Time-Frequency Analysis of Fourier Integral Operators

We use time-frequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames, the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\infty,1}$, for some unimodular Fourier multipliers and metaplectic operators.


Introduction
Fourier Integral Operators (FIOs) are a mathematical tool to study variety of problems arising in partial differential equations. Originally introduced by Lax [33] for the construction of parametrices in the Cauchy problem for hyperbolic equations, they have been widely employed to represent solutions to Cauchy problems, in the framework of both pure and applied mathematics (see, e.g.,the papers [5,6,12,13,27,30], the books [31,35,36] and references therein). In particular, they were employed by Helffer and Robert [28,29] to study the spectral property of a class of globally elliptic operators, generalizing the harmonic oscillator of the Quantum Mechanics. The Fourier Integral operators we work with, possess a phase function similar to those of [28,29]. A simple example is the resolvent of the Cauchy problem for the Schrödinger equation with a quadratic Hamiltonian.
For a given function f on R d the Fourier Integral Operator (FIO) T with symbol σ and phase Φ on R 2d can be formally defined by The phase function Φ(x, η) is smooth on R 2d , fulfills the estimates (1) |∂ α z Φ(z)| ≤ C α , |α| ≥ 2, z ∈ R 2d , and the nondegeneracy condition The symbol σ on R 2d satisfies |∂ α z σ(z)| ≤ C α , |α| ≤ N, a.e. z ∈ R 2d , for a fixed N > 0 (in the sequel we shall work also with rougher symbols).
The first goal of this paper is to rephrase the operator T in terms of timefrequency analysis (see Gröchenig [23] and the next Section 2 for a review of the time-frequency methods.) Denoting T x f (t) = f (t − x), M η f (t) = e 2πiηt f (t), for α, β > 0, g ∈ L 2 (R d ), the set of time-frequency shifts G(g, α, β) = {g m,n := M n T m g} with (m, n) ∈ αZ d ×βZ d , is a Gabor frame if there exist positive constants A, B > 0, such that In Section 3 we show that the matrix representation of a FIO T with respect to a Gabor frame with g ∈ S(R d ) is well-organized (similarly to frames of curvelets and shearlets [6,27]), provided that the symbol σ satisfies the decay estimate for every N > 0 (see Theorem  where χ is the canonical transformation generated by Φ. In the special case of pseudodifferential operators such an almost diagonalization was already obtained in [24,34]. Indeed, notice that pseudodifferential operators correspond to the phase Φ(x, η) = xη and canonical transformation χ(y, η) = (y, η).
As a rilevant byproduct of the results of Section 3, we study the boundedness properties of the operator T on the so-called modulation spaces (Section 4 and 5). To define them, we fix a non-zero Schwartz function g and consider the short-time Fourier Transform V g f of a function f on R d with respect to g which provides a time-frequency representation of f . The (unweighted) modulation space M p,q is the closure of the Schwartz class with respect to the norm (with appropriate modifications when p = ∞ or q = ∞). In particular, when p = q we simply write M p,p = M p , see Subsection 2.2 for exhaustive definitions and properties. These spaces were introduced by Feichtinger [17] and have become canonical for both time-frequency and phase-space analysis [18], most recent employment being the study of PDEs [2,3,38,39,40].
If g ∈ M 1 , and the Gabor frame {T m M n g; (m, n) ∈ αZ d × βZ d } is a tight frame, namely (3) holds with A = B, then it extends to a Banach frame for the modulation spaces M p,q (R 2d ), with the norm equivalence Then, boundedness of the FIO T on M p,q is equivalent to that of the infinite matrix T g m,n , g m ′ ,n ′ on the spaces of sequences l p,q .
