Wigner analysis of fourier integral operators with symbols in the Shubin classes

Abstract

We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes $\Gamma ^m({\mathbb {R}^{2d}})$, with negative order m. The phases considered are the so-called tame ones, which appear in the Schrödinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the Hörmander’s class $S^{0}_{0,0}({\mathbb {R}^{2d}})$. Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order m.

1 Introduction

The protagonist of this study is the Wigner distribution, one of the most popular time-frequency representations. It was introduced by Wigner in 1932 [34] in the framework of Quantum Mechanics and later applied to signal processing and time-frequency analysis by Ville, Cohen and many other authors, see, e.g., [3, 4, 33] and the textbooks [18, 21, 25].

Definition 1.1

Consider $f,g\in L^2(\mathbb {R}^d)$. The cross-Wigner distribution W(f, g) is

$$\begin{aligned} W(f,g)(x,\xi )=\int _{\mathbb {R}^d}f(x+\frac{t}{2})\overline{g(x-\frac{t}{2})}e^{-2\pi i t\xi }\,dt,\quad (x,\xi )\in {\mathbb {R}^{2d}}. \end{aligned}$$

(1)

If $f=g$ we write $Wf:=W(f,f)$, the so-called Wigner distribution of f.

Wigner used the above representation to analyse the action of the Schrödinger propagator. We may extend the Wigner approach in [34] as follows: given a linear operator $T:\,\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$, we consider an operator K on $\mathcal {S}({\mathbb {R}^{2d}})$ such that

$$\begin{aligned} W(Tf,Tg) = KW(f,g), \qquad f,g\in \mathcal {S}(\mathbb {R}^d). \end{aligned}$$

(2)

Its integral kernel k is called the Wigner kernel of T:

$$\begin{aligned} W(Tf,Tg)(z) = \int _{{\mathbb {R}^{2d}}}k(z,w) W(f,g)(w)\,dw,\quad z\in {\mathbb {R}^{2d}},\quad f,g\in \mathcal {S}(\mathbb {R}^d). \end{aligned}$$

(3)

As an elementary example of the effectiveness of the Wigner distribution, consider the Schrödinger propagator $T_\tau$, for a fixed time $\tau \in \mathbb {R}$, of the free particle equation

$$\begin{aligned} T_\tau f(x)=\int _{\mathbb {R}^d}e^{2\pi i( x\xi -\tau \xi ^2)} {\widehat{f}}(\xi )d\xi ,\quad x\in \mathbb {R}^d. \end{aligned}$$

We have

$$\begin{aligned} W(T_\tau f) (x,\xi )=Wf(x-\tau \xi ,\xi ) \end{aligned}$$

with Wigner kernel

$$\begin{aligned} k=\delta _{z-\chi (w)},\quad w,z\in {\mathbb {R}^{2d}}, \end{aligned}$$

(4)

where, if we write $w=(y,\eta )$, then $\chi (y,\eta )=(y+\tau \eta ,\eta )$. This striking result is due to the peculiar action of W on the phase $\Phi (x,\xi )=x\xi -\tau \xi ^2$. It generalizes to quadratic $\Phi (x,\xi )$, corresponding to quadratic Hamiltonians and linear symplectic map $\chi$ in (4), see for example [20].

Our aim is to extend this analysis to more general operators, namely Fourier integral operators of the form

$$\begin{aligned} T_If(x)=\int _{\mathbb {R}^d}e^{2\pi i\Phi (x,\xi )}\sigma (x,\xi ){\hat{f}}(\xi )d\xi , \qquad f\in \mathcal {S}(\mathbb {R}^d), \end{aligned}$$

(5)

with phase $\Phi$ and symbol $\sigma$ in suitable classes. A preliminary step was presented in [12], with T a pseudodifferential operator $\sigma (x,D)$, i.e., $\Phi (x,\xi )=x\xi$ in (5). The case of a quadratic $\Phi$ and a general $\sigma$ was considered in [9] and in [8], where a generalization of (4) was obtained by combining a linear symplectic map $\chi$ with the kernel of a pseudodifferential operator.

In the present paper we focus on the case of nonlinear symplectic mappings $\chi$ corresponding to non-quadratic $\Phi$, which we call tame, see Sect. 2 below for their definition.

As a counterpart of (4) we look for estimates of the type

$$\begin{aligned} |k(z,w) |\lesssim \frac{1}{\langle z-\chi (w)\rangle ^{2N}}, \end{aligned}$$

(6)

where $\langle z\rangle :=(1+|z|^2)^{1/2}$, in the spirit of the estimates for Gabor kernels, which have been widely investigated in the literature, classical references are [2, 10, 15, 16, 26, 27], see also [18, Chapter 5].

There are two obstructions to the validity of (6). The first, evident from (4) and also in the linear case, is that k(z, w) is not point-wise defined for $z=\chi (w)$. This can be easily rephrased by a rescaling of regularity. The second obstruction is of deeper nature, and it concerns only the nonlinear symplectic map $\chi$. In fact, it is well known that the Wigner transform may produce the so-called ghost frequencies. As observed in [9, 13], they are exactly preserved for Schrödinger propagators for linear $\chi$, i.e., quadratic $\Phi$, but this is not the case for nonlinear $\chi$. Namely, highly oscillating terms may appear in the expression of the kernel k(z, w) outside the graph of $z=\chi (w)$.

As a first attempt for eliminating ghost frequencies and re-establishing the validity of (6), we shall consider in the sequel symbols $\sigma$ of low order in Shubin classes [32]. Unluckily, this framework does not allow a direct application to Schrödinger equations, for which we address a future work, following a different smoothing procedure.

Let us outline the contents of the paper. Our starting point, in Sect. 3, will be the following abstract definition, along the lines of [15].

Definition 1.2

Consider a tame symplectic diffeomorphism $\chi$ (cf. Definition 2.3 below). For $N\in \mathbb {N}_+$, $N>d$, we say that the operator K in (2) is in the class FIO($\chi$, N) if its Wigner kernel k in (3) satisfies, for $z=(z_1,z_2)$, $w=(w_1,w_2)\in {\mathbb {R}^{2d}}$,

$$\begin{aligned} |k(z,w) |\lesssim \frac{1}{\langle z-\chi (w)\rangle ^{2N}}. \end{aligned}$$

(7)

Examples of operators which fall in the above class are pseudodifferential operators $\sigma (x,D)$ (the Kohn-Nirenberg form), defined by

$$\begin{aligned} \sigma (x,D) f(x)=\int _{\mathbb {R}^d}e^{2\pi i x \xi }\sigma (x,\xi )\hat{f}(\xi )\,d\xi , \end{aligned}$$

(8)

with a symbol $\sigma$ in the Shubin classes $\Gamma ^m({\mathbb {R}^{2d}})$, $m<-2(d+N)$, whose Wigner kernel $k_\sigma$ satisfies

$$\begin{aligned} |k_\sigma (z,w) |\lesssim \frac{1}{\langle z-w\rangle ^{2N}}. \end{aligned}$$

(9)

Here $\chi =I$, the identity mapping, cf. Section 2 below. More generally, Fourier integral operators of type I (cf. (5)) and II, having symbols in the same Shubin classes above and tame canonical transformations, fall in the class above, as we shall show in Sects. 4 and 5.

Let us state here the preliminary results of Sect. 3, which are the core of this study and may be collected as follows.

Theorem 1.3

(Properties of the class FIO($\chi$, N))

• (i) Boundedness. $T\in FIO(\chi , N)$ is bounded on $L^2(\mathbb {R}^d)$.

• (ii) Algebra Property. If $T_i\in FIO(\chi _i,N)$, $i=1,2$, then $T_1 T_2\in FIO(\chi _1\chi _2,N)$.

• (iii) If $T\in FIO(\chi , N)$ then its adjoint $T^{*}$ is in $FIO(\chi ^{-1}, N)$.

In Sect. 4 we shall show the Fourier integral operators of type I in (5), having symbols in suitable Shubin classes $\Gamma ^m({\mathbb {R}^{2d}})$ and tame phase functions are in the class FIO($\chi$, N).

