quad \text {and} \quad \omega _0(x,\xi ) =\omega _1(x,\phi (x,\xi ))\omega _{2}(x,\phi (x,\xi )), \end{aligned}$$

Also let $a \in {\text {SG}}^{(\omega _1)} _{r_1,1}({\mathbf {R}}^{2d})$ and $b \in {\text {SG}}^{(\omega _2)}_{r_2,1}({\mathbf {R}}^{2d})$. Then

$$\begin{aligned} {\text {Op}}_\varphi (a) \circ {\text {Op}}_\varphi ^{*}(b) = {\text {Op}}(c), \end{aligned}$$

for some $c \in {\text {SG}}^{(\omega _0)}_{r_0,1}({\mathbf {R}}^{2d})$. Furthermore, if $\varepsilon \in (0,1)$, $\chi \in \Xi ^\Delta (\varepsilon )$, $c_0(x,y,\xi )= a(x,\xi )b(y,\xi )\chi (x,y)$ and $S_\varphi$ is given by (3.11), then h admits the asymptotic expansion

$$\begin{aligned} c(x,\xi ) \sim \sum _{\alpha } \frac{i^{|\alpha |}}{\alpha !} (D^\alpha _yD^\alpha _\xi (S_\varphi c_0))(x,y,\xi ) \big |_{y=x}. \end{aligned}$$

To formulate the next result we modify the operator $S_\varphi$ in (3.11) such that it fulfills the formulae

$$\begin{aligned} \begin{array}{ll} (S_\varphi f)(x,\xi ,\eta ) = f(\Phi (x,\xi ,\eta ),\xi ,\eta )\cdot \left| \det \Phi '_x (x,\xi ,\eta )\right| \\ \text{ where }\quad \int _0^1 \varphi _\xi '(\Phi (x,\xi ,\eta ), \eta +t(\xi -\eta ))\, dt =x . \end{array} \end{aligned}$$

(3.14)

Theorem 3.13

Let $\rho _j\in [0,1]$, $\varphi \in \mathfrak {F}^r$ and let $\omega _j \in {\mathscr {P}}_{1,\rho _j}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that $\omega _1$ and $\omega _2$ are $(\phi ,1)$-invariant with respect to $\phi :x \mapsto (\varphi ^\prime _\xi )^{-1}(x,\xi )$,

$$\begin{aligned} \quad \rho _0=\min \{ \rho _1,\rho _2,1\} \quad \text {and} \quad \omega _0(x,\xi ) =\omega _1(\phi (x,\xi ),\xi )\omega _{2}(\phi (x,\xi ),\xi ), \end{aligned}$$

Also let $a \in {\text {SG}}^{(\omega _1)} _{1,\rho _1}({\mathbf {R}}^{2d})$ and $b \in {\text {SG}}^{(\omega _2)}_{1,\rho _2}({\mathbf {R}}^{2d})$. Then

$$\begin{aligned} {\text {Op}}_\varphi ^{*}(b) \circ {\text {Op}}_\varphi (a) = {\text {Op}}(c), \end{aligned}$$

for some $c \in {\text {SG}}^{(\omega _0)}_{1,\rho _0}({\mathbf {R}}^{2d})$. Furthermore, if $\varepsilon \in (0,1)$, $\chi \in \Xi ^\Delta (\varepsilon )$, $c_0(x,\xi ,\eta )= a(x,\xi )b(x,\eta )\chi (\xi ,\eta )$ and $S_\varphi$ is given by (3.14), then h admits the asymptotic expansion

$$\begin{aligned} c(x,\xi ) \sim \sum _{\alpha } \frac{i^{|\alpha |}}{\alpha !} (D^\alpha _xD^\alpha _\eta (S_\varphi c_0))(x,\xi ,\eta ) \big |_{\eta =\xi }. \end{aligned}$$

3.5 Elliptic FIOs of Generalized SG Type and Parametrices: Egorov’s Theorem

The results about the parametrices of the subclass of generalized (${\text {SG}}$) elliptic Fourier integral operators are achieved in the usual way, by means of the composition theorems in Sects. 3.3 and 3.4. The same holds for the versions of the Egorov’s theorem adapted to the present situation. The additional conditions, compared with the statements in [16], concern the invariance of the weights, so that the hypotheses of the composition theorems above are fulfilled.

Definition 3.14

Let $r,\rho >0$, $\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, $\varphi \in \mathfrak {F}^r$ and let $a,b\in {\text {SG}}^{(\omega )}_{r,\rho }({\mathbf {R}}^{2d})$. The operators ${\text {Op}}_{\varphi }(a)$ and ${\text {Op}}^{*}_{\varphi }(b)$ are called elliptic, if a and b are ${\text {SG}}$-elliptic (cf. Sect. 1 in [26].)

Lemma 3.15

Let $\phi =(\phi _1,\phi _2)$, where $\phi _2$ and $\phi _1$ are the ${\text {SG}}$ diffeomorphisms in Theorems 3.12 and 3.13, respectively, and let $\omega \in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ be $\phi$-invariant. Also let $a\in {\text {SG}}^{(\omega )}_{1,1}({\mathbf {R}}^{2d})$ be such that ${\text {Op}}_{\varphi }(a)$ is elliptic.

Then the pseudo-differential operators ${\text {Op}}_{\varphi }(a)\circ {\text {Op}}^{*}_{\varphi }(a)$ and ${\text {Op}}^{*}_{\varphi }(a)\circ {\text {Op}}_{\varphi }(a)$ are ${\text {SG}}$ elliptic.

Theorem 3.16

Let $\omega$ be $\phi$-invariant, $\phi =(\phi _1,\phi _2)$, where $\phi _2$ and $\phi _1$ are the ${\text {SG}}$ diffeomorphisms in Theorems 3.12 and 3.13, respectively. Also let $\varphi \in \mathfrak {F}^r$, and let $a\in {\text {SG}}^{(\omega )}_{1,1}({\mathbf {R}}^{2d})$ be ${\text {SG}}$ elliptic. Then ${\text {Op}}_{\varphi }(a)$ and ${\text {Op}}^{*}_{\varphi }(a)$ admit parametrices which are elliptic ${\text {SG}}$ FIOs of type II and type I, respectively.

In the next two results we need the canonical transformation $\phi :(x,\xi )\mapsto (y,\eta )$ generated by the phase function $\varphi$, given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \xi =\varphi ^\prime _x(x,\eta )\\ y=\varphi ^\prime _\xi (x,\eta )=\varphi ^\prime _\eta (x,\eta ). \end{array}\right. } \end{aligned}$$

(3.15)

Theorem 3.17

Let $\phi$ be the canonical transformation (3.15), $\phi _0\, :\, \xi \mapsto (\varphi '_{x})^{-1}(x,\xi )$, $\omega _1,\omega _2\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$ be such that $\omega _1$ is $( \phi _0,2)$-invariant and $\omega _2$ is $\phi$-invariant, and let

$$\begin{aligned} \omega _0(x,\xi )=\omega _1(x,(\varphi '_{x})^{-1}(x,\xi ))^2 \cdot \omega _2(\phi (x,\xi )) . \end{aligned}$$

Also let $a \in {\text {SG}}^{(\omega _1)}_{1,1}({\mathbf {R}}^{2d})$ and $p \in {\text {SG}}^{(\omega _2)}_{1,1}({\mathbf {R}}^{2d})$. Then

$$\begin{aligned} {\text {Op}}_\varphi (a)\circ {\text {Op}}(p)\circ {\text {Op}}_\varphi ^{*}(a) = {\text {Op}}(p_0), \end{aligned}$$

where $p_0\in {\text {SG}}^{(\omega _0)}_{1,1}({\mathbf {R}}^{2d})$ satisfies

$$\begin{aligned} p_0(x,\xi )&= p( \varphi ^\prime _{\xi }(x, \eta ), \eta )\,|a(x,\eta )|^2\, |\det \varphi ^{\prime \prime }_{x\xi }(x,\eta )|^{-1}\!\!\!\!\mod {\text {SG}}^{(\omega _0\cdot \vartheta _{-1,-1})}_{1,1}({\mathbf {R}}^{2d}), \\&\quad \text {with}\quad \eta = (\varphi '_{x})^{-1}(x,\xi ). \end{aligned}$$

Theorem 3.18

Let $\phi$ be the canonical transformation (3.15), $\omega _1,\omega _2\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$ be such that $\omega _2$ is $\phi$-invariant, and let

$$\begin{aligned} \omega _0(x,\xi )=\omega _2(\phi (x,\xi )) . \end{aligned}$$

Also let $a \in {\text {SG}}^{(\omega _1)}_{1,1}({\mathbf {R}}^{2d})$ be elliptic, $p \in {\text {SG}}^{(\omega _2)}_{1,1}({\mathbf {R}}^{2d})$, and let b be chosen such that ${\text {Op}}_\varphi ^{*}(b)$ is a parametrix to ${\text {Op}}_\varphi (a)$. Then

$$\begin{aligned} {\text {Op}}_\varphi (a)\circ {\text {Op}}(p)\circ {\text {Op}}_\varphi ^{*}(b) = {\text {Op}}(p_0), \end{aligned}$$

where $p_0\in {\text {SG}}^{(\omega _0)}_{1,1}({\mathbf {R}}^{2d})$ satisfies

$$\begin{aligned} p_0(x,\xi ) = p( \phi (x,\xi )) \!\!\!\!\mod {\text {SG}}^{(\omega _0\cdot \vartheta _{-1,-1})}_{1,1}({\mathbf {R}}^{2d}). \end{aligned}$$

4 Continuity on Lebesgue and Modulation Spaces

In this section we recall some basic facts about continuity properties for Fourier integral operators when acting on Lebesgue and modulation spaces. We also use the analysis in previous sections in combination with certain lifting properties for modulation spaces in order to establish weighted versions of continuity results for Fourier integral operators on modulation spaces.

4.1 Continuity on Lebesgue Spaces

We start by considering the following result, which, for trivial Sobolev parameters, is related to Theorem 2.6 in [23]. A direct proof of the $L^2({\mathbf {R}}^{d})\rightarrow L^2({\mathbf {R}}^{d})$ boundedness of ${\text {Op}}_\varphi (a)$ for $a\in {\text {SG}}^{0,0}_{1,1}({\mathbf {R}}^{d})$ and a regular phase function $\varphi \in \mathfrak {F}^r$ was given in [16]. A similar argument actually holds for $a\in {\text {SG}}^{0,0}_{r,\rho }({\mathbf {R}}^{d})$, $r,\rho \ge 0$, and is given in [24] (see also [39]). Here $B_r(x_0)$ is the open ball with center at $x_0\in {\mathbf {R}}^{d}$ and radius r.

Theorem 4.1

Let $\sigma _1,\sigma _2\in \mathbf R$, $p\in (1,\infty )$ and $m,\mu \in \mathbf R$ be such that

$$\begin{aligned} m \le -(d-1)\left| \frac{1}{p}-\frac{1}{2} \right| ,\quad \mu \le -(d-1)\left| \frac{1}{p}-\frac{1}{2} \right| +\sigma _1-\sigma _2. \end{aligned}$$

Also let $\varphi \in {\text {SG}}^{1,1}_{1,1} ({\mathbf {R}}^{2d})$ be such that for some constants $c>0$ and $R>0$ and every multi-index $\alpha$ it holds

$$\begin{aligned} |\det \varphi ''_{x , \xi } (x,\xi )|&\ge c,\quad |\partial _x^\alpha \varphi (x, \xi )| \lesssim \langle x\rangle ^{1-|\alpha |}\langle \xi \rangle \\ \langle \varphi '_x (x,\xi )\rangle&\asymp \langle \xi \rangle ,\quad \langle \varphi '_\xi (x,\xi )\rangle \asymp \langle x\rangle , \end{aligned}$$

and

$$\begin{aligned} \varphi (x,t\xi ) = t\varphi (x,\xi ),\quad x,\xi \in {\mathbf {R}}^{d},\ |\xi | \ge R,\ t\ge 1. \end{aligned}$$

If $a\in {\text {SG}}^{m,\mu }_{1,1}({\mathbf {R}}^{2d})$ is supported outside ${\mathbf {R}}^{d}\times B_r(0)$ for some $r>0$, then ${\text {Op}}_\varphi (a)$ extends to a continuous operator from $H^p_{\sigma _1}({\mathbf {R}}^{d})$ to $H^p_{\sigma _2}({\mathbf {R}}^{d})$.

Proof

Let $T = \langle D\rangle ^{\sigma _2}\circ {\text {Op}}_\varphi (a) \circ \langle D\rangle ^{-\sigma _1}$. Since

$$\begin{aligned} \langle D\rangle ^{\sigma _2}: H^p_{\sigma _2}\rightarrow L^p\quad \text {and}\quad \langle D\rangle ^{-\sigma _1}: L^p\rightarrow H^p_{\sigma _1} \end{aligned}$$

are continuous bijections, the result follows if we prove that T is continuous on $L^p$.

By Theorems 3.8 and 3.9 it follows that

$$\begin{aligned} T={\text {Op}}_\varphi (a_1)\mod {\text {Op}}({{\mathscr {S}}}), \end{aligned}$$

where $a_1\in {\text {SG}}^{m,\mu _0}_{1,1}({\mathbf {R}}^{2d})$ with

$$\begin{aligned} \mu _0\le -(d-1)\left| \frac{1}{p}-\frac{1}{2} \right| . \end{aligned}$$

Furthermore, by the symbolic calculus and the fact that a is supported outside ${\mathbf {R}}^{d}\times B_r(0)$ we get

$$\begin{aligned} {\text {Op}}_\varphi (a_1) = {\text {Op}}_\varphi (a_2)\mod {\text {Op}}({{\mathscr {S}}}), \end{aligned}$$

where $a_2\in {\text {SG}}^{m,\mu _0}_{1,1}({\mathbf {R}}^{2d})$ is supported outside ${\mathbf {R}}^{d}\times B_r(0)$. Hence

$$\begin{aligned} T={\text {Op}}_\varphi (a_2)+{\text {Op}}(c), \end{aligned}$$

where $c\in {{\mathscr {S}}}$, giving that ${\text {Op}}(c)$ is continuous on $L^p$.

Since ${\text {Op}}_\varphi (a_2)$ is continuous on $L^p$, by [23, Theorem 2.6] and its proof, the result follows. $\square$

Remark 4.2

Let $\varphi$ be a phase function satisfying the hypotheses of Theorem 4.1, $s_1,s_2,\sigma _1,\sigma _2\in {\mathbf {R}}^{}$, $p\in (1,\infty )$, and assume that $a\in {\text {SG}}^{m,\mu }_{1,1}({\mathbf {R}}^{2d})$ is supported outside ${\mathbf {R}}^{d}\times B_r(0)$ for some $r>0$, with $m,\mu \in {\mathbf {R}}^{}$ satisfying

$$\begin{aligned} m \le -(d-1)\left| \frac{1}{p}-\frac{1}{2} \right| +s_1-s_2 , \quad \mu \le -(d-1)\left| \frac{1}{p}-\frac{1}{2}\right| +\sigma _1-\sigma _2. \end{aligned}$$

Then ${\text {Op}}_\varphi (a)$ extends to a continuous operator from $H^p_{s_1,\sigma _1}({\mathbf {R}}^{d})$ to $H^p_{s_2,\sigma _2}({\mathbf {R}}^{d})$, which follows by similar arguments as in the proof of Theorem 4.1 (see [23]).

