Sharp results for the Weyl product on modulation spaces
aa r X i v : . [ m a t h . F A ] A p r SHARP RESULTS FOR THE WEYL PRODUCT ONMODULATION SPACES
ELENA CORDERO, JOACHIM TOFT, AND PATRIK WAHLBERG
Abstract.
We give sufficient and necessary conditions on theLebesgue exponents for the Weyl product to be bounded on modu-lation spaces. The sufficient conditions are obtained as the restric-tion to N = 2 of a result valid for the N -fold Weyl product. As abyproduct, we obtain sharp conditions for the twisted convolutionto be bounded on Wiener amalgam spaces. Introduction
In the paper we prove necessary and sufficient conditions for the Weylproduct to be continuous on modulation spaces, and for the twistedconvolution to be continuous on Wiener amalgam spaces. We relax thesufficient conditions in [27] and we prove that the obtained conditionsare also necessary.The Weyl calculus is a part of the theory of pseudo-differential op-erators. For an appropriate distribution a (the symbol) defined on thephase space T ∗ R d ≃ R d , the Weyl operator Op w ( a ) is a linear mapbetween spaces of functions or distributions on R d . (See Section 1 fordefinitions.) Weyl operators appear in various fields. In mathematicalanalysis they are used to represent linear operators, in particular linearpartial differential operators, acting between appropriate function anddistribution spaces. Weyl operators also appear in quantum mechan-ics where a real-valued observable a in classical mechanics correspondsto the self-adjoint Weyl operator Op w ( a ) in quantum mechanics. Forthis reason Op w ( a ) is often called the Weyl quantization of a . In time-frequency analysis pseudo-differential operators are used as models ofnon-stationary filters.In the Weyl calculus operator composition corresponds on the sym-bol level to the Weyl product , or the twisted product , denoted by .This means that the Weyl product a a of appropriate functions ordistributions a and a satisfiesOp w ( a a ) = Op w ( a ) ◦ Op w ( a ) . Key words and phrases.
Weyl product, modulation spaces, twisted convolution,sharpness. MSC 2010 codes: 35S05,42B35,44A35,46E35,46F12. basic problem is to find conditions that are necessary or sufficientfor the bilinear map ( a , a ) a a (0.1)to be well-defined and continuous. Here we investigate these questionswhen the factors belong to modulation spaces, a family of Banachspaces of distributions which appear in time-frequency analysis, har-monic analysis and Gabor analysis.The modulation spaces were introduced by Feichtinger [6], and theirtheory was further developed and generalized by Feichtinger and Gr¨oche-nig [8–10, 15] into the theory of coorbit spaces.The modulation space M p,q ( ω ) ( R d ), where p, q ∈ [1 , ∞ ] and ω is aweight on R d × R d ≃ R d , consists of all tempered distributions, orultra-distributions, on R d , whose short-time Fourier transforms have fi-nite L p,q ( ω ) ( R d ) norm. Thus the Lebesgue exponents p and q , and aboveall the weight ω , give a scale of function spaces M p,q ( ω ) with respect tophase space concentration. The definition of modulation spaces resem-bles that of Besov spaces, and narrow embeddings between modulationand Besov spaces have been found (cf. [14, 26, 32, 38, 45, 47, 51, 52]).Depending on the assumptions on the weights, the modulation spacesare subspaces of the tempered distributions or ultra-distributions (cf.[2, 34, 35, 49, 50]).Since the early 1990s modulation spaces have been used in the theoryof pseudo-differential operators (cf. [39]). Sj¨ostrand [36] introduced themodulation space M ∞ , ( R d ), which contains non-smooth functions,as a symbol class. He proved that M ∞ , corresponds to an algebra of L -bounded operators,Gr¨ochenig and Heil [16, 20] proved that each operator with symbolin M ∞ , is continuous on all modulation spaces M p,q , p, q ∈ [1 , ∞ ].This extends Sj¨ostrand’s L -continuity result since M , = L . Somegeneralizations to operators with symbols in unweighted modulationspaces were obtained in [21, 45], and [46, 48, 50] contain extensions toweighted modulation spaces.Concerning the algebraic properties of the Weyl calculus (cf. [11, 13,28]) with respect to modulation spaces, Sj¨ostrand’s results [36,37] wererefined in [42], and new results were found by Labate [30], Gr¨ochenigand Rzeszotnik [23], and by Holst and two of the authors [27].Our main result in this paper is a multi-linear version of a gener-alization of [27, Theorem 0.3 ′ ] which concerns sufficient conditions forcontinuity of the Weyl product on modulation spaces. We also provethat the sufficient conditions are necessary in the bilinear case, for acertain family of weight functions.The Weyl product (0.1) is continuous S s ( R d ) × S s ( R d )
7→ S s ( R d ),where S s ( R d ) denotes the Gelfand–Shilov space of order s , for every s ≥
0. In order to explain our extension of this result to modulation paces, we introduce the H¨older–Young exponent function R ( p ) = X j =0 p j − , p = ( p , p , p ) ∈ [1 , ∞ ] (0.2)and consider weights ω j , j = 0 , ,
2, in P E ( R d ), the set of moderateweights on R d . We suppose that C ≤ ω ( X + X , X − X ) Y j =1 ω j ( X j + X j − , X j − X j − ) ,X , X , X ∈ R d , (0.3)holds for some C > M p,q ( ω ) ( R d ),as opposed to M p,q ( ω ) ( R d ), is the modulation space defined with the sym-plectic Fourier transform instead of the usual Fourier transform. Theorem 0.1.
Let p j , q j ∈ [1 , ∞ ] , j = 0 , , , and suppose max ( R ( q ′ ) , ≤ min j =0 , , (cid:18) p j , q ′ j , R ( p ) (cid:19) . (0.4) Let ω j ∈ P E ( R d ) , j = 0 , , , and suppose (0.3) holds. Then the map (0.1) from S / ( R d ) × S / ( R d ) to S / ( R d ) extends uniquely to acontinuous map from M p ,q ( ω ) ( R d ) × M p ,q ( ω ) ( R d ) to M p ′ ,q ′ (1 /ω ) ( R d ) . This result is the restriction to N = 2 of a multi-linear result treatingthe Weyl product of N factors a . . . a N proved in Section 2 (see The-orem 0.1 ′ ). Theorem 0.1 extends all results in the literature, familiar tous, on the Weyl product acting on modulation spaces, in particular [27,Theorem 0.3 ′ ] and its slight extension [49, Theorem 6.4]. In Section 4we present a table which explains the difference between [27, Theo-rem 0.3 ′ ] and Theorem 0.1 in the important cases when the Lebesgueexponents p j , q j belong to { , , ∞} .In Section 2 we also present a parallel result to Theorem 0.1 ′ onsufficient conditions for continuity of the Weyl product on modulationspaces. It gives continuity in certain cases not covered by Theorem0.1 ′ with N >
2, e. g. when several of the Weyl operators are Hilbert–Schmidt operators (cf. Theorem 2.9). Section 2 ends with a continuityresult for the twisted convolution on Wiener amalgam spaces (cf. The-orem 2.12).In Section 3 we prove that Theorem 0.1 is sharp with respect to theconditions on the Lebesgue exponents p j and q j , for triplets ( ω , ω , ω )of polynomially moderate weights that are interrelated in a certainway (see (3.1)) which implies that (0.3) is automatically satisfied. Thesharpness means that (0.4) must hold when the map (0.1) from S / × / to S / is extendable to a continuous map from M p ,q ( ω ) × M p ,q ( ω ) to M p ′ ,q ′ (1 /ω ) (cf. Theorem 3.1).1. Preliminaries
In this section we introduce notation and discuss the background onGelfand–Shilov spaces, pseudo-differential operators, the Weyl product,twisted convolution and modulation spaces. Most proofs can be foundin the literature and are therefore omitted.Let 0 < h, s ∈ R be fixed. The space S s,h ( R d ) consists of all f ∈ C ∞ ( R d ) such that k f k S s,h ≡ sup | x β ∂ α f ( x ) | h | α | + | β | α ! s β ! s is finite, with supremum taken over all α, β ∈ N d and x ∈ R d .The space S s,h ⊆ S ( S denotes the Schwartz space) is a Banachspace which increases with h and s . Inclusions between topologicalspaces are understood to be continuous. If s > /
2, or s = 1 / h is sufficiently large, then S s,h contains all finite linear combinations ofHermite functions. Since the space of such linear combinations is densein S , it follows that the topological dual ( S s,h ) ′ ( R d ) of S s,h ( R d ) is aBanach space which contains S ′ ( R d ).The Gelfand–Shilov spaces S s ( R d ) and Σ s ( R d ) (cf. [12]) are the in-ductive and projective limits, respectively, of S s,h ( R d ), with respect tothe parameter h . Thus S s ( R d ) = [ h> S s,h ( R d ) and Σ s ( R d ) = \ h> S s,h ( R d ) , (1.1)where S s ( R d ) is equipped with the the strongest topology such that theinclusion map from S s,h ( R d ) into S s ( R d ) is continuous, for every choiceof h >
0. The space Σ s ( R d ) is a Fr´echet space with seminorms k · k S s,h , h >
0. We have Σ s ( R d ) = { } if and only if s > /
2, and S s ( R d ) = { } if and only if s ≥ /