Whence, the estimates (4) readily give (see Theorem 4.1 for a more general version): Theorem 2. For N > d, 1 ≤ p < ∞, the Fourier integral operator T , with symbol σ and phase Φ as above, extends to a continuous operator on M p . (In the case p = ∞, the space M ∞ is replaced by the closure of the Schwartz function with respect to · M ∞ ).
The continuity property of a FIO T on M p,q , with p = q, fails in general. Indeed, an example is provided by the operator T f (x) = e πi|x| 2 f (x), corresponding to Φ(x, η) = xη + |x| 2 /2, σ ≡ 1, which is bounded on M p,q if and only if p = q (see Proposition 7.1).
Hence, we introduce a new condition on the phase Φ, namely that the map x −→ ∇ x Φ(x, η) has a range of finite diameter, uniformly with respect to η, that allows us to get the boundedness on M p,q (Theorem 5.2): Theorem 3. For N > d, 1 ≤ p, q < ∞, the Fourier integral operator T , with symbol σ and phase Φ as above, and such that extends to a continuous operator on M p,q . (In the case p = ∞ or q = ∞, the space M p,q is replaced by the closure of the Schwartz function with respect to · M p,q ).
As a particular case, we recapture recent boundedness results of unimodular Fourier multipliers [2] (see Example 5.3). With respect to Theorem 2, here the proof is more delicate and combines the estimate (4) with a generalized version of Schur's Test (Proposition 5.1).
To have a simple idea of the possible applications of Theorems 1, 2 and 3, consider the Cauchy problem where H is the Weyl quantization of a quadratic form on R d ×R d (see, e.g., [10,21]). Simple examples are H = − 1 4π ∆ + π|x| 2 , or H = − 1 4π ∆ − π|x| 2 (see [4]). The solution to (5) is a one-parameter family of FIOs: with symbol σ ≡ 1 and a phase given by a quadratic form Φ(x, η), satisfying trivially the preceding assumptions (1) and (2) (see [21] for details). We address to Section 7 for a debited study of such operators.
Finally, Section 6 presents a variant of Theorem 1, cf. (39), and a generalization of Theorem 2, cf. Theorem 6.1, to the case of FIOs T with symbols in the modulation space M ∞,1 . This generalizes the known boundedness results on M p of pseudodifferential operators with symbols in M ∞,1 [25], and intersects a previous result of Boulkhemair [5] on L 2 boundedness of FIOs. We address also to recent contribution [8], where the continuity and Schatten-von Neumann properties of similar operators when acting on L 2 are proved.
Notation. We define |t| 2 = t · t, for t ∈ R d , and xy = x · y is the scalar product on R d .
The Schwartz class is denoted by S(R d ), the space of tempered distributions by S ′ (R d ). We use the brackets f, g to denote 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 ). The Fourier transform is normalized to bef(η) = F f (η) = f (t)e −2πitη dt, the involution g * is g * (t) = g(−t) and the inverse Fourier transform isf (η) = F −1 f (η) =f (−η).
The spaces l p,q µ = l q l p µ , with weight µ, are the Banach spaces of sequences {a m,n } m,n on some lattice, such that We denote by c 0 the space of sequences vanishing at infinity. Throughout the paper, we shall use the notation A B to indicate A ≤ cB for a suitable constant The STFT V g f is defined on many pairs of Banach spaces. For instance, it maps We now recall the following inequality [23,Lemma 11.3.3], which is useful when one needs to change windows. Lemma 2.1. Let g 0 , g 1 , γ ∈ S(R d ) such that γ, g 1 = 0 and let f ∈ S ′ (R d ). Then, for all (x, η) ∈ R 2d .

