Trace operators on Wiener amalgam spaces

0.15 MB·

by  Jayson Cunanan

TRACE OPERATORS ON WIENER AMALGAM SPACES
JAYSONCUNANANAND YOHEITSUTSUI
Abstract. The paper deals with trace operators of Wiener amalgam spaces using
frequency-uniformdecompositionoperatorsandmaximalinequalities,obtainingsharp
6
results. Additionally, we provide the embeddings between standard and anisotropic
1
0 Wiener amalgam spaces.
2
n 1. Introduction
a
J The aim of this paper is to study the trace problem: What can be said about the
1 trace operator T,
2 T :f(x) → f(x¯,0), x¯ = (x ,x ,...,x ),
1 2 n−1
as a mappingfrom Wp,q(Rn) to Wp,q(Rn−1). We note that for a tempered distribution f
] s s
A defined on Rn,f(x,0) has no straightforward meaning and the question is how to define
F the trace for a class of tempered distributions. One can resort to the Schwartz function
h. φ, which has a pointwise trace φ(x¯,0). It can be extended to (quasi-)Banach function
t spaces which contain the Schwartz space S as a dense subspace.
a
m Our setting is on Wiener amalgam spaces. These spaces, together with modulation
spaces, were introduced by Feichtinger [5, 6, 7] in the 80’s and are now widely used func-
[
tion spaces for various problems in PDE and harmonic analysis [1, 2, 3, 11, 16]. They
2
resemble Triebel-Lizorkin spaces in the sense that we are taking Lp(ℓq) norms, but differ
v
4 with the decomposition operator being used. Instead of the dyadic decomposition op-
4 erators ∆ ∼ F−1χ F used for Triebel-Lizorkin spaces, Wiener amalgam spaces
k {ξ:|ξ|∼2k}
2
use frequency uniform decomposition operators (cid:3) ∼ F−1χ F, where Q denotes a
5 k Qk k
0 unit cube with center k and ∪k∈ZnQk =Rn.
1. The concept of trace operator plays an important role in studying the existence and
0 uniqueness of solutions to boundary value problems, that is, to partial differential equa-
6
tions with prescribed boundary conditions [4, 14]. The trace operator makes it possible
1
to extend the notion of restriction of a function to the boundary of its domain to ”gen-
:
v
eralized” functions in various function spaces with regulariy. Now, we give a formal
i
X definition for the trace operators.
r Definition 1.1. LetX andY bequasi-Banach functionspaces definedon Rn andRn−1,
a
respectively. Assume that the Schwartz class S is dense in X. Denote
T :f(x)→ f(x¯,0), f ∈ S.
Assuming that there exist a constant C > 0 such that
||Tf|| ≤ C||f|| , ∀f ∈ S,
Y X
2010 Mathematics Subject Classification. 46E35.
Key words and phrases. Wiener amalgam spaces, trace operators, maximal inequalities.
1
2 JAYSONCUNANANANDYOHEITSUTSUI
one can extend T : X → Y by the density of S in X and we write f(x¯,0) = Tf, which
is said to be the trace of f ∈ X. Moreover, if there exist a continuous linear operator
