Modulation Spaces on Locally Compact Abelian Groups Hans …...Modulation Spaces on Locally Compact Abelian Groups Hans G. Feichtinger 1 Introduction The modulation spaces Ms p;q(R

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. Feichtinger,with only the obvious typos being corrected, one additional section and minorextra footnotes. Only few symbols have been changed to more standard ones,e.g. for the translation operator (which was Ly, following Hans Reiter) hasbeen replaced by Ty, and instead of K(G) we write Cc(G) now. We hopethat by adding comments about recent papers on modulation spaces andpublications which have appeared in the mean-time, as well as updates tothe bibliography the reader will find this “new edition” interesting. Of coursethe page numbers differ slightly from those in the original report (it was 52pages long), but the numbering system of theorems and remarks has beenpreserved in the present version (so that one may refer to the results of thispaper, which is now better accessible, in the same way as to the originalreport.)

Modulation Spaces on Locally Compact Abelian Groups

Hans G. Feichtinger

1 Introduction

The modulation spaces M sp,q(R

m), s ∈ R, 1 ≤ p, q ≤ ∞ to be discussedin this paper are Banach spaces of tempered distributions σ on Rm, whichare characterized by the behaviour of the convolution product Mtg ∗ σ inLp(Rm), for t → ∞ (g ∈ S(Rm)). As will be shown this family behaves(with respect to various properties) very much like the well known familyof Besov spaces Bs

p,q(Rm) (cf. [32], [34], [41], [44], [47]) concerning duality,

interpolation, embedding and trace theorems, or the Fourier transformation.Furthermore, the classical potential spaces Ls

2(Rm) = Hs(Rm) as well as the

remarkable Segal algebra S0(Rm) (see [18]) may be considered as particular

modulation spaces.

In order to give a more precise definition of the modulation spacesM s

p,q(Rm) fix any test function g ∈ S(Rm), g 6= 0, and write Mtg for the

(oscillating) function x 7→ exp(2πi〈x, t〉) g(x), x, t ∈ Rm. The convolutionproduct Mtg∗σ is then well defined for any tempered distribution σ ∈ S ′(Rm)and we may set for 1 ≤ q < ∞:

M sp,q(R

m) := {σ | σ ∈ S ′(Rm), Mtg ∗ σ ∈ Lp(Rm) for each t ∈ Rm, and

‖σ|Msp,q‖ :=

[∫

Rm‖Mtg ∗ σ‖q

p (1 + |t|)sqdt]1/q

< ∞ }.

1

2 Hans G. Feichtinger

The necessary modification for q = ∞ is obvious. It is then possible to showthat different test functions define the same spaces and equivalent norms,and that one obtains a family of Banach spaces which is essentially closedwith respect to duality and complex interpolation.

There is not only a formal similarity in the results concerning modulationspaces and Besov spaces. In fact, one can say that an element σ of a Besovspace Bs

p,q(Rm) is characterized by the behaviour of Mρg ∗ σ in Lp(Rm), for

ρ →∞, where now the ”deformation” of the test function consists in a suit-able dilation (Mρg(x) := ρmg(ρx), ρ > 0).1 Such characterizations can befound in the work of Calderon, Torchinsky and others (cf. [9], [43], [39], §8and elsewhere). It is also possible to describe Besov spaces by dyadic decom-positions of the Fourier transforms of their elements (the dyadic structurehas to do very much with dilations). Such characterizations, going essen-tially back to Hormander, have been used successfully by Peetre, Triebel andmany 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-sitions of the Fourier transforms of their elements. Since such decompositionscorrespond to ”uniform” coverings, obtained by translation (and translationof g corresponds to multiplication of g with a character), these are in our sit-uation the natural analogues to the dyadic decompositions mentioned above.The fact that Banach spaces of distributions characterized by uniform de-compositions have been treated in detail in earlier papers by the author underthe 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-hensive class of Banach spaces of distributions on locally compact groups, it ispossible to define modulation spaces for a class of solid BF -spaces (B, ‖ · ‖B)(including Lp, 1 ≤ p ≤ ∞) on locally compact abelian groups (among themRm). In fact, the general modulation spaces M(B,Lq

v)(G) consist of those(ultra) distributions σ on G, for which t → ‖Mtg ∗ σ‖B, t ∈ G, satisfies aweighted q-integrability condition. Using a suitable general Fourier transformthe relevant facts concerning modulation spaces (in the generality just de-scribed) can be drawn from corresponding properties of Wiener-type spaces ofthe form W (FGB,Lq

v) (G). The results concerning the spaces M sp,q(R

m) can

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 choosenis not only justified by the degree of generality obtained, but also by thefact that direct proofs for the spaces M s

p,q(Rm) would not have been much

shorter, but probably less transparent.

The paper is organized as follows. §2 contains the basic notations andfacts, from harmonic analysis and concerning Banach spaces of distributionson locally compact abelian groups. In section 3 various information concern-ing Wiener-type spaces are collected, mainly for later use in the treatment ofmodulation spaces. In particular, weighted versions of the Hausdorff-Youngtheorem for Wiener-type spaces are derived, and some information concern-ing maximal functions are proved. The results of this section allow us tointroduce modulation spaces in §4 in full generality. In this part the in-dependence of modulation spaces from irrelevant parameters (or auxiliaryexpressions, such as the test functions involved) is shown, and various equiv-alent characterizations of these spaces (discrete and continuous versions ofthe norm, atomic representations, norms involving maximal functions) aregiven. Furthermore, several basic properties of modulation spaces, e.g. con-cerning the density of test function, duality, interpolation, convolution, arederived. In section 5 a general trace theorem is established. The last sectiongives information concerning the modulation spaces M s

p,q(Rm) as described

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 alsoillustrating the abstract results given in the earlier parts of this paper. Thepaper concludes with an outlook on further generalizations, related subjectsand further possible applications.

2 Notations, Generalities.

In the sequel G denotes a lca (locally compact abelian) group, with the Haarmeasure dx. We shall be mainly interested in non-compact and non-discretegroups such as Rm, m ≥ 1. The Lebesgue spaces with respect to dx aredenoted by (Lp, ‖·‖p) for 1 ≤ p ≤ ∞, as usual. The translation operators Ty :Tyf(x) := f(x− y) act isometrically on (Lp, ‖ · ‖p). For 1 ≤ p ≤ ∞ the spaceCc(G) (of continuous, compactly supported complex-valued functions on G)is a dense subspace of Lp(G), and the Banach dual of Lp(G) can be identified


with Lp′(G), where1

p+

1

p′= 1. (L1(G), ‖ · ‖1) is considered as a Banach

algebra with respect to convolution, given by f ∗ g(x) =∫

Gf(x − y)g(y)dy

for f, g ∈ Cc(G).

A strictly positive, locally bounded and measurable function w on G willbe called a weight function if w(x) ≥ 1 and w(x + y) ≤ w(x)w(y) for allx, y ∈ G. Then L1

w(G) := {f | fw ∈ L1(G)} is a Banach algebra underconvolution, called Beurling algebra (cf. [38] III, §7 and VI, §3), with thenorm ‖f‖1,w := ‖fw‖1. We restrict our attention to symmetric weights:w(−x) = w(x).

The dual group G of a lca group consists of all continuous characterst : G → U (homomorphisms into the unit circle). We write 〈t, x〉 or 〈x, t〉 fort(x). Recall that G is itself a lca group, and that G may always be identifiedwith (G)∧ in a natural way (due to the Pontrjagin duality theorem). Recallthat (Rm)∧ ∼= Rm as a topological group, since the continuous characters on

Rm are exactly of the form x 7→ exp

[2πi(

m∑

i=1

xiti)

], x, t ∈ Rm.

The Fourier transform f of f ∈ L1(G) is given by

f(t) :=∫

Gf(x)〈x, t〉dx

The Fourier transformation FG : f 7→ f defines an injective, involutiveBanach algebra homomorphism from L1(G) into C0(G) (considered as aninvolutive, pointwise algebra, with complex conjugation, i.e. (f)− = (f ∗)∧,for f ∗(x) = f(−x)). Consequently, given any symmetric weight function won G, the space Aw(G) := {f | f ∈ L1

w(G)} is a self-adjoint Banach algebraof continuous functions on G under pointwise multiplication, if it is endowedwith the norm ‖f‖Aw := ‖f‖1,w.

We shall be exclusively interested in weights satisfying the so-calledBD-condition (Beurling-Domar’s non-quasianalyticity condition, cf. [38],VI, §3), i.e. only weights satisfying

(BD)∞∑

n=1

n−2 log w(nx) < ∞ for all x ∈ G


will be of interest for us. Typical examples of such weights on Rm are thoseof the form ws : x 7→ (1 + |x|)s, s ≥ 0 or wa,d: x 7→ exp(a|x|d), for a ≥ 0,d ∈ (0, 1) (cf. [11], [2], [28], [45], [14] for explanations concerning suchweights). According to the fundamental work of Domar, the algebra Aw(G)is a regular algebra of continuous functions under this condition. It is a evena Wiener algebra in the sense of Reiter (cf. [38], Chap. II).

Since (Mtf)∧ = Ttf , and because the mapping t 7→ Mtf is continuousfrom G into L1

w(G) (as a consequence of the density of Cc(G) in L1w(G)) it

turns out that (Aw, ‖ · ‖Aw) is a ’nice’ Banach algebra in the sense usedin [6], [22]. Among others it is then possible to define Wiener type spacesW (B, C) on G (cf. §3) for any Banach space (B, ‖·‖B) which is in standardsituation with respect to Aw(G), i.e. for spaces satisfying the followingthree conditions:

i) (Aw)0 ↪→ B ↪→ (Aw)′0

(here (Aw)0 := Aw ∩ Cc(G) is considered as a topological vector spacewith respect to its natural inductive limit topology, and (Aw)

′0 denotes

the topological dual; ↪→ indicates continuous embeddings).

ii) (B, ‖·‖B) is a Banach module (with respect to pointwise multiplication)over Aw(G), i.e. ‖hf‖B < ‖h‖Aw‖f‖B for h ∈ Aw(G), f ∈ B.

iii) (B, ‖ · ‖B) is a Banach module (with respect to convolution on G) overa Beurling algebra L1

w(G) (again we restrict our attention to weights wsatisfying (BD)) 2.

Assumption i) as well as many typical examples on G = Rm = G willjustify our speaking of Banach spaces of distributions (on G). We writeB1 ↪→ B2 for continuous embeddings of topological vector spaces. For Ba-nach spaces in standard situation inclusions are automatically continuousby the closed graph theorem. Hence their (complete) norm is uniquely de-termined up to equivalence. If such a space is translation or characterinvariant (i.e. TyB ⊆ B for y ∈ G, or MtB ⊆ B for all t ∈ G, respectively),

2The symbol for the weight function w on G is choosen to make clear that it is definedon the dual group. It should not be confused with the Fourier transform of w, which is notused anywhere in this paper (and would not make sense in the classical sense, but only asa distribution).


the operators Ty and Mt act boundedly on B, and we write |‖Ty|‖B and|‖Mt|‖B for the corresponding operators norms.

The most interesting class of such spaces for us will be the Fourier trans-forms of weighted, solid (e.g. rearrangement invariant) BF -spaces on G.Recall that a Banach space (B, ‖ ‖B) is called a BF -space on G if it iscontinuously embedded into the space L1

loc(G) of locally integrable functions

on G (endowed with the family of seminorms f 7→∫

K|f(x)|dx, K ⊆ G com-

pact). B is solid if g ∈ L1loc(G), f ∈ B and |g(x)| ≤ |f(x)| l.a.e. (locally

almost everywhere) implies g ∈ B and ‖g‖B ≤ ‖f‖B (equivalently: if B is

a (pointwise) module over L∞(G)). B is called rearrangement invariantif |{x | |g(x)| ≥ α}| = |{x | |f(x)| ≥ α}| for all α > 0 (here the outer | · |indicates: Haar measure of the corresponding set) implies ‖g‖B = ‖f‖B. It isclear that such spaces are isometrically translation invariant, i.e. satisfy‖Tyf‖B = ‖f‖B for all f ∈ B, y ∈ G. Moreover, they have continuoustranslation (i.e. lim

y→0‖Tyf − f‖B = 0 for all f ∈ B) if Cc(G) is a dense

subspace of (B, ‖ · ‖B) (cf. [31], [15]).

