Modulation Spaces on Locally Compact Abelian Groups Hans G. Feichtinger TECHNICAL REPORT, University Vienna, January 1983 AMS-Classifications: 46E25, 43A15, 26B35, 42B25, 42B10, 43A25, 46E15, 46J15, 46M15 This is a literal reproduction of the 1983 report [55] by Hans G. Fe- ichtinger, with only the obvious typos being corrected. Furthermore the symbol for the translation operator (which was L , following Hans Reiter) y has been changed to T , and instead of K(G) we write C (G). We will add y c some information about later results and updates on the bibliography in an last chapter of this paper. Of course the page numbers differ slightly from those in the original report (it was 52 pages long), but the numbering system of theorems and remarks has been preserved in the present version. Modulation Spaces on Locally Compact Abelian Groups Hans G. Feichtinger 1 Introduction The modulation spaces Ms (Rm),s ∈ R,1 ≤ p,q ≤ ∞ to be discussed p,q in this paper are Banach spaces of tempered distributions σ on Rm, which are characterized by the behaviour of the convolution product M g ∗ σ in t Lp(Rm), for t → ∞ (g ∈ S(Rm)). As will be shown this family behaves (with respect to various properties) very much like the well known family of Besov spaces Bs (Rm) (cf. [32], [34], [41], [44], [47]) concerning duality, p,q interpolation, embedding and trace theorems, or the Fourier transformation. Furthermore, the classical potential spaces Ls(Rm) = Hs(Rm) as well as the 2 remarkable Segal algebra S (Rm) (see [18]) may be considered as particular 0 modulation spaces. InordertogiveamoreprecisedefinitionofthemodulationspacesMs (Rm) p,q fix any test function g ∈ S(Rm), g (cid:54)= 0, and write M g for the (oscillating) t function x (cid:55)→ exp(2πi(cid:104)x,t(cid:105)) g(x), x,t ∈ Rm. The convolution product M g ∗σ is then well defined for any tempered distribution σ ∈ S(cid:48)(Rm) and t we may set for 1 ≤ q < ∞: Ms (Rm) := {σ | σ ∈ S(cid:48)(Rm), M g ∗σ ∈ Lp(Rm) for each t ∈ Rm, p,q t (cid:183)(cid:90) (cid:184) 1/q and (cid:107)σ| (cid:107) := (cid:107)M g ∗σ(cid:107)q (1+|t|)sqdt < ∞ }. Mps,q Rm t p 1 2 Hans G. Feichtinger The necessary modification for q = ∞ is obvious. It is then possible to show that different test functions define the same spaces and equivalent norms, and that one obtains a family of Banach spaces which is essentially closed with respect to duality and complex interpolation. There is not only a formal similarity in the results concerning modulation spaces and Besov spaces. In fact, one can say that an element σ of a Besov space Bs (Rm) is characterized by the behaviour of M g ∗σ in Lp(Rm), for p,q ρ ρ → ∞, where now the ”deformation” of the test function consists in a suit- able dilation (M g(x) := ρmg(ρx), ρ > 0).1 Such characterizations can be ρ found in the work of Calderon, Torchinsky and others (cf. [9], [43], [39], §8 and elsewhere). It is also possible to describe Besov spaces by dyadic decompositions of the Fourier transforms of their elements (the dyadic structure has to do very much with dilations). Such characterizations, going essentially back to Hörmander, have been used successfully by Peetre, Triebel and many others (cf. [7], [34], [41], [43] for the basic results, and [32], [41] for the ”classical” characterizations). Our approach to modulation spaces will be through ”uniform” decompo- sitionsoftheFouriertransformsoftheirelements. Sincesuchdecompositions correspond to ”uniform” coverings, obtained by translation (and translation of gˆ corresponds to multiplication of g with a character), these are in our situation the natural analogues to the dyadic decompositions mentioned above. The fact that Banach spaces of distributions characterized by uniform de- compositionshavebeentreatedindetailinearlierpapersbytheauthorunder the name of Wiener type spaces (cf. [20], [21], [23]), will be of great use here. Moreover, since Wiener type spaces are well defined for a