Dipartimento Di Matematica Politecnico Di Torino

Dipartimento di Matematica

E. CORDERO, F. NICOLA

BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES

Rapporto interno N. 23, luglio 2008

Politecnico di Torino

Corso Duca degli Abruzzi, 24-10129 Torino-Italia BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES

ELENA CORDERO AND FABIO NICOLA

Abstract. It is known that Fourier integral operators arising when solving Schr¨odinger-type operators are bounded on the modulation spaces p,q, for 1 p = q , provided their symbols belong to the Sj¨ostrand clasMs M ∞,1. Ho≤wever, they≤ ∞generally fail to be bounded on p,q for p = q. In this paper we study several additional conditions, to be imposeMd on the ph6 ase or on the symbol, which guarantee the boundedness on p,q for p = q, and between p,q q,p, 1 q < p . We also study similarMproblems6 for operators actMing on→WienerM amalgam≤ sp≤ac∞es, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.

1. Introduction The paper is concerned with the study of Fourier integral operators (FIOs) de- ﬁned by

(1.1) T f(x) = e2πiΦ(x,η)σ(x, η)fˆ(η)dη, Rd Z for, say, f (Rd). The functions σ and Φ are called symbol and phase, respec- ∈ S 2πixη tively. Here the Fourier transform of f is normalized to be fˆ(η) = f(x)e− dx. If σ L∞ and the phase Φ is real, the integral converges absolutely and deﬁnes a ∈ R function in L∞. The phase function Φ(x, η) fulﬁlls the following properties: 2d (i) Φ ∞(R ); ∈ C (ii) there exist constants Cα > 0 such that (1.2) ∂αΦ(x, η) C , α Z2d, α 2; | | ≤ α ∀ ∈ + | | ≥ (iii) there exists δ > 0 such that ∂2Φ (1.3) det δ (x, η) R2d. ∂x ∂η (x,η) ≥ ∀ ∈ i l

2000 Mathematics Subje ct Classiﬁcati on. 35S30 ,47G30. Key words and phrases. Fourier integral operators, modulation spaces, Wiener amalgam spaces, short-time Fourier transform, Sj¨ostrand’s algebra. 1 2 ELENA CORDERO AND FABIO NICOLA

Note that our phases differ from those (positively homogeneous of degree 1 in η) of FIOs arising in the solution of hyperbolic equations (see, e.g., [10, 20, 23, 24]). Indeed, FIOs are a mathematical tool to study a variety of problems in partial differential equations, and our FIOs arise naturally in the study of the Cauchy problem for Schrödinger-type operators (see, e.g., [6, 8, 9, 15, 18, 19]). Basic examples of phase functions within the class under consideration are quadratic forms in the variables x, η (see Example 5.3 below). Continuing the study pursued in [8], we focus on boundedness results for these operators, when acting on two classes of Banach spaces, widely used in time-frequency analysis, known as modulation spaces and Wiener amalgam spaces, denoted by M p,q and W ( Lp, Lq), respectively, with 1 p, q . To be definite, we recall the definitionF of these spaces, introduced by≤H. Feic≤h∞tinger (see [11, 17] and Section 2 below for details). 2d 2d In short, given a positive weight function m on R , with m 0(R ), we say that a temperate distribution f belongs to M p,q(Rd), 1 p,∈q S , if its m ≤ ≤ ∞ short-time Fourier transform (STFT) Vgf(x, η), defined in (2.3) below, fulfills V f(x, η)m(x, η) Lp,q(R2d) = Lq(Rd, Lp(Rd)), with the norm g ∈ η x (1.4) f p,q := (V f)m q p < . k kMm k g kLηLx ∞ Here g is a non-zero (so-called window) function in (Rd), which in (2.3) is first translated and then multiplied by f to localize f nearS any point x. Changing g (Rd) produces equivalent norms. Taking the two norms above in the converse or∈deSr yields the norm in W ( Lp, Lq)(Rd): F (1.5) f W ( Lp,Lq) := Vgf Lp Lq < . k k F k k x η ∞ p,q Rd p q Rd Rd The spaces m ( ) and ( L , L )( ) are defined as the closure of ( ) in the p,q Rd M p q W F S Mm ( ) and W ( L , L ) norm, respectively. For heuristic purposes, distributions in M p,q, as well asFin W ( Lq, Lp), may be regarded as functions which are locally in Lq and decay at infiFnity like functions in Lp. Among their properties, we highligF ht the important relation W ( Lp, Lq) = (M p,q). The action on the spaces p :=F p,p of FIOsF as above already appeared in [5, 8] (see also [1, 2, 4, 23]).MIt is aMbasic result that FIOs with symbols in the ,1 2d p d Sjöstrand class M ∞ (R ) extend to bounded operators on (R ), 1 p . Applications to issues of classical analysis were also given Min [10]. Mor≤eov≤er,∞it was observed in [8] that boundedness generally fails on the spaces p,q(Rd), with p = q, although it can hold under an additional condition on the phaseM function. The6 present paper is devoted to a more systematic study of the conditions which guarantee the boundedness on p,q(Rd), for p = q. Our first result is in fact a generalizatM ion of [2,6 Theorem 5] and [8, Theorem 5.2] to the case of rougher symbols. BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 3

Theorem 1.1. Consider a phase function Φ satisfying (i), (ii), and (iii), and a ,1 2d symbol σ M ∞ (R ). Suppose, in addition, that ∈

(1.6) sup xΦ(x, η) xΦ(x0, η) < . x,x0,η Rd |∇ − ∇ | ∞ ∈ Then, the corresponding Fourier integral operator T extends to a bounded operator on p,q(Rd), for every 1 p, q . M ≤ ≤ ∞ The condition (1.6) is seen to be essential for the conclusion to hold. In fact it was shown in [8, Proposition 7.1] that the pointwise multiplication operator by π x 2 x 2 | | e− | | (which has phase Φ(x, η) = xη 2 and symbol σ 1) is not bounded on any p,q, with p = q (see also Theorem− 6.1 below). ≡ If Mwe drop the 6 condition (1.6), we need some further decay condition on the symbol, as explained by the next result. R s1 s2 For s1, s2 , we deﬁne the weight function vs1,s2 (x, η) := x η , (x, η) R2d. ∈ h i h i ∈

Theorem 1.2. Consider a phase Φ satisfying (i), (ii) and (iii), and a symbol ,1 2d ∞ R R σ Mvs ,s 1( ), s1, s2 . ∈ 1 2 ⊗ ∈ (i) Let 1 p . If s , s 0, T extends to a bounded operator on p(Rd). ≤ ≤ ∞ 1 2 ≥ M (ii) Let 1 q < p . If s > d 1 1 , s 0, T extends to a bounded ≤ ≤ ∞ 1 q − p 2 ≥ operator on p,q(Rd). M (iii) Let 1 p < q . If s 0, s > d 1 1 , T extends to a bounded ≤ ≤ ∞ 1 ≥ 2 p − q operator on p,q(Rd). In all cases,M

(1.7) T f p,q . σ ∞,1 f p,q . Mv k kM k k s1,s2 ⊗1 k kM Although we do not exhibit a complete set of counterexamples for the thresholds arising in Theorem 1.2, some examples are given in Section 6, and show that the thresholds are in fact the expected ones (see Remark 6.5). Results for boundedness of FIOs between weighted modulation spaces are attained as well (see Section 4). We also turn our attention to the boundedness of FIOs as above from the modulation space p,q(Rd) into q,p(Rd). This study is mostly suggested by the spe- cial case of metaMplectic operatoMrs (corresponding to a quadratic phase and symbol σ 1), which was investigated in detail in [6]; see also Example 5.3 below. ≡ Theorem 1.3. Let 1 q < p . Consider a phase Φ(x, η) satisfying (i), (ii) and (iii). Moreover, assume≤ one≤of∞the following conditions: 4 ELENA CORDERO AND FABIO NICOLA

,1 2d (a) the symbol σ M ∞ (R ) and, for some δ > 0, ∈ ∂2Φ (1.8) det δ (x, η) R2d, ∂x ∂x (x,η) ≥ ∀ ∈ i l

,1 2d 1 1 (b) the symbol σ M ∞ (R ), with s > d . vs,0 1 q p ∈ ⊗ − Then, the corresponding FIO T extends to a bounde d operator p,q(Rd) q,p(Rd). M → M The additional assumptions (a) or (b) are essential to guarantee the boundedness. A counterexample in this connection is given in Proposition 6.7 below. Moreover, even under those conditions, T generally fails to be bounded between p,q q,p if q > p; see Proposition 6.6 below. M → M For the sake of brevity, in this Introduction we only established our results for FIOs acting on modulation spaces. In the subsequent sections we shall provide corresponding results for Wiener amalgam spaces (Corollaries 3.9 and 5.2). As in [8], the proof of our results relies on a formula expressing the Gabor matrix of the FIO T in terms of the STFT of its symbol σ (see (3.3) below). Here the novelty is provided by the combination of this formula with Schur-type tests and the uncertainty principle, in the form of Bernstein’s inequality and some generalizations. The paper is organized as follows. In Section 2 we prove some preliminary results of classical analysis and we also collect the basic deﬁnitions and properties of modulation and Wiener amalgam spaces. In Section 3 we recall from [8] a useful formula for the Gabor matrix of the operator T , and use it to prove Theorems 1.1 and 1.2. In Section 4 we study the action of a FIO T on weighted modulation spaces. In Section 5 we prove Theorem 1.3. Finally, in Section 6 some examples related to the Schr¨odinger operators are exhibited: they reveal to be useful tests for the sharpness of the above results.