In the sequel we shall be mainly interested in weighted Lp-spaces, or moregenerally in solid BF -spaces on G which are of the form Bm := {f | fm ∈ B},with norm ‖f‖B,m := ‖fm‖B. Here we assume that B is a solid, isometricallytranslation invariant BF -space on G, containing Cc(G) as a subspace, andthat 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 functionw. We shall call m w-moderate in this case, and consider again only weightssatisfying (BD). For the norm of Ty on Bm we then have |‖Ty|‖Bm ≤ w(y) fory ∈ G. If, furthermore, (B, ‖·‖B) contains Cc(G) as a dense subspace, Cc(G)is also dense in (Bm, ‖ · ‖B,m), and therefore Bm has continuous translationin this case. Applying vector-valued integration one derives therefrom thatBm is a Banach convolution module over the Beurling algebra L1

w(G). In thiscase, (Bm, ‖ · ‖B,m) is an admissible BF -space in the sense of §4 below.

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], [34].


Occasionally it will be convenient to write ‖f |B‖ instead of ‖f‖B. Positiveconstants are denoted by C, C1, C

′,... The same symbol may denote differentconstants at different places.

3 Some results on Wiener-type spaces

(Equivalent characterizations, dependence on test functions,a Hausdorff-Young theorem, a maximal function theorem)

Since many of the basic properties of modulation spaces to be discussedbelow are immediate consequences of the corresponding properties of Wiener-type spaces (as introduced in [20]), we shall recall shortly some facts aboutthis family spaces. We shall also prove several new results on Wiener-typespaces in this section, which are of interest for themselves, but which willserve as auxiliary assertions for the main results of this paper.

Given a Banach space (B, ‖ · ‖B) of distributions on a locally compactgroup G which is in standard situation w.r.t. Aw(G) = F [L1

w(G)], anda continuous, moderate function v on G we can describe the Wiener-typespace W (B, Lq

v)(G) as follows:Fixing any ’test function’ g ∈ Aw(G) ∩ Cc(G), g 6= 0, we have:

W (B,Lqv) := {f | f ∈ Bloc, F (g) : t 7→ ‖(Ttg)f‖B ∈ Lq

v(G)},

and ‖f | W (B, Lqv)‖ :=

(∫

G|F (g)(t)|qvq(t)

)1/q

, for 1 ≤ q < ∞

(or supt∈G |F (g)(t)|v(t) for q = ∞).

Here Bloc is the set of all distributions (members of the dual of Aw(G) ∩Cc(G)) which belong to B locally, and hence (Ttg)f belongs to B for allt ∈ G. Using this definition (slightly different but much more convenientthan the original one given in [20]) one shows that different test functionsg 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 (ψi)i∈I a bounded, uniform partition of unity inAw(G) if there exists some relatively compact set Q ⊆ G such that


i) supi∈I

‖ψi‖Aw(G) < ∞;

ii) sup ψi ⊆ ti + Q for i ∈ I;

iii) supi∈I

|{j | (ti + Q) ∩ (tj + Q) 6= ∅}| < ∞.

Then f ∈ Bloc belongs to W (B, Lqv) if and only if

‖f |D(Q, B, lqv)‖ :=

(∑

i∈I

‖fψi‖qBv(ti)

q

)1/q

< ∞

(for q = ∞ one has to take supi∈I

‖fψi‖Bv(ti)).

Such discrete norms are already implicitly in [20] (cf. Theorem 2, and Remark4) and they are treated in detail in [23]. There one can also find a (general)result concerning the equivalence of discrete and continuous (cf. [56]) norms,including the above assertion ([23], Theorem 4.3). Concerning Wiener-typespaces one has as special cases of theorems proved in [23] (e.g. Theorem 2.8)also the following assertion on duality:

If q < ∞ and if translation is continuous in B for any f ∈ B with compactsupport, then Aw(G)∩Cc(G) is dense in BA := {f | f = limα fα in B, suppfα

compact in G}, and W (B, Lqv) = W (BA, Lq

v). Therefore Aw(G) ∩ Cc(G) isdense in W (B, Lq

v) (cf. [20], Theorem 1) and

W (B, Lqv)′ = W (B′, Lq′

1/v).

Using either the discrete representation f =∑

i∈I

fψi, or a continuous variant

of it (described in [21]), one shows that these spaces behave nicely withrespect to complex interpolation (c.f. [21], [23]).

Concerning the independence of the norm from the particular choice of g,g 6= 0, we give a slight extension of known results, which will be very usefulfor us later (cf. [16] for hints in this direction).

Proposition 3.1 Given any w-moderate, continuous function v on G, anynon-zero function g ∈ W (Aw, L1

w) is an admissible test function for W (B, Lqv)

as above (i.e. the norm obtained by using g is still equivalent to the normsconsidered above).


Proof. For g, h ∈ W (Aw, L1w) ⊆ Aw one has of course

F (hg)(t) ≤ ‖h‖AwF (g)(t)

for t ∈ G, by the Aw-module structure of B. Given g it will be sufficientto choose h ∈ Aw ∩ Cc(G) in order to have ‖F (h1)‖q,v ≤ ‖h‖Aw‖F (g)‖q,v for

h1 := hg ∈ Aw ∩ Cc(G).

In order to obtain the converse estimate we may use the representationg =

∑

i∈I

gψi, for which we have

∑

i∈I

‖gψi‖Aww(ti) < ∞.

Then one has for h2 ∈ Aw ∩ Cc(G) satisfying h2(t) = 1 on Q : g =∑

i∈I

gψi

(Ttih2). It follows therefrom

F (g) ≤ ∑

i

‖gψi‖AwF (Ttih2) ≤ ∑

i

‖gψi‖AwT−tiF(h2).

The norm of Tt on Lqv being bounded by w(t) we obtain therefrom

‖F (g)‖q,v ≤(∑

i

w(ti)‖gψi‖Aw

)‖F (h2)‖q,v.

Since it is already known that the norms f 7→ ‖F (h1)‖q,v and f 7→ ‖F (h2)‖q,v

are equivalent, the proof is complete (cf. [20], Theorem 1).

As a preparation for the general Hausdorff-Young type equalities (i.e.statements concerning the isomorphism of certain Wiener-type spaces relatedto Lp-spaces under the Fourier transformation) to be proved below we givethe following lemma:

Lemma 3.2 Let (B, ‖ · ‖B) be a solid, translation invariant BF -space on alc group G (containing Cc(G) as a dense subspace). Then one has:

a) There exists a weight function w on G such that

W (C0, L1w) ↪→ B ↪→ W (M, L∞1/w).

If translation is isometric in B one may take w ≡ 1.


b) If the following condition is satisfied (Beurling-Domar):

(BD)∞∑

n=1

n−2 log |‖Tny|‖B < ∞ for all y ∈ G,

it follows therefrom that ΛKw (G) ↪→ B ↪→ [ΛK

w (G)]′, where

ΛKw (G) = {f | f ∈ L1

w(G), suppf compact in G},endowed with the natural inductive limit topology.

Proof. By the assumptions we have Cc(G) ⊆ B ⊆ L1loc(G). The closed

graph theorem implies that we have: Given a compact set Q ⊆ G thereexists CQ > 0 such that

C−1Q ‖f‖∞ ≤ ‖f‖B ≤ CQ‖f‖1 for f ∈ Cc(G), suppf ⊆ Q.

On the other hand, the density of Cc(G) in B implies that translation iscontinuous. Therefore y 7→ |‖Ty|‖B is an upper semicontinuous (hencemeasurable) and submultiplicative function on G. Consequently w : w(x) :=max(1, |‖Lx|‖B) defines a weight function on G (cf. [15]). Observing that anyf has a representation f =

∑

i

Tti(T−ti(fψi)) with suppT−ti(fψi) ⊆ Q, one

can see that assertion a) follows from the above estimate (for the ’atoms’).In order to prove b) suppose now that w satisfies (BD). Then Aw(G) :=

{f | f ∈ L1w(G)} is a regular Banach algebra on G, and Aw(G) ∩ Cc(G) is

dense in Cc(G) and Aw(G). Since Theorem 3 of [20] implies that W (C0, L1w)

is a dense ideal in L1w one can derive therefrom that one has

Aw(G) ∩ Cc(G) ⊆ F [W (C0, L1)], or ΛKw ↪→ W (C0, L1

w)

(cf. Theorem 5 of [20]), as a dense subspace. Applying the duality formula

(Theorem 2.8 of [23]) we obtain W (M, L∞1/w) ↪→ (ΛKw )′, as was required for

the second inclusion.

Remark 3.1 It would have been sufficient as well (in the above lemma) tosuppose (instead of density of Cc(G) in B) that B is a closed subspace of adual D′ of a solid space D containing Cc(G) as a dense subspace. In fact,as the supremum of a family of continuous functions, y 7→ |‖Ty|‖B is stillmeasurable in that case.


Remark 3.2 It will be relevant for obtaining some of the results below intheir ’natural’ generality to observe that it would have been sufficient aboveto suppose that B is a space in ’standard situation’ with respect to some ’nice’homogeneous Banach algebra (A, ‖ · ‖A) on G, as considered in §3 of [6] (cf.also [23], [20]). Under this assumption one would obtain:

W (A,L1w) ↪→ B ↪→ W (A′, L∞1/w).

Assuming again (BD) and a somewhat strengthened version of Theorem 5of [20] gives us assertion b) from above as well.

Remark 3.3 For (B, ‖ · ‖B) as in Lemma 3.2 the convolution of f ∈ Bwith g ∈ W (A, L1

w) is well defined and gives a continuous function. In fact,Theorem 3 of [20] implies:

g ∗ f ∈ W (A,L1w) ∗ W (A′, L∞1/w) ⊆ W (Cb, L∞1/w) = Cb

1/w(G).

Since the choice A = Aw(G) = {f | f = h, h ∈ L1w(G)} = FL1

w(G)will be the most natural choice we shall now shortly collect some informationconcerning the Wiener-type space W (FL1

w, L1w)(G). As will be seen these

spaces are weighted variants of the Segal algebra S0(G) as treated in [18].

Proposition 3.3 Let w and w be weight functions on G and G respectively,both satisfying the Beurling-Domar condition. Then the Wiener-type spacesW (Aw, L1

w)(G) and W (Aw, L1w)(G) are well defined and the Fourier trans-

form FG : L1(G) → A(G) establishes an isomorphism between these spaces.

Proof. (a) The spaces L1w(G) and L1

w(G) are Banach convolution alge-bras, and as solid BF -spaces they may of course be considered as point-wise A(G) or A(G)-modules respectively. Since the Fourier transforma-tion interchanges the role of convolution and multiplication it is clear thatAw(G) = FG

(L1

w(G))

and Aw(G) = FG [L1w(G)] are pointwise (nice) Banach

algebras (because (BD) is satisfied), as well as L1-modules (i.e. homoge-neous Banach spaces). Consequently they are in ’standard situation’ overthemselves and the corresponding Wiener-type spaces are well defined.

(b) In order to show the claimed isomorphism it will be sufficient (forreasons of symmetry) to verify the inclusion FG (W (Aw, L1

w)) ⊆ W (Aw, L1w).


Using the discrete characterization of Wiener-type spaces we can find for anyf ∈ W a sequence (fn)n≥1 in Aw(G), a relatively compact set Q ⊆ G and asequence (yn)n≥1 in G such that suppfn ⊆ Q,

f =∑

n≥1

Tynfn, and

∑

n≥1

‖fn‖Aww(yn) ≤ C‖f | W (Aw, L1

w)‖.

