Characterization of smooth symbol classes by Gabor matrix decay
aa r X i v : . [ m a t h . F A ] F e b CHARACTERIZATION OF SMOOTH SYMBOL CLASSES BYGABOR MATRIX DECAY
FEDERICO BASTIANONI AND ELENA CORDERO
Abstract.
For m ∈ R we introduce the symbol classes S m , m ∈ R , consistingof smooth functions σ on R d such that | ∂ α σ ( z ) | ≤ C α (1 + | z | ) m/ , z ∈ R d ,and we show that can be characterized by an intersection of different types ofmodulation spaces. In the case m = 0 we recapture the H¨ormander class S , that can be obtained by intersection of suitable Besov spaces as well. Suchspaces contain the Shubin classes Γ mρ , 0 < ρ ≤
1, and can be viewed as theirlimit case ρ = 0. We exhibit almost diagonalization properties for the Gabormatrix of τ -pseudodifferential operators with symbols in such classes, extendingthe characterization proved by Gr¨ochenig and Rzeszotnik in [22]. Finally, wecompute the Gabor matrix of a Born-Jordan operator, which allows to provenew boundedness results for such operators. Introduction and results
Modulation spaces were originally introduced by Feichtinger [16] in 1983 and haverevealed to be very useful in many different frameworks, which include harmonicanalysis, quantum mechanics, pseudodifferential and Fourier integral operators,partial differential equations (we refer the reader to Section 2 for their definitionsand main properties).Several authors have studied inclusion relations of such spaces with other classicalfunction spaces such as Besov, Triebel-Lizorkin Gelfand-Shilov spaces [23, 31, 35,40]. In particular, when they are considered as symbol classes for pseudodifferentialor Fourier integral operators, their relationship with classical symbol spaces suchas the H¨ormander classes or the Shubin-Sobolev spaces has been investigated inmany contributions (see e.g., [4, 10, 25, 36] and the references therein).In 1994 Sj¨ostrand [29] introduced the first symbol class via time-frequency con-centration on the phase-space, the Sj¨ostrand class, which later revealed to be atype of modulation space. This rough symbol class have been inspired manyworks on pseudodifferential operators with symbols in modulation spaces (see, e.g.,
Mathematics Subject Classification.
Key words and phrases.
Time-frequency analysis, modulation spaces, Gabor matrix, pseudo-differential operators, Gabor frames. [2, 3, 10, 20, 21, 32, 33, 26, 34, 35] and the book [10]). The contributions are somany that it is not possible to cite them all.In [30] Sj¨ostrand continued his study on pseudodifferential operators with roughsymbols and he also considered the symbol class object of our study. Namely, for m ∈ R , let us define(1) S m ( R d ) = { σ ∈ C ∞ ( R d ) : | ∂ α σ ( z ) | ≤ C α h z i m , α ∈ N d , z ∈ R d } . Notice that this is a special instance of the class S ( w ) introduced in [30, Formula(3 . m = 0 we recapture the standard H¨ormander class S , ( R d ): pseudodiffe-rential operators with these symbols are an algebra which is closed under inversion.This claim was originally proved by Beals in [1] and later recaptured by Gr¨ochenigand Rzeszotnik in [22], using time-frequency analysis; key tool was the almostdiagonalization property of the related Gabor matrix.We continue this spirit of investigation and present a characterization of pseu-dodifferential operators with symbols in S m ( R d ) in terms of the decay propertiesof the related Gabor matrix. Let us introduce the main features of this work.For τ ∈ [0 , τ -Wigner distribution is the time-frequency represen-tation defined by(2) W τ ( f, g )( x, ω ) = Z R d e − πiyω f ( x + τ y ) g ( x − (1 − τ ) y ) dy, f, g ∈ S ( R d ) . Given any tempered distribution σ ∈ S ′ ( R d ), the τ -pseudodifferential operatorOp τ ( σ ) can be introduced weakly as(3) h Op τ ( σ ) f, g i = h σ, W τ ( g, f ) i , f, g ∈ S ( R d ) . The Weyl form Op W ( σ ) of a pseudodifferential operator can be recaptured when τ = 1 /
2, the Kohn-Nirenberg case Op KN ( σ ) corresponds to τ = 0.Given z = ( x, ω ) ∈ R d , we define the related time-frequency shift acting on afunction or distribution f on R d as(4) π ( z ) f ( t ) = e πiωt f ( t − x ) , t ∈ R d . Let us recall the definition of a Gabor frame. Given a lattice Λ = A Z d , with A ∈ GL (2 d, R ), and a non-zero window function g ∈ L ( R d ), we define the Gaborsystem : G ( g, Λ) = { π ( λ ) g : λ ∈ Λ } . The Gabor system G ( g, Λ) is called a Gabor frame, if there exist constants
A, B > A k f k ≤ X λ ∈ Λ |h f, π ( λ ) g i| ≤ B k f k , ∀ f ∈ L ( R d ) . Fix g ∈ S ( R d ) \ { } . The Gabor matrix of a linear continuous operator T from S ( R d ) to S ′ ( R d ) is defined to be(6) h T π ( z ) g, π ( u ) g i , z, u ∈ R d . This Gabor matrix can be viewed as the kernel of an integral operator, cf. Section2 for details.For τ ∈ [0 , T τ ( z, u ) = ((1 − τ ) z + τ u , τ z + (1 − τ ) u ) , z = ( z , z ) , u = ( u , u ) ∈ R d . We possess all the instruments for the characterization of S m ( R d ): Theorem 1.1.
