Step multipliers, Fourier step multipliers and multiplications on quasi-Banach modulation spaces

We prove the boundedness of a general class of multipliers and Fourier multipliers, in particular of the Hilbert transform, on quasi-Banach modulation spaces. We also deduce boundedness for multiplications and convolutions for elements in such spaces.


Introduction
In the paper we deduce mapping properties of step multipliers and Fourier step multipliers when acting on quasi-Banach modulation spaces. Some parts of our investigations are based on certain continuity properties for multiplications and convolutions for elements in such spaces, deduced in Section 3, and which might be of independent interests.
The Hilbert transform, i. e. multiplication by the signum function on the Fourier transform side, is frequently used in mathematics, science and technology. In physics it can be used to secure causality. For example, in optics, the refractive index of a material is the frequency response of a causal system whose real part gives the phase shift of the penetrating light and the imaginary part gives the attenuation. The relationship between the two are given by the Hilbert transform. Consequently, knowledge of one is sufficient to retrieve the other.
An inconveniently property with the Hilbert transform concerns lack of continuity when acting on commonly used spaces. For example, it is wellknown that the Hilbert transform is continuous on L 2 , but fails to be continuous on L p for p = 2 as well as on S . (See [22] and Section 1 for notations.) A pioneering contribution which drastically improve the situation concerns [23], where K. Okoudjou already in his thesis showed that the Hilbert transform is continuous on the modulation space M p,q when p ∈ (1, ∞) and q ∈ [1, ∞]. The result is surprising because M p,q is rather close to L p when q stays between p and p ′ (see e. g. [8,29]).
(3) Our analysis also include continuity properties for the modulation spaces W p,q (ω) (R d ). In similar ways as in [3], we use Gabor analysis for modulation spaces to show these properties. In [3], the continuity for Fourier step multipliers are obtained by a convenient choice of Gabor atoms in terms of Fourier transforms of second order B-splines. This essentially transfer the critical continuity questions to a finite set of discrete convolution operators acting on ℓ p , with dominating operator being the discrete Hilbert transform. The choice of Gabor atoms then admit precise estimates of the appeared convolution operators.
In our situation the B-splines above are insufficient, because B-splines lack in regularity, and when p approaches 0, unbounded regularity on the Fourier transform of the Gabor atoms are required. In fact, in order to obtain continuity for weighted modulation spaces with general moderate weights in the momentum variables, it is required that the Fourier transform of Gabor atoms obey even stronger regularities of Gevrey types.
In Section 4 we obtain some further extensions and deduce precise estimates of the Fourier multipliers in (0.1), where more restrictive a 0 should belong to ℓ q (bZ) for some q ∈ (0, ∞]. In the end we are able to prove that the Fourier multiplier in (0.1) is continuous from M p,q 1 to M p,q 2 when p ∈ (1, ∞) and q 1 , q 2 ∈ (0, ∞] satisfy More generally, in Section 4 we generalize the continuity properties for the step and Fourier step multiplier results in Section 2 with more general slope step multiplier and Fourier slope step multipliers.
Multiplier functions in Section 2.
Multiplier functions in Section 4.
An important ingredient for the proofs of the latter extension is multiplication and convolution properties for M p,q (ω) and W p,q (ω) spaces, given in Section 3.
Proposition 0.1. Let p j , q j ∈ (0, ∞], j = 0, 1, 2, Then Similar result holds for W p,q spaces. The general multiplication and convolution properties in Section 3 also overlap with results by Bastianoni, Cordero and Nicola in [1], by Bastianoni and Teofanov in [2], and by Guo, Chen, Fan and Zhao in [21]. The multiplication relation in Proposition 0.1 for p j , q j ≥ 1 was obtained already in [8] by Feichtinger. It is also obvious that the convolution relation was well-known since then (though a first formal proof of this relation seems to be given first in [30]). In general, these convolution and multiplication properties follow the rules which goes back to [8] in the Banach space case and to [14] in the quasi-Banach case. See also [11] and [26] for extensions of these relations to more general Banach function spaces and quasi-Banach function spaces, respectively.