For the Fourier transform f = FGf of f this implies:

‖f | W (Aw, L1w)‖ ≤ ∑

n

‖(Tynfn)∧ | W (Aw, L1w)‖

≤ ∑n

w(yn)‖fn | W (Aw, L1w)‖ =: (∗).

The last estimate follows from the formula (Tyf)∧ = Myf , where My denotes

multiplication with the character y ∈ (G)∧ = G, and the fact that theboundedness of My on W (Aw, L1

w) follows from its boundedness on Aw(G),where one has

‖Myf‖Aw = ‖Tyf‖1,w ≤ w(y)‖f‖1,w = w(y)‖f‖Aw .

In order to continue the estimate let us choose some g ∈ W (Aw, L1w), satis-

fying g(x) ≡ 1 on Q. This is possible because w satisfies (BD), W (Aw, L1w)

is a dense Banach ideal in L1w(G). Consequently one has

fn = (g · fn)∧ = g ∗ fn.

Applying now the general convolution formula (Theorem 3 of [20]) for Wiener-type spaces one obtains:

(∗) ≤ ∑n

w(yn) ‖fn ∗ g | W (Aw, L1w)‖

≤ ∑n

w(yn) ‖g | W (Aw, L1w)‖ ‖fn‖1,w

≤ ‖g | W (Aw, L1w)‖ ∑

n

w(yn) ‖fn‖Aw

≤ C1 ‖f | W (Aw, L1w)‖.


The proof is now complete.

By duality we obtain immediately the following result:

Corollary 3.4 Let w, w be as above. Then the Fourier transform extends toan isomorphism between W (F(L∞1/w), L∞1/w) and W (F(L∞1/w), L∞1/w)).

Proof. In view of the fact that the dual of a Wiener-type space may beobtained (under the assumption that the test functions are dense in the space,which is the case for W (Aw, L1

w), see [23], Theorem 2.8) by taking the dualspace in each component one has W (F(L1

w), L1w)′ = W (F(L∞1/w), L∞1/w)). The

extended Fourier transform being defined by transposition, it is clear thatthe isomorphism of the preduals induced by the Fourier transformation givesrise (in fact: extends) to an isomorphism of the dual spaces. This gives thecorollary.

Applying now complex interpolation we obtain the following general re-sult:

Theorem 3.5 Let w, w be weight functions on G and G, both satisfying theBeurling-Domar condition. Furthermore, let α, β ∈ R, and p ∈ (1,∞) begiven. Then the Fourier transform FG extends to an isomorphism betweenW (F(Lp

wα), Lpwβ) and W (F(Lp

wβ), Lpwα).

Proof. First we observe that the spaces involved are well defined (cf. [22],§2). In view of the above results and the characterization of interpolationspaces within the family of Wiener-type spaces (cf. [21]), it will be sufficientto show how the spaces in question are obtained by complex interpolation.Assuming for the moment that α is positive and that β is negative (thearguments for the other cases being similar). In fact, choosing θ = 1/p wehave

W (FLpwα , Lp

wβ) =[W (F(L1

wαp , L1),W (FL∞, L∞1/wβp′ )][θ]

and consequently

FGW (. . .) = [FGW (. . .),FGW (. . .)][θ] =

= [W (FL1, L1wαp),W (FL∞

1/wβp′ , L∞)][θ] = W (FLp

wβ , Lpwα),


as was claimed. In fact, the use of the above results is justified since wαp andwβp′ are again weight functions satisfying (BD).

The case p = 2 of the above theorem will be of special interest, as itgives information concerning potential spaces and their pointwise multipliers(cf. [10], p. 8/9 for related assertions). Writing L2

s for the space of Besselpotentials (= F−1L2

s(Rm)) we have:

Corollary 3.6 For s ∈ R we have

L2s(R

m) = W (L2s, L

2)(Rm).

For s > m/2 it follows that a (continuous) function h defines a pointwisemultiplier if and only if h ∈ W (L2

s, L∞).

Proof. We have for ws with ws(x) = (1 + |x|2)s/2

L2s = FRm [L2

ws(Rm)] = FRm [W (FL2, L2

ws)] = W (FL2

ws, L2) = W (L2

s, L2).

Since L2ws

(Rm) is a Banach convolution algebra for s > m/2 (in this caseL2

ws↪→ L1(Rm), cf. [15]) the second assertion follows from Cor.2.14 of [23].

Further complex interpolation between the result established above andthe Hausdorff-Young inequality proved in [21] (Theorem 3.2) yields:

Theorem 3.7 Let w, w, α, β and p be as in Thm.3.5. Then for r ∈ (1, p] :

FG [W (F(Lpwα), Lr

wβ)(G)] ⊆ W (F(Lrwβ), Lp

wα)(G).

Proof. Starting with the observation that one has for θ ∈ (0, 1), satisfying1/r = (1− θ)/p + θ.

W (F(Lpwα), Lr

wβ) = [W (F(Lpw1

), Lpw2

),W (FLp, L1)][θ],

where w1 = wα/(1−θ), and w2 = wβ/θ. The proof is straightforward.

We conclude this section with two typical results concerning Wiener-typespaces (based on methods developed already in [20]). The first one will beof importance in §5, and the second one will give the possibility of usingcertain maximal functions on arbitrary lca groups. We show the equivalenceof certain norms, without making use of any direct analogue of the Hardy-Littlewood maximal function theorem.


Proposition 3.8 For any admissible solid BF -space (B, ‖ · ‖B) on a lcagroup, and any compact set Q ⊆ G there exists C ′ = C ′(Q, B) such that forany f ∈ B satisfying suppf ⊆ t + Q for some t ∈ G the following is true:

‖f |B‖ ≤ ‖f |W (L∞, B)‖ ≤ C ′‖f |B‖.

Proof. We observe that by the convolution theorem for Wiener-type spaces(i.e. by Theorem 3 of [20]) W (C0, L1

w) is a dense Banach ideal in L1w(G)

(with respect to convolution; otherwise expressed: an abstract Segal algebrain L1

w(G)). Since we have a symbolic calculus in Beurling algebras (i.e.local inversion is possible in Aw(G), [11], or [38], Chap. VI, §3) the algebraFGW (C0, L1

w) coincides locally with Aw(G). The regularity of Aw(G) nowimplies the existence of g ∈ W (C0, L1

w), such that g has compact supportand satisfies g(t) = 1 for all t ∈ Q. Consequently (Mtg)∧(t) = Ttg(t) = 1for all t ∈ t + Q, and thus Mtg ∗ f = f for f as in the lemma. Anotherapplications of Theorem 3 of [20] implies then:

f = Mtg ∗ f ∈ W (C0, L1w) ∗B ⊆ W (C0, L1

w) ∗W (L1, B) ⊆ W (C0, B),

and

‖f |W (C0, B)‖ ≤ ‖Mtg|W (C0, L1w)‖ ‖f |W (L1, B)‖ ≤ ‖g|W (C0, L1

w)‖ ‖f |B‖.The first estimate being obvious the proof is complete.

We come now to the definition of maximal functions on lca groups:

Definition 3.1 Given a continuous function f on G, and a continuous weightfunction w on G, we set:

f#(w)(x) := supy∈G

|f(y)|w(x− y)

.

f#(w) will be called the w-maximal function of f.

Theorem 3.9 Let (B, ‖·‖B) be an admissible, solid BF -space on a lca group(which is a Banach module over L1

w(G) for convolution), and let w be acontinuous weight function on G, such that ww−1 ∈ L1(G). Then thereexists for any compact set Q ⊆ G some C = C(B, Q, w) > 0 such that onehas for all f ∈ B, with suppf ⊆ t + Q (for some t ∈ G):

‖f#(w)‖B ≤ C‖f‖B.


Proof. Let Q0 ⊆ G be any symmetric (Q0 = −Q0), open and relativelycompact subset of G, and let (yi)i∈I be a discrete family in G, such thatQ := (yi +Q0)i∈I is a uniform (hence admissible, cf. [23]) covering of G. Theexistence of such families is shown in [16]. Since w−1(z) ≤ Cww−1(yi) forz ∈ yi + Q0, and all i ∈ I, it is clear that one has

f#(w)(x) ≤ supi∈I

maxz∈yi+Q0

|f(x− z)|w−1(z)

≤ Cw supi∈I

w−1(yi) maxt∈Q0

|Tyif(x− t)|.

Writing f∞ for the function, given by f∞(x) := maxt∈Q0

|f(x− t)| and observing

that (Tyf)∞ = Ty(f∞) one has then

f#(w)(x) ≤ Cw

∑

i∈I

w−1(yi)Tyif∞(x).

Since the norm of Ty on B is less than w(y), we infer from the solidity of B:

‖f#(w)‖B ≤ Cw

∑

i∈I

w−1(yi)‖Tyif∞‖B

≤ Cw(∑

i∈I

w−1(yi)w(yi))‖f∞‖B.

Observing now that the above sum is finite (e.g. by showing that the functionw′ :=

∑

i∈I

w−1(yi)w(yi)cyi+ Q0 is dominated by a scalar multiple of w−1w,

which has been assumed to be integrable) we have by Proposition 3.7:

‖f#(w)‖B ≤ CwC ′′w‖f∞‖B

= CwC ′′w‖f‖W (L∞,B) ≤ CwC ′′

wC ′‖f‖B,

and the proof of the theorem is complete.

Remark 3.4 On G = Rm the weight functions ws, given by ws(x) = (1 +|x|)s are of course the most interesting ones. For rearrangement invariantspaces B on G, such as Lp-spaces or Lorentz- and Orlicz-spaces on Rm theabove theorem is then applicable to show that

‖f#(ws)‖B ≤ Cs‖f‖B,


for any s > m, and all f ∈ B having a fixed (prescribed) diameter of theirspectrum (i.e. of the support of their Fourier transforms, considered as atempered distributions on Rm). Taking the effect of dilations in Rm on thenorm of f in Lp(Rm) and on the size of suppf into account one can alsoderive estimates for C as a function of the diameter of Q (for B = Lp(Rm),and w = ws as above).

Remark 3.5 The above proof shows that at least some non-trivial estimatesinvolving maximal functions (even on Rm) can be obtained using neitherestimates for some gradients nor the Hardy-Littlewood maximal functions.On the other hand, our results are available for arbitrary lca groups.

4 Modulation spaces and basic properties

In this section the general definition of modulation spaces and some oftheir basic properties are given. We shall assume throughout this section that(B, ‖ ·‖B) is an admissible, solid BF -space on a lca group G, i.e. that thereis a weight function w on G, satisfying the (BD)-condition, and such that|‖Ty|‖B ≤ w(y) for y ∈ G, and such that (B, ‖ · ‖B) is a Banach convolutionmodule over L1

w(G). This is the case, for example, if Cc(G) is dense in a

translation invariant, solid BF -space (B, ‖·‖B) and∞∑

n=1

n−2 log |‖Tny|‖B < ∞for any y ∈ G. For such spaces we define now:

Definition 4.1 Let G be a locally compact Abelian group, and let w and w becontinuous weight functions on G and G respectively, both satisfying (BD).Given (B, ‖ · ‖)B as described above, and a continuous w-moderate functionv on G we define for 1 ≤ q ≤ ∞, fixing any

k ∈ ΛKw (G) = {f | f ∈ L1

w(G), suppf compact in G}, k 6= 0 :

M(B, Lqv) := {σ ∈ (ΛK

w )′, Mtk ∗ σ ∈ B for all t ∈ G,

and t 7→ ‖Mtk ∗ σ‖B belongs to Lqv(G)},


and the expressions

‖f |M(B, Lqv)‖ :=

[∫

G‖Mtk ∗ σ‖q

Bvq(t)dt]1/q

for 1 ≤ q < ∞, and

‖f |M(B,L∞v )‖ := supt∈G

‖Mtk ∗ σ‖Bv(t)

are the natural norms on M(B, Lqv). The space M(B, Lq

v) will be calledmodulation space, derived from B, with degree of smoothness Lq

v.

