Modulation spaces Mp,q for 0
Transcript of Modulation spaces Mp,q for 0
JOURNAL OF c© 2006, Scientific Horizon
FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net
Volume 4, Number 3 (2006), 329-341
Modulation spaces Mp,q for 0 < p, q ��� ∞
Masaharu Kobayashi
(Communicated by Hans Triebel)
2000 Mathematics Subject Classification. 42B35, 46A16, 46E30, 46E35.
Keywords and phrases. Modulation spaces, quasi-Banach spaces, Lp estimates
for convolutions, density of the rapidly decreasing functions.
Abstract. The purpose of this paper is to construct modulation spacesMp,q(Rd) for general 0 < p, q � ∞ , which coincide with the usual modulationspaces when 1 � p, q � ∞ , and study their basic properties including their
completeness. Given any g ∈ S(Rd) such that supp �g ⊂ {ξ | |ξ| � 1} and�k∈Zd
�g(ξ − αk) ≡ 1, our modulation space consists of all tempered distributions
f such that the (quasi)-norm
‖f‖Mp,q[g]
:=
��Rd
��Rd
��f ∗ �Mωg�(x)
��pdx
� qp
dω
� 1q
is finite.
1. Introduction
The purpose of this paper is to construct modulation spaces Mp,q(Rd)for general 0 < p, q � ∞ , which coincide with the usual modulationspaces when 1 � p, q � ∞ , and study their basic properties.
330 Modulation spaces
By the usual modulation space Mp,q(Rd), we mean the space of alltempered distributions f ∈ S′(Rd) for which the norm
‖f‖Mp,q =( ∫
Rd
( ∫Rd
|Vgf(x, ω)|pdx) q
p
dω
) 1q
is finite, where Vgf(x, ω) is the Short-Time Fourier Transform of f withrespect to a general window g ∈ S(Rd) defined by
Vgf(x, ω) = 〈f,MωTxg〉 =∫Rd
f(t)g(t− x)e−2πiω·tdt.
Most of the researches on modulation spaces have been restricted to thecase 1 � p, q � ∞. (See Grochenig [3].) The reason is that in the casep < 1, the estimate of the Lp -norm of the convolution is not easy.
To estimate Lp -norm of convolution of two functions, Y. Galperin and S.Samarah [2] use the fact that if f is a function on Rd that has polynomialgrowth, |f(t)| = O(|t|N ), then Bargmann transform Bf(z)
Bf(z) = 2d/4
∫Rd
f(t)e2πt·z−πt2−π2 z2
dt
is an entire function on Cd . For such f , they develop a theory of modulationspaces Mp,q(Rd) for the case 0 < p, q � ∞ . But for general f ∈ S′(Rd),whether or not Bf(z) is an entire function is non-trivial and is not provedin the paper, and completeness of the space Mp,q(Rd) is not proved. InH. Triebel [6] modulation spaces are studied along the lines of the theoryof Besov spaces, but f is restricted to those functions in S(Rd) that thesupport of f is compact. The completeness is not proved, either.
In this article we use only a simple, but key, lemma (see Lemma 2.6)to estimate convolutions, and define modulation spaces Mp,q(Rd) for0 < p, q � ∞ and prove its completeness, in particular.
2. Basic definition
Let S(Rd) be the Schwartz space of all complex-valued rapidly decreasinginfinitely differentiable functions on Rd with the topology defined by thesemi-norms
pM (ϕ) = supx∈Rd
(1 + |x|)M∑
|α|�M
|∂αϕ(x)|, M = 1, 2, · · ·
M. Kobayashi 331
for ϕ ∈ S(Rd). And let S′(Rd) be the topological dual of S(Rd).The Fourier transform is f(ω) =
∫f(t)e−2πit·ωdt , and the inverse Fourier
transform is f(t) = f(−t). We define for 0 < p <∞
‖f‖Lp =( ∫
Rd
|f(x)|pdx) 1
p
and ‖f‖L∞ = ess. supx∈Rd |f(x)| . We use 〈f, g〉 to denote the extension toS′(Rd) × S(Rd) of the inner product 〈f, g〉 =
∫f(t)g(t)dt on L2(Rd).
Definition 2.1. If f ∈ S′(Rd) and g ∈ S(Rd), we define the convolutionf ∗ g by
f ∗ g(x) = 〈f, g(x− ·) 〉 =∫Rd
f(t)g(x− t)dt, x ∈ Rd.