In Section 3 we extend the multiplication and convolution results in [1,2,21] to allow more general weights as well as finding multi-linear versions. We stress that the results in Section 3 hold true for general moderate weights, while corresponding results in [21] are formulated only for polynomially moderate weights which also should be split, i. e. of the form ω(x, ξ) = ω 1 (x)ω 2 (ξ). In Section 3 we also carry out questions on uniqueness for extensions of multiplications and convolutions from the Gelfand-Shilov space Σ 1 (R d ), to the involved modulation spaces. Note that Σ 1 (R d ) is dense in S (R d ) and is contained in all modulation spaces with moderate weights (see e. g. [31]). On the other hand, in contrast to [21], we do not deduce any sharpness for our results.
The analysis to show Proposition 0.1 is more complex compared to the restricted case when p j , q j ≥ 1, because of absence of local-convexity of involved spaces when some of the Lebesgue exponents are smaller than one. In fact, the desired estimates when p j , q j ≥ 1 can be achieved by straightforward applications of Hölder's and Young's inequalities. For corresponding estimates in Proposition 0.1 ′ , some additional arguments seems to be needed. In our situation we discretize the situations in similar ways as in [1] by using Gabor analysis for modulation spaces, and then apply some further arguments, valid in non-convex analysis. This approach is slightly different compared to what is used in [21] which follows the discretization technique introduced in [36], and which has some traces of Gabor analysis.
A non-trivial question concerns wether the multiplications and convolutions in Propositions 0.1 and 0.1 ′ are uniquely defined or not. If p j , q j < ∞, j = 1, 2, then the uniqueness is evident because the Schwartz space is dense in M p j ,q j . In the case p 1 , q 1 < ∞ or p 2 , q 2 < ∞, the uniqueness in Proposition 0.1 follows from the first case, duality and embedding properties for quasi-Banach modulation spaces into Banach modulation spaces. The uniqueness in 0.1 ′ then follows from the uniqueness in Proposition 0.1 and the fact that M p,q increases with p and q.
A critical situation appear when p 1 + q 1 = p 2 + q 2 = ∞. Then S is neither dense in M p 1 ,q 1 nor in M p 2 ,q 2 . For the multiplications in Propositions 0.1, the uniqueness can be obtained by suitable approaches based on the so-called narrow convergence, which is a weaker form of convergence compared to norm convergence (see [28,29,31]). However, for the convolution in Propositions 0.1, we are not able to show any uniqueness of these extensions in this critical situation.
The paper is organized as follows. In Section 1 we present well-known properties of Gelfand-Shilov spaces, modulation spaces, multipliers and Fourier multipliers. In Section 2 we deduce continuity properties for step and Fourier step multipliers when acting on (quasi-Banach) modulation spaces. Then we establish convolution and continuity properties for quasi-Banach modulation spaces in Section 3. In Section 4 we show how the multiplication and convolution results in Section 3 can be used to generalize the continuity results in Section 2, to more general slope step multiplier and Fourier slope step multipliers. Finally we present a proof of a multi-linear convolution result in Appendix A.

Acknowledgement
The idea of the paper appeared when I supervised Nils Zandler-Andersson for his bachelor degree (see [37]). In those thesis, Mr. Andersson deduced some extensions of the multiplier results in [3] to certain quasi-Banach modulation spaces (see [37,Theorem 4.16]). I am also grateful to Elena Cordero and Nenad Teofanov for reading the paper and giving valuable comments, leading to improvements of the content.

Preliminaries
In this section we recall some facts on Gelfand-Shilov spaces, modulation spaces, discrete convolutions, step and Fourier step multipliers. After explaining some properties of the Gelfand-Shilov spaces and their distribution spaces, we consider a suitable twisted convolution and recall some facts on weight functions and mixed norm spaces. Thereafter we consider classical modulation spaces, which are more general compared Feichtinger in [8] in the sense of more general weights as well as we permit the Lebesgue exponents to belong to the full interval (0, ∞] instead of [1, ∞]. Here we also recall some facts on Gabor expansions for modulation spaces. Then we collect some facts on discrete convolution estimates on weighted ℓ p spaces with the exponents in the full interval (0, ∞]. We finish the section by giving the definition of step and Fourier step multipliers.

