Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces

In this paper, we investigate bi-parameter modulation spaces on the product of two Euclidean spaces \begin{document}$ \mathbb{R}^{n} $\end{document} and \begin{document}$ \mathbb{R}^{m} $\end{document} via uniform decompositions of each factor. A molecular decomposition of these bi-parameter spaces are given, which generalizes the related single-parameter result of Kobayashi and Sawano [ 33 ]. Furthermore, we prove the boundedness of a class of Fourier multipliers on bi-parameter modulation spaces, generalizing the results of Benyi et al. [ 2 ] and Feichtinger and Narimani [ 17 ].

1. Introduction and statement of main results. Modulation spaces, introduced by Feichtinger [15] in the early 1980s, were originally used to measure smoothness of functions and to analyze local properties of frequency space. During the last decades, modulation spaces have been studied systematically; see, for instance, [2,16,17,19,20,33,35,49] and the references therein. These spaces have turned out to be useful in the study of pseudo-differential operators, partial differential equations, signal analysis and quantum mechanics. For instance, by using modulation spaces, Sjöstrand [47] and Tachizawa [50] generalized the Calderón-Vaillancourt theorem; see also the work of Gröchenig and Heil [20]. Moreover, Tomita [51] generalized the Hörmander multiplier theorem by using modulation spaces. In recent years, modulation spaces were also applied to study the global well-posedness of solutions to partial differential equations [3,53].
The main purpose of this paper is to investigate bi-parameter modulation spaces, in particular, to derive a molecular decomposition for these spaces and to prove the boundedness of a class of Fourier multipliers on these spaces. Recall that there are two equivalent definitions for the classical (single-parameter) modulation spaces. One definition makes use of the short-time Fourier transform and window functions (see, e.g., [15,19]), while the other one is based on a uniform decomposition of the Euclidean spaces. To recall the latter definition, we introduce some notation. For any ξ ∈ R n , we denote ξ := (1 + |ξ| 2 ) 1/2 = (1 + |ξ 1 | 2 + · · · + |ξ n | 2 ) 1/2 . Let Q 0 = {ξ ∈ R n : −1/2 ≤ ξ i < 1/2 for i = 1, · · · , n} be the unit cube in R n and let Q k := k + Q 0 for k ∈ Z n . Then the Q k 's are pairwise disjoint and we have R n = ∪ k∈Z n Q k , which is called the uniform decomposition of R n . Let {ϕ k } k∈Z n be a sequence of Schwartz functions and assume that it is exclusive (see Section 2 for definition). Define the operators 2 k f := F −1 (ϕ k Ff ), k ∈ Z n . Then, for s ∈ R and 0 < p, q ≤ ∞, the (single-parameter) modulation space M s p,q (R n ) is defined as the collection of all distributions f ∈ S (R n ) for which It is well known that the scale of M s p,q (R n ) is independent of the choice of the exclusive sequence {ϕ k } k∈Z n .
Bi-parameter modulation spaces were first introduced by Kobayashi, Sugimoto and Tomita [34], Indeed, they introduced a more general notion, which is called biparameter α-modulation spaces and denoted by BM s,α p,q (R n × R m ), for 0 ≤ α ≤ 1, s = (s 1 , s 2 ) ∈ R × R and 1 ≤ p, q ≤ ∞. When α = 0, these spaces reduce to the bi-parameter modulation spaces BM s p,q (R n × R m ). Recently, Xu and Huang [54] extend the definition of bi-parameter α-modulation spaces BM s,α p,q (R n × R m ) to the case 0 < p, q ≤ ∞, and proved the boundedness of bi-parameter pseudo-differential operators on BM s,α p,q (R n × R m ). See also [55] for the study of boundedness of biparameter fractional integrals on bi-parameter modulation spaces.
Let us recall the definition of bi-parameter modulation spaces. More details will be given in Section 2. We fix two exclusive sequences {ϕ k } k∈Z n and {ψ k } k ∈Z m with respect to the uniform decompositions of R n and R m , respectively. Define the operators 2 k,k f : where k := (1 + |k 1 | 2 + · · · + |k n | 2 ) 1/2 . In Section 2 we shall show that the scale of BM s p,q (R n × R m ) is independent of the choice of {ϕ k } k∈Z n and {ψ k } k ∈Z m , as long as they are exclusive.
One main objective of the present paper is to derive a molecular decomposition for the bi-parameter modulation spaces BM s p,q (R n × R m ). We point out that various definitions of molecules of (single-parameter) modulation spaces has been introduced in [1,21,33]. Our definition of molecules for bi-parameter modulation spaces is inspired by [33].
for all α ∈ Z n + and β ∈ Z m + such that |α| + |β| ≤ K. For 0 < p, q ≤ ∞ and a sequence {f j } of functions on R n × R m , we define Definition 1.2 (Sequence space m p,q ). Let 0 < p, q ≤ ∞. Given a sequence λ = λ k,l : k = (k, k ) ∈ Z n × Z m , l = (l, l ) ∈ Z n × Z m of complex numbers, we define where R n = ∪ l∈Z n Q l and R m = ∪ l ∈Z m Q l are uniform decompositions of R n and R m , respectively, and (χ Q l ⊗ χ Q l )(ξ, ξ ) := χ Q l (ξ)χ Q l (ξ ).
Our result concerning molecular decomposition for bi-parameter modulation spaces can be stated as follows. Theorem 1.3. Let s = (s 1 , s 2 ) ∈ R × R and 0 < p, q ≤ ∞.
(i) For every f ∈ BM s p,q (R n × R m ), there exist a sequence λ = {λ k,l } k,l∈Z n ×Z m ∈ m p,q and a sequence {Ψ k,l } k,l∈Z n ×Z m , where each Ψ k,l is an (s; k, l)-molecule, such that and where C is a constant independent of f .
(ii) Conversely, if λ = {λ k,l } k,l∈Z n ×Z m ∈ m p,q and {Ψ k,l } k,l∈Z n ×Z m is a sequence such that each Ψ k,l is an (s; k, l)-molecule with K, N, N in Definition 1.1 satisfying K > 4 max(n, m) max(1, 1/p) + 2 max(|s 1 |, |s 2 |) + 2, N > n/ min{p, 1} and N > m/ min{p, 1}, then the sum k,l∈Z n ×Z m λ k,l Ψ k,l converges in S (R n × R m ), and We also consider Fourier multipliers for bi-parameter modulation spaces. Bényi, Grafakos, Gröchenig and Okoudjou [2] studied a class of Fourier multipliers of the form where b > 0 and {c j } j∈Z is a sequence of bounded complex numbers. They proved that if m(ξ) is as above then for 1 < p < ∞ and 1 ≤ q ≤ ∞, one has . Feichtinger and Narimani [17] reproved this result using a different approach. In the present paper, we generalize this result to the bi-parameter setting. For the sake of simplicity we only formulate our result in the case n = m = 1, i.e., for bi-parameter modulation spaces on R × R.
Then, for any sequence {c j } of bounded complex numbers, the function m : R × R → C given by The rest of this paper is organized as follows. In Section 2 we give the definition of bi-parameter modulation spaces and show that they are well-defined. In Section 3 we present some fundamental properties of bi-parameter modulation spaces. Sections 4 and 5 are devoted to the proofs of Theorems 1.1 and 1.2, respectively.
Notation. We use Ff (or f ) and F −1 f (or f ∨ ) to denote the Fourier transform and the inverse Fourier transform of a function or distribution f , respectively. The letter C will denote positive constants, which may vary at every occurrence. If a and b are two quantities (typically nonnegative), we use a b or b a to denote that there exists a positive constant C such that a ≤ Cb. By writing a ∼ b we mean that a b a.