We define the translation and the modulation operators by
Txf(t) = f(t− x), and Mωf(t) = e2πiω·tf(t) (x, ω ∈ Rd),
respectively. The following Lemmas are useful.
Lemma 2.2. For g ∈ S(Rd) , k ∈ Zd and x ∈ Rd we have(Tkg
)∨(x) =(Mkg
)(x).
Lemma 2.3. Let Γ be a compact subset of Rd and f ∈ S′(Rd) . Ifsupp f ⊂ Γ then for any ξ0 ∈ Rd , we have
(1) supp(Mξ0f
) ⊂ ξ0 + Γ.
This is employed to prove that the constant C in the estimate (3) belowdepends only on Γ.
Definition 2.4. Let 0 < p � ∞ , and Γ be a compact subset of Rd .Then Lp
Γ is defined by
(2) LpΓ = {f ∈ S′(Rd) | ∃ξ0 ∈ Rd, supp f ⊂ ξ0 + Γ, ‖f‖Lp <∞}.
By the Paley-Wiener-Schwartz theorem LpΓ consists of entire analytic
functions.
Theorem 2.5. Let Γ be a compact subset of Rd and let 0 < p � q � ∞.
Then there exists a positive constant C (which depends only on the diameterof Γ and p) such that
(3) ‖f‖Lq � C‖f‖Lp
holds for all f ∈ LpΓ .
332 Modulation spaces
Proof. If supp f ⊂ Γ, (3) is just the famous Nikol’skij-Triebel inequality(see [5] Theorem 1.4.1). Let f ∈ Lp
Γ . Then we can find ξ0 ∈ Rd such thatsupp f ⊂ ξ0 + Γ. By lemma 2.3,
supp(M−ξ0f
)⊂ Γ and ‖f‖Lp = ‖M−ξ0f‖Lp .
So we have
‖f‖Lq = ‖M−ξ0f‖Lq � C‖M−ξ0f‖Lp = C‖f‖Lp. �From Theorem 2.5 we have the following key Lemma.
Lemma 2.6. Let 0 < p � 1 and Γ,Γ′ be compact subsets of Rd . Thenthere exists a positive constant C (which depends only on the diameters ofΓ,Γ′ and p ) such that
(4) ‖ |f | ∗ |g| ‖Lp � C‖f‖Lp‖g‖Lp
holds for all f ∈ LpΓ and all g ∈ Lp
Γ′ .
Proof. Step 1. First we assume f ∈ LpΓ, g ∈ Lp
Γ′ be such that
supp f ⊂ Γ, supp g ⊂ Γ′.
Then for a.e. x ∈ Rd , f(y)g(x− y) ∈ Lp (as a function of y ) and
supp(f(·)g(x− ·))⊂ Γ − Γ′.
Here we used the notation −A = {x | − x ∈ A} . By Theorem 2.5
|f | ∗ |g|(x) =( ∫
Rd
|f(y)g(x− y)|dy)
� C
( ∫Rd
|f(y)g(x− y)|pdy) 1
p
,
where C depends only on Γ,Γ′ and p . Taking the Lp -norm of both sides,we have (4).
Step 2. Let f ∈ LpΓ, g ∈ Lp
Γ′ . Then there exist ξ0, ξ′0 ∈ Rd such that
supp f ⊂ ξ0 + Γ and supp g ⊂ ξ′0 + Γ′. Applying Step.1 to M−ξ0f and
M−ξ′0g instead of f and g respectively, we have (4). �
M. Kobayashi 333
3. Modulation spaces
Definition 3.1. For α > 0 we define Φα(Rd) as follows:
Φα(Rd)
:={g ∈ S(Rd)
∣∣∣∣ supp g ⊂ {ξ | |ξ| � 1}, and∑
k∈Zd
g(ξ−αk) ≡ 1, ∀ξ ∈ Rd
}.
In the following, we choose a sufficiently small α > 0 so that the functionspace Φα(Rd) is not empty.
Definition 3.2. Given a g ∈ Φα(Rd), and 0 < p, q � ∞ , we define themodulation space Mp,q
[g] (Rd) to be the space of all tempered distributionsf ∈ S′(Rd) such that the (quasi)-norm
(5) ‖f‖Mp,q[g]
:=( ∫
Rd
(∫Rd
∣∣f ∗ (Mωg
)(x)
∣∣pdx) qp
dω
) 1q
is finite.