quite compre- hensiveclassofBanachspacesofdistributionsonlocallycompactgroups, itis possible to define modulation spaces for a class of solid BF-spaces (B,(cid:107)·(cid:107) ) B (including Lp, 1 ≤ p ≤ ∞) on locally compact abelian groups (among them Rm). In fact, the general modulation spaces M(B,Lp)(G) consist of those v ˆ (ultra) distributions σ on G, for which t → (cid:107)M g ∗ σ(cid:107) , t ∈ G, satisfies a t B weightedq-integrabilitycondition. UsingasuitablegeneralFouriertransform the relevant facts concerning modulation spaces (in the generality just described)canbedrawnfromcorrespondingpropertiesofWiener-typespacesof the form W(F B,Lq) (Gˆ). The results concerning the spaces Ms (Rm) can G v p,q 1The notation is due to H. Reiter [38]. Modulation Spaces on Locally Compact Abelian Groups 3 then be obtained as special cases of general results. The approach choosen is not only justified by the degree of generality obtained, but also by the fact that direct proofs for the spaces Ms (Rm) would not have been much p,q shorter, but probably less transparent. The paper is organized as follows. §2 contains the basic notations and facts, from harmonic analysis and concerning Banach spaces of distributions on locally compact abelian groups. In section 3 various information concerning Wiener-type spaces are collected, mainly for later use in the treatment of modulation spaces. In particular, weighted versions of the Hausdorff-Young theorem for Wiener-type spaces are derived, and some information concerning maximal functions are proved. The results of this section allow us to introduce modulation spaces in §4 in full generality. In this part the independence of modulation spaces from irrelevant parameters (or auxiliary expressions, such as the test function involved) is shown, and various equivalent characterizations of these spaces (discrete and continuous versions of the norm, atomic representations, norms involving maximal functions) are given. Furthermore, several basic properties of modulation spaces, e.g. concerning the density of test function, duality, interpolation, convolution, are derived. In section 5 a general trace theorem is established. The last section gives information concerning the modulation spaces Ms (Rm) as described p,q at the beginning. The facts concerning these spaces are obtained by special- ization from the general principles to be found in sections 4 and 5, thus also illustrating the abstract results given in the earlier parts of this paper. The paper concludes with an outlook on further generalizations, related subjects and further possible applications. 2 Notations, Generalities. In the sequel G denotes a lca (locally compact abelian) group, with the Haar measure dx. We shall be mainly interested in non-compact and non-discrete groups such as Rm, m ≥ 1. The Lebesgue spaces with respect to dx are denotedby(Lp,(cid:107)·(cid:107) ) for 1 ≤ p ≤ ∞, as usual. The translation operators T : p y T f(x) := f(x−y) act isometrically on (Lp,(cid:107)·(cid:107) ). For 1 ≤ p ≤ ∞ the space y p C (G) (of continuous, compactly supported complex-valued functions on G) c is a dense subspace of Lp(G), and the Banach dual of Lp(G) can be identified 4 Hans G. Feichtinger 1 1 with Lp(cid:48)(G), where + = 1. (L1(G),(cid:107) · (cid:107) ) is considered as a Banach p p(cid:48) 1 (cid:90) algebra with respect to convolution, given by f ∗g(x) = f(x−y)g(y)dy G for f,g ∈ C (G). c A strictly positive, locally bounded and measurable function w on G will be called a weight function if w(x) ≥ 1 and w(x+y) ≤ w(x)w(y) for all x,y ∈ G. Then L1(G) := {f | fw ∈ L1(G)} is a Banach algebra under w convolution, called Beurling algebra (cf. [38] III, §7 and VI, §3), with the norm (cid:107)f(cid:107) := (cid:107)fw(cid:107) . We restrict our attention to symmetric weights: 1,w 1 w(−x) = w(x). ˆ The dual group G of a lca group consists of all continuous characters t : G → U (homomorphisms