After the preliminaries given in sections 2 and 3 it is clear that we havethe following basic results:

Theorem 4.1 A) The spaces M(B, Lqv), 1 ≤ q ≤ ∞, are Banach spaces

(of distributions on G) with respect to their natural norms, and theembeddings ΛK

w ↪→ M(B, Lqv) ↪→ (ΛK

w )′ ↪→ M(B, Lqv) are continuous

(where each of these spaces being endowed with its natural topology).

B) The spaces M(B, Lqv) do not depend on the particular choice of the test

functions k ∈ Cc(G) ∩ Aw(G) (i.e. different test functions define thesame space and equivalent norms). Furthermore these spaces do notdepend on the weights w, w (satisfying BD) involved in the definition.

C) For any K ⊆ G, compact, the norms of B and these of M(B,Lqv) are

equivalent on {f | f ∈ B, suppf ⊆ K}.D) If Cc(G) is a dense subspace of (B, ‖ · ‖B), then ΛK

w is a dense subspaceof M(B, Lq

v) for 1 ≤ q < ∞, and translation is continuous in M(B, Lqv)

in this case.

E) For B and q as in D) above the dual space of M(B, Lqv) can be identified

with M(B′, Lq′1/v).

Proof. A+B) Since w satisfies (BD) there exists k 6= 0, k ∈ Aw(G)∩Cc(G).Applying the extended Fourier transform (which establishes an isomorphismbetween ΛK

w and (Aw)0 = Aw ∩Cc(G), and between (ΛKw )′ and (Aw)′0 respec-

tively) it is clear that FG[M(B, Lqv)] coincides with the Wiener-type space


W (FB,Lqv) on G. In fact, by the convolution theorem one has FG(Mtk∗σ) =

(Ttk)σ, where k belongs to

FG(Aw)0 ⊂ FG(W (FL1w, L1

w)) = W (FL1w, L1

w)

by Proposition 3.3. But then the norm of t 7→ ‖(Ttk)σ‖FB = ‖Mtk∗f‖B in Lqv

actually defines a norm on W (FB, Lqv), for any such k, and different choices

give equivalent norms by Proposition 3.1 (and our assumption concerning v).The continuity of the embedding

ΛKw ↪→ M(B,Lq

v) ↪→ (ΛKw )′

then follows from the embeddings (cf. [20], Theorem 1)

(Aw)0 ↪→ W (FB, Lqv) ↪→ (Aw)′0.

In a similar way the completeness of M(B,Lqv) follows from that of Wiener

type spaces.

The verification of C) is left to the reader. For the proof of D) observe thatAw(G)∩Cc(G) is dense in Cc(G) (by (BD)), and thus in B as well. It followstherefrom that W (Aw, L1

w) is a dense subspace of B (apply remark 3.1, i.e.a modification of Lemma 3.2, to FB, and use Proposition 3.3 then). ButW (Aw, L1

w) is a dense essential Banach ideal of L1w(G), and therefore contains

the space {f | f ∈ L1w, suppf compact } = ΛK

w as a dense subspace (cf.[39], Chapter VI, §2.2 for related assertions concerning Segal algebras=dense,essential Banach ideals of L1(G), i.e. for the case w ≡ 1). Combining thesefacts with Theorem 1, iv) of [20] the density of ΛK

w in M(B,Lqv) follows. Due

to the density of ΛKw and the continuity of translation in (ΛK

w , ‖·‖B) the proofof D) is complete.

Finally, the characterization of M(B, Lqv)′ is a consequence of the density

of ΛKw in M(B, Lq

v) which allows (combined with the Fourier transform) toapply Theorem 2.8 of [23].

Remark 4.1 It is clear that equivalent w-moderate functions v and v′ (i.e.having both quotients bounded) define the same space and equivalent norms(e.g. (1 + |x|)s ∼ (1 + |x|2)s/2 on Rm).


Remark 4.2 If (B, ‖ · ‖B) as in D) above has isometric translation, thenM(B, Lq

v), 1 ≤ q < ∞ is a homogeneous Banach space of locally integrablefunctions. In particular, the spaces M(L1, Lq

v), 1 ≤ q < ∞, are Segal Alge-bras (related spaces are discussed in [24]).

Remark 4.3 Our definition is related to Calderon’s characterization of Besovspaces (cf. [9], [43] and others). There is also some analogue to Calderon’srepresentation formula, which reads in our setting (with a suitable interpre-tation of the vector-valued integral)

σ =∫

GMtk ∗ σdt,

for k being suitably normed (cf. [21]). We shall not use this formula here.

There is also a discrete characterization of modulation spaces (very usefulin practice), making use of a uniform decomposition of the Fourier transformsof their elements.

Corollary 4.2 Given any bounded, uniform partition of unity Ψ = (ψi)i∈I inAw(G), (satisfying suppψi ⊆ ti+Q0, for some relatively compact set Q0 ⊆ G)we may characterize M(B, Lq

v) as follows:

M(B, Lqv) = {σ | σ ∈ (ΛK

w )′, F−1G [ψiFGσ] ∈ B for all i ∈ I,

‖σ‖ := [∑

i∈I

‖F−1G [ψi(FGσ)]‖q

B vq(ti)]1/q < ∞},

(with a supremum for q = ∞), and ‖ · ‖ defines an equivalent norm onM(B, Lq

v).

Writing ϕi := F−1G ψi and applying (the maximal function) Theorem 3.9

we obtain another equivalent norm (cf. [46], §2.1.2 for a related result forBesov spaces).

Corollary 4.3 Let (ϕi)i∈I be as above, then for any weight function w on Gsatisfying wv−1 ∈ L1(G) the expression

[∑

i∈I

‖(ϕi ∗ σ)#(w)‖qB vq(ti)

]1/q

(or the corresponding supremum) defines an equivalent norm on M(B, Lqv).


There is also an ’atomic’ characterization of M(B,Lqv), showing that mod-

ulation spaces on R might be considered as special cases of spaces as definedby Goldman ([26]). In fact, applying Corollary 2.6 of [23] we have:

Corollary 4.4 Given any uniform covering Q = (ti + Q0)i∈I of G, then onehas f ∈ M(B, Lq

v) (1 ≤ q < ∞) if and only if there is a family (fi)i∈B in B,such that suppfi ⊆ ti + Q0, f =

∑

i∈I

fi (as a distribution), and

[∑

i∈I

‖fi‖qB vq(ti)

]1/q

< ∞.

Moreover, the infimum over all such expressions defines an equivalent normon M(B, Lq

v) (again with obvious modifications for q = ∞).

The isomorphism to Wiener-type spaces given by the Fourier transformalso allows to write down the result on complex interpolation immediately.as a typical application let us describe the case B = Lp

m(G), for w-moderate,continuous functions mi on G, i = 1, 2.

Theorem 4.5 Given 1 ≤ p1, q1 < ∞, 1 ≤ p2, q2 ≤ ∞, and θ ∈ (0, 1) onehas (

M(Lp1m1

, Lq1v1

),M(Lp2m2

, Lq2v2

))

[θ]= M(Lp

m, Lqv),

where

1

p=

1− θ

p1

+θ

p2

,1

q=

1− θ

q1

+θ

q2

, m = m1−θ1 mθ

2, v = v1−θ1 vθ

2.

Proof. cf. [21], or [23].

For the treatment of various questions (some of which are discussed in[6] and [19], [22] respectively) it is also of interest to check that modulationspaces have a double module structure. In fact, they are even Banach spacesin ’standard situation’ as treated in [6]:

Theorem 4.6 A) For any solid, admissible BF -space on a lca group Gand v as in Definition 4.1 and 1 ≤ q ≤ ∞, the space M(B, Lq

v)(G) isa Banach module over Aw(G) with respect to pointwise multiplication,as well as a Banach module over L1

w(G) with respect to convolution.


B) Furthermore one has the inclusions

(Aw)0 ↪→ M(B,Lqv) ↪→ (Aw)′0.

Therefore, both algebras having bounded approximate units (and allproperties required further), M(B,Lq

v) is a Banach space in standardsituation in the sense of [6], §3.

C) If Cc(G) is dense in M(B,Lqv) and if q < ∞, then (Aw)0 is a dense

subspace of M(B,Lqv).

Proof.

A) (B, ‖ · ‖B) being a Banach module over L1w(G) it is clear that we have

the following estimate:

‖Mtk ∗ g ∗ f‖B ≤ ‖g‖1,w‖Mtk ∗ f‖B.

This implies that M(B,Lqv) is a module over L1

w(G) as well. The C0(G)(hence A(G)-) module structure corresponds to a L1(G)-convolutionstructure on FB. By Theorem 3 of [20] we have a W (L1, L1

w) = L1w(G)

convolution structure on W (FB,Lqv). Applying the inverse Fourier

transform the pointwise Aw-structure on M(B, Lqv) follows.

B) The arguments for these assertions are very similar to those in the proofof Theorem 4.1 (cf. also [6]) and therefore left to the reader (see alsoRemark 3.1, for W (FB, Lq

v)).

C) There are various ways of verifying the density of (Aw)0 = Aw ∩Cc(G).Either it follows from the density of W (FL1, L1

w) in W (FB, Lqv) for

1 ≤ q < ∞ (observe that one has

F−1G [W (FL1, L1

w)] ∩ Cc(G) = Aw ∩ Cc(G),

cf. Theorem 4.1), or it follows from the fact that M(B,Lqv) is an essen-

tial Banach module with respect to both actions defined (by Theorem4.2 of [6]). That the actions are essential follows in turn from the conti-nuity of y → Tyf and t → Mtf from G and G to M(B,Lq

v) respectively,for all f ∈ M(B,Lq

v), by vector-valued integration.


Corollary 4.7 (Compactness criterion) Let (B, ‖ ·‖B) be a solid, admis-sible BF -space, containing Cc(G) as a dense subspace. Then one has for1 ≤ q < ∞:

A bounded subset S ⊆ M(B, Lqv) is relatively compact if and only if for

any ε > 0 there exist compact sets K, K in G and G respectively, such thatfor any f ∈ M(B,Lq

v) there exists k1, k2 ∈ M(B, Lqv) such that suppk1 ⊆ K,

suppk2 ⊆ K, and

‖(f − ki) | M(B, Lqv)‖ < ε for i = 1, 2.

Proof. In view of Theorem 4.6 above the main result (Theorem 2.2) of [25] isapplicable. The description of tightness and equicontinuity of S in M(B,Lq

v)given above is justified by Propositions 2.3 and 2.4 of [22].

Remark 4.4 If Cc(G) is dense in (B, ‖ · ‖B) the closure of Aw ∩ Cc(G) inM(B,L∞v ) coincides with

M(B,C0v ) := {σ | σ ∈ M(B,L∞v ), lim

t→∞ ‖Mtk ∗ σ‖B · v(t) = 0}.

The dual of M(B, C0v ) coincides of course with M(B′, L1

1/v), cf. again [23],Theorem 2.8.

Remark 4.5 At the end of this section we mention that the solidity of theoriginal space (B, ‖ · ‖B) is by no means relevant (although certainly natural)for the possibility of constructing the modulation spaces M(B, Lq

v) and show-ing their basic properties (as above). It would be sufficient to have a pointwiseAw(G)-module (with w a weight on G, satisfying (BD)) on a Banach space(B, ‖ · ‖B) in standard situation.

Remark 4.6 In view of the above remark it would be possible to ask aboutreiterations of the construction method yielding modulation spaces. At leastfor solid, admissible BF-spaces (B, ‖ · ‖B) we can state the following result(which is easily verified using the discrete description of Corollary 4.2):

M(M(B,Lrv1

), Lqv2

) = M(B,Lqv1v2

).


5 The trace theorem

We shall prove in this section a trace theorem for modulation spaces in theperhaps most general reasonable way, i.e. for spaces of the form M(Bm, Lq

v),where B is a rearrangement-invariant solid BF -space. In fact, such spacesare defined on arbitrary measure spaces, and in particular B(G) and B(H)are related in a canonical way, if H is a closed subgroup of a locally compactgroup G (typical examples are of course the Lp-spaces, or Lorentz-spacesL(p, q), defined for any group).

