Almost diagonalization of $\tau $-pseudodifferential operators with ...

arXiv:1802.10314v1 [math.FA] 28 Feb 2018

ALMOST DIAGONALIZATION OF τ -PSEUDODIFFERENTIAL OPERATORS WITH SYMBOLS IN WIENER AMALGAM AND MODULATION SPACES ELENA CORDERO, FABIO NICOLA, AND S. IVAN TRAPASSO Abstract. In this paper we focus on the almost-diagonalization properties of τ pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. Such spaces are nowadays called modulation spaces as well. A particular example is provided by the Sj¨ ostrand class, for which Gr¨ochenig [Gr¨o06] exhibits the almost diagonalization of Weyl operators. We shall show that such result can be extended to any τ -pseudodifferential operator, for τ ∈ [0, 1]. This is not surprising, since the mapping that goes from a Weyl symbol to a τ -symbol is bounded in the Sj¨ ostrand class. What is new and quite striking is the almost diagonalization for τ -operators with symbols in weighted Wiener amalgam spaces. In this case the diagonalization depends on the parameter τ . In particular, we have an almost diagonalization for τ ∈ (0, 1) whereas the cases τ = 0 or τ = 1 yield only to weaker results. As a consequence, we infer boundedness, algebra and Wiener properties for τ -pseudodifferential operators on Wiener amalgam and modulation spaces.

1. Introduction It is widely known that the techniques of Time-frequency Analysis revealed to be very fruitful when dealing with diverse problems in quite different fields, from signal processing to harmonic analysis, but also from PDE’s to quantum mechanics (see, e.g., [BDDO10, CN08, DdGP14, dG17, RWZ16, STW11, STW11]). Besides the manifold achievements, Gabor analysis is a fascinating discipline in itself: the study of related functional-analytic problems is an inexhausted source of challenges and it happens to cast new light on established results, eventually. As an example, Gr¨ochenig employed these methods in his work [Gr¨o06], almost diagonalizing Weyl operators with symbols in the Sj¨ostrand class via Gabor frames. This result has provided new insights on a number of well-known outcomes previously obtained by 2010 Mathematics Subject Classification. 47G30,35S05,42B35,81S30. Key words and phrases. Time-frequency analysis, τ -Wigner distribution, τ -pseudodifferential operators, almost diagonalization, modulation spaces, Wiener amalgam spaces. 1

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ELENA CORDERO, FABIO NICOLA, AND S. IVAN TRAPASSO

Sj¨ostrand within the realm of classical analysis [Sj¨o94, Sj¨o95] and, more importantly, has lead to far-reaching generalizations. In this work we extend part of the results obtained in [Gr¨o06] and derive a number of relevant consequences on the boundedness of pseudodifferential operators. In particular, our study has been carried out within the unifying framework offered by τ -representations and τ -pseudodifferential operators. τ -pseudodifferential operators can be either defined as a quantization rule or by means of the related time-frequency representation (cf. [BDDO10, BCDDO10, BDDOC10]). In the following section we give a more detailed account, but here we simply recall the latter. For τ ∈ [0, 1], the (cross-)τ -Wigner distribution is given by Z (1) Wτ (f, g)(x, ω) = e−2πiyζ f (x + τ y)g(x − (1 − τ )y) dy, f, g ∈ S(Rd ), Rd

whereas the τ -pseudodifferential operator is defined by

(2)

hOpτ (a)f, gi = ha, Wτ (g, f )i,

f, g ∈ S(Rd ).

For τ = 1/2 we recapture the Weyl operator studied in [Gr¨o06], if τ = 0 the operator is called the Kohn-Nirenberg operator OpKN , if τ = 1 the related operator is named operator “with right symbol”. For what concerns the symbol classes, we consider Banach spaces allowing a suitable measure of the time-frequency decay of distributions, i.e. modulation spaces. These function spaces were introduced by Feichtinger in the ’80s (cf. [Fei81, Fei83]) and demonstrated to be the optimal setting for a large class of problems related to harmonic analysis and PDE’s (cf. [dG11, dGGR16, WHHG11]). To fix notation, we write a point in phase space (in time-frequency space) as z = (x, ω) ∈ R2d , and the corresponding phase-space shift (time-frequency shift) acting on a function or distribution as (3)

π(z)f (t) = e2πiωt f (t − x),

t ∈ Rd .