Gelfand-Shilov spaces and their distribution spaces. For any
, of Roumieu and Beurling types respectively, are the inductive and projective limits of S σ s,h (R d ) with respect to h > 0 (see e. g. [15]). It follows that We remark that Σ σ s (R d ) = {0}, if and only if s + σ > 1, and S σ s (R d ) = {0}, if and only if s + σ ≥ 1, and that The Gelfand-Shilov distribution spaces (S σ s ) ′ (R d ) and (Σ σ s ) ′ (R d ), of Roumieu and Beurling types respectively, are the (strong) duals of S σ s (R d ) and Σ σ s (R d ), respectively. It follows that if (S σ s,h ) ′ (R d ) is the L 2 -dual of S σ s,h (R d ) and s + σ ≥ 1 (s + σ > 1), then (S σ s ) ′ (R d ) ((Σ σ s ) ′ (R d )) can be identified with the projective limit (inductive limit) of (S σ s,h ) ′ (R d ) with respect to h > 0. It follows that for such choices of s and σ. (See [24].) We remark that when s 1 < s 2 , σ 1 < σ 2 and s 1 + σ 1 ≥ 1.
For convenience we set S s = S s s and Σ s = Σ s s . The Gelfand-Shilov spaces are invariant under several basic transformations. For example they are invariant under translations, dilations and under (partial) Fourier transformations. In fact, let F be the Fourier transform which takes the form Here · , · denotes the usual scalar product on R d . The map F extends uniquely to homeomorphisms on and to a unitary operator on L 2 (R d ).
There are several characterizations of Gelfand-Shilov spaces and their distribution spaces (cf. [6,7,33] and the references therein). For example, it follows from [6,7] that the following is true. Here g(θ) h(θ), θ ∈ Ω, means that there is a constant c > 0 such that g(θ) ≤ ch(θ) for all θ ∈ Ω. Proposition 1.1. Let f ∈ S ′ (R d ) and s, σ > 0. Then the following conditions are equivalent: ( σ for some r > 0 (for every r > 0); Gelfand-Shilov spaces and their distribution spaces can also be characterized by estimates on their short-time Fourier transforms Let φ ∈ S s (R d ) (φ ∈ Σ s (R d )) be fixed. Then the short-time Fourier transform of f ∈ S ′ s (R d ) (of f ∈ Σ ′ s (R d )) with respect to φ is defined by We observe that [34]). If in addition f ∈ L p (R d ) for some p ∈ [1, ∞], then In the next lemma we present characterizations of Gelfand-Shilov spaces and their distribution spaces in terms of estimates on the short-time Fourier transforms of the involved elements. The proof is omitted, since the first part follows from [20], and the second part from [31,33].
Then the following is true: for some r > 0 (for every r > 0); for every r > 0 (for some r > 0).
We also need the following. Here the first part is a straight-forward consequence of the definitions, and the second part follows from the first part and duality. The details are left for the reader.
Then the following is true: The same holds true with S s or S in place of Σ s at each occurrence.

1.2.
A suitable twisted convolution. Let f be a distribution on R d , φ, φ j , j = 1, 2, 3, be suitable test functions on R d , and let F and G be a pair of suitable distribution/test function on R 2d . Then the twisted convolution F * V G of F and G is defined by The convolution above should be interpreted as when F belongs to a distribution space on R 2d and G belongs to the corresponding test function space. By straight-forward computations it follows that and φ j ∈ Σ s (R d ) and φ ∈ Σ s (R d ) \ 0, j = 1, 2, 3, then it follows by straight-forward applications of Parseval's formula that ( 1.9) and that if then when F ∈ Σ ′ s (R 2d ). We observe that (See e. g. Chapters 11 and 12 in [17]. ) We On the other hand, suppose that (1.13) holds and let f be given by (1.14). Then V φ f = P φ F = F, and the asserted equivalence follows.
We notice that the same holds true with S s or S in place of Σ s at each occurrence.