To be more precise, we assume that a lca group G is of the form G =H1 × H2, where H1, H2 are considered as closed subgroups of G. Then therestriction operator R1 : Cb(G) → Cb(H1) (R2 denoted restriction to H2)maps Aw ∩Cc(G) into Aw ∩Cc(H1), where w(z) := inf

y∈H2

w(z, y) (this follows

from [38], III, 7.13 via the Fourier transformation), and it will be shownto extend to a continuous and surjective linear mapping between suitablepairs of modulation spaces on G and H1 respectively, if we suppose thatthe degree of smoothness of the domain is high enough (i.e. that Lq

v(G) issmall enough). In particular, we shall have to show that there exist suitableextension operators in order to establish the surjectivity in each case.

The proof of our main theorem will be obtained by a series of auxiliaryresults. The basic idea is to prove it first for (smooth) function havingcompact spectrum (supp f) and of ”pasting together” the results for these”atoms”. Again, heavy use of methods from the theory of Wiener-type spaceswill be made. But also ideas, developed for the study of the ”canonicalmapping” TH : L1(H ′ × H) → L1(H ′) (cf. [38], III and VIII) as operatoracting on weighted Lp-spaces (cf. [14]) will be used at least indirectly.

The main theorem of this section reads as follows:

Theorem 5.1 (Trace theorem) Assume that G is of the form G = H1 ×H2, for H1, H2 being locally compact abelian groups. Let m be a w-moderatefunction, and v a w-moderate function on G and G respectively (with w,w satisfying (BD)). Furthermore, let B(G) and B(H) be rearrangementinvariant, solid BF -spaces, corresponding to each other in a canonical way.Assume that (R2v)−1 ∈ Lq′(H2) (R2: restriction to H2), and define v〈q〉 and


m1 by m1(z) := R1m(z) = m(z, 0), and

v〈q〉(t) =[∫

H2

v(t, s)−q′dH2s]−1/q′

.

Then the restriction mapping R1 : Cb(G) → Cb(H1) (R1f(x) := f(x, 0))induces a continuous, surjective mapping from M(Bm, Lq

v)(G) ontoM(Bm1 , Lq

v〈q〉). For q = 1 one has to set v〈1〉(t) := infs∈H2v(t, s).

Remark 5.1 As usual, one must agree to obtain R1f via regularizations,if f happens to be not continuous near H1 ⊆ G. In our proof R1f willarise as a convergent sum in M(Bm1 , L

qv〈q〉). It would be possible as well to

set R1f := limα

R1(eα ∗ f) (the second R1 being ordinary restriction), for a

suitable, smooth approximation (eα)α∈I to the identity (e.g. various classical’means’ for the applications).

As already mentioned the general trace theorem will be based on a seriesof auxiliary results which we shall state first. In order not to interrupt thegeneral flow of ideas we shall give the proof of Theorem 5.1 immediately afterthis series of lemmata, which will be proved afterwards.

Lemma 5.2 For any compact set Q ⊆ G there exists C ′′ = C ′′(Q, m) > 0such that

‖R1f | Bm1(H1)‖ ≤ C ′′‖f | Bm‖for all f ∈ Bm(G) satisfying supp f ⊆ t + Q for some t ∈ G.

Lemma 5.3 There exists some weight function w2 on H2, satisfying theBeurling-Domar condition, and C1 > 0 such that f ⊗ g ∈ Bm(G) for f ∈Bm1(H1) and g ∈ W (Cb, L1

w2)(H2) and

‖f ⊗ g | Bm(G)‖ ≤ C1‖f | Bm1‖ ‖g | W (Cb, L1w2

)‖

Lemma 5.4 For any pair (Q1, Q2) of open, relatively compact subsets of H1

and H2 respectively, there exists C > 0 such that for any f ∈ Bm1(H) withsuppf ⊆ t1 + Q for some t1 ∈ H1 there exists an extension F ∈ Bm(G)satisfying

‖F | Bm(G)‖ ≤ C‖f | Bm1(H1)‖


andsuppF ⊆ (t1 + Q1)× Q2.

Lemma 5.5 Let Q1, Q2 be open, relatively compact subsets of H1 and H2

respectively, and let (tl)l∈L and (sr)r∈R be (uniformly discrete) families inH1 and H2 respectively, such that the families (tl + Q1)l∈L and (sr + Q2)r∈R

are uniform (hence admissible) coverings (of bounded height) for H1 and H2

respectively. Then there exists for q ∈ (1,∞] a positive numbers γ > 0, suchthat for each l ∈ L there is a finite set El ⊆ R, and δl ≤ γ, such that

2δl

[∑

r∈R

v−q′(tl, sr)

]−1/q′

≥ δl

∑

r∈El

v−q′(tl, sr)

−1/q′

= v〈q〉(tl).

Proof of the Theorem. A) (restriction theorem)

We start by observing that the assumptions imply that v〈q〉, as defined,is again a moderate function (on H2) and that the corresponding weight (wewrite w1) satisfies the condition (BD) as well. The same can be said concern-ing m1 (and the corresponding weight w1 = R1w). Therefore M(Bm1 , L

qv〈q〉)

is at least well defined.

We start by observing that there are bounded, uniform partitions of unity(ψ1

l )l∈L and (ψ2r)r∈R in Aw1(H1) and Aw2(H2) respectively, (w2 : = R2w)

satisfying

supp ψ1l ⊆ tl + Q1 for all l ∈ L and supp ψ2

r ⊆ sr + Q2 for all r ∈ R.

Then the family {ψ1l ⊗ ψ2

r}(l,r)∈L×R defines a bounded, uniform partition in

Aw(G). It will be convenient to set ψl,r : = ψl ⊗ ψr, ϕl : = F−1H1

ψl, andϕl,r : = F−1

G ψl,r. Recall that we have then

‖f |M(Bm, Lqv)(G)‖ = [

∑

l,r

‖ϕl,r ∗ f |Bm(G)‖qvq(tl, sr)]1/q,

which is equal to the norm of α : = (αl,r)(l,r)∈L×R, given by αl,r : =‖ϕl,r ∗ f |Bm)(G)‖, in a weighted lq -space on L × R (this space is de-noted by lqv in a suggestive way). What we have to do is to give an es-timate for β = (βl)l∈L, given by βl := ‖ϕl ∗ R1f |Bm1‖. Since one has


FH1(R1g) = TH2(FGg) for g ∈ Cc(G) (cf. [38], V, 5.4) it is clear by a limiting

argument that one has for f ∈ M(Bm, Lqv)(G)

ϕm ∗R1(ϕl,r ∗ f) = 0 if ψmψl = 0,

i.e. only m ∈ L(l) : = {n | n ∈ L, ψnψl 6= 0} is of relevance.

Therefore one has for each l ∈ L

ϕl ∗R1f = ϕl ∗R1(∑n,r

ϕn,r ∗ f) =∑n,r

ϕl ∗ [R1(ϕn,r ∗ f)]

=∑r

∑n∈L(l)

ϕl ∗ [R1(ϕn,r ∗ f)].

Taking into account the uniform structure of our covering of G and themoderateness of v it is clear that there exists C1 > 0 such that for all y ∈ H2,l ∈ L,

C−11 v(tn, y) ≤ v(tl, y) ≤ C1v(tn, y) whenever n ∈ L(l).

It follows therefrom that the weighted lq−space lqv is regular (in the senseof [23], with respect to the uniform covering of G{(tl, sr) + Q1 × Q2}), andtherefore we can give estimate of the form

‖α∗|lqv‖ ≤ C2‖α|lqv‖ = C2‖f |M(Bm, Lqv)‖,

given by α∗l,r : =∑

n∈L(l)

|αn,r|. Applying now the norm of Bm1(H1) to the above

identity one obtains:

βl ≤ ∑r∈R

∑n∈L(l)

‖ϕl‖1,w1‖R1(ϕn,r ∗ f)|Bm1(H1)‖by Lemma 5.2

≤ C3(supl∈L ‖ϕl‖1,w1)∑r

∑n∈L(l)

‖ϕn,r ∗ f |Bm(G)‖

≤ C4∑

r∈Rα∗l,r and by Holder’s inequality

≤ C4[∑

r∈Rα∗ql,rv

q(tl, sr)]1/q[

∑r∈R

v(tl, sr)−q′ ]1/q′

by Lemma 5.5≤ C5[

∑r∈R

α∗ql,rvq(tl, sr)]

1/qv〈q〉−1(tl).


Consequently one has (by the estimate for α∗)

‖R1f |M(Bm1 , Lqv〈q〉)‖ = [

∑l∈L

(βlv〈q〉(tl))q]1/q

≤ C5[∑l∈L

∑r∈R

(α∗ql,rv(tl, sr))q]1/q

≤ C2C5‖f |M(Bm, Lqv)‖, q.e.d.

B) (extension theorem) It will be shown that an extension operatorcan be obtained by combining Lemma 5.5 with the corresponding ’atomic’extension given by Lemma 5.4. In fact, by Corollary 4.4 it will be sufficientto find, given f ∈ M(Bm1 , L

qv〈q〉)(H1), a family (Fl,r)(l,r)∈L×R in the vector-

valued sequence space lqv(Bm(G)), such that

supp Fl,r ⊆ (tl + Q1)× (sr + Q2), and∑

l,r

R1Fl,r = f

(the distribution F : =∑l,r

Fl,r is then the required extension for f).

We start by observing that Lemma 5.4 gives for each l ∈ L someFl ∈ Bm(G) satisfying R1Fl = ϕl ∗ f, suppFl ⊆ (tl + Q1) × Q2 and‖Fl|Bm‖ ≤ C‖f |Bm1‖. Unfortunately the degree of smoothness of H :=

∑l Fl

is not good enough in general. However, for El, δl as in Lemma 5.5 one has

f ∗ ϕl = R1[v〈q〉q′(tl)δ−q′l

∑

r∈El

v−q′(tl, sr)Fl].

It is then possible to define

Fl,r : = δ−q′l v〈q〉q′(tl)v−q′(tl, sr)M(0,sr)Fl

for r ∈ El, and Fl,r : = 0 for r 6∈ El. It is then clear that one has

supp Fr,l = supp T(0,sr)Fl = (0, sr) + [(tl + Q1)× Q2],

as we claimed. It is also clear that F : =∑

l,r

Fl,r is well defined as a distribu-

tion, and that R1(M(0,sr)Fl) = R1Fl implies

R1F =∑

l,r

R1Fl,r =∑

l

R1(Fl) =∑

l

ϕl ∗ f = f.


Finally, the required norm estimate follows from Lemmas 5.4 and 5.5 (usingthe equality q + q′ = qq′):

∑l,r

vq(tl, sr)‖Fl,r|Bm‖q ≤ ∑l

∑r∈El

vq(tl, sr)[v−1(tl, sr)v〈q〉(tl)δ−1

l ]q′q · ‖Fl|Bm‖

=∑l[

∑r∈El

v−q′(tl, sr)]v〈q〉q′q(tl)δ−q′ql · ‖Fl|Bm‖

=∑l

δq′l v〈q〉−q′(tl)v〈q〉q′+q(tl)δ

−q′−ql · ‖Fl|Bm‖

≤ ∑l

δ−ql v〈q〉q(tl)Cq‖f ∗ ϕl|Bm1‖

≤ (γ−1C)q ∑l

v〈q〉q(tl)‖f ∗ ϕl|Bm1‖

≤ C6‖f |M(Bm1 , Lqv〈q〉‖q, q.e.d.

The necessary modifications for q = 1 are left to the reader (part B) is mucheasier in that case).