Modulation
Spaces. The modulation space norms are a measure of the joint time-frequency distribution of f ∈ S ′ . For their basic properties we refer, for instance, to [23, and the original literature quoted there. For the quantitative description of decay properties, we use weight functions on the time-frequency plane. In the sequel v will always be a continuous, positive, even, submultiplicative weight function (in short, a submultiplicative weight), hence For our investigation of FIOs we will mostly use the polynomial weights defined Given a non-zero window g ∈ S(R d ), µ ∈ M v , and 1 ≤ p, q ≤ ∞, the modulation space

Wiener amalgam spaces.
For a detailed treatment we refer to [15,14,16,20,22]. Let g ∈ D(R 2d ) be a test function that satisfies (k,l)∈Z 2d T (k,l) g ≡ 1. Let X(R 2d ) be a Banach space of functions invariant under translations and with the property that D ·X ⊂ X, e.g., L p , F L p , or L p,q . Then the Wiener amalgam space W (X, L p,q µ ) with local component X and global component L p,q µ is defined as the space of all functions or distributions for which the norm It can be shown that different choices of g ∈ D generate the same space and yield equivalent norms. In the sequel we shall use the inclusions relations between Wiener amalgam spaces: if B 1 ֒→ B 2 and C 1 ֒→ C 2 , We now recall the following regularity property of the STFT [9, Lemma 4.1]: We also give a slight generalization of [23, Proposition 11.1.4] and its subsequent Remark.
is any function everywhere defined on R 2d and lower semi-continuous, then the restriction . Proof. One uses the arguments of [23, Proposition 11.1.4] and its subsequent Remark. We just shall highlight the key points that make those arguments to work under our assumptions.
First of all, if (r, s) ∈ Z 2d and X is separated, then the number of sampling points of X in (r, s) + [0, 1] 2d is bounded independently of (r, s).

2.4.
Gabor frames. Fix a function g ∈ L 2 (R d ) and a lattice Λ = αZ d × βZ d , for α, β > 0. For (m, n) ∈ Λ, define g m,n := M n T m g. The set of time-frequency shifts G(g, α, β) = {g m,n , (m, n) ∈ Λ} is called Gabor system. Associated to G(g, α, β) we define the coefficient operator C g , which maps functions to sequences as follows: and the Gabor frame operator The set G(g, α, β) is called a Gabor frame for the Hilbert space L 2 (R d ) if S g is a bounded and invertible operator on L 2 (R d ). Equivalently, C g is bounded from Gabor frame for L 2 (R d ), then the so-called dual window γ = S −1 g g is well-defined and the set G(γ, α, β) is a frame (the so-called canonical dual frame of G(g, α, β)). Every f ∈ L 2 (R d ) posseses the frame expansion with unconditional convergence in L 2 (R d ), and norm equivalence: This result is contained in [23, Proposition 5.2.1]. In particular, if γ = g and g L 2 = 1 the frame is called normalized tight Gabor frame and the expansion (9) reduces to If we ask for more regularity on the window g, then the previous result can be extended to suitable Banach spaces, as shown below [19,26].
We also establish the following properties. Denote byM p,q µ the closure of the Schwartz class in M p,q µ . Hence,M p,q µ = M p,q µ if p < ∞ and q < ∞. Also, denote bỹ l p,q µ the closure of the space of eventually zero sequences in l p,q µ . Hencel p,q µ = l p,q µ if p < ∞ and q < ∞. Theorem 2.3. Under the assumptions of Theorem 2.2, for every 1 ≤ p, q ≤ ∞ the operator C g is continuous fromM p,q µ intol p,q µ , whereas the operator D g is continuous froml p,q µ intoM p,q µ . Proof. Since C g is continuous from M p,q µ into l p,q µ it suffices to verify that, if f is a Schwartz function then C g (f ) ∈l p,q µ . This follows from the fact that C g (f ) ∈ l 1 µ . Similarly, for D g it suffices to verify that, if c is any eventually zero sequence, then D g (c) ∈M p,q µ . This is true because D g (c) ∈ M 1 µ .

