Sawanoman2.dvi


CUBO A Mathematical Journal

Vol.12, No¯ 03, (187–202). October 2010

Modulation Spaces

with

Aloc∞ -Weights

YOSHIHIRO SAWANO

Department of Mathematics, Kyoto University,

Kyoto, 606-8502, Japan,

email: yosihiro@math.kyoto-u.ac.jp

ABSTRACT

In this paper we describe the function space M
s,w
p,q with w ∈ A

loc
∞ together with some related

results of weighted modulation spaces.

RESUMEN

En este artículo describimos el espacio de la funciones M
s,w
p,q con w ∈ A

loc
∞ junto con algunos

resultados relacionados a espacios de modulación con peso.

Key words and phrases: Modulation spaces, Exponential weights.

Math. Subj. Class.: 41A17, 42B35.



188 Yoshihiro Sawano CUBO
12, 3 (2010)

1 Modulation Spaces

Modulation spaces, which were initiated by Feichtinger in 1983 (see [5]), were investigated

for the purpose of measuring smoothness of functions and distributions in a way other than

Besov spaces. Besov spaces as well as Triebel-Lizorkin spaces are very close to Sobolev spaces

and are used in partial differential equations. These spaces are defined by way of dilations.

Feichtinger took full advantage of the group structure of Rn. Recall that Rn carries the struc-

ture of a Lie group not with dilation but with addition. Therefore, it seems natural that we

consider the translation.

The goal of the present paper is to combine the results in [17, 21]. The main results of

[21] can be summarized as follows : Quite a few of the results of usual modulation spaces

M sp,q carries over to the A
loc
∞ -weighted cases with 0 < p, q ≤ ∞. In the present paper we shall

establish the following results on modulation spaces. To describe the result, we make a setup.

Assume that W : Rn → (0,∞) is a measurable function with Aloc∞ condition: There exists
1 < P < ∞ such that W satisfies the Aloc

P
condition

sup
Q:cube

(

1

|Q|

ˆ

Q

W (x) dx
) (

1

|Q|

ˆ

Q

W (x)−
1

P−1 dx

)
1

P−1
< ∞. (1.1)

Suppose that the parameters p, q, s satisfy

0 < p < ∞, 0 < q < ∞, s ∈ R. (1.2)

Fix a window function ϕ ∈ C∞c so that it satisfies the non-degenerate condition:
ˆ

Rn

ϕ(x) dx 6= 0, supp(ϕ) ⊂ [−1, 1]n. (1.3)

We write ϕm,x(z) = exp(2πim · z)ϕ(z − x) for m ∈ Zn and x ∈ Rn. We define

∥

∥

∥ f : M
s,W
p,q

∥

∥

∥

g
>

(

∑

m∈Zn
〈m〉q

(
ˆ

R
n

|〈f ,ϕm,x〉|pW (x) dx
)

q
p

)
1
q

(1.4)

for f ∈ C∞c , where we write 〈x〉 =
√

1 +|x|2.

Theorem 1. Assume (1.1) and (1.2). Then different choices of admissible ϕ satisfying (1.3)

will yield equivalent norms. That is, if ϕ1,ϕ2 satisfy (1.3), then the norm equivalence
∥

∥

∥ f : M
s,W
p,q

∥

∥

∥

ϕ1
≃

∥

∥

∥ f : M
s,W
p,q

∥

∥

∥

ϕ2
(1.5)

holds for f ∈ C∞c (R
n).



CUBO
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Modulation Spaces with Aloc∞ -Weights 189

In view of (1.5), we shall write
∥

∥

∥ f : M
s,W
p,q

∥

∥

∥ instead of
∥

∥

∥ f : M
s,W
p,q

∥

∥

∥

g
.

As for this (new) modulation norm
∥

∥

∥ f : M
s,W
p,q

∥

∥

∥, we have the following quantitiave infor-

mation.

Lemma 1. There exist C > 0 and N ∈ N depending only on W and p, q, s such that

|〈f ,ψ〉| ≤ C
∥

∥

∥ f : M
s,W
p,q

∥

∥

∥

∑

|α|≤N
sup
x∈Rn

e
N|x||∂αψ(x)| (1.6)

holds for all ψ ∈ C∞c .

Denote by M s,Wp,q the (abstract) completion of C
∞
c with

∥

∥

∥ f : M
s,W
p,q

∥

∥

∥

g
. In view of (1.6), we

see that M s,Wp,q is a subset of D
′ satisfying

|〈f ,ϕ〉| ≤ C
∑

|α|≤N
sup
x∈Rn

e
N|x||∂αϕ(x)| (1.7)

holds for all ϕ ∈ C∞c .