into the unit circle). We write (cid:104)t,x(cid:105) or (cid:104)x,t(cid:105) for ˆ t(x). Recall that G is itself a lca group, and that G may always be identified with (Gˆ)∧ in a natural way (due to the Pontrjagin duality theorem). Recall that (Rm)∧ ∼= Rm as a topological group, since the continuous characters on (cid:34) (cid:35) (cid:88)m Rm are exactly of the form x (cid:55)→ exp 2πi( x t ) , x,t ∈ Rm. i i i=1 The Fourier transform fôf f ∈ L1(G) is given by (cid:90) ˆ f(t) := f(x)(cid:104)x,t(cid:105)dx G ˆ The Fourier transformation F : f (cid:55)→ f defines an injective, involutive Ba- G nach algebra homomorphism from L1(G) into C0(Gˆ) (considered as an involutive, pointwise algebra, with complex conjugation, i.e. (fˆ)− = (f∗)∧, for f∗(x) = f(−x)). Consequently, given any symmetric weight function w on G, the space A (Gˆ) := {fˆ| f ∈ L1(G)} is a self-adjoint Banach algebra of w w ˆ continuous functions on G under pointwise multiplication, if it is endowed ˆ with the norm (cid:107)f(cid:107) := (cid:107)f(cid:107) . Aw 1,w We shall be exclusively interested in weights satisfying the so-called BD-condition (Beurling-Domar’s non-quasianalyticity condition, cf. [38], VI, §3), i.e. only weights satisfying (cid:88)∞ (BD) n−2logw(nx) < ∞ for all x ∈ G n=1 Modulation Spaces on Locally Compact Abelian Groups 5 will be of interest for us. Typical examples of such weights on Rm are those of the form w : x (cid:55)→ (1+|x|)s, s ≥ 0 or w¯ : x (cid:55)→ exp(a|x|d), for a ≥ 0, d ∈ s a,d (0,1) (cf. [11], [2], [28], [45], [14] for explanations concerning such weights). ˆ According to the fundamental work of Domar, the algebra A (G) is a regular w algebra of continuous functions under this condition. It is a even a Wiener algebra in the sense of Reiter (cf. [38], Chap. II). Since (M f)∧ = T fˆ, and because the mapping t (cid:55)→ M f is continuous t t t from Gˆ into L1(G) (as a consequence of the density of C (G) in L1(G)) it w c w turns out that (A ,(cid:107)· (cid:107) ) is a ’nice’ Banach algebra in the sense used w Aw in [6], [22]. Among others it is then possible to define Wiener type spaces ˆ W(B,C) on G (cf. §3) for any Banach space (B,(cid:107)·(cid:107) ) which is in standard B ˆ situation with respect to A (G), i.e. for spaces satisfying the following w three conditions: i) (A ) (cid:44)→ B (cid:44)→ (A )(cid:48) w 0 w 0 ˆ (here (A ) := A ∩C (G) is considered as a topological vector space w 0 w c with respect to its natural inductive limit topology, and (A )(cid:48) denotes w 0 the topological dual; (cid:44)→ indicates continuous embeddings). ii) (B,(cid:107)·(cid:107) )isaBanachmodule(withrespecttopointwisemultiplication) B ˆ ˆ over A (G), i.e. (cid:107)hf(cid:107) < (cid:107)h(cid:107) (cid:107)f(cid:107) for h ∈ A (G), f ∈ B. w B Aw B w ˆ iii) (B,(cid:107)·(cid:107) ) is a Banach module (with respect to convolution on G) over B a Beurling algebra L1(Gˆ) (again we restrict our attention to weights wˆ wˆ satisfying (BD)). Assumption i) as well as many typical examples on Gˆ = Rm = G will ˆ justify our speaking of Banach spaces of distributions (on G). We write B1 (cid:44)→ B2 for continuous embeddings of topological vector spaces. For Ba- nach spaces in standard situation inclusions are automatically continuous by the closed graph theorem. Hence their (complete) norm is uniquely de- termined up to equivalence. If such a space is translation or character ˆ invariant (i.e. T B ⊆ B for y ∈ G, or M B ⊆ B for all t ∈ G, respectively), y t the operators T and M act boundedly on B, and we write |(cid:107)T |(cid:107) and y t y B |(cid:107)M |(cid:107) for the corresponding operators norms. t B The most interesting class of such spaces for us will be the Fourier transforms of weighted, solid (e.g. rearrangement invariant) BF-spaces on G. 6 Hans G. Feichtinger Recall that a Banach space (B,(cid:107) (cid:107) ) is called a BF-space on G if it is B continuously embedded into the space