The set of moderate weights on R d is denoted by P E (R d ), and if s > 0, then P E,s (R d ) is the set of all moderate weights ω 0 on R d such that (1.15) holds for v(y) = e r|y| 1 s for some r > 0. We also let P σ E,s (R 2d ) be the set of all weights ω such that In particular, P E,s (R d ) = P E (R d ) when s ≤ 1 (see [19]). For any weight ω on R 2d and for every p, q ∈ (0, ∞], we set In similar ways, let Ω 1 , Ω 2 be discrete sets and ℓ ′ 0 (Ω 1 × Ω 2 ) consists of all formal (complex-valued) sequences c = {c(j, k)} j∈Ω 1 ,k∈Ω 2 . Then the discrete Lebesgue spaces  [10] for definition of more general modulation spaces.) . Remark 1.6. Modulation spaces possess several convenient properties. In fact, let p, q ∈ (0, ∞], ω ∈ P E (R 2d ) and φ ∈ Σ 1 (R d ) \ 0. Then the following is true (see [8, 10-12, 14, 17] and their analyses for verifications): , and different choices give rise to equivalent quasi-norms; • the spaces M p,q (ω) (R d ) and W p,q (ω) (R d ) are quasi-Banach spaces which increase with p and q, and decrease with ω. If in addition p, q ≥ 1, then they are Banach spaces.
If in addition p, q < ∞, then the duals of M p,q (ω) (R d ) and Gabor expansions for modulation spaces. A fundamental property for modulation spaces is that they can be discretized in convenient ways by Gabor expansions. For fundamental contributions, see e. g. [5, 9, 11-14, 16, 17, 20] and the references therein. Here we present a straight way to obtain such expansions in the case when we may find compactly supported Gabor atoms.
Hence, by periodization it follows from Fourier analysis that where which is the Gabor expansion of f with respect to the Gabor pair (φ, ψ) and lattice Λ, i. e. with respect to the Gabor atom φ and the dual Gabor atom ψ. Here the series converges in (S σ s ) ′ (R d ). By duality and the fact that A combination of these expansions show that where A = (a(j, K)) j,k∈Λ is the Λ × Λ-matrix, given by when j = (j, ι) and k = (k, κ). (1.24) By the Gabor analysis for modulation spaces we get the following. We refer to [9, 11-14, 16, 17, 32] for details.
Then the following is true: .
The same holds true with W p,q (ω) and ℓ p,q * ,(ω) in place of M p,q (ω) respectively ℓ p,q (ω) at each occurrence.
for any choice of σ > 1. In this situation, it is not possible to find compactly supported elements in Gabor pairs which can be used for expanding all elements in . For a general weight ω ∈ P E (R 2d ) which is moderated by the submultiplicative weight v ∈ P E (R 2d ), we may always find a lattice Λ ∈ R d and a Gabor pair (φ, ψ) such that and for some constant C, where the series convergence with respect to the weak * topology in Theorem S] and some further comments in [32]. See also [11][12][13] for more facts.) In such approach we still have that if p, q ∈ (0, ∞], then ) and in addition p, q < ∞, then the series in (1.25) quasi-norm).
Next we discuss extended Hölder and Young relations for multiplications and convolutions on discrete Lebesgue spaces. Here the involved weights should satisfy or and it is convenient to make use of the functional The Hölder and Young conditions on Lebesgue exponent are then (1) if (1.36) holds true, then the map (a 1 , . . . , a N ) → a 1 · · · a N from ℓ 0 (Λ) × · · · × ℓ 0 (Λ) to ℓ 0 (Λ) extends uniquely to a continuous map , and (2) if (1.37) holds true, then the map (a 1 , . . . , a N ) → a 1 * · · · * a N from ℓ 0 (Λ) × · · · × ℓ 0 (Λ) to ℓ 0 (Λ) extends uniquely to a continuous map from The assertion (1) in Proposition 1.10 is the standard Hölder's inequality for discrete Lebesgue spaces. The assertion (2) in that proposition is the usual Young's inequality for Lebesgue spaces on lattices in the case when p 1 , . . . , p N ∈ [1, ∞]. In order to be self-contained we give a proof when p 1 , . . . , p N are allowed to belong to the full interval (0, ∞] in Appendix A.

1.7.
Step and Fourier step multipliers. Let b ∈ R d + be fixed, Λ b be the lattice given by (1.43) Q b be the b-cube, given by and a 0 ∈ ℓ ∞ (Λ b ). Then we let the Fourier step multiplier M F,b,a 0 (with respect to b and a 0 ) be defined by where M b,a 0 is the multiplier Here χ Ω is the characteristic function of Ω.