In the present paper we shall prove the molecular decomposition suitable for M s,Wp,q .

Definition 1 (Molecule, Atom). Let s ∈ R.

1. Suppose that K , N ∈ N are large enough and fixed. A CK -function τ : Rn → C is said to be
an (s; m, l)-molecule, if it satisfies

|∂α(e−im·xτ(x))| ≤ 〈m〉−s e−N|x−l|, x ∈ Rn

for |α| ≤ K .

2. Suppose that K , N ∈ N are large enough and fixed. A CK -function τ : Rn → C is said to be
an (s; m, l)-atom, if it satisfies

|∂α(e−im·xτ(x))| ≤ 〈m〉−sχl+[−2,2]n , x ∈ Rn

for |α| ≤ K .

3. Also set

M
s := {{Ψs

ml
}m,l∈Zn : each Ψ

s
ml

is an (s; m, l)-molecule}

A
s := {{as

ml
}m,l∈Zn : each a

s
ml

is an (s; m, l)-atom}.

Next, we introduce a sequence space m p,q to describe the condition of the coefficients of

the molecular decomposition.



190 Yoshihiro Sawano CUBO
12, 3 (2010)

Definition 2 (Sequence space m p,q ). Let 0 < p, q ≤ ∞. Given a sequence λ = {λml }m,l∈Zn ,
define

‖λ : mWp,q‖ >





∑

m∈Zn

{

ˆ

Rn

∣

∣

∣

∣

∣

∑

l∈Zn
λmlχl+[0,1]n (x)

∣

∣

∣

∣

∣

p

W (x) dx

}

q
p





1
q

.

Here a natural modification is made when p and/or q is infinite. mWp,q is the set of doubly

indexed sequences λ = {λml }m,l∈Zn for which the quasi-norm ‖λ : mWp,q‖ is finite.

With these definitions in mind, we present a typical result in [21].

Theorem 2. Assume (1.1) and (1.2).

1. If λ = {λml }m,l∈Zn ∈ m
s,W
p,q and {Ψ

s
ml

}m,l∈Zn ∈ M s, then

f :=
∑

m,l∈Zn
λml ·Ψsml (1.8)

converges unconditionally in the topology of M
s,W
p,q .

2. There exists {as
ml

}m,l∈Zn ∈ A s such that any f ∈ M
s,W
p,q admits the following decomposition:

f =
∑

m,l∈Zn
λml · asml , (1.9)

where λ = {λml }m,l∈Zn satisfies

‖λ : ms,Wp,q ‖ ≤ C ‖f : M
s,W
p,q ‖ (1.10)

for some C > 0 independent of f .

In the early 90’s, more and more applications were found out. For example, time-fre-

quency analysis, which is a branch of signal analysis, deals with the translation and the

modulation, so that modulation spaces come into play naturally.

Also, it is remarkable that modulation spaces are applied effectively to the pseudo-

differential operators by Sjöstrand, Tachizawa and many researchers [12, 14, 15, 19, 22, 23,

24, 25]. Modulation spaces are applicable to various partial differential equations. For exam-

ple, Baoxiang and Chunyan used modulation spaces to investigate the KdV equation (see [3]).

Recently modulation spaces can be applied even to the modeling of wireless channels and the

quantum mechanics [2].

Now we describe the organization of this paper. In Section 2 we describe other weighted

modulation spaces and compare them with ours. Section 3 is devoted to establishing Theorem

1 as well as Lemma 1. Section 4 is intended as the proof of Theorem 2. In Section 5 we

consider the weighted modulation space M s,Wp∞ . Finally in Section 6 we present some examples.



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Modulation Spaces with Aloc∞ -Weights 191

2 Various Weighted Modulation Spaces

Based on the standard notation of signal analysis, we adopt the following notations.

Ta f (x) := f (x − a), Mb f (x) := e
ib·x

f (x), a, b ∈ Rn, f ∈ S ′.

Fix ϕ ∈ C∞c be a positinve non-zero function. Then define

‖f : M sp,q‖ >
(

ˆ

Rn

(
ˆ

Rn

|〈f , M y Txϕ〉|p dx
)

q
p

〈y〉s q d y
)

1
q

for s ∈ R and 1 ≤ p, q ≤ ∞. Denote by M sp,q the set of all tempered distributions f ∈ S
′ for

which the norm is finite. An important observation is that the function space M sp,q does not

depend on the specific choices of g. For more details we refer to [11, 18].