T−1 : Y → X such that TT−1 is the identity operator on Y, then T is said to be a
trace-retraction from X onto Y.
For (α-) modulation spaces, Besov spaces and Tribel-Lizorkin spaces, trace theorems
havebeenextensivelystudied[9,13,14]. Feichtinger, HuangandWang[9]consideredthe
trace theorems on anisotropic modulation spaces Mp,q,r with 0 < p,q,r < ∞,s ∈ R and
s
they obtainedTMp,q,p∧q∧1(Rn)= Mp,q(Rn−1). In[10,15], wefindthatfor0 < p,q ≤ ∞,
s s
and s−1/p > (n−1)(1/p−1), we have TBp,q(Rn) = Bp,q (Rn−1) and TFp,q(Rn) =
s s−1/p s
Fp,p (Rn−1) (the case F∞,q is ommited). The use of atoms as a framework in studying
s−1/p
trace problems can be found in [15] and the references within.
Our main results are the following.
Theorem 1.1. Let n ≥ 2,0 < p,q < ∞,s ∈R. Then
T : f(x)→ f(x¯,0), x¯ = (x ,x ,...,x )
1 2 n−1
is a trace-retraction from Wp,q,1∧q(Rn) to Wp,q(Rn−1).
s s
In view of the embedding in Theorem 2.1 (II-ii), we immediately have the following
corollary.
Corollary 1.1. Let n≥ 2,0 < p,q < ∞,s ≥ 0. Then for any ǫ > 0
T : Wp,q (Rn)→ Wp,q(Rn−1).
s+ 1 −1+ǫ s
1∧q q
Weremarkthatourresultshowsindependenceofp.Thisisduethepointwiseestimates
we were able to prove in Section 3. An interesting observation is that, the trace theorem
of Triebel-Lizorkin spaces stated above, shows independence in q. This difference might
be due to the decomposition operators used in the norm of each function spaces.
The paper is organised as follows: In Section 2, the embeddings between standard
and anisotropic Wiener amalgam spaces are given. We also define notations, function
spaces and some Lemmas to be used throughout this paper. In Section 3, we prove our
main result, Theorem 1.1 and the sharpness of Corollary 1.1.
2. Preliminaries
Notations. The Schwartz class of test functions on Rn shall be denoted by S :=
S(Rn) and its dual, the space of tempered distributions, by S′ := S′(Rn). The Lp(Rn)
norm is given by ||f||Lp= ( Rn|f(x)|p dx)1/p whenever 1 ≤ p < ∞, and ||f||L∞=
ess.supx∈Rn|f(x)|. The FourieRr transform of a function f ∈ S(Rn) is given by
Ff(ξ) = fˆ(ξ) = e−i2πx·ξf(x) dx
Rn
Z
which is an isomorphism of the Schwartz space S(Rn) onto itself that extends to the
tempered distributions S′(Rn) by duality. The inverse Fourier transform is given by
F−1f(x) = fˇ(x) = ei2πξ·xf(ξ) dξ. Given 1 ≤ p ≤ ∞, we denote by p′ the conjugate
Rn
exponent of p (i.e. 1/p + 1/p′ = 1). We use the notation u . v to denote u ≤ cv
R
TRACE OPERATORS ON WIENER AMALGAM SPACES 3
for a positive constant c independent of u and v. We write a ∧ b := min(a,b) and
a∨b := max(a,b). We now define the function spaces in this paper.
Let η : R → [0,1] be a smooth bump function satisfying
1, |ξ|≤ 1
η(ξ) := smooth, 1 < |ξ|≤ 2