When considering symbol functions, we shall work with time-frequency shifts π(z, ζ), with z, ζ ∈ R2d (the variables are doubled). Consider a Schwartz function g ∈ S(R2d ) \ {0}. We define the modulation space M ∞,1 (R2d ) (or Sj¨ostrand class) as the space of tempered distributions σ ∈ S ′ (R2d ) such that Z sup |hσ, π(z, ζ)gi|dζ < ∞. R2d z∈R2d

For the properties of such space and its weighted versions we refer to the next section. Here the main focus is on symbols in the so-called Wiener amalgam space W (F L∞ , L1 )(R2d ) = F M ∞,1 (R2d ), where F is the Fourier transform. Heuristically, symbols in W (F L∞ , L1 )(R2d ) display locally a regularity of the type F L∞ (R2d ) and globally decay as functions in L1 (R2d ). For instance, the δ distribution (in S ′ (R2d ))

ALMOST DIAGONALIZATION OF τ -PSEUDODIFFERENTIAL OPERATORS

3

belongs to W (F L∞ , L1 )(R2d ). Nowadays this class can be referred to as modulation space as well (although in a generalized sense, as suggested by Feichtinger in his retrospective [Fei06]), see also [MP13]. The central issue of our work is the approximate diagonalization of τ -pseudodifferential operators. This is a well-known problem, studied in several contexts of harmonic analysis: classical references are [Mey90, RT98]. See also, for the more general framework of Fourier integral operators, [GL08, CGNR13] and references therein. In somewhat heuristic terms, the choice of certain type of symbols assures that these operators preserve the time-frequency localization, since their kernel with respect to continuous or discrete time-frequency shifts satisfies a convenient decay condition. To be precise, thanks to a covariance property for the τ -Wigner distribution under the action of time-frequency shifts, we are able to prove the following claim, which extends [Gr¨o06, Thereom 3.2] proved for τ = 21 , i.e., for the Weyl quantization (see the subsequent Theorem 4.1 for the weighted version). First, we recall the definition of a Gabor frame. Let Λ = AZ2d , with A ∈ GL(2d, R), be a lattice of the time-frequency plane. The set of time-frequency shifts G(ϕ, Λ) = {π(λ)ϕ : λ ∈ Λ} for a non-zero ϕ ∈ L2 (Rd ) (the so-called window function) is named a Gabor system. The set G(ϕ, Λ) is a Gabor frame, if there exist constants A, B > 0 such that X |hf, π(λ)ϕi|2 ≤ Bkf k22 , ∀f ∈ L2 (Rd ). (4) Akf k22 ≤ λ∈Λ

Our result is the following:


Theorem. Fix a
non-zero window ϕ ∈ S(Rd ) Rd such that G (ϕ, Λ) is a Gabor frame for L2 Rd . For any τ ∈ [0, 1], the following properties are equivalent:
2d (i) σ ∈ M ∞,1 R
.
′ 2d (ii) σ ∈ S R and there exists a function Hτ ∈ L1 R2d such that |hOpτ (σ) π (z) ϕ, π (w) ϕi| ≤ Hτ (w − z) , ∀w, z ∈ R2d .
(iii) σ ∈ S ′ R2d and there exists a sequence hτ ∈ ℓ1 (Λ) such that |hOpτ (σ) π (µ) ϕ, π (λ) ϕi| ≤ hτ (λ − µ) ,

∀λ, µ ∈ Λ.

This kind of control is a characterizing property of symbols in those spaces, in the following sense.
d Corollary. Under the hypotheses of the previous Theorem, assume that T : S R →
′ d S R is continuous and satisfies one of the following conditions: (i) |hT π (z) ϕ, π (w) ϕi| ≤ H (w − z) , ∀w, z ∈ R2d for some H ∈ L1 . (ii) |hT π (µ) ϕ, π (λ) ϕi| ≤ h (λ − µ) , ∀λ, µ ∈ Λ for some h ∈ ℓ1 .
Therefore, T = Opτ (σ) for some symbol σ ∈ M ∞,1 R2d and some τ ∈ [0, 1].