In general by weighted modulation norm we mean the following norm given by

‖f : Mvp,q‖ >
(

ˆ

Rn

(ˆ

Rn

|〈f , M y Txϕ〉|p v(x, y) dx
)

q
p

d y

)
1
q

.

Note that M sp,q is recovered by setting v(x, y) = 〈y〉
s q. There are many important classes of

weights.

Definition 3.

1. A weight v : R2n → [0,∞) is said to be a submultiplicative, if there exists a constant C > 0
such that v(x + y) ≤ C v(x) v( y) for all x, y ∈ R2n.

2. A weight v : R2n → [0,∞) is said to be subconvolutive, if v−1 ∈ L1(R2n) and v−1∗v−1 ≤ c v−1

for some constant c > 0.

3. A weight v : R2n → [0,∞) is said to satisfy the Gelfand-Raikov-Shilov condition, if

lim
n→∞

v(n x)
1
n = 1

for all x 6= 0.

4. A weight v : R2n → [0,∞) is said to satisfy the Beurling-Domar condition, if
∞
∑

j=1

log v(n x)

n
< ∞.

5. A weight v : R2n → [0,∞) is said to satisfy the logarithmic integral condition, if
ˆ

|x|≥1

log v(x)

|x|n+1
dx < ∞.



192 Yoshihiro Sawano CUBO
12, 3 (2010)

Example 1.

1. The function eα|x| with α ≥ 0 is a submultiplicative weight. Similarly 〈x〉α with α ≥ 0 is
a submultiplicative weight.

2. The function 〈x〉n+ε is a subconvolutive weight.

We refer to [7] for more details of the submultiplicative, moderate and subconvolutive

weights not only on Rn but also on locally compact aberian groups.

Proposition 1. [13] The Bourling-Domar condition is stronger than the Gelfand-Raikov-

Shilov condition.

Proof. This is just an easy consequence of the fact that the limit of a positive summable

sequence is zero.

In the present paper, we consider weights of the form

v(x, y) = W (x)〈y〉s,

where s ∈ R and W belongs to the class Aloc∞ described just below. As the example W (x) =
|x|α, α > −n shows, it can happen that v fails the submultiplicative condition or subconvolutive
condition. Another similar example shows that v does not necessarily satisfy the Bouring-

Domar condition.

Before we go further, we recall the definition of Alocp -weights. In the sequel by a “weight",

we mean a non-negative measurable function W ∈ L1
l oc

satisfying 0 < W < ∞ for a.e. and we
define the local maximal operator Mloc by

M
loc

f (x) := sup
x∈Q

Q : cube,|Q|≤1

1

|Q|

ˆ

Q

|f ( y)| d y.

Let 1 ≤ p < ∞. Then we define

A
loc
p (W )=























ess. sup
x∈Rn

MlocW (x)

W (x)
if p = 1

sup
Q : cube
|Q|≤1

(ˆ

Q

W (x)
dx

|Q|

)

·
(ˆ

Q

W (x)
1

1−p
dx

|Q|

)p−1
if 1 < p < ∞.

The quantity Alocp (W ) is called the A
loc
p -norm of W . Then it is easy to see that

A
loc
p (W ) ≤ A

loc
q (W ), 1 ≤ q ≤ p < ∞.



CUBO
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Modulation Spaces with Aloc∞ -Weights 193

The class Alocp of weights is the set of all weights W for which the norm A
loc
p (W ) is finite. We

also define

A
loc
∞ :=

⋃

1≤p<∞
A

loc
p .

We remark that |x|−n+ε ∈ Aloc1 for all 0 < ε < n and that e
α|x| ∈ Aloc1 for all α ≥ 0.

Let W be a weight. Then we define

‖f : LWp ‖ >
(ˆ

Rn

|f (x)|pW (x) dx
)

1
p

, 1 ≤ p < ∞.

Here and below we assume that W ∈ Aloc
P

with 1 ≤ P < ∞ for the sake of definiteness.

3 Proof of Theorem 1

Now we prove Theorem 1 and Lemma 1. Before we prove Theorem 1, we first establish an

auxiliary result (Proposition 2) and then we prove Theorem 1. Proposition 2 will be strength-

ened after we prove Lemma 1.

