# Bilinear Localization Operators on Modulation Spaces

Authors: Nenad Teofanov
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We introduce a class of bilinear localization operators and show how to interpret them as bilinear Weyl pseudodifferential operators. Such interpretation is well known in linear case whereas in bilinear case it has not been considered so far. Then we study continuity properties of both bilinear Weyl pseudodifferential operators and bilinear localization operators which are formulated in terms of a modified version of modulation spaces.


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## Body

## 1. Introduction

Localization operators were introduced by Berezin in the study of general Hamiltonians related to quantization problem in quantum mechanics [1]. In signal analysis they are related to the localization technique proposed by Slepian-Polak-Landau; see, for example, a survey article [2]. Thereafter, a more detailed study of localization in phase space together with basic facts on localization operators and their applications in optics and signal analysis was given by Daubechies in [3]. That paper initiated further study of the topic. Daubechies studied localization operators Aaφ=Aaφ1,φ2 on L2(Rd) (see Section 2 for the definition) with Gaussian windows (1)φ1t=φ2t=π-1/4exp⁡-t22,radialsymbola∈L1Rd.Such operators are named Daubechies’ operators afterwards. The eigenfunctions of Daubechies’ operators are Hermite functions: (2)hnt=-1nπ-1/42nn!-1/2exp⁡t22exp⁡-t2n,t∈Rd,n=0,1,…,and the eigenvalues can be explicitly calculated. This is an important issue in applications (cf. [4]).For related inverse problem in the case of simply connected localization domainΩ, we refer to [5] where it is proved that if one of the eigenfunctions of Daubechies’ operator is a Hermite function, then Ω is a disc centered at the origin.An exposition of different quantization theories and connection between localization operators and Toeplitz operators is given in, for example, [6, 7]. The problem of quantization served as a motivation for the study of localization operators on Lp(G), 1≤p≤∞, where G is a locally compact group; see [8]. There one can also find a product formula and Schatten-von Neumann properties of localization operators; see also [9].Since the beginning of the XXI century, localization operators in the context of modulation spaces were studied by many authors (cf. [10–23]). See also the references given there.Our results are related to, but different from, investigations given in [15] due to the difference in definitions of bilinear localization operators; see Remark 2. Moreover, instead of standard modulation spaces Mp,q observed in [15], continuity properties are here formulated in terms of a modified version of modulation spaces denoted by Ms,tp,q; see Definition 5. Also, in contrast to [15] in this paper we do not observe multilinear version of localization operators, since we use sharp convolution estimates for modulation spaces given in [24]; see Theorems 8 and 9. These results are well suited for the study of bilinear operators, but their extension to multilinear case is a challenging problem.One of the key ingredients in our investigations is the interpretation of bilinear localization operators as bilinear Weyl pseudodifferential operators,ΨDO for short. Indeed, results on multilinear Kohn-Nirenberg ΨDOs from [10–12] served as a background for the continuity properties of multilinear Kohn-Nirenberg localization operators given in [15]. Here we consider the so-called Weyl correspondence instead and obtain continuity properties (Theorems 15 and 16) analogous to [15, Theorems 4.5 and 4.8] when restricted to bilinear operators.The paper can be summarized as follows. We first introduce bilinear localization operators, bilinear WeylΨDOs, and bilinear Wigner transform and show how they are connected; see Section 2. In particular, we prove that bilinear localization operator can be interpreted as bilinear ΨDO, Theorem 4. In Section 3 we recall the definition and basic properties of modulation spaces and introduce their convenient modified version Ms,tp,q (see Definition 5 for details). Then we recall convolution estimates from [24] which will be used in the proof of main results of the paper given in Section 4. These results are boundedness of bilinear ΨDOs on Ms,tp,q (Theorem 14) and sufficient and necessary conditions for boundedness of bilinear localization operators on Ms,tp,q (Theorems 15 and 16, resp.).Notation. The Schwartz class is denoted by S(Rd) and the space of tempered distributions by S′(Rd). We use the brackets 〈f,g〉 to denote the extension of the inner product 〈f,g〉=∫‍f(t)g(t)¯dt on L2(Rd) to any pair of dual spaces. The Fourier transform is normalized to be f^(ω)=Ff(ω)=∫‍f(t)e-2πitωdt. The involution f∗ is f∗(·)=f(-·)¯, and the convolution of f and g is given by f∗g(x)=∫‍f(x-y)g(y)dy, when the integral exists.We denote by〈·〉s the polynomial weights (3)x,ωs=1+x2+ω2s/2,x,ω∈R2d,s∈R,and 〈x〉=〈1+|x|2〉1/2, when x∈Rd.We use the notationA≲B to indicate that A≤cB for a suitable constant c>0, whereas A≍B means that c-1A≤B≤cA for some c≥1.


