Characterization of boundedness on weighted modulation spaces of τ -Wigner distributions
aa r X i v : . [ m a t h . F A ] N ov CHARACTERIZATION OF BOUNDEDNESS ON WEIGHTEDMODULATION SPACES OF τ -WIGNER DISTRIBUTIONS WEICHAO GUO, JIECHENG CHEN, DASHAN FAN, AND GUOPING ZHAO
Abstract.
This paper is devoted to give several characterizations on a more general levelfor the boundedness of τ -Wigner distributions acting from weighted modulation spaces toweighted modulation and Wiener amalgam spaces. As applications, sharp exponents areobtained for the boundedness of τ -Wigner distributions on modulation spaces with powerweights. We also recapture the main theorems of Wigner distribution obtained in [8, 4]. Asconsequences, the characterizations of the boundedness on weighted modulation spaces ofseveral types of pseudodifferential operators are established. In particular, we give the sharpexponents for the boundedness of pseudodifferential operators with symbols in Sj¨ostrand’sclass and the corresponding Wiener amalgam spaces. INTRODUCTION
The study of cross-Wigner distribution has a long history. It was first introduced in 1932in E.Wigner’s ground-breaking paper [27], and then introduced in 1948 by J.Ville [25] in thefield of signal analysis. Let us recall the definition as follows.Given two functions f , f ∈ L ( R d ), the cross-Wigner distribution W ( f , f ) is defined by W ( f , f )( x, ξ ) := ˆ R d f ( x + t f ( x − t e − πit · ξ dt. Let T s be the symmetric coordinate change defined by T s F ( x, t ) = F ( x + t , x − t , and let F be the partial Fourier transform in the second variable defined by F F ( x, ξ ) = ˆ R d F ( x, t ) e − πit · ξ dt. The cross-Wigner distribution can be written as W ( f , f ) = F T s ( f ⊗ ¯ f ) . For f = f = f , W f = W ( f, f ) is simply called the Wigner distribution of f . For simplicity,in the remaining part of this paper, we omit the word “cross” no matter whether f = f ornot.As an important time-frequency representation, the Wigner distribution is closed related tothe short-time Fourier transform (STFT) defined by V g f ( x, ξ ) := ˆ R d f ( t ) g ( t − x ) e − πit · ξ dt, f, g ∈ L ( R d ) . Mathematics Subject Classification.
Key words and phrases.
Wigner distribution, pseudodifferential operator, modulation space, Wiener amalgamspace.
In fact, a direct calculation shows that W ( f, g )( x, ξ ) = 2 d e πix · ξ V I g f (2 x, ξ ) , f, g ∈ L ( R d ) , where I g ( t ) = g ( − t ).On the other hand, the pseudodifferential operator in the Weyl form, i.e., the Weyl operator L σ with symbol σ ∈ S ′ ( R d ) can be defined by means of duality pairing between the symboland the Wigner distribution: h L σ f, g i = h σ, W ( g, f ) i , f, g ∈ S ( R d ) . The localization operator A ϕ ,ϕ a with symbol a ∈ S ′ ( R d ), analysis window ϕ ∈ S ( R d ), andsynthesis window function ϕ ∈ S ( R d ) can be regarded as the Weyl operator whose symbol isthe convolution of a with the Wigner distribution of the windows ϕ and ϕ : A ϕ ,ϕ a = L a ∗ W ( ϕ ,ϕ ) . For τ ∈ [0 , τ -Wignerdistribution of f , f ∈ L ( R d ) is defined by W τ ( f , f )( x, ξ ) := ˆ R d f ( x + τ t ) f ( x − (1 − τ ) t ) e − πiξ · t dt. For f = f = f , W τ f := W τ ( f, f ) is simply called the τ -Wigner distribution of f . Forsimplicity, we omit the word “cross” in the remaining part of this paper.Note that τ -Wigner distribution is a generalization of the Wigner distribution. Varying theparameter τ , W τ ( f , f ) is a family of time-frequency representations:For τ = 1 / W / ( f , f ) becomes the Wigner distribution W ( f , f ).For τ = 0, W ( f , f ) coincides with the Rihaczek distribution R ( f , f ): W ( f , f )( x, ξ ) = R ( f , f )( x, ξ ) = e − πix · ξ f ( x ) ˆ f ( ξ ) . For τ = 1, W ( f , f ) coincides with the conjugate Rihaczek distribution R ∗ ( f , f ): W ( f , f )( x, ξ ) = R ∗ ( f , f )( x, ξ ) = R ( f , f )( x, ξ ) = e πix · ξ f ( x ) ˆ f ( ξ ) . By the Rihaczek distribution, the famous Kohn-Nirenberg operator K σ with symbol σ canbe defined weakly as: h K σ f, g i = h σ, R ( g, f ) i , f, g ∈ S ( R d ) , σ ∈ S ′ ( R d ) . In general, for τ ∈ [0 , τ -operators or Shubin operators [21] can be defined as h OP τ ( σ ) f, g i = h σ, W τ ( g, f ) i , f, g ∈ S ( R d ) . Note that OP ( σ ) coincides with the Khon-Nirenberg operator K σ , OP / ( σ ) is just the Weyloperator L σ . As the adjoint operator of OP ( σ ), OP ( σ ) is also called anti-Kohn-Nirenbergoperator.According to the above relations, the boundedness of several important operators has di-rect connections with the corresponding boundedness of τ -Wigner distributions. Hence, it isimportant to establish the boundedness results of τ -Wigner distribution on function spaces.Among them, the boundedness acting on modulation spaces has its important position, sinceit has closed relationship with time-frequency analysis.Modulation spaces were invented by H. Feichtinger [11] in 1983. Nowadays, they have beenfully recognized as the “right” function spaces for time-frequency analysis. More precisely, OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 3 modulation spaces are defined by measuring the decay and integrability of the STFT as fol-lowing: M p,qm ( R d ) = { f ∈ S ′ ( R d ) : V g f ∈ L p,qm ( R d ) } endowed with the obvious (quasi-)norm, where L p,qm ( R d ) are weighted mixed-norm Lebesguespaces with the weight m , see Section 2 for more details. We use M p,qm ( R d ) to denote the S ( R d )closure in M p,qm ( R d ).For the power weights v s,t ( z ) = h z i s h z i t , v s ( z ) = h z i s = (1 + | z | ) s/ s, t ∈ R , z = ( z , z ) ∈ R d , the problem for the boundedness of τ -Wigner distribution acting from weighted modulationspaces to weighted modulation spaces (BMM) is to find the full range of exponents of p i , q i , p, q ∈ (0 , ∞ ], s i , t i ∈ R , i = 1 , W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ M p,q ⊗ v s ( R d ) , that is, k W τ ( f , f ) k M p,q ⊗ vs . k f k M p ,q vs ,t · k f k M p ,q vs ,t , f, g ∈ S ( R d ) , where we write (1 ⊗ v s )( z, ζ ) := v s ( ζ ) for ( z, ζ ) ∈ R d .Note that, to avoid the fact that S ( R d ) is not dense in some endpoint spaces, such as M p,qm with p = ∞ or q = ∞ , we only consider the action of τ -Wigner distribution on Schwartzfunction spaces. Similarly, we only consider the action of τ -operator OP τ ( σ ) on Schwartzfunction spaces.This problem restricted to s = s i = t i = 0, namely, W : M p ,q ( R d ) × M p ,q ( R d ) −→ M p,q ( R d ) (1.1)was studied by Toft [24, Theorem 4.2], and then refined very recently by Cordero-Nicola [8,Theorem 1.1] and Cordero [4, Theorem 3.2]. In [8, 4], the authors find the sharp conditionsfor (1.1) of exponents p i , q i , p, q ∈ (0 , ∞ ], i = 1 ,
2. Under the same conditions, they also obtainthe following estimate: W : M p ,q v | s | ( R d ) × M p ,q v s ( R d ) −→ M p,q ⊗ v s ( R d ) . In the present paper, our first major goal is to consider BMM problem on a more generallevel. For suitable weight function m, m , m on R d (see Section 2 for more precise definitionsof weights), our first main theorem shows that BMM can be characterized by the correspondingconvolution inequalities of discrete mixed-norm spaces. Theorem 1.1. (First characterization of BMM) Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , , τ ∈ [0 , .Suppose that m, m i ∈ P ( R d ) , i = 1 , . We have W τ : M p ,q m ( R d ) × M p ,q m ( R d ) −→ M p,q ⊗ m ( R d ) if and only if for all ~a,~b , k ( a k ,k b n − k ,n − k ) k l p,q ⊗ mJ . k ~a k l p ,q m ( Z d ) k ~b k l p ,q I m ( Z d ) . (1.2) In particular, for p < ∞ , this is equivalent to l p /p,q /pm p ( Z d ) ∗ l p /p,q /p I m p ( Z d ) ⊂ l q/p,q/pm pJ ( Z d ) . (1.3) Here we write I m ( z ) = m ( − z ) , m J ( z ) = m ( J z ) = m ( z , − z ) for z ∈ Z d , where J is thesymplectic matrix (see Section 2). WEICHAO GUO, JIECHENG CHEN, DASHAN FAN, AND GUOPING ZHAO
Furthermore, for submultiplicative weight m and variable-separable weights m and m ,namely, m ( z + n , z + n ) . m ( z , z ) m ( n , n ) , m = ω ⊗ µ , m = ω ⊗ µ , our second main theorem shows that BMM can be further characterized by some convolutionor embedding inequalities of discrete norm spaces. Theorem 1.2. (Second characterization of BMM) Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , , τ ∈ [0 , . Suppose that m ∈ P ( R d ) is submultiplicative, ω i , µ i ∈ P ( R d ) , i = 1 , . We have W τ : M p ,q ω ⊗ µ ( R d ) × M p ,q ω ⊗ µ ( R d ) −→ M p,q ⊗ m ( R d ) if and only if l p /pω p ( Z d ) ∗ l p /p I ω p ( Z d ) ⊂ l q/p I m pβ ( Z d ) , l q /pµ p ( Z d ) ∗ l q /p I µ p ( Z d ) ⊂ l q/pm pα ( Z d ) , p < ∞ , (1.4) l p ω ( Z d ) , l p I ω ( Z d ) ⊂ l q I m β ( Z d ) , l q µ ( Z d ) , l q I µ ( Z d ) ⊂ l qm α ( Z d ) , p ≥ q. (1.5) Here, we write m α ( z ) = m ( z , and m β ( z ) = m (0 , z ) for z , z ∈ R d . As an application, we return to the case of power weights. Our third characterization showsthat in this case BMM can be characterized by some convolution or embedding inequalities ofdiscrete norm spaces with power weights. Some further characterizations of exponents will beshown in Section 5.