4.2 Continuity on Modulation Spaces

Next we consider continuity properties on modulation spaces. The following result extends Theorem 1.2 in [13]. Here we let $M^\infty _{0,(\omega )} ({\mathbf {R}}^{d})$ be the completion of ${\mathscr {S}}({\mathbf {R}}^{d})$ under the norm $\Vert \, \cdot \, \Vert _{M^\infty _{(\omega )}}$. We also say that a (complex-valued) Gauss function $\Psi$ is non-degenerate, if $|\Psi |$ tends to zero at infinity.

Theorem 4.3

Let $m,\mu \in \mathbf R$ and $1\le p<\infty$ be such that

$$\begin{aligned} m\le -d\left| \frac{1}{2}-\frac{1}{p} \right| ,\quad \mu \le -d\left| \frac{1}{2}-\frac{1}{p} \right| , \end{aligned}$$

and let $\omega _j\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that

$$\begin{aligned} \omega _0(x,\xi ) \lesssim \frac{\omega _1(\varphi ^\prime _\xi (x,\xi ),\xi )}{\omega _2(x,\varphi ^\prime _x(x,\xi ) )}\langle x\rangle ^m \langle \xi \rangle ^\mu . \end{aligned}$$

Also let $a\in {\text {SG}}^{(\omega _0)}_{1,1}({\mathbf {R}}^{2d})$ and $\varphi \in \mathfrak {F}^r$, and assume that $\omega _j$ is $(\phi _j,j)$-invariant, $j=1,2$, with $\phi _1:x\mapsto \varphi ^\prime _\xi (x,\xi )$ and $\phi _2:\xi \mapsto \varphi ^\prime _x(x,\xi )$. Then ${\text {Op}}_\varphi (a)$ is uniquely extendable to a continuous map from $M^p_{(\omega _1)}({\mathbf {R}}^{d})$ to $M^p_{(\omega _2)}({\mathbf {R}}^{d})$ and from $M^\infty _{0,(\omega _1)}({\mathbf {R}}^{d})$ to $M^\infty _{0,(\omega _2)}({\mathbf {R}}^{d})$.

Proof

Let $\Psi$ be a Gaussian, and let $T_1$ and $T_2$ be the (Toeplitz) operators, defined by the formulas

$$\begin{aligned} (T_1f,g) = (\omega _1^{-1}V_\Psi f,V_\Psi g)\quad \text {and}\quad (T_2f,g) = (\omega _2V_\Psi f,V_\Psi g). \end{aligned}$$

Then it follows from Theorem 1.1 in [34] that $T_1$ and $T_2$ on ${\mathscr {S}}$ are uniquely extendable to continuous bijections between $M^{p}$ to $M^{p}_{(\omega _1)}$, and from $M^{p}_{(\omega _2)}$ to $M^{p}$. Since ${{\mathscr {S}}}$ is dense in $M^p_{(\omega _j)}$ and in $M^\infty _{(\omega _j)}$, the result follows if we prove

$$\begin{aligned} \Vert (T_2 \circ {\text {Op}}_\varphi (a)\circ T_1)f\Vert _{M^p}\lesssim \Vert f\Vert _{M^p},\quad f\in {{\mathscr {S}}}. \end{aligned}$$

For some non-degenerate Gauss function $\Phi$ which depends on $\Psi$ we have

$$\begin{aligned} T_j = {\text {Op}}( a_j ),\ j=1,2,\quad \text {where}\quad a_1= ((\omega _1)^{-1})*\Phi \quad \text {and}\quad a_2 = \omega _2*\Phi . \end{aligned}$$

Furthermore, using the fact that $\omega _j\in {\mathscr {P}}_{1,1}$, it follows by straight-forward computations that $a_1\in {\text {SG}}^{(1/\omega _1)}_{1,1}$ and $a_2\in {\text {SG}}^{(\omega _2)}_{1,1}$.

By using these facts in combination with Theorems 3.8 and 3.9, we get

$$\begin{aligned} T_2 {\text {Op}}_\varphi (a)\circ T_1 =T_2\circ ({\text {Op}}_\varphi (h_1) +S_1) = {\text {Op}}_\varphi (h_2)+ S_2+T_2\circ S_1, \end{aligned}$$

for some operators $S_j\in {\text {Op}}({{\mathscr {S}}})$, $j=1,2$, where

$$\begin{aligned} h_1\in {\text {SG}}^{(\omega _0/\widetilde{\omega } _1)}_{1,1}\quad \text {and}\quad h_2\in {\text {SG}}^{(\omega _0\widetilde{\omega } _2/\widetilde{\omega } _1)}_{1,1}\subseteq {\text {SG}}^{m,\mu }_{1,1}, \end{aligned}$$

$\widetilde{\omega }_1(x,\xi )=\omega _1(\varphi ^\prime _\xi (x,\xi ),\xi )$, $\widetilde{\omega }_2(x,\xi )=\omega _2(x,\varphi ^\prime _x(x,\xi ) )$. Since

$$\begin{aligned} T_2\circ S_1 \in {\text {Op}}({\text {SG}}^{(\omega _2)}_{1,1})\circ {\text {Op}}({\mathscr {S}}) \subseteq {\text {Op}}({\mathscr {S}}), \end{aligned}$$

it follows that

$$\begin{aligned} T_2 \circ {\text {Op}}_\varphi (a)\circ T_1 = {\text {Op}}_\varphi (h_2)+S_0, \end{aligned}$$

where $S_0\in {\text {Op}}({{\mathscr {S}}})$, giving that $S_0$ is continuous on $M^p$. Furthermore, the fact that $h_2\in {\text {SG}}^{m,\mu }_{1,1}$ and Theorem 1.2 in [13] imply that

$$\begin{aligned} \Vert {\text {Op}}_\varphi (h_2)f\Vert _{M^{p}}\lesssim \Vert f\Vert _{M^{p}},\quad f\in {{\mathscr {S}}}. \end{aligned}$$

This gives the result. $\square$

Remark 4.4

Let $\rho \in [1,2]$, $p,q\in [1,\infty ]$ and $t\in \mathbf R$. A Fourier integral operator which frequently appears in the literature is the continuous map from ${\mathscr {S}}({\mathbf {R}}^{d})$ to $L^\infty ({\mathbf {R}}^{d})$, given by ${\text {Op}}_{\varphi (t)} (a)$, with symbol $a(x,\xi )=1$ and a family of phase function $\varphi (t)$, parameterized by t, given by $\varphi (t,x,\xi )=it|\xi |^\rho$. That is, ${\text {Op}}_{\varphi (t)} (a)$ is the operator

$$\begin{aligned} f\mapsto (2\pi )^{-d/2}\int _{{\mathbf {R}}^{d}} e^{it|\xi |^\rho }\widehat{f}(\xi )e^{i\langle x,\xi \rangle }\, d\xi , \end{aligned}$$

for admissible f.

We remark that in [2] it is proved that ${\text {Op}}_{\varphi (t)}(a)$ is uniquely extendable to a continuous map on $M^{p,q}({\mathbf {R}}^{d})$. In particular, Theorem 4.3 holds for $\omega _0=\omega _1=\omega _2$, without the loss of regularity, imposed by the conditions on m and $\mu$. We also remark that the latter result was proved in the case $\rho =2$ already in [42]. (See also [12, 45] and the references therein for other related results and approaches.)

The continuity properties of ${\text {SG}}$ pseudo-differential operators on modulation spaces, as well as the propagation of the global wave-front sets under their action, shortly recalled in the next Sect. 5, motivate the next definition, originally given in [26].

Definition 4.5

Let $r,\rho \in [0,1]$, $t\in \mathbf R$, and let ${\mathcal B}$ be a topological vector space of distributions on ${\mathbf {R}}^{d}$ such that

$$\begin{aligned} {\mathscr {S}}({\mathbf {R}}^{d})\subseteq {\mathcal B}\subseteq {\mathscr {S}}'({\mathbf {R}}^{d}) \end{aligned}$$

with continuous embeddings. Then ${\mathcal B}$ is called ${\text {SG}}$ -admissible (with respect to $(r, \rho )$) when ${\text {Op}}_t(a)$ maps ${\mathcal B}$ continuously into itself, for every $a\in {\text {SG}}^{0,0}_{r,\rho }({\mathbf {R}}^{d})$. If ${\mathcal B}$ and ${\mathcal C}$ are ${\text {SG}}$-admissible with respect to $(r, \rho )$, and $\omega _0\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, then the pair $({\mathcal B},{\mathcal C})$ is called ${\text {SG}}$ -ordered (with respect to $(r, \rho ,\omega _0)$), when the mappings

$$\begin{aligned} {\text {Op}}_t(a)\, :\, {\mathcal B}\rightarrow {\mathcal C}\quad \text {and}\quad {\text {Op}}_t(b)\, :\, {\mathcal C}\rightarrow {\mathcal B}\end{aligned}$$

are continuous for every $a\in {\text {SG}}^{(\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$ and $b\in {\text {SG}}^{(1/\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$.

The following definition, which extends Definition 4.5 to the case of generalized Fourier integral operators, is justified by Theorems 4.1 and 4.3.

Definition 4.6

Let $\varphi \in {\text {SG}}^{1,1}_{1,1}({\mathbf {R}}^{2d})$ be a regular phase function, and ${\mathcal B}$, ${\mathcal B}_1$, ${\mathcal B}_2$, ${\mathcal C}$, ${\mathcal C}_1$, ${\mathcal C}_2$, be ${\text {SG}}$-admissible with respect to r, $\rho$ and d. Also let $\omega _0,\omega _1,\omega _2\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, and $\Omega \subseteq {\mathbf {R}}^{d}$ be open. Then the pair $({\mathcal B},{\mathcal C})$ is called weakly-I ${\text {SG}}$ -ordered (with respect to $(r,\rho ,\omega _0,\varphi ,\Omega )$), when the mapping

$$\begin{aligned} {\text {Op}}_\varphi (a)\, :\, {\mathcal B}\rightarrow {\mathcal C}\end{aligned}$$

is continuous for every $a\in {\text {SG}}^{(\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$ which is supported outside ${\mathbf {R}}^{d}\times \Omega$. Similarly, the pair $({\mathcal B},{\mathcal C})$ is called weakly-II ${\text {SG}}$ -ordered (with respect to $(r,\rho ,\omega _0,\varphi ,\Omega )$), when the mapping

$$\begin{aligned} {\text {Op}}_\varphi ^{*}(b)\, :\, {\mathcal C}\rightarrow {\mathcal B}\end{aligned}$$

is continuous for every $b\in {\text {SG}}^{(\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$ which is supported outside $\Omega \times {\mathbf {R}}^{d}$. Furthermore, $({\mathcal B}_1, {\mathcal C}_1, {\mathcal B}_2, {\mathcal C}_2)$ are called ${\text {SG}}$ -ordered (with respect to r, $\rho$, $\omega _1$, $\omega _2$, $\varphi$, and $\Omega$), when $({\mathcal B}_1,{\mathcal C}_1)$ is a weakly-I ${\text {SG}}$-ordered pair with respect to $(r,\rho ,\omega _1,\varphi ,\Omega )$, and $({\mathcal B}_2,{\mathcal C}_2)$ is a weakly-II ${\text {SG}}$-ordered pair with respect to $(r,\rho ,\omega _2,\varphi ,\Omega )$.

Remark 4.7

Let $\sigma _1$, $\sigma _2$, p, m and $\mu$ be the same as in Theorem 4.1. Then it follows from [26, Remark 1.9], and Theorem 4.1, Remark 4.2, and Theorem 4.3, that the following is true.

• (1) $(H^p_{\sigma _1,\sigma _2},H^p_{\sigma _1-\mu ,\sigma _2-m})$ are weakly-I ${\text {SG}}$-ordered with respect to $(r,\rho ,\omega _0,\varphi ,\Omega )$, when

  $$\begin{aligned} \omega _0(x,\xi ) = \langle x\rangle ^{m}\langle \xi \rangle ^{\mu }\quad \text {and}\quad \Omega = B_\varepsilon (0), \varepsilon >0\text{. } \end{aligned}$$

• (2) If p, m, $\mu$ and $\omega _j$, $j=0,1,2$ are the same as in Theorem 4.3, then it follows that $(M^p_{(\omega _1)},M^p_{(\omega _2)})$ are weakly-I ${\text {SG}}$-ordered with respect to $(r,\rho ,\omega _0,\varphi ,\emptyset )$. If, in addition, $\varphi (x,\xi )=\langle x,\xi \rangle$ and ${\mathscr {B}}$ is an invariant BF-space, then $(M(\omega _1,{\mathscr {B}}), M(\omega _2,{\mathscr {B}}))$ are ${\text {SG}}$-ordered with respect to $\omega _0$.

5 Propagation Results for Global Wave-Front Sets and Generalised FIOs of ${\text {SG}}$ Type

We first recall the definition of the global wave-front sets, given in [26]. The content of Sect. 5.1 again comes from [27]. In Sect. 5.2 we prove our main results about the propagation of singularities in the ${\text {SG}}$ context, under the action of the Fourier integral operators described above.

5.1 Global Wave-Front Sets

Here we recall the definition given in [26] of global wave-front sets for temperate distributions with respect to appropriate Banach or Fréchet spaces and state some of their properties (see also [27]). First we recall the definitions of the sets of characteristic points. Notice that if $a\in {\text {SG}}^{(\omega _0 )}_{r,\rho }({\mathbf {R}}^{2d})$, then

$$\begin{aligned} |a(x,\xi )|\lesssim \omega _0 (x,\xi ). \end{aligned}$$

On the other hand, a is invertible, in the sense that 1 / a is a symbol in ${\text {SG}}^{(1/\omega _0 )}_{r,\rho }({\mathbf {R}}^{2d})$, if and only if

$$\begin{aligned} \omega _0 (x,\xi )\lesssim |a(x,\xi )|. \end{aligned}$$

(5.1)

We need to deal with the situations where (5.1) holds only in certain (conic-shaped) subset of ${\mathbf {R}}^{d} \times {\mathbf {R}}^{d}$. Here we let $\Omega _m$, $m=1,2,3$, be the sets

$$\begin{aligned} \Omega _1&= {\mathbf {R}}^{d}\times ({\mathbf {R}}^{d}\setminus 0),\quad \Omega _2 = ({\mathbf {R}}^{d} \setminus 0)\times {\mathbf {R}}^{d}, \nonumber \\ \Omega _3&= ({\mathbf {R}}^{d} \setminus 0)\times ({\mathbf {R}}^{d} \setminus 0), \end{aligned}$$

(5.2)

Definition 5.1

Let $r,\rho \ge 0$, $\omega _0\in {\mathscr {P}} _{r,\rho } ({\mathbf {R}}^{2d})$, $\Omega _m$, $m=1,2,3$ be as in (5.2), and let $a\in {\text {SG}}^{(\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$.

• (1) a is called locally or type-1 invertible with respect to $\omega _0$ at the point $(x_0,\xi _0)\in \Omega _1$, if there exist a neighbourhood X of $x_0$, an open conical neighbourhood $\Gamma$ of $\xi _0$ and a positive constant R such that (5.1) holds for $x\in X$, $\xi \in \Gamma$ and $|\xi |\ge R$.

