Wigner Analysis of Fourier Integral Operators with symbols in the Shubin classes

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 Schr\"odinger 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 H\"ormander'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 $m$.


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 [35] in the framework of Quantum Mechanics and later applied to signal processing and timefrequency analysis by Ville, Cohen and many other authors, see, e.g., [3,4,34] and the textbooks [19,25,26].
If f = g we write W f := W (f, f ), the so-called Wigner distribution of f .Wigner used the above representation to analyse the action of the Schrödinger propagators.We may extend the Wigner approach in [35] as follows: given a linear operator T : S(R d ) → S ′ (R d ), we consider an operator K on S(R 2d ) such that (2) W (T f, T g) = KW (f, g), f, g ∈ S(R d ).
Its integral kernel k is called the Wigner kernel of T : (3) k(z, w)W (f, g)(w) dw, z ∈ R 2d , f, g ∈ S(R d ).
As an elementary example of the effectiveness of the Wigner distribution, consider the Schrödinger propagator T τ , for a fixed time τ ∈ R, of the free particle equation We have where, if we write w = (y, η), then χ(y, η) = (y+τ η, η).So we recapture exactly the inertial first Newton's law, from a probabilistic point of view.This striking result is due to the peculiar action of W on the phase Φ(x, ξ) = xξ −τ ξ 2 .It generalizes to quadratic Φ(x, ξ), corresponding to quadratic Hamiltonians and linear symplectic map χ in (4), see for example [24].
Our aim is to extend this analysis to more general operators, namely Fourier integral operators of the form (5) T with phase Φ and symbol σ in suitable classes.A preliminary step was presented in [13], with T a pseudodifferential operator σ(x, D), i.e., Φ(x, ξ) = xξ in (5).The case of a quadratic Φ and a general σ was considered in [9] and in [8], where a generalization of (4) was obtained by combining a linear symplectic map χ with the kernel of a pseudodifferential operator.
In the present paper we focus on the case of nonlinear symplectic mappings χ corresponding to non-quadratic Φ, which we call tame, see Section 2 below for their definition.
As a counterpart of (4) we look for estimates of the type 2N , in the spirit of the estimates for Gabor kernels, which have been widely investigated in the literature, classical references are [2,11,16,17,27,28], see also [19,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 = χ(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 χ.In fact, it is well known that the Wigner transform may produce the so-called ghost frequencies.As observed in [9,14], they are exactly preserved for Schrödinger propagators for linear χ, i.e., quadratic Φ, but this is not the case for nonlinear χ.Namely, highly oscillating terms may appear in the expression of the kernel k(z, w) outside the graph of z = χ(w).
As a first attempt for eliminating ghost frequencies and re-establishing the validity of (6), we shall consider in the sequel symbols σ of low order in Shubin classes [33].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 Section 3, will be the following abstract definition, in the lines of [16].
Examples of operators which fall in the above class are pseudodifferential operators σ(x, D) (the Kohn-Nirenberg form), defined by Here χ = 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 Sections 4 and 5.
Let us state here the preliminary results of Section 3, which are the core of this study and may be collected as follows.
In Section 4 we shall show the Fourier integral operators of type I in (5), having symbols in suitable Shubin classes Γ m (R 2d ) and tame phase functions are in the class FIO(χ, N).
The last Section 5 is devoted to the L 2 -adjoint of the FIO I in (5), which can be written explicitly in the form 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 F IO(χ, N) as well.We underline that these results are valid for the whole class of tame phase Φ defined in Subsection 2.3, of particular interest is the case Φ non quadratic which gives rise to nonlinear symplectic transformations χ, 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 [21,29,36,37,38], see also the recent contribution [20].