3.1 An auxiliary result on maximal operators

We write

pN (ψ) >
∑

α∈Zn+,|α|≤N
sup
x∈Rn

e
N|x||∂αψ(x)|

for ψ ∈ C∞c .

Proposition 2. Let k ∈ Z, N > 0 and 0 < η ≤ 1. Then we have

sup
ψ∈C∞c

pN (ψ)≤1

|Mkψ∗ f (x)|
η ≤ c

∑

l∈Z

ˆ

Rn

|Mlϕ∗ f (x − y)|η

〈k − l〉Nη eNη|y|
d y (3.11)

for all f ∈ C∞c .

Proof. First let us consider the case when η = 1. Note that
∑

l∈Zn
F ϕ(x + l)2

= (2π)−
n
2

∑

l∈Zn
F [ϕ∗ϕ](x + l)

= (2π)−
n
2

∑

m∈Zn

(

ˆ

Rn

∑

l∈Zn
F [ϕ∗ϕ]( y + l) exp(−2πi y · m) d y

)

exp(2πix · m)



194 Yoshihiro Sawano CUBO
12, 3 (2010)

>
∑

m∈Zn
ϕ∗ϕ(−2πm) exp(2πix · m) ≡ ϕ∗ϕ(0)

from the Poisson summation formula. Consequently we obtain

Mkψ∗ f = cn
∑

l∈Z
Mkψ∗ Mlϕ∗ Mlϕ∗ f . (3.12)

Now we shall estimate each summand. First of all, a repeated integration by parts yields that

for all N > 0 there exists c = cN > 0 such that

|Mkψ∗ Mlϕ( y)| ≤ c〈k − l〉−N e−N|y|.

As a consequence we obtain

|Mkψ∗ Mlϕ∗ Mlϕ∗ f (x)| ≤ c〈k − l〉−N
ˆ

Rn

e
−N|y||Ml ϕ∗ f (x − y)| d y.

Inserting (3.12), we obtain the result when η = 1. Namely we have proved

|Mkψ∗ f (x)| ≤ c
∑

l∈Z
〈k − l〉−N

ˆ

Rn

e
−N|y||Ml ϕ∗ f (x − y)| d y (3.13)

up to this point. Of course, the constant c > 0 does depend on N. Now we pass to the case
when 0 < η < 1. We define

MN,k f (x) := sup
ψ∈C∞c , pN (ψ)≤1

y∈R, l∈Z

|Mlψ∗ f (x − y)|
〈k − l〉N eN|y|

.

Then from (3.13) we deduce

MN,k f (x) ≤ c sup
y∈R
l∈Z

(

1

〈k − l〉N eN|y|
∑

m∈Z

ˆ

|Mmϕ∗ f (x − y − z)|
〈m − l〉N eN|z|

d y

)

≤ c sup
y∈R

(

∑

m∈Z

ˆ

|Mmϕ∗ f (x − y − z)|
〈m − k〉N eN|y+z|

d z

)

≤ c MN,k f (x)1−η
∑

m∈Z

ˆ

|Mmϕ∗ f (x − y)|η

〈m − k〉Nη eNη|y|
d y.

Here we have used the Peetre inequality 〈a + b〉 ≤
p

2〈a〉·〈b〉. As a result, we obtain

|Mkψ∗ f (x)|η ≤ MN,k f (x)η ≤ c
∑

m∈Z

ˆ

|Mmϕ∗ f (x − y)|η

〈m − k〉Nη eNη|y|
d y,

since MN,k f (x) < ∞.



CUBO
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Modulation Spaces with Aloc∞ -Weights 195

Proposition 3. Let W ∈ Aloc
P

and F : Rn → [0,∞) a measurable function. Then we have

{

ˆ

R
n

(
ˆ

R
n

F(x − y)η
d y

eBη|y|

)
p
η

W (x) dx

}
1
p

≤ C
(
ˆ

R
n

F(x)pW (x) dx
)

1
p

(3.14)

for all p > Pη and B ≫ 1.

Proof. By replacing p/η with p, we can assume that η = 1 and p > P. Let ℓ ≥ 1. We denote
χr =

χ(−r,r)n

rn
. Then define Mloc≤ℓ f (x) > sup

r≤ℓ
χr ∗|f |(x). Then there exists α > 0 such that

(
ˆ

R
n

M
loc
≤ℓ f (x)

p
dx

)
1
p

≤ eαℓ
(
ˆ

R
n

|f (x)|p dx
)

1
p

. (3.15)

Indeed, this inequality is true for ℓ = 1 by the definition of Aloc
P

. Since χr ∗χ1 ≥ χr+1 for r ≥ 1,
we have

M
loc
≤k ≤ (M

loc
≤1 )

k.