• (2) a is called Fourier-locally or type-2 invertible with respect to $\omega _0$ at the point $(x_0,\xi _0)\in \Omega _2$, if there exist an open conical neighbourhood $\Gamma$ of $x_0$, a neighbourhood X of $\xi _0$ and a positive constant R such that (5.1) holds for $x\in \Gamma$, $|x|\ge R$ and $\xi \in X$.

• (3) a is called oscillating or type-3 invertible with respect to $\omega _0$ at the point $(x_0,\xi _0)\in \Omega _3$, if there exist open conical neighbourhoods $\Gamma _1$ of $x_0$ and $\Gamma _2$ of $\xi _0$, and a positive constant R such that (5.1) holds for $x\in \Gamma _1$, $|x|\ge R$, $\xi \in \Gamma _2$ and $|\xi |\ge R$.

If $m\in \{ 1,2,3\}$ and a is not type-m invertible with respect to $\omega _0$ at $(x_0,\xi _0)\in \Omega _m$, then $(x_0,\xi _0)$ is called type- m characteristic for a with respect to $\omega _0$. The set of type-m characteristic points for a with respect to $\omega _0$ is denoted by ${\text {Char}}_{(\omega _0)}^m(a)$.

The (global) set of characteristic points (the characteristic set), for a symbol $a\in {\text {SG}}^{(\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$ with respect to $\omega _0$ is defined as

$$\begin{aligned} {\text {Char}}(a)={\text {Char}}_{(\omega _0)}(a)={\text {Char}}^1_{(\omega _0)}(a)\bigcup {\text {Char}}^2 _{(\omega _0)}(a)\bigcup {\text {Char}}^3_{(\omega _0)}(a). \end{aligned}$$

Remark 5.2

In the case $\omega _0=1$ we exclude the phrase “with respect to $\omega _0$” in Definition 5.1. For example, $a\in {\text {SG}}^{0,0}_{r,\rho }({\mathbf {R}}^{2d})$ is type-1 invertible at $(x_0,\xi _0) \in {\mathbf {R}}^{d}\times ({\mathbf {R}}^{d}\backslash {0})$ if $(x_0,\xi _0)\notin {\text {Char}}^1 _{(\omega _0)}(a)$ with $\omega _0=1$. This means that there exist a neighbourhood X of $x_0$, an open conical neighbourhood $\Gamma$ of $\xi _0$ and $R>0$ such that (5.1) holds for $\omega _0=1$, $x\in X$ and $\xi \in \Gamma$ satisfies $|\xi |\ge R$.

In the next definition we introduce different classes of cutoff functions (see also Definition 1.9 in [25]).

Definition 5.3

Let $X\subseteq {\mathbf {R}}^{d}$ be open, $\Gamma \subseteq {\mathbf {R}}^{d}\setminus 0$ be an open cone, $x_0\in X$ and let $\xi _0\in \Gamma$.

• (1) A smooth function $\varphi$ on ${\mathbf {R}}^{d}$ is called a cutoff (function) with respect to $x_0$ and X, if $0\le \varphi \le 1$, $\varphi \in C_0^\infty (X)$ and $\varphi =1$ in an open neighbourhood of $x_0$. The set of cutoffs with respect to $x_0$ and X is denoted by ${\mathscr {C}}_{x_0}(X)$ or ${\mathscr {C}}_{x_0}$.

• (2) A smooth function $\psi$ on ${\mathbf {R}}^{d}$ is called a directional cutoff (function) with respect to $\xi _0$ and $\Gamma$, if there is a constant $R>0$ and open conical neighbourhood $\Gamma _1\subseteq \Gamma$ of $\xi _0$ such that the following is true:

  • $0\le \psi \le 1$ and ${\text {supp}}\psi \subseteq \Gamma$;

  • $\psi (t\xi )=\psi (\xi )$ when $t\ge 1$ and $|\xi |\ge R$;

  • $\psi (\xi )=1$ when $\xi \in \Gamma _1$ and $|\xi |\ge R$.

  The set of directional cutoffs with respect to $\xi _0$ and $\Gamma$ is denoted by ${\mathscr {C}}^{{\text {dir}}}_{\xi _0}(\Gamma )$ or ${\mathscr {C}}^{{\text {dir}}}_{\xi _0}$.

Remark 5.4

Let $X\subseteq {\mathbf {R}}^{d}$ be open and $\Gamma ,\Gamma _1,\Gamma _2 \subseteq {\mathbf {R}}^{d}\backslash {0}$ be open cones. Then the following is true.

• (1) if $x_0\in X$, $\xi _0\in \Gamma$, $\varphi \in {\mathscr {C}} _{x _0}(X)$ and $\psi \in {\mathscr {C}} ^{{{\text {dir}}}} _{\xi _0}(\Gamma )$, then $c_1=\varphi \otimes \psi$ belongs to ${\text {SG}}^{0,0}_{1,1}({\mathbf {R}}^{2d})$, and is type-1 invertible at $(x_0,\xi _0)$;

• (2) if $x_0\in \Gamma$, $\xi _0\in X$, $\psi \in {\mathscr {C}} ^{{{\text {dir}}}} _{x_0}(\Gamma )$ and $\varphi \in {\mathscr {C}} _{\xi _0}(X)$, then $c_2=\varphi \otimes \psi$ belongs to ${\text {SG}}^{0,0}_{1,1}({\mathbf {R}}^{2d})$, and is type-2 invertible at $(x_0,\xi _0)$;

• (3) if $x_0\in \Gamma _1$, $\xi _0\in \Gamma _2$, $\psi _1\in {\mathscr {C}} ^{{{\text {dir}}}} _{x_0}(\Gamma _1)$ and $\psi _2\in {\mathscr {C}} ^{{{\text {dir}}}} _{\xi _0}(\Gamma _2)$, then $c_3=\psi _1 \otimes \psi _2$ belongs to ${\text {SG}}^{0,0}_{1,1}({\mathbf {R}}^{2d})$, and is type-3 invertible at $(x_0,\xi _0)$.

The next proposition shows that ${\text {Op}}_t(a)$ for $t\in \mathbf R$ satisfies convenient invertibility properties of the form

$$\begin{aligned} {\text {Op}}_t(a)\circ {\text {Op}}_t(b) = {\text {Op}}_{t}(c) +{\text {Op}}_{t}(h), \end{aligned}$$

(5.3)

outside the set of characteristic points for a symbol a. Here ${\text {Op}}_t(b)$, ${\text {Op}}_t(c)$ and ${\text {Op}}_t(h)$ have the roles of “local inverse”, “local identity” and smoothing operators respectively. From these statements it also follows that our set of characteristic points in Definition 5.1 are related to those in [19, 35]. We let $\mathbb I_m$, $m=1,2,3$, be the sets

$$\begin{aligned} \mathbb I_1 \equiv [0,1]\times (0,1],\, \mathbb I_2 \equiv (0,1]\times [0,1],\, \mathbb I_3 \equiv (0,1]\times (0,1] = \mathbb I_1\cap \mathbb I_2. \end{aligned}$$

(5.4)

which will be useful in the sequel.

Proposition 5.5

Let $m\in \{1,2,3\}$, $(r,\rho )\in \mathbb I_m$, $\omega _0\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$ and let $a\in {\text {SG}}^{(\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$. Also let $\Omega _m$ be as in (5.2), $(x_0,\xi _0)\in \Omega _m$, and let $(r_0,\rho _0)$ be equal to (r, 0), $(0,\rho )$ and $(r,\rho )$ when m is equal to 1, 2 and 3, respectively. Then the following conditions are equivalent:

• (1) $(x_0,\xi _0)\notin {\text {Char}}_{(\omega _0)}^m (a)$;

• (2) there is an element $c\in {\text {SG}}^{0,0}_{r,\rho }$ which is type-m invertible at $(x_0,\xi _0)$, and an element $b\in {\text {SG}}^{(1/\omega _0)}_{r,\rho }$ such that $ab=c$;

• (3) (5.3) holds for some $c\in {\text {SG}}^{0,0}_{r,\rho }$ which is type-m invertible at $(x_0,\xi _0)$, and some elements $h\in {\text {SG}}^{-r_0,-\rho _0}_{r,\rho }$ and $b\in {\text {SG}}^{(1/\omega _0)}_{r,\rho }$;

• (4) (5.3) holds for some $c_m\in {\text {SG}}^{0,0}_{r,\rho }$ in Remark 5.4 which is type-m invertible at $(x_0,\xi _0)$, and some elements h and $b\in {\text {SG}}^{(1/\omega _0)}_{r,\rho }$, where $h\in {\mathscr {S}}$ when $m\in \{ 1,3\}$ and $h\in {\text {SG}}^{-\infty ,0}$ when $m=2$. Furthermore, if $t=0$, then the supports of b and h can be chosen to be contained in $X\times {\mathbf {R}}^{d}$ when $m=1$, in $\Gamma \times {\mathbf {R}}^{d}$ when $m=2$, and in $\Gamma _1\times {\mathbf {R}}^{d}$ when $m=3$.

We now introduce the complements of the wave-front sets. More precisely, let $\Omega _m$, $m\in \{ 1,2,3\}$, be given by (5.2), ${\mathcal B}$ be an ${\text {SG}}$-admissible (Banach or Fréchet) space, and let $f\in {\mathscr {S}}'({\mathbf {R}}^{d})$. We recall that ${\mathscr {S}}({\mathbf {R}}^{d})\subseteq {\mathcal B}\subseteq {\mathscr {S}}'({\mathbf {R}}^{d})$ by the definitions, and that ${\mathscr {S}}({\mathbf {R}}^{d})$, Sobolev–Kato spaces and, more generally, modulation spaces, are ${\text {SG}}$-admissible, see [26, 27]. Then the point $(x_0,\xi _0)\in \Omega _m$ is called type- m regular for f with respect to ${\mathcal B}$, if

$$\begin{aligned} {\text {Op}}(c_m)f\in {\mathcal B}, \end{aligned}$$

(5.5)

for some $c_m$ in Remark 5.4. The set of all type-m regular points for f with respect to ${\mathcal B}$, is denoted by $\Theta ^m_{{\mathcal B}}(f)$.

Definition 5.6

Let $m\in \{ 1,2,3\}$, $\Omega _m$ be as in (5.2), and let ${\mathcal B}$ be an ${\text {SG}}$-admissible (Banach or Fréchet) space.

• (1) The type- m wave-front set of $f\in {\mathscr {S}}'({\mathbf {R}}^{d})$ with respect to ${\mathcal B}$ is the complement of $\Theta ^m_{{\mathcal B}}(f)$ in $\Omega _m$, and is denoted by ${\text {WF}}^m_{{\mathcal B}}(f)$;

• (2) The global wave-front set ${\text {WF}}_{{\mathcal B}}(f)\subseteq ({\mathbf {R}}^{d}\times {\mathbf {R}}^{d})\backslash {0}$ is the set

  $$\begin{aligned} {\text {WF}}_{{\mathcal B}}(f) \equiv {\text {WF}}^1_{\mathcal B}(f) \bigcup {\text {WF}}^2_{\mathcal B}(f) \bigcup {\text {WF}}^3_{\mathcal B}(f). \end{aligned}$$

The sets ${\text {WF}}^1_{{\mathcal B}}(f)$, ${\text {WF}}^2_{{\mathcal B}}(f)$ and ${\text {WF}}^3_{{\mathcal B}}(f)$ in Definition 5.6, are also called the local, Fourier-local and oscillating wave-front set of f with respect to ${\mathcal B}$.

Remark 5.7

Let $\Omega _m$, $m=1,2,3$ be the same as in (5.2).

• (1) If $\Omega \subseteq \Omega _1$, and $(x_0,\xi _0)\in \Omega \ \Longleftrightarrow \ (x_0,\sigma \xi _0)\in \Omega$ for $\sigma \ge 1$, then $\Omega$ is called 1-conical;

• (2) If $\Omega \subseteq \Omega _2$, and $(x_0,\xi _0)\in \Omega \ \Longleftrightarrow \ (sx_0,\xi _0)\in \Theta ^2_{{\mathcal B}}(f)$ for $s \ge 1$, then $\Omega$ is called 2-conical;

• (3) If $\Omega \subseteq \Omega _3$, and $(x_0,\xi _0) \in \Omega \ \Longleftrightarrow \ (sx_0,\sigma \xi _0)\in \Omega$ for $s,\sigma \ge 1$, then $\Omega$ is called 3-conical.

By (5.5) and the paragraph before Definition 5.6, it follows that if $m=1,2,3$, then $\Theta ^m_{\mathcal B}(f)$ is m-conical. The same holds for ${\text {WF}}^m_{\mathcal B}(f)$, $m=1,2,3$, by Definition 5.6, noticing that, for any $x_0\in {\mathbf {R}}^{r} \setminus \{0\}$, any open cone $\Gamma \ni x_0$, and any $s>0$, ${\mathscr {C}} ^{{{\text {dir}}}} _{x_0}(\Gamma ) ={\mathscr {C}} ^{{{\text {dir}}}} _{s x_0}(\Gamma )$. For any $R>0$ and $m\in \{1,2,3\}$, we set

$$\begin{aligned} \Omega _{1,R}&\equiv \{ \, (x,\xi )\in \Omega _1\, ;\, |\xi |\ge R\, \} , \quad \Omega _{2,R} \equiv \{ \, (x,\xi )\in \Omega _2\, ;\, |x |\ge R\, \} , \\ \Omega _{3,R}&\equiv \{ \, (x,\xi )\in \Omega _3\, ;\, |x|, |\xi |\ge R\, \} \end{aligned}$$

Evidently, $\Omega _{m, R}$ is m-conical for every $m\in \{ 1,2,3\}$.

The next result describes the relation between “regularity with respect to ${\mathcal B}$ ” of temperate distributions and global wave-front sets.

Proposition 5.8

Let ${\mathcal B}$ be ${\text {SG}}$-admissible, and let $f\in {{\mathscr {S}}}^\prime ({\mathbf {R}}^{d})$. Then

$$\begin{aligned} f\in {\mathcal B}\quad \Longleftrightarrow \quad {\text {WF}}_{{\mathcal B}}(f)=\emptyset . \end{aligned}$$

For the sake of completeness, we recall that microlocality and microellipticity hold for these global wave-front sets and pseudo-differential operators in ${\text {Op}}({\text {SG}}^{(\omega _0)}_{r,\rho })$, see [26]. This implies that operators which are elliptic with respect to $\omega _0\in {\mathscr {P}}_{\rho ,\delta }({\mathbf {R}}^{2d})$ when $0< r ,\rho \le 1$ preserve the global wave-front set of temperate distributions. The next result is an immediate corollary of microlocality and microellipticity for operators in ${\text {Op}}({\text {SG}}^{(\omega _0)}_{r,\rho })$:

Proposition 5.9

Let $m\in \{ 1,2,3\}$, $(r,\rho )\in \mathbb I_m$, $t\in \mathbf R$, $\omega _0\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, $a\in {\text {SG}}^{(\omega _0)}_{r,\rho } ({\mathbf {R}}^{2d})$ be ${\text {SG}}$-elliptic with respect to $\omega _0$ and let $f\in {{\mathscr {S}}}'({\mathbf {R}}^{d})$. Moreover, let $({\mathcal B}, {\mathcal C})$ be an ${\text {SG}}$-ordered pair with respect to $\omega _0$. Then

$$\begin{aligned} {\text {WF}}^{m} _{\mathcal C} ({\text {Op}}_t(a)f)&= {\text {WF}}^{m}_{\mathcal B} (f). \end{aligned}$$

5.2 Action of Generalised FIOs of SG Type on Global Wave-Front Sets

We let $\phi$ be the canonical transformation of $T^{*} {\mathbf {R}}^{d}$ into itself generated by the phase function $\varphi \in \mathfrak {F}^r$. This means that $\phi = (\phi _1,\phi _2)$ is the smooth function on $T^{*}{\mathbf {R}}^{d}$ into itself, defined by the relations

$$\begin{aligned} (x,\xi )=\phi (y,\eta ) \Longleftrightarrow {\left\{ \begin{array}{ll} y = \varphi ^\prime _{\xi } (x,\eta ) = \varphi ^\prime _{\eta } (x,\eta ), \\ \xi = \varphi ^\prime _x (x, \eta ), \end{array}\right. } \end{aligned}$$

(5.6)

As we have seen in Sect. 3.5, such transformations appear in the Egorov’s theorem, through which we prove Theorems 5.14 and Corollaries 5.15 and 5.16 below. This justifies the following definition of admissibility of phase functions. Namely, the latter are required to generate transformations of the type (5.6) which “preserve the shape” of the different kinds of neighborhoods appearing in the Definition 5.1 of the set of characteristic points.