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ELENA CORDERO, FABIO NICOLA, AND S. IVAN TRAPASSO

A parallel pattern can be traced for symbols in W (F L∞ , L1 )(R2d ) although it is somewhat more involved because of the peculiar way τ comes across. This is not surprising if we notice that Weyl operators with symbols in L1 (R2d ) ⊂ W (F L∞ , L1 )(R2d ) are bounded on L2 (Rd ) (cf. [WHHG11]) but neither KohnNireberg operators (τ = 0) nor operators “with right symbol”(τ = 1) are, cf. [Bou95, Bou97a, Bou97b], see also [CDNT18]. For τ ∈ (0, 1), we introduce the symplectic matrix  τ  Id×d 0d×d 1−τ (5) Uτ = − ∈ Sp (2d, R) 1−τ 0d×d Id×d τ (see the next section for details). Our result is as follows:
Theorem. Fix non-zero window ϕ ∈ S Rd . For any τ ∈ (0, 1), the following properties are equivalent:
(i) σ ∈ W (F L
∞ , L1 ) R2d .
(ii) σ ∈ S ′ R2d and there exists a function Hτ ∈ L1 R2d such that |hOpτ (σ) π (z) ϕ, π (w) ϕi| ≤ Hτ (w − Uτ z) ,

∀w, z ∈ R2d .

Another difference from the Sj¨ostrand symbol class concerns the discrete almost diagonalization, recovered only if τ = 12 , see the subsequent Corollary 4.4. This kind of control on the kernels paves the way for proving relevant properties of τ -operators. Here we mainly focus on boundedness results in different settings, where Wiener amalgam and modulation spaces play the role of symbol class and domain/target. In particular, we underline the connection with the theory of Fourier integral operators (FIOs): we have that, under suitable conditions, τ -pseudodifferential operators admit a representation as type I FIOs, thus inheriting a number of features (including boundedness, algebra and Wiener properties) established in previous papers by two of the authors - see for instance [CNR09, CGNR13, CGNR14]. In addition, we point out that even when this overlapping does not hold, the insightful proof strategies developed there can still be exploited. Apart from the technical assumptions on weights which will be specified later, our results can be collected in Table 1. It is important to emphasize again the singular behaviour showed by the endpoints τ = 0 and τ = 1, which is essentially due to weaker versions of the previous theorems, see Propositions 5.8 and 5.9 in the sequel. We finally remark that, as said earlier, the results on approximate diagonalization allow a potentially large number of directions to be investigated, even in a more abstract framework (cf. for instance [GR08]). We wish to explore some of these routes in subsequent papers.

ALMOST DIAGONALIZATION OF τ -PSEUDODIFFERENTIAL OPERATORS

Symbol - R2d

τ




(0, 1) σ ∈ W (F L∞ , L1 ) σ ∈ M ∞,1

[0, 1]

5


Boundedness - Rd , 1 ≤ p, q ≤ ∞ Opτ (σ) : M p,q → M p,q

Opτ (σ) : W (F Lp , Lq ) → W (F Lp , Lq )

(0, 1) σ ∈ W (F L∞ , L1 )

Opτ (σ) : W (F Lp , Lq ) → W (F Lp , Lq )

0

σ ∈ W (F L∞ , L1 )

OpKN (σ) : M 1,∞ → M 1,∞

1

σ ∈ W (F L∞ , L1 ) Op1 (σ) : W (F L1 , L∞ ) → W (F L1 , L∞ ) Table 1. Boundedness results proved in the paper.

2. Preliminaries Notation. We define t2 = t · t, for t ∈ Rd , and xy = x · y is the scalar product on Rd . The Schwartz class is denoted by S(Rd ), the space of tempered distributions by S ′ (Rd ). We use the brackets hf, gi to denote the extension to S ′ (Rd ) × S(Rd ) of the R inner product hf, gi = f (t)g(t)dt on L2 (Rd ). The Fourier transform of a function f on Rd is normalized as Z F f (ξ) = e−2πixξ f (x) dx. Rd

The modulation Mω and translation Tx operators are defined as Mω f (t) = e2πitω f (t) ,

Tx f (t) = f (t − x) .