Consider g ∈ S ( R d ) \ { } and a lattice Λ such that G ( g, Λ) is aGabor frame for L (cid:0) R d (cid:1) . Fix m ∈ R . For any τ ∈ [0 , , the following propertiesare equivalent: ( i ) σ ∈ S m (cid:0) R d (cid:1) . ( ii ) σ ∈ S ′ (cid:0) R d (cid:1) and for every s ≥ there exists a constant C s > such that (8) |h Op τ ( σ ) π ( z ) g, π ( u ) g i| ≤ C s hT τ ( z, u ) i m h u − z i s , ∀ u, z ∈ R d . ( iii ) σ ∈ S ′ (cid:0) R d (cid:1) and for every s ≥ there exists a constant C s > such that (9) |h Op τ ( σ ) π ( µ ) g, π ( λ ) g i| ≤ C s hT τ ( µ, λ ) i m h λ − µ i s , ∀ λ, µ ∈ Λ . Let us stress that the positive constants C s appearing in (8) and (9) are inde-pendent of τ . For the H¨ormander class S ( R d ) = S , ( R d ), the Gabor matrixcharacterization for Weyl operators was shown by Gr¨ochenig and Rzeszotnik in[22, Theorem 6.2] (see also [27]).The central role in the proof of the result above is the characterization of the class S m ( R d ) by an intersection of weighted modulation spaces (in particular, weightedSj¨ostrand classes): S m ( R d ) = \ s ≥ M ∞h·i − m ⊗h·i s ( R d ) = \ s ≥ M ∞ , h·i − m ⊗h·i s ( R d ) , cf. Lemma 2.2 and Remark 2.5 below (see also Lemma 2.4 for the intersection ofquasi-Banach modulation spaces). FEDERICO BASTIANONI AND ELENA CORDERO
For the special case m = 0, the H¨ormander class S ( R d ) = S , ( R d ) can alsobe represented as the intersection of Besov spaces and H¨older-Zygmund classes: S , ( R d ) = \ s ≥ C s ( R d ) = \ s ≥ B ∞ , s ( R d ) = \ s ≥ M ∞ ⊗h·i s ( R d ) = \ s ≥ M ∞ , ⊗h·i s ( R d ) , cf. Lemma 2.3, which extends the characterization in [22].Observe that S m contains the Shubin classes Γ mρ , 0 < ρ ≤
1, defined as [28]Γ mρ ( R d ) = { σ ∈ C ∞ ( R d ) : | ∂ α σ ( z ) | ≤ C α h z i m − ρ | α | , α ∈ N d , z ∈ R d } , and can be viewed as their limit case ρ = 0. The Shubin classes enjoy a symboliccalculus very useful when dealing with the corresponding pseudodifferential opera-tors. This is not the case of S m ( R d ). Hence, the characterization in Theorem 1.1might be an instrument to infer boundedness, composition, inversion properties ofthe corresponding operators in suitable function spaces, such as the modulationones. Subsection 3.1 contains an application in terms of boundedness results of τ -operators.As a by product, Theorem 1.1 allows to compute the Gabor matrix decay ofa Born-Jordan operator. We present some continuity properties of the latter onweighted modulation spaces, extending the work [7].This study paves the way to other possible investigations. For instance, whenthe symbol σ on R d satisfies a Geverey-type regularity of order s > | ∂ α σ ( z ) | . M ( z ) C | α | ( α !) s , α ∈ N d , z ∈ R d , with M any possible v -moderate weight (see Section 2 for its definition). Thesesymbols were applied in [12] to investigate the sparsity of the Gabor-matrix rep-resentation of Fourier integral operators. In this case we conjecture that the rightmodulation spaces to be considered are of the type M ∞ M ⊗ e − ǫ |·| /s ( R d ).Eventually, one might extend the characterization exhibited in Theorem 1.1 toFourier integral operators of Schr¨odinger-type with symbols in S m and suitablephases as in [9]. This will be the object of a further work.The paper is organized as follows. In Section 2 we present the function spacesobject of our study. In particular, we focus on modulation spaces and present theproperties needed for our results. We then prove the characterization of the classes S m ( R d ) and in particular of the H¨ormander classes S , ( R d ). Section 3 is devotedto the study of the Gabor matrix for τ -operators and Born-Jordan operators. Asan application, boundedness results on modulation spaces are exhibited.2. Function spaces and preliminaries
In this manuscript ֒ → denotes the continuous embeddings of function spaces.Recall that the conjugate exponent p ′ of p ∈ [1 , ∞ ] is defined by 1 /p + 1 /p ′ = 1. We denote by v a continuous, positive, submultiplicative weight function on R d ,i.e., v ( z + z ) ≤ v ( z ) v ( z ), for all z , z ∈ R d . We say that w ∈ M v ( R d ) if w is apositive, continuous weight function on R d v -moderate : w ( z + z ) ≤ Cv ( z ) w ( z )for all z , z ∈ R d (or for all z , z ∈ Z d ). We will mainly work with polynomialweights of the type(11) v s ( z ) = h z i s = (1 + | z | ) s/ , s ∈ R , z ∈ R d (or Z d ) , where | z | stands for the Euclidean norm of z .Observe that, for s < v s is v | s | -moderate. Moreover, we limit to weights w withat most polynomial growth, that is there exist C > s > w ( z ) ≤ C h z i s , z ∈ R d . We shall work mostly with weights on R d or Z d ; we define ( w ⊗ w )( x, ω ) := w ( x ) w ( ω ), for w , w weights on R d . Spaces of sequences.
For 0 < p ≤ ∞ , w ∈ M v ( Z d ), the space ℓ pw ( Z d ) consistsof all sequences a = ( a k ) k ∈ Z d for which the (quasi-)norm k a k ℓ pw = X k ∈ Z d | a k | p w ( k ) p ! p (with obvious modification for p = ∞ ) is finite.Their main properties can be summarized as follows [18, 19]:(i) Inclusion relations : If 0 < p ≤ p ≤ ∞ , then ℓ p w ( Z d ) ֒ → ℓ p w ( Z d ), for anypositive weight function w on Z d .(ii) Young’s convolution inequality : Consider w ∈ M v ( Z d ), 0 < p, q, r ≤ ∞ with(13) 1 p + 1 q = 1 + 1 r , for 1 ≤ r ≤ ∞ and(14) p = q = r, for 0 < r < . Then for all a ∈ ℓ pw ( Z d ) and b ∈ ℓ qv ( Z d ), we have a ∗ b ∈ ℓ rw ( Z d ), with(15) k a ∗ b k ℓ rw ≤ C k a k ℓ pw k b k ℓ qv , where C is independent of p, q, r , a and b . If w ≡ v ≡
1, then C = 1.(iii) H¨older’s inequality : For any positive weight function w on Z d , 0 < p, q, r ≤∞ , with 1 /p + 1 /q = 1 /r ,(16) ℓ pw ( Z d ) · ℓ q /w ( Z d ) ֒ → ℓ r ( Z d ) . FEDERICO BASTIANONI AND ELENA CORDERO
The so-called translation and modulation operators are defined by T x g ( y ) = g ( y − x ) and M ω g ( y ) = e πiωy g ( y ), respectively. Let S ( R d ) be the Schwartz classand consider g ∈ S ( R d ) a non-zero window function. The the short-time Fouriertransform (STFT) V g f of a function/tempered distribution f in S ′ ( R d ) with respectto the the window g is defined by V g f ( x, ω ) = h f, M ω T x g i = Z e − πiωy f ( y ) g ( y − x ) dy, (i.e., the Fourier transform F applied to f T x g ). Modulation Spaces.
For 1 ≤ p, q ≤ ∞ such spaces were introduced by H.Feichtinger in [16], then extended to 0 < p, q ≤ ∞ by Y.V. Galperin and S.Samarah in [19]. Their main properties and applications are now available inseveral textbooks, see for instance [10]. Definition 2.1.
Fix a non-zero window g ∈ S ( R d ) , a weight w ∈ M v ( R d ) and < p, q ≤ ∞ . The modulation space M p,qw ( R d ) consists of all tempered distributions f ∈ S ′ ( R d ) such that the (quasi-)norm (17) k f k M p,qw = k V g f k L p,qw = Z R d (cid:18)Z R d | V g f ( x, ω ) | p w ( x, ω ) p dx (cid:19) qp dω ! q (obvious changes with p = ∞ or q = ∞ ) is finite. They are quasi-Banach spaces (Banach spaces whenever 1 ≤ p, q ≤ ∞ ), whosenorm does not depend on the window g , in the sense that different non-zero win-dow functions in S ( R d ) yield equivalent norms. Moreover, if 1 ≤ p, q ≤ ∞ , thewindow class S ( R d ) can be extended to the modulation space M , v ( R d ) (so-calledFeichtinger algebra).To be short, we write M pw ( R d ) in place of M p,pw ( R d ) and M p,q ( R d ) if w ≡ g ∈ M v ( R d ) \ { } , f ∈ M p,qw ( R d ), with m ∈ M v ( R d ), then(18) f = 1 k g k Z R d V g f ( z ) π ( z ) g dz , and the equality holds in M p,qm ( R d ). The adjoint operator of V g , defined by V ∗ g F ( t ) = Z R d F ( z ) π ( z ) gdz , maps the mixed-norm space L p,qw ( R d ) into M p,qw ( R d ). In particular, if F = V g f theinversion formula (18) can be rephrased as(19) Id M p,qw = 1 k g k V ∗ g V g . We need to introduce an alternative definition of modulation spaces we shall usein the sequel. For k ∈ Z d , we denote by Q k the unit closed cube centred at k . Thefamily { Q k } k ∈ Z d is a covering of R d . We define | ξ | ∞ := max i =1 ,...,d | ξ i | . Considernow a smooth function ρ : R d → [0 ,
1] satisfying ρ ( ξ ) = 1 for | ξ | ∞ ≤ / ρ ( ξ ) = 0 for | ξ | ∞ ≥ /
4. Define(20) ρ k ( ξ ) = T k ρ ( ξ ) = ρ ( ξ − k ) , k ∈ Z d , that is, ρ k is the translation of ρ at k . By the assumption on ρ , we infer that ρ k ( ξ ) = 1 for ξ ∈ Q k and X k ∈ Z d ρ k ( ξ ) ≥ , ∀ ξ ∈ R d . Denote by(21) σ k ( ξ ) = ρ k ( ξ ) P l ∈ Z d ρ l ( ξ ) , ξ ∈ R d , k ∈ Z d . Observe that σ k ( ξ ) = σ ( ξ − k ) ∈ D ( R d ) and the sequence { σ k } k ∈ Z d is a smoothpartition of unity X k ∈ Z d σ k ( ξ ) = 1 , ∀ ξ ∈ R d . For k ∈ Z d , we define the frequency-uniform decomposition operator by(22) (cid:3) k := F − σ k F . The previous operators allow to introduce an alternative (quasi-)norm on theweighted modulation spaces M p,qh ⊗ w ( R d ), as follows. Proposition 2.2.