L1 (G) of locally integrable functions loc (cid:90) on G (endowed with the family of seminorms f (cid:55)→ |f(x)|dx, K ⊆ G com- K pact). B is solid if g ∈ L1 (G), f ∈ B and |g(x)| ≤ |f(x)| l.a.e. (locally loc almost everywhere) implies g ∈ B and (cid:107)g(cid:107) ≤ (cid:107)f(cid:107) (equivalently: if B is B B a (pointwise) module over L∞(G)). B is called rearrangement invariant if |{x | |g(x)| ≥ α}| = |{x | |f(x)| ≥ α}| for all α > 0 (here the outer | · | indicates: Haar measure of the corresponding set) implies (cid:107)g(cid:107) = (cid:107)f(cid:107) . It is B B clear that such spaces are isometrically translation invariant, i.e. satisfy (cid:107)T f(cid:107) = (cid:107)f(cid:107) for all f ∈ B, y ∈ G. Moreover, they have continuous y B B translation (i.e. lim(cid:107)T f − f(cid:107) = 0 for all f ∈ B) if C (G) is a dense y B c y→0 subspace of (B,(cid:107)·(cid:107) ) (cf. [31], [15]). B In the sequel we shall be mainly interested in weighted Lp-spaces, or more generallyinsolidBF-spacesonGwhichareoftheformB := {f | fm ∈ B}, m withnorm(cid:107)f(cid:107) := (cid:107)fm(cid:107) . HereweassumethatB isasolid, isometrically B,m B translation invariant BF-space on G, containing C (G) as a subspace, and c that m is a moderate, strictly positive and continuous function on G, i.e. which satisfies m(x+y) ≤ w(y)m(x) for x,y ∈ G and some weight function w. Weshallcallmw-moderateinthiscase, andconsideragainonlyweights satisfying(BD). ForthenormofT onB wethenhave|(cid:107)T |(cid:107) ≤ w(y)for y m y Bm y ∈ G. If, furthermore, (B,(cid:107)·(cid:107) ) contains C (G) as a dense subspace, C (G) B c c is also dense in (B ,(cid:107)·(cid:107) ), and therefore B has continuous translation m B,m m in this case. Applying vector-valued integration one derives therefrom that B is a Banach convolution module over the Beurling algebra L1(G). In this m w case, (B ,(cid:107)·(cid:107) ) is an admissible BF-space in the sense of §4 below. m B,m For further generalities concerning harmonic analysis see [29], [40], [38]. For basic results on Euclidean Fourier analysis cf. [28], [41], [44], [2] et al.. For results on homogeneous Banach spaces, quasimeasures, multipliers, and the relevant (elementary) theory of Banach modules see [28], [30], [12], [22], [17], [6]. For generalities on interpolation theory see [1], [9], [44], and [34]. Occasionallyitwillbeconvenienttowrite(cid:107)f|B(cid:107)insteadof(cid:107)f(cid:107) . Positive B constants are denoted by C,C ,C(cid:48),... The same symbol may denote different 1 constants at different places. Modulation Spaces on Locally Compact Abelian Groups 7 3 Some results on Wiener-type spaces (Equivalentcharacterizations, dependenceontestfunctions, aHaus- dorff-Young theorem, a maximal function theorem) Since many of the basic properties of modulation spaces to be discussed belowareimmediateconsequencesofthecorrespondingpropertiesofWiener- type spaces (as introduced in [20]), we shall recall shortly some facts about this family spaces. We shall also prove several new results on Wiener-type spaces in this section, which are of interest for themselves, but which will serve as auxiliary assertions for the main results of this paper. Given a Banach space (B,(cid:107) · (cid:107) ) of distributions on a locally compact B group Gˆ which is in standard situation w.r.t. A (Gˆ) = F[L1(G)], and w w ˆ a continuous, moderate function v on G we can describe the Wiener-type space W(B,Lq)(Gˆ) as follows: v ˆ Fixing any ’test function’ g ∈ A (G)∩C (G), g (cid:54)= 0, we have: w c W(B,Lq) := {f | f ∈ B , F(g) : t (cid:55)→ (cid:107)(T g)f(cid:107) ∈ Lq(Gˆ)}, v loc t B v (cid:181)(cid:90) (cid:182) 1/q and (cid:107)f | W(B,Lq)(cid:107) := |F(g)(t)|qvq(t) , for 1 ≤ q < ∞ v Gˆ (or sup |F(g)(t)|v(t) for q = ∞). t∈Gˆ One shows that different test functions g define the same space (for B,q, and v fixed) and equivalent norms. ([20], Remark 2). Moreover, there is an equivalent ’discrete’ characterization. Let us call a family (ψ ) a bounded, uniform partition of unity in i i∈I ˆ ˆ ˆ A (G) if there exists some relatively compact set Q ⊆ G such that w i) sup(cid:107)ψ (cid:107) < ∞; i Aw(Gˆ) i∈I ˆ ii) supψ ⊆ t +Q for i ∈ I; i i ˆ ˆ iii) sup|{j | (t +Q)∩(t +Q) (cid:54)= ∅}| < ∞. i j i∈I Then one has f ∈ B belongs to W(B,Lq) if and only if loc v 8 Hans G. Feichtinger (cid:195) (cid:33) 1/q (cid:88) (cid:107)f|D(Qˆ,B,lq)(cid:107) := (cid:107)fψ (cid:107)q v(t )q < ∞ v i B i i∈I (for q = ∞ one has to take sup(cid:107)fψ (cid:107) v(t )). i B i i∈I Suchdiscretenormsarealreadyimplicitlyin[20](cf.Theorem2, andRemark 4) and they are treated in detail in [23]. There one can also find a (general) result concerning the equivalence of discrete and continuous norms, including the above assertion ([23], Theorem 4.3). Concerning Wiener-type spaces one has as special cases of theorems proved in [23] (e.g. Theorem 2.8) also the following assertion on duality: Ifq < ∞andiftranslationiscontinuousinB foranyf ∈ B withcompact ˆ ˆ support, then A (G)∩C (G) is dense in B := {f | f = lim f in B, suppf w c A α α α compact in Gˆ}, and W(B,Lq) = W(B ,Lq). Therefore A (Gˆ) ∩ C (Gˆ) is v A v w c dense in W(B,Lq) (cf. [20], Theorem 1) and v W(B,Lq)(cid:48) = W(B(cid:48),Lq(cid:48) ). v 1/v (cid:88) Using either the discrete representation f = fψ , or a continuous variant i i∈I of it (described in [21]), one shows that these spaces behave nicely with respect to complex interpolation (c.f. [21], [23]). Concerning the independence of the norm from the particular choice of g, g (cid:54)= 0, we give a slight extension of known results, which will be very useful for us later (cf. [16] for hints in this direction). ˆ Proposition 3.1 Given any wˆ-moderate, continuous function v on G, any non-zerofunction g ∈ W(A ,L1)isanadmissible testfunctionfor W(B,Lq) w wˆ v as above (i.e. the norm obtained by using g is still equivalent to the norms considered above). Proof. For g,h ∈ W(A ,L1) ⊆ A one has of course w wˆ w F(hg)(t) ≤ (cid:107)h(cid:107) F(g)(t) Aw ˆ for t ∈ G, by the A -module structure of B. Given g it will be sufficient w to choose h ∈ A ∩ C (Gˆ) in order to have (cid:107)F(h1)(cid:107) ≤ (cid:107)h(cid:107) (cid:107)F(g)(cid:107) for w c q,v Aw q,v ˆ h := hg ∈ A ∩C (G). 1 w c Modulation Spaces on Locally Compact Abelian Groups 9 In order to obtain the converse estimate we may use the representation (cid:88) g = gψ , for which we have i i∈I (cid:88) (cid:107)gψ (cid:107) wˆ(t ) < ∞. i Aw i i∈I (cid:88) ˆ ˆ Then one has for h ∈ A ∩ C (G) satisfying h (t) = 1 on Q : g = gψ 2 w c 2 i i∈I (T h ). It follows therefrom ti 2 (cid:88) (cid:88) F(g) ≤ (cid:107)gψi(cid:107)AwF(Ttih2) ≤ (cid:107)gψi(cid:107)AwT−tiF(h2). i i The norm of T on Lq being bounded by wˆ(t) we obtain therefrom t v (cid:195) (cid:33) (cid:88) (cid:107)Fg(cid:107) ≤ wˆ(t )(cid:107)gψ (cid:107) (cid:107)F(h2)(cid:107) . q,v i i Aw q,v i Since it is already known that the norms f (cid:55)→ (cid:107)F(h1)(cid:107) and f (cid:55)→ (cid:107)F(h2)(cid:107) q,v q,v are equivalent, the proof is complete (cf. [20], Theorem 1). As a preparation for the general Hausdorff-Young type equalities (i.e. statementsconcerningtheisomorphismofcertainWiener-typespacesrelated to Lp-spaces under the Fourier transformation) to be proved below we give the following lemma: Lemma 3.2 Let (B,(cid:107)·(cid:107) ) be a solid, translation invariant BF-space on a B lc group G (containing C (G) as a dense subspace). Then one has: c a) There exists a weight function wˆ on G such that W(C0,L1) (cid:44)→ B (cid:44)→ W(M,L∞ ). w¯ 1/w¯ If translation is isometric in B one may take w¯ ≡ 1. b) If the following condition is satisfied (Beurling-Domar): (cid:88)∞ (BD) n−2log|(cid:107)T |(cid:107) < ∞ for all y ∈ G, ny B n=1

Modulation Spaces on Locally Compact Abelian Groups Hans G. Feichtinger TECHNICAL ...

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