The following result exhibits composition and commutation properties of the timefrequency shifts. Lemma 2.1. For x, x′ , ω, ω ′ ∈ Rd , we have (i) Commutation relations for TF-shifts: (6)

Tx Mω = e−2πixω Mω Tx .

(ii) Compositions of TF-shifts: (7)

′

π (x, ω) π (x′ , ω ′ ) = e−2πixω π (x + x′ , ω + ω ′ ) .

More explicitly, ′

(Mω Tx ) (Mω′ Tx′ ) = e−2πixω Mω+ω′ Tx+x′ . Let f ∈ S ′ (Rd ). We define the short-time Fourier transform of f as Z (8) Vg f (x, ω) = hf, π(x, ω)gi = F (f Tx g)(ω) = f (y) g(y − x) e−2πiyω dy. Rd

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ELENA CORDERO, FABIO NICOLA, AND S. IVAN TRAPASSO

Recall the fundamental property of time-frequency analysis (9)

Vg f (x, ω) = e−2πixω Vgˆfˆ (ω, −x) .

Denote by J the canonical symplectic matrix in R2d :   0d×d Id×d J= ∈ Sp (2d, R) , −Id×d 0d×d where the symplectic group Sp (2d, R) is defined by  Sp(2d, R) = M ∈ GL(2d, R) : M ⊤ JM = J .

Observe that, for z = (z1 , z2 ) ∈ R2d , we have Jz = J (z1 , z2 ) = (z2 , −z1 ) , J −1 z = J −1 (z1 , z2 ) = (−z2 , z1 ) = −Jz, and J 2 = −I2d×2d . Lemma 2.2. For any τ ∈ (0, 1), consider the matrix ! q 1−τ 0d×d Id×d τ . (10) Aτ = p τ 0d×d − 1−τ Id×d

The following properties hold:

(1) Aτ ∈ Sp (d, R); in particular, A1/2 = J. −1 (2) A⊤ τ = −A1−τ , Aτ = −Aτ . ⊤ −1 (3) A1−τ Aτ = Aτ Aτ = I2d×2d − Bτ , where   1 I 0 d×d d×d 1−τ . (11) Bτ = 1 I 0d×d τ d×d p p (4) τ (1 − τ ) (Aτ + A1−τ ) = τ (1 − τ )Bτ Aτ = J. Proof. These properties follow by easy computations.



Although Aτ is well defined only for τ ∈ (0, 1), notice that the matrix   0d×d (1 − τ ) Id×d ′ (12) Aτ = −τ Id×d 0d×d is well defined for any τ ∈ [0, 1], though non-invertible and non-symplectic when τ ∈ {0, 1}. In particular: p (1) For τ ∈ (0, 1), A′τ = τ (1 − τ )Aτ . (2) (A′τ )⊤ = −A′1−τ . (3) A′τ + A′1−τ = J. In the following we shall use the short-time Fourier transform of a τ -Wigner distribution, which is contained in [CDNT18].

ALMOST DIAGONALIZATION OF τ -PSEUDODIFFERENTIAL OPERATORS

7



Lemma 2.3. Consider τ ∈ [0, 1], ϕ1 , ϕ2 ∈ S Rd , f, g ∈ S Rd and set Φτ =
Wτ (ϕ1 , ϕ2 ) ∈ S R2d . Then,
VΦτ Wτ (g, f ) (z, ζ) = e−2πiz2 ζ2 Vϕ1 g z − A′1−τ ζ Vϕ2 f (z + A′τ ζ) ,

where z = (z1 , z2 ) , ζ = (ζ1 , ζ2 ) ∈ R2d and the matrix A′τ is defined in (12). In particular,
|VΦτ Wτ (g, f )| = Vϕ1 g z − A′1−τ ζ · |Vϕ2 f (z + A′τ ζ)| .

2.1. Function Spaces. We first need to introduce the weight functions under our consideration. Weight Functions. Following [Gr¨o06, Sec. 2.3], we will work with so-called admissible weight functions, i.e. by v we always denote a non-negative continuous function on R2d such that (1) v (0) = 1 and v is even in each coordinate: v (±z1 , . . . , ±z2d ) = v (z1 , . . . , z2d ) . (2) v is submultiplicative, that is v (w + z) ≤ v (w) v (z)

∀w, z ∈ R2d .