As a consequece, we obtain (3.15).

Once we establish (3.15), (3.14) is an easy consequence of inequality
ˆ

R
n

F(x − y)e−B|y| d y ≤
∞
∑

j=1

ˆ

(−2 j ,2 j )n
F(x − y)e−2

j−1B
d y

≤ 2n
∞
∑

j=1
e
−2 j−1B

M
loc
≤2 j F(x).

The proof is therefore complete.

3.2 Proof of Theorem 1

Let W ∈ Aloc∞ throughout. Then define

‖f m : l q (LWp )‖ >
(

∑

m∈Zn
‖f m : LWp ‖

q

)
1
q

for a family of measurable functions { f m}m∈Zn . Let 0 < p, q ≤ ∞ and s ∈ R. Then the modula-
tion norm (1.4) can be written as

‖f : M s,Wp,q ‖ >
(

∑

m∈Zn
〈m〉qs‖ Mmϕ∗ f : LWp ‖

q

)
1
q

. (3.16)

We are now in the position of establishing Theorem 1.



196 Yoshihiro Sawano CUBO
12, 3 (2010)

By Proposition 2 we have

|Mkϕ2 ∗ f (x)|η ≤ c
∑

l∈Z

ˆ

Rn

|Mlϕ1 ∗ f (x − y)|η

〈k − l〉Nη eBη|y|
d y.

If we invoke Proposition 3, we obtain

‖Mkϕ2 ∗ f ‖LWp ≤ c
∑

l∈Z

1

〈k − l〉Nη
‖Mlϕ1 ∗ f ‖LWp

if η < P/ p, N ≫ 1. Hence it follows that

∑

k∈Zn

(

〈k〉s‖Mkϕ2 ∗ f ‖LWp
)q

≤ c
∑

k∈Zn

(

∑

l∈Z

〈k〉s

〈k − l〉Nη
‖Ml ϕ1 ∗ f ‖LWp

)q

≤ c
∑

l∈Zn

(

〈l〉s‖Mlϕ2 ∗ f ‖LWp
)q

,

which implies ‖f : M s,Wp,q ‖ϕ2 ≤ c‖f : M
s,W
p,q ‖ϕ1 . By symmetry Theorem 1 was proved completely.

3.3 Proof of Lemma 1

Instead of dealing with 〈f ,ψ〉 directly, we have only to deal with ψ∗ f (0), which is justified by
the isomorphism ψ 7→ ψ(−·). Proposition 3 and a normalization yield

|ψ∗ f (0)|η ≤ c pN (ψ)η
∑

l∈Z

ˆ

Rn

|Ml ϕ∗ f ( y)|η

〈l〉Nη eNη|y|
d y

with 0 < η ≪
min( p, P, 1)

2
.

ˆ

Rn

|Ml ϕ∗ f ( y)|η

eNη|y|
d y >

ˆ

Rn

|Ml ϕ∗ f ( y)|ηW ( y)η/p

eNη|y|W ( y)η/p
d y

≤ (‖Mlϕ∗ f ‖LWp )
η ·

(

ˆ

Rn

(

W ( y)−η/p

eNη|y|

)−p/( p−η)

d y

)

p−η
η

.

Since W−
1

P−1 ∈ Aloc∞ , we see that W
η/(p−η)

∈ Aloc∞ . Hence, if we choose s ≫ 1, then we obtain
ˆ

Rn

(e−Nη|y|W ( y)−η/p)−p/( p−η) d y

≤
∞
∑

j=1

ˆ

[−2 j ,2 j ]
e
−2 j−1 N p/( p−η)

W ( y)
η/(p−η)

d y

≤ Cs
∞
∑

j=1
2 jn e−2

j−1 N p/( p−η)
M≤2 j [χ1]( y)

s
W ( y)

η/(p−η)
d y



CUBO
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Modulation Spaces with Aloc∞ -Weights 197

< ∞.

As a consequence, Lemma 1 was proved.

We define Se as the set of all C
∞-functions f for which the norm

pN (ψ) >
∑

α∈Zn+,|α|≤N
sup
x∈Rn

exp(N|x|)|∂αψ(x)|

is finite. S ′e is defined as the topological dual of Se . We remark that S
′
e is a special case of

Gelfand-Shilov spaces (see [16]).