Theorem 1.3. (Third characterization of
BM M ) Assume p, q, p i , q i ∈ (0 , ∞ ] , s i , t i ∈ R , i = 1 , , τ ∈ [0 , . We have W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ M p,q ⊗ v s ( R d ) if and only if l p /pps ( Z d ) ∗ l p /pps ( Z d ) ⊂ l q/pps ( Z d ) , l q /ppt ( Z d ) ∗ l q /ppt ( Z d ) ⊂ l q/pps ( Z d ) , p < ∞ , (1.6) l p s ( Z d ) , l p s ( Z d ) ⊂ l qs ( Z d ) , l q t ( Z d ) , l q t ( Z d ) ⊂ l qs ( Z d ) , p ≥ q. (1.7)It is well known that the boundeness property of τ -pseudodifferential operators with symbolsin modulation spaces are independent with τ ∈ [0 , τ as shown in Theorems 1.1 to 1.3. However,situation changes in the problem of the boundedness of τ -Wigner distribution acting fromweighted modulation spaces to weighted Wiener amalgam spaces (BMW).In this paper, we consider the Wiener amalgam spaces W ( F L p , L qm )( R d ), which are theimage of modulation spaces M p,q ⊗ m ( R d ) under the Fourier transform, see the next section forits precise definition. In contrast with the fruitful works on BMM, there are only few resultsof BMW. In [9, 6], some sufficient conditions of BMW (with weight on the first component ofWiener amalgam space, namely, W ( F L pm , L q )( R d )) was established for τ ∈ (0 , τ = 0 , m, m , m on R d (see Section 2 for more precise definitions of weights),our first main theorem for BMW is as follows. Theorem 1.4. (First characterization of BMW) Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , , τ ∈ [0 , . Suppose that m, m i ∈ P ( R d ) , i = 1 , . Denote e m ( ζ , ζ ) = m ((1 − τ ) ζ , τ ζ ) , f m ( z , z ) = m ( − ττ z , τ − τ z ) . We have W τ : M p ,q m ( R d ) × M p ,q m ( R d ) −→ W ( F L p , L qm )( R d ) OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 5 if and only if for all ~a,~b , k ( a k ,k b n − k ,n − k ) k l p,q ⊗ e m . k ~a k l p ,q m ( Z d ) k ~b k l p ,q f m ( Z d ) τ ∈ (0 , , (1.8) k ( a n ,k b k ,n ) k l p,q ⊗ m . k ~a k l p ,q m ( Z d ) k ~b k l p ,q m ( Z d ) τ = 0 , (1.9) k ( a n ,k b k ,n ) k l p,q ⊗ m . k ~b k l p ,q m ( Z d ) k ~a k l p ,q m ( Z d ) τ = 1 . (1.10) In particular, for p < ∞ , the condition (1.8) is equivalent to l p /p,q /pm p ( Z d ) ∗ l p /p,q /p f m p ( Z d ) ⊂ l q/p,q/p e m p ( Z d ) . As in the case of BMM, if m is submultiplicative, m and m are variable-separable, weobtain a further characterization of BMW. See the definition of l ( p,q ) m in Definition 2.13. Theorem 1.5. (Second characterization of
BM W ) Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , , τ ∈ [0 , . Suppose that m ∈ P ( R d ) is submultiplicative, ω i , µ i ∈ P ( R d ) , i = 1 , . Denote e m ( ζ , ζ ) = m ((1 − τ ) ζ , τ ζ ) , f ω ( z ) = ω ( − ττ z ) , f µ ( z ) = µ ( τ − τ z ) . We have W τ : M p ,q ω ⊗ µ ( R d ) × M p ,q ω ⊗ µ ( R d ) −→ W ( F L p , L qm )( R d ) if and only if l p /pω p ( Z d ) ∗ l p /p f ω p ( Z d ) ⊂ l q/p e m pα ( Z d ) , l q /pµ p ( Z d ) ∗ l q /p f µ p ( Z d ) ⊂ l q/p e m pβ ( Z d ) , p < ∞ , (1.11) l p ω ( Z d ) , l p f ω ( Z d ) ⊂ l q e m α ( Z d ) , l q µ ( Z d ) , l q f µ ( Z d ) ⊂ l q e m β ( Z d ) , p ≥ q, (1.12) for τ ∈ (0 , , and l p ,q ω ⊗ µ ( Z d ) ⊂ l ( q,p ) m α ⊗ ( Z d ) , l p ω ( Z d ) ⊂ l p ( Z d ) , l q µ ( Z d ) ⊂ l qm β ( Z d ) , τ = 0 , (1.13) l p ,q ω ⊗ µ ( Z d ) ⊂ l ( q,p ) m α ⊗ ( Z d ) , l p ω ( Z d ) ⊂ l p ( Z d ) , l q µ ( Z d ) ⊂ l qm β ( Z d ) , τ = 1 . (1.14) Here, we write e m α ( z ) = e m ( z , and e m β ( z ) = e m (0 , z ) for z , z ∈ R d . For the case of power weight, we have following further characterization. See the character-izations of exponents in Section 5.
Theorem 1.6. (Third characterization of
BM W ) Assume p i , q i , p, q ∈ (0 , ∞ ] , s i , t i ∈ R , i = 1 , , τ ∈ [0 , . We have W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ W ( F L p , L qs )( R d ) if and only if l p /pps ( Z d ) ∗ l p /pps ( Z d ) ⊂ l q/pps ( Z d ) , l q /ppt ( Z d ) ∗ l q /ppt ( Z d ) ⊂ l q/ppt ( Z d ) , p < ∞ , (1.15) l p s ( Z d ) , l p s ( Z d ) ⊂ l qs ( Z d ) , l q t ( Z d ) , l q t ( Z d ) ⊂ l qt ( Z d ) , p ≥ q, (1.16) for τ ∈ (0 , , and l q t ( Z d ) , l p s ( Z d ) ⊂ l p ( Z d ) , l p s ( Z d ) , l q s + t ( Z d ) , l q t ( Z d ) ⊂ l qs ( Z d ) , τ = 0 , (1.17) l q t ( Z d ) , l p s ( Z d ) ⊂ l p ( Z d ) , l p s ( Z d ) , l q s + t ( Z d ) , l q t ( Z d ) ⊂ l qs ( Z d ) , τ = 1 . (1.18) WEICHAO GUO, JIECHENG CHEN, DASHAN FAN, AND GUOPING ZHAO
As mentioned in the beginning of this paper, the boundedness property of Wigner distribu-tion has closed connections with some important operators, for which we can deduce fruitfulnew boundedness results from our main Theorems 1.1 to 1.6. Here, we focus on the bounded-ness of pseudodifferential operators with symbols in modulation and Wiener amalgam spaces.Let us mention that the study of pseudodifferential operators has a long history in the fieldof classical harmonic analysis, we refer the reader to the pioneering works of Kohn–Nirenberg[19] and H¨ormander [18]. See also the famous H¨ormander class in [18]. The classical Calderon-Vaillancourt theorem [3] gives the L -boundedness of Kohn-Nirenberg operator with symbolsbelonging to the H¨ormander’s class S , , in which all the derivatives of symbols are requiredto be bounded.In the field of time-frequency analysis, the earliest work of pseudodifferential operators isdue to Sj¨ostrand [22], where the boundednss on L of pseudodifferential operators with symbolsin M ∞ , (Sj¨ostrand’s class) was obtained. Since S , ( M ∞ , , Sj¨ostrand’s result essentiallyextended the Calderon-Vaillancourt theorem. Then, Gr¨ochenig–Heil [15] and Gr¨ochenig [13]extended Sj¨ostrand’s result to the boundedness on all modulation spaces M p,q with 1 ≤ p, q ≤∞ .In this paper, we consider the problems for the boundedness on modulation spaces of pseudo-differential operators with symbols in modulation spaces (BPM) and the boundedness on mod-ulation spaces of pseudodifferential operators with symbols in Wiener amalgam spaces (BPW).By an equivalent characterization between BMM (or BMW) and BPM (or BPW), we giveseveral characterizations for BPM and BPW. See Section 6 for more details.We also point out that our methods and theorems for BPM and BPW can be extended tothe bilinear and even multilinear cases. See [2, 1] for the boundedness on modulation spaces ofmultilinear pseudodifferential operators with symbols in modulation spaces, and see a recentcontribution in [20] for symbols in some modified modulation spaces. We may revisit this topicof multilinear cases in the future.The rest of this paper is organized as follows. In Section 2, we recall some definitions offunction spaces we shall use. We also list some basic time-frequency representations associatedwith Wigner distribution, and recall the Gabor expansion of modulation spaces, which are thekey tools for our first characterizations in Theorems 1.1 and 1.4.Section 3 is devoted to the first characterizations of BMM and BMW. First, Theorem 1.1is proved by the help of the time-frequency tools mentioned in Section 2. Then, we establishthe relations between BMM and BMW in Proposition 3.3. Combining this with Theorem 1.1,we give the proof for the non-endpoint case of Theorem 1.4. Like Theorem 1.1, the endpointcase of 1.4 will be proved directly by the time-frequency tools.In Section 4, under some reasonable assumptions of weights, we give further characterizationsof BMM and BMW. The separation of convolution inequality, i.e. Proposition 4.1, yields theproof for Theorem 1.2 and the non-endpoint case of Theorem 1.5. The separation of mixed-norm embedding inequality, i.e. Proposition 4.2, yields the proof for the endpoint case ofTheorem 1.5.The power weight case will be handled in Section 5. The proof for Theorem 1.3 and the non-endpoint case of Theorem 1.6 follows directly by Theorems 1.2 and 1.5. The proof of endpointcase of Theorem 1.6 follows by the further separation of mixed-norm embedding, namely,Proposition 5.10. We also list Lemmas 5.1, 5.2 and 5.7 for further exponent characterizations.Then, several characterizations of exponents are established for BMM and BMW, see Theorems OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 7 W ( F L , L ∞ )( R d )can be founded in Theorems 6.6, 6.7 and 6.11, and Remarks 6.8 and 6.12 will be preparedfor the comparisons between them. At the end of this section, we give the sharp exponentsfor the boundedness on Sobolev spaces H s of pseudodifferential operators with symbols inWiener amalgam spaces, showing that the boundedness on Sobolev spaces can not happenwith W ( F L ∞ , L )( R d ) symbols. Notations:
Throughout this paper, we will adopt the following notations. Let C be apositive constant that may depend on d, p, q, p i , q i , s i , t i , m, m i , ω i , µ i , ( i = 1 , X . Y denotes the statement that X ≤ CY , and The notation X ∼ Y means the statement X . Y . X . The Schwartz function space is denoted by S ( R d ), and the space of tempereddistributions by S ′ ( R d ). We use the brackets h f, g i to denote the extension to S ′ ( R d ) × S ( R d )of the inner product h f, g i = ´ R d f ( x ) g ( x ) dx for f, g ∈ L ( R d ). We set I f ( x ) = f ( − x ) and D λ f ( x ) = f ( λx ) for λ ∈ R , x ∈ R d .2. PRELIMINARIES
Time-frequency representations.
The translation operator T x and modulation oper-ator M ξ are defined as T x f ( t ) = f ( t − x ) , M ξ f ( t ) = e πitξ f ( t ) . We recall that, as a bilinear map on L ( R d ) × L ( R d ), the STFT V g f can be extended to be amap from S ′ ( R d ) × S ( R d ) into S ′ ( R d ) by V g f ( x, ξ ) = h f, M ξ T x g i . In fact, for f ∈ S ′ ( R d ) and g ∈ S ( R d ), V g f is a continuous function on R d with polyno-mial growth, see [14, Theorem 11.2.3]. The so-called fundamental indentity of time-frequencyanalysis is as follows: V g f ( x, ξ ) = e − πix · ξ V ˆ g ˆ f ( ξ, − x ) , ( x, ξ ) ∈ R d . Next, we calculate the linear transform of STFT.
Lemma 2.1 (Linear transform of STFT) . Assume f, g ∈ L ( R d ) . Let L be a invertible lineartransform on R d . For a function f , denote f L ( x ) := f ( Lx ) . We have V φ L f L ( x, ξ ) = | det( L ) | − V φ f ( Lx, ( L − ) T ξ ) . Proof.
By a direct calculation, we have V φ L f L ( x, ξ ) = ˆ R d f ( Lt ) φ ( Lt − Lx ) e − πiLt · ( L − ) T ξ dt = | det( L ) | − ˆ R d f ( t ) φ ( t − Lx ) e − πit · ( L − ) T ξ dy = | det( L ) | − V φ f ( Lx, ( L − ) T ξ ) . (cid:3) WEICHAO GUO, JIECHENG CHEN, DASHAN FAN, AND GUOPING ZHAO
In the next lemma, we calculate the STFTs of of τ -Wigner distributions, which are the keytools for the estimates of τ -Wigner distributions on modulation spaces. We refer the readersto [6] for the process of calculations. Lemma 2.2 (STFT of τ -Wigner distribution) . Consider τ ∈ [0 , . Let Φ τ = W τ ( φ , φ ) fornonzero functions φ , φ ∈ S ( R d ) . Then the STFT of W τ ( f , f ) with respect to the window Φ τ is given by V Φ τ ( W τ ( f , f ))( z, ζ ) = e − πiz ζ V φ f ( z − τ ζ , z + (1 − τ ) ζ ) V φ f ( z + (1 − τ ) ζ , z − τ ζ ) . In particular, for τ = 0 , V Φ ( W ( f , f ))( z, ζ ) = e − πiz ζ V φ f ( z , z + ζ ) V φ f ( z + ζ , z ) . For τ = 1 , we have V Φ ( W ( f , f ))( z, ζ ) = e − πiz ζ V φ f ( z − ζ , z ) V φ f ( z , z − ζ ) . For τ = , we have V Φ ( W ( f , f ))( z, ζ ) = e − πiz ζ V φ f ( z − ζ , z + ζ V φ f ( z + ζ , z − ζ , = e − πiz ζ V φ f ( z − J ζ ) V φ f ( z + 12 J ζ ) , where J is the canonical symplectic matrix in R d defined by J = (cid:18) d × d I d × d − I d × d d × d (cid:19) . Lemma 2.3 (Connection between STFT and τ -Wigner distribution I) . For τ ∈ (0 , , f , f ∈ L ( R d ) , we have W τ ( f , f )( x, ξ ) = τ − d e πiτ − x · ξ V D − ττ I f f ( 11 − τ x, τ ξ ) . Proof. W τ ( f , f )( x, ξ ) = ˆ R d f ( x + τ t ) f ( x − (1 − τ ) t ) e − πit · ξ dt = ˆ R d f ( x + τ t )( D − ττ I f )( τ t − τ − τ x ) e − πit · ξ dt = τ − d ˆ R d f ( x + t )( D − ττ I f )( t − τ − τ x ) e − πiτ − t · ξ dt = τ − d e πiτ − x · ξ ˆ R d f ( t )( D − ττ I f )( t − − τ x ) e − πiτ − t · ξ dt = τ − d e πiτ − x · ξ V D − ττ I f f ( 11 − τ x, τ ξ ) . (cid:3) Lemma 2.4 (Connection between STFT and τ -Wigner distribution II) . F W τ ( f , f )( z ) = e − πiτz · z V f f ( − J z ) OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 9 Proof.
