The Weyl product on quasi-Banach modulation spaces

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.


Introduction
In this paper we study the Weyl product acting on weighted modulation spaces with Lebesgue parameters in (0, ∞]. We work out conditions on the weights and the Lebesgue parameters that are sufficient for continuity of the Weyl product, and we also prove necessary conditions. The Weyl product or twisted product is the product of symbols in the Weyl calculus of pseudodifferential operators corresponding to operator composition. This means that the Weyl product (a 1 , a 2 ) → a 1 #a 2 of two distributions a 1 , a 2 defined on the phase space T * R d ≃ R 2d is defined by Op w (a 1 #a 2 ) = Op w (a 1 ) • Op w (a 2 ) provided the composition is well defined.
The necessary conditions we deduce are as follows. Suppose (0.1) holds for all a 1 , a 2 ∈ S (R 2d ), for a triple of polynomial type weights ω j , j = 0, 1, 2 interrelated in a certain way, see (3.7). Then which are strictly weaker than the sufficient conditions. Our results for the Weyl product are special cases of results formulated and proved for a family of pseudodifferential calculi parametrized by real matrices A ∈ R d×d . In fact we work with a symbol product indexed by A ∈ R d×d , denoted and defined by where Op A (a) is the A-indexed pseudodifferential operator with symbol a. This family of calculi contains the Weyl quantization as the special case A = 1 2 I. The sufficient conditions and the necessary conditions that we find extend results [7,24] where the same problem was studied for the narrower range of Lebesgue parameters [1, ∞]. In the latter case modulation spaces are Banach spaces, whereas they are merely quasi-Banach spaces if a Lebesgue parameter is smaller than one.
The Weyl product on Banach modulation spaces has been studied in e. g. [7,18,21,24,27,30,31]. In [7] conditions on the Lebesgue parameters were found that are both necessary and sufficient for continuity of the Weyl product, thus characterizing the Weyl product acting on Banach modulation spaces.
One possible reason that we do not obtain characterizations in the full range of Lebesgue parameters (0, ∞] is that new difficulties arise as soon as a Lebesgue parameter is smaller than one. The available techniques are quite different, and many tools that are useful in the Banach space case, e.g. duality and complex interpolation, are not applicable or fraught with subtle difficulties.
Our technique to prove the sufficient conditions consists of a discretization of the Weyl product by means of a Gabor frame. This reduces the continuity of the Weyl product to the continuity of certain infinite-dimensional matrix operators. A similar idea has been developed in [39].
The paper is organized as follows. Section 1 fixes notation and gives the background on Gelfand-Shilov function and distribution spaces, pseudodifferential calculi, modulation spaces, Gabor frames, and symbol product results for Banach modulation spaces.
Section 2 contains the result on sufficient conditions for continuity on quasi-Banach modulation spaces (Theorem 2.1). Section 3 contains the result on necessary conditions for continuity on quasi-Banach modulation spaces (Theorem 3.3). Finally in Appendix we show a Fubini type result for Gelfand-Shilov distributions that is needed in the definition of the short-time Fourier transform of a Gelfand-Shilov distribution.
1. Preliminaries Here f (θ) g(θ) means that f (θ) ≤ cg(θ) holds uniformly for all θ in the intersection of the domains of f and g for some constant c > 0, and we write f ≍ g when f g f . Note that (1.1) implies the estimates If v in (1.1) can be chosen as a polynomial then ω is called polynomially moderate or a weight of polynomial type. We let P(R d ) and P E (R d ) be the sets of all weights of polynomial type and moderate weights on R d , respectively.
If ω ∈ P E (R d ) then there exists r > 0 such that ω is v-moderate for v(x) = e r|x| [19]. Hence by (1.2) for any In the paper v and v j for j ≥ 0 will denote submultiplicative weights if not otherwise stated.
is finite, where the supremum is taken over all α, β ∈ N d and x ∈ R d .
Obviously S s,h is a Banach space which increases with h and s, and it is contained in the Schwartz space S . (Inclusions of function and distribution spaces understand embeddings.) The topological dual (1.4) The topology for S s (R d ) is the strongest topology such that each inclu- The Gelfand-Shilov distribution spaces S ′ s (R d ) and Σ ′ s (R d ) are the projective and inductive limits respectively of The space S ′ s (R d ) is the topological dual of S s (R d ), and if s > 1/2 then Σ ′ s (R d ) is the topological dual of Σ s (R d ) [12]. The action of a distribution f on a test function φ is written f, φ , and the conjugate linear action is written (u, φ) = u, φ , consistent with the L 2 inner product ( · , · ) = ( · , · ) L 2 which is conjugate linear in the second argument.
We use the normalization where · , · denotes the scalar product on R d . The Fourier transform F extends uniquely to homeomorphisms on S ′ (R d ), S ′ s (R d ) and Σ ′ s (R d ), and restricts to homeomorphisms on S (R d ), S s (R d ) and Σ s (R d ), and to a unitary operator on L 2 (R d ).
The symplectic Fourier transform of a ∈ S s (R 2d ) where s ≥ 1/2 is defined by 4 where σ is the symplectic form Since F σ a(x, ξ) = 2 d F a(−2ξ, 2x), the definition of F σ extends in the same way as F .
for fixed x ∈ R d , and therefore its Fourier transform is an element in S ′ s (R d ). The fact that the Fourier transform is actually a smooth function given by the formula (1.5) is proved in Appendix. If , then T extends uniquely to sequentially continuous mappings , and similarly when S s and S ′ s are replaced by Σ s and Σ ′ s , respectively, or by S and S ′ , respectively [6,34].
There are several ways to characterize Gelfand-Shilov function and distribution spaces, for example in terms of expansions with respect to Hermite functions [13,25], or in terms of the Fourier transform and the STFT [5,22,34,38].