(3) v satisfies the Gelfand-Raikov-Shilov (GRS) condition: 1

lim v (nz) n = 1

n→∞

∀z ∈ R2d .

b

Every weight of the form v (z) = ea|z| (1 + |z|)s logr (e + |z|), with real parameters a, r, s ≥ 0 and 0 ≤ b < 1, is admissible. Weights of particular interest are provided by
s (13) vs (z) = hzis = 1 + |z|2 2 , z ∈ R2d , s ≥ 0. Observe that, for s ≥ 0, the weight function vs is equivalent to the submultiplicative weight (1 + | · |)s , that is, there exist C1 , C2 > 0 such that C1 vs (z) ≤ (1 + |z|)s ≤ C2 vs (z),

z ∈ R2d .

A positive, even weight function m on R2d is called v-moderate if m(z1 + z2 ) ≤ Cv(z1 )m(z2 ) for all z1 , z2 ∈ R2d . We simply write Mv for the class of v-moderate weights. Here and elsewhere the conjugate exponent p′ of p ∈ [1, ∞] is defined by 1/p + 1/p′ = 1. Moreover, in order to remain in the framework of tempered distributions, in what follows we shall consider weight functions m on Rd or R2d satisfying the following condition (14)

m(z) ≥ 1,

∀z ∈ Rd

or m(z) & hzi−N .

for a suitable N ∈ N. The same holds for weights on R2d .

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ELENA CORDERO, FABIO NICOLA, AND S. IVAN TRAPASSO

Modulation Spaces. Given a non-zero window g ∈ S(Rd ), a v-moderate weight p,q function m on R2d satisfying (14), and 1 ≤ p, q ≤ ∞, the modulation space Mm (Rd ) 2d consists of all tempered distributions f ∈ S ′ (Rd ) such that Vg f ∈ Lp,q m (R ) (weighted p,q mixed-norm space). The norm on Mm is q/p !1/q Z Z kf kMmp,q = kVg f kLp,q = |Vg f (x, ω)|p m(x, ω)p dx dω . m Rd

Rd

p p,p If p = q, we write Mm instead of Mm , and if m(z) ≡ 1 on R2d , then we write M p,q p p,q p,p and M for Mm and Mm . p,q Then Mm (Rd ) is a Banach space whose definition is independent of the choice of the window g. The class of admissible windows can be extended to Mv1 , as stated below ([Gr¨o06, Thm. 11.3.7]).

Theorem 2.1. Let m be a v-moderate weight satisfying (14) and g ∈ Mv1 \ {0}, p,q then kVg f kLp,q is an equivalent norm for Mm (Rd ). m p,q Hence, given any g ∈ Mv1 (Rd ) and f ∈ Mm we have the norm equivalence

kf kMmp,q ≍ kVg f kLp,q . m

(15)

The previous equivalence will be heavily exploited in this work. We recall the inversion formula for the STFT (see [Gr¨o01, Proposition 11.3.2]): assume g ∈ p,q Mv1 (Rd ) \ {0}, f ∈ Mm (Rd ), with m satisfying (14) then Z 1 Vg f (z)π(z)g dz , (16) f= kgk22 R2d

p,q and the equality holds in Mm (Rd ). The adjoint operator of Vg , defined by Z ∗ Vg F (t) = F (z)π(z)gdz , R2d

2d p,q d maps the Banach space Lp,q m (R ) into Mm (R ). In particular, if F = Vg f the inversion formula (16) reads

(17)

IdMmp,q =

1 V ∗ Vg . kgk22 g

Wiener Amalgam Spaces. Fix g ∈ S(Rd ) \ {0}. Consider even weigh functions u, w on Rd satisfying (14). Then the Wiener amalgam space W (F Lpu , Lqw )(Rd ) is the space of distributions f ∈ S ′ (Rd ) such that !1/q q/p Z Z kf kW (F Lpu ,Lqw )(Rd ) := |Vg f (x, ω)|pup (ω) dω w q (x)dx 0 such that (46)

kOpτ (σ)kMmp,q ≤ Cτ kσkM ∞,1

1⊗v◦J −1

.