Notation. We deﬁne x 2 = x x, for x Rd, where x y = xy is the scalar product Rd | | · ∈ · Rd on . The space of smooth functions with compact support is denoted by 0∞( ), d dC the Schwartz class is (R ), the space of tempered distributions 0(R ). The S 2πitη S Fourier transform is normalized to be fˆ(η) = f(η) = f(t)e− dt. Translation and modulation operators (time and frequencyFshifts) are deﬁned, respectively, by R T f(t) = f(t x) and M f(t) = e2πiηtf(t). x − η 2πixη We have the formulas (Txf)ˆ = M xfˆ, (Mηf)ˆ = Tηfˆ, and MηTx = e TxMη. − The inner product of two functions f, g L2(Rd) is f, g = f(t)g(t) dt, and ∈ h i Rd its extension to 0 will be also denoted by , . The notation A . B means S × S h· ·i R 1 A cB for a suitable constant c > 0, whereas A B means c− A B cA, for≤some c 1. The symbol B , B denotes thecontinuous embedding≤ ≤of the ≥ 1 → 2 BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 5

d space B1 into B2. The open ball in R of center x and radius R will be denoted by B(x, R).

2. Preliminary results 2.1. Bernstein inequalities. The core of our proofs relies on the classical Bern- stein’s inequality (see, e.g., [27]) and some of its generalizations, described in what follows. Recall that the ball of center x Rd and radius R > 0 is denoted by B(x, R). ∈

d Lemma 2.1 (Bernstein’s inequality). Let f 0(R ) such that fˆ is supported in B(0, R), and let 1 p q . Then,∈therS e exists a positive constant C (independent of f, R, p,≤ q),≤such≤that∞ d( 1 1 ) (2.1) f CR p − q f . k kq ≤ k kp We shall use also the following generalizations of the Bernstein’s inequalities. Lemma 2.2. Consider a mapping v : Rd Rd satisfying v(x) . x , x Rd. Then, for any s 0, → | | | | ∀ ∈ ≥ s s sup x − f(v(x)) . x − f(x) dx, x Rd{h i | |} h i | | ∈ Z d for every function f 0(R ), such that fˆ is supported in B(η0, 1), for some η Rd. ∈ S 0 ∈ Proof. Take a Schwartz function g, satisfying gˆ(η) = 1 for η 1. If fˆ is supported ˆ ˆ | | ≤ in B(η0, 1) we have f = fTη0 gˆ. Hence

s s (2.2) x − f(v(x)) x − f(y) g(v(x) y) dy. h i | | ≤ h i | || − | Z Now we have y s . y v(x) s v(x) s . y v(x) s x s, h i h − i h i h − i h i so that s s N s N+s x − g(v(x) y) x − v(x) y − . y − v(x) y − . h i | − | ≤ h i h − i h i h − i Using this inequality in (2.2), with N s, we attain the desired conclusion. ≥ We recall from [27, Proposition 5.5] the following localized version of Bernstein’s inequality. 2 N d Lemma 2.3. Let N 1 be an integer and ϕ(x) = (1+ x )− . For R > 0, x R , v x ≥ d | | ∈ let ϕ (v) = ϕ( − ), v R . There exists C > 0 such that x,R R ∈ N 1 sup f(y) CN µ(B(x, R)))− ϕx,R(v) f(v) dv, y B(x,R) | | ≤ Rd | | ∈ Z 6 ELENA CORDERO AND FABIO NICOLA

d d for every f 0(R ) such that fˆ is supported in B(η0, 1/R), for some η0 R , where µ is the∈ SLebesgue measure. ∈ 2.2. Schur-type tests. The next proposition collects Schur-type tests assuring the q p q Rd p Rd boundedness of integral operators on the mixed-norm spaces LηLx := L ( η; L ( x)), 1 p, q . ≤ ≤ ∞ Proposition 2.4. Consider an integral operator A on R2d, given by

(Af)(x0, η0) = K(x0, η0; x, η)f(x, η) dx dη. ZZ 1 1 1 (i) If K Lη∞Lη0 Lx∞0 Lx, then A is continuous on LηLx∞. ∈ 1 1 1 (ii) If K Lη∞0 LηLx∞Lx0 , then A is continuous on Lη∞Lx. ∈ 1 1 1 1 1 1 (iii) If K Lη∞Lη0 Lx∞0 Lx Lη∞0 LηLx∞Lx0 , and, moreover, K Lx∞0,η0 Lx,η Lx,η∞ Lx0,η0 , ∈ ∩ q p ∈ ∩ then the operator A is continuous on LηLx, for every 1 p, q . 1 1 ≤ 1 ≤ ∞ (iv) If K L∞ 0 L 0 , then A is continuous L L∞ L∞L . ∈ η,η x,x η x → η x Proof. The proof of all items, but (iv), is just a repetition, with obvious changes, of that of [8, Proposition 5.1], where a discrete version was presented. Let us now prove (iv). We have

Af L∞L1 = sup K(x0, η0; x, η)f(x, η) dx dη dx0 η0 x0 k k η0 Rd Rd ∈ Z x0 ZZ

sup K(x0, η0; x, η)f(x, η) dx dx 0 dη ≤ η0 Rd R3d | | ∈ ZZZ x,x0,η

sup sup f(x, η) K(x0, η0; x, η) dx dx0 dη ≤ η0 Rd Rd x Rd | | | | ∈ Z η ∈ ZZ K L∞ L1 f L1 L∞ . ≤ k k η,η0 x,x0 k k η x This concludes the proof.

2.3. Modulation spaces. [11, 12, 13, 14, 17, 26] For s R, we denote by s = (1 + 2)s/2. In what follows we limit ourselves to the ∈class of weight functionsh·i | · | s1 s2 R R2d s1 s2 vs1,s2 (x, η) = x η , si , i = 1, 2, on , or m(x, η, z, ζ) = x η = h i h i R4∈d h i h i (vs1,s2 1)(x, η, z, ζ), on . In order to define such spaces, we make use of the following⊗ time-frequency representation: the short-time Fourier transform (STFT) d Vgf of a function/tempered distribution f 0(R ) with respect to the the window g (Rd) is defined by ∈ S ∈ S 2πiηy (2.3) Vgf(x, η) = f, MηTxg = e− f(y)g(y x) dy, h i Rd − Z BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 7 i.e., the Fourier transform applied to fTxg. Given a non-zero windowFg (Rd), a weight function m as those quoted above, ∈ S p,q Rd and 1 p, q , the modulation space Mm ( ) consists of all tempered distri- ≤ ≤ R∞d p,q R2d butions f 0( ) such that Vgf Lm ( ) (weighted mixed-norm spaces). The ∈p,qS ∈ norm on Mm is q/p 1/p p,q p,q p p f Mm = Vgf Lm = Vgf(x, η) m(x, η) dx dη k k k k Rd Rd | | Z Z ! p (with obvious changes when p = or q = ). If p = q, we write Mm instead p,p R2d ∞ ∞ p,q p p,q p,p of Mm , and if m(z) 1 on , then we write M and M for Mm and Mm , ≡p,q Rd respectively. Then Mm ( ) is a Banach space whose definition is independent of the choice of the window g. For the properties of these spaces we refer to the literature quoted at the beginning of this subsection. p,q Rd Rd p,q We define by m ( ) the closure of ( ) in the Mm -norm. Observe that p,q p,q M S m = Mm , whenever the indices p and q are finite. They enjoy the duality M p,q p0,q0 property: ( m )∗ = 1/m, with 1 p, q , and p0, q0 being the conjugate exponents. M M ≤ ≤ ∞ We recall the inversion formula for the STFT (see e.g. ([17, Corollary 3.2.3]): if 2 d g 2 = 1 and, for example, u L (R ), it turns out k kL ∈