For < p, q ≤ ∞ , h, w ∈ M v ( R d ) have (23) k f k M p,qh ⊗ w ( R d ) ≍ X k ∈ Z d k (cid:3) k f k qL ph w ( k ) q ! q , f ∈ S ′ ( R d ) , with obvious modification for q = ∞ .Proof. The case p, q ≥ < p < < q < < p ≤
1, we consider (cid:3) k f = F − σ k F f = F − σ k T ξ ¯ˆ φ F f, for ξ ∈ Q k , since T ξ ¯ˆ φ = 1 in supp σ k for ξ ∈ Q k . Using Young’s inequality for distributionscompactly supported in the frequencies (see [24, Lemma 2.6], which holds also for L ph , 0 < p ≤
1, with h being v -moderate), for ξ ∈ Q k , we obtain k (cid:3) k f k L ph . kF − σ k k L pv kF − T ξ ¯ˆ φ F f k L ph . kF − T ξ ¯ˆ φ F f k L ph . FEDERICO BASTIANONI AND ELENA CORDERO
The rest of the proof is analogous to the Banach case and we leave the details tothe interested reader.An useful embedding is contained in what follows.
Proposition 2.3.
Given < p , p , q , q ≤ ∞ , with m, s , s in R , one has (24) M p ,q h·i m ⊗h·i s ( R d ) ֒ → M p ,q h·i m ⊗h·i s ( R d ) if and only if (25) p ≤ p and (26) q ≤ q , s ≥ s or q > q , s d + 1 q > s d + 1 q . Proof.
The Banach case when m = 0 was originally shown by H. Feichtinger in[16]. We use similar arguments as in that proof. The discrete modulation normdefined in (23) is given by k f k M p,q h·i m ⊗h·i s ≍ X k ∈ Z d k (cid:3) k f k qL p h·i m h k i sq ! q . The necessity of (25) follows from the fact that F L p is locally contained in F L p if and only if p ≤ p (with strict inclusion if p < p ), cf. [5, 17, 24, 38]. The setof conditions in (26) in turn describes the inclusions between weighted ℓ q spaces: ℓ q h·i s ⊂ ℓ q h·i s if and only if the indices’ relations in (26) are satisfied, cf. for instance[23, Lemma 2.10]. This concludes the proof.We also recall the following inclusion relations, see e.g. [10, Theorem 2.4.17] or[19, Theorem 3.4]: If p ≤ p , q ≤ q and w . w , then(27) M p ,q w ( R d ) ֒ → M p ,q w ( R d ) . Corollary 2.4.
For < q < , d ∈ N + , m, s, r ∈ R , r > s + d (1 − q ) /q , we havethe following continuous embeddings: (28) M ∞ ,q h·i m ⊗h·i r ( R d ) ֒ → M ∞ , h·i m ⊗h·i r ( R d ) ֒ → M ∞ ,q h·i m ⊗h·i s ( R d ) . Proof.
The first embedding is a straightforward application of the inclusion rela-tions in (27). The second one follows by the embedding in Proposition 2.3.
Besov Spaces.
The Besov spaces are denoted by B p,qs ( R d ), 1 ≤ p, q ≤ ∞ , s ∈ R , and defined as follows. Suppose that ψ , ψ ∈ S ( R d ) satisfy supp ψ ⊂ { ω ∈ R d : | ω | ≤ } , supp ψ ⊂ { ω ∈ R d : 1 / ≤ | ω | ≤ } and ψ ( ω ) + P ∞ j =1 ψ (2 − j ω ) = 1 for every ω ∈ R d . Set ψ j ( ω ) := ψ (2 − j ω ), ω ∈ R d . Then the Besov space B p,qs ( R d )consists of all tempered distributions f ∈ S ′ ( R d ) such that the norm(29) k f k B p,qs = ∞ X j =0 jsq kF − ( ψ j F f ) k qp ! /q < ∞ (with usual modifications when q = ∞ ). Besov spaces are generalizations of bothH¨older-Zygmund and Sobolev spaces, see e.g. [38]. Precisely, we recapture theSobolev spaces when p = q = 2, s ∈ R : B , s ( R d ) = H s ( R d ). For s > B ∞ , ∞ s ( R d ) = C s ( R d ), the H¨older-Zygmund classes, whose definition is as follows.For s >
0, we can write s = n + ǫ , with n ∈ N and ǫ <
1. Then C s ( R d ) is the spaceof functions f ∈ C n ( R d ) such that for each multi-index α ∈ N d , with | α | = n , thederivative ∂ α f satisfies the H¨older condition | ∂ α f ( x ) − ∂ α f ( y ) | ≤ K | x − y | ǫ , for asuitable K > B ∞ , ∞ s ( R d ) ֒ → M ∞ ⊗h·i s ( R d ) ֒ → B ∞ s − d ( R d ) , s ∈ R , (31) B ∞ , s + d ( R d ) ֒ → M ∞ , ⊗h·i s ( R d ) ֒ → B ∞ , s ( R d ) , s ∈ R . Gabor analysis of τ -pseudodifferential operators. For any fixed m ∈ R ,the class S m ( R d ) in (1) is a Fr´echet space when endowed with the sequence ofnorms {| · | N,m } N ∈ N ,(32) | σ | N,m := sup | α |≤ N sup z ∈ R d | ∂ α σ ( z ) |h z i − m , N ∈ N . For n ∈ N , m ∈ R \ { } , we define by C nm ( R d ) the space of functions having n derivatives and satisfying (32) for N = n , whereas C n ( R d ) is the space of functionswith n bounded derivatives. Clearly we have the equalities S m ( R d ) = \ n ≥ C nm ( R d ) , m ∈ R \ { } , S ( R d ) = \ n ≥ C n ( R d ) . A characterization of the class S ( R d ) = S , ( R d ) with modulation spaces wasannounced by Toft in [37, Remark 3.1] and proved in [22, Lemma 6.1]. Lemma 2.1.