0, |ξ|≥ 2.
We write for k = (k1,...,kn) and ξ =(ξ1,...,ξn),
φ = η(2(ξ −k )).
ki i i
Put
φ (ξ )...φ (ξ )
(1) ϕ (ξ) = k1 1 kn n , k ∈ Zn.
k
φ (ξ )...φ (ξ )
k∈Zn k1 1 kn n
Definition 2.1 (Wiener amalgPam spaces). For 0 < p,q ≤ ∞, and s ∈ R, the Wiener
amalgam space Wp,q consists ofall tempereddistributionsf ∈ S′ forwhichthefollowing
s
is finite:
(2) ||f||Wsp,q= || ||{hkis(cid:3)kf}||ℓq||Lp,
with (cid:3) f = F−1(ϕ f).
k k
Wenotethat(2)isaquasi-normif0 < p,q ≤ ∞,andnormif1 ≤ p,q ≤ ∞. Moreover,
b
(2) is independent of the choice of ϕ = {ϕk}k∈Zn. We refer the reader to [5, 6, 8] for
equivalent definitions (continuous versions).
We write x¯ = (x ,x ,...,x ) and define the anisotropic Wiener amalgam spaces
1 2 n−1
Wp,q,r by the following norm,
||f||Wsp,q,r(Rn)= ||( ( hk¯isq|(cid:3)kf|q)r/q)1/r||Lp(Rn).
kXn∈Z k¯∈XZn−1
Similarly, for x¯¯ = (x ,x ,...,x ), we define
1 2 n−2
||f||Wsp,q,r,r(Rn)= ||( ( hk¯¯isq|(cid:3)kf|q)r/q)1/r||Lp(Rn).
(kn−X1kn)∈Z2 k¯¯∈XZn−2
p,q p,q,r
Comparingamalgam spaces W with anisotropic amalgam spaces W we see that
s s
p,q p,q,r
W is, but W is not rotational invariant. Using the almost orthogonality of ϕ
s s
p,q,r
we see that the Ws is independent of ϕ. Moreover, recalling that ||f||Wsp,q,r is the
function sequence {(cid:3)kf}k∈Zn equipped with the Lpℓrknℓqk¯ norm, it is easy to see that
Wp,q,r is a quasi-Banach space for any s ∈R,p,q,r ∈(0,∞] and a Banach space for any
s
s ∈ R,1 ≤ p,q,r ≤ ∞. Moreover, the Schwartz space is dense in Wp,q,r if p,q,r < ∞.
s
The proofs are similar to those of amalgam spaces in [5, 6, 8].
We collect properties of Wiener amalgam spaces in the following lemma.
Lemma 2.1. Let p,q,p .q ∈[1,∞] for i = 1,2 and s ∈ R for j = 1,2. Then
i i j
(1) S(Rn)֒→ Wp,q(Rn) ֒→ S′(Rn);
(2) S is dense in Wp,q if p and q < ∞;
(3) If q ≤ q and p ≤ p , then Wp1,q1 ֒→ Wp2,q2;
1 2 1 2
4 JAYSONCUNANANANDYOHEITSUTSUI
p,q p,q
(4) If s ≥ s , then W ֒→ W ;
1 2 s1 s2
1 θ 1−θ 1 θ 1−θ
(5) (Complex interpolation) For 0 < θ < 1. Let = + , = +
p p p q q q
1 2 1 2
and s = θs +(1−θ)s . Then
1 2
[Wp1,q1,Wp2,q2] = Wp,q.
s1 s2 [θ] s
The proofs of these statements can be found in [5, 7, 8, 12].
Theorem 2.1 (Embedding: Wp,q ֒→ Wp,q,r). Let p,q,r ∈ (0,∞] and s≥ 0.
s s′
(I): The case r =q.
(I - i): The case r = q = ∞.
Wp,∞ ֒→ Wp,∞,∞.
s s
(I - ii): The case r = q < ∞.
Wp,q ֒→ Wp,q,q.
s s
(II): The case r < q.
(II - i): The case q = ∞. If s > 1/r, then
Wp,∞ ֒→ Wp,∞,r,
s s′
for any s′ ∈(−∞,s−1/r).
(II - ii): The case q < ∞. If s > (1/r−1/q), then
Wp,q ֒→ Wp,q,r,
s s′
for any s′ ∈(−∞,s−(1/r−1/q)).
(III): The case q < r.
(III - i): The case r = ∞.
Wp,q ֒→ Wp,q,∞.
s s
(III - ii): The case r < ∞.
Wp,q ֒→ Wp,q,r.
s s
Proof. For part (I), it suffice to show the following estimates.
(I-i):
supsuphk¯is|(cid:3) f|≤suphkis|(cid:3) f|.
k k
kn k¯ k
(I - ii):
1/q
1/q
hk¯isq|(cid:3) f|q ≤ hkisq|(cid:3) f|q .
k k
 