(2.4) u = Vgu(x, η)MηTxg dx dη. R2d Z The following inequality, proved in [17, Lemma 11.3.3], is useful for changing windows. d d Lemma 2.5. Let g , g , γ (R ) such that γ, g = 0 and let f 0(R ). Then, 0 1 ∈ S h 1i 6 ∈ S 1 V f(x, η) ( V f V γ )(x, η), | g0 | ≤ γ, g | g1 | ∗ | g0 | |h 1i| for all (x, η) R2d. ∈ The complex interpolation theory for these spaces reads as follows (see, e.g., [14]). Proposition 2.6. Let 0 < θ < 1, p , q [1, ] and m be v-moderate weight j j ∈ ∞ j functions (i.e., mj(x + y) Cv(x)mj(y), v being a submultiplicative weight), j = 1, 2. Set ≤ 1 1 θ θ 1 1 θ θ 1 θ θ = − + , = − + , m = m1− m2, p p1 p2 q q1 q2 then ( p1,q1 (Rd), p2,q2 (Rd)) = p,q(Rd). Mm1 Mm2 [θ] Mm We shall also use the following interpolation result: 8 ELENA CORDERO AND FABIO NICOLA

Proposition 2.7. Let 0 < θ < 1, pj, qj [1, ] and mj be v-moderate weight functions on Rd, j = 1, 2. Then, ∈ ∞ ,1 d ,1 d ,1 d ∞ R ∞ R ∞ R (Mm1 1( ), Mm2 1( ))[θ] = Mm 1( ), ⊗ ⊗ ⊗ 1 θ θ with m = m1− m2. 1 1 1 Proof. By [14], this amounts to computing [` (Lm∞1 ), ` (Lm∞2 )][θ] = ` ([Lm∞1 , Lm∞2 ][θ]), where the equality is due to [3, Theorem 5.1.2]. Using the connection between complex and real interpolation given by [3, Theorem 4.7.2], we have

[Lm∞ , Lm∞ ][θ] = (Lm∞ , Lm∞ )θ, . 1 2 1 2 ∞ Finally, the interpolation theorem of Stein-Weiss [3, Theorem 5.4.1] gives 1 θ θ (Lm∞ , Lm∞ )θ, = Lm∞, m = m1− m2, 1 2 ∞ as desired.

Remark 2.8. We observe that our results are established as the existence of a bounded extension p,q p,˜ q˜ of a class of FIOs T with symbols σ in a weighted ,1 M → M space Mm∞ . It is important to observe that such an extension follows from a uniform estimate of the type Rd (2.5) T f M p˜,q˜ C σ ∞,1 f M p,q , f ( ). k k ≤ k kMm k k ∀ ∈ S Indeed, this estimate shows that T extends to a bounded operator p,q M p,˜ q˜. In order to prove that this extension takes values in p,˜ q˜, it suﬃcesMto ver→ify that T f p,˜ q˜ when f is a Schwartz function. Since thisMis clearly true if the symbol ∈ M d σ itself is a Schwartz function, we approximate, in the 0(R )-topology, a general ,1 S σ M ∞ by a sequence of Schwartz symbols σn satisfying σn ∞,1 C σ ∞,1 ∈ m k kMm ≤ k kMm (cf. the construction in the proof of [17, Proposition 11.3.4]). Let T (n) be the (n) d corresponding FIOs. Then the sequence T f converges in 0(R ) to T f and, by (2.5), it is bounded in the space p,˜ q˜. By Alaoglu’s theorem,S a subsequence is p,˜ q˜ M d weak∗ convergent in . By the uniqueness of the limit in 0(R ), we obtain T f p,˜ q˜. M S Hence∈ M in the subsequent proofs we will prove estimates of the type (2.5). Boundedness results dealing with FIOs having symbols in weighted modulation spaces and acting on unweighted modulation spaces could be rephrased as boundedness results for FIOs with symbols in unweighted spaces and acting on weighted spaces, as explained below. Proposition 2.9. Let T be a FIO with symbol σ and T˜ a FIO with the same phase as T and symbol σ˜(x, η) := x s1 σ(x, η) η s2 , x, η Rd, s R, i = 1, 2. h i h i ∈ i ∈ BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 9

Then, (i) the operator T is bounded from p,q into p,˜ q˜ if and only if the operator T˜ is p,q p,˜ q˜ M M bounded from v into v . M 0,s2 M −s1,0 (ii) It holds true

,1 2d ,1 2d ∞ R R (2.6) σ Mvs ,s 1( ) σ˜ M ∞ ( ). ∈ 1 2 ⊗ ⇐⇒ ∈ Proof. The proof is an immediate consequence of [25, Theorem 2.2, Corollary 2.3]. Indeed, they guarantee that the vertical arrows of the following commutative diagram deﬁne isomorphisms:

p,q T - p,˜ q˜ M 6 M

D s2 pointwise product by x s1 (2.7) .h i h i ? p,q T˜ - p,˜ q˜ v0,s v M 2 M −s1,0

s1 s2 Moreover, the product by the weight function η0(x, η, ζ1, ζ2) := x η , x, η, d 4d ,1 2d h i h i ,1 2d R R ∞ R R ζ1, ζ2 , deﬁned on , is a isomorphism from Mvs ,s 1( ) to M ∞ ( ), ∈ 1 2 ⊗ that is (ii).

2.4. Wiener amalgam spaces. [11, 13, 16] For 1 p , s R, we de- p Rd p Rd p Rd ≤ ≤ ∞ R∈d note by Ls( ) := L s ( ). Let Ls( ) be the space of f 0( ) such that h·i F ∈ S ˆ p Rd Rd f Ls( ). Let g ( ) be a non-zero window function. For 1 p, q, , the ∈ ∈ S p q Rd ≤ p ≤ ∞ Wiener amalgam space W ( Ls1 , Ls2 )( ) with local component Ls1 and global q R F F component Ls2 , si , i = 1, 2, is deﬁned as the space of all functions or distributions for which the∈norm

q/p 1/q p q p ps1 qs2 f W ( Ls ,Ls ) = (f Txg)(η) η dη x dx k k F 1 2 Rd Rd |F · | h i h i Z Z ! p q Rd is ﬁnite. Analogously to modulation spaces, we deﬁne by ( Ls1 , Ls2 )( ) the Rd p q W F closure of ( ) with respect to the W ( Ls1 , Ls2 )-norm. The propSerties of Wiener amalgam sFpaces are similar to those of modulation spaces, since we have the following relation: p,q p q R (2.8) ( v ) = ( Ls , Ls ), si , i = 1, 2. F M s1,s2 W F 1 2 ∈ We recall from [7, Lemma 5.3] the following auxiliary result. 10 ELENA CORDERO AND FABIO NICOLA

π|x|2 d/2 Lemma 2.10. For a, b R, a > 0, set Ga+ib(x) = (a + ib)− e− a+ib , and choose π∈y 2 the Gaussian g(y) = e− | | as window function. Then, for every 1 q, r , it follows that ≤ ≤ ∞ π 2 2 2 \ 2 2 d/4 2 2 [(a(a+1)+b ) η +2bxη+(a+1) x ] (2.9) G T g(η) = ((a + 1) + b )− e− (a+1) +b | | | | , | a+ib x | d 1 1 2 2 2 2 ( q − 2 ) πa|x| ((a + 1) + b ) 2 q − a(a+1)+b (2.10) Ga+ibTxg L = d d e , k kF q 2q (a(a + 1) + b2) 2q and d 1 1 2 2 ( ) ((a + 1) + b ) 2 q − 2 (2.11) Ga+ib W ( Lq,Lr) . F d d ( 1 1 ) k k a 2r (a(a + 1) + b2) 2 q − r p,q R2d R Lemma 2.11. Let σ Mvs ,s 1( ), si , i = 1, 2, and consider its adjoint σ∗ ∈ 1 2 ⊗ ∈ deﬁned by 2d (2.12) σ∗(x, η) = σ(η, x), (x, η) R . ∈ p,q R2d Then, σ∗ Mvs ,s 1( ). ∈ 2 1 ⊗ Proof. Fix a window function g (Rd). By a direct computation, ∈ S 2πi(y1ζ1+y2ζ2) V σ∗(z , z ; ζ , ζ ) = e− σ(y , y ) g(y z , y z ) dy dy g 1 2 1 2 2 1 1 − 1 2 − 2 1 2 Z 2πi(v2ζ1+v1ζ2) = e− σ(v , v ) g(v z , v ζ ) dv dv 1 2 2 − 1 1 − 2 1 2 Z = Vtgσ(z2, z1; ζ2, ζ1) t p,q R2d where we used the notation g(y1, y2) := g(y2, y1). Since σ Mvs ,s 1( ), the ∈ 1 2 ⊗ result immediately follows by the indipendence of the window function for the computation of the modulation space norm.