Write W τ ( f , f )( x, ξ ) = F − ( f ( x − τ · ) f ( x + (1 − τ ) · ))( ξ ) . Then, F W τ ( f , f )( z ) = F ( f ( · − τ z ) f ( · + (1 − τ ) z ))( z )= ˆ R d f ( x − τ z ) f ( x + (1 − τ ) z ) e − πix · z dx = e − πiτz · z ˆ R d f ( x ) f ( x + z ) e − πixz dx = e − πiτz · z V f f ( − z , z ) = e − πiτz · z V f f ( − J z ) . (cid:3) Function spaces.
As we mentioned above, modulation spaces are defined as a measureof the STFT of f ∈ S ′ . In order to draw a more accurate portrait of the decay and summabilityproperties of STFT, modulation space is usually be measured by the weighted norm. Let usrecall the definitions of weights we shall use.A weight function v on R d is called submultiplicative if v ( z + z ) ≤ v ( z ) v ( z ) for all z , z ∈ R d , a weight function m on R d is called v -moderate if m ( z + z ) ≤ Cv ( z ) m ( z ) , z , z ∈ R d . In this paper, we consider the v -moderate weights where v is submultiplicative with polynomialgrowth. We use the notation P ( R d ) to denote the cone of all non-negative functions whichare v -moderate. Similarly, we can define P ( R d ).The weighted mixed-norm spaces used to measure the STFT is defined as following. Definition 2.5 (Weighted mixed-norm spaces.) . Let m ∈ P ( R d ) , p, q ∈ (0 , ∞ ] . Then theweighted mixed-norm space L p,qm ( R d ) consists of all Lebesgue measurable functions on R d suchthat the (quasi-)norm k F k L p,qm ( R d ) = ˆ R d (cid:18) ˆ R d | F ( x, ξ ) | p m ( x, ξ ) p dx (cid:19) q/p dξ ! /q is finite, with usual modification when p = ∞ or q = ∞ . Now, we recall the definition of modulation space.
Definition 2.6.
Let < p, q ≤ ∞ , m ∈ P ( R d ) . Given a non-zero window function φ ∈S ( R d ) , the (weighted) modulation space M p,qm ( R d ) consists of all f ∈ S ′ ( R d ) such that thenorm k f k M p,qm ( R d ) := k V φ f ( x, ξ ) k L p,qm ( R d ) = ˆ R d (cid:18) ˆ R d | V φ f ( x, ξ ) m ( x, ξ ) | p dx (cid:19) q/p dξ ! /q is finite. We write M p,q for modulation space with m ≡ . Recall that the above definition of M p,qm is independent of the choice of window function φ .The readers may see this fact in [14] for the case ( p, q ) ∈ [1 , ∞ ] , and in [12] for full range( p, q ) ∈ (0 , ∞ ] . In particular, we point out that in [12], the author find a admissible windows,denoted by M p,qv , depending on p, q , for the modulation space M p,qm . Denote by M p,qm ( R d ) the S ( R d ) closure in M p,qm ( R d ). Note that M p,qm ( R d ) = M p,qm ( R d ) for p, q = ∞ .Next, we turn to the definition of Wiener amalgam space. As we mentioned in Section 1,in this paper we consider the Wiener amalgam space of the type of the image of modulationspace under Fourier transform. Write k f k F M p,qm ( R d ) = k F − f k M p,qm ( R d ) = k V ˇ φ ˇ f ( x, ξ ) k L p,qm ( R d ) = k V φ f ( ξ, − x ) k L p,qm ( R d ) = k ( V φ f )( J ( x, ξ )) k L p,qm ( R d ) . The Wiener amalgam space can be also defined by the weighted mixed-norm of STFT.
Definition 2.7.
Let < p, q ≤ ∞ , m ∈ P ( R d ) . Given a non-zero window function φ ∈S ( R d ) , the (weighted) Wiener amalgam space F ( M p,qm ) consists of all f ∈ S ′ ( R d ) such that thenorm k f k F M p,qm ( R d ) = k V φ f ( ξ, − x ) k L p,qm ( R d ) = ˆ R n (cid:18) ˆ R n | V φ f ( ξ, − x ) m ( x, ξ ) | p dx (cid:19) q/p dξ ! /q is finite. In particular, for f ∈ S ′ ( R d ), φ ∈ S ( R d ), k f k F Mp,q ⊗ m ( R d ) = ˆ R d (cid:18) ˆ R d | V φ f ( ξ, − x ) | p dx (cid:19) q/p m ( ξ ) q dξ ! /q . Using the notation of Wiener amalgam space in [10], we have F M p,q ⊗ m ( R d ) = W ( F L p , L qm )( R d ) . This representation makes us more clear that f belongs to F M p,q ⊗ m ( R d ) means it lies locallyin F L p ( R d ) and globally in L qm ( R d ).Next, we collect following calculations for the linear transform of modulation and Wieneramalgam spaces. Lemma 2.8.
Let < p, q ≤ ∞ , L be a invertible linear transform on R d , m ∈ P ( R d ) . Fora function f on R d , denote f L ( x ) := f ( Lx ) . We have k f L k M p,qm ( R d ) ∼ k f k M p,q e m ( R d ) , where e m ( x, ξ ) = m ( L − x, L T ξ ) , x, ξ ∈ R d , i = 1 , .Proof. Using Lemma 2.1, we write k f L k M p,qm = k V φ L f L ( x, ξ ) k L p,qm ∼k V φ f ( Lx, ( L − ) T ξ ) k L p,qm ∼ k V φ f ( x, ξ ) k L p,q e m ∼ k f k M p,q e m . (cid:3) Lemma 2.9.
Let < p, q ≤ ∞ , L be a invertible linear transform on R d , m ∈ P ( R d ) . Fora function f on R d , denote f L ( x ) := f ( Lx ) . We have k f L k W ( F L p ,L qm )( R d ) ∼ k f k W ( F L p ,L q e m )( R d ) , where e m ( z ) = m ( L − z ) . OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 11 Proof.
By Lemma 2.1 and Definition 2.7, we have k f L k W ( F L p ,L qm )( R d ) = k V φ L f L ( ζ, − z ) k L p,q ⊗ m ∼k V φ f ( Lζ, − ( L − ) T z ) k L p,q ⊗ m ∼k V φ f ( ζ, − z ) k L p,q ⊗ e m ∼ k f k W ( F L p ,L q e m )( R d ) . (cid:3) Next, we recall a multiplication property of Wiener amalgam space.
Lemma 2.10.
Let < p, q ≤ ∞ , ˙ p = min { p, } . We have W ( F L p , L qm ) · W ( F L ˙ p , L ∞ ) ⊂ W ( F L p , L qm ) . Proof.
Using the relation between modulation and Wiener amalgam space, the desired conclu-sion is equivalent to M p,q ⊗ m ( R d ) ∗ M ˙ p, ∞ ⊂ M p,q ⊗ m ( R d ) . This is a direct conclusion of [16, Theorem 1.3] with the fact l p ∗ l ˙ p ⊂ l p , l qm · l ∞ ⊂ l qm . (cid:3) Lemma 2.11 (Chirp function) . For any λ = 0 , we have G λ ( x, ξ ) = e πiλx · ξ ∈ W ( F L ˙ p , L ∞ )( R d ) . Moreover, for any function F on R d , we have k F G λ k W ( F L p ,L qm )( R d ) ∼ k F k W ( F L p ,L qm )( R d ) . Proof.
Using Lemma 2.9, we only need to consider the case λ = 1, i.e., to verify that G = e πix · ξ ∈ W ( F L ˙ p , L qm ). Denote by g ( x, ξ ) = e − π ( | x | + | ξ | ) the Gaussian function. By thecalculation in [5, Proposition 3.2], we have | V g G ( z, ζ ) | = 2 − d/ e − π | z − ζ | e − π | z − ζ | . By Definition 2.7, we conclude that k G k W ( F L ˙ p ,L ∞ )( R d ) = k V g G ( ζ, − z ) k L ˙ p, ∞ ( R d ) ∼k e − π | ζ + z | e − π | ζ + z | k L ˙ p, ∞ ( R d ) ∼ . This completes the proof of G λ ∈ W ( F L ˙ p , L ∞ ).Moreover, by the multiplication property (Lemma 2.10), we deduce that k F G λ k W ( F L p ,L qm ) . k F k W ( F L p ,L qm ) k G λ k W ( F L ˙ p ,L ∞ ) . k F k W ( F L p ,L qm ) = k F G λ G − λ k W ( F L p ,L qm ) . k F G λ k W ( F L p ,L qm ) . From this, we obtain k F G λ k W ( F L p ,L qm ) ∼ k F k W ( F L p ,L qm ) . (cid:3) In order to measure the summability and decay properties of Gabor coefficients, we recallthe discrete weighted mixed-norm space.
Definition 2.12 (Discrete mixed-norm spaces I) . Let < p, q ≤ ∞ , m ∈ P ( R d ) . The space l p,qm ( Z d ) consists of all sequences ~a = { a k,n } k,n ∈ Z d for which the (quasi-)norm k ~a k l p,qm ( Z d ) = X n ∈ Z d X k ∈ Z d | a k,n | p m ( k, n ) p q/p /q is finite. In Theorem 1.5, we use another type of discrete weighted mixed-norm space as following.
Definition 2.13 (Discrete mixed-norm spaces II) . Let < p, q ≤ ∞ , m ∈ P ( R d ) . The space l ( p,q ) m ( Z d ) consists of all sequences ~a = { a k,n } k,n ∈ Z d for which the (quasi-)norm k ~a k l ( p,q ) m ( Z d ) = X k ∈ Z d X n ∈ Z d | a k,n | q m ( k, n ) q p/q /p is finite. As usual for ω ∈ P ( R d ), the space l pω ( Z d ) consists of all ~b = { b k } k ∈ Z d for which the (quasi-)norm k ~b k l pω ( Z d ) = X k ∈ Z d | b k | p ω ( k ) p /p is finite. For ω = v s , we write l pv s := l ps for simplicity.2.3. Gabor analysis of modulation spaces.
Comparing with the classical definition ofmodulation space in Definition 2.6, or the semi-discrete definition such as in [16, Proposition2.1] in the same way as Besov spaces, the modulation spaces can be also characterized by thesummability and decay properties of their Gabor coefficients, this is an important reason whythe modulation spaces play the central role in the field of time-frequency analysis.We recall some important operators which are the key tools for the discretization of modu-lation spaces.
Definition 2.14.
Assume that g, γ ∈ L ( R d ) and α, β > . The coefficient operator or analysisoperator C α,βg is defined by C α,βg f = ( h f, T αk M βn g i ) k,n ∈ Z d . The synthesis operator or reconstruction operator D α,βg is defined by D α,βγ ~c = X k ∈ Z d X n ∈ Z d c k,n T αk M βn γ. The Gabor frame operator S α,βg,γ is defined by S α,βg,γ f = D α,βγ C α,βg f = X k ∈ Z d X n ∈ Z d h f, T αk M βn g i T αk M βn γ. In order to extend the boundedness result of analysis operator and synthesis operator to themodulation spaces of full range, following admissible window class was introduced in [12].
OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 13 Definition 2.15 (The space of admissible windows) . Assume that m is v -moderate and let < p, q ≤ ∞ . Let r = min { , p } and s = min { , p, q } . For r , s > , denote ω r ,s ( x, ω ) = v ( x, ω ) · (1 + | x | ) r · (1 + | ω | ) s . Define the space of admissible windows M p,qv for the modulation space M p,qm to be M p,qv = [ r >d/rs >d/s ≤ p < ∞ M p ω r ,s . Based on the window class mentioned above, we recall the boundedness of C α,βg and D α,βg ,which works on the full range p, q ∈ (0 , ∞ ]. Lemma 2.16.