!
Xkn Xk¯ Xk
 
(II - i): Let s′ := s−1/r−ε, (ε > 0). We may assume that s′ ≥ 0.
1/r 1/r
suphk¯is′r|(cid:3) f|r ≤ suphkis|(cid:3) f| × suphki−srhk¯is′r
k k
   
Xkn k¯ (cid:18) k (cid:19) Xkn k¯
   
TRACE OPERATORS ON WIENER AMALGAM SPACES 5
The last term is equivalent to
s r 1/r (s−s′)r 1/r
sup 1 ts′ ≤ 1 supt(s′−s′)r < ∞,
t+|m| 1+|m|
m∈Z(cid:18)t≥1 (cid:18) (cid:19) (cid:19) ! m∈Z(cid:18) (cid:19) t≥1 !
X X
here (s−s′)r = 1+εr > 1 and s′ ≥ 0 have been used.
(II - ii): Let s′ := s − (1/r −1/q) − ε, (ε > 0). It suffice to show the embedding
in the case s′ ≥ 0. Remark that q/r ∈ (1,∞) and (q/r)′ = 1/(r(1/r − 1/q)). Let
α := 1−r/q+εr.
1/r
r/q
hk¯is′q|(cid:3) f|q
  k  
Xkn Xk¯ 
 
  1/r
  r/q
= hk¯is′qhk iαq/r|(cid:3) f|q hk i−α
  n k  n 
Xkn Xk¯ 
 
  r/q  1/(q/r)′ 1/r
≤ hk¯is′qhk iαq/r|(cid:3) f|q × hk i−α(q/r)′
 n k   n  
Xkn Xk¯  Xkn
 
 
 1/q 
 
. hk¯is′qhk iαq/r|(cid:3) f|q
n k
!
k
X
1/q
= hkisq|(cid:3) f|q hk¯is′hk iα/rhki−s q
k n
( )
Xk (cid:16) (cid:17)
1/q
≤ suphk¯is′hk iα/rhki−s × hkisq|(cid:3) f|q .
n k
(cid:20) k (cid:21) ( k )
X
ε
Here, we have used α(q/r)′ = 1 + > 1. Because α/r = 1/r − 1/q + ε =
1/r−1/q
s−s′, s−s′ ≥0 and s′ ≥ 0,
hk¯is′hk iα/rhki−s = hkni s−s′ hk¯i s′ . 1.
n
hki hki
(cid:18) (cid:19) (cid:18) (cid:19)
(III - i):
1/q
1/q
sup hk¯isq|(cid:3) f|q ≤ hkisq|(cid:3) f|q .
k k
 
kn Xk¯ Xk !
 
Here, we have used s ≥ 0.
6 JAYSONCUNANANANDYOHEITSUTSUI
(III - ii): Using the embedding ℓq ֒→ ℓr,
1/r
r/q
1/q
hk¯isq|(cid:3) f|q ≤ hk¯isq|(cid:3) f|q
 k  k
 
Xkn Xk¯  Xk !
 
1/q
 
 
≤ hkisq|(cid:3) f|q .
k
!
k
X
In the last inequality, we need s ≥ 0.
(cid:3)
Lemma 2.2 (Triebel, [14]). Let 0 < p < ∞ and 0 < q ≤ ∞. Let Ω = {Ωk}k∈Zn be a
sequence of compact subsets of Rn. Let d be the diameter of Ω . If 0 < r < min(p,q),
k k
then there exist a constant c such that
|f (·−z)|
k
||sup || ≤ c||f ||
z∈Rn 1+|dkz|n/r Lp(ℓq) k Lp(ℓq)
holds for all f ∈ LpΩ(ℓq), where f = {fk}, ||fk||Lp(ℓq)= || ||fk(·)||ℓq||Lp and
LpΩ(ℓq) = {f | f = {fk}k∈Zn ⊂ S′,suppFfk ⊂ Ωk,and ||fk||Lp(ℓq)< ∞}.
Definition 2.2 (Maximal Functions). Let b > 0 and f ∈ S. Then
|(cid:3) f(x−y)|
(3) (cid:3)∗f(x):= sup k x ∈ Rn,k ∈ Zn
k y∈Zn 1+|y|b
n
Proposition 2.1. Let 0 < p < ∞ and 0 <q ≤ ∞, b > . Then
min(p,q)
1/q
(4) || hkisq|(cid:3)∗kf|q ||Lp(Rn)
k∈Zn !
X
and
1/r
r/q
(5) ||  hk¯isq|(cid:3)∗kf|q  ||Lp(Rn)
kXn∈Z k¯∈XZn−1
 