Proposition 4. Proposition 3 remains vaild for f ∈ S ′e .

Proof. Keep to the same notation as Proposition 3. The proof does not undergo any major

change until the end of the proof of Proposition 3. If MN,K f (x) were finite, then we would

obtain

|Mkϕ∗ f (x)|
η ≤ MN,K f (x)η ≤ c

∑

m∈Z

ˆ

|Mmγ∗ f (x − y)|η

〈m − k〉Nη eNη|y|
d y. (3.17)

However, this does not always work because MN,K f (x) can be infinite. We shall show that

(3.17) still holds for all f ∈ S ′e (R) even when MN,K f (x) = ∞. For this purpose let us assume
the most right-hand side (3.17) is finite. Otherwise there is nothing to prove. Assuming that

the most right-hand side (3.17) is finite, we shall establish MN,K f (x) < ∞. Since f ∈ S ′e (R),
there exist N f > 0 such that MN,K f (x) < ∞ for all N ≥ N f . As a consequence (3.17) holds for
such N and N. From this we deduce

|Mkϕ∗ f (x)|
η ≤ c

∑

m∈Z

ˆ

|Mmγ∗ f (x − y)|η

〈m − k〉N f η eN f η|y|
d y. (3.18)

The constant in (3.17) being dependent implicitly on N, c in (3.17) must be dependent on f .

To emphasize this dependence, let us write this constant as c f . Then we have

|Mkϕ∗ f (x)|
η ≤ c f

∑

m∈Z

ˆ

|Mmγ∗ f (x − y)|η

〈m − k〉N f η eN f η|y|
d y

≤ c f
∑

m∈Z

1

〈m − k〉Nη

ˆ

|Mmγ∗ f (x − y)|η

eNη|y|
d y

for all N with N ≤ N f . As a consequence for all N > 0, there exists c f such that

|Mkϕ∗ f (x)|η ≤ c f
∑

m∈Z

ˆ

|Mmγ∗ f (x − y)|η

〈m − k〉Nη eNη|y|
d y.

From the definition of the maximal operator MN,K f (x), we have

MN,K f (x) ≤ c f sup
y∈R

(

∑

m∈Z

ˆ

|Mmγ∗ f (x − y − z)|η

〈k − l〉Nη〈m − l〉Nη eNη(|y|+|z|)
d z

)



198 Yoshihiro Sawano CUBO
12, 3 (2010)

≤ c f
∑

m∈Z

ˆ

|Mmγ∗ f (x − z)|η

〈k − m〉Nη eNη|z|
d z

< ∞.

As a consequence (3.17) holds for all f ∈ S ′e (R).

4 Proof of Theorem 2

A fundamental technique in harmonic analysis is to represent functions or distributions as a

linear combination of functions of an elementary form. We shall investigate the structure of

weighted modulation spaces.

We refer to [1, 4, 6, 8, 9, 10, 15, 20] for the definition of the molecules and atoms for

different modulation spaces.

Now we prove Theorem 2.

1. Let N ∈ N be fixed. An integration by parts yields

〈m〉s
∣

∣

∣

∣

∣

∑

l,m∈Zn
λml Mkϕ∗Ψsml(x)

∣

∣

∣

∣

∣

≤ c
∑

l,m∈Zn

|λml|
〈k − m〉N

exp(−N|x − l|)

≤ c
∞
∑

j=1

∑

l∈Zn

e−N j

〈k − m〉N
M

loc
≤ j

(

∑

m∈Zn
λmlχQm

)

for some constant c depending only on N. As a result, we obtain the desired result by

virtue of (3.15).

2. Note that Mm ∗ ϕ ∗ Mmϕ ∗ ψ = cψ for all ψ ∈ Se , since we have seen that
∑

m∈Zn
F ϕ(ξ +

m)2 =: I 6= 0. We set

aml (x) :=
1

I

ˆ

l+[0,1]n
Mmϕ( y)Mmϕ∗ f (x − y) d y.

Then we have f =
∑

l,m∈Zn
aml in S

′
e . Since

M−m aml (x) =
1

I

ˆ

l+[0,1]n
Mmϕ( y)〈f , exp(−im · ( y +∗))ϕ(x − y −∗)〉 d y,

we have Mm[∂
α(M−m aml )](x) =

1

I

ˆ

l+[0,1]n
Mmϕ( y)Mm[∂

α
ϕ]∗ f (x− y) d y. Therefore, if we

define

λml > sup
x∈l+[−2,2]n

sup
|α|≤M

|∂α(M−m aml )(x)|,



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Modulation Spaces with Aloc∞ -Weights 199

then, by Proposition 3, we have

‖{λml }m,l∈Zn : mWp,q‖ ≤ c‖f : M
s,W
p,q ‖.