2. From now on we assume that s > / s ( R d ), and s ≥ / S s ( R d ).The Gelfand–Shilov distribution spaces S ′ s ( R d ) and Σ ′ s ( R d ) are theprojective and inductive limit respectively of S ′ s ( R d ). This means that S ′ s ( R d ) = \ h> S ′ s,h ( R d ) and Σ ′ s ( R d ) = [ h> S ′ s,h ( R d ) . (1.1) ′ In [12, 29, 33] it is proved that S ′ s ( R d ) is the topological dual of S s ( R d ),and Σ ′ s ( R d ) is the topological dual of Σ s ( R d ).For each ε > s > / S / ( R d ) ⊆ Σ s ( R d ) ⊆ S s ( R d ) ⊆ Σ s + ε ( R d )and Σ ′ s + ε ( R d ) ⊆ S ′ s ( R d ) ⊆ Σ ′ s ( R d ) ⊆ S ′ / ( R d ) . (1.2) he Gelfand–Shilov spaces are invariant under several basic opera-tions, e. g. translations, dilations, tensor products and (partial) Fouriertransformation.We normalize the Fourier transform of f ∈ L ( R d ) as( F f )( ξ ) = b f ( ξ ) ≡ (2 π ) − d/ Z R d f ( x ) e − i h x,ξ i dx, where h · , · i denotes the scalar product on R d . The map F extendsuniquely to homeomorphisms on S ′ ( R d ), S ′ s ( R d ) and Σ ′ s ( R d ), and re-stricts to homeomorphisms on S ( R d ), S s ( R d ) and Σ s ( R d ), and to aunitary operator on L ( R d ).Next we recall some basic facts from pseudo-differential calculus (cf.[28]). Let s ≥ / a ∈ S s ( R d ), and t ∈ R be fixed. The pseudo-differential operator Op t ( a ) defined byOp t ( a ) f ( x ) = (2 π ) − d Z Z R d a ((1 − t ) x + ty, ξ ) f ( y ) e i h x − y,ξ i dydξ (1.3)is a linear and continuous operator on S s ( R d ). For a ∈ S ′ s ( R d ) thepseudo-differential operator Op t ( a ) is defined as the continuous opera-tor from S s ( R d ) to S ′ s ( R d ) with distribution kernel given by K a,t ( x, y ) = (2 π ) − d/ ( F − a )((1 − t ) x + ty, x − y ) . (1.4)Here F F is the partial Fourier transform of F ( x, y ) ∈ S ′ s ( R d ) withrespect to the variable y ∈ R d . This definition generalizes (1.3) and iswell defined, since the mappings F and F ( x, y ) F ((1 − t ) x + ty, y − x ) (1.5)are homeomorphisms on S ′ s ( R d ). The map a K a,t is hence a home-omorphism on S ′ s ( R d ).For any K ∈ S ′ s ( R d + d ), let T K be the linear and continuous map-ping from S s ( R d ) to S ′ s ( R d ) defined by( T K f, g ) L ( R d ) = ( K, g ⊗ f ) L ( R d d ) , f ∈ S s ( R d ) , g ∈ S s ( R d ) . (1.6)It is a well known consequence of the Schwartz kernel theorem that if t ∈ R , then K T K and a Op t ( a ) are bijective mappings from S ′ ( R d ) to the space of linear and continuous mappings from S ( R d )to S ′ ( R d ) (cf. e. g. [28]).Likewise the maps K T K and a Op t ( a ) are uniquely extendableto bijective mappings from S ′ s ( R d ) to the set of linear and continuousmappings from S s ( R d ) to S ′ s ( R d ). In fact, the asserted bijectivity forthe map K T K follows from the kernel theorem [31, Theorem 2.3](cf. [12, vol. IV]). This kernel theorem corresponds to the Schwartzkernel theorem in the usual distribution theory. The other assertionfollows from the fact that a K a,t is a homeomorphism on S ′ s ( R d ). n particular, for each a ∈ S ′ s ( R d ) and t , t ∈ R , there is a unique a ∈ S ′ s ( R d ) such that Op t ( a ) = Op t ( a ). The relation between a and a is given byOp t ( a ) = Op t ( a ) ⇐⇒ a ( x, ξ ) = e i ( t − t ) h D x ,D ξ i a ( x, ξ ) . (1.7)(Cf. [28].) Note that the right-hand side makes sense, since it means b a ( x, ξ ) = e i ( t − t ) h x,ξ i b a ( x, ξ ), and since the map a ( x, ξ ) e it h x,ξ i a ( x, ξ )is continuous on S ′ s ( R d ).Next we discuss the Weyl product, twisted convolution and relatedoperations (see [11, 28]). Let s ≥ / a, b ∈ S ′ s ( R d ). The Weylproduct a b between a and b is the function or distribution whichsatisfies Op w ( a b ) = Op w ( a ) ◦ Op w ( b ), provided the right-hand sidemakes sense as a continuous operator from S s ( R d ) to S ′ s ( R d ).The Wigner distribution is defined by W f,g ( x, ξ ) = F ( f ( x + · / g ( x − · / ξ ) , f, g ∈ S ′ / ( R d ) , and takes the form W f,g ( x, ξ ) = (2 π ) − d/ Z R d f ( x + y/ g ( x − y/ e − i h y,ξ i dy, when f, g ∈ S / ( R d ). The Wigner distribution appears in the Weylcalculus in the formula(Op w ( a ) f, g ) L ( R d ) = (2 π ) − d/ ( a, W g,f ) L ( R d ) ,a ∈ S ′ / ( R d ) , f, g ∈ S / ( R d ) . The Weyl product can be expressed in terms of the symplectic Fouriertransform and the twisted convolution. The symplectic Fourier trans-form of a ∈ S s ( R d ), where s ≥ /
2, is defined by( F σ a )( X ) = π − d Z R d a ( Y ) e iσ ( X,Y ) dY, where σ is the symplectic form σ ( X, Y ) = h y, ξ i − h x, η i , X = ( x, ξ ) ∈ R d , Y = ( y, η ) ∈ R d . We note that F σ = T ◦ ( F − ⊗ F ), when ( T a )( x, ξ ) = 2 d a (2 ξ, x ).The symplectic Fourier transform F σ is continuous on S s ( R d ) andextends uniquely to a homeomorphism on S ′ s ( R d ), and to a unitarymap on L ( R d ), since similar facts hold for F . Furthermore F σ is theidentity operator.Let s ≥ / a, b ∈ S s ( R d ). The twisted convolution of a and b is defined by( a ∗ σ b )( X ) = (2 /π ) d/ Z R d a ( X − Y ) b ( Y ) e iσ ( X,Y ) dY. (1.8)The definition of ∗ σ extends in different ways. For example it extendsto a continuous multiplication on L p ( R d ) when p ∈ [1 , ontinuous map from S ′ s ( R d ) × S s ( R d ) to S ′ s ( R d ). If a, b ∈ S ′ s ( R d ),then a b makes sense if and only if a ∗ σ b b makes sense, and a b = (2 π ) − d/ a ∗ σ ( F σ b ) . (1.9)For the twisted convolution we have F σ ( a ∗ σ b ) = ( F σ a ) ∗ σ b = ˇ a ∗ σ ( F σ b ) , (1.10)where ˇ a ( X ) = a ( − X ) (cf. [43]). A combination of (1.9) and (1.10) gives F σ ( a b ) = (2 π ) − d/ ( F σ a ) ∗ σ ( F σ b ) . (1.11)If e a ( X ) = a ( − X ) then( a ∗ σ a , b ) = ( a , b ∗ σ e a ) = ( a , e a ∗ σ b ) , ( a ∗ σ a ) ∗ σ b = a ∗ σ ( a ∗ σ b ) , for appropriate a , a , b , and furthermore (cf. [27])( a a , b ) = ( a , a b ) = ( a , b a ) . (1.12)Next we turn to the basic properties of modulation spaces, and startby recalling the conditions for the involved weight functions. Let 0 <ω, v ∈ L ∞ loc ( R d ). Then ω is called moderate or v -moderate if ω ( x + y ) . ω ( x ) v ( y ) , x, y ∈ R d . (1.13)Here the notation f ( x ) . g ( x ) means that there exists C > f ( x ) ≤ Cg ( x ) for all arguments x in the domain of f and g . If f . g and g . f we write f ≍ g . The function v is called submultiplicative ifit is even and (1.13) holds when ω = v . We note that if (1.13) holdsthen v − . ω . v. For such ω it follows that (1.13) is true when v ( x ) = Ce c | x | , for some positive constants c and C (cf. [19]). In particular, if ω ismoderate on R d , then e − c | x | . ω ( x ) . e c | x | , for some c > R d is denoted by P E ( R d ). If v in (1.13) can be chosen as v ( x ) = h x i s = (1 + | x | ) s/ for some s ≥ ω is said to be of polynomial type or polynomially moderate. Welet P ( R d ) be the set of all polynomially moderate functions on R d .Let φ ∈ S s ( R d ) \ short-time Fourier transform (STFT) V φ f of f ∈ S ′ s ( R d ) with respect to the window function φ is the Gelfand–Shilov distribution on R d defined by V φ f ( x, ξ ) ≡ F ( f φ ( · − x ))( ξ ) . or a ∈ S ′ / ( R d ) and Φ ∈ S / ( R d ) \ symplectic short-timeFourier transform V Φ a of a with respect to Φ is the defined similarlyas V Φ a ( X, Y ) = F σ (cid:0) a Φ( · − X ) (cid:1) ( Y ) , X, Y ∈ R d . We have V Φ a ( X, Y ) = 2 d V Φ a ( x, ξ, − η, y ) ,X = ( x, ξ ) ∈ R d , Y = ( y, η ) ∈ R d , (1.14)which shows the close connection between V Φ a and V Φ a . The Wignerdistribution W f,φ and V φ f are also closely related.If f, φ ∈ S s ( R d ) and a, Φ ∈ S s ( R d ) then V φ f ( x, ξ ) = (2 π ) − d/ Z f ( y ) φ ( y − x ) e − i h y,ξ i dy and V Φ a ( X, Y ) = π − d Z a ( Z )Φ( Z − X ) e iσ ( Y,Z ) dZ. Let ω ∈ P E ( R d ), p, q ∈ [1 , ∞ ] and φ ∈ S / ( R d ) \ modulation space M p,q ( ω ) ( R d ) consists of all f ∈ S ′ / ( R d ) such that k f k M p,q ( ω ) ≡ (cid:16) Z R d (cid:16) Z R d | V φ f ( x, ξ ) ω ( x, ξ ) | p dx (cid:17) q/p dξ (cid:17) /q (1.15)is finite, and the Wiener amalgam space W p,q ( ω ) ( R d ) consists of all f ∈S ′ / ( R d ) such that k f k W p,q ( ω ) ≡ (cid:16) Z R d (cid:16) Z R d | V φ f ( x, ξ ) ω ( x, ξ ) | q dξ (cid:17) p/q dx (cid:17) /p (1.16)is finite (with obvious modifications in (1.15) and (1.16) when p = ∞ or q = ∞ ). Remark . As follows from Proposition 1.3 (2) below we have thatin fact M p,q ( ω ) ( R d ) contains the superspace Σ ( R d ) of S / ( R d ), and iscontained in the subspace Σ ′ ( R d ) of S ′ / ( R d ), when ω ∈ P E ( R d ).Hence we could from the beginning have assumed that f ∈ Σ ′ ( R d ) in(1.15) and (1.16).On the other hand, in [49], certain weight classes containing P E ( R d )and superexponential weights are introduced. For any s > /
2, thecorresponding families of modulation spaces are large enough to containsuperspaces of S ′ s ( R d ) and subspaces of S s ( R d ).However, we are not dealing with these large families of modulationspaces because we need (1) and (2) in Proposition 1.3 which are notknown to be true for weights of this generality. emark . The literature contains slightly different conventions con-cerning modulation and Wiener amalgam spaces. Sometimes our defi-nition of a Wiener amalgam space is considered as a particular case of ageneral class of modulation spaces (cf. [5–7]). Our definition is adaptedto give the relation (1.19) that suits our purpose to transfer continuityfor the Weyl product on modulation spaces to continuity for twistedconvolution on Wiener amalgam spaces.On the even-dimensional phase space R d we may define modula-tion spaces based on the symplectic STFT. Thus if ω ∈ P E ( R d ), p, q ∈ [1 , ∞ ] and Φ ∈ S / ( R d ) \ symplectic mod-ulation spaces M p,q ( ω ) ( R d ) and Wiener amalgam spaces W p,q ( ω ) ( R d ) areobtained by replacing the STFT a V Φ a by the corresponding sym-plectic version a
7→ V Φ a in (1.15) and (1.16). (Sometimes the word symplectic before modulation space is omitted for brevity.) By (1.14)we have M p,q ( ω ) ( R d ) = M p,q ( ω ) ( R d ) , ω ( x, ξ, y, η ) = ω ( x, ξ, − η, y ) . It follows that all properties which are valid for M p,q ( ω ) carry over to M p,q ( ω ) .From V b φ b f ( ξ, − x ) = e i h x,ξ i V φ f ( x, ξ ) (1.17)it follows that f ∈ W q,p ( ω ) ( R d ) ⇐⇒ b f ∈ M p,q ( ω ) ( R d ) , ω ( ξ, − x ) = ω ( x, ξ ) . In the symplectic situation these formulas read V F σ Φ ( F σ a )( X, Y ) = e iσ ( Y,X ) V Φ a ( Y, X ) (1.18)and F σ M p,q ( ω ) ( R d ) = W q,p ( ω ) ( R d ) , ω ( X, Y ) = ω ( Y, X ) . (1.19)For brevity we denote M p ( ω ) = M p,p ( ω ) , W p ( ω ) = W p,p ( ω ) , and when ω ≡ M p,q = M p,q ( ω ) and W p,q = W p,q ( ω ) . We also let M p,q ( ω ) ( R d ) be thecompletion of S s ( R d ) with respect to the norm k · k M p,q ( ω ) .In the following proposition we list some basic facts on invariance,growth and duality for modulation spaces. Recall that p, p ′ ∈ [1 , ∞ ]satisfy 1 /p + 1 /p ′ = 1. Since our main results are formulated in termsof symplectic modulation spaces, we state the result for them insteadof the modulation spaces M p,q ( ω ) ( R d ). Proposition 1.3.