 
are equivalent norms inWp,q(Rn) and Wp,q,r(Rn), respectively.
s s
The proof is a direct consequence of Lemma 2.2, taking f = (cid:3) f. See also [13,
k k
Proposition].
3. Proof of the main results
First,wenarratetheideaoftheproof. Wegiveanequivalentformulationfor(cid:3) (Tf)(x¯),
k¯
afunctioninRn−1,viasome(cid:3) f(x¯,0)afunctioninRn.Thenwecomputeforpointwise
k¯,l
estimates between the corresponding ℓq norms and ℓr ℓq norms for cases 0< q < 1 and
kn k¯
1 ≤ q < ∞, separately. Finally, taking Lp(Rn−1) norms and using our equivalent norms
in Proposition (2.1), we arrive to our conclusion.
TRACE OPERATORS ON WIENER AMALGAM SPACES 7
We denote F (F−1) the partial (inverse) Fourier transform on x¯ (ξ¯) ∈ Rn−1. Write
x¯ ξ¯
{ϕk¯}k¯∈Zn−1 as versions of (1) in Rn−1. By the support property of ϕk¯, we observe
(cid:3) (Tf)(x¯) = (F−1ϕ F )(Tf)(x¯)
k¯ ξ¯ k¯ x¯
= {F−1ϕ F [(F−1ϕ Ff)(y¯,0)]}(x¯)
ξ¯ k¯ x¯ l
l∈Zn
X
= χ(|k¯−¯l|≤1) F−1ψk¯,lFf (x¯,0)
l∈Zn
X (cid:0) (cid:1)
(6) = χ (cid:3) f(x¯,0),
(|k¯−¯l|≤1) k¯,l
l∈Zn
X
where ψ (ξ) = ϕ (ξ¯)ϕ (ξ),l =(¯l,l ), and (cid:3) f := F−1ψ Ff. Note that the left-hand
k¯,l k¯ l n k¯,l k¯,l
side is a function in Rn−1 while the right-hand side is a function in Rn.
Recall our maximal function (3) and take y = y = ··· = y = 0,y = x we have
1 2 n−1 n n
for |x |≤ 1,
n
(7) |(cid:3) f(x¯,0)|. (cid:3)∗f(x).
k k
Proof of Theorem 1.1. We start by taking the ℓq-norm of (6). We write,
1/q 1/q
q
(8) hk¯isq|(cid:3) (Tf)(x¯)|q = hk¯isq χ (cid:3) f(x¯,0) .
 k¯   (|k¯−¯l|≤1) k¯,l 
k¯∈XZn−1 k¯∈XZn−1 lX∈Zn !
   
For 0 < q < 1, we estimate (8) by
1/q 1/q
hk¯isq|(cid:3) (Tf)(x¯)|q . h¯lisqχ |(cid:3) f(x¯,0)|q
 k¯   (|k¯−¯l|≤1) k¯,l 
k¯∈XZn−1 lX∈Znk¯∈XZn−1
    1/q
(9) = h¯lisq χ |(cid:3) f(x¯,0)|q
 (|k¯−¯l|≤1) k¯,l 
lXn∈Z¯l∈XZn−1 k¯∈XZn−1
 