Hence, it follows that f =
∑

m,l∈Zn
λml ·

aml

λml
is an atomic decomposition of f . This is the

desired result.

5 Weighted Modulation Space M
s,W
p,∞

A minor modification of the results above will yield a theory of the function space M
s,W
p,∞. We

define the function space M s,Wp,∞ as follows :

Definition 4. Let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ R. Assume that W ∈ Aloc∞ . Then define

‖f : M s,Wp,q ‖ >
{

∑

l∈Zn
〈m〉qs

(ˆ

Rn

|Mmϕ∗ f (x)|pW (x) dx
)

q
p

}
1
q

for f ∈ S ′e .

Lemma 2. Let 0 < p < ∞, s ∈ R, W ∈ Aloc∞ . If ε and q satisfy

ε > 0, 0 < q < ∞, qε > n.

then we have M
s,W
p,∞ ,→ M

s−ε,W
p,q .

Proof. This follows from a fundamental inequality

(

∑

m∈Zn
〈m〉−qε|am|q

)
1
q

≤ sup
m∈Zn

|am|
(

∑

m∈Zn
〈m〉−qε

)
1
q

which holds for all complex sequences {am}m∈Zn .

The atomic decomposition theorem can be formulated as follows:

Theorem 3. Let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ R. Assume that W ∈ Aloc∞ .

1. The function space M
s,W
p,q does not depend on the choice of specific ϕ satisfying (1.3).

2. If λ = {λml }m,l∈Zn ∈ m
s,W
p,q and {Ψ

s
ml

}m,l∈Zn ∈ M s0 , then

f :=
∑

m,l∈Zn
λml ·Ψ

s
ml

(5.19)

converges unconditionally in the topology of S ′e .



200 Yoshihiro Sawano CUBO
12, 3 (2010)

3. There exists {as
ml

}m,l∈Zn ∈ A s such that any f ∈ M
s,W
p,q admits the following decomposition:

f =
∑

m,l∈Zn
λml · asml , (5.20)

where λ = {λml }m,l∈Zn satisfies

‖λ : ms,Wp,q ‖ ≤ C ‖f : M
s,W
p,q ‖ (5.21)

for some C > 0 independent of f .

Proof. Almost all the proofs remains unchanged except for the convergence in (5.19). This

will be established by Lemma 2.

6 Examples

Here we shall present some examples of weights.

Example 2. A weight Wa(ξ) = exp(a|ξ|), a ∈ R belongs to the class of our admissible weights.
It is interesting that M

s,Wa
p,q is much larger than M

s
p,q = M

s,W0
p,q for a < 0.

Example 3. If we define W (x) = (1 +|x|2)
a
2 , then M

s,W
2,2 is the weighted Sobolev space.

Proposition 5. Let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ R. If we define W (x) = (1 + |x|2)
a
2 , then

M
s,W
p,q ⊂ S ′.

Proof. In analogy with Proposition 2, we can prove

sup
ψ∈C∞c

qN (ψ)≤1

|Mkψ∗ f (x)|η ≤ c
∑

l∈Z

ˆ

Rn

|Mlϕ∗ f (x − y)|η

〈k − l〉Nη〈y〉Nη
d y (6.22)

for all f ∈ C∞c , where qN (ψ) =
∑

|α|≤N
sup
x∈Rn

〈x〉N|∂αψ(x)|. Therefore, we can proceed as in the proof

of Lemma 1.

References

[1] BALAN, R., CASAZZA, P.G., HEIL, C. AND LANDAU, Z., Density, overcompleteness, and

localization of frames, II. Gabor systems, J. Fourier Anal. Appl., 12 (3), 309–344, 2006.

[2] BÉNYI, A., GRÖCHENIG, K., OKOUDJOU, K.A. AND ROGERS, L.G., Unimodular Fourier

multipliers for modulation spaces (English summary), J. Funct. Anal., 246, no. 2, 366–

384, 2007.



CUBO
12, 3 (2010)

Modulation Spaces with Aloc∞ -Weights 201

[3] BAOXIANG, W. AND CHUNYAN, H., Frequency-uniform decomposition method for the

generalized BO, KdV and NLS equations, J. Differential Equations, 239(2007), no 1,

213–250.