We have the equalities (33) \ n ≥ C n ( R d ) = \ s ≥ M ∞ ⊗h·i s ( R d ) = \ s ≥ M ∞ , ⊗h·i s ( R d ) . Hence S ( R d ) = T s ≥ M ∞ ⊗h·i s ( R d ) = T s ≥ M ∞ , ⊗h·i s ( R d ) . In what follows we extend the previous outcome to all the classes S m ( R d ), m ∈ R . Lemma 2.2.
For m ∈ R , we have the equalities of Fr´echet spaces (34) \ n ≥ C nm ( R d ) = \ n ≥ M ∞h·i − m ⊗h·i n ( R d ) = \ n ≥ M ∞ , h·i − m ⊗h·i n ( R d ) (for m = 0 replace C n ( R d ) by C n ( R d ) ); in the sense that the sequences of norms (35) {| · | n,m } n ∈ N , {k · k M ∞h·i− m ⊗h·i n } n ∈ N , {k · k M ∞ , h·i− m ⊗h·i n } n ∈ N are equivalent, hence they define the same Fr´echet space. In particular, (36) S m ( R d ) = \ n ≥ M ∞h·i − m ⊗h·i n ( R d ) = \ n ≥ M ∞ , h·i − m ⊗h·i n ( R d ) and it is a Fr´echet space with the equivalent sequences of norms in (35) .Proof. Let us show that the sequences of norms {|·| n,m } n ∈ N in (32), {k·k M ∞h·i− m ⊗h·i n } n ∈ N and {k · k M ∞ , h·i− m ⊗h·i n } n ∈ N are equivalent. From now on g will denote a fixed non-zerowindow function in S ( R d ). Step 1.
We show the equivalence of {| · | n,m } n ∈ N and {k · k M ∞h·i− m ⊗h·i n } n ∈ N . Wedraw on the proof of [22, Lemma 6.1]. For f ∈ C nm ( R d ) ( C n ( R d ) if m = 0) and anymulti-index α ∈ N d with | α | ≤ n , we consider the function ∂ α ( f T x ¯ g ). Taking itsFourier transform we get(37) F ( ∂ α ( f T x ¯ g ))( ω ) = (2 πiω ) α F ( f T x ¯ g )( ω ) = (2 πiω ) α V g f ( x, ω ) . In what follows we use the boundedness of F : L ( R d ) → C ( R d ), Peetre’s inequality h x i − m ≤ − m h x − t i | m | h t i − m , and Leibniz’ formula: h x i − m kF ( ∂ α ( f T x ¯ g )) k ∞ ≤ h x i − m k ∂ α ( f T x ¯ g ) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h x i − m X β ≤ α (cid:18) αβ (cid:19) ∂ β f T x ∂ α − β ¯ g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ − m X β ≤ α (cid:18) αβ (cid:19) k ( ∂ β f ) h·i − m k ∞ k ( ∂ α − β ¯ g ) h·i | m | k ≤ − m sup | β |≤ n k ( ∂ β f ) h·i − m k ∞ M α max β ≤ α (cid:18) αβ (cid:19) k ( ∂ α − β ¯ g ) h·i | m | k = C α,g,m | f | n,m , where C g,α,m = 2 − m M α max β ≤ α (cid:0) αβ (cid:1) k ( ∂ α − β ¯ g ) h·i | m | k with M α = { β ∈ N d , β ≤ α } .The estimate above and formula (37) yield(38) sup x ∈ R d | V g f ( x, ω ) |h x i − m ≤ C g,α,m | f | n,m | ω α | − , | ω | 6 = 0 , ∀| α | ≤ n. Now if f ∈ T n ≥ C nm ( R d ) then for every α ∈ N d there exists C = C α > h ω i n ≤ P | α |≤ n c α | ω α | for suitable c α ≥
0, weobtain sup x,ω ∈ R d | V g f ( x, ω ) |h x i − m h ω i n ≤ C | f | n,m , ∀ n ≥ C = C ( n, m ) > k f k M ∞h·i− m ⊗h·i n ≤ C ( n, m ) | f | n,m , for every n ≥ m ∈ R . Step 2.
Consider f ∈ T n ≥ M ∞h·i − m ⊗h·i n ( R d ). For any n ∈ N , we consider l ∈ N such that l > n + d and estimate k f k M ∞ , h·i− m ⊗h·i n = Z R d sup x ∈ R d | V g f ( x, ω ) |h x i − m h ω i n dω ≤ k f k M ∞h·i− m ⊗h·i l Z R d h ω i n − l dω ≤ C k f k M ∞h·i− m ⊗h·i l , where 0 < C = R R d h ω i n − l dω < ∞ , because n − l < − d . We have k f k M ∞h·i− m ⊗h·i l < ∞ since by assumption f ∈ T n ≥ M ∞h·i − m ⊗h·i n ( R d ). This gives that, for every m ∈ R , n ∈ N , there exists l ∈ N such that the following norm-estimate holds k f k M ∞ , h·i− m ⊗h·i n ≤ C k f k M ∞h·i− m ⊗h·i l . Step 3.
Consider f ∈ T n ≥ M ∞ , h·i − m ⊗h·i n ( R d ). Here we take the window g ∈ S ( R d )such that g ( t ) = 1 , for every t ∈ B := { t ∈ R d : | t | ≤ } . By Leibniz’ formula ∂ α ( f T x ¯ g ) = X β ≤ α (cid:18) αβ (cid:19) ∂ β f ∂ α − β T x ¯ g = ∂ α f T x ¯ g + X β<α (cid:18) αβ (cid:19) ∂ β f ∂ α − β T x ¯ g. If t ∈ x + B then the sum over β < α vanishes and(40) ∂ α ( f T x ¯ g )( t ) = ∂ α f T x ¯ g ( t ) = ∂ α f T x χ B ( t ) , where χ B denotes the characteristic function of the set B .We recall that for any s ∈ R , for any x ∈ R d and t ∈ x + B we have C − s h x i s ≤ h t i s ≤ C s h x i s , where C s := 2 s max y ∈ B h y i | s | . The previous inequalities, the equalities in (40) andPeetre’s inequality yield, for t ∈ x + B , | ∂ α f ( t ) |h t i − m = h t i − m | ∂ α f ( t ) T x χ B ( t ) | = h t i − m | ∂ α f ( t ) T x ¯ g ( t ) | = h t i − m | ∂ α ( f ( t ) T x ¯ g ( t )) | ≤ C − m h x i − m | ∂ α ( f ( t ) T x ¯ g ( t )) |≤ C − m h x i − m k ∂ α ( f T x ¯ g ) k ∞ . Now we estimate h x i − m k ∂ α ( f T x ¯ g ) k ∞ . The inversion formula for the Fourier trans-form and the continuity of F − : L ( R d ) → C ( R d ) let us write, for every x ∈ R d , h x i − m k ∂ α ( f T x ¯ g ) k ∞ ≤ (2 π ) | α | h x i − m Z R d |F ( f T x ¯ g )( ω ) ω α | dω ≤ C α h x i − m Z R d |F ( f T x ¯ g )( ω ) |h ω i | α | dω ≤ C α Z R d sup x ∈ R d [ h x i − m |F ( f T x ¯ g )( ω ) | ] h ω i | α | dω ≍ C α k f k M ∞ , h·i− m ⊗h·i| α | . Hence, for every n ∈ N , m ∈ R , | f | n,m = sup | α |≤ n sup t ∈ R d | ∂ α f ( t ) |h t i − m ≤ C − m sup | α |≤ n C α k f k M ∞ , h·i− m ⊗h·i| α | ≤ C k f k M ∞ , h·i− m ⊗h·i n , where in the last inequality we used the continuous embedding M ∞ , h·i − m ⊗h·i n ֒ → M ∞ , h·i − m ⊗h·i | α | , for every | α | ≤ n . Steps 1, 2, and 3 allow us to obtain the equivalenceof the sequences of norms defined in (35) so that the complete spaces in (34) definethe same Fr´echet space. This assertion concludes the proof.For weights h·i s with real exponent s ∈ R we infer the same result, as explainedbelow. Remark 2.5. ( i ) For any s ∈ R , s ≥ , consider its integer part n := [ s ] , so that n ≤ s < n + 1 . The inclusion relations for modulation spaces in (27) give M ∞h·i − m ⊗h·i n +1 ( R d ) ֒ → M ∞h·i − m ⊗h·i s ( R d ) ֒ → M ∞h·i − m ⊗h·i n ( R d ) (and analogously for M ∞ , h·i − m ⊗h·i s ( R d ) ). This yields the equalities (41) \ n ≥ C nm ( R d ) = \ s ≥ M ∞h·i − m ⊗h·i s ( R d ) = \ s ≥ M ∞ , h·i − m ⊗h·i s ( R d ) . In particular, for m = 0 we recapture the outcome of Lemma 2.1. Notice that inthis case the family of norms are no more countable, thus we do not have a Fr´echet space. Similar arguments as in the proof of Lemma 2.2 yield that the family ofnorms {| · | n,m } n ∈ N , {k · k M ∞h·i− m ⊗h·i s } s ≥ , {k · k M ∞ , h·i− m ⊗h·i s } s ≥ are equivalent, thus they define the same topological vector space. For the case m = 0 we can characterize the H¨ormander class S ( R d ) = S , ( R d )by H¨older-Zygmund classes C s ( R d ) = B ∞ , ∞ s ( R d ) and also the Besov spaces B ∞ , s ( R d ). Lemma 2.3.