2.
Step and Fourier step multipliers on modulation spaces In this section we deduce continuity properties for step and Fourier step multipliers on modulation spaces (see Theorems 2.1 and 2.3 below). In contrast to [3], the results presented here permit Lebesgue exponents to be smaller than one We begin with step multipliers when acting on modulation spaces. Here involved Lebesgue exponents should fullfil . Then the following is true: We observe that the conditions on q 2 in Theorem 2.1 implies that q 2 > 1, since otherwise (2.1) should lead to q 1 ≤ min(p, 1), which contradicts the assumptions on q 1 .
We need the following lemma for the proof of Theorem 2.1.
Lemma 2.2. Let p, q ∈ (1, ∞) and θ ∈ (0, 1) be such that and suppose that a = {a(j)} j∈Z d ⊆ C satisfies is uniquely extendable to a continuous mapping from ℓ p (Z d ) to ℓ q (Z d ).
We observe that the conditions in Lemma 2.2 implies that p < q. Lemma 2.2 is a straight-forward consequence of [22,Theorem 4.5.3]. In fact, by that theorem we have for which gives suitable boundedness properties for f → f * h θ 0 . We also have By a straight-forward combination of this estimate with (2.4) we obtain where now * denotes the discrete convolution. The continuity assertions in Lemma 2.2 now follows from and the uniqueness assertions follows from the fact that Proof of Theorem 2.1. By straight-forward computations it follows that if , and similarly with M p,q in place of W p,q at each occurrence. This reduce ourself to the case when b = 1. Let φ, ψ and Λ be the same as in (1.16) where A = (a(j, K)) j,k∈Λ is the matrix with elements By the support properties of φ and ψ we have a(j, k) = 0 when j − k / ∈ Ω 2 , and for j − k ∈ Ω 2 we get where the last three sums are taken over all l ∈ Z d such that l − (j − k) ∈ Ω 3 . We have to estimate when j − k ∈ Ω 2 and l − (j − k) ∈ Ω 3 . By Parseval's formula we get is the sinc function. Since Here h 0 is given by (2.3), and we have used By inserting this into (2.6) we get Hence, If c(j, ι) = |V φ f (j, ι)|, then (2.7) gives (h 0 * c(j + k, · ))(ι), (2.8) and Lemma 2.2 gives By applying the ℓ p (ω 0 ) norm on the last inequality and raise it to the power r = min(1, p), we obtain Here we have used the fact that the number of elements in Ω 2 is equal to 5 d . The asserted continuity in (1) now follows in the case when p < ∞ by combining (2.9) and the facts that .
The uniqueness of the map M b,a 0 on W p,q (ω) (R d ) follows from the fact that finite sequences in (1.22) are dense in W p,q (ω) (R d ) gives. The case when p = ∞ now follows from the case when p = 1 and duality, and (1) follows.
In order to prove (2) we first consider the case when p < ∞. By applying the ℓ p (ω 0 ) norm with respect to the j variable in (2.8), we get . Let p 0 = r −1 q 1 , q 0 = r −1 q 2 and u = r −1 . Then (2.2) holds with p 0 and q 0 in place of p and q, respectively. Hence by applying the ℓ q 0 norm on the last estimates, Lemma 2.2 gives The asserted continuity in (2) now follows in the case when p < ∞ by combining (2.9) and the facts that .
The uniqueness assertions as well as the continuity in the case p = ∞ follow by similar arguments as in the proof of (1). The details are left for the reader.
By the links between M p,q (ω) (R d ) and W p,q (ω) (R d ) via the Fourier transform, explained in Remark 1.6, the following result follows from Theorem 2.1 and Fourier transformation. The details are left for the reader.
Then the following is true:

Multiplications and convolutions of quasi-Banach modulation spaces
In this section we extend the multiplication and convolution properties on modulation spaces in [8,30] to allow the Lebesgue exponents to belong to the full interval (0, ∞] instead of [1, ∞], and to allow general moderate weights. There are several approaches in the case when the involved Lebesgue exponents belong to [1, ∞] (see [4,8,11,21,27,30]). There are also some results when such exponents belong to the full interval (0, ∞] (see [1,2,14,25,26,32]). Here we remark that our results in this section cover several of these earlier results. For example, we observe that Theorem 3.2 below extends [1, Proposition 3.1].