Proof of Lemma 5.2 Since W (Cb, Bm1)(H1) ↪→ Bm1(H1) in view of thesolidity of Bm1(H1) it will be sufficient ( in view of Proposition 3.8) to showthat there exists C1 = C1(Q) > 0 such

‖R1f |W (Cb, Bm1)(H1)‖ ≤ C1‖f |W (Cb, Bm)(G)‖

for f ∈ Bm(G) satisfying suppf ⊆ t + Q for some t ∈ G. The easiest wayto obtain this estimate will be that of describing the norms involved by thenorms of (R1f)∞ and f∞ in Bm1(H1) and Bm(G) respectively, where

(R1f)∞(x) : = maxv∈x+Q1

|R1f(v)|, and

f∞(x, y) : = maxz∈(x+Q1,y+Q2)

|f(z)|,

for compact subsets Q1 and Q2 in H1 and H2 respectively, which may besupposed to satisfy µH1(Q1) = 1 = µH2(Q2), and Q2 = −Q2. It is then clearthat (R1f)∞(x) ≥ α for some α > 0 and x ∈ H1 implies f∞(x, y) ≥ α for ally ∈ Q2. The moderateness of m and the compactness of Q2 imply that there


exists δ > 0 such that m(x, y) ≥ δm(x, 0) = δm1(x) for all y ∈ Q2. SinceµG(Q1 ×Q2) = 1 we find

µH1{x|x ∈ H1, (R1f)∞(x)m1(x) ≥ α} ≤ µG{(x, y)|δ−1f∞(x, y)m(x, y) ≥ α}.

B(G) and B(H) being rearrangement invariant spaces, related in a canonicalway, this implies

‖(R1f)∞|Bm1(H1)‖ ≤ δ−1‖f∞|Bm(G)‖, as required .

Proof of Lemma 5.3 Given the w-moderate function m we write m1 : R1m,w1 := R1w, w2 := R2w. Then it is clear that one has m(x, y) ≤ m1(x)w2(y)for all (x, y) ∈ G, and both weights w1 and w2 satisfy the (BD)-condition.For Q2 given, there is a suitable family (yj)j∈J in H2, (such that (yj+Q2)j∈J isa uniform covering of H2, such that the norm ‖g|W (C0, L1

w2) (H2)‖ may be

replaced by the equivalent expression∑j∈J

αj w2(yj) whenever convenient,

where one defines αj := supy∈yj+Q2|g(y)| (cf. [20], [16]). Since g ≤ ∑

j∈J

αj cyj+Q2

one has

µG{z | fm1 ⊗ cyj+Q2(z) ≥ α} = µH2(Q2)µH1{x | fm1(x) ≥ α}.

One obtains therefrom (using the moderateness of w2):

‖f ⊗ cyj+Q2|Bm1⊗w2(G)‖ ≤ C‖f |Bm1(H1)‖w2(yj)

for some C > 0. Finally one concludes

‖f ⊗ g|Bm(G)‖ ≤ ‖f ⊗ g|Bm1⊗w2(G)‖

≤ (∑j∈J

αjw2(yj))C‖f |Bm1(H1)‖

≤ C1‖f |Bm1(H1)‖ ‖g|W (C0, L1w2

)(H2)‖.

Proof of Lemma 5.4 Let f ∈ Bm1(H1), and Q2 ⊆ H2, open, be given.Since w2 satisfies the (BD)-condition we know that Aw2(H2) is a regular(pointwise) algebra, and it is possible to find g ∈ L1

w2(H2) such that supp g


is a compact subset of Q2. Thus g ∈ ΛKw2

(H2) belongs to any dense ideal ofL1

w2(H2), in particular to W (C0, L1

w2)(H2) (cf. [20], Theorem 3). Applying

now Lemma 5.3 we have for F := f ⊗ g :

‖F |Mm(G)‖ ≤(C1‖g|W (C0, L1

w2)(H2)‖

)‖f |Bm1(H1)‖.

Furthermore, we may assume (after suitable renormalization) that g(0) = 1.Then R1F = f. Since

supp FGf ⊆ supp (FH1f)× supp (FH2g) ⊆ (t1 + Q1)× Q2,

as was we required, the proof is complete.

Proof of Lemma 5.5 Since v−1 is a moderate function, and{(tl, sr) + (Q1 × Q2)} = {Q1,r} is a uniform covering of bounded heightone has

v−1 ∼ ∑

l,r

v−1(tl, sr) cQ1,r ∼ [∑

l,r

v−q′(tl, sr)]−1/q′ .

Applying the definition of v〈q〉 for q > 1 one obtains (for any l0 ∈ L, fixed):

v〈q〉−q′(tl0) ∼ ∫H2

∑l,r

v−q′(tl, sr)cQl,r(tl0 , y)dH2

y

∼ ∫H2

∑r

v−q′(tl0 , sr)cQl0,r(tl0 , y)dH2

y

= µH2(Q2)

∑r

v−q′(tl0 , sr).

The sum being convergent for each l0 ∈ L one can find for each l ∈ L a finiteset El ⊆ R such that

∑

r∈El

v−q′(tl, sr) ≥ 2−q′ ∑

r∈R

v−q′(tl, sr).

and the proof is complete.

As a matter of fact the proof of Theorem 5.1 also gives information con-cerning the behaviour of weighted Wiener-type spaces such asW (Lp, Lq)(H × H ′) under the canonical mapping TH , given by THf(y) =∫H f(x, y)dx, y ∈ H ′, for Cc(H ×H ′) (cf. [38], and [14]). The corresponding

result reads as follows:


Theorem 5.6 Let m be a continuous, moderate function on a locallycompact group G = H × H ′. Assume that (RHm)−1 ∈ Lq′(H) for someq ∈ [1,∞]. Then TH extends to a bounded, linear mapping from W (Lp, Lq

m)(G),1 ≤ p ≤ ∞, onto W (Lp, Lq

m〈q〉)(H′), where m〈q〉 is given by

m〈q〉(y) : = (∫

Hm−q′(x, y)dx)−1/q′

(appropriate modifications for q = ∞ are possible).

We don’t give the proof of this result here because it is very similar (andeven easier) than the proof of our trace theorem. In fact, one needs onlyresults which replace the ’atomic’ results, i.e. Lemma 5.2 and Lemma 5.4.The required assertions are:

i) Given a compact set K ⊆ G, and p ∈ [1,∞] there exists CK > 0and a compact set K ′ ⊆ H ′ = G/H, such that any f ∈ Lp(G) withsuppf ⊆ y + K for some y ∈ G one has suppTHf ⊆ y′ + K ′ for somey′ ∈ H ′, and ‖THf |Lp(H ′)‖ ≤ CK‖f |Lp(G)‖.

ii) The converse of i) is true, i.e. any f1 ∈ Lp(H ′) with supp f1 ⊆ y′ + K ′

for some y′ ∈ H ′ = G/H can be obtained in this way, i.e. f1 = THf,for some f as above, satisfying ‖f |Lp(G)‖ ≤ CK′‖f1|Lp(H ′)‖ for someCK′ > 0.

Assertions i) and ii) can be obtained without difficulties by an application ofthe methods developed in [14].

Remark 5.2 For W 1(G) = W (C0, L1) the closure of Cc(G) in W (L∞, L1)a corresponding result has been proved, using moderate (= functions of trans-lation type) dominants by Burger (see [8], Corollary 4.1).

Remark 5.3 By suitable modifications Theorem 5.6 can be even shown tohold true for arbitrary normal, closed subgroups H of G (cf. [38] VIII and[14] for the necessary modifications).


6 Modulation spaces on Euclidean spaces

In this concluding section the general results given above are used to showthat for a certain family of modulation spaces on Euclidean spaces one hasmany properties resembling those of the family of Besov spaces Bs

p,q(Rm)

(e.g. concerning duality, interpolation, embeddings, traces,...). Our family ofmodulation spaces includes the classical spaces Ls

2(Rm) of Bessel potentials,

the remarkable Segal algebra S0(Rm) (discussed in [18]), as well as the space

of (distributions defining) multipliers on S0(Rm) (cf. Lemma 6.3). In a sense

it is even the smallest of Banach spaces which contains these three spacesand which is closed with respect to duality and complex interpolation.

For convenience we shall write Mtf for the function x 7→ exp(2πi〈x, t〉)f(x)on Rm, t ∈ (Rm), in this section. Furthermore, we write ws for the moderatefunction x 7→ (1 + |x|)s, s ∈ Rm (ws is a weight function for s ≥ 0), and Lq

s

for Lqws

(Rm).

Definition 6.1 Given s ∈ R, 1 ≤ p, q ≤ ∞ we define, fixing anyk ∈ S(Rm), k 6= 0:

M sp,q(R

m) := {σ | σ ∈ S ′(Rm), Mtk ∗ σ ∈ Lp(Rm) for each t ∈ Rm,

and t 7→ ‖Mtk ∗ σ‖p ∈ Lqs(R

m)}.

Writing σ(p) for the control function of σ (with respect to Lp(Rm)),given by σ(p)(t) : = ‖Mtk ∗ σ‖p, the natural norm on M s

p,q(Rm) is given by:

‖f |M sp,q(R

m)‖ :=[∫

Rm|σ(p)(t)|qwq

s(t)dt]1/q

for 1 ≤ q < ∞,

and‖f |M s

p,∞(Rm)‖ : = supt∈Rm

|σ(p)(t)|ws(t).

A first summary of basic results is given in the following theorem:

Theorem 6.1 A) M sp,q(R

m) coincides with the modulation spaceM(Lp, Lq

s) (Rm), and any two norms obtained from different test func-tions k1, k2 ∈ S(Rm) define equivalent norms.


B) There are continuous embeddings S(Rm) ↪→ M sp,q(R

m) ↪→ S ′(Rm).

C) For 1 ≤ p, q < ∞, s ∈ R the space S(Rm) is dense in M sp,q(R

m). Inthat case one has

(M sp,q(R

m))′ = M−sp′,q′(R

m).

For s ≥ 0 the spaces M sp,q(R

m) are homogeneous Banach spaces ofquasi-measures.

D) For 1 ≤ p1, q1 < ∞, 1 ≤ p2, q2 ≤ ∞, s1, s2 ∈ R and θ ∈ (0, 1) one has

(M s1p1,q1

(Rm),M s2p2,q2

(Rm))[θ] = M sp,q(R

m),

with

1

p=

1− θ

p1

+θ

p2

,1

q=

1− θ

p1

+θ

q2

, s = (1− θ)s1 + θs2.

Proof. We discuss only A) and B), the other results being more or less imme-diate consequences of the corresponding results in §4. Due to the invarianceof S and S ′ under the Fourier transform assertion A) may be equivalentlydescribed by the embeddings

S ↪→ W (FLp, Lqs) ↪→ S ′

and the admissibility of k ∈ S(Rm) in the definition of the norm of W (FLp, Lqs).

Since ws is a weight function on Rm for s ≥ 0, it is not difficult to checkthat W (FLp, Lp

s) is a Banach convolution module over L1|s| (cf. [20], Theorem

3). By Proposition 3.1 any k ∈ W (FL1, L1|s|) defines an equivalent norm on

W (FLp, Lqs). An application of Proposition 3.3 yields the admissibility of any

k ∈ W (FL1|s|, L

1) for the definition of M sp,q(R

m). Since it is also obvious that

W (FL1, L1|s|) ↪→ W (FLp, Lq

s) for any p, q ≥ 1, and s ∈ R, it remains to show

in a separate lemma bellow the dense inclusion S ↪→ W (FL1s, L

1s) for any

s ≥ 0. In that fact, the density of S in W (FL1, L1s) can be used to derive

for any s ∈ R :

W (FLp, Lqs) ↪→ W (FL∞, L∞|s|) = W (FL1, L1

|s|)′ ↪→ S ′,

which gives the second required inclusion.


Lemma 6.2 For any s ∈ R one has S(Rm) ↪→ W (FL1, L1s)(R

m) is a densesubspace.