We have the equalities (42) \ s ≥ C s ( R d ) = \ s ≥ M ∞ ⊗h·i s ( R d ) = \ s ≥ M ∞ , ⊗h·i s ( R d ) and the families of norms (43) {k · k B ∞ , ∞ s } s ≥ , {k · k B ∞ , s } s ≥ , {k · k M ∞ ⊗h·i s } s ≥ , {k · k M ∞ , ⊗h·i s } s ≥ are equivalent. In particular, the following equalities of topological vector spaceshold S , ( R d ) = \ s ≥ C s ( R d ) = \ s ≥ B ∞ , s ( R d ) = \ s ≥ M ∞ ⊗h·i s ( R d ) = \ s ≥ M ∞ , ⊗h·i s ( R d ) . Proof.
The proof uses the characterization of S ( R d ) = S , ( R d ) in Lemma 2.2and the inclusion relations in (30) and (31). We leave the details to the interestedreader.Another characterization of the classes S m ( R d ) is by the intersection of quasi-Banach modulations spaces. Lemma 2.4.
For m ∈ R , < q < , the families of (quasi-)norms (44) {k · k M ∞ ,q h·i− m ⊗h·i s } s ≥ , {k · k M ∞ , h·i− m ⊗h·i s } s ≥ are equivalent, providing the equalities of the corresponding topological vector space: (45) \ s ≥ M ∞ ,q h·i − m ⊗h·i s ( R d ) = \ s ≥ M ∞ , h·i − m ⊗h·i s ( R d ) . Thus we can write, for n ∈ N , < q ≤ , S m ( R d ) = \ n ≥ M ∞ ,q h·i − m ⊗h·i n ( R d ) . Proof.
The result is a straightforward consequence of the inclusion relations statedin Corollary 2.4. Gabor matrix decay
Let us first represent the Gabor matrix as a kernel of an integral operator.Consider a linear and bounded operator T from S ( R d ) into S ′ ( R d ). The inversionformula (19) for g ∈ M v ( R d ), k g k = 1 is simply V ∗ g V g = Id. The operator T canbe written as(46) T = V ∗ g V g T V ∗ g V g . The linear transformation V g T V ∗ g is an integral operator with kernel K T given bythe Gabor matrix of T : K T ( u, z ) = h T π ( z ) g, π ( u ) g i , u, z ∈ R d . By definition, V g is bounded from M p,qw ( R d ) to L p,qw ( R d ) and V ∗ g from L p,qw ( R d )to M p,qw ( R d ). Hence the continuity properties of T on modulation spaces can beobtained by the corresponding ones of the operator V g T V ∗ g on mixed-norm L p,qw spaces. These issues will be studied in Proposition 3.7 and Corollary 3.12 and canbe achieved by studying the Gabor matrix decay of T .First, we focus on the characterization of the Gabor matrix of Op τ ( σ ).In what follows we exhibit a series of lemmas which are preliminaries for theGabor matrix decay of Theorem 3.10. We begin with a generalization of [8, Lemma3.2]. Lemma 3.1.
Consider τ ∈ [0 , and define (47) ψ ( t ) := e − πt , Ψ( x, ω ) := e − π ( x + ω ) , Ψ τ := W τ ( ψ, ψ ) , for t, x, ω ∈ R d . Then for v submultiplicative weight on R d there exists a constant C > such that (48) k V Ψ Ψ τ k L v ≤ C, ∀ τ ∈ [0 , . Consequently (49) k Ψ τ k M v ≤ C, ∀ τ ∈ [0 , . Proof.
We first observe that any submultiplicative weight function v on R d cangrow at most exponentially, i.e., there exist C > b > v ( z ) ≤ Ce b | z | , ∀ z ∈ R d , see, e.g. [10, Lemma 2.1.4]. Following the proof in [8] and using (50), we canmajorize in the following manner: k V Ψ Ψ τ k L v ≤ C Z R d e − π z z τ − τ +5 e b | z | I dz dz ≤ C Z R d e − π z z τ − τ +5 e b ( | z | + | z | ) I dz dz , where C > v and I is an integral over R d whichcan be controlled from above by I ≤ ˜ C e π (1 − τ )2 z τ − τ +2)(2 τ − τ +5) + a | − τ | τ − τ +2 | z | e π (1 − τ )2 z τ − τ +2)(2 τ − τ +5) + a | − τ | τ − τ +2 | z | , for some a > C > τ . Hence setting C := C ˜ C we have k V Ψ Ψ τ k L v ≤ C Z R d e − π z τ − τ +5 + π (1 − τ )2 z τ − τ +2)(2 τ − τ +5) + a | − τ | τ − τ +2 | z | + b | z | dz × Z R d e − π z τ − τ +5 + π (1 − τ )2 z τ − τ +2)(2 τ − τ +5) + a | − τ | τ − τ +2 | z | + b | z | dz = C Z R d e − π z τ − τ +5 + π (1 − τ )2 z τ − τ +2)(2 τ − τ +5) + a | − τ | τ − τ +2 | z | + b | z | dz | {z } =: I . The integral I can be controlled as I = Z R d e − π τ − τ +2) − (1 − τ )2(2 τ − τ +2)(2 τ − τ +5) z + (cid:16) a | − τ | τ − τ +2 + b (cid:17) | z | dz ≤ Z R d e − πC z + =: C ≥ z }| { ( aC + b ) | z | dz , being C = min τ ∈ [0 , τ − τ + 2) − (1 − τ ) (2 τ − τ + 2)(2 τ − τ + 5) = 12 , C = max τ ∈ [0 , | − τ | τ − τ + 2 = 12 . Since e − π C z + C | z | → | z | → + ∞ , for ε > R > e − π C z + C | z | < ε for every z / ∈ B R (0). Therefore I ≤ Z R d e − πC z + C | z | dz = Z B R (0) e − πC z + C | z | dz + Z R d \ B R (0) e − πC z + C | z | dz ≤ e C R Z B R (0) e − πC z dz + ε Z R d \ B R (0) e − π C z dz < + ∞ , Hence there exists
C > k V Ψ Ψ τ k L v ≤ C , uniformly w.r.t. τ ∈ [0 , k Ψ τ k M v ≍ k V Ψ Ψ τ k L v concludes the proof. Corollary 3.1.