Definition 5.10

Let $\varphi \in \mathfrak {F}^r$ and let $\phi$ be the canonical transformation (5.6), generated by $\varphi$. Let $m\in \{1,2,3\}$ and $\Omega _m$ be as in (5.2).

• (1) $\varphi$ is called 1-admissible at $(y_0,\eta _0) \in \Omega _1$ if, for every 1-cone $X\times \Gamma$ containing $\phi (y_0,\eta _0)$ and $r>0$, there is a 1-cone $Y\times \Gamma _0$ containing $(y_0,\eta _0)$ and $R>0$ such that

  $$\begin{aligned} \phi (y,\eta ) \in (X\times \Gamma )\bigcap \Omega _{1,r} \quad \text {when}\quad (y,\eta ) \in (Y\times \Gamma _0)\bigcap \Omega _{1,R}\text{; } \end{aligned}$$

• (2) $\varphi$ is called 2-admissible at $(y_0,\eta _0) \in \Omega _2$ if, for every 2-cone $\Gamma \times X$ containing $\phi (y_0,\eta _0)$ and $r>0$, there is a 2-cone $\Gamma _0\times Y$ containing $(y_0,\eta _0)$ and $R>0$ such that

  $$\begin{aligned} \phi (y,\eta ) \in (\Gamma \times X)\bigcap \Omega _{2,r} \quad \text {when}\quad (y,\eta ) \in (\Gamma _0\times Y)\bigcap \Omega _{2,R}\text{; } \end{aligned}$$

• (3) $\varphi$ is called 3-admissible at $(y_0,\eta _0) \in \Omega _3$ if, for every 3-cone $\Gamma _1\times \Gamma _2$ containing $\phi (y_0,\eta _0)$ and $r>0$, there is a 3-cone $\Gamma _{0,1}\times \Gamma _{0,2}$ containing $(y_0,\eta _0)$ and $R>0$ such that

  $$\begin{aligned} \phi (y,\eta ) \in (\Gamma _1\times \Gamma _2) \bigcap \Omega _{3,r} \quad \text {when}\quad (y,\eta ) \in (\Gamma _{0,1}\times \Gamma _{0,2})\bigcap \Omega _{3,R}\text{. } \end{aligned}$$

Furthermore, $\varphi$ is called m -admissible if it is m-admissible at all points $(y,\eta )\in \Omega _m$, and $\varphi$ is called admissible if it is m-admissible for all $m=1,2,3$.

Remark 5.11

Notice that the inverse transformation $\phi ^{-1}$ is defined as in (5.6), by exchanging the role of $(x,\xi )$ and $(y,\eta )$. If $\varphi$ is m-admissible, $m=1,2,3$, then both $\phi$ and $\phi ^{-1}$ satisfy the corresponding property in Definition 5.10.

Remark 5.12

Let $\varphi$ be m-admissible, $m=1,2,3$, and let $\omega _0\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$ be invariant with respect to the canonical transformation (5.6). For any $a\in {\text {SG}}^{(\omega _0)} _{r,\rho }({\mathbf {R}}^{2d})$, setting $\widetilde{\omega }_0= \omega _0\circ \phi$, we have

$$\begin{aligned} (y_0,\eta _0)\in {\text {Char}}^m_{(\widetilde{\omega }_0)}(a\circ \phi ) \quad \Longleftrightarrow \quad (x_0,\xi _0)=\phi (y_0,\eta _0)\in {\text {Char}}^m_{(\omega _0)}(a). \end{aligned}$$

By Remark 5.11, similar properties hold with $\phi ^{-1}$ in place of $\phi$.

Remark 5.13

Let $\varphi _m$, $m=1,2,3$, be phase functions such that

• $\xi \mapsto \varphi _1(x,\xi )$ is homogeneous of order 1 for large $|\xi |$;

• $x \mapsto \varphi _2(x,\xi )$ is homogeneous of order 1 for large |x|;

• $x \mapsto \varphi _3(x,\xi )$ and $\xi \mapsto \varphi _3(x,\xi )$ are homogeneous of order 1 for large |x| and $|\xi |$.

Such phase functions are common in the literature. An example of admissible phase functions, which is not necessarily homogeneous, is given by the so called ${\text {SG}}$ -classical phase functions. Families of such objects, smoothly depending on a parameter $t\in [-T,T]$, $T>0$, are obtained by solving Cauchy problems associated with classical ${\text {SG}}$-hyperbolic systems with diagonal principal part.

In fact, omitting the dependence on the time variable t, an ${\text {SG}}$-classical phase functions $\varphi$ admits expansions in terms which are homogeneous with respect to x, respectively $\xi$, satisfying suitable compatibility relations, see, e.g., [18, 19]. In particular, $\varphi$ admits a principal symbol, given by a triple $(\varphi _1,\varphi _2,\varphi _3)$, that is, it can be written as

$$\begin{aligned} \begin{array}{ll} \varphi (x,\xi )&{}=\chi (\xi )\,\varphi _1(x,\xi )+\chi (x)(\varphi _2(x,\xi ) -\chi (\xi )\,\varphi _3(x,\xi )) \\ &{}\mod {\text {SG}}^{0,0}_{1,1}({\mathbf {R}}^{2d}). \end{array} \end{aligned}$$

(5.7)

In (5.7), $\chi$ is a 0-excision function, while $\chi (\xi )\,\varphi _1(x,\xi )$, $\chi (x)\,\varphi _2(x,\xi )$, $\chi (\xi )\, \chi (x)\varphi _3(x,\xi )\in {\text {SG}}^{1,1}_{1,1}({\mathbf {R}}^{2d})$, where $\varphi _1$ is 1-homogeneous with respect to the variable $\xi$, $\varphi _2$ is 1-homogeneous with respect to the variable x, and $\varphi _3$ is 1-homogeneous with respect to each one of the variables $x,\xi$. Observe that then

$$\begin{aligned} \varphi (x,\xi )&=\chi (\xi )\,\varphi _1(x,\xi )\mod {\text {SG}}^{1,0}_{1,1}({\mathbf {R}}^{2d}), \nonumber \\ \varphi (x,\xi )&=\chi (x)\,\varphi _2(x,\xi )\mod {\text {SG}}^{0,1}_{1,1}({\mathbf {R}}^{2d}),\nonumber \\ \varphi (x,\xi )&=\chi (x)\,\chi (\xi )\,\varphi _3(x,\xi )\mod {\text {SG}}^{0,1}_{1,1}({\mathbf {R}}^{2d})+{\text {SG}}^{1,0}_{1,1}({\mathbf {R}}^{2d}). \end{aligned}$$

(5.8)

The homogeneity of the leading terms in (5.8) implies, in particular,

$$\begin{aligned} \varphi ^\prime _x(x,\xi )&= |\xi |\left[ \varphi ^\prime _{1,x}\left( x,\dfrac{\xi }{|\xi |}\right) +|\xi |^{-1}r_1(x,\xi )\right] \text { for }|\xi |>R, \nonumber \\ \varphi ^\prime _\xi (x,\xi )&= |x|\left[ \varphi ^\prime _{2,\xi }\left( \dfrac{x}{|x|},\xi \right) +|x|^{-1}r_2(x,\xi )\right] \text { for }|x|>R, \nonumber \\ \varphi ^\prime _x(x,\xi )&= |\xi |\left[ \varphi ^\prime _{3,x}\left( \dfrac{x}{|x|},\dfrac{\xi }{|\xi |}\right) + |\xi |^{-1}(r_{31}(x,\xi )+s_{31}(x,\xi ))\right] \nonumber \\&\quad \text {for}\;|x|,|\xi |>R, \nonumber \\ \varphi ^\prime _\xi (x,\xi )&= |x|\left[ \varphi ^\prime _{3,\xi }\left( \dfrac{x}{|x|},\dfrac{\xi }{|\xi |}\right) + |x|^{-1}(r_{32}(x,\xi )+s_{32}(x,\xi ))\right] \nonumber \\&\quad \text {for}\;|x|,|\xi |>R, \end{aligned}$$

(5.9)

with $r_1,r_2,r_{31},r_{32}\in {\text {SG}}^{0,0}_{1,1}({\mathbf {R}}^{2d})$, $s_{31}\in {\text {SG}}^{-1,1}_{1,1}({\mathbf {R}}^{2d})$, $s_{32}\in {\text {SG}}^{1,-1}_{1,1}({\mathbf {R}}^{2d})$. By the properties of generalised ${\text {SG}}$ symbols and (5.9) it is possible to prove that all the conditions in Definition 5.10 are fulfilled.

We can now state the first of our main results concerning the propagation of (global) singularities under the action of the generalised ${\text {SG}}$ FIOs.

Theorem 5.14

Let $\varphi \in \mathfrak {F}^r$ be m-admissible, $m\in \{1,2,3 \}$, $\omega _0\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, $a\in {\text {SG}}^{(\omega _0)}_{r,\rho }({\mathbf {R}}^{2d})$, supported outside ${\mathbf {R}}^{d}\times \Omega$, $\Omega \subseteq {\mathbf {R}}^{d}$ open. Assume that $\omega _0$ is $\phi$-invariant, where $\phi$ is as in Theorem 3.16. Assume also that a is ${\text {SG}}$-elliptic and $({\mathcal B},{\mathcal C})$ is a weakly-I ${\text {SG}}$-ordered pair with respect to $(r,\rho ,\omega _0,\varphi ,\Omega )$. Then

$$\begin{aligned} {\text {WF}}^{m}_{\mathcal C}({\text {Op}}_\varphi (a)f) = \phi ({\text {WF}}^{m}_{\mathcal B}(f)),\quad f \in \mathscr {S}^\prime ({\mathbf {R}}^{d}), \end{aligned}$$

(5.10)

where $\phi$ is the canonical transformation (5.6), generated by $\varphi$.

Proof

We only prove the result for $m=3$. The other cases follow by similar arguments and are left for the reader. Let $(y_{0}, \eta _{0})=\phi ^{-1}(x_{0}, \xi _{0})\in \Theta ^m_{\mathcal B}(f)$, $m\in \{1,2,3\}$, and let $c_m \in {\text {SG}}^{0,0}_{1,1}$ be a symbol as in (5.5) and Remark 5.4 such that ${\text {Op}}(c_m) u \in {\mathcal B}$. Recalling Remark 2.14, the weight $\omega (x,\xi )=\vartheta _{0,0}(x,\xi )=1\in \mathscr {P}_{1,1}$ is invariant with respect to any ${\text {SG}}$ diffeomorphism with ${\text {SG}}^0$ parameter dependence. Let $A={\text {Op}}_\varphi (a)$, $C_m={\text {Op}}(c_m)$, and let B be a parametrix for A. Then for some $q_m$ we have

$$\begin{aligned} Q_m = A\circ C_m \circ B,\quad Q_m={\text {Op}}(q_m), \end{aligned}$$

or equivalently,

$$\begin{aligned} Q_m\circ A = A\circ C_m\mod {\text {Op}}({\text {SG}}^{-\infty ,-\infty )}). \end{aligned}$$

By Theorem 3.18 and (5.6), we have $q_m = c_m \circ \phi ^{-1}\mod {\text {SG}}^{-1,-1}_{1,1}$, which implies $q_m\in {\text {SG}}^{0,0}_{1,1}$. Then $(x_{0}, \xi _{0})\in \Theta ^m_{{\mathcal C}} (Af)$, since $Q_m (A f) \equiv A(C_mf) \in {\mathcal C}$ by the hypotheses on $({\mathcal B},{\mathcal C})$. This means that

$$\begin{aligned} \phi (\Theta ^{m}_{\mathcal B}(f)) \subseteq \Theta ^{m} _{\mathcal C}(Af). \end{aligned}$$

(5.11)

Complementing (5.11) with respect to $\Omega _m$, repeating a similar argument starting from Af, recalling Remark 5.12 and that $\phi$ is a diffeomorphism, we finally obtain (5.10). $\square$

The next result is proved in a similar fashion. In fact, with notation analogous to the one used in the proof of Theorem 5.14, denoting $B={\text {Op}}^{*}_\varphi (b)$, we have that $Q_m=B\circ C_m\circ B^{-1}$ satisfies $\mathrm{Sym}\left( Q_m\right) =c_m\circ \phi$ modulo lower order terms. It is then enough to recall Remark 5.11. Evidently, when one deals with ${\text {SG}}$-ordered spaces $({\mathcal B}_1,{\mathcal C}_1,{\mathcal B}_2, {\mathcal C}_2)$, both (5.10) and (5.12) hold, as stated in Corollary 5.16.

Corollary 5.15

Let $\varphi \in \mathfrak {F}^r$ be m-admissible, $m\in \{1,2,3\}$, $\omega _0\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$, $b\in {\text {SG}}^{(\omega _0)}_{1,1}({\mathbf {R}}^{2d})$, supported outside $\Omega \times {\mathbf {R}}^{d}$, $\Omega \subseteq {\mathbf {R}}^{d}$ open. Assume that $\omega _0$ is $\phi ^{-1}$-invariant, where $\phi$ is as in Theorem 3.16. Assume also that b is ${\text {SG}}$-elliptic and that $({\mathcal B},{\mathcal C})$ is a weakly-II ${\text {SG}}$-ordered pair with respect to $(r,\rho ,\omega _0,\varphi ,\Omega )$. Then

$$\begin{aligned} {\text {WF}}^{m}_{{\mathcal B}}({\text {Op}}_\varphi ^{*}(b)f) = \phi ^{-1}({\text {WF}}^{m}_{{\mathcal C}}(f)),\quad f \in \mathscr {S}^\prime ({\mathbf {R}}^{d}), \end{aligned}$$

(5.12)

with the inverse $\phi ^{-1}$ of the canonical transformation (5.6).