Our definition is close to the original definition of modulation spaces byFeichtinger [1].
Remark. Since f ∈ S′(Rd) and g ∈ S(Rd), we have Vgf(x, ω) =e−2πix·ωf ∗ (Mω g)(x), where g(x) = g(−x). Since the usual modulationspace Mp,q(Rd) (1 � p, q � ∞) is independent of the choice of a windowg ∈ S(Rd)�{0} (see [3, Proposition 11.3.2]), our modulation space coincideswith the usual one if 1 � p, q � ∞ .
Theorem 3.3. Let 0 < p � ∞ , 0 < q � ∞ , then
(6)( ∑
k∈Zd
( ∫Rd
∣∣f ∗ (Mαkg
)(x)
∣∣pdx) qp) 1
q
is an equivalent quasi-norm on Mp,q[g] (Rd) , with modifications if p or
q = ∞ .
Proof. Step 1. We first prove that quasi-norm (5) can be estimatedfrom above by the quasi-norm (6). For this purpose fix an ω ∈ Rd . Thenthere exists k = (k1, · · · , kd) ∈ Zd such that
(7) ω ∈ [αk1, α(k1 + 1)] × · · · × [αkd, α(kd + 1)] = [αk, α(k + 1)].
334 Modulation spaces
Then there exists a positive constant N which depends only on the size ofsupp g , α > 0 and the dimension d , such that
f ∗ (Mωg) = f ∗ (Tω g)∨
= f ∗( ∑
|r|�N
Tα(k+r)g · Tωg
)∨
= f ∗( ∑
|r|�N
Tα(k+r)g
)∨∗ (Tω g
)∨
=∑
|r|�N
(f ∗ (Mα(k+r)g)
)∗ (Mωg).
Taking the Lp -norm on both sides and p-th power, we obtain
‖f ∗ (Mωg)‖pLp =
∫Rd
∣∣∣∣ ∑|r|�N
(f ∗ (Mα(k+r)g)
) ∗ (Mωg)(x)∣∣∣∣p
dx
� C∑
|r|�N
‖(f ∗ (Mα(k+r)g)) ∗ (Mωg)‖p
Lp
� C′ ∑|r|�N
‖f ∗ (Mα(k+r)g)‖pLp .
(8)
Here, for the last inequality we used Young’s theorem for p � 1
‖F ∗G‖Lp � ‖F‖Lp‖G‖L1
and Lemma 2.6 for p � 1. We take the q/p-th power of (8), integrate over[αk, α(k + 1)], and sum over k ∈ Zd , and obtain
∫Rd
‖f ∗ (Mωg)‖qLpdω � C′ ∑
k∈Zd
∫[αk,α(k+1)]
( ∑|r|�N
‖f ∗ (Mα(k+r)g)‖pLp
) qp
dω
� C′′ ∑k∈Zd
‖f ∗ (Mαkg)‖qLp .
Step 2. Fix a k ∈ Zd . Then for all ω ∈ [αk, α(k + 1)] we have
f ∗ (Mαkg) = f ∗ (∑
|r|�N
Tω+αrg · Tαkg)∨
=∑
|r|�N
f ∗ (Mω+αrg) ∗ (Mαkg).
M. Kobayashi 335
Taking p-th power and integrating over Rd , we obtain
‖f ∗ (Mαkg)‖pLp � C
∑|r|�N
‖f ∗ (Mω+αrg)‖pLp .
Similarly we have
‖f ∗ (Mαkg)‖qLp � C
( ∑|r|�N
‖f ∗ (Mω+αrg)‖pLp
) qp
� C′∫
[αk,α(k+1)]
( ∑|r|�N
‖f ∗ (Mω+αrg)‖pLp
) qp
dω
� C′′ ∑|r|�N
∫[αk,α(k+1)]
‖f ∗ (Mω+αrg)‖qLpdω.
Taking sum over k ∈ Zd , we get
∑k∈Zd
‖f ∗ (Mαkg)‖qLp � C′
∫Rd
‖f ∗ (Mωg)‖qLpdω.
So the proof is complete. �In the sequel, we shall not distinguish between equivalent quasi-norms of
a given quasi-normed space.