The last Sect. 5 is devoted to the $L^2$-adjoint of the FIO I in (5), which can be written explicitly in the form

$$\begin{aligned} T_{II}f(x)=\int _{{\mathbb {R}^{2d}}}e^{-2\pi i[\Phi (y,\xi )-x\xi ]}\tau (y,\xi )f(y)dyd\xi , \qquad f\in \mathcal {S}(\mathbb {R}^d), \end{aligned}$$

where $\tau (y,\xi )\in \mathcal {S}'({\mathbb {R}^{2d}})$ is the symbol. Using tools from metaplectic Wigner distributions implemented in [8, 9] we are able to compute the Wigner kernel of the FIOs II above and prove that, under suitable assumptions on their symbols, they belong to $FIO(\chi ,N)$ as well. We underline that these results are valid for the whole class of tame phase $\Phi$ defined in Sect. 2.3, of particular interest is the case $\Phi$ non quadratic which gives rise to nonlinear symplectic transformations $\chi$, which were not treated in [8].

We believe that such theoretical study will pave the way to a better understanding of Wigner kernels for Fourier integral operators, with possible applications to dynamical versions of Hardy’s uncertainty principles [23, 28, 35,36,37], see also the recent contribution [22].

2 Preliminaries

Notation. We define $t^2=t\cdot t$, $t\in \mathbb {R}^d$, and, similarly, $xy=x\cdot y$. The space $\mathcal {S}(\mathbb {R}^d)$ is the Schwartz class and $\mathcal {S}'(\mathbb {R}^d)$ its dual (the space of tempered distributions). The brackets $\langle f,g\rangle$ means the extension to $\mathcal {S}' (\mathbb {R}^d)\times \mathcal {S}(\mathbb {R}^d)$ of the inner product $\langle f,g\rangle =\int f(t){\overline{g(t)}}dt$ on $L^2(\mathbb {R}^d)$ (conjugate-linear in the second component). We define by $Sym(2d,\mathbb {R})$ the group of $2d\times 2d$ real symmetric matrices. A point in the phase space is denoted by $z=(x,\xi )\in {\mathbb {R}^{2d}}$. We call (time-frequency shift) the operators

$$\begin{aligned} \pi (z)f(t) = e^{2\pi i \xi t} f(t-x), \, \quad t\in \mathbb {R}^d. \end{aligned}$$

(10)

$GL(d,\mathbb {R})$ denotes the group of real invertible $d\times d$ matrices.

2.1 The symplectic group $Sp(d,\mathbb {R})$, metaplectic operators and Wigner distributions

The standard symplectic matrix is

$$\begin{aligned} J=\begin{pmatrix} 0_{d\times d}&\quad I_{d\times d}\textrm{i}_{d\times d}&\quad 0_{d\times d}\end{pmatrix}. \end{aligned}$$

(11)

The symplectic group is defined by

$$\begin{aligned} Sp(d,\mathbb {R})=\left\{ \mathcal {A}\in GL(2d,\mathbb {R}):\;\mathcal {A}^T J\mathcal {A}=J\right\} , \end{aligned}$$

(12)

where $\mathcal {A}^T$ is the transpose of $\mathcal {A}$. We have $\det (\mathcal {A})=1$.

For $L\in GL(d,\mathbb {R})$ and $C\in Sym(2d,\mathbb {R})$, define:

$$\begin{aligned} \mathcal {D}_L:=\begin{pmatrix} L^{-1} &{} 0_{d\times d}\\ 0_{d\times d} &{} L^T \end{pmatrix} \qquad \text {and} \qquad V_C:=\begin{pmatrix} I_{d\times d} &{} 0\\ C &{} I_{d\times d} \end{pmatrix}. \end{aligned}$$

(13)

The matrices J, $V_C$, and $\mathcal {D}_L$ generate the group $Sp(d,\mathbb {R})$.

The Schrödinger representation $\rho$ of the Heisenberg group is given by

$$\begin{aligned} \rho (x,\xi ;\tau )=e^{2\pi i\tau }e^{-\pi i\xi x}\pi (x,\xi ), \end{aligned}$$

for all $x,\xi \in \mathbb {R}^d$, $\tau \in \mathbb {R}$. For every $A\in Sp(d,\mathbb {R})$, $\rho _A(x,\xi ;\tau ):=\rho (A (x,\xi );\tau )$ defines another representation of the Heisenberg group that is equivalent to $\rho$, that is, there exists a unitary operator ${\hat{A}}:L^2(\mathbb {R}^d)\rightarrow L^2(\mathbb {R}^d)$ such that

$$\begin{aligned} {\hat{A}}\rho (x,\xi ;\tau ){\hat{A}}^{-1}=\rho (A(x,\xi );\tau ), \qquad x,\xi \in \mathbb {R}^d, \ \tau \in \mathbb {R}. \end{aligned}$$

(14)

This operator is not unique: if ${\hat{A}}'$ is another unitary transformation satisfying (14), then ${\hat{A}}'=c{\hat{A}}$, for some $c\in \mathbb {C}$, with $|c|=1$. The set $\{{\hat{A}}: A\in Sp(d,\mathbb {R})\}$ is a group under operator composition and has the metaplectic group $Mp(d,\mathbb {R})$ as subgroup. It is a realization of the two-fold cover of $Sp(d,\mathbb {R})$. The projection

$$\begin{aligned} \pi ^{Mp}:Mp(d,\mathbb {R})\rightarrow Sp(d,\mathbb {R}) \end{aligned}$$

(15)

is a group homomorphism with kernel $\ker (\pi ^{Mp})=\{-id_{{L^2}},id_{{L^2}}\}$.

Here, if ${\hat{A}}\in Mp(d,\mathbb {R})$, the matrix A will be the unique symplectic matrix satisfying $\pi ^{Mp}({\hat{A}})=A$. Some examples of metaplectic operators we will use in the following are detailed below.

Example 2.1

Consider the matrices J, $\mathcal {D}_L$ and $V_C$ defined in (11) and (13). Then, if we denoted by $\mathcal {F}$ the Fourier transform,

(i):

$\pi ^{Mp}(\mathcal {F})=J$;

(ii):

if $\mathfrak {T}_L:=|\det (L)|^{1/2}\,f(L\cdot )$, then $\pi ^{Mp}(\mathfrak {T}_L)=\mathcal {D}_L$;

The relation between time-frequency shifts and metaplectic operators is the following:

$$\begin{aligned} \pi (\mathcal {A}z) = c_\mathcal {A}\, {\hat{\mathcal {A}}} \pi (z) {\hat{\mathcal {A}}}^{-1}\quad \forall z\in {\mathbb {R}^{2d}}\,, \end{aligned}$$

(16)

with a phase factor $c_\mathcal {A}\in \mathbb {C}, |c_{\mathcal {A}}| =1$ (see, e.g., [20, 24]).

Metaplectic Wigner distributions. In the study of FIOs of type II we will use tools from the theory of metaplectic Wigner distributions. Here we list the basic elements for this study. For ${\hat{\mathcal {A}}}\in Mp(2d,\mathbb {R})$, the metaplectic Wigner distribution associated to ${\hat{\mathcal {A}}}$ is defined as

$$\begin{aligned} W_\mathcal {A}(f,g)={\hat{\mathcal {A}}}(f\otimes {\bar{g}}),\quad f,g\in L^2(\mathbb {R}^d). \end{aligned}$$

(17)

The most important time-frequency representations are metaplectic Wigner distributions. The $\tau$-Wigner distributions, $\tau \in \mathbb {R}$, defined by

$$\begin{aligned} W_\tau (f,g)(x,\xi )=\int _{\mathbb {R}^d} f(x+\tau t)\overline{g(x-(1-\tau )t)}e^{-2\pi i\xi t}dt, \qquad (x,\xi )\in {\mathbb {R}^{2d}},\nonumber \\ \end{aligned}$$

(18)

for $f,g\in L^2(\mathbb {R}^d)$, are metaplectic Wigner distributions. The case $\tau =1/2$ is the cross-Wigner distribution, defined in (1). $\tau$-Wigner distributions are metaplectic Wigner distributions:

$$\begin{aligned} W_\tau (f,g)={\hat{A}}_\tau (f\otimes {\bar{g}}), \end{aligned}$$

with

$$\begin{aligned} A_\tau =\begin{pmatrix} (1-\tau )I_{d\times d} &{} \tau I_{d\times d} &{} 0_{d\times d} &{} 0_{d\times d}\\ 0_{d\times d} &{} 0_{d\times d} &{} \tau I_{d\times d} &{} -(1-\tau )I_{d\times d}\\ 0_{d\times d} &{} 0_{d\times d} &{} I_{d\times d} &{} I_{d\times d}\\ -I_{d\times d} &{} I_{d\times d} &{} 0_{d\times d} &{} 0_{d\times d} \end{pmatrix}. \end{aligned}$$