Almost diagonalization of FIOs
For a given function f on R d the FIO T with symbol σ and phase Φ can be formally defined by (12) T To avoid technicalities we take f ∈ S(R d ) or, more generally, f ∈ M 1 . If σ ∈ L ∞ and the phase Φ is real, the integral converges absolutely and defines a function in L ∞ . Assume that the phase function Φ(x, η) fulfills the following properties: and solve with respect to (x, ξ), we obtain a mapping χ, defined by (x, ξ) = χ(y, η), which is a smooth bilipschitz canonical transformation. This means that χ is a smooth diffeomorphism on R 2d ; -both χ and χ −1 are Lipschitz continuous; -χ preserves the symplectic form, i.e., dx ∧ dξ = dy ∧ dη.
Indeed, under the above assumptions, the global inversion function theorem (see e.g. [32]) allows us to solve the first equation in (15) with respect to x, and substituting in the second equation yields the smooth map χ. The bounds on the derivatives of χ, which give the Lipschitz continuity, follow from the expression for the derivatives of an inverse function combined with the bounds in (ii) and (iii). The symplectic nature of the map χ is classical, see e.g. [7]. Similarly, solving the second equation in (15) with respect to η one obtains the function χ −1 with the desired properties. In this section we prove an almost diagonalization result for FIOs as above, with respect to a Gabor frame. Here we consider the case of regular symbols. In Section 6 we will study the case of symbols in modulation spaces.
Precisely, for a given N ∈ N, we consider symbols σ on R 2d satisfying, for z = (x, η), (16) |∂ α z σ(z)| ≤ C α , a.e. ζ ∈ R 2d , |α| ≤ 2N, here ∂ α z denotes distributional derivatives. Our goal is to study the decay properties of the matrix of the FIO T with respect to a Gabor frame. For simplicity, we consider a normalized tight frame G(g, α, β), with g ∈ S(R d ).
Theorem 3.1. Consider a phase function satisfying (i) and (ii) and a symbol satisfying (16). There exists C N > 0 such that Proof. Recall that the time-frequency shifts interchange under the action of the Fourier transform : Using this properties, we can write Since Φ is smooth, we expand Φ(x, η) into a Taylor series around (m ′ , n) and obtain where the remainder is given by Whence, we can write For N ∈ N, using the identity: we integrate by parts and obtain By means of Leibniz's formula the factor can be expressed as where p(∂ |α| Φ 2,(m ′ ,n) )(z) is a polynomial made of derivatives of Φ 2,(m ′ ,n) of order at most |α|.
We now assume the additional hypothesis (iii) on the phase, and rewrite (17) in a form convenient for the applications to the continuity of FIOs in the next section. We need the following lemma.
Proof. It suffices to prove the following inequalities: We observe that, by (15), we have Hence, we have m = ∇ η (x(m, n), n), so that wehere the last inequality follows from the fact that, for every fixed η, the map x −→ ∇Φ η (x, η) has a Lipschitz inverse, with Lipschitz constant uniform with respect to η. This proves (22). In order to prove (23) we observe that, in view of (25), it turns out where the last inequality follows from the Taylor formula for the function y −→ ξ(y, n), taking into account that the function ξ has bounded derivatives. This proves (23).
Combining the previous lemma with (17) we obtain the following result.
Theorem 3.3. Consider a phase function Φ satisfying (i), (ii), and (iii), and a symbol satisfying (16). Let g ∈ S(R d ). There exists a constant C N > 0 such that where χ is the canonical transformation generated by Φ.
This result shows that the matrix representation of a FIO with respect a Gabor frame is well-organized, similarly to the results recently obtained by [6,27] in terms of shearlets and curvelets frames. More precisely, if σ ∈ S 0 0,0 , namely if (16) is satisfied for every N ∈ N, then the Gabor matrix of T is highly concentrated along the graph of χ.

Continuity of FIOs on M p µ
In this section we study the continuity of FIOs on the modulation spaces M p µ associated with a weight function µ ∈ M vs , s ≥ 0. We need the following preliminary lemma.