Let p, q, p j , q j ∈ [1 , ∞ ] for j = 1 , , and ω, ω , ω , v ∈ P E ( R d ) be such that v = ˇ v , ω is v -moderate and ω . ω . Then thefollowing is true: a ∈ M p,q ( ω ) ( R d ) if and only if (1.15) holds for any φ ∈ M v ) ( R d ) \ . Moreover, M p,q ( ω ) is a Banach space under the norm in (1.15) and different choices of φ give rise to equivalent norms; (2) if p ≤ p and q ≤ q then Σ ( R d ) ⊆ M p ,q ( ω ) ( R d ) ⊆ M p ,q ( ω ) ( R d ) ⊆ Σ ′ ( R d ) . (3) the L inner product ( · , · ) L on S / extends uniquely to a con-tinuous sesquilinear form ( · , · ) on M p,q ( ω ) ( R d ) × M p ′ ,q ′ (1 /ω ) ( R d ) .On the other hand, if k a k = sup | ( a, b ) | , where the supremum istaken over all b ∈ S / ( R d ) such that k b k M p ′ ,q ′ (1 /ω ) ≤ , then k · k and k · k M p,q ( ω ) are equivalent norms; (4) if p, q < ∞ , then S / ( R d ) is dense in M p,q ( ω ) ( R d ) and thedual space of M p,q ( ω ) ( R d ) can be identified with M p ′ ,q ′ (1 /ω ) ( R d ) ,through the form ( · , · ) . Moreover, S / ( R d ) is weakly densein M p ′ ,q ′ ( ω ) ( R d ) with respect to the form ( · , · ) provided ( p, q ) =( ∞ , and ( p, q ) = (1 , ∞ ) ; (5) if p, q, r, s, u, v ∈ [1 , ∞ ] , ≤ θ ≤ , p = 1 − θr + θu and q = 1 − θs + θv , then complex interpolation gives ( M r,s ( ω ) , M u,v ( ω ) ) [ θ ] = M p,q ( ω ) . Similar facts hold if the M p,q ( ω ) spaces are replaced by the W p,q ( ω ) spaces. The proof of Proposition 1.3 can be found in [2, 5, 6, 8–10, 16, 45, 47–49].In fact, (1) follows from Gr¨ochenig’s argument verbatim in [16, Propo-sition 11.3.2 (c)]. Note that the window class M v ) ( R d ) in (1) containsΣ ( R d ), which in turn contains S / ( R d ). Furthermore, if in addition v ∈ P ( R d ), then M v ) ( R d ) contains S ( R d ).The proof of (2) in [16, Chapter 12] is based on Gabor frames andformulated for polynomial type weights P ( R d ). These arguments alsohold for the broader weight class P E ( R d ). Another way to prove thisis by means of [16, Lemma 11.3.3] and Young’s inequality.The assertions (3)–(5) in Proposition 1.3 can be found for more gen-eral weights in Theorem 4.17, and a combination of Theorem 3.4 andProposition 5.2 in [49]. Remark . Let P be the set of all ω ∈ P E ( R d ) such that ω ( X, Y ) = e c ( | X | /s + | Y | /s ) , or some c >
0. (Note that this implies that s ≥ \ ω ∈P M p,q ( ω ) ( R d ) = Σ s ( R d ) , [ ω ∈P M p,q (1 /ω ) ( R d ) = Σ ′ s ( R d ) [ ω ∈P M p,q ( ω ) ( R d ) = S s ( R d ) , \ ω ∈P M p,q (1 /ω ) ( R d ) = S ′ s ( R d ) , and for ω ∈ P Σ s ( R d ) ⊆ M p,q ( ω ) ( R d ) ⊆ S s ( R d ) and S ′ s ( R d ) ⊆ M p,q (1 /ω ) ( R d ) ⊆ Σ ′ s ( R d ) . (Cf. [3, Prop. 4.5], [25, Prop. 4], [34, Cor. 5.2] and [41, Thm. 4.1].See also [49, Thm. 3.9] for an extension of these inclusions to broaderclasses of Gelfand–Shilov and modulation spaces.)We have the following result for the map e it h D x ,D ξ i in (1.7) when thedomains are modulation spaces. We refer to [48, Proposition 1.7] forthe proof (see also [49, Proposition 6.14]). Proposition 1.5.
Let ω ∈ P E ( R d ) , p, q ∈ [1 , ∞ ] , t, t , t ∈ R , andset ω t ( x, ξ, η, y ) = ω ( x + ty, ξ + tη, η, y ) . The map e it h D x ,D ξ i on S ′ / ( R d ) restricts to a homeomorphism from M p,q ( ω ) ( R d ) to M p,q ( ω t ) ( R d ) .In particular, if a , a ∈ S ′ / ( R d ) satisfy (1.7) , then a ∈ M p,q ( ω t ) ( R d ) ,if and only if a ∈ M p,q ( ω t ) ( R d ) . (Note that in the equality of (2) in [49, Proposition 6.14], y and η should be interchanged in the last two arguments in ω .)By Proposition 1.3 (4) we have norm density of S / in M p,q ( ω ) when p, q < ∞ . We may relax the assumptions on p , provided we replace thenorm convergence with narrow convergence. This concept, that allowsus to approximate elements in M ∞ ,q ( ω ) ( R d ) for 1 ≤ q < ∞ , is treatedin [36,45,47], and, for the current setup of possibly exponential weights,in [49]. (Sj¨ostrand’s original definition in [36] is somewhat different.)Narrow convergence is defined by means of the function H a,ω,p ( Y ) ≡ kV Φ a ( · , Y ) ω ( · , Y ) k L p ( R d ) , Y ∈ R d , for a ∈ S ′ / ( R d ), ω ∈ P E ( R d ), Φ ∈ S / ( R d ) \ p ∈ [1 , ∞ ]. Definition 1.6.
Let p, q ∈ [1 , ∞ ], and a, a j ∈ M p,q ( ω ) ( R d ), j = 1 , , . . . .Then a j is said to converge narrowly to a with respect to p, q , Φ ∈S / ( R d ) \ ω ∈ P E ( R d ), if there exist g j , g ∈ L q ( R d ) suchthat:(1) a j → a in S ′ / ( R d ) as j → ∞ ;(2) H a j ,ω,p ≤ g j and g j → g in L q ( R d ) and a. e. as j → ∞ . roposition 1.7. If ω ∈ P E ( R d ) and ≤ q < ∞ then the followingis true: (1) S / ( R d ) is dense in M ∞ ,q ( ω ) ( R d ) with respect to narrow conver-gence; (2) M ∞ ,q ( ω ) ( R d ) is sequentially complete with respect to the topologydefined by narrow convergence.Proof. Assertion (1) is a consequence of [49, Definition 2.12 and The-orem 4.19].To prove (2), let { a n } ∞ n =1 ⊆ S ′ / ( R d ) be a Cauchy sequence withrespect to narrow convergence. This means that( a n − a k , ϕ ) → , n, k → ∞ , ϕ ∈ S / ( R d ) , (1.20)and there exists a sequence { g n } ⊆ L q ( R d ) such that H a n ,ω, ∞ ≤ g n ,and k g n − g k k L q → g n − g k → n, k → ∞ . By(1.20) and the completeness of S ′ / ( R d ) there exists a ∈ S ′ / ( R d )such that a n → a in S ′ / ( R d ) as n → ∞ , and by the completeness of L q ( R d ) there exists g ∈ L q ( R d ) such that g n → g in L q ( R d ) and a. e.as n → ∞ . This shows that conditions (1) and (2) of Definition 1.6 aresatisfied.To show a n → a narrowly as n → ∞ it remains to prove a ∈ M ∞ ,q ( ω ) ( R d ). We have for Y ∈ R d H a,ω, ∞ ( Y ) = k lim n →∞ V Φ a n ( · , Y ) ω ( · , Y ) k L ∞ ≤ lim sup n →∞ kV Φ a n ( · , Y ) ω ( · , Y ) k L ∞ ≤ k lim sup n →∞ |V Φ a n ( · , Y ) | ω ( · , Y ) k L ∞ = H a,ω, ∞ ( Y ) . Since H a,ω, ∞ ( Y ) ≤ lim inf n →∞ H a n ,ω, ∞ ( Y ) , Y ∈ R d , the limit lim n →∞ H a n ,ω, ∞ ( Y ) exists, so for almost all Y ∈ R d it followsthat H a,ω, ∞ ( Y ) = lim n →∞ H a n ,ω, ∞ ( Y ) ≤ lim sup n →∞ g n ( Y ) = g ( Y ) . Since g ∈ L q ( R d ) we conclude that H a,ω, ∞ ∈ L q ( R d ) which meansthat a ∈ M ∞ ,q ( ω ) ( R d ). (cid:3) Continuity for the Weyl product on modulation spaces
In this section we deduce results on sufficient conditions for continu-ity of the Weyl product on modulation spaces, and the twisted convo-lution on Wiener amalgam spaces. The main results are Theorems 0.1 ′ and 2.9 concerning the Weyl product, and Theorem 2.12 concerningthe twisted convolution. he first main result Theorem 0.1 ′ together with Theorem 2.9 isequivalent to Theorem 2.12. In the bilinear case, Theorem 0.1 ′ is thesame as Theorem 0.1 in the introduction, and contains [27, Theorem0.3 ′ ] and Theorem 2.9. On the other hand, in the multi-linear case with N >
2, Theorems 0.1 ′ and 2.9 are distinct results with none of themincluded in the other.When proving Theorem 0.1 ′ we first need norm estimates. Then weprove the uniqueness of the extension, where generally norm approxi-mation not suffices, since the test function space may fail to be dense inseveral of the domain spaces. The situation is saved by a comprehensiveargument based on narrow convergence. First we prove the importantspecial cases Propositions 2.2 and 2.5 and then we state and proveTheorem 0.1 ′ .For N ≥ R N be the H¨older–Young exponent function R N ( p ) = ( N − − N X j =0 p j − ! , p = ( p , p , . . . , p N ) ∈ [1 , ∞ ] N +1 , (0.2) ′ and we consider mappings of the form( a , . . . , a N ) a · · · a N . (0.1) ′ We first show a formula for the STFT of a · · · a N expressed with F j ( X, Y ) = V Φ j a j ( X + Y, X − Y ) . (2.1) Lemma 2.1.