(19)

In particular, we recapture the Wigner case when $\tau =1/2$:

$$\begin{aligned} Wf=W_{1/2}(f,f)={\hat{A}}_{1/2}(f\otimes {\bar{f}}), \quad f\in L^2(\mathbb {R}^d). \end{aligned}$$

(20)

${\hat{A}}_{1/2}$ can be split into the product

$$\begin{aligned} {\hat{A}}_{1/2}=\mathcal {F}_2\mathfrak {T}_L, \end{aligned}$$

(21)

with

$$\begin{aligned} L=\begin{pmatrix}I_{d\times d} &{} \frac{1}{2}I_{d\times d}\\ I_{d\times d} &{} -\frac{1}{2}I_{d\times d}\end{pmatrix}. \end{aligned}$$

Hence

$$\begin{aligned} {\hat{A}}_{1/2}F(x,\xi )=\int _{\mathbb {R}^d}F(x+t/2,x-t/2)e^{-2\pi i\xi t}dt, \qquad F\in \mathcal {S}({\mathbb {R}^{2d}}), \end{aligned}$$

and

$$\begin{aligned} {\hat{A}}_{1/2}^{-1}=\mathfrak {T}_{L^{-1}}\mathcal {F}_2^{-1}, \end{aligned}$$

where

$$\begin{aligned} L^{-1}=\begin{pmatrix} \frac{1}{2}I_{d\times d} &{} \frac{1}{2}I_{d\times d}\\ I_{d\times d} &{} -I_{d\times d} \end{pmatrix}, \end{aligned}$$

so that

$$\begin{aligned} {\hat{A}}_{1/2}^{-1}F(x,\xi )=\int _{\mathbb {R}^d}F(x/2+\xi /2,y)e^{2\pi i(x-\xi ) y}dy, \qquad F\in \mathcal {S}({\mathbb {R}^{2d}}). \end{aligned}$$

(22)

2.2 Shubin and Hörmander classes [18, 30, 32]

In our study we shall consider the following weight functions

$$\begin{aligned} v_s(z)=\langle z\rangle ^s=(1+|z|^2)^{\frac{s}{2}},\quad s\in \mathbb {R}, \end{aligned}$$

(23)

Definition 2.2

Fix $m\in \mathbb {R}$. The Shubin class $\Gamma ^m({\mathbb {R}^{2d}})$ is the set of functions $a\in \mathcal {C}^\infty ({\mathbb {R}^{2d}})$ satisfying

$$\begin{aligned} |\partial ^\alpha _z a(z)|\le C_\alpha v_{m-|\alpha |}(z),\quad z\in {\mathbb {R}^{2d}}, \,\alpha \in \mathbb {Z}^{2d}_+, \end{aligned}$$

for a suitable constant $C_\alpha >0$, where $v_s(z)=\langle z\rangle ^s$ is defined in (23).

The Hörmander class $S^0_{0,0}({\mathbb {R}^{2d}})$, consists of smooth functions $\sigma$ on ${\mathbb {R}^{2d}}$ such that

$$\begin{aligned} |\partial _x^\alpha \partial _\xi ^\beta \sigma (x,\xi )|\le c_{\alpha ,\beta },\quad \alpha ,\beta \in \mathbb {N}^d,\quad x,\xi \in \mathbb {R}^d. \end{aligned}$$

(24)

2.3 Tame phase functions and related canonical transformations

Definition 2.3

We follow the notation of [8, 15]. A real phase function $\Phi (x,\eta )$ is named tame if it satisfies the following properties:

A1.:

$\Phi \in \mathcal {C}^{\infty }({\mathbb {R}^{2d}})$;

A2.:

For $z=(x,\xi )\in {\mathbb {R}^{2d}}$,

$$\begin{aligned} |\partial _z^\alpha \Phi (z)|\le C_\alpha ,\quad |\alpha |\ge 2;\end{aligned}$$

(25)

A3.:

There exists $\delta >0$:

$$\begin{aligned} |\det \,\partial ^2_{x,\eta } \Phi (x,\xi )|\ge \delta . \end{aligned}$$

(26)

Solving the system

$$\begin{aligned} \left\{ \begin{array}{l} y=\Phi _\eta (x,\eta ), \\ \xi =\Phi _x(x,\eta ), \end{array} \right. \end{aligned}$$

(27)

with respect to $(x,\xi )$, one obtains a map $\chi$

$$\begin{aligned} (x,\xi )=\chi (y,\eta ), \end{aligned}$$

(28)

with the following properties:

A4.:

$\chi :{\mathbb {R}^{2d}}\rightarrow {\mathbb {R}^{2d}}$ is a symplectomorphism (smooth, invertible, and preserves the symplectic form in ${\mathbb {R}^{2d}}$, i.e., $dx\wedge d\xi = d y\wedge d\eta$.)

A5.:

For $z=(y,\eta )$,

$$\begin{aligned} |\partial _z^\alpha \chi (z)|\le C_\alpha ,\quad |\alpha |\ge 1;\end{aligned}$$

(29)

A6.:

There exists $\delta >0$:

$$\begin{aligned} |\det \,\frac{\partial x}{\partial y}(y,\eta )|\ge \delta \quad \text{ for }\,\, (x,\xi )=\chi (y,\eta ). \end{aligned}$$

(30)

Conversely, as it was observed in [15], to every transformation $\chi$ satisfying the three hypothesis above corresponds a tame phase $\Phi$, uniquely determined up to a constant.

2.4 Properties of the Wigner Kernel

The Wigner kernel of a continuous, linear operator $T:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ was introduced and studied in [8]. We recall its definition and the properties useful for our framework.

Definition 2.4

The Wigner kernel of a continuous, linear operator $T:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ is the distribution $k\in \mathcal {S}'(\mathbb {R}^{4d})$ satisfying

$$\begin{aligned} \langle W(Tf,Tg),W(u,v)\rangle =\langle k,W(u,v)\otimes \overline{W(f,g)}\rangle , \quad f,g,u,v\in \mathcal {S}(\mathbb {R}^d). \end{aligned}$$

(31)

Observe that if $k\in \mathcal {S}(\mathbb {R}^{4d})$ the integral formula (3) holds true. The results of Theorem 3.3 and 4.3 in [8] can be rephrased as follows:

Theorem 2.5

Consider T as above and let $k_T\in \mathcal {S}'({\mathbb {R}^{2d}})$ be its kernel. There exists a unique distribution $k\in \mathcal {S}'(\mathbb {R}^{4d})$ such that (31) holds. Hence, every continuous linear operator $T:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ has a unique Wigner kernel. Furthermore,

$$\begin{aligned} k=\mathfrak {T}_pWk_T, \end{aligned}$$

(32)

with $\mathfrak {T}_pF(x,\xi ,y,\eta )=F(x,y,\xi ,-\eta )$.

In particular, if $T\in B(L^2(\mathbb {R}^d))$ has Wigner kernel k, then its adjoint $T^*\in B(L^2(\mathbb {R}^d))$ has Wigner kernel ${\tilde{k}}(z,w)= k(w,z)$, $z,w\in {\mathbb {R}^{2d}}$.

3 Properties of FIO($\chi , N$)

This section is devoted to prove Theorem 1.3 in the introduction. This requires several steps, developed in what follows.

Theorem 3.1

An operator $T\in FIO(\chi ,N)$, is bounded on $L^2(\mathbb {R}^d)$.

Proof

For $f\in L^2(\mathbb {R}^d)$ we recall [18, Chapter 1] that the Wigner $Wf\in L^2({\mathbb {R}^{2d}})$ and Moyal’s identity $\Vert Wf\Vert _{L^2({\mathbb {R}^{2d}})}=\Vert f\Vert ^2_{L^2(\mathbb {R}^d)}$.