We can now state the boundedness result for τ -pseudodifferential operators on Wiener amalgam spaces.
Theorem 5.6. Consider m = m1 ⊗ m2 ∈ Mv R2d satisfying (14). For any τ ∈


∞,1 R2d , the operator Opτ (σ) is bounded on W F Lpm1 , Lqm2 Rd [0, 1] and σ ∈ M1⊗v with kOpτ (σ)kW (F Lpm ,Lqm ) ≤ Cτ kσkM ∞,1 , 1

for a suitable Cτ > 0.

2

1⊗v

24

ELENA CORDERO, FABIO NICOLA, AND S. IVAN TRAPASSO

Proof. Consider the following commutative diagram: p,q Mm Rd

O

Op1−τ (σJ )




/

p,q Mm Rd

F −1

W F Lpm1 , Lqm2




F

Rd


Opτ (σ) /






W F Lpm1 , Lqm2

Rd









∞,1 ∞,1 2d by Lemma 5.2. The operator Indeed, since σ ∈ M1⊗v R2d , σJ ∈ M1⊗(v◦J −1 ) R p,q d Op1−τ (σJ ) is bounded on Mm (R ) by virtue of Theorem 5.5 with τ ′ = 1 − τ ∈ [0, 1] and the thesis follows at once thanks to the Lemma 5.1.  The same argument (with obvious modifications) allow to extend the boundedness result for τ -pseudodifferential operators contained in Theorem 5.1 to Wiener amalgam spaces for symbols in suitable Wiener amalgam spaces.
Theorem 5.7. Consider m = m1 ⊗
m2 ∈
Mv R2d satisfying (14). For any τ ∈ (0, 1) and σ ∈ W F L∞ , L1v◦Bτ ◦J −1 R2d , the operator Opτ σ is bounded from  
d

p q p q W F Lm1 , Lm2 R to W F Lm ◦ U −1 , Lm ◦ U −1 Rd , 1 ≤ p, q ≤ ∞, where 1 ( 1−τ ) 2 ( 1−τ ) 1 2 −1 U1−τ




(x) = − 1

τ x, 1−τ

−1 U1−τ




2

(x) = −

1−τ x, τ

x ∈ Rd .

Proof. Consider the following commutative diagram: p,q Mm Rd

O




Op1−τ (σJ )

/

p,q Mm◦U Rd −1 1−τ

F −1


d
Opτ (σ) p q / W F Lm1 , Lm2 R W F Lpm

 −1 )1 (U1−τ

1◦

F




, Lqm ◦(U −1 ) 2 1−τ 2



(Rd )





Indeed, since σ ∈ W F L∞ , L1v◦Bτ ◦J −1 R2d , σJ ∈ W F L∞ , L1v◦Bτ R2d by Lemma 5.2. The operator Op1−τ σJ is bounded by virtue of Theorem 5.1 with τ ′ = 1 − τ ∈ (0, 1) and the thesis follows at once thanks to the Lemma 5.1.  The cases τ = 0 and τ = 1. Theorem 4.5 allows us to obtain some boundedness results for τ -pseudodifferential operators with τ = 0 or τ = 1, having symbols in Wiener amalgam spaces. Proposition 5.8. Assume σ ∈ W (F L∞ , L1 )(Rd ). Then the Kohn-Nirenberg operator OpKN (σ) (τ = 0) is bounded on M 1,∞ (Rd ).

ALMOST DIAGONALIZATION OF τ -PSEUDODIFFERENTIAL OPERATORS

25


Proof. Consider H0 (T0 (w, z)) with H0 ∈ L1 R2d as in Theorem 4.5. The integral operator TH0 with kernel H0 (T0 (w, z)) = H0 (w1 , z2 ) can be written as Z Z Z TH0 F (w) = H0 ◦ T0 (w, z) F (z) dz = H (w1 , z2 ) F (z1 , z2 ) dz1 dz2 . R2d

Rd

Rd



It is immediate to notice that TH0 : L1,∞ R2d
→ L1,∞ R2d is a bounded operator. Then, for a fixed non-zero window g ∈ S Rd , we have that
T = Vg∗ TH0 Vg : M 1,∞ Rd → M 1,∞ (Rd ) is a bounded operator. The claim then follows.