Let Φ j ∈ S / ( R d ) , j = 1 , . . . , N , a k ∈ S ′ / ( R d ) forsome ≤ k ≤ N , and a j ∈ S / ( R d ) for j ∈ { , . . . , N } \ k . Suppose Φ = π ( N − d Φ · · · Φ N and a = a · · · a N . If F j are given by (2.1) then F ( X N , X )= Z · · · Z R N − d e iQ ( X ,...,X N ) N Y j =1 F j ( X j , X j − ) dX · · · dX N − (2.2) with Q ( X , . . . , X N ) = N − X j =1 σ ( X j − X , X j +1 − X ) . Proof.
The result follows in the case N = 2 by letting X = X + X and Y = X − X in [27, Lemma 2.1]. For N > (cid:3) ext we use the previous lemma to find sufficient conditions for theextension of (0.1) ′ to modulation spaces. The integral representationof V Φ a in the previous lemma leads to the weight condition1 . ω ( X N + X , X N − X ) N Y j =1 ω j ( X j + X j − , X j − X j − ) ,X , X , . . . , X N ∈ R d . (0.3) ′ The following result is a generalization of [27, Proposition 0.1].
Proposition 2.2.
Let p j , q j ∈ [1 , ∞ ] , j = 0 , , . . . , N , and suppose R N ( q ′ ) ≤ ≤ R N ( p ) . Let ω j , j = 0 , , . . . , N , and suppose (0.3) ′ holds. Then the map (0.1) ′ from S / ( R d ) × · · · × S / ( R d ) to S / ( R d ) extends uniquely to acontinuous and associative map from M p ,q ( ω ) ( R d ) × · · · × M p N ,q N ( ω N ) ( R d ) to M p ′ ,q ′ (1 /ω ) ( R d ) . The associativity means that for any product (0.1) ′ , where the factors a j satisfy the hypotheses, the subproduct a k a k +1 · · · a k is well defined as a distribution for any 1 ≤ k ≤ k ≤ N , and a · · · a N = ( a · · · a k ) ( a k +1 · · · a N ) , for any 1 ≤ k ≤ N − Lemma 2.3.
Let < q < ∞ , let { f n } n ≥ and { g n } n ≥ be sequences in L q ( R d ) such that lim n →∞ k g n − g k L q ( R d ) = 0 , lim n →∞ g n = g a. e., | f n | ≤ g n , and lim n →∞ f n = f a. e., for some measurable functions f and g . Then f ∈ L q ( R d ) and lim n →∞ k f n − f k L q ( R d ) = 0 . Proof.
The result follows from an argument based on Fatou’s lemmaapplied on Z q g ( x ) q dx = Z lim inf n →∞ (( g n ( x ) + g ( x )) q − | f n ( x ) − f ( x ) | q ) dx. (cid:3) emma 2.4. Let < q ≤ ∞ , let f ∈ L q ′ ( R d ) , and let { g n } n ≥ be asequence in L q ( R d ) such that sup n k g n k L q ( R d ) < ∞ and lim n →∞ g n = g a. e., for some measurable function g . Then g ∈ L q ( R d ) and lim n →∞ Z R d ( g n ( x ) − g ( x )) f ( x ) dx = 0 . Proof.
The result follows from a combination of Egorov’s theorem andthe facts that for any ε > B ⊆ R d such that k f k L q ′ ( R d \ B ) < ε, and lim | E |→ k f k L q ′ ( E ) = 0 , where | E | denotes the volume of the measurable set E ⊆ R d . Thedetails are left for the reader. (cid:3) Proof of Proposition 2.2.
By Proposition 1.3 (2) we may assume that R N ( p ) = R N ( q ′ ) = 0, which will allow us to use H¨older’s and Young’sinequalities.Let a , . . . , a N ∈ S / ( R d ). By replacing X j with X j + X in (0.3) ′ , j = 1 , . . . , N , and then replacing 2 X with X , we get1 . ω ( X N + X , X N ) ω ( X + X , X ) N Y j =2 ω j ( X j + X j − + X , X j − X j − ) ,X , . . . , X N ∈ R d . (2.3)Let Φ j , j = 0 , . . . , N , be as in Lemma 2.1. Set G j ( X, Y ) ≡ |V Φ j a j ( X, Y ) | ω j ( X, Y ) , g j ( Y ) ≡ k G j ( · , Y ) k L pj , for j = 1 , . . . , N , and K ( X , . . . , X N ) = G ( X + X , X ) N Y j =2 G j ( X j + X j − + X , X j − X j − ) . Then Lemma 2.1 gives |V Φ a ( X N + X , X N ) | /ω ( X N + X , X N ) . Z · · · Z R N − d K ( X , . . . , X N ) dX · · · dX N − . aking the L p ′ norm in the first variable gives, using Minkowski’sand H¨older’s inequalities, kV Φ a ( · , X N ) | /ω ( · , X N ) k L p ′ . Z · · · Z R N − d k K ( · , X , . . . , X N ) k L p ′ dX · · · dX N − ≤ Z · · · Z R N − d g ( X ) N Y j =2 g j ( X j − X j − ) dX · · · dX N − = ( g ∗ · · · ∗ g N )( X N ) . Applying the L q ′ norm and using Young’s inequality we get k a k M p ′ ,q ′ /ω . k g k L q · · · k g N k L qN = k a k M p ,q ω · · · k a N k M pN ,qN ( ωN ) . (2.4)The result now follows in the case when p j , q j < ∞ for j = 1 , . . . , N ,from the estimate (2.4) and the fact that S / is dense in M p j ,q j ( ω j ) . Inthe case when at least one p j or q j attain ∞ for some j = 1 , . . . , N ,(2.4) still holds when a j ∈ S / , j = 1 , . . . , N , and the Hahn–Banachtheorem and duality guarantee the existence of a continuous extension.We must prove its uniqueness and associativity. First we observe thatthe assumption R N ( q ′ ) = 0 is equivalent to P Nj =0 /q j = N , so q k = ∞ may hold for at most one k , and in that case q j = 1 must hold for j ∈ { , . . . , N } \ k . If q > q j < ∞ for 1 ≤ j ≤ N . So either theuniqueness concerns the inclusion M p ,q ( ω ) · · · M p N ,q N ( ω N ) ⊆ M p ′ ,q ′ (1 /ω ) , q j < ∞ , ≤ j ≤ N, q > , (2.5)or M p ,q ( ω ) · · · M p N ,q N ( ω N ) ⊆ M p ′ , ∞ (1 /ω ) , q j < ∞ , ≤ j ≤ N, (2.6)or, for a unique k such that 1 ≤ k ≤ N , M p , ω ) · · · M p k , ∞ ( ω k ) · · · M p N , ω N ) ⊆ M p ′ , ∞ (1 /ω ) . (2.7)First we consider (2.5). For all j such that p j < ∞ we may extendthe Weyl product uniquely from a j ∈ S / to a j ∈ M p j ,q j ( ω j ) as in the firstpart of the proof, and for the remaining j we extend the Weyl productfrom a j ∈ S / to a j ∈ M ∞ ,q j ( ω j ) using narrow convergence, as follows. Byinduction it suffices to perform the extension for some j ∈ { , . . . , N } from a j ∈ S / to a j ∈ M ∞ ,q j ( ω j ) .Assume for simplicity that j = 1. We may assume q >
1. In fact,our aim is to prove uniqueness only, so if q = 1 we may by Proposition1.3 (2) consider the first factor a as an element in M ∞ , ˜ q ( ω ) with theexponent q modified as 1 / ˜ q = 1 /q − ε <
1, where ε > hat we still have 1 / ˜ q = 1 /q + ε < q of theexponent q .Take a sequence { a ,n } ∞ n =1 that converges narrowly to a ∈ M ∞ ,q ( ω ) .By Definition 1.6 this means that a ,n → a in S ′ / ( R d ) as n → ∞ ,and the existence of g ,n , g ∈ L q ( R d ) such that kV Φ a ,n ( · , Y ) ω ( · , Y ) k L ∞ ≤ g ,n ( Y )and g ,n → g in L q ( R d ) as well as a. e. as n → ∞ . Set g j ( Y ) = kV Φ j a j ( · , X ) ω j ( · , Y ) k L ∞ , j = 2 , . . . , N, and define a ,n = a ,n a · · · a N . Lemma 2.2 and the definitionsabove yield kV Φ a ,n ( · , X N ) /ω ( · , X N ) k L ∞ . Z · · · Z R N − d g ,n ( X ) N Y j =2 g j ( X j − X j − ) dX · · · dX N − = g ,n ∗ g ∗ · · · ∗ g N ( X N ) . From g ,n → g in L q as n → ∞ and Young’s inequality, we mayconclude that g ,n ∗ g ∗ · · · ∗ g N → g ∗ g ∗ · · · ∗ g N in L q ′ ( R d ).The assumption P Nj =0 /q j = N implies 1 /q j ≥ /q ′ for any 1 ≤ j ≤ N . Due to Proposition 1.3 (2) we may therefore assume that g ∈ L q with 1 /q = 1 /q − /q ′ . Then Young’s inequality guarantees that g ∗ · · · ∗ g N ∈ L q ′ ( R d ). It now follows from Lemma 2.4 that g ,n ∗ g ∗· · · ∗ g N → g ∗ g ∗ · · · ∗ g N a. e. So we have shown that the sequence { a ,n } satisfies condition (2) in Definition 1.6 for the modulation space M ∞ ,q ′ (1 /ω ) ( R d ).Let ϕ ∈ S / ( R d ). Our plan is to show that ( a ,n − a ,k , ϕ ) → n, k → ∞ . Together with the conclusions above this will implythat { a ,n } is a Cauchy sequence with respect to narrow convergence.Proposition 1.7 (2) then guarantees that it has a narrow limit a ∈ M ∞ ,q ′ (1 /ω ) ( R d ), which we use as the definition of a · · · a N . It followsthat the Weyl product extends uniquely from a ∈ S / to a ∈ M ∞ ,q ( ω ) .By Lemma 2.1 we have( a ,n − a ,k , ϕ ) = C Φ ( V Φ ( a ,n − a ,k ) , V Φ ϕ )= C Φ Z · · · Z R N +1) d e iQ ( X ,...,X N ) H n,k ( X , . . . , X N ) dX · · · dX N , (2.8) here H n,k ( X , . . . , X N ) = 4 d V Φ ( a ,n − a ,k )( X + X , X − X ) ×× N