Assume that m is v -moderate, p, q ∈ (0 , ∞ ] , and g belongs to the subclass M p ω r ,s of M p,qv . Then the analysis operator C α,βg is boundedness from M p,qm into l p,q ˜ m , and thesynthesis operator D α,βg is boundedness form l p,q ˜ m into M p,qm for all α, β > , where ˜ m ( k, n ) = m ( αk, βn ) . Now, we recall the main theorem in [12], which extends the Gabor expansion of modulationspaces to the full range 0 < p, q ≤ ∞ . Theorem 2.17. (see [12] ) Assume that m is v -moderate, p, q ∈ (0 , ∞ ] , g, γ ∈ M p,qv , and thatthe Gabor frame operator S α,βg,γ = D α,βγ C α,βg = I on L ( R d ) . Then f = X k ∈ Z d X n ∈ Z d h f, T αk M βn g i T αk M βn γ = X k ∈ Z d X n ∈ Z d h f, T αk M βn γ i T αk M βn g with unconditional convergence in M p,qm if p, q < ∞ , and with weak-star convergence in M ∞ /v otherwise. Furthermore there are constants A, B > such that for all f ∈ M p,qm A k f k M p,qm ≤ X n ∈ Z d X k ∈ Z d |h f, T αk M βn g i| p m ( αk, βn ) p q/p /q ≤ B k f k M p,qm with obvious modification for p = ∞ or q = ∞ . Likewise, the quasi-norm equivalence A ′ k f k M p,qm ≤ X n ∈ Z d X k ∈ Z d |h f, T αk M βn γ i| p m ( kα, nβ ) p q/p /q ≤ B ′ k f k M p,qm holds on M p,qm . The following well known theorem provide a way to find the Gabor frame of L ( R d ). Recallthat k g k W ( L ∞ ,L )( R d ) = P n ∈ Z d k gχ Q + n k L ∞ with Q = [0 , d . Theorem 2.18. (Walnut [26] ) Suppose that g ∈ W ( L ∞ , L )( R d ) satisfying A ≤ X k ∈ Z d | g ( x − αk ) | ≤ B a.e. for constants
A, B ∈ (0 , ∞ ) . Then there exists a constant β depending on α such that G ( g, α, β ) is a Gabor frame of L ( R d ) for all β ≤ β . In order to find the dual window in a suitable function space, the following result is impor-tant.
Theorem 2.19. ( [14] ) Assume that g ∈ M v ( R d ) and that { T αk M βn g } k,n ∈ Z d is a Gabor framefor L ( R d ) . Then the Gabor frame operator S α,βg,g is invertible on M v ( R d ) . As a consequence, S α,βg,g is invertible on all modulation spaces M p,qm ( R d ) for ≤ p, q ≤ ∞ and m ∈ P ( R d ) . First characterizations: discretizations by time-frequency tools
Discretization by Gabor coefficients for BMM.Proposition 3.1.
Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , . For any α > , we have W : M p ,q m ( R d ) × M p ,q m ( R d ) −→ M p,q ⊗ m ( R d ) if and only if k ( a ( k ) b ( k + J n )) k l p,q ⊗ e m ( Z d × Z d ) . k ( a ( k )) k l p ,q f m ( Z d ) · k ( b ( k )) k l p ,q f m ( Z d ) . (3.1) Here, we denote e m ( z ) := m ( αz ) and f m i ( z ) := m i ( αz ) for z ∈ Z d , i = 1 , .Proof. We divide the proof into two parts. “Only if ” part.
Let ϕ be a smooth function supported in ( − α/ , α/ d , satisfying ˆ ϕ ( x ) ≥ ϕ (0) = 1. For any α > ~a = ( a j,l ) j,l ∈ Z d and ~b = ( b j,l ) j,l ∈ Z d , we set f = D α,αϕ ~a = X j,l ∈ Z d a j,l T αj M αl ϕ, f = D α,αϕ ~b = X j,l ∈ Z d b j,l T αj M βl ϕ. Take the window Φ = W ( φ, φ ) with φ ∈ S ( R d ) supported in ( − α/ , α/ d , satisfying ˆ φ ( x ) ≥ φ (0) = 1. By the sampling property of STFT (see Lemma 2.16), k V Φ ( W ( f , f )) | α Z d × α Z d k l p,q ⊗ e m . k V Φ ( W ( f , f )) k L p,q ⊗ m = k W ( f , f ) k M p,q ⊗ m . (3.2)Let us turn to the lower bound estimates of the first term in (3.2). Using Lemma 2.2, we write k V Φ ( W ( f , f )) | α Z d × α Z d k l p,q ⊗ e m = k V φ f ( z , z + ζ ) V φ f ( z + ζ , z ) | α Z d × α Z d k l p,q ⊗ e m = k V φ f ( z ) V φ f ( z + J ζ ) | α Z d × α Z d k l p,q ⊗ e m = k ( V φ f ( αk , αk ) V φ f ( α ( k + n ) , α ( k − n ))) k l p,q ⊗ e m . By the support of φ and ϕ , we estimate | V φ f ( αk , αk ) | by | V φ f ( αk , αk ) | = |h f , T αk M αk φ i| = (cid:12)(cid:12)(cid:12) h X j,l a j,l T αj M αl ϕ, T αk M αk φ i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h X l a k ,l T αk M αl ϕ, T αk M αk φ i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X l a k ,l h M αl ϕ, M αk φ i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X l a k ,l h ϕ, M α ( k − l ) φ i (cid:12)(cid:12)(cid:12) , where the interchange of the order of summation and integration is valid since the number ofterms is finite. Using the nonnegativity of ~a , ~b , ˆ φ and ˆ ϕ , we obtain X l a k ,l h ϕ, M α ( k − l ) φ i = X l a k ,l h ˆ ϕ, T α ( k − l ) ˆ φ i ≥ a k ,k h ˆ ϕ, ˆ φ i & a k ,k . OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 15 From the above two estimates, we get the lower bound of | V φ f ( αk , αk ) | : | V φ f ( αk , αk ) | & a k ,k . A similar calculation yields the lower bound of | V φ f ( α ( k + n ) , α ( k − n )) | : | V φ f ( α ( k + n ) , α ( k − n )) | & b k + n ,k − n . By the lower bounds of | V φ f ( αk , αk ) | and | V φ f ( α ( k + n ) , α ( k − n )) | , we obtain k V Φ ( W ( f , f )) | Z d × Z d k l p,q ⊗ e m = k V φ f ( αk , αk ) V φ f ( α ( k + n ) , α ( k − n )) | Z d × Z d k l p,q ⊗ e m & k ( a k ,k b k + n ,k − n ) k l p,q ⊗ e m = k a ( k ) b ( k + J n ) k l p,q ⊗ e m ( Z d × Z d ) . Using this and (3.2), we obtain the lower estimate of k W ( f , f ) k M p,q ⊗ m : k W ( f , f ) k M p,q ⊗ m & k a ( k ) b ( k + J n ) k l p,q ⊗ e m ( Z d × Z d ) . (3.3)On the other hand, using the boundedness of synthesis operator in Lemma 2.16, we get theupper bound estimates of k f i k M pi,qimi : k f k M p ,q m = k D α,αϕ ~a k M p ,q m . k ~a k l p ,q f m ( Z d ) , k f k M p ,q m = k D α,αϕ ~b k M p ,q m . k ~b k l p ,q f m ( Z d ) . (3.4)Combining the estimates (3.3) and (3.4), we conclude the desired inequality by k a ( k ) b ( k + J n ) k l p,q ⊗ e m ( Z d × Z d ) . k W ( f , f ) k M p,q ⊗ m . k f k M p ,q m k f k M p ,q m . k ~a k l p ,q f m ( Z d ) k ~b k l p ,q f m ( Z d ) . By a standard limiting argument, the above inequality is valid for all sequences ~a and ~b . “If ” part. For any fixed α >
0, if (3.1) is valid, we claim that it also holds for αN with anypositive integer N : that is, (3.1) implies k ( a ( k ) b ( k + J n )) k l p,q ⊗ ee m ( Z d × Z d ) . k ( a ( k )) k l p ,q ff m ( Z d ) · k ( b ( k )) k l p ,q ff m ( Z d ) , where ee m ( z ) = e m ( zN ) = m ( αzN ), ff m i ( z ) = f m i ( zN ) = m i ( αzN ). DenoteΛ = [0 , N ) d ∩ Z d , Γ j,l = ( j, l ) + N Z d , ( j, l ) ∈ Λ . We have Z d = S ( j,l ) ∈ Λ Γ j,l and k ( a ( k ) b ( k + J n )) k l p,q ⊗ ee m ( Z d × Z d ) ∼ X ( j,l ) ∈ Λ k ( a ( k ) b ( k + J n )) k l p,q ⊗ ee m (Γ j,l ) ∼ X ( j,l ) ∈ Λ k a ( N k + j ) b ( N ( k + J n ) + ( j + J l )) ee m ( N n + l ) k l p,q ( Z d × Z d ) ∼ X ( j,l ) ∈ Λ k a ( N k + j ) b ( N ( k + J n ) + ( j + J l )) e m ( n ) k l p,q ( Z d × Z d ) = X ( j,l ) ∈ Λ k a ( N k + j ) b ( N ( k + J n ) + ( j + J l )) k l p,q × e m ( Z d × Z d ) . Then, we continue this estimate by applying (3.1): X ( j,l ) ∈ Λ k a ( N k + j ) b ( N ( k + J n ) + ( j + J l )) k l p,q × e m ( Z d × Z d ) . X ( j,l ) ∈ Λ k a ( N k + j ) k l p ,q f m ( Z d ) k b ( N k + j + J l ) k l p ,q f m ( Z d ) = X ( j,l ) ∈ Λ k a ( N k + j ) f m ( k ) k l p ,q ( Z d ) k b ( N k + j + J l ) f m ( k ) k l p ,q ( Z d ) ∼ X ( j,l ) ∈ Λ k a ( N k + j ) ff m ( N k + j ) k l p ,q ( Z d ) k b ( N k + j + J l ) ff m ( N k + j + J l ) k l p ,q ( Z d ) . X ( j,l ) ∈ Λ k a ( k ) k l p ,q ff m ( Z d ) k b ( k ) k l p ,q ff m ( Z d ) . k a ( k ) k l p ,q ff m ( Z d ) k b ( k ) k l p ,q ff m ( Z d ) , where in the last inequality we use the fact | Λ | < ∞ . The claim follows by the above twoestimates. From this claim, we find that if (3.1) holds for some α >
0, then α can be takento be sufficiently small. Now, we turn to verify the boundedness of W under the assumptionthat (3.1) holds for some α > W ( φ, φ ) ∈ S ( R d ) with φ ∈ S ( R d ). There exists a sufficiently large integer N such that for suitable constants A, B ∈ (0 , ∞ ) A ≤ X k ∈ Z d | Φ( x − αN k ) | ≤ B. Denote e α = αN . Using Theorem 2.18, there exists a constant β = e α/N = αN N with suffi-ciently large integer N such that G (Φ , e α, β ) is a Gabor frame of L ( Z d ). Let Ψ = ( S e α,β Φ , Φ ) − Φbe the canonical dual widow of Φ. Note that Φ ∈ S ⊂ M p,qv , then Definition 2.15 and Theorem2.19 implies that Ψ ∈ M p,qv . By the definitions of Φ and Ψ, we have S e α,β Φ , Ψ = D e α,β Ψ C e α,β Φ = I on L ( R d ). Then Theorem 2.17 yields that f = S e α,β Φ , Ψ f for all f ∈ M p,q ⊗ m . Recalling β = e α/N and using Lemma 2.2, we find that k C e α,β Φ W ( f , f ) k l p,q ⊗ e m = k V Φ ( W ( f , f ))( z, ζ ) | e α Z d × β Z d k l p,q ⊗ e m ≤k V Φ ( W ( f , f ))( z, ζ ) | β Z d × β Z d k l p,q ⊗ e m = k V φ f ( z , z + ζ ) V φ f ( z + ζ , z ) | β Z d × β Z d k l p,q ⊗ e m = k V φ f ( z ) V φ f ( z + J ζ ) | β Z d × β Z d k l p,q ⊗ e m = k V φ f ( βk ) V φ f ( β ( k + J n )) k l p,q ⊗ e m ( Z d × Z d ) . Using the inequality of discrete mixed-norm spaces (3.1) with αN N and Lemma 2.16, wecontinue the above estimate by k ( V φ f ( βk ) V φ f ( β ( k + J n ))) k l p,q ⊗ e m ( Z d × Z d ) . k ( V φ f ( βk )) k l p ,q f m ( Z d ) · k ( V φ f ( βk )) k l p ,q f m ( Z d ) = k V φ f ( z ) | β Z d × β Z d k l p ,q f m k V φ f ( z ) | β Z d × β Z d k l p ,q f m . k f k M p ,q m · k f k M p ,q m . The above two estimates imply that k C e α,β Φ W ( f , f ) k l p,q ⊗ e m . k f k M p ,q m · k f k M p ,q m . OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 17 Hence, k W ( f , f ) k M p,q ⊗ m = k D e α,β Ψ C e α,β Φ W ( f , f ) k M p,q ⊗ m . k C e α,β Φ W ( f , f ) k l p,q ⊗ e m . k f k M p ,q m · k f k M p ,q m . We have now completed the proof. (cid:3)
We are now in a position to give the proof of Theorem 1.1. As we will see, it follows byProposition 3.1 with some changes of variables.