Corollary 3.4. Let 0 < p0 � p1 � ∞ and 0 < q0 � q1 � ∞ . Then
Mp0,q0[g] (Rd) ⊂Mp1,q1
[g] (Rd).
Proof. This follows form the monotonicity of the lp -spaces and Theorem2.5. �
Lemma 3.5. If 0 < p � ∞ , 0 < q � ∞ and g1, g2 ∈ Φα(Rd) , then‖f‖Mp,q
[g1]and ‖f‖Mp,q
[g2]are equivalent quasi-norms on Mp,q(Rd) .
Proof. It is easy to see that both ‖f‖Mp,q[g1]
and ‖f‖Mp,q[g2]
are quasi-norms.In order to prove the equivalence of these two quasi-norms we apply Young’sinequality (or Lemma 2.6 if p � 1). Since Tαkg1 =
∑|r|�N
Tα(k+r)g2 · Tαkg1
we have
f ∗ (Mαkg1) =(f · Tαkg1 ·
∑|r|�N
Tα(k+r)g2
)∨
336 Modulation spaces
= f ∗( ∑
|r|�N
Mα(k+r)g2
)∗ (Mαkg1).
Hence there exists a positive constant c such that
‖f ∗ (Mαkg1)‖Lp � c
∥∥∥∥f ∗( ∑
|r|�N
Mα(k+r)g2
)∥∥∥∥Lp
.
This proves that ‖f‖Mp,q[g1]
can be estimated from above by c‖f‖Mp,q[g2]
. Hencethese two quasi-norms are equivalent to each other. �
Theorem 3.6. We have the continuous imbeddings
(9) S(Rd) ⊂Mp,q(Rd) ⊂ S′(Rd)
for 0 < p � ∞ and 0 < q � ∞.
Proof. To prove the left-hand side of (9), let f ∈ S(Rd) and denote
Δ =d∑
j=1
(∂2/∂2j ). Then for any positive integer L we have
(1 + 4π2|ω|2)L|f ∗ (Mωg)(x)|
=∣∣∣∣∫Rd
f(x− t)g(t)(1 + 4π2|ω|2)Le2πiω·tdt∣∣∣∣
=∣∣∣∣∫Rd
f(x− t)g(t)(1 − Δ)Le2πiω·tdt∣∣∣∣
=∣∣∣∣∫Rd
∑|α+β|�2L
Cα,β∂αf(x− t)∂βg(t)e2πiω·tdt
∣∣∣∣.Therefore
|f ∗ (Mωg)(x)|(1 + |x|2)M (1 + 4π2|ω|2)L � C p2(L+M)(f).
Taking L,M sufficiently large, we have
‖f‖Mp,q � Cp2(L+M)(f).
Next we prove the right-hand side of (9) with q = ∞ . First note thatthere exists a constant N (depending only on the size of supp g , α > 0and dimension d) such that Tαkg =
∑|r|�N
Tα(k+r)g · Tαkg for all k ∈ Zd . If
M. Kobayashi 337
f ∈Mp,∞(Rd) and ψ ∈ S(Rd), then
|〈f, ψ〉| =∣∣∣∣ ∑
k∈Zd
〈(Tαkg · f)∨,∑
|r|�N
(Mα(k+r)g) ∗ ψ〉∣∣∣∣
�∑
k∈Zd
‖f ∗ (Mαkg)‖L∞
∥∥∥∥ ∑|r|�N
(Mα(k+r)g) ∗ ψ∥∥∥∥
L1
.
Recall that both f ∗(Mαkg) and∑
|r|�N
(Mα(k+r)g)∗ψ are analytic functions.
Hence, the last estimate makes sense. By Theorem 2.5 we have
‖f ∗ (Mαkg)‖L∞ � C‖f ∗ (Mαkg)‖Lp.
Hence,
|〈f, ψ〉| � C′‖f‖Mp,∞∑
k∈Zd
∥∥∥∥ ∑|r|�N
(Mα(k+r)g) ∗ ψ∥∥∥∥
L1
.
The last sum can be estimated from above by C‖ψ‖M1,1 . Consequently, ifM is a sufficiently large natural number, we have
|〈f, ψ〉| � C‖f‖Mp,∞pM (ψ)
for any ψ ∈ S(Rd). This proves that Mp,∞(Rd) is continuously embeddedin S′(Rd). �
Theorem 3.7. Mp,q(Rd) is a quasi-Banach space if 0 < p � ∞ and0 < q � ∞ (Banach space if 1 � p � ∞ and 1 � q � ∞ ) .