2.
Definition of bi-parameter modulation spaces. In this section, we introduce bi-parameter modulation spaces on R n ×R m for full range of indices, and show that these spaces are well-defined. It should be pointed out that Xu and Huang introduced the notion of bi-parameter of α-modulation spaces for full range of indices in their recent work [54].
First we make some conventions. For r > 0, k ∈ Z n and k ∈ Z m , set Let Q 0 := Q(1/2) and Q 0 := Q (1/2) be unit cubes centered at the origins of R n and R m , respectively. Let Q k := k + Q 0 for k ∈ Z n and Q k := k + Q 0 for k ∈ Z m . Then we have R n = ∪ k∈Z n Q k and R m = ∪ k ∈Z m Q k , which are called the uniform decompositions of R n and R m , respectively. The Fourier and inverse Fourier transform on R n are defined respetively by In order to introduce bi-parameter modulation spaces, we need the notion of exclusive sequence of functions associated to the uniform decomposition.
Now we are ready to introduce bi-parameter modulation spaces.

QING HONG AND GUORONG HU
Next we show that the bi-parameter modulation spaces introduced above are independent of the choice of the sequences {ϕ k } k∈Z n ∈ Υ n and {ψ k } k ∈Z m ∈ Υ m . For our purpose we need some preparations.
If 0 < p < ∞ and Ω is a compact subset of R n × R m , we denote The following result can be found in [52, §1.5.1].

Lemma 2.3.
Let Ω and Γ be compact subsets of R n × R m , 0 < p ≤ ∞ and p 0 = min{1, p}. Then there exists a constant C > 0 such that Let Ω be a compact subset of R n ×R m . If α 1 > σ p and α 2 > σ p , then there exists a constant C > 0 such that Choose a function θ ∈ S(R n × R m ) such that supp θ ⊆ Γ and θ ≡ 1 on Ω. For any ω ∈ H α 2 (R n × R m ) and f ∈ L p Ω (R n × R m ), from the definition of L p Ω (R n × R m ) we see that F −1 (ωFf ) = F −1 (θωFf ). Hence it follows from Lemma 2.3 that Therefore, it suffices to show F −1 (θω) L p 0 (R n ×R m ) ω H α 2 (R n ×R m ) . To see the latter inequality, we consider the following two cases. Case 1. 0 < p ≤ 1. In this case, p 0 = min(p, 1) = p, thus we need to show By Hölder's inequality, we have

BI-PARAMETER MODULATION SPACES 3109
From this and the fact that α 1 > σ p and α 2 > σ p , it follows that Similar properties are also satisfied by the x ∈ F ν . Therefore, as desired. Case 2. p > 1. In this case p 0 = min(p, 1) = 1. The same argument as in Case 1 yields . Thus the proof of the lemma is complete.
With the above preparation, we are ready to show the well-definedness of our bi-parameter modulation spaces, which is stated in the following proposition.
Moreover, the number of the elements in Λ k,k is uniformly bounded.
By the properties of {ϕ k }, { ϕ k }, {ψ k } and { ψ k } and Lemma 2.4, we have Since Combining (7) and (8), we get Hence, using (6) and the fact that the number of the elements in Λ k,k is uniformly bounded, we further deduce that This verifies the direction " " in (5). By symmetry, the converse direction " " also holds. Thus the proof of Proposition 2.1 is complete.
At the end of this section we list some fundamental properties of bi-parameter modulation spaces. Since the proofs of these properties are analogous to those in the single-parameter setting, we shall skip the details. Proposition 2.2. Let s = (s 1 , s 2 ) ∈ R × R and 0 < p, q ≤ ∞.
(i) BM s p,q (R n ×R m ) is a quasi-Banach space, and a Banach space if 1 ≤ p, q ≤ ∞.