Proof of Theorem 1.1.
Let Φ τ = W τ ( φ, φ ) with nonzero function φ ∈ S ( R d ). Using Lemma2.2, we obtain that k W τ ( f , f ) k M p,q ⊗ m = k V Φ τ W τ ( f , f ) k L p,q ⊗ m = k V φ f ( z − τ ζ , z + (1 − τ ) ζ ) V φ f ( z + (1 − τ ) ζ , z − τ ζ ) k L p,q ⊗ m = k V φ f ( z , z + ζ ) V φ f ( z + ζ , z ) k L p,q ⊗ m = k V Φ W ( f , f ) k L p,q ⊗ m = k W ( f , f ) k M p,q ⊗ m . Hence, the boundedness property: W τ : M p ,q m ( R d ) × M p ,q m ( R d ) −→ M p,q ⊗ m ( R d )is independent with τ ∈ [0 , τ = 0.Write (3.1) with α = 1 by (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a k ,k b k + n ,k − n | p (cid:19) q/p m ( n , n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | b k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q , which is equivalent to (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a k ,k b − n − k ,n − k | p (cid:19) q/p m ( n , n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | b − k , − n m ( k , n ) | p (cid:19) q /p (cid:19) /q , After some change of variables, one can find the following equivalent form: (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a k ,k b n − k ,n − k | p (cid:19) q/p m ( n , − n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | b k ,n m ( − k , − n ) | p (cid:19) q /p (cid:19) /q . This is equivalent to the desired inequality k ( a k ,k b n − k ,n − k ) k l p,q ⊗ mJ . k ~a k l p ,q m ( Z d ) k ~b k l p ,q I m ( Z d ) . In particular, when p < ∞ , this is equivalent to the convolution inequality: l p /p,q /pm p ( Z d ) ∗ l p /p,q /p I m p ( Z d ) ⊂ l q/p,q/pm pJ ( Z d ) . Hence, the conclusion in Theorem 1.1 (with τ = 0) follows directly by Proposition 3.1. (cid:3) Remark 3.2.
The reader may observe that, in the proof of Theorem 1.1, we only use Proposi-tion 3.1 with α = 1 . However, in order to verify Proposition 3.1 with α = 1 , we actually need (3.1) for sufficiently small α . Thus, we would like to keep a stronger version with any α > . Relations between BMM and BMW for τ ∈ (0 , .Proposition 3.3. Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , . For τ ∈ (0 , , we have k W τ ( f , f ) k M p,q ⊗ e mJ − . k f k M p ,q m · k f k M p ,q I f m (3.5) if and only if k W τ ( f , f ) k W ( F L p ,L qm ) . k f k M p ,q m · k f k M p ,q m , (3.6) where e m ( ζ , ζ ) = m ((1 − τ ) ζ , τ ζ ) , f m ( z , z ) = m ( − ττ z , τ − τ z ) .Proof. Using Lemmas 2.3, 2.9 and 2.11, we obtain k W τ ( f , f ) k W ( F L p ,L qm ) ∼k V D − ττ I f f ( 11 − τ x, τ ξ ) k W ( F L p ,L qm ) ∼k V D − ττ I f f k W ( F L p ,L q e m ) , (3.7)where e m ( ζ , ζ ) = m ((1 − τ ) ζ , τ ζ ). On the other hand, recalling W ( F L p , L q e m J − ) = F M p,q ⊗ e m J − ,and using Lemma 2.4 and Lemma 2.11, we obtain k W τ ( f , f ) k M p,q ⊗ e mJ − = k F ( W τ ( f , f )) k W ( F L p ,L q e mJ − ) = k e − πiτu · u V f f ( − J u ) k W ( F L p ,L q e mJ − ) ∼k V f f ( − J u ) k W ( F L p ,L q e mJ − ) . Observing that J − = − J , and applying Lemma 2.9, we continue the above estimate by k W τ ( f , f ) k M p,q ⊗ e mJ − = k V f f ( − J u ) k W ( F L p ,L q e mJ − ) ∼ k V f f k W ( F L p ,L q e m ) . (3.8)Using (3.7) and (3.8), we obtain k W τ ( f , f ) k W ( F L p ,L qm ) ∼ k W τ ( f , D − ττ I f ) k M p,q ⊗ e mJ − . Moreover, by Lemma 2.8, we have k f k M p ,q m = kD τ − τ ID − ττ I f k M p ,q m ∼ kD − ττ I f k M p ,q I f m , where f m ( z , z ) = m ( − ττ z , τ − τ z ).If (3.5) holds, (3.6) follows by k W τ ( f , f ) k W ( F L p ,L qm ) ∼k W τ ( f , D − ττ I f ) k M p,q ⊗ e mJ − . k f k M p ,q m · kD − ττ I f k M p ,q I e m ∼ k f k M p ,q m · k f k M p ,q m . OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 19 Viceversa, if (3.6) holds, then (3.5) follows by k W τ ( f , f ) k M p,q ⊗ e mJ − = k W τ ( f , D − ττ ID τ − τ I f ) k M p,q ⊗ e mJ − ∼k W τ ( f , D τ − τ I f ) k W ( F L p ,L qm ) . k f k M p ,q m · kD τ − τ I f k M p ,q m ∼ k f k M p ,q m k f k M p ,q I f m . (cid:3) By the above proposition, we can prove Theorem 1.4 for τ ∈ (0 , Proof of Theorem 1.4 for τ ∈ (0 , . Observe that ( e m J − ) J = e m and I ( I f m ) = f m . The de-sired conclusion follows by Theorem 1.1 and Proposition 3.3. (cid:3) Discretization by Gabor coefficients for BMW with endpoints τ = 0 , . Proof of Theorem 1.4 for τ = 0 . We divide the proof into two parts. “Only if ” part.
Let f , f , Φ be the same functions in the proof of Theorem 3.1. Write k W ( f , f ) k W ( F L p ,L qm ) = k F − ( W ( f , f )) k M p,q ⊗ m = k V ˇΦ ( F − ( W ( f , f )))( z, ζ ) k L p,q ⊗ m = k V Φ ( W ( f , f ))( ζ, − z ) k L p,q ⊗ m = k V φ f ( ζ , ζ − z ) V φ f ( ζ − z , ζ ) k L p,q ⊗ m = k V φ f ( ζ , z ) V φ f ( z , ζ ) k L p,q ⊗ m . By the same argument in the proof of Proposition 3.1, we have the lower estimate: k W ( f , f ) k W ( F L p ,L qm ) & k V φ f ( n , k ) V φ f ( k , n ) | Z d × Z d k l p,q ⊗ m & k ( a n ,k b k ,n ) k l p,q ⊗ m , and the upper bound estimates k f k M p ,q m . k ~a k l p ,q m ( Z d ) , k f k M p ,q m . k ~b k l p ,q m ( Z d ) . Combin-ing the above two estimates, we deduce that k ( a n ,k b k ,n ) k l p,q ⊗ m . k W ( f , f ) k M p,q ⊗ m . k f k M p ,q m k f k M p ,q m . k ~a k l p ,q m ( Z d ) k ~b k l p ,q m ( Z d ) . Note that, by the same method in the proof of Proposition 3.1, the above inequality is also validif m , m i are replaced by e m and f m i respectively. Here e m ( z ) := m ( zN ), and f m i ( z ) := m i ( zN ), z ∈ R d , N ∈ N , i = 1 , “If ” part. As in the proof of Theorem 3.1, for Φ ∈ S ( R d ) mentioned above, there exists e Ψ ∈ M p,qv , such that S α,β ˇΦ , e Ψ = D α,β e Ψ C α,β ˇΦ = I on M p,q ⊗ m , where α = 1 /N , β = α/N , with somelarge integers N , N .Applying Lemma 2.2, we find that k C α,β ˇΦ F − W ( f , f ) k l p,q ⊗ e m = k V ˇΦ ( F − W ( f , f ))( z, ζ ) | α Z d × β Z d k l p,q ⊗ e m ≤k V ˇΦ ( F − W ( f , f ))( z, ζ ) | β Z d × β Z d k l p,q ⊗ e m = k V φ f ( ζ , z ) V φ f ( z , ζ ) | β Z d × β Z d k l p,q ⊗ e m . k V φ f ( β ( n , k )) V φ f ( β ( k , n )) k l p,q ⊗ e m ( Z d × Z d ) . Using the inequality of discrete mixed-norm spaces and the sampling property of STFT, wecontinue the above estimate by k V φ f ( β ( n , k )) V φ f ( β ( k , n )) k l p,q ⊗ e m ( Z d × Z d ) . k ( V φ f ( βk )) k l p ,q f m ( Z d ) · k ( V φ f ( βk )) k l p ,q f m ( Z d ) = k V φ f ( z ) | β Z d × β Z d k l p ,q f m k V φ f ( z ) | β Z d × β Z d k l p ,q f m . k f k M p ,q m · k f k M p ,q m . The above two estimates imply that k C α,β ˇΦ F − W ( f , f ) k l p,q ⊗ e m . k f k M p ,q m · k f k M p ,q m . Hence, k W ( f , f ) k W ( F L p ,L qm ) = k F − W ( f , f ) k M p,q ⊗ m = k D α,β e Ψ C α,β ˇΦ F − W ( f , f ) k M p,q ⊗ m . k C α,β ˇΦ F − W ( f , f ) k l p,q ⊗ e m . k f k M p ,q m · k f k M p ,q m . We have now completed the proof. (cid:3)
Proof of Theorem 1.4 for τ = 1 . A direct calculation yields that k W ( f , f ) k W ( F L p ,L qm ) = k W ( f , f ) k W ( F L p ,L qm ) = k V ¯Φ ( W ( f , f ))( ζ, − z ) k L p,q ⊗ m = k V Φ ( W ( f , f ))( ζ, z ) k L p,q ⊗ m = k V Φ ( W ( f , f ))( ζ, − z ) k L p,q ⊗ m = k W ( f , f ) k W ( F L p ,L qm ) . The desired result follows by this and the case of τ = 0. (cid:3) Second characterizations: separations of mixed-norm inequalities
Separation of mixed-norm convolution inequality.Proposition 4.1 (Separation of convolution) . Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , . Supposethat m is submultiplicative, m i = ω i ⊗ µ i , i = 1 , . We have k ( a k ,k b n − k ,n − k ) k l p,q ⊗ m ( Z d ) . k ~a k l p ,q m ( Z d ) k ~b k l p ,q m ( Z d ) (4.1) if and only if l p /pω p ( Z d ) ∗ l p /pω p ( Z d ) ⊂ l q/pm pα ( Z d ) , l q /pµ p ( Z d ) ∗ l q /pµ p ( Z d ) ⊂ l q/pm pβ ( Z d ) , p < ∞ , (4.2) l p ω ( Z d ) , l p ω ( Z d ) ⊂ l qm α ( Z d ) , l q µ ( Z d ) , l q µ ( Z d ) ⊂ l qm β ( Z d ) , p ≥ q. (4.3) Here, we denote m α ( z ) = m ( z , and m β ( z ) = m (0 , z ) for z , z ∈ R d .Proof. We divide the proof into two parts. “Only if ” part.
In this part, we separate (4.1) by means of testing it by several constructeddiscrete sequences.
OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 21 Test 1.
Set a k ,n = 0 if n = 0 Z d , and b k ,n = 0 if n = 0 Z d . The inequality (4.1) saysthat (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k , b n − k , | p (cid:19) q/p m ( n , q (cid:19) /q . (cid:18) X k ∈ Z d | a k , m ( k , | p (cid:19) /p (cid:18) X k ∈ Z d | b k , m ( k , | p (cid:19) /p . (4.4)Note that for p < ∞ the above inequality is equivalent to l p /pω p ( Z d ) ∗ l p /pω p ( Z d ) ⊂ l q/pm pα ( Z d ).Moreover, if we further assume b k ,n = 0 for all n , k = 0 Z d , (4.4) implies the embeddingrelation l p ω ( Z d ) ⊂ l qm α ( Z d ). On the other hand, by taking a k ,n = 0 for all k , n = 0 Z d , weget l p ω ( Z d ) ⊂ l qm α ( Z d ). Test 2.