1.
3. An extended family of pseudodifferential calculi. We consider a family of pseudodifferential calculi parameterized by the real d × d matrices, denoted M(d, R) [3,37]. Let s ≥ 1/2, let a ∈ S s (R 2d ) and let A ∈ M(d, R) be fixed. The pseudodifferential operator Op A (a) is the linear and continuous operator is defined as the linear and continuous operator from Here F 2 F is the partial Fourier transform of F (x, y) ∈ S ′ s (R 2d ) with respect to the y variable. This definition makes sense since ). An important special case is A = tI, with t ∈ R and I ∈ M(d, R) denoting the identity matrix. In this case we write Op t (a) = Op tI (a). The normal or Kohn-Nirenberg representation a(x, D) corresponds to t = 0, and the Weyl quantization Op w (a) corresponds to t = 1 2 . Thus a(x, D) = Op 0 (a) = Op(a) and Op w (a) = Op 1/2 (a).
The Weyl calculus is connected to the Wigner distribution with the formula . The following restatement of [37, Proposition 1.1] explains the relations between a 1 and a 2 . (1.10) On the even-dimensional phase space R 2d one may define modulation spaces based on the symplectic STFT. Thus if ω ∈ P E (R 4d ), p, q ∈ (0, ∞] and Φ ∈ S 1/2 (R 2d ) \ 0 are fixed, the symplectic modulation (1.10). It holds (cf. [7]) ) so all properties that are valid for M p,q (ω) carry over to M p,q (ω) . In the following propositions we list some properties of modulation spaces and refer to [8-11, 17, 33] for proofs.
. We will rely heavily on Gabor expansions so we need the following concepts. The operators in Definition 1.4 are well defined and continuous by the analysis in [17,.
Let v, φ and Λ be as in Proposition 1.5.
Then the following is true: (1) The operators with unconditional quasi-norm convergence in M p,q (ω) when p, q < ∞, and with convergence in M ∞ (ω) with respect to the weak * topology otherwise. ( . The series (1.12) are called Gabor expansions of f with respect to φ, ψ and Λ. 8 Remark 1.7. There are many ways to achieve dual frames (1.11) satisfying the required properties in Proposition 1.6. In fact, let v, v 0 ∈ P E (R 2d ) be submultiplicative such that ω is v-moderate and This inclusion is satisfied e.g. for v 0 (x) = e ε|x| with ε > 0. Proposition 1.5 guarantees that for some choice of φ, ψ ∈ M 1 We usually assume that Λ = θZ d , with θ > 0 small enough to guarantee the hypotheses in Propositions 1.5 and 1.6 be fulfilled, and that the window function and its dual belong to M r (v) for every r > 0. This is always possible, in view of Remark 1.7.
We need the following version of Proposition 1.5, which is a consequence of [3, Corollary 3.2] and the Fourier invariance of Σ 1 (R 2d ). (1.13) Then there is a lattice Λ 2 ⊆ R 2d such that is a Gabor frame for L 2 (R 2d ) with canonical dual frame The right-hand side of (1.13) is called the cross-Rihaczek distribution of φ 1 and φ 2 [17]. 1.5. Pseudodifferential operators and Gabor analysis [36]. In order to discuss a reformulation of pseudodifferential operators by means of Gabor analysis, we need the following matrix concepts. Definition 1.10. Let p, q ∈ (0, ∞], θ > 0, let J be an index set and let Λ = θZ d be a lattice, and let ω ∈ P E (R 2d ).
(1) U ′ 0 (J) is the set of all matrices A = (a(j, k)) j,k∈J with entries in C; (2) U 0 (J) is the set of all A = (a(j, k)) j,k∈J ∈ U ′ 0 (J) such that a(j, k) = 0 for at most finitely many (j, k) ∈ J × J; (1.15) The set U p,q (ω, Λ) consists of all matrices A = (a(j, k)) j,k∈Λ such that is finite.
U p,q (ω, Λ) is a quasi-Banach space, and if p, q ≥ 1 it is a Banach space.
If J is an index set then A = (a(j, k)) j,k∈J ∈ U ′ 0 (J) is called properly supported if the sets { j ∈ J ; a(j, k 0 ) = 0 } and { k ∈ J ; a(j 0 , k) = 0 } are finite for every j 0 , k 0 ∈ J. The set of properly supported matrices is denoted U p (J), and evidently U 0 (J) ⊆ U p (J). The sets U 0 (J) and U p (J) are rings under matrix multiplication, and U ′ 0 (J) is a U p (J)module with respect to matrix multiplication.
Then it follows from Propositions 1.5 and 1.6 that provided θ is sufficiently small. By identifying matrices with corresponding linear operators, [36, (j, k)) j,k∈Λ 2 , and the matrix C is defined as and (1.23) 1.6. Composition of pseudodifferential operators with symbols in Banach modulation spaces. We recall algebraic results for pseudodifferential operators with symbols in modulation spaces with Lebesgue exponents not smaller than one [7,24,37].