Corollary 5.16

Let $\varphi \in \mathfrak {F}^r$ be m-admissible, $m\in \{1,2,3\}$, $\omega _1,\omega _2\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$. Moreover, let $a\in {\text {SG}}^{(\omega _1)}_{r,\rho }({\mathbf {R}}^{2d})$, $b\in {\text {SG}}^{(\omega _2)}_{r,\rho }({\mathbf {R}}^{2d})$, with a supported outside ${\mathbf {R}}^{d}\times \Omega$, b supported outside $\Omega \times {\mathbf {R}}^{d}$, respectively, where $\Omega \subseteq {\mathbf {R}}^{d}$ is open. Assume that a and b are ${\text {SG}}$-elliptic and that $({\mathcal B}_1,{\mathcal C}_1,{\mathcal B}_2,{\mathcal C}_2)$ are ${\text {SG}}$-ordered with respect to

$$\begin{aligned} r,\ \rho ,\ \omega _1 ,\ \omega _2 ,\ \varphi \ \text {and}\ \Omega . \end{aligned}$$

If $f \in {{\mathscr {S}}}^\prime ({\mathbf {R}}^{d})$, then

$$\begin{aligned} {\text {WF}}^{m}_{{\mathcal C}_1}({\text {Op}}_\varphi (a)f) = \phi ({\text {WF}}^{m}_{{\mathcal B}_1}(f)), \end{aligned}$$

and

$$\begin{aligned} {\text {WF}}^{m}_{{\mathcal B}_2}({\text {Op}}_\varphi ^{*}(b)f) = \phi ^{-1}({\text {WF}}^{m}_{{\mathcal C}_2}(f)), \end{aligned}$$

with the canonical transformation $\phi$ in (5.6) and its inverse $\phi ^{-1}$, provided that $\omega _1$ and $\omega _2$ satisfy the invariance properties required in Theorem 5.14 and Corollary 5.15, respectively.

The next result generalizes Theorem 5.14 and Corollaries 5.15 and 5.16 to the case where the involved amplitudes are not ${\text {SG}}$-elliptic. In such a situation, the set of admissible phase functions needs to be slightly restricted, similarly to the calculus of Fourier integral operators developed in [36]. Such restriction is not very harmful, since the phase functions we are mostly interested in are those appearing in the next Sect. 5.3, and it can be proved that they fulfill (5.14) below if a sufficiently small “time interval” $J^\prime =[-T^\prime ,T^\prime ]$ is chosen. This can be easily verified by checking the technique of solution of the involved eikonal equations, see, e.g., [15, 17, 18, 36]. Here the symbols satisfy

$$\begin{aligned} {\text {supp}}a\subseteq {\mathbf {R}}^{d}\times \Omega ^\complement , \quad {\text {supp}}b\subseteq \Omega ^\complement \times {\mathbf {R}}^{d} \end{aligned}$$

(5.13)

for suitable open set $\Omega$, where $\Omega ^\complement$ equals ${\mathbf {R}}^{d}\setminus \Omega$, and the phase function satisfies

$$\begin{aligned}&|\langle x\rangle ^{-1+|\alpha |}\langle \xi \rangle ^{-1+|\beta |}D^\alpha _xD^\beta _\xi \kappa (x,\xi )|\le \tau ,\quad x,\xi \in {\mathbf {R}}^{d},\ |\alpha +\beta |\le 2,\nonumber \\&\quad \text {where}\quad \kappa (x,\xi )=\varphi (x,\xi )-\langle x,\xi \rangle \in {\text {SG}}^{1,1}_{1,1}({\mathbf {R}}^{2d}). \end{aligned}$$

(5.14)

Theorem 5.17

Let $\varphi \in \mathfrak {F}^r$ be m-admissible, $m\in \{1,2,3\}$, and fulfill (5.14) for a fixed $\tau \in (0,1)$. Also let $\omega _1\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, $r,\rho \ge 1/2$, and let $a\in {\text {SG}}^{(\omega _1)}_{r,\rho }({\mathbf {R}}^{2d})$ satisfy (5.13). Finally assume that $({\mathcal B}_1,{\mathcal C}_1)$ is a weakly-I ${\text {SG}}$-ordered pair with respect to $(r,\rho ,\omega _1,\varphi ,\Omega )$. Then

$$\begin{aligned} \begin{array}{ll} &{}{\text {WF}}^{m}_{{\mathcal C}_1}({\text {Op}}_\varphi (a)f) \subseteq \Lambda ^m_{{\mathcal B}_1}(f),\\ &{}\Lambda ^m_{{\mathcal B}_1}(f)=\{(x,\xi )=\phi (y,\eta ) ; (y,\eta )\in {\text {WF}}^{m}_{{\mathcal B}_1}(f)\}^{\mathrm {con}_m},\quad f \in {{\mathscr {S}}}^\prime ({\mathbf {R}}^{d}), \end{array} \end{aligned}$$

(5.15)

where $\phi$ is the canonical transformation generated by $\varphi$ in (5.6).

Theorem 5.18

Let $\varphi \in \mathfrak {F}^r$ be m-admissible, $m\in \{1,2,3\}$, and fulfill (5.14) for a fixed $\tau \in (0,1)$. Also let $\omega _2\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, $r,\rho \ge 1/2$, and let $b\in {\text {SG}}^{(\omega _2)}_{r,\rho }({\mathbf {R}}^{2d})$ satisfy (5.13). Finally assume that $({\mathcal B}_2,{\mathcal C}_2)$ is a weakly-II ${\text {SG}}$-ordered pair with respect to $(r,\rho ,\omega _2,\varphi ,\Omega )$ Then

$$\begin{aligned} \begin{array}{ll} &{}{\text {WF}}^{m}_{{\mathcal B}_2}({\text {Op}}_\varphi ^{*}(b)f) \subseteq \Lambda ^{m}_{{\mathcal C}_2}(f)^{*}, \\ &{}\Lambda ^{m}_{{\mathcal C}_2}(f)^{*}=\{(y,\eta )=\phi ^{-1}(x,\xi );(x,\xi ) \in {\text {WF}}^{m}_{{\mathcal C}_2}(f)\}^{\mathrm {con}_m},\quad f \in {{\mathscr {S}}}^\prime ({\mathbf {R}}^{d}), \end{array} \end{aligned}$$

(5.16)

where $\phi ^{-1}$ is the inverse of the canonical transformation $\phi$ in (5.6).

Corollary 5.19

Let $\varphi \in \mathfrak {F}^r$ be m-admissible, $m\in \{1,2,3\}$, and fulfill (5.14) for a fixed $\tau \in (0,1)$. Also let $\omega _1,\omega _2\in {\mathscr {P}}_{r,\rho }({\mathbf {R}}^{2d})$, $r,\rho \ge 1/2$, and $a\in {\text {SG}}^{(\omega _1)}_{r,\rho }({\mathbf {R}}^{2d})$ and $b\in {\text {SG}}^{(\omega _2)}_{r,\rho }({\mathbf {R}}^{2d})$ satisfy (5.13). Assume also that $({\mathcal B}_1,{\mathcal C}_1,{\mathcal B}_2,{\mathcal C}_2)$ are ${\text {SG}}$-ordered with respect to

$$\begin{aligned} r,\ \rho ,\ \omega _1, \omega _2 ,\ \varphi \ \text {and}\ \Omega . \end{aligned}$$

Then both (5.15) and (5.16) hold.

In the results above, $V^{\mathrm {con}_m}$ for $V\subseteq \Omega _m$, is the smallest m-conical subset of $\Omega _m$ which includes V, $m\in \{1,2,3\}$.

We prove only Theorem 5.17. Theorem 5.18 follows by similar arguments and is left for the reader.

Proof of Theorem 5.17

Since here we are dropping the ellipticity hypothesis on the amplitude a, we use only the composition results between generalised SG pseudo-differential operators and Fourier integral operators given in Sect. 3.3. That is, the proofs of the theorem again rely on the generalised ${\text {SG}}$ asymptotic expansions discussed in [22], and on the properties of the admissible phase functions. We now prove (5.15) in detail for the case $m=3$, by showing the opposite inclusion between the complements of the involved sets with respect to $\Omega _3$. In the sequel, we write ${\mathcal B}$ and ${\mathcal C}$ in place of ${\mathcal B}_1$ and ${\mathcal C}_1$, respectively.

Let $(x_0,\xi _0)\notin \Lambda ^3_{\mathcal B}(f)$ for $f\in {\mathcal B}$, and set $2N=\min \{|x_0|,|\xi _0|\}$>0. By its definition in (5.15), $\Lambda ^3_{{\mathcal B}}(f)$ is a closed 3-conical set. Then, choosing $\varepsilon >0$ sufficiently small, it is possible to find a 3-conical set of the form

$$\begin{aligned} \Gamma _{3,x_0,\xi _0}^{4\varepsilon ,4\varepsilon ,N/4} =\left\{ (x,\xi )\in {\mathbf {R}}^{2d};\right. \,&\left| \frac{x}{|x|}-\frac{x_0}{|x_0|}\right| <4\varepsilon ,\\&\left| \left. \frac{\xi }{|\xi |}-\frac{\xi _0}{|\xi _0|}\right| <4\varepsilon , |x|,|\xi |\ge \frac{N}{4} \right\} \end{aligned}$$

such that $\Gamma _{3,x_0,\xi _0}^{4\varepsilon ,4\varepsilon ,N/4} \cap \Lambda ^3_{{\mathcal B}}(f)=\emptyset$. Then, as it is also possible (see Sect. 5.1 above and [14]), pick $q\in {\text {SG}}^{0,0}_{1,1,}$ such that

$$\begin{aligned} {\text {supp}}q\subseteq \Gamma ^{2\varepsilon ,2\varepsilon ,N/2}_{3,x_0,\xi _0} \text { and } (x,\xi )\in \Gamma ^{\varepsilon ,\varepsilon ,N}_{3,x_0,\xi _0}\Rightarrow q(x,\xi )=1. \end{aligned}$$

We now observe that $(y_0,\eta _0)=\phi ^{-1}(x_0,\xi _0)\notin {\text {WF}}^3_{\mathcal B}(f)$, in view of the definition of $\Lambda ^3_{\mathcal B}(f)$. Setting $2\widetilde{N}=\min \{|y_0|,|\eta _0|\}$, we can consider the subset of $\Omega _3$ given by

$$\begin{aligned} W={\text {WF}}^3_{\mathcal B}(f)\cap \Omega _3^{\widetilde{N}}. \end{aligned}$$

W is closed, and, by Remark 5.7 it is 3-conical. Then there exist two 3-conical neighborhoods U, V of W such that $W\subset V\subset U\subset \Omega _3$. For instance, for an arbitrarily small $\tilde{\delta }>0$, one can consider the coverings of W given by

$$\begin{aligned} \widetilde{U}=\bigcup _{(z_0,\zeta _0)\in W} \Gamma ^{4\tilde{\delta },4\tilde{\delta },\widetilde{N}/4}_{3,z_0,\zeta _0}, \quad \widetilde{V}=\bigcup _{(z_0,\zeta _0)\in W} \Gamma ^{2\tilde{\delta },2\tilde{\delta },\widetilde{N}/2}_{3,z_0,\zeta _0}. \end{aligned}$$

By a standard compactness argument on the unit sphere of ${\mathbf {R}}^{d}$, define V and U as suitable finite subcoverings extracted from $\widetilde{V}$ and $\widetilde{U}$, respectively. Since $\Gamma ^{2\tilde{\delta },2\tilde{\delta },\widetilde{N}/4}_{3,z_0,\zeta _0} \subset \Gamma ^{4\tilde{\delta },4\tilde{\delta },\widetilde{N}/4}_{3,z_0,\zeta _0}$, we get $W\subset V \subset U$, as desired. Then take a symbol $\chi \in {\text {SG}}^{0,0}_{1,1}$ such that

$$\begin{aligned}&{\text {supp}}\chi \subset U,(y,\eta )\in V\Rightarrow \chi (y,\eta )=1, \\&{\text {supp}}q\cap \{(x,\xi )=\phi (y,\eta )\;:(y,\eta )\in {\text {supp}}\chi \}=\emptyset , \end{aligned}$$

which is possible by choosing $\tilde{\delta }$ small enough, in view of the hypotheses and of (3) in Definition 5.10. Indeed, we can start from a 3-conical neighbourhood $Z\supset \Lambda ^3_{\mathcal B}(f)\cap \Omega _3^N$, obtained as a finite union of sets of the form $\Gamma _{3,t_0,\tau _0}^{2\varepsilon ,2\varepsilon ,N/2}$, $(t_0,\tau _0)\in \Lambda ^3_{\mathcal B}(f)$, disjoint from $\Gamma ^{4\varepsilon ,4\varepsilon ,N/4}_{3,x_0,\xi _0}$, by choosing $\varepsilon >0$ suitably small. Observing that all the involved sets are 3-conical, it is then possible to choose $\tilde{\delta }$ small enough such that $\phi (U)\subset Z$, and $\chi$ with the desired properties.

Let us now consider

$$\begin{aligned}{}[{\text {Op}}(q)\circ {\text {Op}}_\varphi (a)](f)= & {} [{\text {Op}}(q)\circ {\text {Op}}_\varphi (a)\circ {\text {Op}}(1-\chi )]f\nonumber \\&+\,[{\text {Op}}(q)\circ {\text {Op}}_\varphi (a)\circ {\text {Op}}(\chi )]f. \end{aligned}$$

(5.17)

By Remark 2.14, the weight $\vartheta _{0,0}(x,\xi )=1$ is invariant with respect to any ${\text {SG}}$ diffeomorphism with ${\text {SG}}^0$ parameter dependence.

For $C={\text {Op}}(q)\circ {\text {Op}}_\varphi (a)\circ {\text {Op}}(\chi )$, we apply Theorems 3.7 and 3.8, and find that $C={\text {Op}}_\varphi (c) \mathrm{Mod}{\text {Op}}({\mathscr {S}})$ with

$$\begin{aligned} c(x,\eta )\sim \sum p_{\alpha \beta k l}(x,\eta )&\cdot (\partial ^k_\xi q)(x,\varphi ^\prime _x(x,\eta )) \cdot (\partial ^\alpha _\xi \partial ^\beta _x a)(x,\eta ) \\&\cdot (\partial ^l_x\chi )(\varphi ^\prime _\xi (x,\eta ),\eta )\sim 0, \end{aligned}$$

which implies that $C:{{\mathscr {S}}}^\prime \rightarrow {{\mathscr {S}}}$. In fact, setting $\xi =\varphi ^\prime _x(x,\eta )$, $y=\varphi ^\prime _\xi (x,\eta )$, by (5.6) we have $(x,\xi )=\phi (y,\eta )$, and, by construction, ${\text {supp}}\partial ^k_\xi q$ $\cap \phi ({\text {supp}}\partial ^l_x\chi )=\emptyset$. Now, setting

$$\begin{aligned} \Sigma =\{(y,\eta )\in {\text {supp}}(1-\chi ):|y|,|\eta |\ge \widetilde{N}/2\}, \end{aligned}$$

again by construction we have $\Sigma \cap {\text {WF}}^3_{\mathcal B}(f)=\emptyset$. Then there exist $p\in {\text {SG}}^{0,0}_{1,1}$ such that ${\text {Op}}(p)f\in {\mathcal B}$ and $p(x,\xi )\ge C >0$ on $\Sigma$, and $r,s\in {\text {SG}}^{0,0}_{1,1}$ such that

$$\begin{aligned} {\text {Op}}(r)-{\text {Op}}(s)\circ {\text {Op}}(p):{{\mathscr {S}}}^\prime \rightarrow {{\mathscr {S}}}, \end{aligned}$$

(5.18)

with $r(x,\xi )\equiv 1$ for $|x|,|\xi |\ge \frac{N}{2}$ belonging to a 3-conical neighborhood of $\Sigma$. This can be proved by relying on the concept of ${\text {SG}}$-ellipticity with respect to a symbol (or local md-ellipticity, cf. [14], Ch. 2, §3). We can write

$$\begin{aligned} {\text {Op}}(1-\chi )f&=[{\text {Op}}(1-\chi )\circ {\text {Op}}(1-r)]f \\&\quad +[{\text {Op}}(1-\chi )\circ [{\text {Op}}(r)-{\text {Op}}(s)\circ {\text {Op}}(p))]f \\&\quad +[{\text {Op}}(1-\chi )\circ {\text {Op}}(s)][{\text {Op}}(p)f]. \end{aligned}$$

The first term is in ${{\mathscr {S}}}$, since the symbols of the two operators in the composition have, by construction, disjoint supports. The second term is in ${{\mathscr {S}}}$ as well, by (5.18). The third term is in ${\mathcal B}$, since this is true for ${\text {Op}}(p)f$, ${\text {Op}}(1-\chi )\circ {\text {Op}}(s)={\text {Op}}(\lambda )$, with $\lambda \in {\text {SG}}^{0,0}_{1,1}$, and ${\mathcal B}$ is ${\text {SG}}$-admissible.