Using (3), Definition 1.2, for any $f\in L^2(\mathbb {R}^d)$,

$$\begin{aligned} \Vert Tf\Vert ^2_{L^2(\mathbb {R}^d)}=\Vert W(Tf)\Vert _{L^2({\mathbb {R}^{2d}})}, \end{aligned}$$

and

$$\begin{aligned} \Vert W(Tf)\Vert _{L^2({\mathbb {R}^{2d}})}&\lesssim \left\| \int _{{\mathbb {R}^{2d}}}\frac{1}{\langle z-\chi (w)\rangle ^{2N}}Wf(w)\,dw\right\| _{L^2({\mathbb {R}^{2d}})}\\&\asymp \left\| \int _{{\mathbb {R}^{2d}}}\frac{1}{\langle \chi ^{-1}(z)-w\rangle ^{2N}}Wf(w)\,dw\right\| _{L^2({\mathbb {R}^{2d}})}\\&\lesssim \left\| \left( \frac{1}{\langle \cdot \rangle ^{2N}}*Wf\right) (\chi ^{-1}(z))\right\| _{L^2({\mathbb {R}^{2d}})}\\&\lesssim \left\| \left( \frac{1}{\langle \cdot \rangle ^{2N}}*Wf\right) (z)\right\| _{L^2({\mathbb {R}^{2d}})}\\&\le \left\| \frac{1}{\langle \cdot \rangle ^{2N}}\right\| _{L^1({\mathbb {R}^{2d}})}\Vert Wf\Vert _{L^2({\mathbb {R}^{2d}})}\\&\le C_N \Vert f\Vert ^2_{L^2(\mathbb {R}^d)}, \end{aligned}$$

where in the last row we used Young’s inequality (observe that $N>d$) and, in the last but one, the change of variables $z'=\chi ^{-1}(z)$ which, for any $F\in L^2({\mathbb {R}^{2d}})$,

$$\begin{aligned} \Vert F(\chi ^{-1}\cdot )\Vert ^2_{L^2({\mathbb {R}^{2d}})}&=\int _{{\mathbb {R}^{2d}}}|F(\chi ^{-1}(z))|^2\,dz=\int _{{\mathbb {R}^{2d}}}|F(z')|^2 \det |J\chi (z')|dz'\\&\le C_N\Vert F\Vert ^2_{L^2({\mathbb {R}^{2d}})}, \end{aligned}$$

by (29). Hence $\Vert Tf\Vert _{L^2(\mathbb {R}^d)}\le \sqrt{C_N} \Vert f\Vert _{L^2(\mathbb {R}^d)}$, that is $T\in B(L^2(\mathbb {R}^d))$. $\square$

Theorem 3.2

If $T_i\in FIO(\chi _i,N)$, $i=1,2$, then $T_1 T_2\in FIO(\chi _1\chi _2,N)$.

Proof

Using the Wigner representation in (3) we can write

$$\begin{aligned} W(T_1T_2f,T_1T_2g)(z) = \int _{{\mathbb {R}^{2d}}}k_{I,1}(z,w) W(T_2f,T_2g)(w)\,dw, \end{aligned}$$

(33)

where $k_{I,1}(z,w)$ is the Wigner kernel of the operator $T_1$, satisfying (7) with symplectic transformation $\chi _1$. Similarly,

$$\begin{aligned} W(T_2f,T_2g)(w) = \int _{{\mathbb {R}^{2d}}}k_{I,2}(w,u) W(f,g)(u)\,du, \end{aligned}$$

with $k_{I,2}(w,u)$ being the Wigner kernel of $T_2$ satisfying (7) with symplectic transformation $\chi _2$. Substituting the expression of $W(T_2f,T_2g)(w)$ in (33) we obtain

$$\begin{aligned} W(T_1T_2f,T_1T_2g)(z) = \int _{\mathbb {R}^{4d}} k_{I,1}(z,w) k_{I,2}(w,u) W(f,g)(u)\,du\,dw, \end{aligned}$$

(34)

with $k_{I,i}(z,w)$ satisfying (7), $i=1,2$. Interchanging the integrals in (34) (observe that the assumptions of Fubini Theorem are satisfied) we can write

$$\begin{aligned} W(T_1T_2f,T_1T_2g)(z) = \int _{\mathbb {R}^{2d}}\left( \int _{{\mathbb {R}^{2d}}}k_{I,1}(z,w) k_{I,2}(w,u)\,dw\right) W(f,g)(u)\,du\nonumber \\ \end{aligned}$$

(35)

so that the Wigner kernel $k_{I,1 2}$ of the product $T_1T_2$ is given by

$$\begin{aligned} k_{I,1 2}(z,u):= \int _{{\mathbb {R}^{2d}}}k_{I,1}(z,w) k_{I,2}(w,u)\,dw. \end{aligned}$$

Using the Wigner kernel’s estimates in (7), we obtain

$$\begin{aligned} |k_{I,1 2}(z,u)|&\le \int _{{\mathbb {R}^{2d}}}| k_{I,1}(z,w)| | k_{I,2}(w,u)|dw\\&\lesssim \int _{{\mathbb {R}^{2d}}}\frac{1}{\langle z-\chi _1(w)\rangle ^{2N}\langle w-\chi _2(u)\rangle ^{2N}} dw\\&\asymp \int _{{\mathbb {R}^{2d}}}\frac{1}{\langle z-\chi _1(w)\rangle ^{2N}\langle \chi _1(w)-\chi _1\chi _2(u)\rangle ^{2N}}dw\\&\lesssim \int _{{\mathbb {R}^{2d}}}\frac{1}{\langle z-\chi _1(w)\rangle ^{2N}\langle \chi _1(w)-\chi _1\chi _2(u)\rangle ^{2N}}dw\\&=\int _{{\mathbb {R}^{2d}}}\frac{1}{\langle \chi _1(w)-z\rangle ^{2N}\langle \chi _1\chi _2(u)-\chi _1(w)\rangle ^{2N}}dw\\&=\int _{{\mathbb {R}^{2d}}}\frac{1}{\langle w'-z\rangle ^{2N}\langle \chi _1\chi _2(u)-w'\rangle ^{2N}}|\det J\chi _1^{-1}(w)| dw'\\ \end{aligned}$$

where we used the change of variables $\chi _1(w)=w'$ so that $dw=|\det J\chi _1^{-1}(w)| dw'$ since $|\det J\chi _1^{-1}(w)|\le C$ by (29), we obtain

$$\begin{aligned} |k_{I,1 2}(z,u)|&\lesssim \int _{{\mathbb {R}^{2d}}}\frac{1}{\langle w'-z\rangle ^{2N}\langle \chi _1\chi _2(u)-w'\rangle ^{2N}}dw'\\&=\int _{{\mathbb {R}^{2d}}}\frac{1}{\langle v \rangle ^{2N}\langle \chi _1\chi _2(u)-z-v \rangle ^{2N}}dv\\&=(\langle \cdot \rangle ^{-2N}*\langle \cdot \rangle ^{-2N})(\chi _1\chi _2(u)-z)\\&\lesssim \frac{1}{\langle z-\chi _1\chi _2(u)\rangle ^{2N}} \end{aligned}$$

where in the last row we used the weight convolution property $\langle \cdot \rangle ^s*\langle \cdot \rangle ^s\lesssim \langle \cdot \rangle ^s$ for $s<-2d$ (observe $N>d$). Thus, we obtain the desired estimate

$$\begin{aligned} |k_{I,1 2}(z,u)|\lesssim \frac{1}{\langle z-\chi _1\chi _2(u)\rangle ^{2N}}, \end{aligned}$$

that is $T_1T_2\in FIO(\chi _1\chi _2,N)$. $\square$

Theorem 3.3

If $T\in FIO(\chi ,N)$, then $T^{*}\in FIO(\chi ^{-1},N)$.