, and the inclusion maps are continuous.
(i) If s 1 ≥ r 1 , s 2 ≥ r 2 , p ≤p and q ≤q, then BM s p,q (R n ×R m ) ⊆ BM r p,q (R n ×R m ).
3. Molecular decomposition of bi-parameter modulation spaces. The aim of this section is to prove Theorem 1.3. We follow the idea of Kobayshi and Sawano [33]. For our purpose, we will need the following lemma, which is a bi-parameter analogue of the Theorem in [52, §1.3.1]. Since its proof is parallel to that of Theorem in [52, §1.3.1], we omit the details here.
Let Ω be a compact subset in R n × R m and 0 < r < ∞. Then for all where C is a positive constant depending only on n, m and diam(Ω), and M s is the strong maximal operator defined by (1).
Remark 1. If f ∈ S (R n ×R m ) such that supp Ff ⊂ k +Q(1) × k +Q (1) , and ω, ω are functions exactly the same as in Lemma 3.2, then f has the representation To see this, one only needs to replace ω and ω in the proof of Lemma 3.2 with ω(· − k) and ω(· − k ) respectively, and use the fact that F −1 ω(· − k) = M k F −1 ω.
We will also need the following convolution type lemma, whose proof is standard. Lemma 3.3. Let 0 < p, q ≤ ∞. Let {F k,k } (k,k )∈Z n ×Z m be a sequence of positive measurable functions. Set where δ > 2n max{1, 1/p} and δ > 2m max{1, 1/p}. Then we have With these preparations, we are now in a position to prove Theorem 1.3.
Proof of Theorem 1.3. First we prove (i). Let f ∈ BM s p,q (R n × R m ), {ϕ j } ∈ Υ(n) and {ψ j } ∈ Υ(m). Fix two functions ω, ω which satisfy the hypothesis of Lemma 3.2. Then using that k∈Z n ×Z m ϕ k (ξ)ψ k (ξ ) ≡ 1 and Remark 3.1, we have we have f = k,l∈Z n ×Z m λ k,l Ψ s k,l . Obviously, if C is sufficiently large, then each Ψ s k,l is an (s; k, l)-molecule. Let us show that the coefficients {λ k,l } satisfies (3). Indeed, for fixed k and (x, x ), there is a unique l 0 = (l 0 , l 0 ) ∈ Z n × Z m such that (x, x ) ∈ Q l0 × Q l 0 . This is because {Q l } l∈Z n and {Q l } l ∈Z m are pairwise disjoint sequences of sets. Thus, fixing 0 < r < min(1, p), we have where for the third line we used the fact that (x, x ), (y, y ) ∈ Q l0 × Q l 0 =⇒ (1 + |x − y|) n/r (1 + |x − y |) m/r 1 and for the last line we applied Lemma 3.1. Finally, by the vector-valued inequality for strong maximal function we obtain λ mp,q f BM s p,q (R n ×R m ) .
Next we prove (ii). Let λ = {λ k,l } k,l∈Z n ×Z m ∈ m p,q and let {Ψ k,l } k,l∈Z n ×Z m be a sequence such that each Ψ k,l is a (s; k, l)-molecule with K, N, N in Definition 1.1 satisfying K > 4n max(1, 1/p) + 2|s 1 | + 2, K > 4m max(1, 1/p) + 2|s 2 | + 2, N > n/ min{p, 1} and N > m/ min{p, 1}. Define f = k,l∈Z n ×Z m λ k,l Ψ s k,l . Then we must show that To this end, let k = (k, k ), l = (l, l ), j = (j, j ) ∈ Z n × Z m , and K 0 := [K/4]. Then by elementary Fourier analysis, we have where we used integration by parts and the fact that Note that for any α ∈ Z n + and β ∈ Z m + , where for the last inequality we used the property (iv) in Definition 2.1. Here, L and L are positive integers, which can be chosen arbitrarily large. We now choose L, L such that 2L − N > n and 2L − N > m. Then from the esitmate (14) and the definition of molecules (recall that K 0 = [K/4]), it follows that where for the last estimate we used the elementary inequalities Inserting (15) into (13), and using that we obtain It then follows that To estimate the last sum in (16), we fix a number r such that max{n/N, m/N } < r < min{p, 1}.