## 2. Bilinear Localization Operators

To define localization operators we start with the short-time Fourier transform, a time-frequency representation related to Feichtinger’s modulation spaces [25, 26].LetS(1)(Rd) be the Gelfand-Shilov type space of analytic functions given by (4)f∈S1Rd⟺fxeh·xL∞<∞,f^ωeh·ωL∞<∞,∀h>0.Iff∈S(1)(Rd), then f extends to a holomorphic function f(x+iy) in the strip {x+iy∈Cd:|y|<T} for a suitable T>0 [27, 28]. The dual space of S(1)(Rd) will be denoted by S1′(Rd).Translation and modulation operators are given by(5)Txf·=f·-x,Mωf·=e2πiω·f·x,ω∈Rd.The short-time Fourier transform (STFT in the sequel) off∈S(1)(Rd) with respect to the window g∈S1Rd∖0 is given by (6)Vgfx,ω=f,MωTxg=∫Rdftgt-x¯e-2πiωtdt.The map(f,g)↦Vgf from S(1)(Rd)⊗S(1)(Rd) to S(1)(R2d) extends uniquely to a continuous operator from S1′(Rd)⊗S1′(Rd) to S1′(R2d) by duality.Moreover, for a fixedg∈S(1)(Rd)∖0 the following characterization holds: (7)f∈S1Rd⟺Vgf∈S1R2d.We refer to [20, 21, 29–31] for the proof and more details on STFT in other spaces of Gelfand-Shilov type.The localization operatorAaφ1,φ2 with symbol a∈L2(R2d) and windows φ1,φ2∈L2(Rd) is given by (8)Aaφ1,φ2ft=∫R2dax,ωVφ1fx,ωMωTxφ2tdxdω,f∈L2Rd.Next, we define bilinear localization operators as follows.Definition 1.
Letf1,f2∈S(1)(Rd) and f→=(f1,f2). Thebilinear localization operatorAaφ withsymbola∈S1′(R4d) andwindowφ=(φ→,ϕ→)=(φ1,φ2,ϕ1,ϕ2), where φj,ϕj∈S(1)(Rd), j=1,2, is given by (9)Aaφf→t=∫R2dax,ωVφ1f1x1,ω1Vφ2f2x2,ω2Mω1Tx1ϕ1t1Mω2Tx2ϕ2t2dxdω,where x=(x1,x2), ω=(ω1,ω2), t=(t1,t2), and x1,x2,ω1,ω2,t1,andt2∈Rd.Remark 2.
LetR denote the trace mapping that assigns to each function F defined on R2d a function defined on Rd by the formula (10)R:F⟼Ft1=t2t1,t2∈Rd.Then, RAaφ is the bilinear operator given in [15, Definition 1.1].In order to give an interpretation of bilinear operators in the weak sense, we introduce the following notation. Let there be givenφ1,φ2,f1,andf2∈S(1)(Rd). Then (11)Vφ1⊗φ2f1⊗f2x,ω=∫R2df1t1f2t2Mω1Tx1φ1t1Mω2Tx2φ2t2¯dt1dt2=∫R2df1⊗f2tMω1Tx1φ1⊗Mω2Tx2φ2t¯dt,where x=(x1,x2), ω=(ω1,ω2), t=(t1,t2), and x1,x2,ω1,ω2,t1,andt2∈Rd. Thus, the weak definition of (9) is given by (12)Aaφf→,g→=aVφ1⊗φ2f1⊗f2,Vϕ1⊗ϕ2g1⊗g2=a,Vφ1⊗φ2f1⊗f2¯Vϕ1⊗ϕ2g1⊗g2,and f1,f2,g1,andg2∈S(1)(Rd). The brackets can be interpreted as suitable duality between a pair of dual spaces. Thus, Aaφ is a well-defined continuous operator from S(1)(Rd)⊗S(1)(Rd) to S1′(R4d).Next, we introduce a class of bilinear Weyl pseudodifferential operators and use the Wigner transform to provide appropriate interpretation of bilinear localization operators as bilinear Weyl pseudodifferential operators.Letσ∈S(1)(R2d). Then the Weyl pseudodifferential