Proof

Theorem 3.1 gives that $T\in B(L^2(\mathbb {R}^d))$. Let k be integral kernel of T, then Theorem 2.5 says that the adjoint $T^*\in B(L^2(\mathbb {R}^d))$ has kernel ${\tilde{k}}$ given by

$$\begin{aligned} {\tilde{k}}(z,w)={k(w,z)}. \end{aligned}$$

This means it satisfies (cf. (7))

$$\begin{aligned} |{\tilde{k}}(z,w) |=| k(w,z) |\lesssim \frac{1}{\langle w-\chi (z)\rangle ^{2N}}. \end{aligned}$$

(36)

Since $\chi$ is a bi-Lipschitz transformation, $|w-\chi (z)|\asymp |z-\chi ^{-1}(w)|$ so that $\langle w-\chi (z)\rangle ^{2N}\asymp \langle z-\chi ^{-1}(w)\rangle ^{2N}$ and we obtain

$$\begin{aligned} |{\tilde{k}}(z,w) |\lesssim \frac{1}{ \langle z-\chi ^{-1}(w)\rangle ^{2N}}. \end{aligned}$$

(37)

Hence $T^*\in FIO(\chi ^{-1},N)$, as desired. $\square$

4 FIOs of type I

Here we focus on the analysis of Wigner kernels for FIOs of type I:

$$\begin{aligned} T_If(x)=\int _{\mathbb {R}^d}e^{2\pi i\Phi (x,\xi )}\sigma (x,\xi ){\hat{f}}(\xi )d\xi , \qquad f\in \mathcal {S}(\mathbb {R}^d). \end{aligned}$$

(38)

Recall that the Schwartz Kernel Theorem guarantees that every continuous linear operator $T:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ can be expressed in the form $T=T_I$, with a given phase $\Phi (x,\xi )$ and symbol $\sigma (x,\xi )$ in $\mathcal {S}'({\mathbb {R}^{2d}})$.

These operators have been widely investigated in the framework of PDEs, both from a theoretical and a numerical point of view; the literature is so huge that we cannot report all the results but limit to a very partial list of them, cf. [5,6,7, 14, 17, 19, 31].

If we assume that T is a continuous linear operator $\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ and $\chi$ satisfies conditions A4, A5, and A6 in Definition 2.3 then $T=T_{I,\Phi _\chi ,\sigma }$ (FIO of type I), with symbol $\sigma$ and phase $\Phi _\chi$.

A first result related to the Wigner kernel of a FIO I was obtained in [8, Theorem 5.8]. There, FIOs of type I with symbols in the Hörmander class $S^0_{0,0}({\mathbb {R}^{2d}})$ were considered. Since $\Gamma ^{m}({\mathbb {R}^{2d}})\subset S^0_{0,0}({\mathbb {R}^{2d}})$ whenever $m\le 0$, we can rephrase it in our context as follows.

Theorem 4.1

Let $T_I$ be a FIO of type I defined in (38) with symbol $\sigma \in \Gamma ^{m}({\mathbb {R}^{2d}})$, $m<0$. For $f\in \mathcal {S}(\mathbb {R}^d)$,

$$\begin{aligned} K(W(f,g))(x,\xi )=W(T_If,T_Ig)(x,\xi )=\int _{{\mathbb {R}^{2d}}}k_I(x,\xi ,y,\eta )W(f,g)(y,\eta )dyd\eta ,\nonumber \\ \end{aligned}$$

(39)

with Wigner kernel $k_I$ given by

$$\begin{aligned} k_I(x,\xi ,y,\eta )=\int _{{\mathbb {R}^{2d}}}e^{2\pi i[\Phi _I(x,\eta ,t,r)-(\xi t+ry)]} \sigma _I(x,\eta ,t,r)dtdr, \end{aligned}$$

(40)

and, for $x,\eta ,t,r\in \mathbb {R}^d$,

$$\begin{aligned}{} & {} \Phi _I(x,\eta ,t,r)=\Phi (x+\frac{t}{2},\eta +\frac{r}{2})-\Phi (x-\frac{t}{2},\eta -\frac{r}{2}), \end{aligned}$$

(41)

$$\begin{aligned}{} & {} \sigma _I(x,\eta ,t,r):=\sigma (x+\frac{t}{2},\eta +\frac{r}{2})\overline{\sigma (x-\frac{t}{2},\eta -\frac{r}{2})}. \end{aligned}$$

(42)

We have now all the tools to estimate the Wigner kernel of $T_I$.

Theorem 4.2

Consider $T_I$ the FIO of type I in (38). Fix $N\in \mathbb {N}$, $N>d$, and assume that the symbol $\sigma \in \Gamma ^{m}({\mathbb {R}^{2d}})$, with $m<-2(d+N)$. Let $k_I$ be the associated Wigner kernel in (40). Then,

$$\begin{aligned} |k_I(x,\xi ,y,\eta )|\lesssim \frac{\langle (x,\eta )\rangle ^{2N+m}}{\langle (x,\xi )-\chi (y,\eta )\rangle ^{2N}},\qquad x,\xi ,y,\eta \in {\mathbb {R}^{2d}}. \end{aligned}$$

(43)

Proof

Since $\Phi$ is smooth, we can expand $\Phi (x+\frac{t}{2},\eta +\frac{r}{2})$ and $\Phi (x-\frac{t}{2},\eta -\frac{r}{2})$ into a Taylor series around $(x, \eta )$. Namely,

$$\begin{aligned} \Phi \left( x+\frac{t}{2},\eta +\frac{r}{2}\right) =\Phi (x,\eta )+\frac{t}{2}\Phi _x (x,\eta )+\frac{r}{2} \Phi _\eta (x,\eta ) +\Phi _{2}(x,\eta ,t,r), \end{aligned}$$

(44)

where the remainder $\Phi _{2}$ is given by

$$\begin{aligned} \Phi _{2}(x,\eta ,t,r)=\sum _{|\alpha |=2}\int _0^1(1-\tau )\partial ^\alpha \Phi ((x,\eta )+\tau (t,r)/2)\,d\tau \frac{(t,r)^\alpha }{2^3\alpha !}. \end{aligned}$$

(45)

Similarly,

$$\begin{aligned} \Phi \left( x-\frac{t}{2},\eta -\frac{r}{2}\right) =\Phi (x,\eta )-\frac{t}{2}\Phi _x (x,\eta )-\frac{r}{2} \Phi _\eta (x,\eta ) +\widetilde{\Phi }_{2}(x,\eta ,t,r), \end{aligned}$$

(46)

with $\widetilde{\Phi }_{2}$ defined as

$$\begin{aligned} \widetilde{\Phi }_{2}(x,\eta ,t,r)=\sum _{|\alpha |=2}\int _0^1(1-\tau )\partial ^\alpha \Phi ((x,\eta )-\tau (t,r)/2)\,d\tau \frac{(t,r)^\alpha }{2^3\alpha !}. \end{aligned}$$

(47)

Inserting the phase expansions above in (40) we obtain

$$\begin{aligned} k_I(x,\xi ,y,\eta )=\int _{{\mathbb {R}^{2d}}} e^{-2\pi i[t\cdot (\xi -\Phi _x(x,\eta ))+ r\cdot (y-\Phi _\eta (x,\eta ))]}\tilde{\sigma }(x,\eta ,t,r)\,dtdr \end{aligned}$$

(48)

where $\tilde{\sigma }$ is defined as

$$\begin{aligned} \tilde{\sigma }(x,\eta ,t,r)=e^{2\pi i[\Phi _{2}-\widetilde{\Phi }_{2}](x,\eta ,t,r)} \sigma (x+\frac{t}{2},\eta +\frac{r}{2})\overline{\sigma (x-\frac{t}{2},\eta -\frac{r}{2})}, \end{aligned}$$

(49)

For $N\in \mathbb {N}$, $u=(t,r)\in {\mathbb {R}^{2d}}$, using the identity:

$$\begin{aligned}{} & {} (1-\Delta _u)^Ne^{-2\pi i[(\xi -\Phi _x(x,\eta ), y-\Phi _\eta (x,\eta ))\cdot (t,r)]}\\{} & {} \quad =\langle 2\pi (\xi -\Phi _x(x,\eta ), y-\Phi _\eta (x,\eta ))\rangle ^{2N} e^{-2\pi i [(\xi -\Phi _x(x,\eta ), y-\Phi _\eta (x,\eta ))\cdot (t,r)]}, \end{aligned}$$

we integrate by parts in (48) and obtain

$$\begin{aligned} k_I(x,\xi ,y,\eta )&=\frac{1}{\langle 2\pi (\xi -\Phi _x(x,\eta ), y-\Phi _\eta (x,\eta ))\rangle ^{2N}}\int _{{\mathbb {R}^{2d}}} e^{-2\pi i [(\xi -\Phi _x(x,\eta ), y-\Phi _\eta (x,\eta ))\cdot (t,r)]} \\&\quad \times (1-\Delta _u)^N \tilde{\sigma }(x,\eta ,t,r)\,dtdr.\\ \end{aligned}$$

