6 1 0 2 n a J 1 Multiplication and Composition in Weighted 3 Modulation Spaces ] A F Maximilian Reich and Winfried Sickel . h t a m Abstract. We studytheexistence of theproduct of two weighted mod- [ ulationspaces.Forthispurposewediscusstwodifferentstrategies.The 1 more simple one allows transparent proofs in various situations. How- v ever, our second method allows a closer look onto associated norm in- 0 8 equalities under restrictions in the Fourier image. This will give us the 2 opportunityto treat the boundednessof composition operators. 0 0 Mathematics SubjectClassification (2010). 46E35, 47B38, 47H30. . 2 Keywords. Weighted modulation spaces, short-time Fourier transform, 0 frequency-uniform decomposition, multiplication of distributions, mul- 6 1 tiplication algebras, composition of functions. : v i X r a 1. Introduction Since modulation spaces have been introduced by Feichtinger [7] they have becomecanonicalforbothtime-frequencyandphase-spaceanalysis.However, in recent time modulation spaces have been found useful also in connection with linear and nonlinear partial differential equations, see, e.g., Wang et all [38, 37, 35, 36], Ruzhansky, Sugimoto and Wang [26] or Bourdaud, Reissig, S. [5]. Investigations of partial differential equations require partly different tools than used in time-frequency and phase-space analysis. In particular, Fourier multipliers, pointwise multiplication and composition of functions needtobestudied.Inourcontributionwewillconcentrateonpointwisemul- tiplication and composition of functions. Already Feichtinger [7] was aware of the importance of pointwise multiplication in modulation spaces. In the 2 Reich and Sickel meanwhile several authors have studied this problem, we refer, e.g., to [6], [13], [29] and [30], [32]. In Section 3 we will give a survey about the known results. Therefore we will discuss two different proof strategies. The more simple one, due to Toft [30, 32] and Sugimoto, Tomita and Wang [29], allows transparentproofs in various situations,in particular one can dealwith those situations where the modulation spaces form algebras with respect to pointwise multiplication. As a consequence, Sugimoto et all [29] are able to deal with composition operators on modulation spaces induced by analytic functions. Our second method, much more complicated, allows a closer look onto associated norm inequalities under restrictions in the Fourier image. This will give us the possibility to discuss the boundedness of composition operators on weighted modulation spaces based on a technique which goes back to Bourdaud [3], see also Bourdaud, Reissig, S. [5] and Reich, Reissig, S. [23]. Our approach will allow to deal with the boundedness of nonlinear operators T : g f g without assuming f to be analytic. However, as f 7→ ◦ the case of Ms shows, our sufficient conditions are not very close to the 2,2 necessary conditions. There is still a certain gap. The paper is organized as follows. In Section 2 we collect what is needed about the weighted modulation spaces we are interested in. The next sec- tionisdevotedtothe study ofpointwise multiplication.Inparticular,we are interested in embeddings of the type Ms1 Ms2 ֒ Ms0 , p,q · p,q → p,q where