operator Lσ with the Weyl symbol σ can be defined as the oscillatory integral: (13)Lσfx=∬σx+y2,ωfye2πix-y·ωdydω,f∈S1Rd.This definition extends to each σ∈S1′(R2d), so that Lσ is a continuous mapping from S(1)(R2d) to S1′(R2d). If (14)Wf,gx,ω=∫fx+t2gx-t2¯e-2πiωtdt,f,g∈S1Rd,denotes the Wigner transform, also known as the cross-Wigner distribution, then the following formula holds: (15)Lσf,g=σ,Wg,f,f,g∈S1Rd,for each σ∈S1′(R2d); see, for example, [26, 32, 33].By analogy with (13) we define the bilinear Weyl pseudodifferential operator as follows: (16)Lσf→x=∫R4dσx+y2,ωf1y1f2y2e2πiIx-y·ωdydω,x∈R2d,where σ∈S1′(R4d), f1,f2∈S(1)(Rd), and f→=(f1,f2). Here I denotes the identity matrix in 2d (e.g., if d=2, then I(x-y)·ω=ω1(x1-y1)+ω2(x2-y2)).To give the interpretation of (15) in the context of bilinear ΨDOs we introduce the bilinear version of (14) as follows. Let f1,f2,g1,andg2∈S(1)(Rd), f→=(f1,f2), and g→=(g1,g2). Then thebilinear Wigner transformW(f→,g→) is given by (17)Wf→,g→x,ω=Wf1,g1x1,ω1⊗Wf2,g2x2,ω2=∫R2df1x1+t12g1x1-t12¯f2x2+t22g2x2-t22¯e-2πiIωtdt,and t=(t1,t2)∈Rd×Rd. It is easy to see that W(f→,g→)∈S(1)(R4d).Lemma 3.
Letσ∈S(1)(R4d) and f1,f2,g1,andg2∈S(1)(Rd). Then Lσ given by (16) extends to a continuous map from S(1)(Rd)⊗S(1)(Rd) to S1′(R4d) and the following formula holds: (18)Lσf→,g→=σ,Wg→,f→.Proof.
The proof follows by the straightforward calculation:(19)σ,Wg→,f→=∫R4dσx,ωWf→,g→x,ωdxdω=∫R6dσx,ωf1x1+t12g1x1-t12¯f2x2+t22g2x2-t22¯e-2πiIωtdtdxdω=∫R6dσu+v2,ωf1v1g1u1¯f2v2g2u2¯e-2πiIu-vωdudvdω=σu+v2,ωf1⊗f2ve2πiIu-vω,g1⊗g2u=Lσf→,g→,where we used W(g→,f→)=W(f→,g→)¯ and the change of variables u=x+t/2 and v=x-t/2. This extends to each σ∈S1′(R4d), since W(f→,g→)∈S(1)(R4d) when f1,f2,g1,andg2∈S(1)(Rd).The so-called Weyl connection between the set of linear localization operators and WeylΨDOs is well known; we refer to, for example, [21, 32, 34]. The proof of the following Weyl connection between the set of bilinear localization operators and corresponding bilinear Weyl ΨDOs is based on the kernel theorem for Gelfand-Shilov spaces (see, e.g., [20, 35]) and direct calculations. Since the proof is quite technical we present it in the separate Section 5. The conclusion of Theorem 4 is that, as in the linear case, the bilinear localization operators can be viewed as a subclass of the bilinear Weyl ΨDOs.Theorem 4.
Let there be givena∈S1′(R2d) and let φ→=(φ1,φ2) and ϕ→=(ϕ1,ϕ2), where φj∈S(1)(Rd), j=1,2. Then the localization operator Aaφ is the Weyl pseudodifferential operator with the Weyl symbol (20)σ=a∗Wϕ→,φ→=a∗Wϕ1,φ1⊗Wϕ2,φ2.Therefore, if f→=(f1,f2), f1,f2,∈S1′(Rd), and g→=(g1,g2), g1,g2,∈S(1)(Rd), we have (21)Aaφf→,g→=La∗Wϕ→,φ→f→,g→=La∗Wϕ1,φ1⊗Wϕ2,φ2f→,g→.