We recall that convolutions and multiplications on Σ 1 (R d ) are commutative and associative. That is, for any N ≥ 1, f 1 , . . . , f N ∈ Σ 1 (R d ) and j, k ∈ {1, . . . , N } one has and similarly for convolutions in place of multiplications at each occurrence. Because of possible lacks of density properties, we do not always reach the uniqueness when extending the convolutions and multiplications from the case when each f j belong to Σ 1 (R d ) to the case when each f j belong to suitable modulation spaces. In some cases we manage the uniqueness by replacing the (quasi-)norm convergence by a weaker convergence, the socalled narrow convergence (see [28,29,31]). In the other situations we define multiplications and convolutions in terms of short-time Fourier transforms, in similar ways as in [30].
Then the multiplication f 1 · · · f N can be expressed by We observe that (3.2) is the same as and that we may extract f 0 = f 1 · · · f N by the formula In the same way, let φ 0 , . . . , φ N ∈ Σ 1 (R d ) be fixed such that and let f 1 , . . . , f N , g ∈ Σ 1 (R d ).
Then the convolution f 1 * · · · * f N can be expressed by for every ϕ ∈ Σ 1 (R d ), where F j and Φ are given by (3.3). We observe that (3.7) is the same as (3.8) and that we may extract f 0 = f 1 * · · · * f N from (3.5).
(1) Let φ 0 , . . . , φ N ∈ Σ 1 (R d ) be fixed and such that (3.1) holds, and suppose that the integrand in (3.2) belongs to L 1 (R (N +1)d ) for every be fixed and such that (3.6) holds, and suppose that the integrand in (3.7) belongs to . Next we discuss convolutions and multiplications for modulation spaces, and start with the following convolution result for modulation spaces. Here the conditions for the involved weight functions are given by x, ξ 1 , . . . , ξ N ∈ R d (3.9) or by For multiplications of elements in modulation spaces we need to swap the conditions for the involved Lebesgue exponents compared to (1.39) and (1.40). That is, these conditions become 1 r j , r j = min(1, q j , r) (3.13) and R N (q 1 , . . . , q N ) = R 1,N (q 1 , . . . , q N ). (3.14) Evidently, R r,N (q 1 , . . . , q N ) = R N (q 1 , . . . , q N ) when r ≥ 1.
For the proofs of Theorems 3.2-3.5 we need the following proposition. Here recall [11,13,17,25,26] and Remark 1.4 for some facts concerning the operators P φ and V * φ .
Proposition 3.6. Let p, q ∈ (0, ∞], ω ∈ P E (R 2d ), φ ∈ Σ 1 (R d ) \ 0 and P φ be the projection in Remark 1.4. and For p, q ≥ 1, i. .e. the case when all spaces are Banach spaces, proofs of Proposition 3.6 can be found in e. g. [17] as well as in abstract forms in [11]. In the general case when p, q > 0, proofs of Proposition 3.6 are essentially given in [14,26]. In order to be self-contained we here present a short proof.
Since V φ φ ∈ Σ 1 (R 2d ), Proposition 1.10 gives Theorems 3.2 and 3.3 are Fourier transformations of Theorems 3.4 and 3.5. Hence it suffices to prove the last two theorems.
By applying the ℓ q 0 quasi-norm and using Proposition 1.10 (1) we now get This is the same as .