s ,s ,p and q are given and we are asking for an optimal s . These 1 2 0 results will be applied to problems around the regularity of composition of functions in Section 4. For convenience of the reader we also recall what is known in the more general situation Ms1 Ms2 ֒ Ms0 . p1,q1 · p2,q2 → p,q Special attention will be paid to the algebra property. Here the known sufficient conditions are supplemented by necessaryconditions, see Theorem3.5. Also only partly new is our main result in Section 3 stated in Theorem 3.22. Here we investigate multiplication of distributions (possibly singular) with regular functions (which are not assumed to be C ). Partly we have found ∞ necessary and sufficient conditions also in this more general situation. Fi- nally, Section 4 deals with composition operators. As direct consequences of Multiplication and Composition in Weighted Modulation Spaces 3 the obtained results for pointwise multiplication we can deal with the map- pingsg gℓ,ℓ 2,seeSubsection4.1.InSubsection4.3weshallinvestigate 7→ ≥ g f g,wheref isnotassumedtobeanalytic.Sufficientconditions,either 7→ ◦ in terms of a decay for f or in terms of regularity of f, are given. F Notation We introducesomebasicnotation.As usual,N denotesthenaturalnumbers, N :=N 0 ,ZtheintegersandRtherealnumbers,Creferstothecomplex 0 ∪{ } numbers. For a real number a we put a := max(a,0). For x Rn we + ∈ use x := max x . Many times we shall use the abbreviation j=1,...,n j k k∞ | | ξ :=(1+ ξ 2)21, ξ Rn. h i | | ∈ The symbols c,c ,c , ... ,C,C ,C , ... denote positive constants which are 1 2 1 2 independent of the main parameters involved but whose values may differ from line to line. The notation a.b is equivalent to a Cb with a positive ≤ constant C. Moreover,by writing a b we mean a.b.a. ≍ Let X and Y be two Banach spaces. Then the symbol X ֒ Y indicates → that the embedding is continuous. By (X,Y) we denote the collection of L all linear and continuous operators which map X into Y. By C (Rn) the 0∞ set of compactly supported infinitely differentiable functions f : Rn C → is denoted. Let (Rn) be the Schwartz space of all complex-valued rapidly S decreasinginfinitelydifferentiable functionsonRn.Thetopologicaldual,the class of tempered distributions, is denoted by (Rn) (equipped with the ′ S weak topology). The Fourier transform on (Rn) is given by S ϕ(ξ)=(2π) n/2 eixξϕ(x)dx, ξ Rn. − · F ZRn ∈ The inverse transformation is denoted by 1. We use both notations also − F for the transformations defined on (Rn). ′ S Convention. If not otherwise stated all functions will be considered on the Euclidean n-space Rn. Therefore Rn will be omitted in notation. 2. Basics on Modulation Spaces 2.1. Definitions A general reference for definition and properties of weighted modulation spaces is Gröchenig’s monograph [10, Chapt. 11]. 