## 3. Modulation Spaces

Since we essentially use the convolution estimates for polynomially weighted modulation spaces (Theorems8 and 9), by Theorem 7(3) below it is enough to use the duality between S and S′ instead of the more general duality between S(1) and S1′. We refer to [16, 17] for investigations in the framework of subexponential and superexponential weights and leave the study of bilinear localization operators in that case for a separate contribution.Modulation spaces [25, 26] are defined through decay and integrability conditions on STFT, which makes them suitable for time-frequency analysis and for the study of localization operators in particular. Their definition is given in terms of weighted mixed-norm Lebesgue spaces.In general, a weightw(·) on Rd is a nonnegative and continuous function. By Lwp(Rd) and p∈[1,∞], we denote the weighted Lebesgue space defined by the norm (22)fLwp=fwLp=∫fxpwxpdx1/p,with the usual modification when p=∞. When w(x)=〈x〉t, t∈R, we use the notation Ltp(Rd) instead.Similarly, the weighted mixed-norm spaceLwp,q(R2d), p,q∈[1,∞], consists of (Lebesgue) measurable functions on R2d such that (23)FLwp,q=∫Rd∫RdFx,ωpwx,ωpdxq/pdω1/q<∞,where w(x,ω) is a weight on R2d.In particular, whenw(x,ω)=〈x〉t〈ω〉s, s,t∈R, we use the notation Lwp,q(R2d)=Lt,sp,q(R2d).Now, modulation spaceMs,tp,q(Rd) consists of distributions whose STFT belong to Lt,sp,q(R2d).Definition 5.
Letϕ∈S(Rd)∖0, s,t∈R, and p,q∈[1,∞]. Themodulation spaceMs,tp,q(Rd) consists of all f∈S′(Rd) such that (24)fMs,tp,q≡∫Rd∫RdVϕfx,ωxtωspdxq/pdω1/q<∞(with obvious interpretation of the integrals when p=∞ or q=∞).For the consistency, and according to (11), we denote by Ms,tp,q(R2d) the set of f→=(f1,f2), f1,f2∈S1′(Rd), such that (25)f→Ms,tp,q≡∫R2d∫R2dVφ1⊗φ2f1⊗f2x,ωxtωspdxq/pdω1/q<∞.In special cases we use the usual abbreviations:M0,0p,p=Mp, Mt,tp,p=Mtp, and so on.Remark 6.
Notice that the original definition given in [25] contains more generalsubmultiplicative weights. We restrict ourselves to w(x,ω)=〈x〉t〈ω〉s, s,t∈R, since the convolution and multiplication estimates which will be used later on are formulated in terms of weighted spaces with such polynomial weights. As already mentioned, weights of exponential type growth are used in the study of Gelfand-Shilov spaces and their duals in [16, 29–31]. We refer to [36] for a survey on the most important types of weights commonly used in time-frequency analysis.The following theorem lists some of the basic properties of modulation spaces. We refer to [25, 26] for its proof.Theorem 7.
Letp,q,pj,andqj∈[1,∞] and s,t,sj,andtj∈R, j=1,2. Then (1)
Ms,tp,q(Rd) are Banach spaces, independent of the choice of ϕ∈S(Rd)∖0;(2)
ifp1≤p2, q1≤q2, s2≤s1, and t2≤t1, then (26)SRd⊆Ms1,t1p1,q1Rd⊆Ms2,t2p2,q2Rd⊆S′Rd;(3)
⋂s,tMs,tp,q(Rd)=S(Rd) and ⋃s,tMs,tp,q(Rd)=S′(Rd).Modulation spaces include the following well-known function spaces:(a)
M2(Rd)=L2(Rd).(b)
Mt,02(Rd)=Lt2(Rd).(c)
Sobolev spaces:M0,s2(Rd)=Hs2(Rd)={f∣f^(ω)〈ω〉s∈L2(Rd)}.(d)
Shubin spaces:Ms2(Rd)=Ls2(Rd)∩Hs2(Rd)=Qs(Rd) (cf. [37]).For the results on multiplication and convolution in modulation spaces and in weighted Lebesgue spaces we first introduce theYoung functional: (27)Rp≡2-1p0-1p1-1p2,p=p0,p1,p2∈1,∞3.WhenR(p)=0, the Young inequality for convolution reads as (28)f1∗f2Lp0′≤f1Lp1f2Lp2,fj∈LpjRd,j=1,2.The following theorem is an extension of the Young inequality to the case of weighted Lebesgue spaces and modulation spaces when0≤R(p)≤1/2.Theorem 8.