3. Boundedness of FIOs on p,q M In the following we assume that the phase function Φ satisfies the assumptions (i), (ii) and (iii) in the Introduction, so that we shall repeatedly use the following property. Remark 3.1. It follows from the assumptions (i), (ii) and (iii) and Hadamard’s global inversion function theorem (see, e.g., [22]) that the mappings x Φ(x, η) 7−→ ∇η and η Φ(x, η) 7−→ ∇x BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 11 are global diffeomorphisms of Rd, and their inverse Jacobian determinant are uniformly bounded with respect to all variables. The proof of our results relies on a formula, obtained in [8, Section 6], which expresses the Gabor matrix of the FIO T in terms of the STFT of its symbol σ. Namely, choose a non-zero window function g (Rd), and define ∈ S 2πiΦ 0 (y,ζ) 2d (3.1) Ψ 0 (y, ζ) := e 2,(x ,η) (g¯ gˆ)(y, ζ), (y, ζ) R (x ,η) ⊗ ∈ with 1 α α (y, ζ) (3.2) Φ 0 (y, ζ) = 2 (1 t)∂ Φ((x0, η) + t(y, ζ)) dt . 2,(x ,η) − α! α =2 0 |X| Z Moreover, let g (t) = (M T g) (t), t, x, η Rd. x,η η x ∈ Proposition 3.2. It turns out

(3.3) T g , g 0 0 = V σ((x0, η), (η0 Φ(x0, η), x Φ(x0, η))) . |h x,η x ,η i| | Ψ(x0,η) − ∇x − ∇η | Remark 3.3. Observe that the window Ψ(x0,η) of the STFT above depends on the pair (x0, η). However, the assumptions (1.2) imply that these windows belong to a bounded subset of the Schwartz space, i.e. the corresponding seminorms are uniformly bounded with respect to (x0, η). In this section we focus on the proofs of Theorems 1.1 and 1.2. We ﬁrst prove Theorem 1.1. We need the following auxiliary results. R2d Lemma 3.4. Let Ψ0 ( ) with Ψ0 L2 = 1 and Ψ(x0,η) be deﬁned by (3.1), 2d ∈ Sd k k (x0, η) R , and g (R ). Then, ∈ ∈ S

(3.4) sup VΨ(x0,η) Ψ0(w) dw < . R4d (x0,η) R2d | | ∞ Z ∈ Proof. We shall show that (4d+1) 2d (3.5) V Ψ (w) C w − , (x0, η) R . | Ψ(x0,η) 0 | ≤ h i ∀ ∈ Using the switching property of the STFT: 2πiηx (V g)(x, η) = e− (V f)( x, η), f g − − we observe that V Ψ (w , w ) = V Ψ 0 ( w , w ) , and by the even prop- | Ψ(x0,η) 0 1 2 | | Ψ0 (x ,η) − 1 − 2 | erty of of the weight , relation (3.5) is equivalent to h·i (4d+1) 2d (3.6) V Ψ 0 (w) C w − , (x0, η) R . | Ψ0 (x ,η) | ≤ h i ∀ ∈ R2d R4d Now, the mapping VΨ0 is continuous from ( ) to ( ) (see [17, Chap. 11]), which combined with the Remark (3.3) yieldsS (3.6). S 12 ELENA CORDERO AND FABIO NICOLA

Lemma 3.5. With the notation of Proposition 2.3, for every R > 0 there exists CR such that

ϕx,R(v)f(v)dv dx = CR f(x) dx, ZZ Z for every measurable function f 0. ≥ Proof. The proof is just an application of Fubini’s theorem on the exchange of integrals, since ϕx,R(v)dx = CR is independent of v.

Proposition 3.R6. Let Ψ (R2d) supported in the ball B(0, 1/R). Then 0 ∈ S 1 sup VΨ0 σ(u; z1 + v, z2) CN µ(B(z1, R))− ϕz1,R(v) VΦ0 σ(u; v, z2) dv, v B(0,R) | | ≤ | | ∈ Z for every u R2d, z , z Rd. ∈ 1 2 ∈ Rd Proof. It suﬃces to apply Proposition 2.3 to the function v VΨ0 σ(u; v, z2). Indeed, setting u = (u , u ) Rd Rd, its Fourier transform3has 7−supp→ ort contained 1 2 ∈ × in the ball B(u1, 1/R).

Proof of Theorem 1.1. Since the boundedness on p = p,p, 1 p , was already proved in [8, Theorem 6.1], see also [5, TheorM emM2.1], the≤desired≤ ∞result will follow by interpolation from the cases (p, q) = ( , 1) and (p, q) = (1, ). To prove them, we observe that the inversion formula (2∞.4) for the STFT gives∞

Vg(T u)(x0, η0) = T gx,η, gx0,η0 Vgu(x, η)dx dη. R2d h i Z By Remark 2.8, the desired estimate therefore follows if we prove that the map KT deﬁned by

KT G(x0, η0) = T gx,η, gx0,η0 G(x, η)dx dη R2d h i 1 1Z is continuous on LηLx∞ and Lη∞Lx. By Proposition 2.4 it suﬃces to prove that its integral kernel K (x0, η0; x, η) = T g , g 0 0 T h x,η x ,η i satisﬁes 1 1 (3.7) K L∞L 0 L∞0 L , T ∈ η η x x and 1 1 (3.8) K L∞0 L L∞L 0 . T ∈ η η x x Let us verify (3.7). By (3.3) we have

K (x0, η0; x, η) = V σ(z(x0, η0, x, η)) , | T | | Ψ(x0,η) | BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 13 where

z(x0, η0, x, η) = (x0, η, η0 Φ(x0, η), x Φ(x0, η)). − ∇x − ∇η By Lemma 2.5 for g1 = γ = Ψ0, we have V σ(z) ( V σ V Ψ )(z), z R4d, | Ψ(x0,η) | ≤ | Ψ0 | ∗ | Ψ(x0,η) 0| ∈ so that

VΨ 0 σ(z(x0, η0, x, η)) L∞L1 L∞L1 k (x ,η) k η η0 x0 x

1 ∞ VΨ0 σ(z(x0, η0, x, η) w) L∞L L L1 sup VΨ 0 Ψ0(w) dw η η0 x0 x (x ,η) ≤ k − k (x0,η) R2d | | Z ∈

1 ∞ sup VΨ0 σ(z(x0, η0, x, η) w) L∞L L L1 sup VΨ 0 Ψ0(w) dw. η η0 x0 x (x ,η) ≤ w R4d k − k (x0,η) R2d | | ∈ Z ∈ In view of Lemma 3.4 we are therefore reduced to proving the estimate

1 ∞ ∞,1 VΨ0 σ(z(x0, η0, x, η) w) L∞L L L1 C σ M , k − k η η0 x0 x ≤ k k uniformly with respect to w R4d. Since, ∈

VΨ0 σ(z(x0, η0, x, η) w) sup VΨ(x0,η) σ(u, z˜(x0, η0, x, η, w2) , | − | ≤ u R2d | | ∈ with

4d z˜(x0, η0, x, η, w ) = (η0 Φ(x0, η), x Φ(x0, η)) w , w = (w , w ) R , 2 − ∇x − ∇η − 2 1 2 ∈ we shall prove that

1 ∞ ∞,1 (3.9) sup VΨ0 σ(u, z˜(x0, η0, x, η, w2) L∞L L L1 C σ M , η η0 x0 x k u R2d | |k ≤ k k ∈ uniformly with respect to w R2d. A translation shows that the left-hand side in 2 ∈ (3.9) is indeed independent of w2, and coincides with

sup sup sup VΨ0 σ(u; η0 xΦ(x0, η), x ηΦ(x0, η)) dx dη0. η Rd Rd x0 Rd Rd u R2d | − ∇ − ∇ | ∈ Z η0 ∈ Z x ∈

Here we perform the change of variables x x ηφ(x0, η). The last expression will be 7−→ − ∇

sup sup sup VΨ0 σ(u; η0 xΦ(x0, η), x) dx dη0. ≤ η Rd Rd Rd u R2d x0 Rd | − ∇ | ∈ Z η0 Z x ∈ ∈ Now we observe that

Φ(x0, η) = Φ(0, η) + A(x0, η), ∇x ∇x 14 ELENA CORDERO AND FABIO NICOLA where, by the assumption (1.6), A(x0, η) R for some R > 0. By Proposition 3.6 we can continue the majorization| as | ≤

0 . sup sup ϕη xΦ(0,η),R(v) VΨ0 σ(u; v, x) dv dx dη0. η Rd Rd Rd u R2d Rd −∇ | | ∈ Z η0 Z x ∈ Z v Now we bring the supremum with respect to u inside the more interior integral and perform the change of variables η0 η0 φ(0, η), obtaining 7−→ − ∇x

0 ϕη ,R(v) sup VΨ0 σ(u; v, x) dv dx dη0. ≤ Rd Rd Rd u R2d | | Z η0 Z x Z v ∈ Finally we exchange the two more interior integral and apply Lemma 3.5. The last expression is seen to be

∞,1 = CR sup VΨ0 σ(u; η0, x) dx dη0 = CR σ M . Rd Rd u R2d | | k k Z η0 Z x ∈ We now prove (3.8). Here we will use repeatedly the Remark 3.1. By arguing as above we are reduced to proving that