4 Reich and Sickel Definition 2.1. Let φ be nontrivial. Then the short-time Fourier trans- ∈ S form of a function f with respect to φ is defined as Vφf(x,ξ)=(2π)−n2 f(s)φ(s x)e−is·ξds (x,ξ Rn). ZRn − ∈ The function φ is usually called the window function. For f the ′ ∈ S short-time Fourier transform V f is a continuous function of at most poly- φ nomial growth on R2n, see [10, Thm. 11.2.3]. Definition 2.2. Let 1 p,q . Let φ be a fixed window and assume ≤ ≤ ∞ ∈ S s R.ThentheweightedmodulationspaceMs isthecollectionofallf ∈ p,q ∈S′ such that q 1 f = V f(x,ξ) ξ s pdx pdξ q < Ms φ k k p,q (cid:16)ZRn(cid:16)ZRn| h i | (cid:17) (cid:17) ∞ (with obvious modifications if p= and/or q = ). ∞ ∞ Formally these spaces Ms depend on the window φ. However, for dif- p,q ferent windows φ ,φ the resulting spaces coincide as sets and the norms 1 2 are equivalent, see [10, Prop. 11.3.2]. For that reason we do not indicate the windowinthenotation(we donotdistinguishspaceswhichdiffer onlybyan equivalent norm). Remark 2.3. (i) General references with respect to weighted modulation spaces are Feichtinger [7], Gröchenig [10, Chapt. 11], Gol’dman [9], Guo et all[11],Toft[30],[31],[32],Triebel[34]andWanget.all[38]tomentiononly a few. (ii) There is an important special case. In case of p = q = 2 we obtain Ms = Hs in the sense of equivalent norms, see Feichtinger [7], Gröchenig 2,2 [10, Prop. 11.3.1]. Here Hs is nothing but the standard Sobolev space built on L , at least for s N. In general Hs is the collection of all f such 2 ′ ∈ ∈ S that 1/2 f := (1+ ξ 2)s f(ξ)2dξ < . Hs k k (cid:16)ZRn | | |F | (cid:17) ∞ For us of great use will be another alternative approach to the spaces Ms . This will be more close to the standard techniques used in connection p,q with Besov spaces. We shall use the so-called frequency-uniform decomposition, see , e.g., Wang [37]. Therefore, let ρ : Rn [0,1] be a Schwartz 7→ function which is compactly supported in the cube Q := ξ Rn : 1 ξ 1, i=1,...,n . 0 i { ∈ − ≤ ≤ } Multiplication and Composition in Weighted Modulation Spaces 5 Moreover,we assume 1 ρ(ξ)=1 if ξ , i=1,2,...,n. i | |≤ 2 With ρ (ξ):=ρ(ξ k), ξ Rn, k Zn, it follows k − ∈ ∈ ρ (ξ) 1 for all ξ Rn. k ≥ ∈ k Zn X∈ Finally we define 1 σ (ξ):=ρ (ξ) ρ (ξ) − , ξ Rn, k Zn. k k k ∈ ∈ (cid:16)kX∈Zn (cid:17) The following properties are obvious: 0 σ (ξ) 1 for all ξ Rn; k • ≤ ≤ ∈ suppσ Q := ξ Rn : 1 ξ k 1, i=1,...,n ; k k i i • ⊂ { ∈ − ≤ − ≤ } σ (ξ) 1 for all ξ Rn; k • ≡ ∈ k Zn TX∈here exists a constant C >0 such that σ (ξ) C if max ξ k i=1,...,n i • ≥ | − k 1; i|≤ 2 For all m N there exist positive constants C such that for α m 0 m • ∈ | |≤ sup sup Dασ (ξ) C . k m k Zn ξ Rn | |≤ ∈ ∈ We shall call the mapping (cid:3) f := 1[σ (ξ) f(ξ)](), k Zn, f , k − k ′ F F · ∈ ∈S frequency-uniform decomposition operator. As it is well-knownthere is anequivalentdescriptionofthe modulation spaces by means of the frequency-uniform decomposition operators. Proposition 2.4. Let 1 p,q and assume s R. Then the weighted ≤ ≤ ∞ ∈ modulation space Ms consists of all tempered distributions f such that p,q ∈S′ 1 f = k sq (cid:3) f q q < . k k∗Mps,q h i k k kLp ∞ (cid:16)kX∈Zn (cid:17) Furthermore, the norms f and f are equivalent. k kMps,q k k∗Mps,q We refer to Feichtinger [7] or Wang and Hudzik [37]. In what follows we shall work with both characterizations. In general we shall use the same notation for both norms. Ms k · k p,q 6 Reich and Sickel Lemma 2.5. (i) The modulation space Ms is a Banach space. p,q (ii) Ms is independent of the choice of the window ρ C in the sense of p,q ∈ 0∞ equivalent norms. (iii) Ms is continuously embedded into . p,q S′ (iv) Ms has the Fatou property, i.e., if (f ) Ms is a sequence such p,q m ∞m=1 ⊂ p,q that f ⇀f (weak convergence in ) and m ′ S sup f < , m Ms m N k k p,q ∞ ∈ then f Ms follows and ∈ p,q f sup f < . Ms m Ms k k p,q ≤m N k k p,q ∞ ∈ Proof. For (i), (ii), (iii) we refer to [10]. We comment on a proof of (iv). Therefore we follow [8] and work with the norm . From the assumption we obtain that for all k Zn and k · k∗Mps,q ∈ x Rn, ∈ 1[σ f ](x)=(2π) n/2f (x )(σ ) f(x )(σ )= 1[σ f](x) − k m − m k k − k F F −· → −· F F as m . Fatou’s lemma yields →∞ q 1[σ f](x)pdx p − k |kX|≤N(cid:16)ZRn |F F | (cid:17) q liminf 1[σ f ](x)pdx p . − k m ≤ m→∞ |kX|≤N (cid:16)ZRn |F F | (cid:17) An obvious monotonicity argument completes the proof. (cid:4) 2.2. Embeddings Obviouslythe spacesMs aremonotoneins andq.Butthey arealsomono- p,q tonewithrespecttop.ToshowthiswerecallNikol’skij’sinequality,see,e.g., Nikol’skij [21, 3.4] or Triebel [33, 1.3.2]. Lemma 2.6. Let 1 p q and f be an integrable function with ≤ ≤ ≤ ∞ supp f B(y,r), i.e., the support of the Fourier transform of f is con- F ⊂ tained in a ball with radius r >0 and center in y Rn. Then it holds ∈ kfkLq ≤Crn(p1−q1)kfkLp with a constant C >0 independent of r and y. This implies (cid:3) f c (cid:3) f if p q with c independent of k k k kLq ≤ k k kLp ≤ and f which results in the following corollary (by using the norm ). k · k∗Mps,q Multiplication and Composition in Weighted Modulation Spaces 7 Corollary 2.7. Let s >s, p

n 1 1 • 0 ≤ 1 0− 1 q1 − q0 or p0 p1, s0 =s1 and(cid:16)q0 =q1.(cid:17) • ≤ Remark 2.9. Embeddingsofmodulationspacesaretreatedatvariousplaces, we refer to Feichtinger [7], Wang, Hudzik [37], Cordero,Nicola [6], Iwabuchi [13] and Guo, Fan, Wu and Zhao [11]. The weighted modulation spaces Ms cannot distinguish between p,q boundedness and continuity (as Besov spaces). Let C denote the class of ub all uniformly continuous and bounded functions f : Rn C equipped with → thesupremumnorm.Iff Ms isaregulardistributionitisdetermined(as ∈ p,q a function) almost everywhere. We shall say that f is a continuous function if there is one continuous function g which equals f almost everywhere. Corollary 2.10. Let s R and 1 p,q . Then the following assertions ∈ ≤ ≤ ∞ are equivalent: Ms ֒ L ; • p,q → ∞ Ms ֒ C ; • p,q → ub Ms ֒ M0 ; • p,q → ∞,1 either s 0 and q =1 or s>n/q . ′ • ≥ 8 Reich and Sickel Proof. We shall work with . k · k∗Mps,q Step1.Sufficiency.ByProposition2.8itwillbeenoughtoshowM0 ֒ C . ∞,1 → ub From the definition of M0 it follows that ,1 ∞ (cid:3) f(x) k k Zn X∈ is pointwise convergent (for all x Rn). Furthermore, since (cid:3) f C , k ∞ ∈ ∈ there is a continuous representative in the equivalence class f, given by (cid:3) f(x). In what follows we shall work with this representative. k Zn k ∈ Boundedness of f M0 is obvious, we have P ∈ ∞,1 f(x) = (cid:3) f(x) f . | | | k |≤k kM∞0,1 k Zn X∈ It remains to prove uniform continuity. For fixed ε > 0 we choose N such that (cid:3) f <ε/2. k k kL∞ |kX|>N In case k N we observe that | |≤ (cid:3) f(x) (cid:3) f(y) ((cid:3) f) x y . | k − k |≤k∇ k kL∞| − | It follows from [33, Thm. 1.3.1] that ((cid:3) f) c (M(cid:3) f) k∇ k kL∞ ≤ 1k k kL∞ with a constant c independent of f and k. Here M