Letsj,tj∈R and pj,qj∈[1,∞], j=0,1,2. Assume that 0≤R(p)≤1/2, R(q)≤1, and(29)0≤tj+tk,j,k=0,1,2,j≠k,(30)0≤t0+t1+t2-d·Rp,(31)0≤s0+s1+s2,with strict inequality in (30) when R(p)>0 and tj=d·R(p) for some j=0,1,2.
Then(f1,f2)↦f1∗f2 on C0∞(Rd) extends uniquely to a continuous map from (1)
Lt1p1(Rd)×Lt2p2(Rd) to L-t0p0′(Rd);(2)
Ms1,t1p1,q1(Rd)×Ms2,t2p2,q2(Rd) to M-s0,-t0p0′,q0′(Rd).For the proof we refer to [24]. It is based on the detailed study of an auxiliary three-linear map over carefully chosen regions in Rd; see Sections 3.1 and 3.2 in [24]. This result extends multiplication and convolution properties obtained in [38]. Moreover, the result is sharp in the following sense.Theorem 9.
Letpj,qj∈[1,∞] and sj,tj∈R, j=0,1,2. Assume that at least one of the following statements holds true: (1)
The map(f1,f2)↦f1∗f2 on C0∞(Rd) is continuously extendable to a map from Lt1p1(Rd)×Lt2p2(Rd) to L-t0p0′(Rd).(2)
The map(f1,f2)↦f1∗f2 on C0∞(Rd) is continuously extendable to a map from Ms1,t1p1,q1(Rd)×Ms2,t2p2,q2(Rd) to M-s0,-t0p0′,q0′(Rd);
Then (29) and (30) hold true.


## 4. Continuity Properties

We start estimates of the modulation space norm of the cross-Wigner distribution (cf. [21]). We refer to [39, Theorem 4.2] for more refined estimates, and note that in [40] the sufficient conditions for the continuity of the cross-Wigner distribution on modulation spaces are proved to be necessary too (in the unweighted case). Proposition 10 coincides with certain sufficient conditions from [40, Theorem 1.1] when restricted to R(p)=0, t0=-t1, and t2=|t0|.Proposition 10.
Let the assumptions of Theorem8 hold. If f,g∈S(Rd), then the map (f,g)↦W(g,f) where W is the cross-Wigner distribution given by (14) extends to sesquilinear continuous map from Mt2p2(Rd)×Mt1p1(Rd) to M-t0,01,p0′(R2d).Proof.
We give a short version of the proof for the sake of completeness and refer to [21] for details. Let f,g∈S(Rd). Then (32)Wg,fM-t0,01,p0′≍VWψ1,ψ2Wg,fz,ζζ-t0L1,p0′,where z,ζ∈R2d, since, by Theorem 7(1), modulation spaces are independent of the choice of the window function (from S(R2d)∖0).
From(33)Wf,gx,ω=2de4πix·ωVg∗¯f2x,2ω,f,g∈S1Rd,see [26, Lemma 4.3.1], and from the proof of [26, Lemma 14.5.1 (b)] it follows that (34)VWψ1,ψ2Wg,fz,ζ=e-2πiz2ζ2Vψ1f¯z1+ζ22,z2-ζ12Vψ2gz1-ζ22,z2+ζ12.Hence the norms in (32) are equivalent to (35)∫R2dVψ1f∗Vψ2g¯∗ζ2,-ζ1ζ2,-ζ1-t0p0′dζ1/p0′,where the convolution is obtained from the integration over z and after the change of variables (z1,z2)↦(z1-ζ2/2,z2+ζ1/2); see also [13, Proposition 2.5]. Therefore (36)Wg,fM-t0,01,p0′≲Vψ1f∗Vψ2g∗L-t0p0′≲Vψ1fLt1p1Vψ2gLt2p2,where the last estimate follows from Theorem 8(1).We refer to [10] for the multilinear version of (32), which in turn gives the multilinear version of Corollary 11 (in unweighted case). In fact, from the inspection of the proof of Proposition 10, the definition of W(f→,g→) given by (17), and (25) we conclude the following.Corollary 11.