Set a k ,n = 0 if k = 0 Z d , and b k ,n = 0 if k = 0 Z d . The inequality (4.1) says that (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a ,k b ,n − k | p (cid:19) q/p m (0 , n ) q (cid:19) /q . (cid:18) X n ∈ Z d | a ,n m (0 , n ) | q (cid:19) /q (cid:18) X n ∈ Z d | b ,n m (0 , n ) | q (cid:19) /q . (4.5)If p < ∞ , the above inequality is equivalent to l q /pµ p ( Z d ) ∗ l q /pµ p ( Z d ) ⊂ l q/pm pβ ( Z d ). By taking b k ,n = 0 for all n , k = 0 Z d , (4.4) becomes the embedding relation l q µ ( Z d ) ⊂ l qm β ( Z d ).Similarly, by taking a k ,n = 0 for all k , n = 0 Z d , we get l q µ ( Z d ) ⊂ l qm β ( Z d ).From Tests 1 and 2, we get (4.2) and (4.3). “If ” part. In this part, we consider following cases.
Case 1: p < q . In this case, we will verify (4.2) implies (4.1). Using the Minkowskiinequality and m ( n , n ) . m α ( n ) m β ( n ), we conclude that (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a k ,k b n − k ,n − k | p (cid:19) q/p m ( n , n ) q (cid:19) /q . (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a k ,k b n − k ,n − k | p (cid:19) q/p m α ( n ) q m β ( n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,k b n − k ,n − k | p (cid:19) q/p m α ( n ) q (cid:19) p/q (cid:19) q/p m β ( n ) q (cid:19) /q . Using the convolution inequality l p /pω p ( Z d ) ∗ l p /pω p ( Z d ) ⊂ l q/pm pα ( Z d ), we continue the above estimateby (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,k b n − k ,n − k | p (cid:19) q/p m α ( n ) q (cid:19) p/q (cid:19) q/p m β ( n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d k ( a k ,k ) k k pl p ω k ( b n ,n − k ) n k pl p ω (cid:19) q/p m β ( n ) q (cid:19) /q . Then, the convolution inequality l q /pµ p ( Z d ) ∗ l q /pµ p ( Z d ) ⊂ l q/pm pβ ( Z d ) further implies that (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d k ( a k ,k ) k k pl p ω k ( b n ,n − k ) n k pl p ω (cid:19) q/p m β ( n ) q (cid:19) /q . k ( k ( a k ,n ) k k l p ω ) n k l q µ k ( k ( b k ,n ) k k l p ω ) n k l q µ = k ~a k l p ,q ω ⊗ µ k ~b k l p ,q ω ⊗ µ . Case 2: p ≥ q . In this case, we will verify (4.3) implies (4.1), then the conclusion (4.2) = ⇒ (4.1) follows by the fact that (4.2) implies (4.3) for p < ∞ . By the well known embeddingrelation l q ⊂ l p for p ≥ q , and the submultiplicative property of m , we have (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a k ,k b n − k ,n − k | p (cid:19) q/p m ( n , n ) q (cid:19) /q . (cid:18) X n ,n ,k ,k ∈ Z d | a k ,k | q | b n − k ,n − k | q m ( k , k ) q m ( n − k , n − k ) q (cid:19) /q = (cid:18) X k ,k ∈ Z d | a k ,k | q m ( k , k ) q (cid:19) /q (cid:18) X n ,n ∈ Z d | b n − k ,n − k | q m ( n − k , n − k ) q (cid:19) /q . k ~a k l q,qmα ⊗ mβ k ~b k l q,qmα ⊗ mβ . Using (4.3), i.e., the embedding relations l p ω ( Z d ) , l p ω ( Z d ) ⊂ l qm α ( Z d ) and l q µ ( Z d ) , l q µ ( Z d ) ⊂ l qm β ( Z d ), we obtain the following embedding relations for discrete mixed-norm spaces: l p ,q ω ⊗ µ ( Z d ) ⊂ l q,qm α ⊗ m β ( Z d ) , l p ,q ω ⊗ µ ( Z d ) ⊂ l q,qm α ⊗ m β ( Z d ) . Now, we continue our main estimate by (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a k ,k b n − k ,n − k | p (cid:19) q/p m ( n , n ) q (cid:19) /q . k ~a k l q,qmα ⊗ mβ k ~b k l q,qmα ⊗ mβ . k ~a k l p ,q ω ⊗ µ k ~b k l p ,q ω ⊗ µ . This concludes the proof. (cid:3)
Proof of Theorem 1.2.
This proof follows directly by Theorem 1.1 and Proposition 4.1 withthe fact that ( m J ) α = I m β and ( m J ) β = m α . (cid:3) Separation of mixed-norm embedding inequality.Proposition 4.2 (Separation of embedding) . Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , . Supposethat m is submultiplicative, m i = ω i ⊗ µ i , i = 1 , . We have k ( a n ,k b k ,n ) k l p,q ⊗ m . k ~a k l p ,q m ( Z d ) k ~b k l p ,q m ( Z d ) (4.6) if and only if the following two convolution relations: l p ω ⊂ l p , l q µ ⊂ l qm β , (4.7) l p ,q ω ⊗ µ ⊂ l ( q,p ) m α ⊗ (4.8) hold. Here, we denote m α ( z ) = m ( z , and m β ( z ) = m (0 , z ) for z , z ∈ R d . OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 23 Proof.
We divide the proof into two parts. “Only if ” part.
Write (4.6) by (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a n ,k b k ,n | p (cid:19) q/p m ( n , n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | b k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q . (4.9)The above inequality will be tested by several constructed discrete sequences. Test 1.
Set a k ,n = 0 if ( k , n ) = 0 Z d × Z d , and b k ,n = 0 if n = 0 Z d . The inequality (4.9)says that (cid:18) X k ∈ Z d | b k , | p (cid:19) /p . (cid:18) X k ∈ Z d | b k , m ( k , | p (cid:19) /p , which implies the embedding relation l p ω ⊂ l p in (4.7). Test 2.
Set a k ,n = 0 if ( k , n ) = 0 Z d × Z d , and b k ,n = 0 if k = 0 Z d . The inequality (4.9)says that (cid:18) X n ∈ Z d | b ,n | q m (0 , n ) q (cid:19) /q . (cid:18) X n ∈ Z d | b ,n m (0 , n ) | q (cid:19) /q . This is just the embedding relation l q µ ⊂ l qm β in (4.7). Test 3.
Set b k ,n = 0 if ( k , n ) = 0 Z d × Z d . The inequality (4.9) says that (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a n ,k | p (cid:19) q/p m ( n , q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q . This is just the embedding relation l p ,q ω ⊗ µ ⊂ l ( q,p ) m α ⊗ in (4.8). “If ” part. Recall m ( n , n ) . m α ( n ) m β ( n ). Write (cid:18) X n ,n ∈ Z d (cid:18) X k ,k ∈ Z d | a n ,k b k ,n | p (cid:19) q/p m ( n , n ) q (cid:19) /q . (cid:18) X n ,n ∈ Z d (cid:18) X k ∈ Z d | a n ,k | p X k ∈ Z d | b k ,n | p (cid:19) q/p m α ( n ) q m β ( n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a n ,k | p (cid:19) q/p m α ( n ) q (cid:19) /q (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | b k ,n | p (cid:19) q/p m β ( n ) q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | b k ,n m ( k , n ) | p (cid:19) q /p (cid:19) /q , where in the last inequality we use (4.7) and (4.8). (cid:3) Proof of Theorem 1.5.
Note that e m is submultiplicative if m is submultiplicative. Observethat f m = f ω ⊗ f µ . The case τ ∈ (0 ,
1) follows directly by Theorem 1.4 and Proposition 4.1.The endpoint cases τ = 0 , (cid:3) Third characterizations: applications for power weights
Sharp exponents for convolution inequalities.
Note that Theorem 1.3 is a directconclusion of Theorem 1.2. Observe that in Theorem 1.5, e m ∼ m , f ω ∼ ω and f µ ∼ µ , for m = v s , ω = v s , µ = v t . Then Theorem 1.6 with τ ∈ (0 ,
1) follows by Theorem 1.5. Seethe proof of Theorem 1.6 for τ = 0 , Lemma 5.1. (See [17, Theorem 1.1] ) Suppose ≤ q, q , q ≤ ∞ , s, s , s ∈ R . Then l q s ( R d ) ∗ l q s ( R d ) ⊂ l qs ( R d ) (5.1) if and only if ( q , s ) = ( q, q , q , s, s , s ) satisfies one of the following conditions A i , i =1 , , , . ( A ) s ≤ s , s ≤ s , ≤ s + s , (cid:16) q + sd (cid:17) ∨ < (cid:16) q + s d (cid:17) ∨ (cid:16) q + s d (cid:17) ∨ , q + sd ≤ q + s d , q + sd ≤ q + s d , ≤ q + s d + q + s d , ( q, s ) = ( q , s ) if q + sd = q + s d , ( q, s ) = ( q , s ) if q + sd = q + s d , ( q ′ , − s ) = ( q , s ) if q + s d + q + s d ; (5.2)( A ) ( s = s = s = 0 ,q = q , q = 1 or q = q , q = 1 or q = ∞ , q + q = 1; (5.3)( A ) s ≤ s , s ≤ s , q + q = 1 , s + s = 0 , q + sd < ≤ q + s d , q + s d ; (5.4)( A ) s ≤ s , s ≤ s , ≤ s + s , q + sd = q + s d + q + s d , q ≤ q + q , q + sd < q + s d , q + sd < q + s d , q + sd > ,q = ∞ , q , q = 1 , if s = s or s = s . (5.5) Here, we use the notation a ∨ b = max { a, b } . Lemma 5.2. (see [17, Proposition 2.5] ) Suppose < q, q , q ≤ ∞ . Then l q ( R d ) ∗ l q ( R d ) ⊂ l q ( R d ) (5.6) holds if and only if q ≤ q + 1 q , q ≤ q , q ≤ q . (5.7) Moreover, if (5.7) holds, we have l q | s | ( R d ) ∗ l q s ( R d ) ⊂ l qs ( R d ) . (5.8) OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 25 Proof.
We only point out that (5.8) is a direct conclusion of (5.6) and h j i s . h j − l i | s | h l i s forall s ∈ R . (cid:3) By the above two lemmas, we obtain the following results.
Theorem 5.3.
Suppose that p < ∞ and p ≤ p i , q i , q for i = 1 , , τ ∈ [0 , . We have W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ M p,q ⊗ v s ( R d ) if and only if ( q/p, p /p, p /p, ps, ps , ps ) , ( q/p, q /p, q /p, ps, pt , pt ) ∈ A . Proof.
By Theorem 1.3, we have W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ M p,q ⊗ v s ( R d )if and only if l p /pps ( Z d ) ∗ l p /pps ( Z d ) ⊂ l q/pps , l q /ppt ∗ l q /ppt ⊂ l q/pps . By Lemma 5.1, we find that the above two convolution inequalities are equivalent to( q/p, p /p, p /p, ps, ps , ps ) , ( q/p, q /p, q /p, ps, pt , pt ) ∈ A . (cid:3) Theorem 5.4.
Suppose that p < ∞ and p ≤ p i , q i , q for i = 1 , , τ ∈ (0 , . We have W τ : M p ,q v s ,t × M p ,q v s ,t −→ W ( F L p , L qm ) if and only if ( q/p, p /p, p /p, ps, ps , ps ) , ( q/p, q /p, q /p, ps, pt , pt ) ∈ A . Proof.
As the proof of Theorem 5.3, this is a direct conclusion of Theorem 1.6 and Lemma5.1. (cid:3)
Next, using Lemma 5.2, we recapture the main results of BMM in [8, 4].
Theorem 5.5 (see also [8, 4]) . Let < p, q, p i , q i ≤ ∞ for i = 1 , , τ ∈ [0 , . We have W τ : M p ,q × M p ,q −→ M p,q if and only if p i , q i ≤ q, i = 1 , and p + 1 p ≥ p + 1 q , q + 1 q ≥ p + 1 q . (5.10) Moreover, if (5.9) and (5.10) hold, we have W τ : M p ,q | s | × M p ,q s −→ M p,qs . Proof.
The case p < ∞ follows by Theorem 1.3 and Lemma 5.2. For p = ∞ , the correspondingresults can be verified by Theorem 1.3 and Lemma 5.7. (cid:3) Similarly, we also have following result for BMW.
Theorem 5.6.