By applying the ℓ q 0 /r norm with respect to the λ variable and using Minkowski's and Hölder's inequalities we obtain where c j,ω j (k) = a j,ω j (k, · ) r 1/r ℓ q j /r = a j,ω j (k, · ) ℓ q j . An application of the ℓ p 0 /r quasi-norm on the last inequality and using Proposition 1.10 (2) now gives which is the same as , which in particular shows that f 1 * · · · * f N is well-defined. Since , j = 1, . . . , N, we get (3.20). We need to prove the associativity, symmetry and invariance with respect to φ 0 , . . . , φ N in Definition 3.1. We observe that if r j = max(p j , 1) and s j = q j q , q = min 0≤j≤N (q j ), j = 1, . . . , N, . . , N . By straight-forward computations it follows that if (3.17) or (3.18) hold, then (3.17) respectively (3.18) still hold with r j and s j in place of p j and q j , respectively, j = 1, . . . , N , for some r 0 , s 0 ∈ [1, ∞]. This reduce ourself to the case when p j , q j ∈ [1, ∞] for every j = 0, . . . , N , in which case all modulation spaces are Banach spaces. We observe that Lemmas 5. 2-5.4 and their proofs in [30] still hold true when ω j are allowed to belong to the class P E (R 2d ), provided the involved window functions χ j belong to Σ 1 (R d ), and all distributions are allowed to belong to Σ ′ 1 instead of S ′ . The the associativity, symmetric assertions and invariant properties with respect to the choice of φ 0 , . . . , φ N in Definition 3.1 now follows from these modified Lemmas 5. 2-5.4 in [30] and their proofs. This gives the results.
Remark 3.7. Suppose that p 1 , q j and ω j are the same as in Theorems 3.2-3.5, and that p j + q j = ∞ for at most one j ∈ {1, . . . , N }. Then it follows that extensions of the mappings (f 1 , . . . , f N ) → f 1 · · · f N and (f 1 , . . . , f N ) → In fact, by the proof of Theorem 3.8 below, we may assume that p j , q j ≥ 1 for every j. If p j , q j < ∞ for every j ∈ {1, . . . , N }, then the uniquenesses follow from (3.15), (3.16), (3.19), (3.20) and the fact that For the general situation, the assertion follows from the previous case and duality.
Evidently, Theorems 3.2-3.5 show that multiplications and convolutions on Σ 1 (R d ) can be extended to involve suitable quasi-Banach modulation spaces. Remark 3.7 shows that in most situations, these extensions from products on Σ 1 (R d ) are unique. For the multiplication and convolution mappings in Theorems 3.2 and 3.5 we can say more.
Theorem 3.8. Let ω j ∈ P E (R 2d ) and p j , q j ∈ (0, ∞] and j ∈ {0, . . . , N }. Then the following is true: (1) if (3.9), (3.11) and (3.13) hold, then (3.20) holds; The problems with uniqueness in Theorem 3.8 appear when one or more Lebesgue exponents are equal to infinity, since Σ 1 (R d ) fails to be dense in corresponding modulation spaces. In these situations we shall use narrow convergence, introduced in [28], and is a weaker form of convergence than the norm convergence.
Then f j is said to converge to f narrowly as j → ∞, if the following conditions are fulfilled: The following result is a special case of Theorem 4.17 in [31]. The proof is therefore omitted. Proposition 3.10. Let ω ∈ P E (R 2d ) and p, q ∈ [1, ∞] be such that q < ∞.
(ω) (R d ) with respect to the narrow convergence. We also need the following generalization of Lebesgue's theorem, which follows by a straight-forward application of Fatou's lemma. as j tends to infinity, and that |f j | ≤ g j for every j ∈ N. Then f j → f in in L 1 (dµ) as j tends to infinity.
Remark 3.12. The narrow convergence is especially interesting when p = ∞.
converges to f narrowly as j → ∞, and let H f,ω,∞ be the same as in Definition 3.9. Then we may choose these f j such that (See [31,Theorem 4.17] and its proof.) It is then possible to apply Lemma 3.11 in integral expressions containing V φ f j (x, ξ) and V φ f (x, ξ) and perform suitable limit processes.
Proof of Theorem 3.8. Since (2) is the Fourier transform of (1), it suffices to prove (1). The existence of the extension follows from Theorem 3.2. Since M p,q (ω) (R d ) increases with p and q, we may assume that equality is attained in (3.11) and that p 0 = · · · p N = ∞. By replacing q j with r j = max(1, q j ), it follows from (3.11) that for r 0 = q 0 q ≥ 1 and some r 0 ≥ 1, (ω j ) (R d ) are such that g 1 equals g 2 as elements in M ∞,r j (ω j ) (R d ). Then g 1 is also equal to g 2 as elements in M ∞,q j (ω j ) (R d ). Hence it suffices to prove the uniqueness of the product (ω j ) (R d ), j = 1, . . . , N , when additionally q j ≥ 1, i. e., 1 In particular, all involved modulation spaces are Banach spaces. Let j 0 ∈ {1, . . . , N } be chosen such that q j ≤ q j 0 for every j ∈ {1, . . . , N }. Then j < ∞ when j = j 0 .