By all the considerations above, the mapping properties of ${\text {Op}}_\varphi (a)$, the fact that also ${\mathcal C}$ is ${\text {SG}}$-admissible and that $q\in {\text {SG}}^{0,0}_{1,1}$, we get

$$\begin{aligned}{}[{\text {Op}}(q)\circ {\text {Op}}_\varphi (a)]f&=[{\text {Op}}(q)\circ {\text {Op}}_\varphi (a)\circ {\text {Op}}(1-\chi )]f\mod {{\mathscr {S}}}\\&=[{\text {Op}}(q)\circ {\text {Op}}_\varphi (a)]\underbrace{[{\text {Op}}(1-\chi )f]}_{\in {\mathcal B}}\mod {{\mathscr {S}}}\end{aligned}$$

giving that

$$\begin{aligned}{}[{\text {Op}}(q)\circ {\text {Op}}_\varphi (a)]f\in {\mathcal C}, \end{aligned}$$

which proves $(x_0,\xi _0)\notin {\text {WF}}^3_{\mathcal C}({\text {Op}}_\varphi (a)f)$, and the claim. We observe that (5.15) for the case $m=1$ can be proved by the same argument used, e.g., in [36], Chap. 10, Sect. 3. The case $m=2$ of (5.15) can then be obtained in a completely similar fashion, by exchanging the role of variable and covariable. The details are left for the reader. $\square$

Remark 5.20

As it was observed in [26], there is a simple and useful relation between the global wave-front set of f and of $\widehat{f}$. Namely, with $m, n\in \{ 1,2,3 \}$ such that n equals 2, 1 and 3, when m equals 1, 2 and 3, respectively, we have

$$\begin{aligned} T \big ({\text {WF}}^{m}_{{\mathcal B}}(f) \big ) = {\text {WF}}^{n}_{{\mathcal B}_T}(\widehat{f}), \end{aligned}$$

where ${\mathcal B}=M(\omega ,{\mathscr {B}})$, the torsion T is given by $T(x,\xi )=(-\xi ,x)$, and ${\mathscr {B}}_T=\{ \, F\circ T=T^{*}F\, ;\, F\in {\mathscr {B}}\, \} ,$ $\omega _T=\omega \circ T$, ${\mathcal B}_T=M(\omega _T,{\mathscr {B}}_T)$. Notice that ${{\mathscr {F}}}$ is bijective and continuous, together with its inverse, from ${\mathcal B}_T$ onto ${\mathcal B}$.

It is also immediate to obtain a similar relation among the wave-front sets of f and $\check{f}$, where $\check{f}=f\circ R$ is the pull-back of f under the action of the reflection $R(y)=-y$. Indeed, since obviously, for any $a\in {\text {SG}}^{m,\mu }_{r,\rho }$ and $f\in {{\mathscr {S}}}^\prime$,

$$\begin{aligned} {\text {Op}}(a)\check{f}=[{\text {Op}}({\check{a}})f]\,\check{}, \end{aligned}$$

it follows, for $m\in \{1,2,3\}$,

$$\begin{aligned} R({\text {WF}}^m_{\mathcal B}(f))={\text {WF}}^m_{\check{{\mathcal B}}}(\check{f}), \end{aligned}$$

where $\check{{\mathscr {B}}}=\{ \, R^{*}F\, ;\, F\in {\mathscr {B}}\, \}$, $\check{{\mathcal B}}=M(\omega _R,\check{{\mathscr {B}}})$, where $\omega _R=\omega \circ R$. Notice that, in many cases, $\check{{\mathcal B}}$=${\mathcal B}$. For instance, this is true for all the functional spaces considered in Sect. 4, and, in general, for all $M(\omega ,\mathscr {B})$ such that $\check{\mathscr {B}}=\mathscr {B}$ and $\omega$ is even. Similarly to the above, $R^{*}$ is bijective and continuous, together with its inverse, from ${\mathcal B}$ onto $\check{{\mathcal B}}$.

By (3.6), rewritten as

$$\begin{aligned} {\text {Op}}^{*}_{-\varphi ^{*}}(a^{*})f=({{\mathscr {F}}}\circ {\text {Op}}_\varphi (a)\circ {{\mathscr {F}}}^{-1}f)\,{\check{}}, \quad f\in {{\mathscr {S}}}^\prime , \end{aligned}$$

and the above definitions of ${\mathcal B}_T$ and $\check{{\mathcal B}}$, it also follow that, if $({\mathcal B},{\mathcal C})$ is weakly-I ${\text {SG}}$-ordered with respect to $(r,\rho ,\omega _0,\varphi ,\Omega )$, we find that, for any $a\in {\text {SG}}^{(\omega _0)}_{r,\rho }$, supported outside ${\mathbf {R}}^{d}\times \Omega$, ${\text {Op}}^{*}_{-\varphi ^{*}}(a^{*}):{\mathcal B}_T\rightarrow \check{{\mathcal C}}_T$ continuously, that is, $(\check{{\mathcal C}}_T,{\mathcal B}_T)$ is weakly-II ${\text {SG}}$-ordered with respect to $(r,\rho ,\omega _0,\varphi ,\Omega )$.

5.3 Applications to SG-Hyperbolic Problems

In this subsection we apply the results obtained above to the ${\text {SG}}$-hyperbolic problems considered in [18, 19], to which we refer for the details omitted here. We show how, under natural conditions, the singularities described by the generalised wave-front sets ${\text {WF}}^m_{\mathcal B}(g)$, $m=1,2,3$, for a scalar- or vector-valued initial data $g\in {\mathcal B}$, propagate to the solution $f(t)=f(t,\, \cdot \, )\in {\mathcal B}$, $t\in [-T,T]$. More precisely, the points of ${\text {WF}}^m_{\mathcal C}(f(t))$ lie on bicharacteristics curves determined by ${\text {WF}}^m_{\mathcal B}(g)$, $m=1,2,3$, and by the phase functions of the Fourier operators ${\text {Op}}_{\varphi _k (t)}(a_k(t))$, $k=1, \ldots , \mu$, such that, modulo smooth remainders (see below),

$$\begin{aligned} f(t)=({\text {Op}}_{\varphi _1 (t)}(a_1(t))(t)+\cdots +{\text {Op}}_{\varphi _\mu (t)}(a_\mu (t))(t))g. \end{aligned}$$

Notice that the hyperbolic operators involved in such Cauchy problems arise naturally as local representations of (modified) wave operators of the form $L=\Box _\mathfrak {g}-V$, with a suitable potential V and the D’Alembert operator $\Box _\mathfrak {g}$, on manifolds of the form ${\mathbf {R}}^{}_t\times M_x$, equipped with a hyperbolic metric $\mathfrak {g}=\mathrm {diag}(-1,\mathfrak {h})$, where $\mathfrak {h}$ is a suitable Riemannian metric on the manifold with ends M. In this way,

$$\begin{aligned} L=\Box _\mathfrak {g}-V=-\partial _t^2+\Delta _\mathfrak {h}-V=-\partial _t^2+P, \end{aligned}$$

where $\Delta _\mathfrak {h}$ is the Laplace-Beltrami operator on M associated with the metric $\mathfrak {h}$ and we have set $P=\Delta _\mathfrak {h}-V$. In the following Example 5.21, we show that this indeed occurs, considering a rather simple situation with $\dim M=2$.

Example 5.21

Assume $\dim M=2$ and consider, as local model of one of the “ends” of M, the cylinder in ${\mathbf {R}}^{3}$ given by $u^2+v^2=1$, $z>1$, that is, the manifold $M_\infty =S^1\times (1,+\infty )$. First, we have to equip $M_\infty$ with a $\mathscr {S}$-structure, namely, an ${\text {SG}}$-compatible atlas (see [14, 40]). This can be easily accomplished here, by choosing a standard product atlas on $S^1\times (1,+\infty )$, identifying $S^1$ with the unit circle in ${\mathbf {R}}^{2}$ centred at the origin, as we now explain. With coordinates (u, v) on ${\mathbf {R}}^{2}$, set

$$\begin{aligned} \Omega '_1:= & {} S^1 \backslash \{(0,1)\}, \,\,\,\,\,\,\, \Omega '_2 := S^1 \backslash \{(0,-1)\},\\&\nu '_1 : \Omega '_1 \rightarrow {\mathbf {R}}^{}:(u,v) \mapsto \frac{u}{1-v},\\&\nu '_2 : \Omega '_2 \rightarrow {\mathbf {R}}^{}:(u,v) \mapsto \frac{u}{1+v} \end{aligned}$$

It is immediate to show that $(\nu '_1)^{-1} : \nu '_1(\Omega '_1) \rightarrow \Omega '_1 \subset S^1$ is

$$\begin{aligned} t \mapsto \left( \frac{2t}{1+t^2}, -\frac{1-t^2}{1+t^2} \right) , \end{aligned}$$

and $(\nu '_2)^{-1} : \nu '_2(\Omega '_2) \rightarrow \Omega '_2 \subset S^1$ is

$$\begin{aligned} t \mapsto \left( \frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2} \right) , \end{aligned}$$

so that, for $\displaystyle t \in \nu '_2(\Omega '_1 \bigcap \Omega '_2)=(-\infty ,0)\cup (0,+\infty )$, we find

$$\begin{aligned} \nu _{12}^\prime (t) = \nu '_1((\nu '_2)^{-1}(t)) = \nu '_1 \left( \frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2} \right) = \frac{1}{t}. \end{aligned}$$

Now set

$$\begin{aligned} \Omega _1 := \Omega '_1 \times (1,+\infty ), \,\,\,\,\,\,\, \Omega _2 := \Omega '_2 \times (1,+\infty ), \end{aligned}$$

define $\nu _1 : \Omega _1 \rightarrow U_1 \subset {\mathbf {R}}^{2}$ by

$$\begin{aligned} (u,v,z) \mapsto (\nu '_1(u,v) , 1) \frac{z}{\sqrt{1 + (\nu '_1(u,v))^2}} = \left( \frac{u}{1-v} , 1\right) \frac{z}{\sqrt{1 + \dfrac{u^2}{(1-v)^2}}}, \end{aligned}$$

and $\nu _2 : \Omega _2 \rightarrow U_2 \subset {\mathbf {R}}^{2}$ by

$$\begin{aligned} (u,v,z) \mapsto (\nu '_2(u,v) , 1) \frac{z}{\sqrt{1 + (\nu '_2(u,v))^2}} = \left( \frac{u}{1+v} , 1\right) \frac{z}{\sqrt{1 + \dfrac{u^2}{(1+v)^2}}}. \end{aligned}$$

Again, it is easy to obtain the expressions of $\nu _1^{-1} : U_1 \rightarrow \Omega _1$ and $\nu _2^{-1} : U_2 \rightarrow \Omega _2$, and to prove that, with coordinates $x=(x_1,x_2)$ on ${\mathbf {R}}^{2}$,

$$\begin{aligned} \nu _{12}(x_1,x_2)&= \nu _1((\nu _2)^{-1}(x_1,x_2)) = \left( \frac{x_2}{x_1},1\right) x_1 =(x_2 , x_1), \end{aligned}$$

which shows that the atlas $\{(\Omega _j, \nu _j), j=1,2\}$ defines a $\mathscr {S}$-structure on $M_\infty$, since $\langle \nu _{12}(x)\rangle =\langle x\rangle$ (see again, e.g., [14, 40]). Next, for any $\mu >0$, define a metric $\mathfrak {h}'$ on $\{(u,v,z) \in {\mathbf {R}}^{3} : z > 1\}$ by

$$\begin{aligned} (\mathfrak {h}'_{ij}) := \left( \begin{array}{ccc} \displaystyle \frac{z^2}{4 \langle z \rangle ^\mu } &{} 0 &{} 0 \\ 0 &{} \displaystyle \frac{z^2}{4 \langle z \rangle ^\mu } &{} 0 \\ 0 &{} 0 &{} \displaystyle \frac{1}{ \langle z \rangle ^\mu } \end{array} \right) . \end{aligned}$$

With $x \in U_1$, and denoting by $J_1$ the Jacobian matrix of $\nu _1^{-1}$, it turns out that the pull-back metric $\mathfrak {h} := (\nu _1^{-1})^{*} \mathfrak {h}'$ on $M_\infty$ is given by

$$\begin{aligned} (\mathfrak {h}_{ij}) = J_1 \, ((\nu _1^{-1})^{*}\mathfrak {h}'_{ij})|_{U_1}\, J_1^t = \left( \begin{array}{cc} \displaystyle \frac{1}{ \langle x \rangle ^\mu } &{} 0 \\ 0 &{} \displaystyle \frac{1}{ \langle x \rangle ^\mu } \\ \end{array} \right) . \end{aligned}$$

In the same way, one can show that the metric $\mathfrak {h}$ has the same local expression for $x \in U_2$. Finally, let us compute the Laplace-Beltrami operator on $M_\infty$ associated with $\mathfrak {h}$ in the chosen local coordinates. We have, of course, $(\mathfrak {h}^{ij}) = \mathrm {diag}(\langle x \rangle ^\mu ,\langle x \rangle ^\mu )$ and $\sqrt{|\det \mathfrak {h}|} = \langle x \rangle ^{-\mu }$, thus, for any $f \in C^\infty (M_\infty )$,

$$\begin{aligned} \begin{array}{ll} \Delta _\mathfrak {h} f &{} = \frac{1}{\langle x \rangle ^{-\mu }} \sum _{i,j = 1}^2 \frac{\partial }{\partial x^j} \left( \langle x \rangle ^{-\mu } \mathfrak {h}^{ij} \frac{\partial f}{\partial x^i}\right) \\ &{} = \langle x \rangle ^{\mu } \sum _{i,j=1}^2 \frac{\partial }{\partial x^j} \left( \delta ^{ij} \frac{\partial f}{\partial x^i}\right) \\ &{} = \langle x \rangle ^{\mu } \sum _{i=1}^2 \frac{\partial ^2 f}{\partial x_i^2} = \langle x \rangle ^{\mu } \left( \frac{\partial ^2}{\partial x_1^2} + \frac{\partial ^2}{\partial x_2^2}\right) f, \end{array} \end{aligned}$$

that is

$$\begin{aligned} \Delta _\mathfrak {h} = \langle x\rangle ^\mu \Delta , \end{aligned}$$

where $\Delta$ is the standard Laplacian on ${\mathbf {R}}^{2}$. Choosing $V(x)=\langle x\rangle ^\mu$, the local symbol of $P=\Delta _\mathfrak {h}-V$ is

$$\begin{aligned} p(x,\xi ) = - \langle x\rangle ^\mu \langle \xi \rangle ^2 = -(1+x_1^2+x_2^2)^{\frac{\mu }{2}} (1+\xi _1^2 + \xi _2^2), \end{aligned}$$

(5.19)

which obviously belongs to ${\text {SG}}^{2,\mu }_{1,1}({\mathbf {R}}^{2}\times {\mathbf {R}}^{2})$ and is ${\text {SG}}$-elliptic. In [20], the spectral theory for elliptic self-adjoint operators, generated by local symbols with (different) orders $m,\mu >0$, has been considered. On the other hand, the case $\mu =2$ is of special interest in the context of the ${\text {SG}}$-hyperbolic operators (see below), since then we have that L, in local coordinates, is given by

$$\begin{aligned} \begin{array}{ll} L&{}=\Box _\mathfrak {g}-V=-\partial _t^2+\Delta _\mathfrak {h}-V \\ &{}=D_t^2-P=D_t^2-\langle \, \cdot \, \rangle ^2\langle D\rangle ^2. \end{array} \end{aligned}$$

(5.20)

In the sequel of this subsection, the subscript “$\mathrm {cl}$” denotes the subclasses of ${\text {SG}}$ symbols which are classical, see [18]. Notice that the symbol (5.19) actually belongs to ${\text {SG}}^{2,\mu }_{1,1,\mathrm {cl}}({\mathbf {R}}^{2}\times {\mathbf {R}}^{2})$. We first need to recall some definitions and results, mainly taken from [15, 16, 18].