Let < p, q, p i , q i ≤ ∞ for i = 1 , , τ ∈ (0 , . We have W τ : M p ,q × M p ,q −→ W ( F L p , L q ) if and only if p i , q i ≤ q, i = 1 , and p + 1 p ≥ p + 1 q , q + 1 q ≥ p + 1 q . (5.12) Moreover, if (5.11) and (5.12) hold, we have W τ : M p ,q | s | × M p ,q s −→ W ( F L p , L qs ) . Sharp exponents for embedding relations.
In order to get the sharp exponents ofembedding relations mentioned in Theorems 1.3 and 1.6, we recall the following lemma.
Lemma 5.7 (Sharpness of embedding, discrete form) . Suppose < q, q , q ≤ ∞ , s, s , s ∈ R .Then l q v s ( R d ) ⊂ l q v s ( R d ) holds if and only if ( s ≤ s q + s d < q + s d or ( s = s q = q . Using this lemma, we conclude the sharp exponent characterizations for BMM and BMW.
Theorem 5.8.
Suppose p ≥ q , τ ∈ [0 , . We have W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ M p,q ⊗ v s ( R d ) if and only if s ≤ s , s , t , t , q + sd < p + s d or ( q, s ) = ( p , s ) , q + sd < p + s d or ( q, s ) = ( p , s ) , q + sd < q + t d or ( q, s ) = ( q , t ) , q + sd < q + t d or ( q, s ) = ( q , t ) . Proof.
By Theorem 1.3, we have W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ M p,q ⊗ v s ( R d )if and only if l p s ( Z d ) , l p s ( Z d ) ⊂ l qs ( Z d ) , l q t ( Z d ) , l q t ( Z d ) ⊂ l qs ( Z d ) . Then, the desired conclusion follows by Lemma 5.7. (cid:3)
Theorem 5.9.
Suppose p ≥ q , τ ∈ (0 , . We have W τ : M p ,q v s ,t × M p ,q v s ,t −→ W ( F L p , L qm ) if and only if s ≤ s , s , t , t , q + sd < p + s d or ( q, s ) = ( p , s ) , q + sd < p + s d or ( q, s ) = ( p , s ) , q + sd < q + t d or ( q, s ) = ( q , t ) , q + sd < q + t d or ( q, s ) = ( q , t ) . OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 27 Proof.
As the proof of Theorem 5.3, this is a direct conclusion of Theorem 1.6 and Lemma5.7. (cid:3)
Following proposition is prepared for further separation of the endpoint cases of BMW withpower weights, i.e., for the proof of Theorem 1.6 with τ = 0 , Proposition 5.10.
Assume p , q , p, q ∈ (0 , ∞ ] , s, t ∈ R . Then l p ,q ⊂ l ( q,p ) v s,t (5.13) holds if and only if the following three embedding relations: l p ⊂ l qv s , l q ⊂ l pv t , l q ⊂ l qv s + t . (5.14) Proof.
We divide this proof into two parts. “Only if ” part.
Write (5.13) by (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | p h n i tp (cid:19) q/p h k i sq (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | p (cid:19) q /p (cid:19) /q . (5.15)Then, we text the above inequality by several constructed sequences. Test 1.
Set a k,n = 0 if n = 0 Z d . The inequality (5.15) becomes (cid:18) X k ∈ Z d | a k, | q h k i sq (cid:19) /q . (cid:18) X k ∈ Z d | a k, | p (cid:19) /p . This implies the embedding relations l p ⊂ l qv s . Test 2.
Set a k,n = 0 if k = 0 Z d . The inequality (5.15) becomes (cid:18) X n ∈ Z d | a ,n | p h n i tp (cid:19) /p . (cid:18) X n ∈ Z d | a ,n | q (cid:19) /q . We obtain the embedding relation l q ⊂ l pv t . Test 3.
Set a k,n = 0 if k = n . From the inequality (5.15) we have (cid:18) X k ∈ Z d | a k,k | q h k i ( s + t ) q (cid:19) /q . (cid:18) X k ∈ Z d | a k,k | q (cid:19) /q . This is just the embedding relation l q ⊂ l qv s + t . “Only if ” part. Applying Lemma 5.7 to the embedding relations l p ⊂ l qv s and l q ⊂ l pv t ,we obtain s ≤ t ≤
0. From this, we consider following three cases.
Case 1: s = 0 . In this case, the embedding relations (5.14) can be written as l p ⊂ l q , l q ⊂ l pv t , l q ⊂ l qv t . Correspondingly, our target is to verify l p ,q ⊂ l ( q,p ) v ,t . If q/p ≥
1, the Minkowski inequality implies that (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | p h n i tp (cid:19) q/p (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | q (cid:19) p/q h n i tp (cid:19) /p . Then, we use the embedding relations l p ⊂ l q , l q ⊂ l pv t to deduce that (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | q (cid:19) p/q h n i tp (cid:19) /p . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | p (cid:19) q /p (cid:19) /q . The desired conclusion follows by the above two estimates.If q/p <
1, we use the embedding relations l q ⊂ l p to deduce that (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | p h n i tp (cid:19) q/p (cid:19) /q . (cid:18) X k ∈ Z d X n ∈ Z d | a k,n | q h n i tq (cid:19) /q . Then, the embedding relations l p ⊂ l q , l q ⊂ l qv t further implies that (cid:18) X k ∈ Z d X n ∈ Z d | a k,n | q h n i tq (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | p (cid:19) q /p (cid:19) /q . The desired conclusion follows by the above two estimates.
Case 2: t = 0 . In this case, the embedding relations (5.14) can be written as l p ⊂ l qv s , l q ⊂ l p , l q ⊂ l qv s . Correspondingly, our target is to verify l p ,q ⊂ l ( q,p ) v s, . If p /q ≥
1, we use the embedding relations l p ⊂ l qv s , l q ⊂ l p to deduce that (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | p (cid:19) q/p h k i sq (cid:19) /q . (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | q (cid:19) p /q (cid:19) /p . Then, we use the Minkowski inequality to continue this estimate by (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | q (cid:19) p /q (cid:19) /p . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | p (cid:19) q /p (cid:19) /q . The desired conclusion follows by the above two estimates.If p /q <
1, we use the embedding relations l q ⊂ l qv s , l q ⊂ l p to deduce that (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | p (cid:19) q/p h k i sq (cid:19) /q . (cid:18) X k ∈ Z d X n ∈ Z d | a k,n | q (cid:19) /q . Then, the well known embedding l p ⊂ l q yields that (cid:18) X k ∈ Z d X n ∈ Z d | a k,n | q (cid:19) /q . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | p (cid:19) q /p (cid:19) /q . The desired conclusion follows by the above two estimates.
Case 3: s, t < . In this case, by Lemma 5.7 and the embedding relations (5.14), we obtainthat 1 q + sd < p , p < q − td , q + sd < q − td . From this, there exists a sufficiently small constant ǫ > q + sd + ǫ < p , p < q − td − ǫ, q + sd + ǫ < q − td − ǫ. OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 29 Set 1 ρ := max { q + sd + ǫ, } , r := 1 q − td − ǫ. We have ρ < r and the following relations:1 ρ + sd < ρ < p , p + td < r + td < q . From this and Lemma 5.7, we obtain the following embedding relations l p ⊂ l ρ ⊂ l qv s , l q ⊂ l rv t ⊂ l pv t . Using these embedding relations and the Minkowski inequality with ρ < r , we deduce that (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | p h n i tp (cid:19) q/p h k i sq (cid:19) /q . (cid:18) X k ∈ Z d (cid:18) X n ∈ Z d | a k,n | r h n i tr (cid:19) ρ/r (cid:19) /ρ . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | ρ (cid:19) r/ρ h n i tr (cid:19) /r . (cid:18) X n ∈ Z d (cid:18) X k ∈ Z d | a k,n | p (cid:19) q /p (cid:19) /q . This is the desired conclusion. (cid:3)
Proof of Theorem 1.6.
The case τ ∈ (0 ,
1) follows directly by Theorem 1.5. For τ = 0, observethat l p ,q v s ,t ⊂ l ( q,p ) v s ⊗ ⇐⇒ l p ,q ⊂ l ( q,p ) v s − s , − t . Then, Proposition 5.10 tells us l p ,q ⊂ l ( q,p ) v s − s , − t ⇐⇒ l p ⊂ l qs − s , l q ⊂ l p − t , l q ⊂ l qs − s − t , which is equivalent to l p s ⊂ l qs , l q t ⊂ l p , l q s + t ⊂ l qs . Hence, we have l p ,q v s ,t ⊂ l ( q,p ) v s ⊗ ⇐⇒ l p s ⊂ l qs , l q t ⊂ l p , l q s + t ⊂ l qs . Combining this with Theorem 1.5, we obtain the desired conclusion. The case τ = 1 followsby the same argument of τ = 0, we omit the detail. (cid:3) Using Lemma 5.7 and Theorem 1.6, we obtain the following exponents characterization forBMW with endpoints.
Theorem 5.11.
Assume p i , q i , p, q ∈ (0 , ∞ ] , i = 1 , , τ = 0 , . We have W τ : M p ,q v s ,t ( R d ) × M p ,q v s ,t ( R d ) −→ W ( F L p , L qs )( R d ) if and only if ( τ = 0) s ≤ s , t , ≤ t , s , p ≤ q + t d or ( q , t ) = ( p, , p ≤ p + s d or ( p , s ) = ( p, , q + sd ≤ p + s d or ( q, s ) = ( p , s ) , q + sd ≤ q + s + t d or ( q, s ) = ( q , s + t ) , q + sd ≤ q + t d or ( q, s ) = ( q , t ) , and ( τ = 1) s ≤ s , t , ≤ t , s , p ≤ q + t d or ( q , t ) = ( p, , p ≤ p + s d or ( p , s ) = ( p, , q + sd ≤ p + s d or ( q, s ) = ( p , s ) , q + sd ≤ q + s + t d or ( q, s ) = ( q , s + t ) , q + sd ≤ q + t d or ( q, s ) = ( q , t ) . Complements: pesudodifferential operators on modulation spaces
Relations between BMM (BMW) and BPM (BPW).
In order to give characteri-zations of BPM and BPW, we would like to first establish some equivalent relations associatedwith BMM and BMW. We point out that these equivalent relations follows by the dual ar-guments of function spaces, which have been wildly used before, for instance, one can see theproof of [8, Theorem 5.1] for this direction.
Proposition 6.1.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ ∈ [0 , . Then the followingstatements are equivalent: ( i ) ∀ σ ∈ M p,q ⊗ m ( R d ) = ⇒ OP τ ( σ ) : M p ,q m ( R d ) −→ M p ,q m ( R d ) , ( ii ) k OP τ ( σ ) f k M p ,q m ( R d ) . k σ k M p,q ⊗ m ( R d ) k f k M p ,q m ( R d ) , f ∈ S ( R d ) , ( iii ) W τ : M p ′ ,q ′ m − ( R d ) × M p ,q m ( R d ) −→ M p ′ ,q ′ ⊗ m − ( R d ) . Proof. ( ii ) = ⇒ ( i ) is clear. In order to verify ( i ) = ⇒ ( ii ), we apply the Closed Graph Theoremas in [7, Proposition 4.7]. The map acting from M p,q ⊗ m into B ( M p ,q m , M p ,q m ) is defined by P : σ −→ OP τ ( σ ) . Given any sequence of pairs that ( σ n , OP τ ( σ n )) tends to ( σ, T ) in the topology of graph of P ,for any f, g ∈ S ( R d ) we have h OP τ ( σ ) f, g i = h σ, W τ ( g, f ) i = lim n →∞ h σ n , W τ ( g, f ) i = lim n →∞ h OP τ ( σ n ) f, g i = h T f, g i . From this, we obtain T = OP τ ( σ ). Then the graph of P is closed. Hence, P is bounded, i.e.,(ii) is valid.Next, we turn to the proof of ( ii ) ⇐⇒ ( iii ). This follows by a standard dual argument. If(ii) holds, for any f, g ∈ S ( R d ) and σ ∈ M p,q ⊗ m , we have |h W τ ( f, g ) , σ i| = |h f, OP τ ( σ ) g i| . k f k M p ′ ,q ′ m − k OP τ ( σ ) g k M p ,q m . k f k M p ′ ,q ′ m − k σ k M p,q ⊗ m k g k M p ,q m . Then, the duality of modulation spaces implies (iii).Viceversa, if (iii) holds, for any f, g ∈ S ( R d ) and σ ∈ M p,q ⊗ m , we have |h OP τ ( σ ) f, g i| = |h σ, W τ ( g, f ) i| . k σ k M p,q ⊗ m k W τ ( g, f ) k M p ′ ,q ′ ⊗ m − . k σ k M p,q ⊗ m k g k M p ′ ,q ′ m − k f k M p ,q m . Then, (iii) follows by the duality of modulation spaces. (cid:3)
OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 31 By a similar argument, we give following equivalent relations between BMW and BPW.