Y j =2 F j ( X j , X j − ) ! V Φ ϕ ( X N + X , X N − X ) . By the narrow convergence we have V Φ a ,n → V Φ a pointwise as n → ∞ , which implies thatlim n,k →∞ H n,k ( X , . . . , X N ) = 0 , ( X , . . . , X N ) ∈ R N +1) d . (2.9)If we define G ( X, Y ) = |V Φ ϕ ( X + Y, X − Y ) | ω ( X + Y, X − Y ) , then | H n,k | . K n,k , where K n,k ( X , . . . , X N ) ≡ ( g ,n ( X − X ) + g ,k ( X − X )) N Y j =2 g j ( X j − X j − ) ! | G ( X N , X ) | . By Young’s inequality and the assumption g ,n → g in L q , K n,k hasa limit in L ( R N +1) d ), denoted K . By the assumption g ,n → g a. e., K n,k → K a. e. as n, k → ∞ . Hence (2.8), (2.9) and Lemma 2.3 implythat ( a ,n − a ,k , ϕ ) → n, k → ∞ .By the same arguments it follows that the integral formula (2.2)holds for the extension for almost all ( X N , X ) ∈ R d . This finishes theproof of the uniqueness of the extended Weyl product inclusion (2.5).The uniqueness in the cases (2.6) and (2.7) follow from the unique-ness in the case (2.5) and duality.It remains to prove the asserted associativity, and first we need toprove that any subproduct of a · · · a N is well defined. We observethat (0.3) ′ can be written as1 . ω ( X N + X , X N − X ) ϑ ( X , . . . , X k ) ϑ ( X k , . . . , X N )for ϑ ( X , . . . , X k ) = k Y j =1 ω j ( X j + X j − , X j − X j − )and ϑ ( X k , . . . , X N ) = N Y j = k +1 ω j ( X j + X j − , X j − X j − ) , and any 1 ≤ k ≤ N −
1. If ϑ is defined by ϑ ( X k + X , X k − X ) ≡ inf ϑ ( X , . . . , X k ) here the infimum is taken over all X , . . . , X k − ∈ R d , it follows from(0.3) ′ that1 . ϑ ( X k + X , X k − X ) − k Y j =1 ω j ( X j + X j − , X j − X j − ) ,X , X , . . . , X k ∈ R d , and1 . ω ( X N + X , X N − X ) ϑ ( X k + X , X k − X ) N Y j = k +1 ω j ( X j + X j − , X j − X j − ) ,X , X k , , X k +1 , . . . , X N ∈ R d . Note that ϑ ∈ P E ( R d ) by the assumptions.It now follows from the first part of the proof that a · · · a k ∈ M r,s ( ϑ ) and b a k +1 · · · a N ∈ M p ′ ,q ′ (1 /ω ) , when a j ∈ M p j .q j ( ω j ) , b ∈ M r,s ( ϑ ) , and r, s ∈ [1 , ∞ ] are defined by1 r = k X j =1 p j and 1 s ′ = k X j =1 q ′ j . This shows that a · · · a k and a k +1 · · · a N are well-defined as ele-ments in appropriate modulation spaces.The asserted associativity now follows from the density argumentsin the proof of the uniqueness, and the fact that the Weyl product isassociative on S / . (cid:3) For appropriate weights ω the space M ω ) ( R d ) consists of sym-bols of Hilbert–Schmidt operators acting between certain modulationspaces (cf. [48, 50]). The following proposition, with p j = q j = 2 for j = 0 , . . . , N , is a manifestation of the fact that Hilbert–Schmidt op-erators are closed under composition. The result in that special caseis a consequence of [27, Proposition 0.2], which concerns N = 2, with p j = q j = 2, j = 0 , ,
2, and induction. The general result relaxesthe assumption on the exponents, and is an essential step towards theimprovement Theorem 0.1 ′ below. Proposition 2.5.
Let p j , q j ∈ [1 , ∞ ] , j = 0 , , . . . , N , and suppose max ( R N ( q ′ ) , ≤ min j =0 , ,...,N (cid:18) p j , p ′ j , q j , q ′ j , R N ( p ) (cid:19) . (2.10) et ω j ∈ P E ( R d ) , j = 0 , , . . . , N , and suppose (0.3) ′ holds. Thenthe map (0.1) ′ from S / ( R d ) × · · · × S / ( R d ) to S / ( R d ) extendsuniquely to a continuous and associative map from M p ,q ( ω ) ( R d ) × · · · × M p N ,q N ( ω N ) ( R d ) to M p ′ ,q ′ (1 /ω ) ( R d ) .Proof. First we prove the result for p j = q j = 2 for all 0 ≤ j ≤ N . Let a j ∈ S / , j = 1 , . . . , N , and let G j ( X, Y ) = | F j ( X, Y ) ω j ( X + Y, X − Y ) | , j = 1 , . . . , N, where F j are given by (2.1). Lemma 2.1 and repeated application ofH¨older’s inequality give | F ( X N , X ) | /ω ( X N + X , X N − X ) . Z · · · Z R N − d N Y j =1 G j ( X j , X j − ) ! dX · · · dX N − ≤ k G ( · , X ) k L ( R d ) k G N ( X N , · ) k L ( R d ) N − Y j =2 k G j k L ( R d ) . Taking the L ( R d ) norm gives k a k M /ω . N Y j =1 k G j k L ≍ N Y j =1 k a j k M ωj ) . The claim follows from this estimate and the fact that S / is dense in M ω j ) .The proof of the general case is based on multi-linear interpolationbetween the case p j = q j = 2 for 0 ≤ j ≤ N and Proposition 2.2.More precisely, by Proposition 2.2 and the first part of this proof wehave M r ,s ( ω ) · · · M r N ,s N ( ω N ) ⊆ M r ′ ,s ′ (1 /ω ) and M , ω ) · · · M , ω N ) ⊆ M , /ω ) , when r j , s j ∈ [1 , ∞ ], j = 0 , , . . . , N , and R N ( s ′ ) ≤ ≤ R N ( r ) . (2.11)By multi-linear interpolation, using [1, Theorem 4.4.1] and Proposi-tion 1.3 (5), we get M p ,q ( ω ) · · · M p N ,q N ( ω N ) ⊆ M p ′ ,q ′ (1 /ω ) (2.12)when 1 p j = 1 − θr j + θ q j = 1 − θs j + θ , ≤ θ ≤ , (2.13) j = 0 , , . . . , N .Suppose p j , q j ∈ [1 , ∞ ], j = 0 , , . . . , N , satisfy (2.10). We haveto show that there exist 0 ≤ θ ≤ r j ∈ [1 , ∞ ] and s j ∈ [1 , ∞ ], = 0 , , . . . , N , such that (2.11) and (2.13) are satisfied, after which(2.12) follows by multi-linear interpolation. Our plan is to first find anappropriate θ , and then find r and s with the right properties.We have r j ∈ [1 , ∞ ] if and only if0 ≤ − θr j = 1 p j − θ ≤ − θ, i. e. θ/ ≤ min(1 /p j , /p ′ j ), and likewise s j ∈ [1 , ∞ ] if and only if θ/ ≤ min(1 /q j , /q ′ j ). Since R N ( q ′ ) ≤ / θ ∈ [0 ,
1] such that R N ( q ′ ) ≤ θ ≤ min j =0 , ,...,N (cid:18) q j , q ′ j , p j , p ′ j , R N ( p ) (cid:19) again by the assumption (2.10). With such a choice of θ we have r j , s j ∈ [1 , ∞ ] for j = 0 , , . . . , N , and R N ( q ′ ) ≤ θ/ ≤ R N ( p ) . This gives R N ( r ) = 11 − θ (cid:18) R N ( p ) − θ (cid:19) ≥ R N ( s ′ ) = 11 − θ (cid:18) R N ( q ′ ) − θ (cid:19) ≤ − θ (cid:18) θ − θ (cid:19) = 0 . Hence (2.11) and (2.13) are satisfied, and (2.12) follows. Thus (2.10)implies (2.12).It remains to prove the associativity. If R N ( q ′ ) ≤ ≤ R N ( p ) theassociativity follows from Proposition 2.2, and if p j = q j = 2, j =0 , . . . , N , the associativity follows from the associativity of the Weylproduct on S / , and the fact that S / is dense in M , ω j ) for every j .The associativity now follows in general from the fact that the generalcase is an interpolation between the latter two cases. (cid:3) Remark . A crucial step in the proof is the fact that (2.10) impliesthat θ and r , s ∈ [1 , ∞ ] N +1 can be chosen such that (2.11) and (2.13)holds. On the other hand, by straight-forward computations it followsthat if (2.11) and (2.13) are fulfilled, then (2.10) holds. Remark . We note that Proposition 2.5 extends [27, Theorem 0.3 ′ ].The latter result asserts that if N = 2, R ( p ) = R ( q ′ ) , q , q ≤ q ′ , ≤ R ( p ) ≤ p j , q j , p ′ , q ′ ≤ − R ( q ′ ) , j = 1 , , (2.14)and ω j , j = 0 , ,
2, are weights that satisfy (0.3), then the map (0.1)extends to a continuous map from M p ,q ( ω ) × M p ,q ( ω ) to M p ′ ,q ′ (1 /ω ) . We claim hat (2.14) implies (2.10) when N = 2, which means that Proposition2.5 extends [27, Theorem 0.3 ′ ].In fact, by the last inequality in (2.14) we get R ( q ′ ) ≤ p ′ j , q ′ j , p , q , j = 1 , . A combination of these inequalities gives R ( q ′ ) = R ( p ) ≤ min j =0 , , (cid:18) p j , p ′ j , q j q ′ j (cid:19) , and it follows that the hypothesis (2.10) in Proposition 2.5 is fulfilledfor N = 2.Next we prove that the conclusion of Proposition 2.5 holds underassumptions that are weaker than (2.10). The following lemma showsthat we may omit the condition R N ( q ′ ) ≤ min ≤ j ≤ N (1 /q j ) in (2.10). Lemma 2.8.