Fix G ∈ S ( R d ) and consider an even, submultiplicative weight v on R d . Let Ψ τ be the function defined in (47) . Then there exists a constant C > such that (51) k V Ψ τ G k L v ≤ C, ∀ τ ∈ [0 , . Proof.
The claim is a straightforward consequence of Lemma 3.1, the switchingproperty of the STFT (see, e.g., [10, Lemma 1.2.3]) V Ψ τ G ( z, u ) = e − πzu V G Ψ τ ( − z, − u ) , ( z, u ) ∈ R d . and the even property of the weight v .In the following lemma we summarize [8, Lemma 2.5, Lemma 2.6, Corollary 2.7].For τ ∈ (0 , A τ := " d q − ττ I d − p τ − τ I d d . Lemma 3.2.
Let f, g ∈ S ( R d ) , τ ∈ [0 , and define Φ τ := W τ ( g, g ) . Consider z = ( z , z ) , u = ( u , u ) ∈ R d . If τ ∈ (0 , , then V Φ τ W τ ( f, f )( z, u ) = e − πz u V g f ( z − τ u , z + (1 − τ ) u ) V g f ( z + (1 − τ ) u , z − τ u )= e − πz u V g f ( z + p τ (1 − τ ) A Tτ u ) V g f ( z + p τ (1 − τ ) A τ u ); , (53) where A Tτ stands for the transpose of A τ .If τ = 1 , then (54) V Φ W ( f, f )( z, u ) = e − πz u V g f ( z − u , z ) V g f ( z , z − u ); If τ = 0 , then (55) V Φ W ( f, f )( z, u ) = e − πz u V g f ( z , z + u ) V g f ( z + u , z ) . Remark 3.2.
Notice that, for u = ( u , u ) ∈ R d and τ ∈ (0 , , we have (56) | p τ (1 − τ ) A τ u | = | − τ u | + | (1 − τ ) u | ≤ | u | + | u | = | u | . Therefore (57) h p τ (1 − τ ) A τ u i s ≤ h u i s , ∀ τ ∈ (0 , , ∀ u ∈ R d , ∀ s ≥ . Lemma 3.3.
Consider f, g ∈ S ( R d ) \ { } , τ ∈ [0 , , v a submultiplicative weighton R d satisfying (12) , and define Φ τ := W τ ( g, g ) . Then there exists a constant C > such that (58) k V Φ τ W τ ( f, f ) k L v ≤ C, ∀ τ ∈ [0 , . Proof.
We divide the proof in three cases: τ ∈ (0 , τ = 1 and τ = 0.For τ ∈ (0 ,
1) we apply (53), the change of variable y = z + p τ (1 − τ ) A τ u , sothat z + p τ (1 − τ ) A Tτ u = y − J u , the submultiplicativity of v as well as its grow condition (12), and finally Remark 3.2: k V Φ τ W τ ( f, f ) k L v = Z R d Z R d | V g f ( y − J u ) V g f ( y ) | v ( y − p τ (1 − τ ) A τ u, u ) dydu ≤ Z R d Z R d | V g f ( y − J u ) V g f ( y ) | v ( y, v ( − p τ (1 − τ ) A τ u, v (0 , u ) dydu . Z R d Z R d | V g f ( y − J u ) V g f ( y ) |h y i s h− p τ (1 − τ ) A τ u i s h u i s dydu ≤ Z R d Z R d | V g f ( y − J u ) V g f ( y ) |h y i s h u i s h u i s dydu = Z R d ( | V g f | ∗ | V g f h·i s | ) ( J u ) h u i s du = k| V g f | ∗ | V g f h·i s |k L h·i s < + ∞ . The convergence is due to the fact that f, g ∈ S ( R d ), therefore V g f ∈ S ( R d ).For τ = 1 we apply (54) and the change of variable y = z , y = z − u ; arguingas in the previous stage we obtain the result. In detail, k V Φ W ( f, f ) k L v = Z R d Z R d | V g f ( y − J u ) V g f ( y ) | v ( y , y + u , u , u ) dydu ≤ Z R d Z R d | V g f ( y − J u ) V g f ( y ) | v ( y, v (0 , , u , v (0 , u ) dydu . Z R d Z R d | V g f ( y − J u ) V g f ( y ) |h y i s h u i s h u i s dydu ≤ Z R d Z R d | V g f ( y − J u ) V g f ( y ) |h y i s h u i s h u i s dydu = Z R d ( | V g f | ∗ | V g f h·i s | ) ( J u ) h u i s du = k| V g f | ∗ | V g f h·i s |k L h·i s < + ∞ . The case τ = 0 follows the same argument as before via (55).In each case we found the same upper bound which does not depend on τ ∈ [0 , Corollary 3.3.
Consider τ ∈ [0 , , v a submultiplicative weight on R d satisfying (12) , g ∈ S ( R d ) \ { } and define Φ τ := W τ ( g, g ) . Then there exists a constant C > such that (59) k Φ τ k M v ≤ C, ∀ τ ∈ [0 , . Proof.
Fix a window G ∈ S ( R d ) \ { } and consider the functions ψ and Ψ τ definedin (47). Using the change-window property of the STFT (see, e.g., [10, 1.2.29]),Moyal’s formula for τ -Wigner distributions (see, e.g., [10, Corollary 1.3.28]) andYoung’s inequality for mixed-normed spaces (see, e.g., [10, Theorem 2.2.3]), k V G Φ τ k L v ≤ |h Ψ τ , Ψ τ i| k V Ψ τ Φ τ ∗ V G Ψ τ k L v ≤ k ψ k − L k V Ψ τ Φ τ k L v k V G Ψ τ k L v . The desired result follows now by Lemma 3.1, 3.3 and the fact that k Φ τ k M v ≍ k V G Φ τ k L v , where the constants involved do not depend on τ ∈ [0 , Corollary 3.4.
Consider τ ∈ [0 , , v an even submultiplicative weight on R d satisfying (12) , G ∈ S ( R d ) \ { } , g ∈ S ( R d ) \ { } and define Φ τ := W τ ( g, g ) .Then there exists a constant C > such that (60) k V Φ τ G k L v ≤ C, ∀ τ ∈ [0 , . Proof.
The proof is a straightforward consequence of Corollary 3.3 and the switch-ing property of the STFT, cf. the proof of Corollary 3.1.
Proposition 3.5.