Let the assumptions of Theorem8 hold. If f1,f2,g1,andg2∈S(Rd),f→=(f1,f2), and g→=(g1,g2), then the map (f→,g→)↦W(g→,f→), where W is the cross-Wigner distribution given by (17) which extends to a continuous map from Mt2p2(Rd)×Mt1p1(Rd) to M-t0,01,p0′(R2d).Remark 12.
Proposition10, when restricted to s=-t0=t1=t2≥0, p=p0′=p2∈[1,∞], and p1=1 gives (37)Wφ2,φ1Ms,01,p≲φ1Ms1φ2Msp,that is, [13, Proposition 2.5] (with a slightly different notation).
For a certain choice of the parameters (in particular whenp0′≠p2) we obtain estimates sharper than (37). In particular, if p1=1, p2=∞, and p≥2, we obtain (38)Wφ2,φ1M-t0,01,p≲φ1Mt11φ2Mt2∞,with 0<t0+t1+t2-d·R(p) when p=∞ and tj=d·R(p) for some j=0,1,2.Next we prove an extension of [26, Theorem 14.5.2] to the bilinear Weyl ΨDOs. Recall, if σ∈M∞,1(R2d) is the Weyl symbol of Lσ, then [26, Theorem 14.5.2] says that Lσ is bounded on Mp,q(Rd), 1≤p,q≤∞. This result has a long history starting from the Calderon-Vaillancourt theorem on boundedness of the pseudodifferential operators with smooth and bounded symbols on L2(Rd), [41]. It is generalized by Sjöstrand in [42] where M∞,1 is used as appropriate symbol class. Sjöstrand’s results were thereafter extended in [22, 26, 39, 43–45]. Moreover, we refer to [10–12] for the multilinear Kohn-Nirenberg ΨDOs.To deal with duality whenpq=∞ we observe that, by a slight modification of [10, Lemma 2.2], the following is true.Lemma 13.
LetL0(R4d) denote the space of bounded, measurable functions on R4d which vanish at infinity and put (39)M0,qR2d=f∈M∞,qR2d∣Vφ1⊗φ2f1⊗f2∈L0R4d,1≤q<∞,Mp,0R2d=f∈Mp,∞R2d∣Vφ1⊗φ2f1⊗f2∈L0R4d,1≤p<∞,M0,0R2d=f∈M∞,∞R2d∣Vφ1⊗φ2f1⊗f2∈L0R4d,equipped with the norms of M∞,q, Mp,∞, and M∞,∞, respectively. Then, (a)
M0,q is M∞,q-closure of S in M∞,q, hence is a closed subspace of M∞,q. Likewise for Mp,0 and M0,0.(b)
The following duality results hold for1≤p,q<∞: M0,q′=M1,q′, Mp,0′=Mp′,1, and M0,0′=M1,1.From now on, we will use these duality relations in the casesp=∞ and/or q=∞ without further explanations.Theorem 14.
Letσ∈M∞,1(R4d) and let Lσ be given by (16). The operator Lσ is bounded from Mp,q(R2d) to Mp,q(R2d), 1≤p,q≤∞, with a uniform estimate Lσop≤σM∞,1 for the operator norm.Proof.
Note that the integrals here below are well defined and that the order of integration is irrelevant. We have(40)Lσf→,g→≤VΦσL∞,1R8dVΦWg→,f→L1,∞R8d=σM∞,1R4dsupζ∈R4d∫R4dVΦWg→,f→z,ζdz,where Φ∈S(R2d)∖0 is a window function. By slight modifications of the proof of [26, Theorem 14.5.2] we obtain the following estimate: (41)∫R4dVΦWg→,f→z,ζdz≤f→Mp,qR2dg→Mp′,q′R2d,uniformly in ζ∈R4d. Therefore, (42)Lσf→,g→≤σM∞,1R4df→Mp,qR2dg→Mp′,q′R2d,for all g→∈Mp′,q′(R2d). It follows that Lσf→∈Mp,q(R2d) and (43)Lσf→Mp,qR2d≤σM∞,1R4df→Mp,qR2d.Hence the operator Lσ is bounded on Mp,q(R2d) and Lσop≤σM∞,1 as claimed.Finally, we use the relation between the Weyl pseudodifferential operators and the localization operators (Theorem4) and the convolution estimates for modulation spaces (Theorem 8) to obtain continuity results for Aaφ for different choices of windows and symbols.Theorem 15.