Let N ≥ , x j ∈ [0 , , j = 0 , . . . , N and consider theinequalities: (1) ( N − − N X k =0 x k − ! ≤ min ≤ j ≤ N x j ; (2) x j + x k ≤ , for all k = j ; (3) ( N − − N X k =0 x k − ! ≤ min ≤ j ≤ N (1 − x j ) .Then (1) = ⇒ (2) = ⇒ (3) . If N = 2 then (1) and (2) are equivalent.Proof. Assume that (1) holds but (2) fails. Then x j + x k > j = k . By renumbering we may assume that x ≤ x j for every j , andthat x + x >
1. Then (1) gives( N − x ≤ N X k =2 x k < N X k =0 x k − ≤ ( N − x which is a contradiction. Hence the assumption x + x > x j ≤ − x k , k = j. This gives X j = k x j ≤ N (1 − x k ) ⇐⇒ N X j =0 x j − ≤ ( N − − x k )for any k , so (3) holds. inally, if j = k , N = 2 and (2) holds, then x j + x k ≤
1, which gives x j + x k + x l − ≤ x l , l = 0 , ,
2. In particular, X k =0 x k − ≤ x l , l = 0 , , , and (1) follows. (cid:3) The next result is one of two principal theorems on sufficient condi-tions for continuity. It shows that one can eliminate some conditionson the Lebesgue exponents in Proposition 2.5. In particular the resultextends [27, Theorem 0.3 ′ ] in view of Remark 2.7. Theorem 0.1 ′ . Let p j , q j ∈ [1 , ∞ ] , j = 0 , , . . . , N , and suppose max ( R N ( q ′ ) , ≤ min j =0 , ,...,N (cid:18) p j , q ′ j , R N ( p ) (cid:19) . (0.4) ′ Let ω j ∈ P E ( R d ) , j = 0 , , . . . , N , and suppose (0.3) ′ holds. Thenthe map (0.1) ′ from S / ( R d ) × · · · × S / ( R d ) to S / ( R d ) extendsuniquely to a continuous and associative map from M p ,q ( ω ) ( R d ) × · · · × M p N ,q N ( ω N ) ( R d ) to M p ′ ,q ′ (1 /ω ) ( R d ) .Proof. We may assume that R N ( q ′ ) > ′ imply R N ( q ′ ) ≤ min j =0 , ,...,N (cid:18) q j , q ′ j (cid:19) . (2.15)Hence, if r is defined by 1 r ≡ R N ( q ′ ) , then r ≥ M u ,q ( ω ) · · · M u N ,q N ( ω N ) ⊆ M u ′ ,q ′ (1 /ω ) , when u j ∈ [1 , ∞ ] for 0 ≤ j ≤ N and1 r ≤ min j =0 , ,...,N (cid:18) u j , u ′ j , q ′ j , R N ( u ) (cid:19) . (2.16)Due to Proposition 1.3 (2) the result follows if we can prove that p j ≤ u j for some u j ∈ [1 , ∞ ], j = 0 , . . . , N , that satisfy (2.16). We claim that u j = max( p j , r ′ ), j = 0 , . . . , N , satisfy (2.16).To wit, for such a choice we have1 u ′ j = max (cid:18) p ′ j , r (cid:19) ≥ r , j = 0 , . . . , N, and 1 u j = min (cid:18) p j , r ′ (cid:19) ≥ r , j = 0 , . . . , N, (2.17) here (2.17) follows from r ≥ p j ≤ r .Let I be the set of all j such that r ′ ≤ p j . If I = { , , . . . , N } the result follows from Proposition 2.5. Therefore we may assume thatthere exists k ∈ { , , . . . , N } such that k / ∈ I . Then u k = r ′ , and (2.17)gives( N − R N ( u ) = 1 u k + X j = k u j − r ′ + X j = k u j − − r + X j = k u j ≥ − r + Nr = N − r . Hence R N ( u ) ≥ r and the continuity assertion follows.The uniqueness and associativity follows from Proposition 2.5 andthe inclusions above. (cid:3) In the next section we prove that Theorem 0.1 ′ is sharp for N = 2with respect to the conditions on the Lebesgue exponents. On the otherhand, for N ≥
3, the result cannot be sharp. In fact, Theorem 0.1 ′ with N = 2 gives in particular that every unweighted modulation space M p,q is an M ∞ , -module. This property combined with the fact that M , isan algebra under the Weyl product give the inclusion M ∞ , M , M , ⊆ M , . (2.18)Theorem 0.1 ′ does however not contain this inclusion.The next result gives another sufficient condition for the map (0.1) ′ to be continuous that contains the inclusion (2.18). In the bilinear case N = 2 the result follows from Theorem 0.1 ′ , because of the sharpnessof the latter result in that case. Theorem 2.9.
Let p j , q j ∈ [1 , ∞ ] , j = 0 , , . . . , N , and suppose R N ( p ) ≥ and q ′ j ≤ p j ≤ . (2.19) Let ω j ∈ P E ( R d ) , j = 0 , , . . . , N , and suppose (0.3) ′ holds. Thenthe map (0.1) ′ from S / ( R d ) × · · · × S / ( R d ) to S / ( R d ) extendsuniquely to a continuous and associative map from M p ,q ( ω ) ( R d ) × · · · × M p N ,q N ( ω N ) ( R d ) to M p ′ ,q ′ (1 /ω ) ( R d ) . The proof is by induction over N , and we need the existence of certainintermediate weights. The following lemma guarantees the existence ofsuch weights. emma 2.10. Let ω , . . . , ω N ∈ P E ( R d ) satisfy (0.3) ′ . Then thereexists a weight ϑ ∈ P E ( R d ) such that . ω ( X + X , X − X ) ω N ( X + X , X − X ) ϑ ( X + X , X − X ) , X , X , X ∈ R d , . ϑ ( X N − + X , X N − − X ) N − Y j =1 ω j ( X j + X j − , X j − X j − ) ,X , . . . , X N − ∈ R d . (2.20) Proof.
Let X = X N − + X , Y = X N − − X , define the linear mappingsfrom R d to R d given by T j,k ( X, Y, Z ) = (cid:18) X + ( − j Y Z, ( − k (cid:18) X + ( − j Y − Z (cid:19)(cid:19) , for j, k = 1 ,
2, and set H ( X , . . . , X N − ) ≡ N − Y j =2 ω j ( X j + X j − , X j − X j − ) ,H ( X , X N − , X, Y ) ≡ ω (cid:0) T , ( X, Y, X ) (cid:1) ω N − (cid:0) T , ( X, Y, X N − ) (cid:1) and H ( X N , X, Y ) ≡ ω (cid:0) T , ( X, Y, X N ) (cid:1) ω N (cid:0) T , ( X, Y, X N ) (cid:1) . Then (0.3) ′ is equivalent to( H ( X , X N − , X, Y ) H ( X , . . . , X N − )) − . H ( X N , X, Y ) . The left hand side is independent of X N and the right hand side isindependent of X , . . . , X N − .If we define ϑ ( X, Y ) ≡ sup X ,...,X N − ∈ R d ( H ( X , X N − , X, Y ) H ( X , . . . , X N − )) − then ϑ ( X N − + X , X N − − X ) . H ( X N , X N − + X , X N − − X ) ,X , X N − , X N ∈ R d , and (2.20) holds.It remains to show ϑ ∈ P E ( R d ). For ε > ϑ ( Z + Z , Y + Y ) ≤ ( H ( X , X N − , Z + Z , Y + Y ) H ( X , . . . , X N − )) − + ε or some choice of X , . . . , X N − . Since each ω j is moderate we have( H ( X , X N − , Z + Z , Y + Y )) − ≤ C ( H ( X , X N − , Z , Y )) − v ( Z , Y ) , for C > v ∈ L ∞ loc ( R d ), whichdepends on ω and ω N − .This estimate yields ϑ ( Z + Z , Y + Y ) ≤ C ( H ( X , X N − , Z , Y ) H ( X , . . . , X N − )) − v ( Z , Y ) + ε ≤ Cϑ ( Z , Y ) v ( Z , Y ) + ε. Since ε > C does not depend on ε , it follows that ϑ is v -moderate, and we may conclude that ϑ ∈ P E ( R d ). (cid:3) Proof of Theorem 2.9.
By Proposition 1.3 (2) we may assume q ′ j = p j , j = 0 , . . . , N .We start by proving the result for N = 2. Assume (2.19) for N = 2.Then for every fixed j ∈ { , , } we get R ( p ) = R ( q ′ ) = X k =0 p k − ≤ p j . The continuity statement now follows from Theorem 0.1 ′ .Next we perform the induction step. We assume that N ≥ N . In particular we assume the inclusion M r ,r ′ ( ω ) · · · M r N − ,r ′ N − ( ω N − ) ⊆ M r ′ ,r (1 /ϑ ) (2.21)whenever r j ≥ N − X j =0 r j ≥ , and ( ϑ, ω , . . . , ω N − ) ∈ P E ( R d ) N satisfy (0.3) ′ .We now distinguish two cases.In the first case we suppose that1 p + 1 p N ≤
12 or 1 p j + 1 p j +1 ≤
12 for some j ∈ { , . . . , N − } .(2.22)By (1.12) and duality it suffices to consider the case when the firstinequality in (2.22) holds. We define r as1 r = 1 p + 1 p N ≤ , nd the result follows if we prove the inclusions M p ,p ′ ( ω ) · · · M p N − ,p ′ N − ( ω N − ) ⊆ M r ′ ,r (1 /ϑ ) (2.23)and M r ′ ,r (1 /ϑ ) M p N ,p ′ N ( ω N ) ⊆ M p ′ ,p (1 /ω ) , (2.24)where ϑ is chosen according to Lemma 2.10.Since1 r + N − X k =1 p k = N X k =0 p k ≥ r , p j ≤ , j = 1 , . . . , N − , the inclusion (2.23) follows from the induction assumption (2.21).By letting s = p , s = r ′ , s = p N , it follows from the choice of r that R ( s ) = 0, and the inclusion (2.24) follows from Proposition2.2, since ( ω , /ϑ, ω N ) satisfy (0.3) ′ for N = 2 by Lemma 2.10. Theinduction step is now complete in the first case by combining (2.23)and (2.24).It remains to consider the second case where (2.22) does not hold.Therefore suppose that1 p + 1 p N >
12 and 1 p j + 1 p j +1 > , (2.25)for all j = 0 , . . . , N −
1. In particular we have1 p + 1 p N + 12 − > , so by the result for N = 2 we have the inclusion M , /ϑ ) M p N ,p ′ N ( ω N ) ⊆ M p ′ ,p (1 /ω ) . (2.26)Since N ≥
3, (2.25) implies1 p + 1 p + · · · + 1 p N − + 12 − > , and the induction hypothesis (2.21) thus gives M p ,p ′ ( ω ) · · · M p N − ,p ′ N − ( ω N − ) ⊆ M , /ϑ ) . (2.27)Combining (2.26) and (2.27) gives the induction step in the secondcase. The induction step is thus complete so the continuity statementholds for any integer N ≥ p j = ∞ then by the assumptions q j = 1, and a factor a j ∈ M ∞ , ω j ) can be approximated narrowly byelements in S / . If p j < ∞ then the assumption 2 ≤ q ′ j implies that afactor a j ∈ M p j ,q j ( ω j ) can be approximated in norm by elements in S / . (cid:3) e may use (1.7) and Proposition 1.5 to extend Theorems 0.1 ′ and2.9 to concern not only the Weyl product but general products arisingin the pseudo-differential calculi (1.3) indexed by t ∈ R . More precisely,for every t ∈ R , the t product with N factors( a , . . . , a N ) a t · · · t a N (2.28)from S / ( R d ) × · · · × S / ( R d ) to S / ( R d ) is defined by the formulaOp t ( a t · · · t a N ) = Op t ( a ) ◦ · · · Op t ( a N ) . By (1.7) we have a t · · · t a N = e it h D x ,D ξ i (( e − it h D x ,D ξ i a ) · · · ( e − it h D x ,D ξ i a N )) ,t = 12 − t. If we combine this relation with Proposition 1.5, Theorems 0.1 ′ and2.9, we get the following result. The condition on the weight functionsis 1 . ω ( T t ( X N , X )) N Y j =1 ω j ( T t ( X j , X j − )) , X , . . . , X N ∈ R d , (2.29)where T t ( x, ξ, y, η ) = ( tx + (1 − t ) y, (1 − t ) ξ + tη, η − ξ, x − y ) ,x, ξ, y, η ∈ R d . (2.30) Theorem 2.11.