The factor

$$\begin{aligned} (1-\Delta _u)^N \tilde{\sigma }(z,u),\quad z=(x,\eta ),\,\,u=(t,r) \end{aligned}$$

can be expressed as

$$\begin{aligned} e^{2\pi i[\Phi _{2}-\widetilde{\Phi }_{2}](z,u)}\sum _{|\alpha |+|\beta |+|\gamma |\le 2N} C_{\alpha ,\beta \gamma }p( \partial _u^{|\alpha |}(\Phi _{2}-\widetilde{\Phi }_{2})_z(u) (\partial _u^\beta \sigma )(z+u/2)(\partial _u^\gamma \sigma )(z-u/2), \end{aligned}$$

where $p(\partial _u^{|\alpha |} (\Phi _{2}-\widetilde{\Phi }_{2})_z)(u)$ is a polynomial made of derivatives w.r.t. u of $\Phi _{2}-\widetilde{\Phi }_{2}$ of order at most $|\alpha |$. By assumption,

$$\begin{aligned} |(\partial _u^\beta \sigma )(z+u/2)(\partial _u^\gamma \sigma )(z-u/2)|\lesssim \langle z+u/2\rangle ^{m-|\beta |} \langle z-u/2\rangle ^{m-|\gamma |}, \end{aligned}$$

which implies

$$\begin{aligned} |(1-\Delta _u)^N \tilde{\sigma }(z,u)|&\lesssim \sum _{|\alpha |+|\beta |+|\gamma |\le 2N} \langle u/2\rangle ^{|\alpha |}\langle z-u/2\rangle ^{m-|\beta |} \langle z+u/2\rangle ^{m-|\gamma |}\\&\lesssim \sum _{|\beta |+|\gamma |\le 2N} \langle u/2\rangle ^{2N-|\beta |-|\gamma |}\langle z-u/2\rangle ^{m-|\beta |} \langle z+u/2\rangle ^{m-|\gamma |}\\&\lesssim C_N\langle z-u/2\rangle ^{2N+m} \langle z+u/2\rangle ^{2N+m}. \end{aligned}$$

Using the change of variables $u'=u/2-z$, $du=2^{2d}du'$,

$$\begin{aligned} \int _{{\mathbb {R}^{2d}}}\langle z-u/2\rangle ^{2N+m} \langle z+u/2\rangle ^{2N+m}du= & {} 2^{2d}\int _{{\mathbb {R}^{2d}}}\langle u'\rangle ^{2N+m}\langle (-2z)-u'\rangle ^{2N+m}\\\lesssim & {} \langle z\rangle ^{2N+m} \end{aligned}$$

where, for $v_s=\langle \cdot \rangle ^s$, we used the weight convolution property $v_s*v_s\lesssim v_s$, for $s<2d$, cf. [25, Lemma 11.1.1]. Hence,

$$\begin{aligned} |k_I(x,\xi ,y,\eta )|&\le \frac{1}{\langle 2\pi (\xi -\Phi _x(x,\eta ), y-\Phi _\eta (x,\eta ))\rangle ^{2N}}\int _{{\mathbb {R}^{2d}}} |(1-\Delta _u)^N \tilde{\sigma }(x,\eta ,t,r)|\,dtdr\\&\lesssim \frac{\langle z\rangle ^{2N+m}}{\langle 2\pi (\xi -\Phi _x(x,\eta ), y-\Phi _\eta (x,\eta ))\rangle ^{2N}}\\&\asymp \frac{\langle z\rangle ^{2N+m}}{\langle \chi _1(y,\eta )-x, \chi _2(y,\eta )-\xi \rangle ^{2N}}. \end{aligned}$$

This gives the claim. $\square$

As a consequence,

Corollary 4.3

Under the assumptions of Theorem 4.2, the estimate (7) holds true, hence $T_I\in FIO(\chi ,N)$.

Proof

It is an immediate consequence of Theorem 4.2, since $2N+m<0$ so that $\langle (x,\eta )\rangle ^{2N+m}\le 1$, for every $x,\eta \in \mathbb {R}^d$. $\square$

5 FIOs of Type II

In this section we focus on the $L^2$-adjoint of a FIO of type I, which is a FIO of type II, written formally as

$$\begin{aligned} T_{II}f(x)=\int _{{\mathbb {R}^{2d}}}e^{-2\pi i[\Phi (y,\xi )-x\xi ]}\tau (y,\xi )f(y)dyd\xi , \qquad f\in \mathcal {S}(\mathbb {R}^d). \end{aligned}$$

(50)

First, we shall work with symbols $\tau$ in the Hörmander class $S^0_{0,0}({\mathbb {R}^{2d}})$, referring to [1] for their $L^2$-boundedness.

Proposition 5.1

Consider a FIO of type II as in (50), with $\tau \in S^0_{0,0}({\mathbb {R}^{2d}})$. Then, for all $f,g\in \mathcal {S}(\mathbb {R}^d)$,

$$\begin{aligned} (T_{II}f\otimes {\bar{g}})(x_1,x_2)=T_2(f\otimes {\bar{g}})(x_1,x_2), \end{aligned}$$

$x=(x_1,x_2)\in {\mathbb {R}^{2d}}$, where $T_2$ is the FIO of type II given by

$$\begin{aligned} T_2F(x)=\int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi _2(y,\xi )-x\xi ]}\tau _2(y,\xi )F(y)dyd\xi , \qquad F\in \mathcal {S}({\mathbb {R}^{2d}}), \end{aligned}$$

$y=(y_1,y_2)$, $\xi =(\xi _1,\xi _2)\in {\mathbb {R}^{2d}}$, and $\Phi _2$ is the tame phase on $\mathbb {R}^{4d}$ given by

$$\begin{aligned} \Phi _2(y,\xi )=\Phi (y_1,\xi _1)+y_2\xi _2; \end{aligned}$$

(51)

whereas the symbol $\tau _2$ is in $S^{0}_{0,0}(\mathbb {R}^{4d})$ and given by

$$\begin{aligned} \tau _2(y,\xi )=\tau (y_1,\xi _1)\otimes 1(y_2,\xi _2). \end{aligned}$$

(52)

Proof

Let f, g be in $\mathcal {S}(\mathbb {R}^d)$. Using the Fourier inversion formula on g:

$$\begin{aligned} g(x_2)=\int _{{\mathbb {R}^{2d}}}g(y_2)e^{2\pi i(x_2-y_2)\xi _2}dy_2d\xi _2 \end{aligned}$$

we can write

$$\begin{aligned} \begin{aligned} (T_{II}f\otimes {\bar{g}})(x_1,x_2)&=\Big (\int _{{\mathbb {R}^{2d}}}e^{-2\pi i[\Phi (y_1,\xi _1)-x_1\xi _1]}\tau (y_1,\xi _1)f(y_1)dy_1d\xi _1\Big )\overline{g(x_2)}\\&=\int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi (y_1,\xi _1)+y_2\xi _2-(x_1\xi _1+x_2\xi _2)]}f(y_1)\overline{g(y_2)}\\&\quad \times \tau (y_1,\xi _1)dy_1dy_2d\xi _1d\xi _2. \end{aligned} \end{aligned}$$

Observing that

$$\begin{aligned} x_1\xi _1+x_2\xi _2=x \xi , \end{aligned}$$

we can write

$$\begin{aligned} \Phi (y_1,\xi _1)+y_2\xi _2=\Phi _2(y_1,y_2,\xi _1,\xi _2), \end{aligned}$$

which is (51). Note that $\Phi _2 \in \mathcal {C}^{\infty }(\mathbb {R}^{4d})$ and satisfies (25) and (26), since

$$\begin{aligned} \partial ^2_{y,\xi }\Phi _2=\begin{pmatrix} \partial ^2_{y_1,y_1}\Phi &{} 0_{d\times d} &{} \partial ^2_{y_1,\xi _1}\Phi &{} 0_{d\times d} \\ 0_{d\times d} &{} 0_{d\times d} &{} 0_{d\times d} &{} I_{d\times d}\\ \partial ^2_{y_1,\xi _1}\Phi &{} 0_{d\times d} &{} \partial ^2_{\xi _1,\xi _1}\Phi &{} 0_{d\times d}\\ 0_{d\times d} &{}I_{d\times d} &{} 0_{d\times d} &{} 0_{d\times d} \end{pmatrix}. \end{aligned}$$

(53)

This means that $\Phi _2$ is a tame phase function.