Definition 5.22

Let $J=[-T,T] \subset {\mathbf {R}}^{}$, $T > 0$, and consider the linear operator

$$\begin{aligned} L = D_{t}^\nu + {\text {Op}}(p_1(t))\,D_{t}^{\nu -1} + \cdots + {\text {Op}}(p_\nu (t)), \end{aligned}$$

(5.21)

with $p_{j} = p_{j}(t,x,\xi ) \in C^\infty (J,{\text {SG}}^{1,1}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d}))$. Let

$$\begin{aligned} l(x,\xi ,t,\tau ) = \tau ^\nu + q_{1}(t,x,\xi )\tau ^{\nu -1} + \cdots + q_{\nu }(t,x,\xi ) \end{aligned}$$

be the principal symbol of L, with $q_{j} \in C^\infty (J,{\text {SG}}^{j,j}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d}))$ such that $q_j(t)=q_{j}(t,\, \cdot \, )$ is the principal symbol of $p_j(t)=p_j(t,\, \cdot \, )$, in the sense of (5.7). L is called ${\text {SG}}$-classical hyperbolic with constant multiplicities if the characteristic equation

$$\begin{aligned} \tau ^\nu + q_{1}(t,x,\xi )\tau ^{\nu -1} + \cdots + q_{\nu }(t,x,\xi ) = 0 \end{aligned}$$

(5.22)

has $\mu \le \nu$ distinct real roots $\tau _{j} = \tau _{j}(t,x,\xi ) \in C^\infty (J,{\text {SG}}^{1,1}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d}))$ with multiplicities $l_{j}$, $1\le l_j\le \nu$, $j=1,\ldots ,\mu$, which satisfy, for a suitable $C > 0$ and all $t\in J$, $x,\xi \in {\mathbf {R}}^{d}$, $|x|+|\xi |\ge R>0$,

$$\begin{aligned} \tau _{j+1}(t,x,\xi ) - \tau _{j}(t,x,\xi ) \ge C \langle \xi \rangle \langle x\rangle , j = 1, \ldots , \mu - 1. \end{aligned}$$

(5.23)

L is called strictly hyperbolic if it is hyperbolic with constant multiplicities and the multiplicity of all the $\tau _{j}$, $j=1,\ldots ,\nu =\mu$, is equal to 1.

A standard strategy to solve the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} L f(t) = 0, &{} t \in J, \\ D_{t}^k f(0) = g_k, &{} k=0,\ldots ,\nu -1, \end{array} \right. \end{aligned}$$

(5.24)

for L hyperbolic with constant multiplicities and initial data $g_k$, $k=0,\ldots ,\nu -1$, chosen in appropriate functional spaces, is to show that this is equivalent to solving, modulo smooth elements, a Cauchy problem for a first order system

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\partial F}{\partial t}(t) - iK(t)\,F(t) = 0, t \in J,\\ F(0) = G, \end{array} \right. \end{aligned}$$

with a coefficient matrix K of special form. In our case, one obtains that $K = {\text {Op}}((k_{ij}(t,x,D))_{i,j})$, is a $\mu \nu \times \mu \nu$ matrix of ${\text {SG}}$ pseudo-differential operators with symbols $k_{ij} \in C^\infty (J, {\text {SG}}^{1,1}_{1,1,\mathrm {cl}})$. Under suitable assumptions, see [18, 19], the principal part $k_{1}$ of $k = k_{1} + k_{0}$, $k_{j} \in C^\infty (J,{\text {SG}}^{j,j}_{1,1,\mathrm {cl}})$, $j=0,1$, turns out to be diagonal, so that the system will be symmetric, cfr. [14–16]. This implies that the corresponding Cauchy problem is well-posed. One of the main advantages for using this algorithm is the following Proposition 5.23, which is an adapted version of the Mizohata Lemma of Perfect Factorization, proved in [17] for the general ${\text {SG}}$ symbols (see also the references quoted therein).

Proposition 5.23

Let L be an ${\text {SG}}$-classical hyperbolic linear operator with constant multiplicities $l_{j}$, $j=1,\ldots ,\mu \le \nu$, as in Definition 5.22. Then it is possible to factor L as

$$\begin{aligned} L = L_{\mu } \cdots L_{1} + \sum _{s=1}^\nu {\text {Op}}(r_{s}(t)) \, D_{t}^{\nu -s} \end{aligned}$$

with $L_{j}= (D_{t} - {\text {Op}}(\tau _j(t)))^{l_{j}} + \sum _{k=1}^{l_{j}} {\text {Op}}(s_{jk}(t)) \, (D_{t} - {\text {Op}}(\tau _j(t)))^{l_{j}-k}$ and

$$\begin{aligned}&s_{jk} \in C^\infty (J, {\text {SG}}^{k-1,k-1}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d})), r_{s} \in C^\infty (J,{{\mathscr {S}}}({\mathbf {R}}^{2d})),\\&j=1, \ldots , \mu , k = 1, \ldots , l_{j}, s= 1, \ldots , \nu . \end{aligned}$$

The following corollary, also obtained in [17], follows by means of a reordering of the roots $\tau _{j}$ of the principle symbol of L.

Corollary 5.24

Let $c_{j}$, $j=1, \ldots , \mu$, denote the reorderings of the $\mu$-tuple $(1, \ldots ,$ $\mu )$ given by

$$\begin{aligned} c_{j}(i)= & {} \left\{ \begin{array}{ll} j + i &{} \text{ for } j + i \le \mu \\ j + i - \mu &{} \text{ for } j + i > \mu , \end{array} \right. \\ i, j= & {} 1, \ldots , \mu , \end{aligned}$$

that is, $c_{1} = (2, \ldots , \mu , 1)$, ..., $c_{\mu } = (1, \ldots , \mu )$. Then, under the same hypotheses of Proposition 5.23, we have

$$\begin{aligned} L = L^{(m)}_{c_{m}(\mu )} \cdots L^{(m)}_{c_{m}(1)} + \sum _{s=1}^\nu {\text {Op}}(r^{(m)}_{s}(t)) D_{t}^{\nu -s} \end{aligned}$$

with $L^{(m)}_{j}= (D_{t} - {\text {Op}}(\tau _j(t)))^{l_{j}} + \sum _{k=1}^{l_{j}} {\text {Op}}(s^{(m)}_{jk}(t)) \, (D_{t} - {\text {Op}}(\tau _j(t)))^{l_{j}-k}$ and

$$\begin{aligned}&s^{(m)}_{jk} \in C^\infty (J, {\text {SG}}^{k-1, k-1}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d})), r^{(m)}_{s} \in C^\infty (J,{{\mathscr {S}}}({\mathbf {R}}^{2d})), \\&m,j=1, \ldots , \mu , k = 1, \ldots , l_{j}, s= 1, \ldots , \nu . \end{aligned}$$

Definition 5.25

We say that an ${\text {SG}}$-classical hyperbolic operator L is of Levi type if it satisfies the ${\text {SG}}$-Levi condition¹

$$\begin{aligned} s^{(m)}_{jk} \in C^\infty (J, {\text {SG}}^{0,0}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d})),\quad m,j=1, \ldots , \mu , k=1, \ldots , l_{j}. \end{aligned}$$

(5.25)

Theorem 5.26 below gives the well-posedness for the Cauchy problem (5.24) and the propagation results of the global wave-front sets ${\text {WF}}^m_{\mathcal C}(f(t))$, $m=1,2,3$, for the corresponding solution f(t), under natural assumptions on the ${\text {SG}}$-admissible initial data spaces ${\mathcal B}_k$, $k=0,\ldots ,\nu -1$, and the ${\text {SG}}$-admissible solution space ${\mathcal C}$, see below for the precise statement. It immediately follows by the analysis of ${\text {SG}}$-classical hyperbolic Cauchy problems in [18], by Sect. 4 and by Theorem 5.14.

We here consider an ${\text {SG}}$-classical hyperbolic operator L with constant multiplicities and of Levi type, and denote by $l = \max \{ l_{1}, \ldots , l_{\mu } \}$ the maximum multiplicity of the distinct real roots $\tau _{j}$, $j=1,\ldots ,\mu$, of the characteristic equation (5.22). Then, as proved in [15, 18], for any choice of initial data $g_k\in {{\mathscr {S}}}^\prime ({\mathbf {R}}^{d})$, $k=0,\ldots ,\nu -1$, the Cauchy problem (5.24) admits a unique solution $f \in C(J^\prime ,{{\mathscr {S}}}^\prime ({\mathbf {R}}^{d}))$, $J^\prime =[-T^\prime ,T^\prime ]$, $0<T^\prime \le T$. Collecting the initial conditions in the vector

$$\begin{aligned}&g = \left( \begin{array}{c} g_{0} \\ g_{1} \\ \vdots \\ g_{\nu -1} \end{array} \right) , \end{aligned}$$

the solution f is given by

$$\begin{aligned} f(t) = ({\text {Op}}_{\varphi _{1}(t)}(a_{1}(t)) + \cdots + {\text {Op}}_{\varphi _{\mu }(t)}(a_{\mu }(t))) g, \end{aligned}$$

where each ${\text {Op}}_{\varphi _{j}(t)}(a_{j}(t))$ is a type I FIO with regular phase function $\varphi _{j} \in C^\infty (J^\prime , \mathfrak {F}^r)\cap C^\infty (J^\prime ,{\text {SG}}^{1,1}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d}))$, solution of the eikonal equation associated with $\tau _{j}$, and vector-valued amplitude functions $a_{j} = (a_{j0}, \ldots , a_{j\nu -1})$ with $a_{jk} \in C^\infty (J^\prime ,{\text {SG}}^{l-k-1,l-k-1}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d}))$, $j = 1, \ldots , \mu$, $k=0, \ldots , \nu -1$.

Theorem 5.26

Let L be as above, and let $g_k\in {\mathcal B}_k$, $k=0,\ldots ,\nu -1$, with the $\nu$-tuple of ${\text {SG}}$-admissible spaces $({\mathcal B}_0,\ldots ,{\mathcal B}_{\nu -1})$. Also assume that the ${\text {SG}}$-admissible space ${\mathcal C}$ is such that $({\mathcal B}_k,{\mathcal C})$, $k=0,\ldots ,\nu -1$, are weakly-I ${\text {SG}}$-ordered pairs with respect to

$$\begin{aligned} 1, 1, \langle x\rangle ^{l-k-1} \langle \xi \rangle ^{l-k-1}, \varphi _k(t) \quad \text {and}\quad \Omega . \end{aligned}$$

Then the Cauchy problem (5.24) is well-posed with respect to $({\mathcal B}_0,\ldots ,{\mathcal B}_{\nu -1})$ and ${\mathcal C}$, $u\in C(J^\prime ,{\mathcal C})$, and

$$\begin{aligned} {\text {WF}}^m_{\mathcal C}( f(t) ) \subseteq \bigcup _{j=1}^\mu \bigcup _{k=0}^{\nu -1} (\phi _{j}(t)({\text {WF}}^m_{{\mathcal B}_k} (g_k)))^{\mathrm {con}_m} , \quad m=1,2,3, \end{aligned}$$

(5.26)

where $\phi _j(t)$ is the canonical transformation (5.6) associated with the phase function $\varphi _{j}(t)$.

Corollary 5.27

Assume that the hypotheses of Theorem 5.26 hold. Then ${\text {WF}}^m_{\mathcal C}(f(t))$, $t\in J^\prime$, $m=1,2,3$, consists of arcs of bicharacteristics, generated by the phase functions $\varphi _j(t)$ and emanating from points belonging to ${\text {WF}}^m_{{\mathcal B}_k} (g_k)$, $k=0,\ldots ,\nu -1$.

5.4 Examples

We conclude with some examples where our propagation of singularities results can be applied. We initially look at the first order Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} (D_t+{\text {Op}}(p_1(t)) ) f(t)=0, &{} t\in [-T,T], \\ u|_{t=0}=g. \end{array} \right. \end{aligned}$$

(5.27)

In (5.27) we assume that $p_1(t)$ is a family of classical ${\text {SG}}$ symbols of order (1,1) depending smoothly on t. The hyperbolicity condition means that its principal symbol $q_1(t)$ such that $p_1(t)-q_1(t)\in {\text {SG}}^{0,0}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d})$, is real-valued. We then have the representation of the solution to (5.27) in the form $f(t)={\text {Op}}_{\varphi (t)}(a(t))g$. Theorem 4.1, Remark 4.2 and Theorem 4.3 describe the loss of regularity and weight for the solutions in the corresponding functional settings.