Proof. (cf. [35] for s = 0). One has S(Rm) ↪→ L1s(R

m) for s ∈ R, and forany suitable partial differential operator D (satisfying (1 + D)−1 ∈ L1(Rm),for example D = ∂2m/∂x2

1...∂x2m)

‖f |FL1(Rm)‖ ≤ CD(‖f‖1 + ‖Df‖1) for any f ∈ S(Rm)

Since (for any fixed ϕ0 ∈ D(Rm), ϕ0 6= 0), one has

‖f |W (FL1, Ls1)‖ ∼ ∫

Rm‖(Txϕ0)f‖FL1ws(x)dx

≤ ∫Rm

‖(Txϕ0)f‖1ws(x)dx +∫

Rm‖D[(Txϕ0)f ]‖1ws(x)dx,

and since the first expression is equivalent to ‖f‖1,ws (cf. [20], Remark 3) oneonly has to care for an estimate of the second term. Applying the product ruleone has (for suitably chosen sequences (Di

j)kj=1, i = 1, 2 of partial differential

operators):3

Observing that ϕj := D1jϕ0 is a test function and that D2

jf ∈ S(Rm) ⊆L1

s(Rm) for 1 ≤ j ≤ k one concludes

∫

Rm‖(Txϕj)D

2jf‖1ws(x)dx ∼ ‖D2

jf‖1,ws < ∞ for 1 ≤ j ≤ k,

which implies altogether the finiteness of ‖f |W (FL1, L1s)‖. The continuity of

translation in W (FL1, L1s) (cf. [20], Theorem 1) implies that f = limα eα ∗ f

for f ∈ FL1 ⋂Cc(R

m), for any L1|s| - bounded approximate unit (eα)α∈I in

D(Rm). Since D ∗ Cc(Rm) ⊆ D ⊆ S and since FL1 ⋂

Cc(Rm) is dense in

W (FL1, L1s) the density of S(Rm) is proved.

Before proceeding further let us mention that some of the spaces arisingin our family of modulation spaces coincide with very useful spaces that canalready be found in the literature (cf. [32], [41], [34], [44]-[47] and elsewhere).

3D[(Txϕ0)f ] =k∑

j=1

D1j (Txϕ0)D2

j f =k∑

j=1

Tx(D1j ϕ0)D2

j f.


Lemma 6.3 a) The spaces M s2,2(R

m) coincide with the spaces L(Rm) =Hs(Rm) of Bessel potentials on Rm;

b) The spaces M0p,p coincide with W (FLp, Lp). They arise as complex in-

terpolation spaces between M01,1(R

m), the Segal algebra discussed in

[18], and its dual S′0(R

m) = M0∞,∞(Rm) (cf. [17]).

c) The spaces of all distributions defining bounded convolution operators(=multipliers) on S0(R

m) coincides with M01,∞(Rm).

Proof.

a) follows from the chain FL2s = L2

s = W (FL2, L2s);

b) is a consequence of the invariance of the spaces W (FLp, Lp) under theFourier transform (cf. [21], Theorem 3.2) and the definition of modu-lation spaces;

c) is a transcription of the assertion that pointwise multipliers ofFS0(R

m) = S0(Rm) are exactly multiplications with elements of

W (FL1, L∞) (cf [23], Corollary 2.14).

The general trace theorem in §5, specialized to the family of modulationspaces considered in this section gives:

Theorem 6.4 For 1 ≤ p, q ≤ ∞, k, n ∈ N with k < m and s > k/q′ onehas

RRm−k(M sp,q(R

m)) = M s−k/q′p,q (Rm)

Proof. This result follows from Theorem 5.1, combined with the fact thatTRkws ∼ ws−k (cf. [14], §3).

Remark 6.1 From [14], combined with Theorem 5.1, one can also draw in-formation concerning traces of more refined scales of modulation spaces de-fined by means of weights ws,r : ws,r(x) = (1 + |x|)s logr(1 + |x|). Here thecritical index s = k/q

′(and r sufficiently large) is perhaps of particular in-

terest. Another result on restrictions concerns the case p = q = 1, s = 0,proved in [18] (Theorem 7.C).


Concerning embeddings between different metrics we can give the followingresult:

Proposition 6.5 Given 1 ≤ p1, p2, q1, q2 ≤ ∞, and s1, s2 ∈ R one has

M s1p1,q1

(Rm) ↪→ M s2p2,q2

if and only if

p1 ≤ p2 and

{q1 ≤ q2 and s1 ≥ s2, orq1 > q2 and s1 > s2 + 1/q2 + 1/q1(> s2)

Each nontrivial inclusion is a proper one, i.e. two modulation spaces coincideif and only if the corresponding parameters are all equal (i.e. p1 = p2, q1 = q2

and s1 = s2).

Proof. The necessity of p1 ≤ p2 follows from the fact that FLp1 is locallycontained in FLp2 if and only if p1 ≤ p2 (and strict if p1 < p2. cf. [25],[3]). Thesecond set of conditions in turn describes the inclusions between weightedlq spaces (involved in the discrete description of M s

p,q(Rm), cf. Corollary

4.2, v = ws). Since strict inclusion in the ’global’ components gives strictinclusions for the corresponding Wiener-type spaces (hence the modulationspaces; cf. Corollaries 4.2 and 4.4) the last assertion can easily be verified.

Remark 6.2 We do not give a comparison with the family of Besov spacesBs

p,q(Rm), because such results will be given elsewhere in detail (cf. [27]).

Comparisons with potential spaces are given below.

That the modulation spaces under consideration behave quite nicely underthe transformation (= convolution) by Bessel potentials is the content of thenext result (cf. [46], [41], [31] for conventions):

Theorem 6.6 For 1 ≤ p, q ≤ ∞, s1, s2 ∈ R given, the Bessel poten-tial with kernel Gs, s = s2 − s1, characterized by its Fourier transformGs(t) = (1 + 4π2|t|2)−s/2, gives rise to an isomorphism between M s1

p,q(Rm)

and M s2p,q(R

m). More precisely

M s2p,q(R

m) = Gs ∗M s1p,q(R

m), and

‖Gs ∗ f |M s2p,q‖ ∼ ‖f |M s1

p,q‖.


Proof. Although it would be possible to give a direct proof we prefer (forreasons of shortness) to make use of the corresponding ’lifting’ property forBesov spaces; Gs ∗Br

p,q = Bs+rp,q , with equivalence of the corresponding norms

(cf. [46], 2.6.2). Using this fact it follows that one has for any compact setQ ⊆ Rm and for any f ∈ Lp(Rm), suppf ⊆ t + Q :

‖Gs ∗ f |M s2p,q‖ ∼ (1 + |t|)s2‖Gs ∗ f‖p ∼ ‖Gs ∗ f |Bs2

p,q‖

∼ ‖f |Bs2−sp,q ‖ ∼ (1 + |t|)s2−s‖f‖p ∼ ‖f |M s1

p,q‖.

In view of Corollaries 4.2 or 4.4 the proof of the theorem is complete.

Remark 6.3 Theorem 6.6 gives an alternative approach to the spaces M sp,q

(Rm) for s ≥ 0. By defining first M0p,q(R

m) := F−1(W (FLp, Lq)) one coulddefine M s

p,q := Gs ∗ M0p,q. Observe, that it would be sufficient to take the

Fourier transform in the sense of S ′0(Rm) (cf. [17]), because W (FLp, Lq) ⊆

S ′0. This opens the way to generalizations to lca groups.

Proposition 6.7 For q1 ≤ min(p, p′) and q2 ≥ max(p, p′) one has

M sp,q1

(Rm) ↪→ Lps(R

m) ↪→ M sp,q2

(Rm).

Proof. In view of Theorem 6.6 it will be sufficient to check the result fors = 0. For 1 ≤ p ≤ 2 we have by Theorem 3.2 of [21] (or Theorem 3.5 above)and the ordinary Hausdorff-Young inequality:

M0p,q1

= F [W (FLp, Lp)] = W (FLp, Lp) ⊆ W (Lp′ , Lp) ⊆ Lp,

and on the other hand one has FLp ⊆ W (FLp, Lp′) for 1 ≤ p ≤ 2 (checkfirst p = 1, 2 and interpolate), and therefore Lp ⊆ FW (FLp, Lp′) = M0

p,q2.

The result for p ≤ 2 then follows by dualization.

There is also a variant of Sobolev’s embedding theorem for modulationspaces (cf. [32], [41], V.2.2, [34] §4 or [44], §2.8).

Proposition 6.8 For s > m/q′ one has M sp,q(R

m) ↪→ C0(Rm).


Proof. For s > m/q′ one has Lqs(R

m) ↪→ L1(Rm) by Holder’s inequality.Therefore (one has even) by Theorem 3.2 of [21]

M sp,q = F [W (FLp, Lq

s)] ⊆ F [W (FLp, L1)] ⊆ W (FL1, Lp) ⊆ W (C0, Lp) ⊆ C0.

Some of the modulation spaces are also pointwise Banach algebras (for relatedassertions concerning Besov spaces or Sobolev spaces cf. for example [46],§2.6.2).

Proposition 6.9 For 1 ≤ p ≤ 2 and s > m/q′, or s ≥ 0 and q = 1, thespaces M s

p,q(Rm) are Banach algebras with respect to pointwise multiplication.

Proof. For s > m/q′ we have Lqs(R

m) ↪→ L1(Rm). Since ws is a weaklysubadditive function on Rm this implies that Lq

s(Rm) is a Banach convolution

algebra (see [14], Corollary 3.8).

Thus Theorem 3 of [20] implies

W (FLp, Lqs) ∗W (FLp, Lq

s) ⊆ W (L1, Lqs) ∗W (FLp, Lp

s) ⊆ W (FLp, Lqs).

The case q = 1, s = 0 is easily checked separately. Applying the Fouriertransformation one obtains the required results.

Remark 6.4 The above result even remains true for 2 ≤ p ≤ ∞. In fact, theconvolution of σ1, σ2 ∈ FL∞ (well defined as a product of pseudomeasures)belongs to FL∞. By interpolation with the corresponding result for p = 2 thegeneral assertion follows.

Another result involving pointwise multiplications concerns the compact-ness of pointwise multipliers between modulation spaces. We only give atypical result in this direction:

Proposition 6.10 For p ≤ 2, s ∈ R, any f ∈ M|s|2,1 defines a compact (point-

wise) multiplier from M sp,q into M r

1,q, for any r < s.

Proof. Applying Theorem 3 of [20] and Plancherel’s theorem (implyingL2 ∗ L2 ⊆ FL1) one obtains for g ∈ M s

p,q :

F(fg) = (Ff) ∗ (Fg) ∈ W (FL2, L1|s|) ∗W (FLp, Lq

s)

⊆ W (L2, L1|s|) ∗W (L2, Lq

s) ⊆ W (FL1, Lqs).


It follows therefrom that f induces a compact multiplication operator fromM s

p,q into M s1,q. Since f can be approximated in M s

2,1 by elements with com-pact support (cf. Theorem 4.6.C) the tightness of S in M s

1,q,4 for any bounded

subset S ⊆ M sp,q is obvious. Since the unit ball of M s

1,q is equicontinuous inM r

1,q for r < s (the unit ball of W (FL1, Lrq) being tight in W (FL1, Lr

q) forr < s) the assumptions of the compactness criterion (Corollary 4.7) are sat-isfied, and the proof is complete.

Remark 6.5 The above result applies in particular to any f ∈ L2t , with

t > s + m/2.

Concerning multipliers between modulation spaces we shall describe thesecases where the space of multipliers itself coincides with some modulationspace. Writing HG(B1, B2) for the space of all multipliers (i.e. boundedlinear operators commuting with all translation operators Ty, y ∈ G) formB1 to B2 we have:

Theorem 6.11 For 1 ≤ p, r, t ≤ ∞, and s, s′ ∈ R one has

HRm(M s1,r,M

s′p,t) = M s′−s

p,q , for1

q= max

(0,

1

t− 1

r

).

Proof. Via the Fourier transform we may identify the space of multiplierswith the space of pointwise multipliers from W (FL1, Lr

s) to W (FLp, Ls′t ).