To prove the theorem we prepare the following lemma.
Definition 3.8. For 0 < p, q � ∞ we define Lp,q(Rd × Zd) as follows:
Lp,q(Rd × Zd)
:={f : Rd × Zd → C is measurable and ‖f‖Lp,q := ‖ ‖f‖Lp‖lq <∞
}.
This is a quasi-normed space and hence there is an 0 < r <∞ such that
(10)∥∥∥∥
n∑l=1
fl
∥∥∥∥r
Lp,q
� 2n∑
l=1
‖fl‖rLp,q .
(See Kothe [4] p.162.)
338 Modulation spaces
Lemma 3.9. Let 0 < p, q � ∞ . If {fn(x, k)}∞n=1 is a sequence inLp,q(Rd × Zd) such that
(11)∑n∈N
‖fn‖rLp,q <∞.
Then the series
(12) f(x, k) =∑n∈N
fn(x, k)
converges absolutely for a.e. x ∈ Rd and for all k ∈ Zd and is inLp,q(Rd × Zd) and
(13) ‖f1 + · · · + fn − f‖Lp,q → 0, as n→ ∞.
In particular, Lp,q(Rd × Zd) is complete.
Proof. When either p or q = ∞ , the following inequalities give the proofwith r = min{q, 1} or min{p, 1} .∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣∣ ∑
n∈N
|fn(x, k)|∣∣∣∣∣∣∣∣L∞
∣∣∣∣∣∣∣∣r
lq�
∣∣∣∣∣∣∣∣ ∑
n∈N
‖fn(x, k)‖L∞
∣∣∣∣∣∣∣∣r
lq
�∑n∈N
‖fn(x, k)‖rLp,q , if q � p = ∞,
∣∣∣∣∣∣∣∣ ∑
n∈N
|fn(x, k)|∣∣∣∣∣∣∣∣r
Lp
�∑n∈N
‖fn(x, k)‖rLp
�∑n∈N
(sup
k‖fn(x, k)‖Lp
)r, if p < q = ∞.
Let 0 < p, q <∞ and define
gn(x, k) :=n∑
l=1
|fl(x, k)| and g(x, k) := limn→∞ gn(x, k) =
∞∑n=1
|fn(x, k)|.
By (11) and by the Beppo-Levi theorem
‖g‖rLp,q = lim
n→∞ ‖gn‖rLp,q � lim
n→∞ 2n∑
l=1
‖fl‖rLp,q = 2
∞∑n=1
‖fn‖rLp,q <∞.
This proves that g ∈ Lp,q . Consequently, for a.e. x and for all k , theseries (12) converges absolutely. Since |f(x, k)|p � g(x, k)p ∈ L1(Rd) and‖f(x, k)‖q
Lp � ‖g(x, k)‖qLp ∈ l1(Zd), we conclude that f ∈ Lp,q(Rd × Zd).
M. Kobayashi 339
Similarly, since |f1 + · · ·+ fn − f | �∞∑
l=n+1
|fn| � g ∈ Lp,q , we have (13) by
a repeated application of the Lebesgue dominant convergence theorem.To prove the completeness of Lp,q let {hn}∞n=1 be a Cauchy sequence in
Lp,q . Then we can choose a subsequence {hnk}∞k=1 so that
‖hnk+1 − hnk‖r
Lp,q � 2−(k+1) for all k ∈ N.
From above argument, {hnk}∞k=1 converges to f = limk→∞ hnk
in Lp,q .Since {hn}∞n=1 is a Cauchy sequence, {hnk
} also converges to f in Lp,q . �
Proof of Theorem 3.7. Let {fn}∞n=1 be a sequence in Mp,q(Rd) such that∑n∈N
‖fn‖rMp,q <∞.