Composition of pseudodifferential operators with symbols in quasi-Banach modulation spaces
In this section we deduce a composition result for pseudodifferential operators with symbols in modulation spaces with Lebesgue parameters in (0, ∞]. If A ∈ M(d, R) then the map The following result is the principal result of this paper. It concerns sufficient conditions for the unique extension of (2.1) to symbols in quasi-Banach modulation spaces.
Since the assertion holds true for A 1,k and A 2,l in place of A 1 and A 2 , it follows from the latter estimate that is uniquely defined and that (2.7) holds for A m ∈ U pm,qm (ω m , Λ), m = 1, 2.
We also need the following result on the composition of the analysis operator and the synthesis operator defined by two Gabor systems.
It remains to prove the claimed uniqueness of the extension. If (2.5) holds then M p j ,q j (ω j ) ⊆ M ∞,1 (ω j ) , j = 1, 2, and M p 0 ,q 0 (ω 0 ) ⊆ M ∞,1 (ω 0 ) . Then the claim follows from the uniqueness of the extension which is proved in [7,Theorem 2.11]. Suppose (2.6) holds. Then the same argument applies if q ≤ 1, and if p ≥ 1 then the claim is a consequence of the uniqueness of the extension which is again proved in [7,Theorem 2.11]. Suppose p < 1 < q. If q 1 , q 2 ≥ 1 then the uniqueness follows again from the uniqueness of (2.17). If q 1 ≥ 1 > q 2 then it follows from the uniqueness of (2.17) with q 2 replaced by 1, and analogously for q 2 ≥ 1 > q 1 . Finally if q 1 , q 2 < 1 then the uniqueness follows from the uniqueness of (2.16).
Let p, q ∈ (0, ∞] and set r = min (1, p, q). A particular case of Theorem 2.1 is the inclusion where the weights ω 0 , ω 2 ∈ P E (R 4d ) satisfy and T A is defined by (2.3).
We also note that M p,q (ω) is an algebra under the product # A provided p, q ∈ (0, ∞], q ≤ min(1, p) and ω ∈ P E (R 4d ) satisfies

Necessary conditions
In this final section we show that some of the sufficient conditions in Theorem 2.1 are necessary. We need the following lemma that concerns Wigner distributions.
where c(κ) ≥ 0 for all κ ∈ Λ, and finally let Proof. By replacing Λ by a sufficiently dense lattice Λ 0 , containing Λ and letting c(κ) = 0 when κ ∈ Λ 0 \ Λ, we reduce ourselves to a situation where the hypothesis in Proposition 1.6 is fulfilled. Hence we may assume that (1.11) are dual frames for L 2 (R d ).
In order to prove (3.2), set a = W f,φ ∈ S ′ (R 2d ). Since M p,q is increasing with respect to p and q, it suffices to intersect in (3.2) over 0 < p ≤ min(1, q). We have , and By straightforward computations we get This gives Hence V Φ a(x, ξ, η, y) If q < ∞ we get, in the third inequality using p ≤ 1, The preceding lemma is needed in the proof of Theorem 3.3 below on necessary conditions for continuity. We aim at conditions on the exponents p j , q j , j = 0, 1, 2, that are necessary for to hold for all a, b ∈ S (R 2d ), for certain weight functions ω j , j = 0, 1, 2. We restrict to weights of polynomial type.