(3.10) sup sup sup VΨ0 σ(u; η0 xΦ(x0, η), x ηΦ(x0, η)) dx0 dη η0 Rd Rd x Rd Rd u R2d | − ∇ − ∇ | ∈ Z η ∈ Z x0 ∈ C σ ∞,1 . ≤ k kM By performing the change of variables x0 ηΦ(x0, η) and a subsequent translation, the left-hand side in (3.10) is seen t7−o→be∇

. sup sup sup VΨ0 σ(u; η0 xΦ(B(x0 + x, η), η), x0) dx0 dη, η0 Rd Rd x Rd Rd u R2d | − ∇ | ∈ Z η ∈ Z x0 ∈ for a suitable function B coming from the inverse change of variables. Now, by the assumption (1.6) we have Φ(B(x0 + x, η), η) = Φ(0, η) + A0(x, x0, η), with ∇x ∇x A0(x, x0, η) R for a suitable R > 0. It turns out that the last expression is | | ≤

sup sup sup VΨ0 σ(u; η0 xΦ(0, η) + v, x0) dx0 dη. ≤ η0 Rd Rd Rd u R2d v Rd | − ∇ | ∈ Z η Z x0 ∈ ∈ By the Proposition 3.6 this is

0 . sup sup ϕη xΦ(0,η),R(v) VΨ0 σ(u; v, x0) dv dx0 dη. η0 Rd Rd Rd u R2d Rd −∇ | | ∈ Z η Z x0 ∈ Z v Now we perform the change of variable η xΦ(0, η), and a subsequent translation, obtaining 7−→ ∇

. sup ϕη,R(v) VΨ0 σ(u; v, x0) dv dx0 dη. Rd Rd u R2d Rd | | Z η Z x0 ∈ Z v BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 15

Finally we can bring the supremum with respect to u inside the more interior integral, exchange the integrals with respect to v and η0 and apply Lemma 3.5, obtaining

∞,1 CR sup VΨ0 σ(u; η, x0) dx0 dη = CR σ M , ≤ Rd Rd u R2d | | k k Z η Z x0 ∈ as desired We now prove Theorem 1.2. To this end, we ﬁrst consider the cases (p, q) = ( , 1) (Theorem 3.7) and (p, q) = (1, ) (Theorem 3.8). ∞ ∞ Theorem 3.7. Consider a phase Φ satisfying (i), (ii) and (iii), and a symbol ,1 2d ∞ R σ Mvs,0 1( ), with s > d. Then the corresponding FIO T extends to a bounded ∈ ⊗ ,1 d ,1 d operator ∞ (R ) ∞ (R ). M → M Proof. By arguing as at the beginning of the proof of Theorem 1.1, we see that it suﬃces to prove the estimate

∞,1 sup sup VΨ0 σ(x0, η; η0 xΦ(x0, η), x ηΦ(x0, η)) dx dη0 C σ . Mv ⊗1 η Rd Rd x0 Rd Rd | − ∇ − ∇ | ≤ k k s,0 ∈ Z η0 ∈ Z x The left-hand side of this estimate is seen to be

s sup sup sup x0 − d d 0 d d 2d ≤ η R R x R R (u1,u2) R h i ∈ Z η0 ∈ Z x ∈ s u V σ(u , u ; η0 Φ(x0, η), x Φ(x0, η)) dx dη0, × |h 1i Ψ0 1 2 − ∇x − ∇η | which coincides with

s s sup sup sup x0 − u1 VΨ0 σ(u1, u2; η0 xΦ(x0, η), x) dx dη0 d d 0 d d 2d η R R x R R (u1,u2) R h i |h i − ∇ | ∈ Z η0 ∈ Z x ∈ s s sup sup u1 sup x0 − VΨ0 σ(u1, u2; η0 xΦ(x0, η), x) dx dη0 d d d 2d 0 d ≤ η R R R (u1,u2) R h i x R h i | − ∇ | ∈ Z η0 Z x ∈ ∈ Now we apply Lemma 2.2 to the function

f(ζ) := V σ(u , u ; η0 Φ(0, η) ζ, x) Ψ0 1 2 − ∇x − with v(x0) = xΦ(x0, η) xΦ(0, η) (so that f(v(x0)) = VΨ0 σ(u1, u2; η0 xΦ(x0, η), x)). The assumptio∇ns are indeed−∇ satisﬁed uniformly with respect to all par−∇ameters; in particular

v(x0) = Φ(x0, η) Φ(0, η) C x0 | | |∇x − ∇x | ≤ | | 16 ELENA CORDERO AND FABIO NICOLA

2d for every (x0, η) R by (ii). It follows that w∈e can continue our majorization as

s s . sup sup u1 x0 − VΨ0 σ(u1, u2; η0 xΦ(0, η) x0, x) dx0 dx dη0 d d d 2d d η R R R (u1,u2) R h i R h i | − ∇ − | ∈ Z η0 Z x ∈ Z x0 s s sup x0 − sup u1 VΨ0 σ(u1, u2; η0 xΦ(0, η) x0, x) dx dη0 dx0. d d d d 2d ≤ η R R h i R R (u1,u2) R h i | − ∇ − | ∈ Z x0 Z η0 Z x ∈

By performing the translation η0 η0 xΦ(0, η) x0 and using the integrability condition s > d one sees that this7−→last express−∇ ion is−

s ∞,1 = C sup u1 VΨ0 σ(u1, u2; η0, x) dx dη0 = C σ M . d d 2d vs,0⊗1 R R (u1,u2) R h i | | k k Z η0 Z x ∈ This concludes the proof.

Theorem 3.8. Consider a phase Φ satisfying (i), (ii), and (iii) and a symbol ,1 2d ∞ R σ Mv0,s 1( ), with s > d. Then, the corresponding FIO T extends to a bounded ∈ ⊗ 1, d 1, d operator ∞(R ) ∞(R ). M → M Proof. By arguing as at the beginning of the proof of Theorem 3.7, we see that it suﬃces to prove the estimate

∞,1 sup sup VΨ0 σ(x0, η; η0 xΦ(x0, η), x ηΦ(x0, η)) dx0 dη C σ . Mv ⊗1 η0 Rd Rd x Rd Rd | − ∇ − ∇ | ≤ k k 0,s ∈ Z η ∈ Z x0 The left-hand side of this estimate is seen to be

s sup sup sup η − 0 d d d d 2d ≤ η R R x R R (u1,u2) R h i ∈ Z η ∈ Z x0 ∈ s u V σ(u , u ; η0 Φ(x0, η), x Φ(x0, η)) dx0 dη. × |h 2i Ψ0 1 2 − ∇x − ∇η |

By performing the change of variables x0 ηΦ(x0, η) (see Remark 3.1) and a subsequent translation, we obtain 7−→ ∇

s sup sup sup η − 0 d d d d 2d ≤ η R R x R R (u1,u2) R h i ∈ Z η ∈ Z x0 ∈ s u V σ(u , u ; η0 Φ(B(x + x0, η), η), x0) dx0 dη, × |h 2i Ψ0 1 2 − ∇x | BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 17 for a suitable function B(x0, η) coming from the inverse change of variables. Bring- ing the supremum over η0 inside,

s (3.11) sup sup η − d d d 2d ≤ R x R R (u1,u2) R h i Z η ∈ Z x0 ∈ s u2 sup VΨ0 σ(u1, u2; η0 xΦ(B(x + x0, η), η), x0) dx0 dη. × h i η0 Rd | − ∇ | ∈ Using Bernstein’s inequality (2.1):

sup f(η0) . f(η0) dη0, η0 Rd | | Rd | | ∈ Z for f(η0) = V σ(u , u ; η0 Φ(B(x+x0, η), η), x0), and the translation invariance Ψ0 1 2 −∇x of the Lebesgue measure dη0, the expression (3.11) is less than

s s η − dη sup u2 VΨ0 σ(u1, u2; η0, x0) dx0 dη0. d d d 2d R h i R R (u1,u2) R h i | | Z η Z x0 Z η0 ∈ Using the integrability condition s > d, this last expression is equal to

s ∞,1 C sup u2 VΨ0 σ(u1, u2; η0, x0) dx0 dη = C σ M , d d 2d v0,s⊗1 R R (u1,u2) R h i | | k k Z x0 Z η0 ∈ as desired.