Since HRm(L1, Lp) ∼= Lp(Rm) for 1 < p ≤ ∞, and since HRm(L1, L1) ∼=M(Rm) still implies that any Fourier multiplier (h = µ ∈ FM(Rm)) belongslocally to FL1(Rm), the required follows from Theorem 2.11 of [23] andsimple results concerning ’coordinatewise’ multipliers of weighted sequencesspaces.

Remark 6.6 One can even show for any s ∈ R that HRm(M s1,q, L

1) ∼= M−s1,q′

for 1 ≤ q < ∞, and

HRm(L1,M s1,q)

∼= M s1,q for 1 < q ≤ ∞ or q = 1 and s > m.

4hence in Mr1,q for r ≥ s.


Proofs will be given elsewhere.

We conclude this section with few results concerning the behaviour ofmodulation spaces under the Fourier transform. Again we state only typicalspecial cases.

Proposition 6.12 A) For any p, 1 ≤ p ≤ ∞ , the Fourier transformmaps M0

p,p(Rm) onto itself.

B) For 1 ≤ p ≤ 2 one has F [M sp,q(R

m)] ⊆ W (Lp′ , Lqs)(R

m).

Proof. Assertion A) is just Theorem 3.2 of [21], or a transcription of a specialcase of Theorem 3.5 above. B) follows easily from the ordinary Hausdorff-Young inequality which implies W (FLp, Lq

s) ↪→ W (Lp′ , Lqs) for 1 ≤ p ≤ 2.

Remark 6.7 Recall that A) is obtained from the invariance of S0(Rm) =

M01,1(R

m) by duality and interpolation. Theorem 3.5 also shows that thespaces M(Lp

w, Lpw) are invariant as well.

Remark 6.8 It is worth mentioning that assertions similar to A) are notavailable for Besov spaces. Otherwise expressed, modulation spaces showmore symmetry with respect to the Fourier transformation than do Besovspaces.

Concluding remarks

1) It is also possible to consider modulation spaces for 0 < p, q < 1 (theseare quasi-Banach spaces). Furthermore it is possible to consider notonly ’uniform’ variants of Besov spaces, but it makes also sense toconsider ’uniform’ variants of Triebel’s F s

p,q -spaces. Some results inthis direction are given in [48]. There also a trace theorem for F-spacesis given.

2) A general method of constructing Banach spaces of distributions de-fined by quite arbitrary decompositions of the Fourier transforms oftheir elements is given [23]. (cf. also [26]). The results of this papergive some idea of the joint aspects of Besov spaces and modulation


spaces. It also contains various examples, including a family of ’inter-mediate’ smoothness, spaces in between these two ’extreme’ functionalclasses. (cf. also [24]).

3) At least a good deal of the results of §6 also applies, suitable modified,to the corresponding spaces over p− adic fields or Vilenkin-groups. Thecorresponding theory of Lipschitz- or Besov spaces and even potentialspaces is described in Taibleson’s book [42], and in the work of Bloom,Onneweer, Quek and Yap, for example (only to mention[4,5,33,36,37]).We even dare to say that modulation spaces are perhaps more naturalas a mean of describing smoothness of functions on general lca groups,at least if no dilation is available.

4) The family of modulation spaces also seems to be a reasonable tool forthe description of pseudodifferential operators. We hope to come backto this subject elsewhere.

5) The choice v(t) = exp(s|t|d), 0 < d < 1, s ∈ R, leads to Banach spacesof ultradistributions on Rm (cf. [2], [45]) (i.e. tempered distributionsare no more sufficient in that case). Furthermore, our presentationincludes isotropic as well as anisotropic spaces, depending on the factwhether v is radially symmetric or not (e.g. v = ⊗m

i=1vi, which shouldlead to spaces related to Besov spaces with dominating mixed derivates)( see [32], [47]).

6) Although our general results are formulated for general lca groups wedid not make use of structure theory for these groups. On the otherhand, our approach makes clear that it is possible to introduce on Rm areasonable family of Banach spaces of smooth functions or distributionswithout recurrence to the differentiability or the dilation structure ofthe Euclidean n− space.


7 “Historical” comments, recent literature

Up to this paragraph the paper is the unchanged copy of the original re-port written in 1982/83, which was submitted twice, but not accepted forpublication.5 Only a few symbols where exchanged by those used by mostauthors nowadays. The motivation for publishing this report now was thefact that it has been cited at various places, hence should be available forthe sake of future reference in that particular form. Moreover, there is gener-ally increasing interest in modulation spaces (at least 4 of the contributionsin [75] make explicit use of modulation spaces). In fact, different authorsin the treatment of diverse mathematical problems repeatedly confirm thatmodulation spaces constitute the family of Banach spaces of distributionswhich frequently appear as the appropriate choice for the description of thetime-frequency behaviour of functions, distributions or operators.

The reason why this report was not published earlier lies partially in theauthors point of view that most of the properties of modulation spaces wheresubsumed already in the general theory of coorbit spaces as developed latertogether with K. Grochenig (described in [65, 66] and subsequent papers).However, in retrospect one has to admit that the knowledge of modulationspaces in conjunction with the constructions of wavelets (occuring in thelate 80’s) was very important in order to develop the general coorbit the-ory. At this point the positive influence of the work of A. Grossmann (e.g.,[91, 90]) has to be acknowledged. He already took a strong group-theoreticalpoint of view, showing that the continuous wavelet transform and the shorttime Fourier transforms are special cases of (square) integrable group repre-sentations. In conjunction with the theory of Wiener amalgam spaces (see[20, 21, 93]) over the corresponding lc. groups 6 it was possible to develop aunified approach to atomic decomposition results (in [64, 65, 66]).

Moreover, thinking of the situation around 1986-89, let us recall thatthis was the time when the “wave of wavelets” took off, inspired by theconstruction of orthogonal wavelet systems by Y. Meyer and his coauthors(see [97, 98, 99, 101, 100, 102] for a list of very early publications). The author

5Most likely the main reason for these rejections was the fact that the manuscript wasreferring to various - at that time - unpublished papers (such as [2, 22, 23, 27, 48]).

6The term Wiener-type spaces as used in the “original part of this paper” has beenreplaced subsequently by the more specific term “Wiener amalgam spaces”.


remembers a very inspiring, short visit to Yves Meyer in Paris, in February1987, which was another strong motivation to work in that direction. Keepingup with those new developments appeared to be more exciting then rewritingold notes.

The fact that the 1983–version of this paper contains a substantial amountof still unpublished results (e.g. , the trace theorems) is more importantreason for our aim to publish it than the “historical aspects” are. Only thepaper [57] has been destilled from the results of the old report. It describesthe generalized Fourier transform (in the context of [ultra-] distributions),proposing a collection of Fourier invariant Banach spaces of distributions.This level of generality subsequently becomes the basis for the generalizationof irregular sampling algorithms to Banach spaces of band-limited functionsover general lca. groups (cf. [73]).

From the very beginning modulation spaces have been treated in analogyto the corresponding theory of Besov spaces, essentially by replacing thedyadic decompositions of unity by uniform ones. One of the reasons to dothis was the construction of “smoothness spaces” over general lca. groupswhich don’t have dilation. This is of course possible by replacing dilation by“frequency shifts” (or modulation operators). Starting from this analogy itwas natural to go first for an atomic approach corresponding to the Frazier-Jawerth expansions for Besov-Triebel-Lizorkin spaces (cf. [78, 79]). This wasdone in [58]. Once Daubechies, Jaffard and Journe published the constructionof Wilson basis [54] it was become that they are unconditional bases for thefamily of modulation spaces (cf. [68] or [104]).

In the mid eighties the connection to time-frequency analysis, the roleof the Schrodinger representation of the reduced Heisenberg group in thiscontext, and the relation to Gabor’s idea of a “series expansion of arbitraryfunction in terms of time-frequency shifted copies of a template” becametransparent. At this point the author has to thank his colleagues from thetechnical university Vienna (in particular Franz Hlawatsch) for pointing outthe relevance of Gabor’s work [80] in the present context. As a consequence,contacts to A.J.E.M. Janssen has intensified very much over the years. Hewas the first (e.g. in [95]) to analyze the informal classical approach suggestedby D. Gabor in a strict mathematical way by using (tempered) distributions.Of course, the work of I. Daubechies (on localization operators) also has to


be acknowledged here (see [53]).

With the appearance of wavelets, which where shown to be unconditionalbases for the whole family of Besov-Triebel-Lizorkin spaces (cf. [82]) it be-came immediately clear that there must be a strong analogy between thosefunction spaces and their unconditional bases and what is nowadays known asBanach frames of Gabor type for modulation spaces. As already mentionedthe unifying approach of the group theoretical framework, covering bothcases (as well as others) was developed in the late eighties by Feichtinger andGrochenig, who introduced coorbit spaces with respect to a given irreducible,integrable group representation (cf. [64, 65, 66, 83]). Only subsequently ithas been realized that the atomic decompositions which are obtained in thecase by using the reduced Heisenberg group and the Schrodinger representa-tion, or in the terminology of engineers a collection of time-frequency shiftsof a given “father wavelet” (or atom, such as the Gaussian), is exactly whatD.Gabor has proposed in his paper of 1946, at least in spirit (see [67] for atranslation between the two contexts).

Nowadays one may recommend K. Grochenig’s book [85], maybe in con-junction with G. Folland’s book [76], as an introduction to this branch ofmathematical analysis. Grochenig’s book offers - to mathematicians at least- an optimal (self-contained) introduction into the field, covering the fine de-tails of Gabor Analysis and the use of modulation spaces in this context, up topseudo-differential operators. The relevance of modulation spaces in this con-text became evident in [86] where Grochenig and Heil demonstrate that themodulation spaces can be used in the generalization of the classical Calderon-Vaillancourt theorem. A whole stream of publications is presently followingthis direction, starting from the pioneering work of Tachizawa [109, 110, 107],up to the recent papers by Heil and Grochenig [87] in this volume giving alsothe overview over the present state. There is also work concerning pseudodif-ferential operators on modulation spaces by Benyi and Okoudjou [49], Hoganand Lakey [94], Labate [96], Pilipovic and Teofanov [104, 106, 105] and Toft[113], for example. Again one can point to Grochenig’s book [85], Chap.14,as a suitable introduction to the field.

Given the interest in modulation spaces it is natural that there is anincreasing interest in embedding results between modulation spaces and other(maybe more well known) Banach spaces of functions. The first sharp results


in this direction appeared already in PhD thesis of Peter Grobner (Vienna,1992, [27]). More recent results are obtained by K. Okoudjou [103] and Toft[112, 113], but one should also mention [86] and [94] in this context.

Modulation spaces are also the “appropriate class of function spaces”when it comes to the description of the boundedness properties of Gabormultipliers (depending on the ingredients), cf. [72] and the references giventhere. They also take their role in the description of quantization of TF-expansions ([115]) or (in an ongoing project) to describe functions of variableband-width.

Final remark: With the time going on (and the report reproduced abovehaving been not available to many authors in the past) there has been somediffuseness in the use of the word “modulation spaces” recently. The twomain interpretations are either that modulation spaces are defined as “Wieneramalgams on the Fourier transform side” (which was the original viewpoint)or (what we suggest to become the standard interpretation for the future) tocall “modulation space” a space which arises as coorbit space with respect tothe Schrodinger representation of the reduced Weyl-Heisenberg group, whichis equivalent to say that members of a modulation space are characterizedby the fact that their short-time Fourier transforms (say with respect to theGaussian window) belong to some solid and translation invariant Banachspace of measurable functions over the time-frequency plane (a continueddiscussion on such matters is planned on the authors Web-pages, cf. [61]).

Acknowledgement: The author would like the editors of this volume fortheir patience and the willingness to accept this long report, and give in-terested readers access to it in this form after 20 years. My thanks alsogo to Nenad Teofanov from the University of Novi Sad who was very help-ful converting the old manuscript into TEX format and giving it a pleasantoutlook.


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Modulation Spaces on Locally Compact Abelian Groups Hans …...Modulation Spaces on Locally Compact Abelian Groups Hans G. Feichtinger 1 Introduction The modulation spaces Ms p;q(R