Preliminaries
Notation.We define t 2 = t•t, t ∈ R d , and, similarly, xy = x•y.The space S(R d ) is the Schwartz class and S ′ (R d ) its dual (the space of temperate distributions).The brackets f, g means the extension to S ′ (R d ) × S(R d ) of the inner product f, g = f (t)g(t)dt on L 2 (R d ) (conjugate-linear in the second component).A point in the phase space is denoted by z = (x, ξ) ∈ R 2d .We call (timefrequency shift) the operators (10) π(z)f (t) = e 2πiξt f (t − x), t ∈ R d .
GL(d, R) denotes the group of real invertible d × d matrices.
2.1.The symplectic group Sp(d, R), metaplectic operators and Wigner distributions.The standard symplectic matrix is The symplectic group is defined by where A T is the transpose of A. We have det(A) = 1.
For L ∈ GL(d, R) and C ∈ Sym(2d, R), define: The matrices J, V C , and D L generate the group Sp(d, R).
The Schrödinger representation ρ of the Heisenberg group is given by ρ(x, ξ; τ ) = e 2πiτ e −πiξx π(x, ξ), another representation of the Heisenberg group that is equivalent to ρ, that is, there exists a unitary operator Â : This operator is not unique: if Â′ is another unitary transformation satisfying ( 14 Here, if Â ∈ Mp(d, R), the matrix A will be the unique symplectic matrix satisfying π M p ( Â) = A. Some examples of metaplectic operators we will use in the following are detailed below.
Example 2.1.Consider the matrices J, D L and V C defined in (11) and (13).Then, if we denoted by F the Fourier transform, (i) The relation between time-frequency shifts and metaplectic operators is the following: with a phase factor c A ∈ C, |c A | = 1 (see, e.g., [23,24]).

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 Â ∈ Mp(2d, R), the metaplectic Wigner distribution associated to Â is defined as The most important time-frequency representations are metaplectic Wigner distributions.The τ -Wigner distributions, τ ∈ R, defined by ), are metaplectic Wigner distributions.The case τ = 1/2 is the cross-Wigner distribution, defined in (1).τ -Wigner distributions are metaplectic Wigner distributions: In particular, we recapture the Wigner case when τ = 1/2: Â1/2 can be split into the product , where 2.2.Shubin and Hörmander classes [33], [31], [19].In our study we shall consider the following weight functions The Hörmander class S 0 0,0 (R 2d ), consists of smooth functions σ on R 2d such that (24) Tame phase functions and related canonical transformations.Definition 2.3.We follow the notation of [16,8].A real phase function Φ(x, η) is named tame if it satisfies the following properties: with respect to (x, ξ), one obtains a map χ with the following properties: A4. χ : R 2d → R 2d is a symplectomorphism (smooth, invertible, and preserves the symplectic form in R 2d , i.e., dx ∧ dξ = dy ∧ dη.) A5.For z = (y, η), Conversely, as it was observed in [16], to every transformation χ satisfying the three hypothesis above corresponds a tame phase Φ, uniquely determined up to a constant.

2.4.
Properties of the Wigner Kernel.The Wigner kernel of a linear bounded operator T : S(R d ) → S ′ (R d ) was introduced and studied in [8].We recall its definition and the properties useful for our framework.
Observe that if k ∈ S(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 ∈ S ′ (R 2d ) be its kernel.There exists a unique distribution k ∈ S ′ (R 4d ) such that (31) holds.Hence, every bounded linear operator T : S(R d ) → S ′ (R d ) has a unique Wigner kernel.Furthermore,