Proposition 5.9. Assume σ ∈ W (F L∞ , L1 )(Rd ). Then the operator “with right symbol” Op1 (σ) (τ = 1) is bounded on W (F L1 , L∞ )(Rd ).
Proof. Again, we apply Theorem 4.5 and consider H1 (T1 (w, z)) with H1 ∈ L1 R2d . The integral operator TH with kernel H1 (T1 (w, z)) = H1 (z1 , w2 ) can be written as Z Z Z TH1 F (w) = H1 ◦ T1 (w, z) F (z) dz = H1 (z1 , w2 ) F (z1 , z2 ) dz1 dz2 . R2d

Rd

Rd


2d
1 1 It is immediate to notice that TH1 : L∞ R → L
∞ R is a bounded z1 Lz2 w1 Lw2 d operator. Then, for a fixed non-zero window g ∈ S R , we have that



T = Vg∗ TH1 Vg : W F L1 , L∞ Rd → W F L1 , L∞ Rd



2d

is a bounded operator. This concludes the proof.



The consequences of the almost diagonalization of τ -operators are manifold. We notice that the results of this section can be extended by interpolation to symbols in W (F Lp , Lq )(R2d ), following the pattern of [CN16]. This subject with be further investigated in a subsequent paper. 6. Algebra and Wiener properties The connection with the theory Fourier integral operators established in the previous section allows to investigate further properties of τ -operators. First of all, notice that for any τ1 , τ2 ∈ (0, 1), ! τ1 τ2 I 0 d×d d×d (1−τ1 )(1−τ2 ) . Uτ1 Uτ2 = (1−τ1 )(1−τ2 ) Id×d 0d×d τ1 τ2 In particular, Uτ U1−τ = U1−τ Uτ = I2d×2d . Therefore, composition properties of operators in the class F IO (A, vs ) (see [CGNR14, Theorems 3.4] and Theorem 4.1) yield the following result.

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ELENA CORDERO, FABIO NICOLA, AND S. IVAN TRAPASSO


Theorem 6.1 (Algebra property). For any a, b ∈ W F L∞ , L1vs and τ ∈ (0, 1), ∞,1 there exists a symbol c ∈ M1⊗v such that s Opτ (a) Op1−τ (b) = Op1/2 (c) . Remark 6.2. On the other hand, for any choice of τ1 , τ2 ∈ (0, 1), there is no τ ∈ (0, 1) such that Uτ1 Uτ2 = Uτ . This immediately implies that there is no τ -quantization
rule such that composition of τ -operators with symbols in W F L∞ , L1vs has symbol
∞,1 and in the same class. In a similar fashion, given a ∈ W F L∞ , L1vs , b ∈ M1⊗v s τ, τ0 ∈ (0, 1), we have Opτ0 (b) Opτ (a) = Opτ (c1 ) , Opτ (a) Opτ0 (b) = Opτ (c2 ) ,
∞ 1 for some c1 , c2 ∈ W F L , Lvs . This means that, for fixed quantization rules τ, τ0 ,
∞,1 under the the amalgam space W F L∞ , L1vs is a bimodule over the algebra M1⊗v s laws

∞,1 ∞ 1 ∞ 1 : (b, a) 7→ c1 , → W F L , L × W F L , L M1⊗v v v s s s

∞,1 ∞ 1 ∞ 1 W F L , Lvs × M1⊗vs → W F L , Lvs : (a, b) 7→ c2 , with c1 and c2 as before.

In conclusion, we provide a result whose proof easily follows by [CGNR14, Theorem 3.7] and by noticing that Uτ−1 = U1−τ for any τ ∈ (0, 1).
Theorem 6.3 (Wiener property).
For any τ ∈ (0, 1) and a ∈ W F L∞ , L1vs such that Opτ (a) is invertible on L2 Rd , we have Opτ (a)−1 = Op1−τ (b)


for some b ∈ W F L∞ , L1vs .

7. Acknowledgments

The first and second authors were partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). References [BCDDO10]

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` di Torino, Dipartimento di Matematica, Dipartimento di Matematica, Universita via Carlo Alberto 10, 10123 Torino, Italy E-mail address: [email protected] Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail address: [email protected] Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail address: [email protected]