Proof of Theorem 1.2. Item (i) was already proved in [8, Theorem 6.1], see also [5, Theorem 2.1]. Item (ii). Consider the bilinear mapping d 2d d (3.12) (f, σ) (R ) 0(R ) T f 0(R ), ∈ S × S 7−→ ∈ S which associates every function f and symbol σ with the range T f of the corresponding FIO T in (1.1). Then, Theorem 3.7 says that the mapping in (3.12) ,1 d ,1 2d ,1 d R ∞ R R extends to a continuous mapping from ∞ ( ) Mvs,0 1( ) into ∞ ( ), M × ⊗ M with s > d. Moreover, thanks to Item (i), the mapping in (3.12) extends to a p d ,1 2d p d bounded mapping from (R ) M ∞ (R ) into (R ). Using the complex interpolation properties Mfor modula× tion spaces in ProMpositions 2.6 and 2.7, the mapping in (3.12) extends to a bounded mapping between ,1 d p d ,1 2d ,1 2d ,1 d p d R 2 R ∞ R R R 2 R ( ∞ ( ), ( ))[θ] (Mvs,0 1( ), M ∞ ( ))[θ] ( ∞ ( ), ( ))[θ], M M × ⊗ −→ M M for every 1 p2 , 0 < θ < 1, s > d. For 1/p = θ/p2, 1/q = 1 θ + θ/p2, we therefore ha≤ve ≤ ∞ − p,q d ,1 d p,q d R ∞ R R ( ) Mv 1 1 1( ) ( ), q p. M × s( q − p ),0⊗ −→ M ≤ 18 ELENA CORDERO AND FABIO NICOLA

The proof of Item (iii) uses the same complex interpolation arguments as before, ,1 d 1, d replacing the continuity of the FIO T on ∞ (R ), by that on ∞(R ) (Theo- rem 3.8). M M An alternative proof of Items (ii) (iii), which does not refer to Proposition − ,1 2d 2.7 and uses FIOs with simbols in the unweighted modulation space M ∞ (R ), is as follows. By Proposition 2.9, the conclusion in Theorem 1.2 is equivalent to saying that any FIO T˜, with phase Φ satisfying (i), (ii) and (iii) and symbol ,1 R2d p,q Rd p,q Rd σ˜ M ∞ ( ), extends to a bounded operator v ( ) v ( ), for s1, ∈ M 0,s2 → M −s1,0 s2 as in the statement, and moreover

˜ p,q ∞,1 p,q T f v C σ˜ M f v . k kM −s1,0 ≤ k k k kM 0,s2 Since this was already proved for (p, q) = ( , 1), for all 1 p = q , and for (p, q) = (1, ), the desired result follows fro∞m Proposition 2≤.6. ≤ ∞ ∞ Theorem 1.2 has the following counterpart in the framework of Wiener amalgam spaces. Of course, we are interested in ( Lp, Lq), with p = q, since ( Lp, Lp) = p. W F 6 W F M Corollary 3.9. Consider a phase Φ and a symbol σ as in Theorem 1.2. (i) Let 1 q < p . If s > d 1 1 , s 0, T extends to a bounded operator ≤ ≤ ∞ 1 q − p 2 ≥ on ( Lp, Lq)(Rd). W F (ii) Let 1 p < q . If s 0, s > d 1 1 , T extends to a bounded ≤ ≤ ∞ 1 ≥ 2 p − q operator on ( Lp, Lq)(Rd). In all cases,W F

(3.13) T f ( Lp,Lq) . σ ∞,1 f ( Lp,Lq). Mv k kW F k k s1,s2 ⊗1 k kW F Proof. The result easily follows from Theorem 1.2. Indeed, consider an operator T , with symbol σ and phase Φ, satisfying the assumptions of Corollary 3.9. Con- jugating with the Fourier transform yields the operator 1 T˜f(x) = T − f(x) F ◦ ◦ F p,q 1 p q Since = − ( L , L ), it suﬃces to prove that T˜ extends to a bounded operatorM on Fp,q. WByFduality and an explicit computation this is equivalent to verifying thatMthe operator

2πiΦ(η,x) T˜∗f(x) = e− σ(η, x)fˆ(η) dη Rd Z p0,q0 ,1 2d ∞ R extends to a bounded operator on . Since σ Mvs ,s 1( ) implies that M ∈ 1 2 ⊗ ,1 2d ∞ R σ∗(x, η) = σ(η, x) Mvs ,s 1( ) by Lemma 2.11, the desired result is attained ∈ 2 1 ⊗ from Theorem 1.2. BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 19

4. Boundedness of FIOs on weighted p,q M Thanks to the commutativity of the diagram (2.7), the results of Theorem 1.2 may be equivalently stated as the action of a FIO T on weighted modulation spaces. Namely, we have the following result.

Theorem 4.1. Consider a phase Φ satisfying (i), (ii) and (iii), and a symbol ,1 2d σ M ∞ (R ). ∈ (i) If 1 q < p , s < d 1 1 , and s 0, then T extends to a bounded ≤ ≤ ∞ 1 − q − p 2 ≥ p,q Rd p,q Rd operator from v ( ) to v ( ). M 0,s2 M s1,0 (ii) If 1 p < q , s 0, and s > d 1 1 , T extends to a bounded ≤ ≤ ∞ 1 ≤ 2 p − q p,q Rd p,q Rd operator from v ( ) to v ( ). M 0,s2 M s1,0 In both cases,

p,q ∞,1 p,q T f v . σ M f v . k kM s1,0 k k k kM 0,s2 One can easily rephrase the results of Corollary 3.9 in term of weighted Wiener amalgam spaces. We now study the weighted cases not contained above.

Theorem 4.2. Consider a phase Φ satisfying (i), (ii) and (iii), and a symbol ,1 2d σ M ∞ (R ). Then, the corresponding FIO T extends to a bounded operator betwe∈ en the following spaces:

,1 d ,1 d 1, d 1, d ∞ (R ) ∞ (R ), ∞(R ) ∞(R ), M → Mv0,s M → Mvs,0 with s < d, and its norm is bounded from above by C σ M ∞,1 , for a suitable C > 0. − k k

,1 d ,1 d Proof. To prove the boundedness of T from ∞ (R ) to ∞ (R ), we have to M Mv0,s show that the integral kernel

s K (x0, η0; x, η) = T g , g 0 0 η0 T h x,η x ,η ih i

1 1 satisﬁes KT Lη∞Lη0 Lx∞0 Lx. The arguments are similar to those of Theorem 3.7, we sketch the∈m for sake of clarity. The quantity

s sup sup VΨ0 σ(x0, η; η0 xΦ(x0, η), x ηΦ(x0, η)) η0 dxdη0 η Rd Rd x0 Rd Rd | − ∇ − ∇ |h i ∈ Z η0 ∈ Z x 20 ELENA CORDERO AND FABIO NICOLA can be controlled from above by

s C sup sup sup VΨ0 σ(u1, u2; η0 xΦ(x0, η), x) η0 dx dη0, d d 0 d d 2d η R R x R R (u1,u2) R | − ∇ |h i ∈ Z η0 ∈ Z x ∈ s C sup sup sup sup VΨ0 σ(u1, u2; ζ, x) η0 dx dη0 d d 0 d d 2d d ≤ η R R x R R (u1,u2) R ζ R | |h i ∈ Z η0 ∈ Z x ∈ ∈ s C0 sup sup sup VΨ0 σ(u1, u2; ζ, x) dζ η0 dx dη0, d d 0 d d 2d d ≤ η R R x R R (u1,u2) R R | | h i ∈ Z η0 ∈ Z x ∈ Z ζ where in the last estimate we applied Bernstein’s inequality (2.1) to the function f(ζ) = V σ(u , u ; ζ, x). The last estimate, since s < d, is dominated from above Ψ0 1 2 − by σ M ∞,1 , as desired. k k 1, 1, Similarly, to show T : ∞ ∞, is equivalent to proving that the integral M → Mvs,0 kernel s KT (x0, η0; x, η) = T gx,η, gx0,η0 x0 1 1 h ih i satisﬁes KT Lη∞0 LηLx∞Lx0 . The arguments are quite similar to those of Theorem 3.8. Again, ∈we sketch the proof for sake of clarity. By performing the change of variables η Φ(x0, η), the quantity 7−→ ∇x s sup sup VΨ0 σ(x0, η; η0 xΦ(x0, η), x ηΦ(x0, η)) x0 dη dx0 η0 Rd Rd Rd x Rd | − ∇ − ∇ |h i ∈ Z x0 Z η ∈ is less than

s sup sup sup VΨ0 σ(u1, u2; η, x ηΦ(x0, B(x0, η + η0)) x0 dx0 dη, 0 d d d d 2d η R R R x R (u1,u2) R | − ∇ |h i ∈ Z x0 Z η ∈ ∈ for a suitable function B(x0, η) coming from the inverse change of variables. The result is attained using Bernstein’s inequality (2.1): supx Rd f(x) . Rd f(x) dx, for f(x) = V σ(u , u ; η, x), and the condition s < d. ∈ | | | | Ψ0 1 2 − R Since the boundedness of the FIO T on p is provided by Theorem 1.2, the complex interpolation with the preceding resultsM yields: Theorem 4.3. Consider a phase Φ satisfying (i), (ii) and (iii), a symbol σ ,1 d ∈ M ∞ (R ). (i) If 1 q < p , s 0, and s < d 1 1 , then T extends to a bounded ≤ ≤ ∞ 1 ≥ 2 − q − p p,q Rd p,q Rd operator from v ( ) to v ( ). M s1,0 M 0,s2 (ii) If 1 p < q , s < d 1 1 , and s 0, then T extends to a bounded ≤ ≤ ∞ 1 − p − q 2 ≥ p,q Rd p,q Rd operator from v ( ) to v ( ). M 0,s2 M s1,0 In both cases the norm of T is bounded from above by C σ M ∞,1 , for a suitable C > 0. k k BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 21

The results for Wiener amalgam spaces are obtained by similar arguments as those in Corollary 3.9 and left to the reader.