The product f 1 · · · f N is uniquely defined and can be obtained through . . such that f j,k converges to f j narrowly as k tends to infinity, and that (3.31) holds with f j and f j,k in place of f and f j , respectively. Then it follows by replacing f j by f j,k when j = j 0 in (3.2) and applying Lemma 3.11 on the integral in (3.2) that ℓ(ϕ) ≡ lim k→∞ (f 1,k · · · f N,k , ϕ) exists and defines an element in f ∈ Σ ′ 1 (R d ). This shows that the only possibility to define f 1 · · · f N in a continuous way is to put f 1 · · · f N = f , and the asserted uniqueness follows.

Extensions and variations
In this section we extend the results on step and Fourier step multipliers to certain so-called curve step and Fourier curve step multipliers. That is a generalized form of of step and Fourier step multipliers, where the constants a 0 (j) in the definition of M b,a 0 and M F,b,a 0 are replaced by certain nonconstant functions or even distributions. In the end we are able to generalize Theorems 2.1 and 2.3 to such multipliers. These achievements are based on Hölder-Young relations for multiplications and convolutions in Section 3. In the case of trivial weights and all modulation spaces are Banach spaces, our results are similar to [3,Theorem 6] and [27,Proposition 4.12].
The multipliers and Fourier multipliers which we consider are given in the following.
Then the multiplier is called slope step multiplier with respect to b and a 0 . The Fourier multiplier is called slope step Fourier multiplier with respect to b and a 0 .
First we perform some studies of where ψ ∈ S (R d ) is suitable. The conditions on the sequence (4.1) that we have in mind are that for fixed ω 0 ∈ P E (R d ) and p ∈ (0, ∞], the functions should belong to L p (R d ) for every α ∈ N d , or that for some or for every h > 0, the function should belong to L p (R d ).
Proposition 4.2. Let b ∈ R d + be fixed, Λ b be given by (1.43), s, h 0 , σ > 0, (4.1) be a sequence of functions on C ∞ (R d ), ψ ∈ S (R d ), and let T ψ a 0 , b a 0 ,α and b a 0 ,h be given by (4.2)-(4.4) when α ∈ N d and h > 0. Then the following is true: . . , 0) ∈ N d , then the series in (4.2) is locally uniformly convergent and defines an element in C(R d ); Proof. We only prove (1) and (4). The other assertions follow by similar arguments and are left for the reader.
which shows that (4.2) is locally uniformly convergent. Since a 0 (j, · ) and ψ( · − j) are continuous functions, it follows that T ψ a 0 in (4.2) is continuous.
Next suppose additionally that ψ ∈ Σ σ s (R d ) and consider f = T ψ a 0 . For every α ∈ N d , ε > 0 and r > 0, we have and the result follows.
Proof. We only prove (2). The other assertions follow by similar arguments and are left for the reader.
and a 1 * · · · * a N ℓ ∞ ≤ a 1 ℓ 1 · · · a j 0 ℓ ∞ · · · a N ℓ 1 ≤ a 1 ℓ p 1 · · · a j 0 ℓ ∞ · · · a N ℓ p N , and the result follows in this case as well. It remains to prove the result when p j < 1 for at least one j ∈ I N and that p j 0 < ∞, giving that ℓ 0 is dense in ℓ p j for every j ∈ I N . Hence the result follows if we prove (1.42) when a j ∈ ℓ 0 .
Next suppose that N ≥ 3, and that the result holds for less numbers of factors in the convolution. Since the convolution is commutative, we may assume that p N < 1 is the smallest number in I N . Then p N ≤ p 0 , and (1.40) is the same as By the induction hypothesis we get a 1 * · · · * a N −1 ℓ p 0 ≤ a 1 ℓ p 1 · · · a N −1 ℓ p N−1 .