Finally, since both $\tau$ and the function constantly equal to 1 are in the Hörmander class $S^{0}_{0,0}({\mathbb {R}^{2d}})$, it immediately follows that $\tau _2$ belongs to $S^{0}_{0,0}(\mathbb {R}^{4d})$.

$\square$

Using the same arguments as in the previous proposition, one can prove the issue below.

Proposition 5.2

Under the same assumptions of Proposition 5.1, for all $f,g\in \mathcal {S}(\mathbb {R}^d)$,

$$\begin{aligned} (f\otimes \overline{T_{II}g})(x_1,x_2)=T_2'(f\otimes \bar{g})(x_1,x_2), \end{aligned}$$

$x=(x_1,x_2)\in {\mathbb {R}^{2d}}$, where the operator $T_2'$ is the FIO of type II:

$$\begin{aligned} T_2'F(x)=\int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi _2'(y,\xi )-x\xi ]}\tau _2'(y,\xi )F(y)dyd\xi , \qquad F\in \mathcal {S}({\mathbb {R}^{2d}}), \end{aligned}$$

$y=(y_1,y_2)$, $\xi =(\xi _1,\xi _2)\in {\mathbb {R}^{2d}}$ and $\Phi _2'$ is the tame phase

$$\begin{aligned} \Phi _2'(y,\xi )=-\Phi (y_2,-\xi _2)+y_1\xi _1, \end{aligned}$$

whereas the symbol $\tau _2\in S^{0}_{0,0}(\mathbb {R}^{4d})$ is given by

$$\begin{aligned} \tau _2'(y,\xi )=1(y_1,\xi _1)\otimes \overline{\tau (y_2,-\xi _2)}. \end{aligned}$$

Next, we study the composition of the FIOs $T_2$ and $T_2'$.

Proposition 5.3

Consider the FIOs $T_2$ and $T_2'$ defined in Propositions 5.1 and 5.2. Then their product $T_2T_2'$ can be written as the following FIO of type II:

$$\begin{aligned} T_2T_2'F(x)=\int _{\mathbb {R}^{4d}}e^{-2\pi i{\mathbf \Phi }(y,\xi )-x\xi } \mathcal {T}(y,\xi ) F(y)\,dyd\xi ,\quad x\in {\mathbb {R}^{2d}}, \end{aligned}$$

(54)

for every $F\in \mathcal {S}({\mathbb {R}^{2d}})$, with tame phase on $\mathbb {R}^{4d}$:

$$\begin{aligned} {\mathbf \Phi }(y_1,y_2,\xi _1,\xi _2)=\Phi (y_1,\xi _1)-\Phi (y_2,-\xi _2) \end{aligned}$$

(55)

and symbol

$$\begin{aligned} \mathbf {\mathcal {T}}(y_1,y_2,\xi _1,\xi _2)=\tau (y_1,\xi _1)\overline{\tau (y_2,-\xi _2)}\in S^0_{0,0}(\mathbb {R}^{4d}). \end{aligned}$$

(56)

Proof

Let $F\in \mathcal {S}({\mathbb {R}^{2d}})$. We compute

$$\begin{aligned}&T_2T_2' F(x_1,x_2)\\&=\int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi (y_1,\xi _1)+y_2\xi _2-x_1\xi _1-x_2\xi _2]}\tau (y_1,\xi _1)T_2'F(y_1,y_2)dy_1dy_2d\xi _1d\xi _2\\&=\int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi (y_1,\xi _1)+y_2\xi _2-x_1\xi _1-x_2\xi _2]}\tau (y_1,\xi _1)\int _{\mathbb {R}^{4d}}e^{-2\pi i[-\Phi (z_2,-\eta _2)+z_1\eta _1-y_1\eta _1-y_2\eta _2]}\\&\qquad \times \,\overline{\tau (z_2,-\eta _2)}F(z_1,z_2)dz_1dz_2d\eta _1d\eta _2dy_1dy_2d\xi _1d\xi _2\\&=\int _{\mathbb {R}^{8d}}e^{-2\pi i[\Phi (y_1,\xi _1)-\Phi (z_2,-\eta _2)+z_1\eta _1+y_2\xi _2-x_1\xi _1-x_2\xi _2-y_1\eta _1-y_2\eta _2]}\\&\qquad \times \,\tau (y_1,\xi _1)\overline{\tau (z_2,-\eta _2)}F(z_1,z_2)dzd\eta dy dx\xi . \end{aligned}$$

Using the well-known formulae

$$\begin{aligned} \int _{{\mathbb {R}^{2d}}}e^{-2\pi i\eta _1(z_1-y_1)}e^{-2\pi i\xi _2(y_2-x_2)}d\eta _1d\xi _2dz_1dy_2=\delta _{y_1}(z_1)dz_1\delta _{x_2}(y_2)dy_2, \end{aligned}$$

we obtain

$$\begin{aligned} T_2T_2'F(x_1,x_2)&=\int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi (y_1,\xi _1)-\Phi (z_2,-\eta _2)-x_1\xi _1-x_2\eta _2]}\tau (y_1,\xi _1)\overline{\tau (z_2,-\eta _2)}\\&\quad \times \,F(y_1,z_2)dy_1dz_2d\xi _1d\eta _2, \end{aligned}$$

which is (54). It is straightforward to check that the phase ${\mathbf \Phi }$ in (55) is tame, that is, it satisfies the properties of Definition 2.3.

Since $\tau \in S^{0}_{0,0}(\mathbb {R}^d)$, then $\mathcal {T}\in S^{0}_{0,0}(\mathbb {R}^{4d})$. $\square$

Theorem 5.4

Consider the type II FIO $T_{II}$ in (50), with symbol $\tau \in S^0_{0,0}({\mathbb {R}^{2d}})$ and tame phase $\Phi$. Then

$$\begin{aligned} W(T_{II}f,T_{II}g)(x,\xi )=\int _{{\mathbb {R}^{2d}}}k_{II}(x,\xi ,s,z)W(f,g)(s,z)dsdz,\quad f,g\in \mathcal {S}(\mathbb {R}^d), \end{aligned}$$

where

$$\begin{aligned} k_{II}(x,\xi ,y,\eta )&=\int _{{\mathbb {R}^{2d}}}e^{-2\pi i[\Phi (y+\frac{r}{2},\xi +\frac{t}{2})-\Phi (y-\frac{r}{2},\xi -\frac{t}{2})]} e^{2\pi i(tx+r\eta )}\\&\qquad \times \, \tau (y+\frac{r}{2},\xi +\frac{t}{2})\overline{\tau (y-\frac{r}{2},\xi -\frac{t}{2})}\nonumber dt dr. \end{aligned}$$

(57)

Proof

Consider $f\in \mathcal {S}(\mathbb {R}^d)$ and use Proposition 5.3 and Remark 22 to compute

$$\begin{aligned}&W(T_{II}f,T_{II}g)(x,\xi )\\&={\hat{A}}_{1/2}T_2T_2'{\hat{A}}_{1/2}^{-1}W(f,g)(x,\xi )\\&=\int _{\mathbb {R}^d}(T_2T_2'{\hat{A}}_{1/2}^{-1}W(f,g))(x+\frac{t}{2},x-\frac{t}{2})e^{-2\pi i\xi t}dt\\&=\int _{\mathbb {R}^d}\left( \int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi (y_1,\eta _1)-\Phi (y_2,-\eta _2)-(x+t/2)\eta _1-(x-t/2)\eta _2]}\tau (y_1,\eta _1)\overline{\tau (y_2,-\eta _2)}\right. \\&\qquad \times \,\left. ({\hat{A}}_{1/2}^{-1}W(f,g))(y_1,y_2)dy_1dy_2d\eta _1d\eta _2\right) e^{-2\pi i\xi t}dt\\&=\int _{\mathbb {R}^d}\left( \int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi (y_1,\eta _1)-\Phi (y_2,-\eta _2)-(x+t/2)\eta _1-(x-t/2)\eta _2]}\tau (y_1,\eta _1)\overline{\tau (y_2,-\eta _2)}\right. \\&\qquad \times \left. \left( \int _{\mathbb {R}^d}W(f,g)(y_1/2+y_2/2,z)e^{2\pi i(y_1-y_2)z}dz\right) dy_1dy_2d\eta _1d\eta _2\right) e^{-2\pi i\xi t}dt\\&=\int _{\mathbb {R}^{6d}}e^{-2\pi i[\Phi (y_1,\eta _1)-\Phi (y_2,-\eta _2)-x\eta _1-\frac{t}{2}\eta _1-x\eta _2+\frac{t}{2}\eta _2-y_1z+y_2z+\xi t]}\tau (y_1,\eta _1)\overline{\tau (y_2,-\eta _2)}\\&\qquad \times \,W(f,g)(y_1/2+y_2/2,z)dzdy_1dy_2d\eta _1d\eta _2dt. \end{aligned}$$