Consider ≤ p, q ≤ ∞ , τ ∈ [0 , , w ∈ M v ( R d ) , with v evenand satisfying (12) , G ∈ S ( R d ) \ { } , g ∈ S ( R d ) \ { } and define Φ τ := W τ ( g, g ) .Then there exists A = A ( v, g, G ) > , B = B ( v, g, G ) > such that (61) A k V G σ k L p,qw ≤ k V Φ τ σ k L p,qw ≤ B k V G σ k L p,qw , for every τ ∈ [0 , and σ ∈ M p,qw ( R d ) .Proof. Let Ψ τ = W τ ( ψ, ψ ), and ψ be as in (47). Using the change-window propertyof the STFT (see, e.g., [10, 1.2.29]), Moyal’s formula for τ -Wigner distributions (see,e.g., [10, Corollary 1.3.28]) and Young’s inequality for mixed-normed spaces (see,e.g., [10, Theorem 2.2.3]), and Corollary 3.4: k V Φ τ σ k L p,qw ≤ |h Ψ τ , Ψ τ i| k| V Ψ τ σ | ∗ | V Φ τ Ψ τ |k L p,qw ≤ k ψ k − L k V Ψ τ σ k L p,qw k V Φ τ Ψ τ k L v ≤ C k ψ k − L k V Ψ τ σ k L p,qw ≤ C k ψ k − L |h G, G i| k| V G σ | ∗ | V Ψ τ G |k L p,qw ≤ C k ψ k − L k G k − L k V G σ k L p,qw k V Ψ τ G k L v ≤ ˜ C k ψ k − L k G k − L k V G σ k L p,qw , with ˜ C > τ . Similarly, k V G σ k L p,qw ≤ ˜ C k ψ k − L k g k − L k V Φ τ σ k L p,qw . The choice A := ˜ C − k ψ k L k g k L , B := ˜ C k ψ k − L k G k − L let us conclude the proof.Finally, we need the following result for τ -pseudodifferential operators [13, Lemma4.1]. Lemma 3.4.
Fix a window g ∈ S ( R d ) \{ } and define Φ τ = W τ ( g, g ) for τ ∈ [0 , .Then, for σ ∈ S ′ (cid:0) R d (cid:1) , (62) |h Op τ ( σ ) π ( z ) g, π ( u ) g i| = | V Φ τ σ ( T τ ( z, u ) , J ( u − z )) | . where z = ( z , z ) , u = ( u , u ) , the operator T τ is defined in (7) and J is given by J ( z ) = ( z , − z ) . We are ready to state the characterization of τ -operators with symbols in M ∞h·i − m ⊗h·i s ( R d ). Theorem 3.6.
Consider g ∈ S ( R d ) \ { } and a lattice Λ ⊂ R d such that G ( g, Λ) is a Gabor frame for L (cid:0) R d (cid:1) . For τ ∈ [0 , , let T τ be the linear transformationdefined in (7) . For any s, m ∈ R , the following properties are equivalent: ( i ) σ ∈ M ∞h·i − m ⊗h·i s (cid:0) R d (cid:1) . ( ii ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a constant C > such that (63) |h Op τ ( σ ) π ( z ) g, π ( u ) g i| ≤ C hT τ ( z, u ) i m h u − z i s , ∀ u, z ∈ R d . ( iii ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists C > such that (64) |h Op τ ( σ ) π ( µ ) g, π ( λ ) g i| ≤ C hT τ ( µ, λ ) i m h λ − µ i s , ∀ λ, µ ∈ Λ . Proof.
The proof follows the pattern of the corresponding one for Weyl operatorswith symbols in weighted Sj¨ostrand’s classes [20, Theorem 3.2].( i ) ⇒ ( ii ) This implication comes straight forward from the characterization (62)and Proposition 3.5 since the weight h·i − m ⊗ h·i s is moderate with respect to theeven submultiplicative weight v = 2 | m | h·i | m | ⊗ | s | h·i | s | , as can be seen from Petre’s inequality. Hence: |h Op τ ( σ ) π ( z ) g, π ( u ) g i| = | V Φ τ σ ( T τ ( z, u ) , J ( u − z )) |≤ k V Φ τ σ k L ∞h·i− m ⊗h·i s hT τ ( z, u ) i m h u − z i − s ≤ B k σ k M ∞h·i− m ⊗h·i s hT τ ( z, u ) i m h u − z i − s . (65)( ii ) ⇒ ( i ) Consider the change of variables y = T τ ( z, u ) and t = J ( u − z ), so that(66) ( z ( y, t ) = y − U τ J − tu ( y, t ) = y + ( I d − U τ ) J − t , U τ z := (cid:20) τ I d
00 (1 − τ ) I d (cid:21) z = T τ (0 , z )and u ( y, t ) − z ( y, t ) = J − t . Hence using (62) and (63): k σ k M ∞h·i− m ⊗h·i s ≍ sup y,t ∈ R d | V Φ τ σ ( y, t ) | h y i − m h t i s = sup y,t ∈ R d |h Op τ ( σ ) π ( z ( y, t )) g, π ( u ( y, t )) g i| h y i − m h t i s ≤ C. We remark that, due to Proposition 3.5, the constants involved in the above equiv-alence do not depend on τ ∈ [0 , ii ) ⇔ ( iii ) The argument is well known and we refer to [9, Theorem 3.1] and [20,Theorem 3.2].The proof of the characterization of the symbol classes S m ( R d ) claimed in The-orem 1.1, can be inferred easily from the result above. Proof of Theorem 1.1.
The proof is a direct application of the characterization ofthe classes S m ( R d ) presented in (36) and Theorem 3.10.The following issue is an improvement of [6, Theorem 2.4] and relies on the newcharacterization of S m ( R d ) proved in Lemma 2.2. Proposition 3.5.
Consider g ∈ S ( R d ) \ { } , m ∈ R and σ ∈ S m (cid:0) R d (cid:1) . For any n ∈ N there exists C = C ( n ) > , which does not depend on σ or τ , such that (67) |h Op τ ( σ ) π ( z ) g, π ( u ) g i| ≤ C | σ | n,m hT τ ( z, u ) i m h u − z i n , ∀ τ ∈ [0 , , ∀ u, z ∈ R d . Proof.
Using the characterization of the H¨ormander classes S m ( R d ) in (36) weinfer that σ ∈ M ∞h·i − m ⊗h·i n ( R d ) and, for any n ∈ N , the norm estimate in (39) saysthat there exists C = C ( n, m ) such that(68) k σ k M ∞h·i− m ⊗h·i n ≤ C ( n, m ) | σ | n,m , where C ( n, m ) > σ . For z, w ∈ R d we can use the majorizationin (65) and the norm estimate in (68) which yield |h Op τ ( σ ) π ( z ) g, π ( u ) g i| = | V Φ τ σ ( T τ ( z, u ) , J ( u − z )) |≤ C | σ | n,m hT τ ( z, u ) i m h u − z i n , that is the desired result.For s ∈ [0 , + ∞ ) \ N , the estimate reads as follows. Proposition 3.6.
Consider g ∈ S ( R d ) \ { } , τ ∈ [0 , , m ∈ R and σ ∈ S m (cid:0) R d (cid:1) .For any s ∈ [0 , + ∞ ) \ N there exists C = C ( s, m ) > , which does not depend on σ or τ , such that (69) |h Op τ ( σ ) π ( z ) g, π ( u ) g i| ≤ C | σ | n +1 ,m hT τ ( z, u ) i m h u − z i s , ∀ u, z ∈ R d , where n = [ s ] is the integer part of s .Proof. The result is attained by the the same argument as Proposition 3.5 andtheinclusion relations between modulation spaces in (27).3.1.
Boundedness results.
As an application of the Gabor matrix decay we proveboundedness properties of such operators on weighted modulation spaces. Wenotice that for the quasi-Banach cases the discrete characterization in (64) is veryhandy.