Lemma 4.1. Consider a lattice Λ and an operator K defined on sequences as
Then K is continuous on l p (Λ) for every 1 ≤ p ≤ ∞ and moreover maps the space c 0 (Λ) of sequences vanishing at infinity into itself.
Proof. The first part is the classical Schur's test (see e.g. [23, Lemma 6.2.1]). The second part follows in this way. Since we know that K is continuous on l ∞ and the space of eventually zero sequences is dense in c 0 , it suffices to verify that K maps every eventually zero sequence in c 0 . This follows from the fact that any eventually zero sequence belongs to l 1 and therefore, since K is continuous on l 1 , is mapped in l 1 ֒→ c 0 .
We can now state our result.
Proof. We first prove that We see at once that, since σ ∈ L ∞ , T defines a bounded operator from M 1 into L ∞ ֒→ M ∞ . Hence, for all f ∈ S(R d ), we have T f ∈ M ∞ and Theorem 2.2 shows that f M p On the other hand, the expansion (10) holds for f with convergence in M 1 . Therefore T f = m,n f, g m,n T g m,n with convergence in M ∞ . Hence, Therefore we are reduced to proving that the matrix operator (27) {c m,n } −→ m,n∈Z d T g m,n , g m ′ ,n ′ c m,n is bounded from l p µ•χ into l p µ . This follows from Schur's test (Lemma 4.1) if we prove that, upon setting K m ′ ,n ′ ,m,n = T g m,n , g m ′ ,n ′ µ(m ′ , n ′ )/µ(χ(m, n)), we have (28) K m ′ ,n ′ ,m,n ∈ l ∞ m,n l 1 m ′ ,n ′ , and (29) K m ′ ,n ′ ,m,n ∈ l ∞ m ′ ,n ′ l 1 m,n . In view of (26) we have , n)) . (30) is bounded because µ is v s -moderate, so we deduce (28). Finally, since χ is a bilipschitz function we have (29) follows as well. The case p = ∞ follows analogously by using Theorem 2.3 (with p = q = ∞), and the last part of the statement of Lemma 4.1.

Now, the last quotient in
Remark 4.2. Theorem 4.1 with v ≡ 1 gives, in particular, continuity on the unweighted modulation spaces M p . If moreover p = 2, we recapture the classical L 2 -continuity result by Asada and Fujiwara [1]. Also, Theorem 4.1 applies to µ = v t , with |t| ≤ s. In that case we obtain continuity on M vt , because v t • χ ≍ v t .

Continuity of FIOs on M p,q
In this section we study the continuity of FIOs on modulation spaces M p,q possibly with p = q. As shown in Section 7, under the assumptions of Theorem 4.1 such operators may fail to be bounded when p = q. The counterexample is given by the phase Φ(x, η) = xη + |x| 2 /2, and symbol σ = 1, which does not yield a bounded operator on M p,q , except for the case p = q. Here the obstruction is essentially due to the fact that the map x −→ ∇ x Φ(x, η) has unbounded range. Indeed we will show, for general phases, that if such a map has range of finite diameter, uniformly with respect to η, then the corresponding operator is bounded on all M p,q . To this end we need the following result.
the operator K is continuous on l p,q = l q n l p m for every 1 ≤ p, q ≤ ∞. (iv) Assume the hypotheses in (iii). Then K is continuous on alll p,q , 1 ≤ p, q ≤ ∞.
Recall thatl p,q is the closure of the space of eventually zero sequences in l p,q .