Let p j , q j ∈ [1 , ∞ ] , j = 0 , , . . . , N , be as in Theorems0.1 ′ or 2.9. Let t ∈ R , ω j ∈ P E ( R d ) , j = 0 , , . . . , N , and suppose (2.29) and (2.30) hold. Then the map (2.28) from S / ( R d ) × · · · ×S / ( R d ) to S / ( R d ) extends uniquely to a continuous and associativemap from M p ,q ( ω ) ( R d ) × · · · × M p N ,q N ( ω N ) ( R d ) to M p ′ ,q ′ (1 /ω ) ( R d ) . Finally we prove a continuity result for the twisted convolution. Themap (0.1) ′ is then replaced by( a , a , . . . , a N ) a ∗ σ a ∗ σ · · · ∗ σ a N . (2.31)The following result follows immediately from Theorem 0.1 ′ and The-orem 2.9. Here the condition (0.3) ′ is replaced by1 . ω ( X N − X , X N + X ) N Y j =1 ω j ( X j − X j − , X j + X j − ) ,X , X , . . . , X N ∈ R d . (2.32) heorem 2.12. Let p j , q j ∈ [1 , ∞ ] , j = 0 , , . . . , N , and suppose that max ( R N ( p ′ ) , ≤ min j =0 , ,...,N (cid:18) p ′ j , q j , R N ( q ) (cid:19) or R N ( q ) ≥ and p ′ j ≤ q j ≤ . Suppose ω j ∈ P E ( R d ) , j = 0 , , . . . , N , satisfy (2.32) . Then the map (2.31) from S / ( R d ) ×· · ·×S / ( R d ) to S / ( R d ) extends uniquely toa continuous and associative map from W p ,q ( ω ) ( R d ) ×· · ·× W p N ,q N ( ω N ) ( R d ) to W p ′ ,q ′ (1 /ω ) ( R d ) . Necessary boundedness conditions
In this section we prove necessary conditions for continuity of theWeyl product when N = 2 and certain polynomially moderate weighttriplets that satisfy (0.3).More precisely, for weights of the form ω ( X, Y ) = ϑ ( X + Y ) ϑ ( X − Y ) , ω ( X, Y ) = ϑ ( X − Y ) ϑ ( X + Y ) ,ω ( X, Y ) = ϑ ( X − Y ) ϑ ( X + Y ) , (3.1)where ϑ j ∈ P ( R d ), j = 0 , ,
2, we have the following result. Notethat the necessary condition (3.2) equals the sufficient condition (0.4)of Theorem 0.1.
Theorem 3.1.
Let p j , q j ∈ [1 , ∞ ] , ϑ j ∈ P ( R d ) , and define ω j ∈ P ( R d ) by (3.1) for j = 0 , , . If the map (0.1) from S / ( R d ) ×S / ( R d ) to S / ( R d ) is extendable to a continuous map from M p ,q ( ω ) ( R d ) × M p ,q ( ω ) ( R d ) to M p ′ ,q ′ (1 /ω ) ( R d ) , then max( R ( q ′ ) , ≤ min j =0 , , (cid:18) p j , q ′ j , R ( p ) (cid:19) . (3.2) Remark . The conditions (3.1) on the weights appear naturally whenWeyl operators with symbols in modulation spaces act on modula-tion spaces. For example, if (3.1) is fulfilled, p, q ∈ [1 , ∞ ] and a ∈ M ∞ , ω ) ( R d ), then Op w ( a ) is continuous from M p,q ( ϑ ) ( R d ) to M p,q ( ϑ ) ( R d )(cf. [49, Theorem 6.2]).We need some preparations for the proof. The first result is a reduc-tion to trivial weights. For ω ∈ P ( R d ) we use the notation S ( ω ) ( R d )for the symbol space of all a ∈ C ∞ ( R d ) such that ( ∂ α a ) /ω ∈ L ∞ forall α ∈ N d . emma 3.3. Let ϑ, ϑ , ϑ ∈ P ( R d ) , ω , ω ∈ P ( R d ) and let p, q ∈ [1 , ∞ ] . There exist b j ∈ S ( ϑ j ) ( R d ) and c j ∈ S (1 /ϑ j ) ( R d ) such that Op w ( b j ) ◦ Op w ( c j ) = Op w ( c j ) ◦ Op w ( b j ) is the identity map on S ′ ( R d ) , for j = 1 , , and the following holds: (1) Op w ( b j ) is continuous and bijective from M p,q ( ϑ ) ( R d ) to M p,q ( ϑ/ϑ j ) ( R d ) with inverse Op w ( c j ) , j = 1 , ; (2) if ω ( X, Y ) . ω ( X, Y ) /ϑ ( X + Y ) , then the map (0.1) on S ( R d ) extends uniquely to a continuous map from M p,q ( ω ) ( R d ) × S ( ϑ ) ( R d ) to M p,q ( ω ) ( R d ) ; (3) if ω ( X, Y ) . ω ( X, Y ) /ϑ ( X − Y ) , then the map (0.1) on S ( R d ) extends uniquely to a continuous map from S ( ϑ ) ( R d ) × M p,q ( ω ) ( R d ) to M p,q ( ω ) ( R d ) ; (4) if ω ( X, Y ) = ϑ ( X − Y ) /ϑ ( X + Y ) , then the map a b a c is continuous on S ( R d ) , and extends uniquely to a continuousand bijective map from M p,q ( ω ) ( R d ) to M p,q ( R d ) .Proof. The assertion (1) follows immediately from [24, Theorem 3.1].In order to prove (2) and (3), we first use the assumption that ω and ϑ are moderate, which gives ω ( X, Y ) . ω ( X − Y + Z, Z ) h Y − Z i N ,ω ( X, Y ) . ω ( X + Z, Y − Z ) h Z i N , ϑ ( X + Y ) . h Y − Z i N ϑ ( X + Z ) , ϑ ( X − Y ) . h Z i N ϑ ( X − Y + Z ) , for some N ≥
0. The assumption in (2) leads to ω ( X, Y ) . ω ( X − Y + Z, Z ) h Y − Z i N ϑ ( X + Z ) , (3.3)and the assumption in (3) gives ω ( X, Y ) . h Z i N ϑ ( X − Y + Z ) ω ( X + Z, Y − Z ) . (3.4)If we set ω ( X, Y ) = h Y i N /ϑ ( X ) then Theorem 0.1 with (3.3) and(3.4), respectively, shows that the map (0.1) from S × S to S extendsuniquely to a continuous map from M p,q ( ω ) × M ∞ , ω ) and M ∞ , ω ) × M p,q ( ω ) ,respectively, to M p,q ( ω ) . The assertions (2) and (3) now follow from S ( ϑ ) ⊆ M ∞ , ω ) , which is a consequence of [24, Proposition 2.7 (3)].Finally (4) follows by a straight-forward combination of (1)–(3). (cid:3) emma 3.3 shows that for weights ω j , j = 0 , ,
2, satisfying (3.1) wehave k a b k M p ′ ,q ′ /ω . k a k M p ,q ω k b k M p ,q ω , a, b ∈ S / ( R d ) , if and only if k a b k M p ′ ,q ′ . k a k M p ,q k b k M p ,q , a, b ∈ S / ( R d ) , thus reducing the problem to the case of trivial weights.It remains to show Theorem 3.1 with trivial weights. By the last partof Lemma 2.8, the Condition (3.2) is equivalent to the inequalities1 ≤ p + 1 p + 1 p , (3.5)1 ≤ q j + 1 q k , ≤ j = k ≤ , (3.6)2 − q − q − q ≤ p j , j = 0 , , , (3.7)3 ≤ q + 1 q + 1 q + 1 p + 1 p + 1 p , (3.8)which we prove in the following sequence of lemmas.The first lemma shows that (3.5) and (3.6) are necessary for therequested continuity. Lemma 3.4.
Let ≤ p j , q j ≤ ∞ for j = 0 , , , and suppose k a b k M p ′ ,q ′ . k a k M p ,q k b k M p ,q for every a, b ∈ S / ( R d ) . (3.9) Then (3.5) and (3.6) hold.Proof.
First we observe that the assumption (3.9) combined with (1.12),duality and a b = b a (cf. [27]) imply that k a b k M r ′ ,s ′ . k a k M r ,s k b k M r ,s for every a, b ∈ S / ( R d ) , when r j = p σ ( j ) and s j = q σ ( j ) , where σ is any permutation { , , } .By [27, Corollary 3.4] we therefore have1 ≤ q j + 1 q k , ≤ j = k ≤ , i. e. (3.6).In order to show (3.5), we consider the family of functions a λ ( X ) = e − λ | X | , X ∈ R d , λ >
0. Straight-forward computations show that(cf. [27, Proposition 3.1]) k a λ k /d M r,s = π r + s − r − /r s − /s λ − /r (1 + λ ) /r +1 /s − (3.10)and a λ a λ ( X ) = (1 + λ ) − d exp (cid:18) − λ λ | X | (cid:19) . (3.11) his gives k a λ a λ k /d M r,s = π /r +1 /s − r − /r s − /s (1 + λ ) − /s (2 λ ) − /r (1 + λ ) /r +1 /s − . (3.12)From the assumption (3.9) we obtain π /p ′ +1 /q ′ − p ′ − /p ′ q ′ − /q ′ (1+ λ ) − /q ′ (2 λ ) − /p ′ (1+ λ ) /p ′ +1 /q ′ − ≤ Cπ /p +1 /q − p − /p q − /q λ − /p (1 + λ ) /p +1 /q − × π /p +1 /q − p − /p q − /q λ − /p (1 + λ ) /p +1 /q − . Letting λ → (cid:3) Next we introduce more general Gaussians of the form a λ,µ ( x, ξ ) = e − λ | x |− µ | ξ | , x, ξ ∈ R d , λ, µ > . (3.13)We consider a λ ,µ a λ ,µ , where λ j , µ j > λ µ = λ µ , (3.14)so that ν ≍ ν when ν = 1 + λ µ = 1 + λ µ , ν = 1 + λ µ + λ µ . (3.15)The first part of the following result follows by straight-forward com-putations and (1.9). The other statements follow from the first partand [45, Lemma 1.8]. Lemma 3.5.