Proposition 6.2.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ ∈ [0 , . Then the followingstatements are equivalent: ( i ) ∀ σ ∈ F M p,q ⊗ m ( R d ) = ⇒ OP τ ( σ ) : M p ,q m ( R d ) −→ M p ,q m ( R d ) , ( ii ) k OP τ ( σ ) f k M p ,q m ( R d ) . k σ k F M p,q ⊗ m ( R d ) k f k M p ,q m ( R d ) , f ∈ S ( R d ) , ( iii ) W τ : M p ′ ,q ′ m − ( R d ) × M p ,q m ( R d ) −→ F M p ′ ,q ′ ⊗ m − ( R d ) . Sharp exponents of BPM and BPW.
By propositions 6.1 and 6.2, the estimatesof the τ -Wigner distribution can be translated in ones into the corresponding results for τ -operators. Following, we collect some important cases, the interested reader can deduce theresults they need.Following result is a direct conclusion by Proposition 6.1 and Theorem 5.5. Theorem 6.3.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ ∈ [0 , . We have ∀ σ ∈ M p,q ( R d ) = ⇒ OP τ ( σ ) : M p ,q ( R d ) −→ M p ,q ( R d ) if and only if p , q , p ′ , q ′ ≤ q ′ , i = 1 , and p + 1 p ′ ≥ p ′ + 1 q ′ , q + 1 q ′ ≥ p ′ + 1 q ′ . Next, we want to establish the sharp exponents for the boundedness on modulation spaceswith power weights of pseudodifferential operators with symbols in Sj¨ostrand’s class. Beforethis, we first give following characterization of BPM, which can be directly deduced by Propo-sition 6.1 and Theorem 1.3.
Theorem 6.4.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ ∈ [0 , . We have ∀ σ ∈ M p,q ⊗ v s ( R d ) = ⇒ OP τ ( σ ) : M p ,q v s ,t ( R d ) −→ M p ,q v s ,t ( R d ) if and only if l p ′ /p ′ − p ′ s ( Z d ) ∗ l p /p ′ p ′ s ( Z d ) ⊂ l q ′ /p ′ − p ′ s ( Z d ) , l q ′ /p ′ − p ′ t ( Z d ) ∗ l q /p ′ p ′ t ( Z d ) ⊂ l q ′ /p ′ − p ′ s ( Z d ) , p > , (6.1) l p ′ − s ( Z d ) , l p s ( Z d ) ⊂ l q ′ − s ( Z d ) , l q ′ − t ( Z d ) , l q t ( Z d ) ⊂ l q ′ − s ( Z d ) , p ≤ q. (6.2)We recall a special case of Lemma 5.1 as follows. Lemma 6.5.
Suppose ≤ q , q ≤ ∞ , s , s ∈ R . Then l q s ( R d ) ∗ l q s ( R d ) ⊂ l ∞ ( R d ) (6.3) if and only if s = s = 0 , q + 1 q = 1 or ≤ s , s , < q + s d + q + s d , ( q , s ) = ( ∞ , if q + s d = 0 , ( q , s ) = ( ∞ , if q + s d = 0 . (6.4)Now, we are in a position to give the sharp exponents of BPM with Sj¨ostrand’s class. Theorem 6.6.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ ∈ [0 , . We have ∀ σ ∈ M ∞ , ( R d ) = ⇒ OP τ ( σ ) : M p ,q v s ,t ( R d ) −→ M p ,q v s ,t ( R d ) (6.5) if and only if s = s = 0 , p = p or s ≤ ≤ s , p + s d < p + s d , ( p , s ) = ( ∞ , if p + s d = 0 , ( p , s ) = (1 , if p + s d = 1 , (6.6) and t = t = 0 , q = q or t ≤ ≤ t , q + t d < q + t d , ( q , t ) = ( ∞ , if q + t d = 0 , ( q , t ) = (1 , if q + t d = 1 . (6.7) In particular, when s i = t i = 0 , i = 1 , , we have (6.5) holds if and only if p ≤ p , q ≤ q . (6.8) Proof.
Using Theorem 6.4, we obtain that (6.5) holds if and only if l p ′ − s ( Z d ) ∗ l p s ( Z d ) ⊂ l ∞ ( Z d ) , l q ′ − t ( Z d ) ∗ l q t ( Z d ) ⊂ l ∞ ( Z d ) . Then, Lemma 6.5 tells us that the above two convolution inequalities are equivalent to (6.6)and (6.7). (cid:3)
Observe that Wiener amalgam space W ( F L q , L p ) has the same local regularity with M p,q .Moreover, for p > q they have the following inclusion relations: M p,q ( W ( F L q , L p ) , M q,p ) W ( F L p , L q ) . So, there is a natural question that for p > q whether the boundedness of pseudodifferentialoperator with symbols in M p,q or W ( F L p , L q ) can be preserved with symbols in W ( F L q , L p )or M q,p respectively. With the help of our full characterizations of BMM and BMW, one canfind that the answer is negative unless the trivial case happen, i.e., p = q . Here, we onlygive a detailed comparison for the Sj¨ostrand’s class M ∞ , and the corresponding larger space W ( F L , L ∞ ). Let us being with the sharp exponents for the non-endpoint case of BPW. Theorem 6.7.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ ∈ (0 , . We have ∀ σ ∈ W ( F L , L ∞ )( R d ) = ⇒ OP τ ( σ ) : M p ,q v s ,t ( R d ) −→ M p ,q v s ,t ( R d ) (6.9) if and only if s ≤ ≤ s , < p + s d or ( p , s ) = (1 , , p + s d < or ( p , s ) = ( ∞ , , and t ≤ ≤ t , < q + t d or ( q , t ) = (1 , , q + t d < or ( q , t ) = ( ∞ , . OUNDEDNESS ON WEIGHTED MODULATION SPACES OF τ -WIGNER DISTRIBUTIONS 33 In particular, when s i = t i = 0 , i = 1 , , we have (6.9) holds if and only if p = q = 1 , p = q = ∞ . (6.10) Proof.
By Proposition 6.2 and Theorem 1.6, we conclude that (6.9) holds if and only if l p ′ − s ( Z d ) , l p s ( Z d ) ⊂ l ( Z d ) , l q ′ − t ( Z d ) , l q t ( Z d ) ⊂ l ( Z d ) . Then, the desired conclusion follows by Lemma 5.7. (cid:3)
Remark 6.8.
Comparing (6.10) with (6.8) , we find that the range of exponents for BPW ( τ ∈ (0 , ) with symobls in W ( F L , L ∞ )( R d ) is strictly small than that for BPM with Sj¨ostrand’sclass. Next, we handle that endpoint case of BPW. We first give following characterization for theendpoint cases of BPW, which can be directly deduced by Proposition 1.6 and Proposition 6.2.
Theorem 6.9.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ = 0 , . We have ∀ σ ∈ W ( F L p , L qs )( R d ) = ⇒ OP τ ( σ ) : M p ,q v s ,t ( R d ) −→ M p ,q v s ,t ( R d ) if and only if l q ′ − t ( Z d ) , l p s ( Z d ) ⊂ l p ′ ( Z d ) , l p ′ − s ( Z d ) , l q ′ − ( s + t ) ( Z d ) , l q t ( Z d ) ⊂ l q ′ − s ( Z d ) , τ = 0 , (6.11) l q t ( Z d ) , l p ′ − s ( Z d ) ⊂ l p ′ ( Z d ) , l p s ( Z d ) , l q s + t ( Z d ) , l q ′ − t ( Z d ) ⊂ l q ′ − s ( Z d ) , τ = 1 . (6.12)Then, corresponding to Theorem 5.6, we establish the sharp exponents for the endpointcases of BPW with constant weights. Theorem 6.10.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ = 0 , . We have ∀ σ ∈ W ( F L p , L q )( R d ) = ⇒ OP τ ( σ ) : M p ,q ( R d ) −→ M p ,q ( R d ) (6.13) if and only if q ′ , p ≤ p ′ , p ′ , q ′ , q ≤ q ′ , τ = 0 , (6.14) q , p ′ ≤ p ′ , p , q , q ′ ≤ q ′ , τ = 1 . (6.15) Proof.
It follows by Theorem 6.9 that (6.13) is equivalent to l q ′ ( Z d ) , l p ( Z d ) ⊂ l p ′ ( Z d ) , l p ′ ( Z d ) , l q ′ ( Z d ) , l q ( Z d ) ⊂ l q ′ ( Z d ) , τ = 0 ,l q ( Z d ) , l p ′ ( Z d ) ⊂ l p ′ ( Z d ) , l p ( Z d ) , l q ( Z d ) , l q ′ ( Z d ) ⊂ l q ′ ( Z d ) , τ = 1 . Then, the desired conditions can be deduced by Lemma 5.7. (cid:3)
Now, we return to BPW with symbols in W ( F L , L ∞ )( R d ). Theorem 6.11.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ = 0 , . We have ∀ σ ∈ W ( F L , L ∞ )( R d ) = ⇒ OP τ ( σ ) : M p ,q v s ,t ( R d ) −→ M p ,q v s ,t ( R d ) (6.16) if and only if ( τ = 0) s , t ≤ ≤ s , t , < p + s d or ( p , s ) = ( ∞ , , p + s d < or ( p , s ) = ( ∞ , , < q + t d or ( q , t ) = (1 , , q + t d < or ( q , t ) = (1 , , q + s + t d < or ( q , s , t ) = ( ∞ , , and ( τ = 1) s , t ≤ ≤ s , t , < p + s d or ( p , s ) = (1 , , p + s d < or ( p , s ) = (1 , , < q + t d or ( q , t ) = ( ∞ , , q + t d < or ( q , t ) = ( ∞ , , < q + s + t d or ( q , s , t ) = (1 , , . In particular, when s i = t i = 0 , i = 1 , , we have (6.16) holds if and only if q = 1 , p = q = ∞ , τ = 0 , (6.17) q = ∞ , p = q = 1 , τ = 1 . (6.18) Proof.
It follows from Theorem 6.9 that (6.16) is equivalent to l q ′ − t ( Z d ) , l p s ( Z d ) ⊂ l ∞ ( Z d ) , l p ′ − s ( Z d ) , l q ′ − ( s + t ) ( Z d ) , l q t ( Z d ) ⊂ l ( Z d ) , τ = 0 , (6.19) l q t ( Z d ) , l p ′ − s ( Z d ) ⊂ l ∞ ( Z d ) , l p s ( Z d ) , l q s + t ( Z d ) , l q ′ − t ( Z d ) ⊂ l ( Z d ) . τ = 1 . (6.20)Then, the desired conclusion follows from Lemma 5.7. (cid:3) Remark 6.12.
Comparing (6.17) , (6.18) with (6.8) , one can find that the range of exponentsfor BPW ( τ = 0 , ) with symobls in W ( F L , L ∞ )( R d ) is strictly small than that for BPMwith Sj¨ostrand’s class. Finally, we consider the boundedness on Sobolev spaces H s of pseudodifferential operatorswith symbols in Wiener amalgam spaces W ( F L p , L q )( R d ). Note that M , s = H s . Theorem 6.13.
Assume ≤ p, q, p i , q i ≤ ∞ , i = 1 , , τ = 0 , . We have ∀ σ ∈ W ( F L p , L q )( R d ) = ⇒ OP τ ( σ ) : M , v ,t ( R d ) −→ M , v ,t ( R d ) (6.21) if and only if t ≤ ≤ t , p, q ≤ , τ = 0 , . Proof.
Recall that M , s = H s , by Theorem 6.9, we conclude that (6.21) is equivalent to l − t ( Z d ) , l ( Z d ) ⊂ l p ′ ( Z d ) , l ( Z d ) , l − t ( Z d ) , l t ( Z d ) ⊂ l q ′ ( Z d ) , τ = 0 ,l t ( Z d ) , l ( Z d ) ⊂ l p ′ ( Z d ) , l ( Z d ) , l t ( Z d ) , l − t ( Z d ) ⊂ l q ′ ( Z d ) , τ = 1 , which implies the desired conclusion by Lemma 5.7. (cid:3) Remark 6.14.
From Theorem 6.13, for τ = 0 , , we observe that for any s ≥ , there exists asymbol σ ∈ W ( F L ∞ , L )( R d ) such that the corresponding pseudodifferential operators OP τ ( σ ) are not bounded from L ( R d ) to H s ( R d ) . Acknowledgements.
This work was partially supported by the National Natural ScienceFoundation of China (Nos. 11701112, 11601456, 11671414, 11771388).
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