Proof. (i) We have
Since the statement holds for p = q by the classical Schur's test, and for (p, q) = (1, ∞) and (p, q) = (∞, 1) by the items (i) and (ii), it follows by complex interpolation (see (3) on page 128 and (15) on page 134 of [37]) that it holds for all (p, q), except possibly in the cases q = ∞, 1 < p < ∞. For these cases we argue by duality as follows.
In order to prove the continuity of K on l ∞ l p , it suffices to verify that for any sequences c = (c m,n ) ∈ l ∞ n l p m and d = ( whereK is the operator with matrix kernelK m,n,m ′ ,n ′ = |K m ′ ,n ′ ,m,n |. Since it satisfies the same assumptions as K, it is continuous on l 1 l p ′ , which gives (32). (iv) Since K is continuous on l p,q and by the definition ofl p,q , it suffices to verify that K maps every eventually zero sequence inl p,q . This follows from the fact that K maps every eventually zero sequence in l 1 ֒→l p,q , because K is bounded on l 1 .
Theorem 5.2. Consider a phase function Φ satisfying (i), (ii), and (iii), and a symbol satisfying (16), with N > d. Suppose, in addition, that Then the corresponding Fourier integral operator T extends to a bounded operator on M p,q for every 1 ≤ p, q < ∞ and onM p,q if p = ∞ or q = ∞.
Proof. By arguing as in the proof of Theorem 4.1, it suffices to prove the continuity on l p,q = l q n l p m if p < ∞ and q < ∞, orl p,q if p = ∞ or q = ∞, of the operator where T m ′ ,n ′ ,m,n = T g m,n , g m ′ ,n ′ . By applying 5.1, it suffices to verify that (34) {T m ′ ,n ′ ,m,n } ∈ l ∞ n l 1 {T m ′ ,n ′ ,m,n } ∈ l ∞ n ′ l 1 n l ∞ m l 1 m ′ , because we already see from (26) and (31) that {T m ′ ,n ′ ,m,n } ∈ l ∞ m,n l 1 m ′ ,n ′ ∩ l ∞ m ′ ,n ′ l 1 m,n . Let us now prove (34). It follows from (17) and (23) that By (15) we have so that the hypothesis (33) yields ξ(m, n) = ξ(0, n) + O(1).
This concludes the proof.

Modulation spaces as symbol classes
In what follows we shall rephrase the quantity | T g m,n , g m ′ ,n ′ | in terms of the STFT of the symbol σ, without assuming the existence of derivatives of σ. This will be applied to prove the continuity of FIOs with symbols in M ∞,1 on modulation spaces M p .
The same arguments as in Theorem 3.1 yield the equality Expanding the phase Φ into a Taylor series around (m ′ , n) we obtain where the remainder Φ 2,(m ′ ,n) is given by (18). Inserting this expansion in the integrals above, we can write Defining (38) Ψ (m ′ ,n) (x, η) := e 2πiΦ 2,(m ′ ,n) (x,η) (ḡ ⊗ĝ)(x, η), and computing the modulus of the left-hand side of (37), we are led to Observe that the window Ψ (m ′ ,n) of the STFT above depends on the pair (m ′ , n). We now study the continuity problem of T when the symbol σ is in the modulation space M ∞,1 . By arguing as in the proof of Theorem 4.1, it suffices to prove the continuity on l p if 1 ≤ p < ∞ and onl ∞ = c 0 , of the operator (27). In wiew of Schur's test (Lemma 4.1) and (39), it suffices to prove the following result. Proposition 6.2. Consider a phase function Φ satisfying (i) and (ii) and (iii) and a symbol σ ∈ M ∞,1 . If we set (40) z m,n,m ′ ,n ′ := ((m ′ , n), (n ′ −∇ x Φ(m ′ , n), m−∇ η Φ(m ′ , n))), m, m ′ ∈ αZ d , n, n ′ ∈ βZ d , then, We need the following lemma. Lemma 6.1. Let Ψ 0 ∈ S(R 2d ) with Ψ 0 L 2 = 1 and Ψ (m ′ ,n) be defined by (38), with (m ′ , n) ∈ Λ = αZ d × βZ d , and g ∈ S(R d ). Then, Proof of Lemma 6.1. We shall show that Using the switching property of the STFT: , and by the even property of of the weight · , relation (44) is equivalent to Now, the mapping V Ψ 0 is continuous from S(R 2d ) to S(R 4d ) (see [23,Chap. 11]). This means that there exists M ∈ N, K > 0, such that for every (m ′ , n) ∈ Λ. We now claim that Ψ (m ′ ,n) ∈ S(R 2d ) uniformly with respect to (m ′ , n). This is proved as follows: the function e 2πiΦ 2,(m ′ ,n) (x,η) is in C ∞ (R 2d ) and possesses derivatives dominated by powers (x, η) k , k ∈ N, uniformly with respect to (m ′ , n), due to (13); since (ḡ ⊗ĝ) ∈ S(R 2d ), it follows that Ψ (m ′ ,n) ∈ S(R 2d ), with semi-norms uniformly bounded: uniformly with respect to (m, n), w 2 , because the mapping x −→ ∇ η Φ(x, η) has an inverse that is Lipschitz continuous, thanks to (13) and (14). On the other hand, Hence, X is separated uniformly with respect to (m, n), w 2 . Now, we apply Proposition 2.1 (with p = q = 1) to the function which is lower semi-continuous, being V Ψ 0 σ continuous. We obtain If the symbol σ is in M ∞,1 , by Lemma 2.2 the STFT V Ψ 0 σ belongs to the Wiener amalgam space W (F L 1 , L ∞,1 ), and The first inequality is due to F L 1 ֒→ C and the inclusion relations between Wiener amalgam spaces. Combining this inequality with (47) we obtain (46), uniformly with respect to (m, n) and w 2 , that is (41). The estimate (42) is obtained by similar arguments.
Remark 6.3. We observe that the continuity on M 2 = L 2 of FIOs as above, with symbols in M ∞,1 , was already proved in [5] by other methods.