Let r, s ∈ [1 , ∞ ] , let λ, µ, λ j , µ j > , j = 1 , , satisfy (3.14) , and define ν and ν by (3.15) . Then a λ ,µ a λ ,µ ( x, ξ ) = (2 π ) − d/ ν − d e − ( λ + λ ) | x | /ν e − ( µ + µ ) | ξ | /ν , k a λ,µ k /d M r,s = c r,s ( λµ ) − / r (cid:0) (1 + λ )(1 + µ ) (cid:1) (1 /r +1 /s − / , and k a λ ,µ a λ ,µ k /d M r,s = C r,s ν − /s (cid:0) ( λ + λ )( µ + µ ) (cid:1) − / (2 r ) (cid:0) ( ν + λ + λ )( ν + µ + µ ) (cid:1) (1 /r +1 /s − / , for some constants c r,s , C r,s > which only depend on r, s . Lemma 3.5 is used in the proof of the following result, which showsthat (3.8) is a consequence of the requested continuity.
Lemma 3.6. If (3.9) holds then (3.8) is true. roof. Let λ = λ = 1 /µ = 1 /µ = λ >
1. Then ν = 2, and Lemma3.5 gives k a λ ,µ a λ ,µ k M p ′ ,q ′ ≍ λ d (1 /p ′ +1 /q ′ − / , and k a λ j ,µ j k M pj,qj ≍ λ d (1 /p j +1 /q j − / , j = 1 , . The assumption (3.9) together with λ → + ∞ give1 p ′ + 1 q ′ − ≤ p + 1 q − p + 1 q − , which is the same as (3.8). (cid:3) Finally we show that (3.9) implies (3.7).
Lemma 3.7. If (3.9) holds then (3.7) is true.Proof. By duality it suffices to show that (3.9) implies that1 p ′ + 1 q ′ ≤ q + 1 q or 1 p ′ + 1 q ′ ≤ q + 1 q . The proof is a contradictary argument. We assume that (3.9) holds,1 p ′ + 1 q ′ > q + 1 q (3.16)and 1 p ′ + 1 q ′ > q + 1 q . (3.17)This will lead to a contradiction which shows that (3.9) implies1 p ′ + 1 q ′ ≤ q + 1 q , i.e. (3.7) with j = 0. The cases j = 1 , M p ,q ( R d ) M p ,q ( R d ) ⊆ M p ′ ,q ′ ( R d ) . (3.18)From (3.17) we may conclude that there exists ε > p + 1 q + 1 q − q ′ < − εd . (3.19)The rest of the proof is an adaptation of the proof of [27, Theorem3.6] (see also the proof of [22, Theorem 5]).Let 0 ≤ g ∈ C ∞ ( R d ) \ /
4. For n ∈ Z d we set d n = d n,ε = ( n = 0 | n | − ( d + ε ) n = 0 , (3.20)so that { d n } ∈ l . We also set α n = d /p n , β n = d /q n , γ n = d /q n , η n = d /q n , nd we let τ n be the operator τ n f = f ( · − n ). Our plan is to use thefamily of functions on R d f = X n α n τ n g, f = f ,N = X | n |≤ N β n τ n g,f = X n γ n τ n g, f = X n η n τ n g, (3.21)to construct an element b ∈ M p ,q ( R d ) and a sequence { a N } in S ( R d ) such that { a N } is uniformly bounded in M p ,q ( R d ) but { a N b } is not a bounded sequence in M p ′ ,q ′ ( R d ). This is the desired contra-diction to (3.18).By [27, Remark 1.3] we know that the sequence { f ,N } ⊆ S ( R d ) isuniformly bounded in M q , ( R d ), and that f ∈ M p , ( R d ) , b f ∈ F M q , ( R d ) ⊆ M ,q ( R d ) ,f ∈ M q , ( R d ) ⊆ M q ( R d ) . In a moment we will prove that if we choose ϕ ∈ C ∞ ( R d ) \ a N and b as a N = W ϕ,f ,N and Op w ( b ) h = f · ( f ∗ h ) , h ∈ C ∞ ( R d ) , (3.22)then k a N k M p ,q ≤ C, b ∈ M p ,q ( R d ) andOp w ( b ) f = X n λ n τ n g , where g = g · ( g ∗ g ) , and λ n ≥ C | n | − d (1 /q +1 /p − /q ′ ) − ε (1 /q +1 /p +1 /q ) , (3.23)for some constant C > N . We note that g ∗ g is supported in a ball with center at the origin and radius 1 / a N b ) is a bounded sequence in M p ′ ,q ′ ( R d ),which implies that Op w ( a N b ) is a uniformly bounded sequence ofcontinuous operators from M q ( R d ) to M q ′ ( R d ). In fact, (3.6) gives2 /q ′ ≤ /q + 1 /q which combined with (3.16) yield 1 /q ′ < /p ′ .Now [21, Theorem 7.1] or [45, Theorem 4.3] together with Proposition1.3 give the assertion. (See also [4, Theorem 1.1].)On the other hand, since f ∈ M q ( R d ) and Op w ( b ) f = f · ( f ∗ f ),we get Op w ( a N b ) f = (Op w ( W ϕ,f ))( f · ( f ∗ f )) , which by [27, Lemma 1.6] givesOp w ( a N b ) f = (2 π ) − d/ ( f · ( f ∗ f ) , f ) ϕ. (3.24) ow (3.22), (3.23) and the fact that ( g , g ) > f · ( f ∗ f ) , f ) ≥ C X | n |≤ N λ n β n ≥ C ′ X | n |≤ N | n | − d (1 /q +1 /q +1 /p − /q ′ ) − ε (1 /q +1 /q +1 /p +1 /q ) , which gives, using (3.19),( f · ( f ∗ f ) , f ) ≥ C ′ X | n |≤ N | n | − d . (3.25)Consequently, since the right-hand side can be made arbitrarily largeby increasing N , we have obtained a contradiction to the uniformlyboundedness of Op w ( a N b ) as a sequence of operators from M q ( R d ) to M q ′ ( R d ). Hence our assumption, contrary to the statement, is wrong,and the result follows.It remains to prove (3.23). From the assumptions we have that ϕ ∈ C ∞ ( R d ) ⊆ M ( R d ) and f ∈ M q , ( R d ). From [45, Theorem 4.1] itfollows that a N = W ϕ,f is uniformly bounded in M ,q ( R d ), and like-wise in M p ,q ( R d ). We have Op w ( b ) = Op ( c ) with c = f ⊗ b f andOp ( c ) is a pseudo-differential operator of Kohn–Nirenberg type, i. e. t = 0 in (1.3). Since f , b f ∈ M p ,q ( R d ), it follows that c ∈ M p ,q ( R d ).By [27, Remark 1.5] we then obtain b ∈ M p ,q ( R d ) = M p ,q ( R d ).In order to prove the last part of (3.23) we note that f ∗ f = X n µ n τ n ( g ∗ g ) , where { µ n } is the discrete convolution between { γ n } and { η n } , i. e. µ n = X k γ n − k η k . By Young’s inequality it follows that ( µ n ) ∈ l r , where 1 /q + 1 /q =1 + 1 /r . Here (3.6) guarantees that r ∈ [1 , ∞ ].From the support properties of g and g ∗ g , it follows that f · ( f ∗ f ) = X n λ n τ n g where λ n = α n µ n . We have to estimate λ n . For any n ∈ Z d , letΩ n = { k ∈ Z d ; | k | ≤ | n | , k = 0 , k = n } . We have µ n = X k γ n − k η k ≥ X k ∈ Ω n | n − k | − ( d + ε ) /q | k | − ( d + ε ) /q ≥ C | n | d (1 − /q − /q ) − ε (1 /q +1 /q ) , or some C >
0. Hence λ n = α n µ n ≥ C | n | − ( d + ε ) /p | n | d (1 − /q − /q ) − ε (1 /q +1 /q ) = C | n | − d (1 /q +1 /p − /q ′ ) − ε (1 /q +1 /p +1 /q ) . This proves (3.23) and the result follows. (cid:3)
Proof of Theorem 3.1.
By Lemma 3.3 we may assume trivial weights.By Lemmas 3.4, 3.6 and 3.7, the inequalities (3.5)–(3.8) hold. ThusLemma 2.8 shows that (3.2) holds true. (cid:3) Some particular cases
We list in a table some special cases of the inclusion results for theWeyl product on unweighted modulation spaces. More precisely, wecompare the inclusion results in Theorem 0.1 and [27, Theorem 0.3 ′ ]when the exponents belong to p j , q j ∈ { , , ∞} . The table illustratesthe generality of Theorem 0.1 as compared to [27, Theorem 0.3 ′ ]. In factthe latter result gives no inclusions in modulation spaces for several ofthe studied cases.On the other hand [27, Theorem 0.3 ′ ] combined with Proposition 1.3(2) gives the inclusions in Theorem 0.1 for these particular cases.A corresponding table can be made for the twisted convolution act-ing on Wiener amalgam spaces W p,q ( ω ) , provided the involved Lebesgueexponents p j , q j are interchanged (cf. Theorem 2.12). In the literatureit is common that the norm in the Wiener amalgam space W p,q ( ω ) is de-fined with reversed order of the Lebesgue exponents, i. e. the norm isdefined by k f k W p,q ( ω ) ≡ (cid:16) Z R d (cid:16) Z R d | V φ f ( x, ξ ) ω ( x, ξ ) | p dξ (cid:17) q/p dx (cid:17) /q , instead of (1.16) (cf. [5–7]).With this definition of the Wiener amalgam spaces, the table for thetwisted convolution is the same as the table below, if the Weyl product is replaced by the twisted convolution ∗ σ , and the modulation spaces M p,q are replaced by W p,q Finally we remark that the relations in the table are valid for weightedspaces provided the involved weights satisfy (0.3). heorem 0.1: [27, Theorem 0.3 ′ ]: No. M p , q M p , q M p ′ , q ′ M p ′ , q ′ M , M , M , — M , M , M , — M , M , ∞ M , ∞ — M , M , M , — M , M , M , — M , M , ∞ M , ∞ — M p,q M ∞ , M p,q M p,q M , M ∞ , M , M , M , M ∞ , ∞ M , ∞ M , ∞ M , M , M , , M , ∞ M , , M , ∞ M , M , M , — M , M , M , — M , M , M , , M , ∞ — M , M ∞ , M , ∞ M , ∞ M , M , ∞ M , ∞ — M , M , M , M , M , M , M , M , M , M , ∞ M , ∞ M , ∞ M , M ∞ , M , M , M , M ∞ , ∞ M , ∞ M , ∞ M , M ∞ , M , ∞ M , ∞ M ∞ , M ∞ , M ∞ , ∞ M ∞ , ∞ eferences [1] J. Bergh and J. L¨ofstr¨om, Interpolation Spaces, An Introduction , Springer-Verlag, Berlin Heidelberg New York, 1976.[2] E. Cordero,
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Department of Mathematics, University of Turin, Via Carlo Al-berto 10, 10123 Torino (TO), Italy.
E-mail address : [email protected] Department of Mathematics, Linnæus University, V¨axj¨o, Sweden
E-mail address : [email protected] Department of Mathematics, Linnæus University, V¨axj¨o, Sweden
E-mail address : [email protected]@lnu.se