5. Boundedness of FIOs p,q q,p, p q M → M ≥ In this section we shall prove the boundedness of an operator T between p,q q,p, namely Theorem 1.3. As a byproduct, conditions for the boundednessM fro→m M( Lp, Lq) to ( Lq, Lp) are attained as well (Corollary 5.2). By using complex WinterpF olation, asWinFthe proof of Theorem 1.2, we see that it suffices to prove the desired results for (p, q) = ( , 1): ∞ Theorem 5.1. Consider a phase Φ(x, η) satisfying (i) and (ii). Moreover, assume one of the following conditions: ,1 2d (a) the symbol σ M ∞ (R ) and, for some δ > 0, ∈ ∂2Φ (5.1) det δ (x, η) R2d, ∂xi∂xl (x,η) ≥ ∀ ∈ ,1 2d (b) the symbol σ M ∞ (R ), with s > d. vs,0 1 ∈ ⊗ ,1 d 1, d Then, the corresponding FIO T extends to a bounded operator ∞ (R ) ∞(R ). M → M Proof. We argue as in the first part of the proof of Theorem 1.1. We see that it suffices to prove that the map KT defined by

KT G(x0, η0) = T gx,η, gx0,η0 G(x, η)dx dη R2d h i Z 1 1 is continuous LηLx∞ Lη∞Lx. By Proposition 2.4 it suﬃces to prove that its integral kernel → K (x0, η0; x, η) = T g , g 0 0 T h x,η x ,η i satisﬁes 1 (5.2) K L∞ 0 L 0 . T ∈ η,η x,x (a) The same arguments as in the proof of Theorem 1.1 show that it is enough to verify the estimate

(5.3) sup sup VΨ0 σ(u; η0 xΦ(x0, η), x ηΦ(x0, η)) dx dx0 (η,η0) R2d R2d u R2d | − ∇ − ∇ | ∈ ZZ x,x0 ∈ C σ ∞,1 . ≤ k kM To this end, we ﬁrst perform the translation x x ηΦ(x0, η) in the left-hand side of (5.3), obtaining → − ∇

sup sup VΨ0 σ(u; η0 xΦ(x0, η), x) dx dx0. (η,η0) R2d R2d u R2d | − ∇ | ∈ ZZ x,x0 ∈ 22 ELENA CORDERO AND FABIO NICOLA

Then we perform the change of variables x0 xΦ(x0, η) (followed by a translation) which by Remark 3.1 is a global diﬀeomo7−→ rphism∇ of Rd, and whose inverse Jacobian determinant is uniformly bounded with respect to all variables. Hence the last expression is seen to be

∞,1 . sup VΨ0 σ(u; x0, x) dx dx0 = σ M . Rd Rd u R2d | | k k Z x0 Z x ∈ (b) Similar arguments as above show that the result is attained once we prove the estimate

(5.4) sup VΨ0 σ(x0, η; η0 xΦ(x0, η), x ηΦ(x0, η)) dx dx0 (η,η0) R2d R2d | − ∇ − ∇ | ∈ ZZ x,x0 C σ M ∞,1 . ≤ k k vs,0⊗1

For s 0, and performing the change of variables x x ηΦ(x0, η), the left-hand≥ side above is controlled by 7−→ − ∇

s s sup x0 − sup u1 VΨ0 σ(u1, u2; η0 xΦ(x0, η), x) dx dx0 0 2d 2d 2d ≤ (η,η ) R R h i (u1,u2) R h i | − ∇ | ∈ ZZ x,x0 ∈

s s sup x0 − sup u1 sup VΨ0 σ(u1, u2; η0 xΦ(x0, η), x) dx dx0. 0 d 2d 2d d ≤ η R R h i (u1,u2) R h i η R | − ∇ | ∈ ZZ x,x0 ∈ ∈ Now we apply Lemma 2.2 to the function

f(ζ) := V σ(u , u ; η0 Φ(x0, 0) ζ, x) Ψ0 1 2 − ∇x − with v(η) = xΦ(x0, η) xΦ(x0, 0) (so that f(v(η)) = VΨ0 σ(u1, u2; η0 xΦ(x0, η), x)). The assumptio∇ ns are indeed−∇ satisﬁed uniformly with respect to all par−∇ameters; in 2d particular v(η) C η , for every (x0, η) R by (ii). Continuing| our|ma≤ jor|izatio| ns, we obtain ∈

s s . sup sup u1 x0 − VΨ0 σ(u1, u2; η0 xΦ(x0, 0) η, x) dη dx0 dx 0 d d d 2d d η R R R (u1,u2) R h i R h i | − ∇ − | ∈ Z η0 Z x ∈ Z x0 s s sup x0 − sup u1 VΨ0 σ(u1, u2; η, x) dη dx dx0 0 d d d d 2d ≤ η R R h i R R (u1,u2) R h i | | ∈ Z x0 Z η Z x ∈ where in the last row we perform the translation (up to a sign) η η0 7−→ − Φ(x0, 0) η and using the integrability condition s > d one sees that this last ∇x − expression is dominated by σ M ∞,1 . k k vs,0⊗1 From Theorem 1.3 and using the same arguments as in Corollary 3.9, one can prove the following result. BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 23

Corollary 5.2. Let 1 q < p . Consider a phase Φ(x, η) satisfying (i), (ii) and (iii). Moreover, assume≤ one≤of∞the following conditions: ,1 2d (a) the symbol σ M ∞ (R ) and, for some δ > 0, ∈ ∂2Φ (5.5) det δ (x, η) R2d, ∂η ∂η (x,η) ≥ ∀ ∈ i l or ,1 2d 1 1 ∞ R (b) the symbol σ Mv0,s 1( ), with s > d q p . ∈ ⊗ − Then, the corresponding FIO T extends to a bounde doperator ( Lp, Lq)(Rd) ( Lq, Lp)(Rd). W F → W F Example 5.3. (Metaplectic operators) Consider the particular case of quadratic phases, namely phases of the type 1 1 (5.6) Φ(x, η) = Ax x + Bx η + Cη η + η x x η, 2 · · 2 · 0 · − 0 · d where x0, η0 R , A, C are real symmetric d d matrices and B is a real d d nondegenerate∈ matrix. × × It is easy to see that, if we take the symbol σ 1 and the phase (5.6), the corresponding FIO T is (up to a constant factor) a me≡ taplectic operator. This can be seen by means of the easily verified factorization 1 (5.7) T = M U D − U T , η0 A BF C F x0 πiAx x πiCη η where UA and UC are the multiplication operators by e · and e · respectively, and DB is the dilation operator f f(B ). Each of the factors is (up to a constant factor) a metaplectic operator (see7→e.g. ·the proof of [21, Theorem 18.5.9]), so T is. We refer to [6, Proposition 2.7 (ii)] and [8, Section 7] for details about the symplectic matrix which yields such an operator. We see that the assumptions (i) and (ii) in the Introduction are clearly satisfied, whereas the hypotheses (iii) in the Introduction, (1.8) and (5.5) are equivalent to det B = 0, det A = 0 and det C = 0 respectively. In particular, the first part of Corollar6 y 5.2 generalizes6 [6, Theo6rem 4.1].