Then by Lemma 3.9, the series∑n∈N
(fn ∗ (Mαkg)(x)
)
converges absolutely for a.e. x ∈ Rd and for all k ∈ Zd and is inLp,q(Rd × Zd). And by Theorem 3.6, we have f =
∑n∈N
fn converges in
S′(Rd). Since for all x ∈ Rd and k ∈ Zd
f ∗ (Mαkg)(x) =⟨ ∑
n∈N
fn,Mαkg(x− ·)⟩
=∑n∈N
〈fn,Mαkg(x− ·)〉
=∑n∈N
(fn ∗ (Mαkg)(x)
),
we have f ∈Mp,q(Rd). So Mp,q(Rd) is complete. �
Theorem 3.10. If 0 < p <∞ and 0 < q <∞ , then S(Rd) is dense inMp,q(Rd) .
Proof. Let f ∈Mp,q(Rd) and set
fn =∑|k|�n
f ∗ (Mαkg), n ∈ N.
340 Modulation spaces
Of course, fn ∈Mp,q(Rd). Then, it follows that
(f − fn) ∗ (Mαlg)(x) =∑|k|>n
f ∗ (Mαkg) ∗ (Mαlg)(x)
=∑
r∈Λl,n
f ∗ (Mα(l+r)g) ∗ (Mαlg)(x)
where
Λl,n :={r ∈ Zd
∣∣∣∣ |r + l| > n and supp(Tα(l+r)g) ∩ supp(Tαlg) �= ∅}.
Note that
|Λl,n| � ∃N, (∀l ∈ Zd, ∀n ∈ N).
Moreover∣∣∣∣∣∣∣∣ ∑
r∈Λl,n
f ∗ (Mα(l+r)g) ∗ (Mαlg)(x)∣∣∣∣∣∣∣∣Lp
� C
∣∣∣∣∣∣∣∣ ∑
r∈Λl,n
f ∗ (Mα(l+r)g)∣∣∣∣∣∣∣∣Lp
‖Mαlg‖Lmin{1,p}
and thus
‖f − fn‖qMp,q =
∑l∈Zd
∣∣∣∣∣∣∣∣ ∑
r∈Λl,n
f ∗ (Mα(l+r)g)∣∣∣∣∣∣∣∣q
Lp
� C′ ∑|r|>n−N
∣∣∣∣∣∣∣∣f ∗ (Mαrg)
∣∣∣∣∣∣∣∣q
Lp
→ 0, (n→ ∞).
Hence fn approximates f in Mp,q(Rd). Let ϕ ∈ S(Rd) with ϕ(0) = 1and supp ϕ ⊂ {ξ | |ξ| � 1} . Let (fn)δ(x) = ϕ(δx)fn(x) with 0 < δ < 1then (fn)δ ∈ S(Rd) and this is an approximation of fn in Mp,q(Rd). Thisproves that S(Rd) is dense in Mp,q(Rd). �
Lemma 3.11. Let Γ be a compact subset of Rd . If β > 0 is sufficientlysmall (depending on Γ) , then we have equivalency
( ∑k∈Zd
|f(βk)|p) 1
p
∼ ‖f‖Lp, f ∈ LpΓ, 0 < p � ∞.
Proof. See for instance [5] Theorem 1.4.1. �
M. Kobayashi 341
Combining Theorem 3.3 and Lemma 3.11, we have the followingCorollary.
Corollary 3.12. Let 0 < p � ∞ , 0 < q � ∞ . If β > 0 is sufficientlysmall then
(14)( ∑
k∈Zd
( ∑l∈Zd
∣∣f ∗ (Mαkg
)(βl)
∣∣p) qp) 1
q
is an equivalent quasi-norm on Mp,q(Rd) .
References
[1] H. G. Feichtinger, A new class of function spaces, In : Proc. Conf.“Constructive Function Theory”, Kiew, 1983, (1984).
[2] Y. Galperin and S. Samarah, Time-frequency analysis on modulationspaces Mp,q
m , 0 < p, q ≤ ∞ , Appl. Comput. Harmon. Anal., 16 (1)(2004), 1–18.
[3] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser,2001.
[4] G. Kothe, Topologische Lineare Raume I, Zweite Auflage, Springer,1966.
[5] H. Triebel, Theory of Function Spaces, Birkhauser, Boston, 1983.
[6] H. Triebel, Modulation spaces on the euclidean n-space, Z. Anal.Anwendungen, 2(5) (1983), 443–457.
Department of MathematicsTokyo University of ScienceKagurazaka 1-3Shinjuku-kuTokyo 162-8601Japan(E-mail : [email protected])
(Received : September 2005 )
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