Example 5.28

Let $1<p<\infty$ and $g\in H^p_{\sigma _p,\sigma _p}({\mathbf {R}}^{d})$, where we have set $\sigma _p=(d-1)\left| \dfrac{1}{p}-\dfrac{1}{2}\right|$. Assume also that $p_1(t,x,\xi )=\langle c_1(t,x),\xi \rangle +c_0(t)$, so that $q_1(t,x,\xi )=\langle q_{01}(t,x),\xi \rangle$. We restrict here to large frequencies, as in [23], choosing a 0-excision function $\chi \in C^\infty ({\mathbf {R}}^{d})$ such that $\chi (\xi )=1$ for $|\xi |\ge 2\varepsilon$, for some sufficiently large $\varepsilon >0$. Then Theorem 4.1 and Remark 4.2 imply that for each $t\in [-T^\prime ,T^\prime ]$, $0<T^\prime \le T$, the solution f of the Cauchy problem (5.27) satisfies $\chi (D)f(t)\in L^p({\mathbf {R}}^{d})$. Moreover, for every $s,\sigma \in {\mathbf {R}}^{}$, there are $C_T>0$ and $0<T^\prime \le T$ such that

$$\begin{aligned} \Vert \chi (D)f(t)\Vert _{H^p_{s,\sigma }({\mathbf {R}}^{d})}\le C_T \Vert g\Vert _{H^p_{s+\sigma _p,\sigma +\sigma _p}({\mathbf {R}}^{d})}, \end{aligned}$$

for all $t\in [-T^\prime ,T^\prime ]$ and all $g\in H^p_{s+\sigma _p,\sigma +\sigma _p}({\mathbf {R}}^{d})$. Finally, since the hypotheses of Theorem 5.26 are satisfied, with ${\mathcal C}=H^p_{s,\sigma }({\mathbf {R}}^{d})$, ${\mathcal B}_0=H^p_{s+\sigma _p,\sigma +\sigma _p}({\mathbf {R}}^{d})$, $r=\rho =1$, $k=l=1$, $\varphi (t)$, and $\Omega =B_\varepsilon (0)$, we have

$$\begin{aligned} {\text {WF}}^m_{\mathcal C}( \chi (D)f(t) ) \subseteq (\phi (t)({\text {WF}}^m_{{\mathcal B}_0} (g)))^{\mathrm {con}_m}, \quad m=1,2,3, \end{aligned}$$

where $\phi (t)$ is the canonical transformation (5.6) associated with the phase function $\varphi (t)$, which turns out to be (positively) homogeneous with respect to the covariable (since this is true for $q_1(t)$, see, e.g., [36]).

Example 5.29

Let $s,\sigma \in \mathbf R$ and $1\le p<\infty$ be such that

$$\begin{aligned} s\le -d\left| \frac{1}{2}-\frac{1}{p} \right| ,\quad \sigma \le -d\left| \frac{1}{2}-\frac{1}{p} \right| , \end{aligned}$$

and let $\omega _j\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$, $j=1,2$, be such that

$$\begin{aligned} \omega _2(x,\varphi ^\prime _x(x,\xi ) ) \lesssim \omega _1(\varphi ^\prime _\xi (x,\xi ),\xi )\cdot \langle x\rangle ^s \langle \xi \rangle ^\sigma . \end{aligned}$$

Also assume that $\omega _j$ is $(\phi _j,j)$-invariant, $j=1,2$, with $\phi _1:x\mapsto \varphi ^\prime _\xi (x,\xi )$ and $\phi _2:\xi \mapsto \varphi ^\prime _x(x,\xi )$. This clearly holds true, in particular, for the trivial weights $\omega _j(x,\xi )=1$ and for product type weights $\omega _j(x,\xi )=\langle x\rangle ^{s_j}\,\langle \xi \rangle ^{\sigma _j}$, with appropriate choices of $s_j,\sigma _j\in {\mathbf {R}}^{}$, $j=1,2$.

Theorem 4.3 implies that, if $g\in M^p_{(\omega _1)}({\mathbf {R}}^{d})$. Then the solution f(t) of the Cauchy problem (5.27) satisfies $f(t)\in M^p_{(\omega _2)}({\mathbf {R}}^{d})$, for each $t\in [-T^\prime ,T^\prime ]$, $0<T^\prime \le T$. Moreover, there are $C_T>0$, $0<T^\prime \le T$ such that

$$\begin{aligned} \Vert f(t)\Vert _{M^p_{(\omega _2)}({\mathbf {R}}^{d})}\le C_T \Vert g\Vert _{M^p_{(\omega _1)}({\mathbf {R}}^{d})}, \end{aligned}$$

for all $t\in [-T^\prime ,T^\prime ]$ and all $g\in M^p_{(\omega _1)}({\mathbf {R}}^{d})$. Finally, since the hypotheses of Theorem 5.26 are satisfied, with ${\mathcal C}=M^p_{(\omega _2)}({\mathbf {R}}^{d})$, ${\mathcal B}_0=M^p_{(\omega _1)}({\mathbf {R}}^{d})$, $r=\rho =1$, $k=l=1$, $\varphi (t)$, and $\Omega =\emptyset$, we have

$$\begin{aligned} {\text {WF}}^m_{\mathcal C}( f(t) ) \subseteq (\phi (t)({\text {WF}}^m_{{\mathcal B}_0} (g)))^{\mathrm {con}_m}, m=1,2,3, \end{aligned}$$

with the canonical transformation $\phi (t)$ associated with $\varphi (t)$ in (5.6). Completely similar results hold true when $M^p_{(\omega _j)}({\mathbf {R}}^{d})$ is replaced by $M^\infty _{0,(\omega _j)}({\mathbf {R}}^{d})$, $j=1,2$.

As a first example involving second order SG-hyperbolic operators, we consider a variant of (5.20), choosing $V\equiv 0$ and focusing again on large frequencies, by adding a correction term of the form $\langle \, \cdot \, \rangle ^2(1-\widetilde{\chi }(D)^2)|D|^2$, with a 0-excision function $\widetilde{\chi }$, that is

$$\begin{aligned} \left\{ \begin{array}{ll} L_{\widetilde{\chi }}f(t)=(D_t^2-\langle \, \cdot \, \rangle ^2\,(\widetilde{\chi }(D)|D|)^2)f(t)=0, &{} t\in [-T,T], \\ f|_{t=0}=g_0,\\ D_tf|_{t=0}=g_1. \end{array} \right. \end{aligned}$$

(5.28)

In this case, the results on SG-hyperbolic operators stated in Sect. 5.3 cannot be applied directly, since (5.23) does not hold for the two roots $\tau _{1,2}(x,\xi )=\mp \langle x\rangle \widetilde{\chi }(\xi )|\xi |$. Nevertheless, we can anyway switch from the Cauchy problem (5.28) to an equivalent first order $2\times 2$ system, setting $F_1(t)=f(t)$, $F_2=\langle D\rangle ^{-1}\langle \, \cdot \, \rangle ^{-1}D_tf(t)=\langle D\rangle ^{-1}\langle \, \cdot \, \rangle ^{-1}D_tF_1(t)$, namely,

(5.29)

with

$$\begin{aligned} G=\begin{pmatrix} g_0 \\ \langle D\rangle ^{-1}\langle \, \cdot \, \rangle ^{-1}g_1 \end{pmatrix}, \quad k_1(x,\xi )= \begin{pmatrix} 0 &{} \langle x\rangle \langle \xi \rangle \\ \langle x\rangle (\widetilde{\chi }(\xi )|\xi |)^2)\langle \xi \rangle ^{-1} &{} 0 \end{pmatrix}, \end{aligned}$$

and $k_0$ a matrix of SG symbols of order 0, 0. The principal part $k_1$ of the coefficient matrix has, of course, real distinct eigenvalues $\tau _{1,2}$ in the region $|\xi |\ge c >0$. The theory developed in [14], Ch. 6, shows that the system (5.29) can be symmetrized and its principal part diagonalized, modulo order 0, 0 operators. By a variant of the computations in [15], in the region $|\xi |\ge c>0$ it can also be perfectly diagonalized, that is, the two equations can be decoupled, up to smoothing operators. Summing up, in the case of high frequencies, the Fourier integral operator method can be applied to (5.29), with the consequences described in the next Example 5.30.

Example 5.30

Let $1<p<\infty$, $s,\sigma \in {\mathbf {R}}^{}$, $g_0\in H^p_{s+1+\sigma _p,\sigma +1+\sigma _p}({\mathbf {R}}^{d})$, and $g_1\in H^p_{s+\sigma _p,\sigma +\sigma _p}({\mathbf {R}}^{d})$. Choose a 0-excision function $\chi$ as in Example 5.28 with $\varepsilon >0$ sufficiently large. Then the solution f(t) of (5.28) satisfies $\chi (D)f(t)\in H^p_{s+1,\sigma +1}({\mathbf {R}}^{d})$, $t\in [-T^\prime ,T^\prime ]$, $0<T^\prime \le T$. Moreover, (5.26) holds with $\mu =2$, $\nu =1$, ${\mathcal C}=H^p_{s+1,\sigma +1}({\mathbf {R}}^{d})$, ${\mathcal B}_0=H^p_{s+1+\sigma _p,\sigma +1+\sigma _p}({\mathbf {R}}^{d})$ and ${\mathcal B}_1=H^p_{s+\sigma _p,\sigma +\sigma _p}({\mathbf {R}}^{d})$, namely

$$\begin{aligned} {\text {WF}}^m_{\mathcal C}( \chi (D)f(t) )\! \subseteq \bigcup _{j=1}^2\! \left[ (\phi _{j}(t)({\text {WF}}^m_{{\mathcal B}_0} (g_0)))^{\mathrm {con}_m} \cup (\phi _{j}(t)({\text {WF}}^m_{{\mathcal B}_1} (g_1)))^{\mathrm {con}_m} \right] \!, \end{aligned}$$

$m=1,2,3$, where $\phi _j(t)$ is the canonical transformation (5.6) generated by the phase function $\varphi _{j}(t)$, solution to the eikonal equation associated with $\tau _{1,2}$, which turn out to be positively homogeneous with respect to the coverable (since this holds for $\tau _{1,2}$).

Consider now the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} Lf(t)=(D_t^2-\langle \, \cdot \, \rangle ^2\langle D\rangle ^2)f(t)=0, &{} t\in [-T,T], \\ f|_{t=0}=g_0,\\ D_tf|_{t=0}=g_1, \end{array} \right. \end{aligned}$$

(5.30)

involving the operator (5.20). First of all, we prove that L is ${\text {SG}}$-classical strictly hyperbolic. Indeed, $p_2(x,\xi )=\langle x\rangle ^2\langle \xi \rangle ^2\in {\text {SG}}^{2,2}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d})$, with, for instance,

$$\begin{aligned} q_2(x,\xi )=\chi (\xi )|\xi |^2\langle x\rangle ^2+\chi (x)(\langle \xi \rangle ^2|x|^2-\chi (\xi )|x|^2\cdot |\xi |^2)\ge 0, \end{aligned}$$

for a fixed 0-excision function $\chi$. The characteristic equation $\tau ^2-q_2(x,\xi )=0$ has then two real distinct solutions, namely

$$\begin{aligned} \tau _{1,2}(x,\xi )=\mp \sqrt{q_2(x,\xi )}=\mp \tau _0(x,\xi )\in {\text {SG}}^{1,1}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d}), \end{aligned}$$

(5.31)

such that

$$\begin{aligned} \tau _2(x,\xi )-\tau _1(x,\xi )=2\tau _0(x,\xi )\gtrsim \langle x\rangle \langle \xi \rangle \text { for } |x|+|\xi |\ge R, \end{aligned}$$

with a suitable $R>0$. Writing, as above, $L_j=D_t-{\text {Op}}(\tau _j)+{\text {Op}}(s_{j})$, $j=1,2$, we find

$$\begin{aligned} L_2L_1&=D_t^2-{\text {Op}}(\tau _0)^2+({\text {Op}}(s_1)+{\text {Op}}(s_2))D_t+{\text {Op}}(s_2){\text {Op}}(s_1) \\&=D_t^2-{\text {Op}}(q_2)+{\text {Op}}(a)+({\text {Op}}(s_1)+{\text {Op}}(s_2))D_t+{\text {Op}}(s_2){\text {Op}}(s_1), \end{aligned}$$

where

$$\begin{aligned} a(x,\xi )\sim \sum _\alpha \dfrac{i^{|\alpha |}}{\alpha !}D^\alpha _\xi \tau _0(x,\xi ) D^\alpha _x\tau _0(x,\xi ) \in {\text {SG}}^{0,0}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d}). \end{aligned}$$

Proposition 5.23 implies that there exist $s_1,s_2\in {\text {SG}}^{0,0}_{1,1,\mathrm {cl}}({\mathbf {R}}^{2d})$, $s_1=s_0=-s_2+r_1$, $r_1\in {{\mathscr {S}}}({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} {\text {Op}}(a)-{\text {Op}}(s_0)^2={\text {Op}}(q_2-p_2)+{\text {Op}}(r_2), \quad r_2\in {{\mathscr {S}}}({\mathbf {R}}^{2d}), \end{aligned}$$

so that L also satisfies the Levi condition.

Example 5.31

Let $s,\sigma \in \mathbf R$ and $1\le p<\infty$ be such that

$$\begin{aligned} s\le -d\left| \frac{1}{2}-\frac{1}{p} \right| ,\quad \sigma \le -d\left| \frac{1}{2}-\frac{1}{p} \right| , \end{aligned}$$

and let $\omega _j\in {\mathscr {P}}_{1,1}({\mathbf {R}}^{2d})$, $j=0,1,2$, be such that, for $k=1,2$,

$$\begin{aligned} \begin{array}{ll} \omega _0(x,\varphi ^\prime _{kx}(x,\xi ) ) &{}\lesssim \omega _1(\varphi ^\prime _{k\xi }(x,\xi ),\xi )\cdot \langle x\rangle ^{s} \langle \xi \rangle ^{\sigma }, \\ \omega _0(x,\varphi ^\prime _{kx}(x,\xi ) ) &{}\lesssim \omega _2(\varphi ^\prime _{k\xi }(x,\xi ),\xi )\cdot \langle x\rangle ^{s+1} \langle \xi \rangle ^{\sigma +1}, \end{array} \end{aligned}$$

(5.32)

Also assume that each $\omega _j$, $j=0,1,2$, is invariant with respect to the ${\text {SG}}$-diffeomorphims appearing in (5.32), generated by the phase functions $\varphi _j(t)$, solutions of the eikonal equations associated with $\tau _j$, $j=1,2$, given by (5.31).

Theorem 4.3 implies that, if $g_0\in M^p_{(\omega _1)}({\mathbf {R}}^{d})$, $g_1\in M^p_{(\omega _2)}({\mathbf {R}}^{d})$, then the solution f(t) of the Cauchy problem (5.27) satisfies $f(t)\in M^p_{(\omega _0)}({\mathbf {R}}^{d})$, for each $t\in [-T^\prime ,T^\prime ]$, $0<T^\prime \le T$. Moreover, setting ${\mathcal C}=M^p_{(\omega _0)}({\mathbf {R}}^{d})$, ${\mathcal B}_0=M^p_{(\omega _1)}({\mathbf {R}}^{d})$, ${\mathcal B}_1=M^p_{(\omega _2)}({\mathbf {R}}^{d})$, the inclusion (5.26) holds true with $\mu =2$, $\nu =1$, namely

$$\begin{aligned} {\text {WF}}^m_{\mathcal C}( f(t) ) \subseteq \bigcup _{j=1}^2 \left[ (\phi _{j}(t)({\text {WF}}^m_{{\mathcal B}_0} (g_0)))^{\mathrm {con}_m} \cup (\phi _{j}(t)({\text {WF}}^m_{{\mathcal B}_1} (g_1)))^{\mathrm {con}_m} \right] , \end{aligned}$$

$m=1,2,3$, with the canonical transformations $\phi _j(t)$ generated by the phase functions $\varphi _j(t)$, $j=1,2$. A completely similar result holds true when $M^p_{(\omega _j)}({\mathbf {R}}^{d})$ is replaced by $M^\infty _{0,(\omega _j)}({\mathbf {R}}^{d})$, $j=0,1,2$.

Finally, we recall that for certain types of choices of the symbol and the amplitude, we may avoid to have losses of regularity of the solution f(t) compared to the initial data (cf. [2, 12, 42, 45] and Remark 4.4).