The change of variables $y_1/2+y_2/2=s$ gives

$$\begin{aligned}&W(T_{II}f,T_{II}g)(x,\xi )\\&=2^d\int _{\mathbb {R}^{6d}}e^{-2\pi i[\Phi (2s-y_2,\eta _1)-\Phi (y_2,-\eta _2)-x\eta _1-\frac{t}{2}\eta _1-x\eta _2+\frac{t}{2}\eta _2-(2s-y_2)z+y_2z+\xi t]}\\&\qquad \quad \times \tau (2s-y_2,\eta _1)\overline{\tau (y_2,-\eta _2)}W(f,g)(s,z)dzdsdy_2d\eta _1d\eta _2dt. \end{aligned}$$

Next, observing that

$$\begin{aligned} \int _{\mathbb {R}^d} e^{-2\pi i(\frac{\eta _2}{2}-\frac{\eta _1}{2}+\xi )t}dt&=\int _{\mathbb {R}^d} e^{-2\pi i(\eta _2-\eta _1+2\xi )\frac{t}{2}}dt =2^{d}\int _{\mathbb {R}^d} e^{-2\pi i(\eta _2-\eta _1+2\xi )t'}dt',\\&=2^d \int _{\mathbb {R}^d} e^{-2\pi i\eta _2 t'}M_{\eta _1-2\xi }1(\eta _2)dt'=2^d T_{\eta _1-2\xi }\hat{1}(\eta _2)\\&=2^d T_{\eta _1-2\xi }\delta (\eta _2). \end{aligned}$$

we obtain

$$\begin{aligned} W(T_{II}f,T_{II}g)(x,\xi )&=2^{2d}\int _{\mathbb {R}^{4d}}e^{-2\pi i[\Phi (2s-y_2,\eta _1)-\Phi (y_2,2\xi -\eta _1)+2(\xi -\eta _1)x+2(y_2-s)z]}\\&\quad \times \, \tau (2s-y_2,\eta _1)\overline{\tau (y_2,2\xi -\eta _1)}W(f,g)(s,z)dzdsdy_2d\eta _1\\&=\int _{{\mathbb {R}^{2d}}}k_{II}(x,\xi ,s,z)W(f,g)(s,z)dsdz, \end{aligned}$$

where

$$\begin{aligned} k_{II}(x,\xi ,s,z)&=2^{2d}\int _{{\mathbb {R}^{2d}}}e^{-2\pi i[\Phi (2s-y_2,\eta _1)-\Phi (y_2,2\xi -\eta _1)+2(\xi -\eta _1)x+2(y_2-s)z]}\\&\qquad \quad \times \, \tau (2s-y_2,\eta _1)\overline{\tau (y_2,2\xi -\eta _1)} dy_2d\eta _1. \end{aligned}$$

Next, we make the change of variables $s-y_2=r/2$ and $\xi -\eta _1=-t/2$ so that

$$\begin{aligned} k_{II}(x,\xi ,s,z)&=\int _{{\mathbb {R}^{2d}}}e^{-2\pi i[\Phi (s+\frac{r}{2},\xi +\frac{t}{2})-\Phi (s-\frac{r}{2},\xi -\frac{t}{2})]} e^{2\pi i(tx+rz)}\\&\quad \times \, \tau (s+\frac{r}{2},\xi +\frac{t}{2})\overline{\tau (s-\frac{r}{2},\xi -\frac{t}{2})} dt dr, \end{aligned}$$

which is (57). $\square$

Theorem 5.5

Consider $T_{II}$ the FIO of type II in (50). Fix $N\in \mathbb {N}$, $N>d$, and assume that the symbol $\tau \in \Gamma ^m({\mathbb {R}^{2d}})$, with $m<-2(d+N)$. Let $k_{II}$ be the associated Wigner kernel, given by (57). Then,

$$\begin{aligned} |k_{II}(x,\xi ,y,\eta )|\lesssim \frac{\langle (y,\xi ) \rangle ^{2N+m}}{\langle (y,\eta )-\chi (x,\xi ) \rangle ^{2N}}, \qquad x,\xi ,y,\eta \in \mathbb {R}^d. \end{aligned}$$

(58)

Proof

We follow the pattern of the proof of Theorem 4.2. By (57) and using the Taylor expansions in (45) and (47) we obtain

$$\begin{aligned} \begin{aligned} k_{II}(x,\xi ,y,\eta )&=\int _{{\mathbb {R}^{2d}}}e^{2\pi i[r\cdot (\eta -\Phi _y(y,\xi ))+t\cdot (x-\Phi _\xi (y,\xi ))]}\tilde{\tau }(y,\xi ,r,t)drdt, \end{aligned} \end{aligned}$$

(59)

where

$$\begin{aligned} \tilde{\tau }(y,\xi ,r,t)=e^{-2\pi i[\Phi _2-\tilde{\Phi }_2](y,\xi ,r,t)}\times \tau (y+\frac{r}{2},\xi +\frac{t}{2})\overline{\tau (y-\frac{r}{2},\xi -\frac{t}{2})}, \end{aligned}$$

and the reminders are given by:

$$\begin{aligned} \Phi _2(y,\xi ,r,t)=\sum _{|\alpha |=2}\int _0^1(1-\tau )\partial ^\alpha \Phi ((y,\xi )+\tau (r,t)/2)d\tau \frac{(r,t)^\alpha }{2^3\alpha !} \end{aligned}$$

and

$$\begin{aligned} \tilde{\Phi }_2(y,\xi ,r,t)=\sum _{|\alpha |=2}\int _0^1(1-\tau )\partial ^\alpha \Phi ((y,\xi )-\tau (r,t)/2)d\tau \frac{(r,t)^\alpha }{2^3\alpha !}. \end{aligned}$$

Again, for $N\in \mathbb {N}$ and setting $u=(r,t)\in {\mathbb {R}^{2d}}$, we have:

$$\begin{aligned} (1-\Delta _u)^N&e^{2\pi i(\eta -\Phi _y(y,\xi ),x-\Phi _\xi (y,\xi ))\cdot (r,t)}\\&=\langle 2\pi (\eta -\Phi _y(y,\xi ),x-\Phi _\xi (y,\xi )) \rangle ^{2N}e^{2\pi i(\eta -\Phi _y(y,\xi ),x-\Phi _\xi (y,\xi ))\cdot (r,t)}. \end{aligned}$$

Integrating by parts in (59), we get:

$$\begin{aligned} k_{II}(x,\xi ,y,\eta )&=\frac{1}{\langle 2\pi (\eta -\Phi _y(y,\xi ),x-\Phi _\xi (y,\xi )) \rangle ^{2N}}\int _{{\mathbb {R}^{2d}}}e^{2\pi i(\eta -\Phi _y(y,\xi ),x-\Phi _\xi (y,\xi ))\cdot (r,t)}\\&\qquad \times (1-\Delta _u)^N\tilde{\tau }(y,\xi ,r,t)drdt. \end{aligned}$$

The same estimates of Theorem 4.2 yield to:

$$\begin{aligned} |k_{II}(x,\xi ,y,\eta )|&\le \frac{1}{\langle 2\pi (\eta -\Phi _y(y,\xi ),x-\Phi _\xi (y,\xi )) \rangle ^{2N}}\int _{{\mathbb {R}^{2d}}}|(1-\Delta _u)^N\tilde{\tau }(y,\xi ,r,t)|drdt\\&\asymp \frac{\langle (y,\xi ) \rangle ^{2N+m}}{\langle (y,\eta )-\chi (x,\xi )) \rangle ^{2N}}. \end{aligned}$$

$\square$

From [12] we deduce

Corollary 5.6

Under the assumptions of Theorem 5.5, the estimate (7) holds true, hence $T_{II}\in FIO(\chi ,N)$.

Proof

It follows from (58), since $2N+m<0$ so that $\langle (y,\xi )\rangle ^{2N+m}\le 1$, for every $y,\xi \in \mathbb {R}^d$. $\square$