6. Some counterexamples related to the Schrodin¨ ger multiplier In this section, we study the action of the Scrodinger¨ multiplier

1 iπ 2 f − e |·| fˆ , 7−→ F iπ η 2 p q that is the multiplier with symbol e | | , on the Wiener amalgam spaces W ( L , Ls). Equivalently, we study the action of the pointwise multiplication operatorF iπ x 2 d (6.1) Af(x) = e | | f(x), x R ∈ 24 ELENA CORDERO AND FABIO NICOLA on the modulation spaces p,q . This will provide useful examples to test the Mv0,s sharpness of the thresholds arising in our results. Notice that A is the FIO with η 2 | | phase Φ(x, η) = xη + 2 and symbol σ 1. In particular, it satisﬁes (1.8). It was shown in [8, Proposition 7.1] tha≡ t A is bounded on p,q (if and) only if p = q (for a proof of the positive results see [2, 6, 8]). It is naturaMl to wonder whether for suitable negative values of s it is bounded as an operator p,q p,q . The M → Mv0,s next result deals with the optimal range of s for this to happen. Proposition 6.1. If the operator A in (6.1) is bounded as an operator p,q(Rd) p,q (Rd), for some 1 p, q , s R, then p q and s dM(1/q 1/p→). Mv0,s ≤ ≤ ∞ ∈ ≥ ≤ − − Moreover, if p = and q < , then s < d/q. ∞ ∞ − Proof. We will prove later on that p q. Let us verify now the remaining conditions. We test the estimate ≥

(6.2) Af M p,q . f M p,q k k v0,s k k πλ x 2 on the family of functions fλ(x) = e− | | , λ > 0. Applying Lemma 2.10 with a = λ, b = 0 we see that d p + (6.3) fλ M p,q Ga+ib W ( Lp,Lq) λ− 2 , as λ 0 . k k k k F → On the other hand, with the notation of Lemma 2.10, we have

p,q p q Afλ Mv Ga+ib W ( L ,Ls) k k 0,s k k F with a = λ, b = 1. We now estimate this last expression. We see that, by (2.10),

On the other hand, using polar coordinates one sees that, as µ 0+, → 1 if sq > d − x qs dx log µ if sq = d 1  µ 2 x 1 | | | d + sq | − Z ≤| |≤ µ 2 2 if sq < d. − 1 1 Secondly, assume s > 0 in (6.4). Since 1+ µ− 2 x µ− 2 x , we have | | ≥ | | d s p q 2q 2 Ga+ib W ( L ,Ls) & µ− − . k k F Hereby, we deduce, as λ 0+, → d s λ− 2q − 2 if sq > d d s 1 − (6.5) Afλ M p,q & λ− 2q − 2 log λ q if sq = d k k v0,s  | | − 1 if sq < d. − + By combining this estimate with(6.2) and (6.3) and letting λ 0 we deduce the desired conclusion, when q < . An easy modiﬁcation of the→ above arguments yields also the case p = q = .∞ Let us now prove the conditio∞ n p q. By duality, it suﬃces to prove that if the inequality ≥ d 0 0 0 0 R (6.6) Af M p ,q C f M p ,q , f ( ), k k ≤ k k v0,s ∀ ∈ S holds true for some s R, then p q. To see this, we test the estimates on the ∈ ≥ Schwartz functions fλ, 0 < λ < 1, already used before. It follows from (6.5), in particular, that

d 2q0 (6.7) Af p0,q0 & λ− , 0 < λ 1. k λkM ≤ On the other hand, by (2.10) with a = λ, b = 0,

d 2 2p0 πν x Ga+ibTxg Lp0 λ− e− | | , 0 < λ 1, k kF ≤ where ν = λ/(λ2 + λ) 1 for 0 < λ 1. Hence ≤ d 2p0 p0,q0 . − fλ M λ , 0 < λ 1. k k v0,s ≤ + This estimate, combined with (6.6) and (6.7) and letting λ 0 yields p0 q0, namely p q. → ≤ ≥ Here is a related example.

1, d Example 6.2. Let us show the unboundedness of the operator A between M ∞(R ) ,q d and M ∞ (R ), for every q < . Consider the tempered distribution δ, deﬁned by δ, ϕ = ϕ¯(0), for every ϕ ∞(Rd). Then, for a ﬁxed non-zero window g (Rd), h i ∈ S ∈ S 26 ELENA CORDERO AND FABIO NICOLA

1, 2d 1, d the STFT Vgδ(x, η) = δ, MηTxg = g( x) L ∞(R ), that is, δ M ∞(R ). Now, h i − ∈ ∈ πi 2 ,q 2d (6.8) V (Aδ)(x, η) = δ, e− |·| M T g = g( x) / L∞ (R ), g h η x i − ∈ for every q < . ∞ d This also shows the unboundedness of the operator A between M ∞(R ) and Rd Mv∞0,s ( ), for every s > 0. Indeed, using the inclusion relations

,q 1 M ∞ M ∞ , s > , v0,s ⊂ q we see that, if A were bounded between M ∞ and Mv∞0,s for some s > 0, then the ,q operator A would be bounded between M ∞ and M ∞ as well, and this is false, as shown by (6.8). Proposition 6.3. If the operator A in (6.1) is bounded as an operator p,q(Rd) p,q (Rd) for some 1 p, q , s R, then s d(1/q 1/p). MMoreover,→if Mvs,0 ≤ ≤ ∞ ∈ ≤ − − p = and q < , then s < d/q. ∞ ∞ − We need the following elementary result 1 Lemma 6.4. Let a , a , C R with 0 < a C , C− a C . Then 1 2 0 ∈ 1 ≤ 0 0 ≤ | 2| ≤ 0 2 a1 η+a2x s s e− | | η dη & x , Rd h i h i Z where the constant implicit in the notation & only depends on d, C0 and s.

2 a1 η+a2x 1 1 Proof. Let us observe that e− | | e− for η B( a x, a− ). Hence ≥ ∈ − 2 1 2 a1 η+a2x s s s s e− | | η dη & y a2x dy & a2x y −| |dy, Rd h i B(0,a−1)h − i h i B(0,a−1)h i Z Z 1 Z 1 which gives the conclusion by the assumptions on a1 and a2.

Proof of Proposition 6.3. In view of what we proved in Proposition 6.1, it suﬃces to verify that if the estimate d p,q p,q R Af v C f , f ( ), k kM s,0 ≤ k kM ∀ ∈ S πλ x 2 holds true, then (6.2) holds as well, at least for all f(x) = fλ(x) = e− | | , 0 < λ < 1. Hence we are left to prove that

p,q p,q Afλ v & Afλ v , k kM s,0 k kM 0,s or equivalently that

p q p q Ga+ib W ( Ls ,L ) & Ga+ib W ( L ,Ls), k k F k k F BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON MODULATION SPACES 27 with a = λ, b = 1 (see the notation of Lemma 2.10). To this end, observe that by (2.9) we have, for p < , ∞ p Ga+ibTxg Ls k kF 1 2 π(a+1)|x| pπ 2 2 2 p 2 2 d/4 2 2 2 2 [(a +b +a) η +2bxη] ps = ((a + 1) + b )− e− (a+1) +b e− (a+1) +b | | η dη h i Z 1 2 πa|x| pπ a(a+1)+b2η+ b x 2 p 2 2 d/4 2 (a+1)2+b2 √ 2 ps = ((a + 1) + b )− e− a(a+1)+b e− | √a(a+1)+b | η dη h i Z Since a = λ 1 and b = 1 an application of Lemma 6.4 gives ≤ πλ|x|2 1 s s p − λ +λ+1 p Ga+ibTxg Ls & e x Ga+ibTxg L x , k kF h i k kF h i where for the last estimate we used (2.10). Similarly one treats the case p = . ∞ Remark 6.5. By duality, the assumption in Theorem 6.1 could be rephrased as the boundedness of A itself between p0,q0 (Rd) p0,q0 (Rd). Hence, as a con- Mv0,−s → M sequence of Proposition 2.9 and Proposition 6.1, we see that the threshold for s2 arising in Theorem 1.2 (iii) (with s1 = 0) is essentially sharp; namely, it is sharp for p = 1, whereas when p > 1 only the case of equality remains open. Similarly, Proposition 6.3 shows that the threshold for s1 arising in Theorem 1.2 (ii) (with s2 = 0) is essentially sharp. The following result shows that the conclusion in Theorem 1.3 generally fails if q > p. Proposition 6.6. If the operator A in (6.1) is bounded as an operator p,q(Rd) q,p(Rd) for some 1 p, q , then q p. M → M ≤ ≤ ∞ ≤ Proof. We test the estimate d Af q,p . f p,q , f (R ), k kM k kM ∀ ∈ S πλ x 2 on the family of Schwartz functions fλ(x) = e− | | , λ > 0. Applying Lemma 2.10 with a = λ, b = 0 we see that

d d q (6.9) fλ M p,q Ga+ib W ( Lp,Lq) λ 2 − 2 as λ + . k k k k F → ∞ On the other hand, with the notation of Lemma 2.10 with a = λ, b = 1, we have

d d 2p 2 Afλ M q,p Ga+ib W ( Lq,Lp) λ − as λ + . k k s k k F → ∞ Hence it turns out q p. ≤ Finally we present a counterexample related to Theorem 1.3. 28 ELENA CORDERO AND FABIO NICOLA

1 iπ 2 Proposition 6.7. The Schr¨odinger multiplier f − e |·| fˆ in not bounded 7−→ F as an operator p,q(Rd) q,p(Rd) if p = q. M → M 6 Proof. It suﬃces to prove that the pointwise multiplication operator A in (6.1) is not bounded as an operator ( Lp, Lq)(Rd) ( Lq, Lp)(Rd) if p = q. We test the estimate W F → W F 6 d Af ( Lq,Lp) C f ( Lp,Lq), f (R ), k kW F ≤ k kW F ∀ ∈ S πλ x 2 on the family of Schwartz functions fλ(x) = e− | | , λ > 0. By applying Lemma 2.10 with a = 1/λ and b = 0 we obtain

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Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy 30 ELENA CORDERO AND FABIO NICOLA

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail address: [email protected] E-mail address: [email protected]