[4] BAOXIANG, W., LIFENG, Z. AND BOLING, G., Isometric decomposition operators, func-

tion spaces Eλp,q and applications to nonlinear evolution equations, J. Funct. Anal.,

233(1), 1–39, 2006.

[5] FEICHTINGER, H., Modulation spaces on locally compact abelian groups, Technical re-

port, University of Vienna.

[6] FEICHTINGER, H., Atomic characterization of modulation spaces through Gabor-type

representation, In Proc. Conf. Constructive Function Theory, Edmonton, July (1989),

113–126.

[7] FEICHTINGER, H., Gewichtsfunktionen auf lokalkompakten Gruppen,

Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, II, 188 (8-810), 451–471, 1979.

[8] FEICHTINGER, H. AND GRÖCHENIG, K., GABOR WAVELETS AND THE HEISENBERG

GROUP: GABOR EXPANSIONS AND SHORT TIME FOURIER TRANSFORM FROM THE

GROUP THEORETICAL POINT OF VIEW, In Charles K. Chui, editor, Wavelets :A tutorial

in theory and applications, 359–398, Academic Press, Boston, MA, 1992.

[9] FEICHTINGER, H. AND GRÖCHENIG, K., Gabor frames and time-frequency analysis of

distributions, J. Functional. Anal., 146(2) (1997), 464–495.

[10] GALPERIN, Y.V. AND SAMARAH, S., Time-frequency analysis on modulation spaces

M
p,q
m , 0 < p, q ≤ ∞, Appl. Comput. Harmon. Anal., 16 (2004), 1–18.

[11] GRÖCHENIG, K., Foundations of Time-Frequency Analysis, Applied and Numerical Har-

monic Analysis. Birkhäuser Boston, Inc., Boston, MA, 2001.

[12] GRÖCHENIG, K., Time-frequency analysis of Sjöstrands class, Revista Mat. Iberoam., 22

(2), 703–724, (2006), arXiv:math.FA/0409280v1.

[13] GRÖCHENIG, K., Weight functions in time-frequency analysis. (English summary)

Pseudo-differential operators: partial differential equations and time-frequency anal-

ysis, Fields Inst. Commun., bf52, Amer. Math. Soc., Providence, RI, (2007), 343–366.

[14] GRÖCHENIG, K. AND HEIL, C., Modulation spaces and pseudo-differential operators,

Integral Equations Operator Theory, 34, 439–457, 1999.

[15] GRÖCHENIG, K. AND RZESZOTNIK, Z., Almost diagonalization of pseudodifferential op-

erators, Ann. Inst. Fourier, (2008), to appear.



202 Yoshihiro Sawano CUBO
12, 3 (2010)

[16] HASUMI, M., Note on the n-dimension Tempered Ultradistributions, Tohoku Math. J.,

13, 94–104, 1961.

[17] IZUKI, M. AND SAWANO, Y., Greedy bases in weighted modulation spaces, to appear in

J. Nonlinear Analysis Series A: Theory, Methods and Applications.

[18] KOBAYASHI, M., Modulation spaces M p,q for 0 < p, q ≤ ∞, J. Function Spaces Appl, 4(3)
(2006), 329–341.

[19] KOBAYASHI, M. AND SAWANO, Y., Molecular decomposition of the modulation spaces

M p,q and its application to the pseudo-differential operators, to appear in Osaka Math-

ematical Journal.

[20] SAWANO, Y., Atomic decomposition for the modulation space M sp,q with 0 < p, q ≤ ∞, s ∈
R, Proceedings of A. Razmadze Mathematical Institute, 145, 63–68, 2007.

[21] SAWANO, Y., Weighted modulation space M sp,q (w) with w ∈ A
loc
p , J. Math. Anal. Appl,

345, 615–627, 2008.

[22] SJÖSTRAND, J., An algebra of pseudodifferential operators, Math. Res. Lett., 1, no.2,

185–192, 1994.

[23] SJÖSTRAND, J., Wiener type algebras of pseudodifferential operators, In Séminaire sur

les équations aux dérivées partielles, 1994–1995, pages Exp. No. IV, 21. École Polytech.,

Palaiseau, 1995.

[24] SJÖSTRAND, J., Pseudodifferential operators and weighted normed symbol spaces,

Preprint, 2007. arXiv:0704.1230v1.

[25] TACHIZAWA, K., The boundedness of pseudodifferential operators on modulation spaces,

Math. Nachr., 168, 263–277, 1994.