The case of quadratic phases: metaplectic operators
In this section we briefly discuss the particular case of quadratic phases, namely phases of the type It is easy to see that, if we take the symbol σ ≡ 1 and the phase (48), the corresponding FIO T is (up to a constant factor) a metaplectic operator. This can be seen by means of the easily verified factorization where U A and U C are the multiplication operators by e πiAx·x and e πiCη·η respectively, and D B is the dilation operator f → f (B·). Each of the factors is (up to a constant factor) a metaplectic operator (see e.g. the proof of [31, Theorem 18.5.9]), so T is.
The corresponding canonical map, defined by (15), is now an affine symplectic map. For the benefit of the reader, some important special cases are detailed in the table below.
operator phase Φ(x, η) canonical transformation T x 0 (x − x 0 ) · η χ(y, η) = (y + x 0 , η) M η 0 (η + η 0 ) · x χ(y, η) = (y, η + η 0 ) D B Bx · η χ(y, η) = (B −1 y, t Bη) U A x · η + 1 2 Ax · x χ(y, η) = (y, η + Ax) However one should observe that there are metaplectic operators, as the Fourier transform, which cannot be expressed as FIOs of the type (12). Metaplectic operators are known to be bounded on M p vs , see e.g. [23,Proposition 12.1.3]. This also follows from Theorem 4.1. Indeed, since χ is a bilipschitz function, we have v s • χ ≍ v s . Also, Theorem 5.2 applies to quadratic phases whose affine symplectic map χ is (up to translations on the phase space) defined by an upper-triangular matrix, which happens precisely when A = 0. Indeed, we obtain the map χ by solving The phase condition (14) here becomes det B = 0, so that B is an invertible matrix and x = B −1 y − B −1 Cη + B −1 x 0 . Whence, the mapping χ : (y, η) −→ (x, ξ) is given by When A = 0 the phase Φ satisfies (33) and, consequently, the corresponding operators are bounded on all M p,q . This can also be verified by means of the factorization (49) (with A = 0). Indeed the continuity of the operators M η 0 , T x 0 and D B is easily seen, whereas that of the Fourier multiplier F −1 U C F was shown, e.g., in [25, Lemma 2.1].
On the other hand, generally the metaplectic operators are not bounded on M p,q if p = q. An example is given by the Fourier transform itself (see [14]). An example which instead falls in the class of FIOs considered here is the following one.
Proposition 7.1. The multiplication U I d is unbounded on M p,q , for every 1 ≤ p, q ≤ ∞, with p = q.
As λ → 0, we have so that, if we assume Uf M p,q ≤ C f M p,q , then 1/p − 1/q ≥ 0, that is p ≤ q. Moreover, the same argument applies to the adjoint operator U * f (x) = e −πi|x| 2 f (x). Now we show that p = q. By contradiction, if U were bounded on M p,q , with p < q, its adjoint U * would satisfy with q ′ < p ′ , which is a contradiction to what just proved.