Properties of FIO(χ, N)
This section is devoted to prove Theorem 1.3 in the introduction.This requires several steps, developed in what follows.
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 ′ = χ −1 (z) which, for any F ∈ L 2 (R 2d ), Proof.Using the Wigner representation in (3) we can write where k I,1 (z, w) is the Wigner kernel of the operator T 1 , satisfying (7) with symplectic transformation χ 1 .Similarly, with k I,2 (w, u) being the Wigner kernel of T 2 satisfying (7) with symplectic transformation χ 2 .Substituting the expression of W (T 2 f, T 2 g)(w) in (33) we obtain 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 so that the Wigner kernel k I,12 of the product T 1 T 2 is given by Using the Wigner kernel's estimates in (7), we reckon where we used the change of variables χ 29), we obtain where in the last row we used the weight convolution property • s * • s • s for s < −2d (observe N > d).Thus, we obtain the desired estimate Proof.Theorem 3.1 gives that T ∈ B(L 2 (R d )).Let k be integral kernel of T , then Theorem 2.5 says that the adjoint T * ∈ B(L 2 (R d )) has kernel k given by k(z, w) = k(w, z).This means it satisfies (cf.( 7)) 2N and we obtain Hence T * ∈ F IO(χ −1 , N), as desired.

FIOs of type I
Here we focus on the analysis of Wigner kernels for FIOs of type I: Recall that he Schwartz' Kernel Theorem guarantees that every continuous linear operator T : S(R d ) → S ′ (R d ) can be expressed in the form T = T I , with a given phase Φ(x, ξ) and symbol σ(x, ξ) in S ′ (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.[5,6,7,15,18,22,32].
If we assume that T is a continuous linear operator S(R d ) → S ′ (R d ) and χ satisfies conditions A4, A5, and A6 in Definition 2.3 then T = T I,Φχ,σ (FIO of type I), with symbol σ and phase Φ χ .
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 (R 2d ) were considered.Since Γ m (R 2d ) ⊂ S 0 0,0 (R 2d ) whenever m ≤ 0, we can rephrased it in our context as follows.
Theorem 4.1.Let T I be a FIO of type I defined in (38) with Wigner kernel k I given by e 2πi[Φ I (x,η,t,r)−(ξt+ry)] σ I (x, η, t, r)dtdr, and, for x, η, t, r ∈ R d , 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 ∈ N, N > d, and assume that the symbol σ ∈ Γ m (R 2d ), with m < −2(d + N).Let k I be the associated Wigner kernel in (40).Then, Proof.Since Φ is smooth, we can expand Φ(x + t 2 , η + r 2 ) and Φ(x − t 2 , η − r 2 ) into a Taylor series around (x, ξ).Namely, where the remainder Φ 2 is given by Inserting the phase expansions above in (40) we obtain where σ is defined as For N ∈ N, u = (t, r) ∈ R 2d , using the identity: we integrate by parts in (48) and obtain The factor Using the change of variables where, for v s = • s , we used the weight convolution property v s * v s v s , for s < 2d, cf.[26,Lemma 11.1.1].Hence, This gives the claim.
Observing that ) and satisfies ( 25) and ( 26), since This means that Φ 2 is a tame phase function.
Finally, since both τ and the function constantly equal to 1 are in the Hörmander class S 0 0,0 (R 2d ), it immediately follows that τ 2 belongs to S 0 0,0 (R 4d ).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 ∈ , where the operator T ′ 2 is the FIO of type II: . 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 2 T ′ 2 can be written as the following FIO of type II: e −2πiΦ(y,ξ)−xξ T (y, ξ)F (y) dydξ, x ∈ R 2d , for every F ∈ S(R 2d ), with tame phase on R 4d : and symbol Proof.Let F ∈ S(R 2d ).We compute Using the well-known formulae we obtain which is (54).It is straightforward to check that the phase Φ in (55) is tame, that is, it satisfies the properties of Definition 2.3.Since τ ∈ S 0 0,0 (R d ), then T ∈ S 0 0,0 (R 4d ).
), then Â′ = c Â, for some c ∈ C, with |c| = 1.The set { Â : A ∈ Sp(d, R)} is a group under operator composition and has the metaplectic group Mp(d, R) as subgroup.It is a realization of the two-fold cover of Sp(d, R).The projection(15)

Definition 2 . 4 .
The Wigner kernel of T : S(R d ) → S ′ (R d ) linear and bounded operator is the distribution k