QING HONG AND GUORONG HU
This is possible since N > n/ min{p, 1} and N > m/ min{p, 1}. Then we note that Using the facts that and |Q l × Q l | = 1, we infer 1 2 µn 2 νm Inserting this into (17) and using that N r − n > 0 and N r − m > 0 we obtain Meanwhile, by the elementary inequality j s k −s ≤ ( Inserting (19) and (18) into (16), and using Lemma 3.3 (taking into account that 2K 0 −|s 1 | > 2n max(1, 1/p) and 2K 0 −|s 2 | > 2m max(1, 1/p)) and the vector-valued inequality for strong maximal function, we obtain Then there is a constant C such that for all f ∈ L p (R × R), Hence, it suffices to show that there is a constant C such that for all k, k ∈ Z, To this end, we fix arbitrary k, k ∈ Z. By the support conditions of ϕ k and ψ k , we see that there exist finitely many integers j 1 , j 2 · · · j N such that Moreover, the integer N is independent of k and k . Thus, it suffices to show that there is a constant C such that for any j ∈ Z, Fix j ∈ Z arbitrarily, and write R j = [a j , b j ) × [d j , e j ). Then we can write χ Rj as χ Rj (ξ, ξ ) = χ [aj ,bj ) (ξ)χ [dj ,ej ) (ξ ) Obviously, γ 1 (ξ, ξ ), γ 2 (ξ, ξ ), γ 3 (ξ, ξ ), γ 4 (ξ, ξ ) satisfy (20). Hence, applying Lemma 4.1 yields (21). This completes the proof of Theorem 1.4.