Let the assumptions of Theorem8 hold and let φj,ϕj∈S′(Rd)j=1,2, and φ→=(φ1,φ2),ϕ→=(ϕ1,ϕ2). If φ→∈Mt1p1(R2d), ϕ→∈Mt2p2(R2d), and a∈Mu,v∞,r(R4d), where 1≤r≤p0, u≥t0, and v≥dR(p) with v>dR(p) when R(p)>0, then Aaφ is bounded on Mp,q(R2d) for all 1≤p,q≤∞, and the operator norm satisfies the uniform estimate (44)Aaφop≲aMu,v∞,rφ→Mt1p1ϕ→Mt2p2.Proof.
Letφ→∈Mt1p1(R2d) and ϕ→∈Mt2p2(R2d). From Corollary 11 it follows that (45)Wϕ→,φ→=Wϕ1,φ1⊗Wϕ2,φ2∈M-t0,01,p0′R4d.
Now, the calculation ofa∗W(ϕ→,φ→) from the proof of Theorem 4 together with Theorem 8(2) implies that (46)a∗Wϕ→,φ→∈Mp~,1R4d,p~≥2,if the involved parameters fulfill the conditions of the theorem.
On the one hand, for the Lebesgue parameters it is easy to see thatp~≥2 is equivalent to R(p)=R(p,∞,1)∈[0,1/2] and that 1≤r≤p0 is equivalent to Rq=R∞,r,p0′≤1. On the other hand, by the choice of the weight parameters u and v it follows that (47)a∗Wϕ1,φ1⊗Wϕ2,φ2∈Mp~,1R4d,p~≥2.
In particular, ifp~=∞ then a∗(W(ϕ→,φ→))∈M∞,1(R4d).
Putσ=a∗W(ϕ→,φ→). Then Theorems 14, 4, and 8 imply that (48)Aaφop=Lσop≤σM∞,1≤aMu,v∞,rφ→Mt1p1ϕ→Mt2p2,and the Theorem is proved.We remark that a modification of Theorem15 can be obtained by using [26, Theorem 14.5.6] instead. That result allows symbols σ from weighted modulation spaces. We leave for the reader to check how to change the conditions on weight parameters in Theorem 15 in that case.We finish with a necessary condition. The proof of Theorem16 is a slight modification of the proof [13, Theorem 4.3], and we leave it for the reader. Here below ·S∞ denotes the norm of a bounded operator.Theorem 16.
Let the assumptions of Theorem8 hold and let a∈S′(R4d). If there exists a constant C=C(a)>0 depending only on the symbol a and (49)AaφS∞≤Cφ→Mt1p1ϕ→Mt2p2for all φ→=(φ1,φ2) and ϕ→=(ϕ1,ϕ2), with φj,ϕj∈S(Rd)j=1,2, then a∈Mt0,0∞,1(R2d).


## 5. The Proof of Theorem4

Note first that the integrals here below are absolutely convergent and that changing the order of integration is allowed. Moreover, certain oscillatory integrals are meaningful inS1′(Rd) in a suitable interpretation. For example, if δ denotes the Dirac distribution, then the Fourier inversion formula in the sense of distributions gives ∫‍e2πixωdω=δ(x), wherefrom ∬‍ϕ(x)e2πi(x-y)ωdxdω=ϕ(y), when ϕ∈S(1)(Rd).We first rewrite (12) in the form of a kernel operator. (50)Aaφf→,g→=∬R4dax,ω∫Rdf1t1Mω1Tx1φ1¯t1dt1∫Rdf2t2Mω2Tx2φ2¯t2dt2·∫Rdg1¯s1Mω1Tx1ϕ1s1ds1∫Rdg2¯s2Mω2Tx2ϕ2s2ds2dxdω=∬R2d∫R2df1t1f2t2g1¯s1g2¯s2∬R4dax,ω·Mω1Tx1φ1¯t1Mω2Tx2φ2¯t2Mω1Tx1ϕ1s1Mω2Tx2ϕ2s2dxdωdtds=k,f1⊗f2¯⊗g1⊗g2,where the kernel k=k(t,s), t=(t1,t2), s=(s1,s2), and t1,t2,s1,s2∈Rd, is given by (51)kt,s=kt1,t2,s1,s2=∫R4dax,ωMω1Tx1φ1¯t1Mω2Tx2φ2¯t2Mω1Tx1ϕ1s1Mω2Tx2ϕ2s2dxdω=∫R4dax,ωMω1Tx1φ1⊗Mω2Tx2φ2¯tMω1Tx1ϕ1⊗Mω2Tx2ϕ2sdxdω.Next, we calculate the convolutiona∗(W(ϕ1,φ1)⊗W(ϕ2,φ2)). We will use W(f2,f1)=W(f1,f2)¯ and the following covariance property of the Wigner transform: (52)WTx1Mω1ϕ1,Tx1Mω1φ1p1,q1=∫RdTx1Mω1ϕ1p1+t2Tx1Mω1φ1p-t2¯e-2πiq1tdt=∫Rde2πip1+t/2ω1ϕ1p1+t2-x1e-2πip1-t/2ω1φ1p1-t2-x1¯e-2πiq1tdt=∫Rdϕ1p1-x1+t2φ1p1-x1-t2¯e-2πiq1-ω1tdt=Wϕ1,φ1p1-x1,q1-ω1.Letp=(p1,p2), q=(q1,q2), and p1,p2,q1,andq2∈Rd. Then (53)a∗Wϕ1,φ1⊗Wϕ2,φ2p,q=∫R4dax,ωWϕ1,φ1p1-x1,q1-ω1Wϕ2,φ2p2-x2,q2-ω2dxdω=∫R2dax,ωWTx1Mω1ϕ1,Tx1Mω1φ1p1,q1·WTx2Mω2ϕ2,Tx2Mω2φ2p2,q2dxdω=∫R4dax,ω∬R2dTx1Mω1ϕ1p1+t12Tx1Mω1φ1¯p1-t12e-2πiq1t1·Tx2Mω2ϕ2p2+t22Tx2Mω2φ2¯p2-t22e-2πiq2t2dt1dt2dxdω=∫R4dax,ω∬RdMω1Tx1ϕ1p1+t12Mω1Tx1φ1¯p1-t12e-2πiq1t1·Mω2Tx2ϕ2p2+t22Mω2Tx2φ2¯p2-t22e-2πiq2t2dt1dt2dxdω,where we have used the commutation relation TxMω=e-2πix·ωMωTx.Now,(54)La∗Wϕ→,φ→f→,g→=La∗Wϕ1,φ1⊗Wϕ2,φ2f→,g→=a∗Wϕ1,φ1⊗Wϕ2,φ2,Wg→,f→=∫R4dax,ω∫R4d∫R4dMω1Tx1ϕ1p1+t12Mω1Tx1φ1¯p1-t12·Mω2Tx2ϕ2p2+t22Mω2Tx2φ2¯p2-t22e-2πiIt-sq·g1¯p1+s12g2¯p2+s22f1p1-s12f2p2-s22dtdsdpdqdxdω=∫R4dax,ω∫R2d∫R4dMω1Tx1ϕ1p1+t12Mω1Tx1φ1¯p1-t12·Mω2Tx2ϕ2p2+t22Mω2Tx2φ2¯p2-t22δt1-s1δt2-s2·g1¯p1+s12g2¯p2+s22f1p1-s12f2p2-s22dtdsdpdxdω=∫R4dax,ω∫R2d∫R2dMω1Tx1ϕ1p1+t12Mω1Tx1φ1¯p1-t12·Mω2Tx2ϕ2p2+t22Mω2Tx2φ2¯p2-t22g1¯p1+t12g2¯p2+t22·f1p1-t12f2p2-t22dtdpdxdω,where the interpretation of oscillatory integrals is used as mentioned in the beginning of this Section. Finally, the change of variables (55)p1+t12=v1,p1-t12=u1,p2+t22=v2,p2-t22=u2gives (56)La∗Wϕ1,φ1⊗Wϕ2,φ2f→,g→=∫R4d∫R4dax,ωMω1Tx1φ1⊗Mω2Tx2φ2¯uMω1Tx1ϕ1⊗Mω2Tx2ϕ2v·f1⊗f2ug1⊗g2¯vdudvdxdω=k,f1⊗f2¯⊗g1⊗g2,where the kernel k is given by (51). The theorem follows from the uniqueness of the kernel.



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*Source: 7560870-2018-01-15.xml*