denotes the Hardy- 1 Littlewoodmaximalfunction. Inthe quotedreferencethe assumption(cid:3) f k ∈ is used. A closer look at the proof shows that (cid:3) f Lℓoc satisfying S k ∈ 1 (cid:3) f(x) dx c (1+ k )N, k Zn, k 2 | | ≤ | | ∈ ZQk for some N N is sufficient. Since (cid:3) f L this is obvious. Consequently k ∈ ∈ ∞ we obtain (cid:3) f(x) (cid:3) f(y) c (M(cid:3) f) x y c (cid:3) f x y | k − k | ≤ 1k k kL∞| − |≤ 1k k kL∞| − | c f x y , ≤ 2k kL∞| − | where in the last step we used the standard convolution inequality g h g f . This implies uniform continuity of (cid:3) f and k ∗ kL∞ ≤ k kL1k kL∞ k Multiplication and Composition in Weighted Modulation Spaces 9 therefore of (cid:3) f. In particular, we find k N k | |≤ P f(x) f(y) = ((cid:3) f(x) (cid:3) f(y)) k k | − | − (cid:12)(cid:12)kX∈Zn (cid:12)(cid:12) (cid:12) ((cid:3) f(x) + (cid:3) f(y)(cid:12))+c f x y 1 ≤ | k | | k | 2k kL∞| − | |kX|>N |kX|≤N ε+c f x y (2N +1)n. ≤ 2k kL∞| − | Choosing δ =(c f (2N +1)n) 1ε we arrive at 2k kL∞ − f(x) f(y) <2ε if x y <δ. | − | | − | Step 2. Necessity. Let ψ be a real-valued function such that ψ(0) = 1 ∈ S and supp ψ ξ : max ξ <ε with ε<1/2. j F ⊂{ j=1,...,n | | } We define f by f(ξ):= a ψ(ξ k). k F F − k Zn X∈ Clearly, (cid:3) f(x)=a eikxψ(x), k Zn. k k ∈ Substep 2.1. Lets=0and1 p .Theaboveargumentsimplyf M0 ≤ ≤∞ ∈ p,q if and only if (a ) ℓ . On the other hand, k k q ∈ f(x)=ψ(x) a eikx (2.1) k k Zn X∈ which implies that f is unbounded in 0 if a = . Choosing k Zn k ∞ ∈ P (k log(2+k )) 1 if k N, k=(k ,0,... 0); 1 1 − 1 1 a := ∈ k ( 0 otherwise; then f M0 L , q >1, follows. ∈ p,q\ ∞ Substep 2.2. Let 1 p and q = . Then we choose a := k n. It k − ≤ ≤ ∞ ∞ h i follows f Mn but f(0)=+ . ∈ p,∞ ∞ Substep 2.3. Let 1 p , 10, we ′ ≤ ≤∞ ∞ choose k n log k (1+δ)/q if k >0; − − a := h i h i | | k ( 0 otherwise. 10 Reich and Sickel It follows f = ψ k nq+nq/q′(log k ) (1+δ) k kMpn,/qq′ k kLp h i− h i − k>0 |X| = ψ k n(log k ) (1+δ) < . k kLp h i− h i − ∞ |kX|>0 On the other hand we have f(0)= k n log k (1+δ)/q = − − h i h i ∞ k>0 |X| if (1+δ)/q 1. Hence, for choosing δ =q 1 the claim follows. (cid:4) ≤ − Remark 2.11. Sufficientconditionsforembeddingsofmodulationspacesinto spacesofcontinuousfunctionscanbefoundatseveralplaces,inparticularin Feichtinger’s original paper [7]. We did not find references for the necessity. 3. Pointwise Multiplication in Modulation Spaces We are interested in embeddings of the type Ms1 Ms2 ֒ Ms0 , p,q · p,q → p,q where s ,s ,p and q are given and we are asking for an optimal s . These 1 2 0 resultswillbeappliedinconnectionwithourinvestigationsonthe regularity of compositions of functions in Section 4. However, several times we shall deal with the slightly more general problem 1 1 1 Ms1 Ms2 ֒ Ms0 , = + . p1,q · p2,q → p,q p p p 1 2 In view of Corollary 2.7 this always yields 1 1 1 Ms1 Ms2 ֒ Ms0 , + . p1,q · p2,q → p,q p ≤ p p 1 2 Forconvenienceofthereaderwealsorecallwhatisknowninthemoregeneral situation Ms1 Ms2 ֒ Ms0 . p1,q1 · p2,q2 → p,q At first we shall deal with the algebra property. Afterwards we turn to the existence of the product in more general situations.

Multiplication and Composition in Weighted Modulation Spaces

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