Proposition 3.7.
Consider τ ∈ [0 , , m ∈ R , σ ∈ S m ( R d ) , < p, q ≤ ∞ . Then Op τ ( σ ) , from S ( R d ) to S ′ ( R d ) , extends uniquely to a bounded operator Op τ ( σ ) : M p,q h·i r + m + | m | ( R d ) → M p,q h·i r ( R d ) , for every r ∈ R .Proof. Case p, q ≥
1. Fix g ∈ S ( R d ) with k g k L = 1. We use the representa-tion (46) for T = Op τ ( σ ) and prove the continuity of the operator V g T V ∗ g from L p,q h·i r + m + | m | ( R d ) into L p,q h·i r ( R d ). Consider f ∈ M p,q h·i r + m + | m | ( R d ). The Gabor matrixdecay in Theorem 3.10 let us estimate k Op τ ( σ ) f k M p,q h·i r ≍ k V g (Op τ ( σ ) f ) k L p,q h·i r ≤ C s (cid:13)(cid:13)(cid:13)(cid:13)Z R d hT τ ( z, · ) i m h· − z i s | V g f ( z ) | dz (cid:13)(cid:13)(cid:13)(cid:13) L p,q h·i r . Using Petre’s inequality,(70) hT τ ( z, u ) i m . h ((1 − τ ) z , τ z ) i | m | h ( τ u , (1 − τ ) u ) i m ≤ h z i | m | h u i m , ∀ u, z ∈ R d , so that k Op τ ( σ ) f k M p,q h·i r ≤ C s (cid:13)(cid:13)(cid:13)(cid:13) h·i s ∗ | V g f |h·i | m | (cid:13)(cid:13)(cid:13)(cid:13) L p,q h·i r + m . We choose s > d + | r + m | and use Young’s inequality for mixed-normed spacesto get the claim: k Op τ ( σ ) f k M p,q h·i r . k V g f h·i | m | k L p,q h·i r + m ≍ k V g f k L p,q h·i r + m + | m | . Case p < or q <
1. In this case we choose g ∈ S ( R d ) and a lattice Λ suchthat G ( g, Λ) is a Gabor frame for L ( R d ). Define t := min { p, q } and choose s > (2 d + | r + m | ) /t . Using the discrete characterization in (64) and (70), as well asYoung’s convolution inequality in [18, Theorem 3.1], we obtain the result. Namely, k Op τ ( σ ) f k M p,q h·i r ≍ k V g (Op τ ( σ ) f ) k ℓ p,q h·i r (Λ) ≤ C s (cid:13)(cid:13)(cid:13)(cid:13) h·i s ∗ | V g f |h·i | m | (cid:13)(cid:13)(cid:13)(cid:13) ℓ p,q h·i r + m (Λ) ≤ C s (cid:13)(cid:13)(cid:13)(cid:13) h·i s (cid:13)(cid:13)(cid:13)(cid:13) ℓ t h·i| r + m | (Λ) (cid:13)(cid:13) V g f h·i | m | (cid:13)(cid:13) ℓ p,q h·i r + m (Λ) ≤ ˜ C s k f k M p,q h·i r + m + | m | . The proof is completed.
Remark 3.8. (i) For m ≤ , then σ ∈ S ( R d ) = S , ( R d ) and we recapture thecontinuity of Op τ ( σ ) : M p,q h·i r ( R d ) → M p,q h·i r ( R d ) . This was already shown in [35] for p, q ≥ , for the quasi-Banach cases see [36] .(ii) For p = q = 2 we have the continuity between the Shubin-Sobolev spaces Q r + m + | m | ( R d ) and Q r ( R d ) . Corollary 3.9.
Consider τ ∈ [0 , , m, r ∈ R , σ ∈ S m ( R d ) , < p, q ≤ ∞ . Let k Op τ ( σ ) k denote the norm of Op τ ( σ ) in B ( M p,q h·i r + m + | m | ( R d ) , M p,q h·i r ( R d )) . Then thereexists a constant C > such that (71) k Op τ ( σ ) k ≤ C, ∀ τ ∈ [0 , . Proof.
The claim is evident from proof of Proposition 3.7. Born-Jordan operators.
The Born-Jordan operator with symbol σ ∈ S ′ ( R d )can be defined as h Op BJ ( σ ) f, g i = h f, W BJ ( g, f ) i , f, g ∈ S ( R d ) , where the Born-Jordan distribution W BJ ( g, f ) is W BJ ( g, f ) = Z W τ ( g, f ) dτ, see, e.g., the textbook [14]. Theorem 3.10.
Consider g ∈ S ( R d ) \ { } . If σ ∈ S m (cid:0) R d (cid:1) then for every s ≥ there exists a constant C = C s > such that (72) |h Op BJ ( σ ) π ( z ) g, π ( u ) g i| ≤ C h z i | m | h u i m h u − z i − s , ∀ u, z ∈ R d . Proof.
For σ ∈ S ′ ( R d ), Op BJ ( σ ) is linear and continuous from S ( R d ) into S ′ ( R d ),see [15]. For z, u ∈ R d , σ ∈ S m ( R d ) and g ∈ S ( R d ) we compute h Op BJ ( σ ) π ( z ) g, π ( u ) g i = h σ, W BJ ( π ( z ) g, π ( u ) g ) i = Z R d σ ( y ) Z W τ ( π ( z ) g, π ( u ) g )( y ) dτ dy =: I. From [15, Proposition 2.2, Remark 2.3] we have that the mapping R × S ( R d ) × S ( R d ) → S ( R d ) , ( t, ϕ, ψ ) W t ( ϕ, ψ )is continuous and locally uniformly bounded. Thus W BJ ( ϕ, ψ ) ∈ S ( R d ) and theintegral I is absolutely convergent, so that I = Z Z R d σ ( y ) W τ ( π ( z ) g, π ( u ) g )( y ) dydτ = Z h Op τ ( σ ) π ( z ) g, π ( u ) g i dτ. Using Theorem 1.1 and (70) we can proceed as follows: | I | ≤ Z |h Op τ ( σ ) π ( z ) g, π ( u ) g i| dτ ≤ C s Z h z i | m | h u i m h u − z i s dτ = C s h z i | m | h u i m h u − z i s . This concludes the proof.
Remark 3.11.
Arguing as in Theorem 3.10, we may discretize the Gabor matrixdecay in (72) as follows: consider g ∈ S ( R d ) \ { } and a lattice Λ in R d such that G ( g, Λ) is a Gabor frame for L (cid:0) R d (cid:1) . If σ ∈ S m (cid:0) R d (cid:1) then for every s ≥ thereexists a constant C = C s > such that |h Op BJ ( σ ) π ( µ ) g, π ( λ ) g i| ≤ C h z i | m | h u i m h λ − µ i − s , ∀ λ, µ ∈ Λ . Corollary 3.12.
Consider m ∈ R , σ ∈ S m ( R d ) , < p, q ≤ ∞ . Then Op BJ ( σ ) ,from S ( R d ) to S ′ ( R d ) , extends uniquely to a bounded operator Op BJ ( σ ) : M p,q h·i r + m + | m | ( R d ) → M p,q h·i r ( R d ) , for every r ∈ R .Proof. The proof goes as the one for Proposition 3.7, using the decay for Gabormatrix of Op BJ ( σ ) found in Theorem 3.10. Acknowledgements
The authors would like to thank Professors Fabio Nicola and Luigi Rodino forfruitful conversations and comments.
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