Note that χ |(cid:3) f(x¯,0)|q= n−1|(cid:3) f(x¯,0)|q, where e is the jth
k¯∈Zn−1 (|k¯−¯l|≤1) k¯,l j=1 l±ej,l j
column of the identity matrix. In the sequel, it suffice to consider only the case j = 1.
P P
Moreover, we write (cid:3) f := (cid:3) f for some ψ satisfying (1). Using (7) we have,
l l±e1,l l
e 1/q 1/q
(10) h¯lisq|(cid:3) f(x¯,0)|q . h¯lisq|(cid:3)∗f(x¯,x )|q .
 l±e1,l   l n 
lXn∈Z¯l∈XZn−1 lXn∈Z¯l∈XZn−1
   e 
Combining(9)and(10), thentaking theLp(Rn−1)-norm andraisingtop-thpower gives,
1/q
||f(x¯,0)||p . || h¯lisq(cid:3)∗qf(x¯,x ) ||p
Wsp,q(Rn−1) l∈Zn l n ! Lp(Rn−1)
X
e
8 JAYSONCUNANANANDYOHEITSUTSUI
Integrating over x ∈ [0,1],
n
1/q
||f(x¯,0)||Wsp,q(Rn−1) . || h¯lisq(cid:3)∗lqf(x¯,xn) ||Lp(Rn)
l∈Zn !
X
. ||f||Wsp,q,q(Rne).
Note that the last inequality follows from Proposition 2.1.
For 1 ≤ q ≤ ∞, we use Minkowski’s inequality to give an upper bound of (8) as
follows,
1/q q 1/q
 hk¯isq|(cid:3)k¯(Tf)(x¯)|q .  hk¯isq χ(|k¯−¯l|≤1)(cid:3)k¯,lf(x¯,0) 
k¯∈XZn−1 k¯,¯l∈XZn−1 lXn∈Z
    1/q
.  hk¯isqχ(|k¯−¯l|≤1)|(cid:3)k¯,lf(x¯,0)|q
lXn∈Z k¯,¯l∈XZn−1
 1/q 
(11) . h¯lisq|(cid:3)∗f(x¯,x )|q .
 l n 
lXn∈Z ¯l∈XZn−1
 e 
Repeating the arguments above on (11) gives us the estimate,
||f(x¯,0)||Wsp,q(Rn−1). ||f||Wsp,q,1(Rn).
Hence, we arrive to our desired estimates.
Let η′ ∈ S(R) be a function with suppη′ ⊂ (−1/4,1/4) and (F−1)η′(0) = 1. For
ξn
any f ∈ Wp,q(Rn−1), we define g(x) = (T−1f)(x) := (F−1)η′(x ) f(x¯). We easily
s ξn n
see that g(x¯,0) = f(x¯) and (cid:3)kg = 0 when |kn|≥ 3. hMoreover, wei can decompose
(cid:3) g = (cid:3) f ·(cid:3) (F−1η′) due to the way ϕ is defined in (1). Now we do an estimate,
k k¯ kn ξn k
1/q∧1
q∧1/q
||g||Wsp,q,q∧1(Rn) = ||  hk¯isq|(cid:3)kg|q  ||Lp(Rn)
kXn∈Z k¯∈XZn−1
 
 
 1/q  1/q∧1
= || hk¯isq|(cid:3)k¯f|q  |(cid:3)kn(Fξ−n1η′)|1∧q ||Lp(Rn)
k¯∈XZn−1 |kXn|≤2
. ||f||Wsp,q(Rn−1).   
Thus, T−1 : Wp,q(Rn−1)→ Wp,q,q∧1(Rn).
s s
(cid:3)
TRACE OPERATORS ON WIENER AMALGAM SPACES 9
1/q 1/q
0 0
1 1
1/q′ 1/q′ 1/p−1/q
0 1 1/p 0 1 1/p
Figure 1. Comparison between the critical regularity index s for
TWp,q(Rn)= Wp,q(Rn−1) (left) and TMp,q(Rn) = Mp,q(Rn−1) (right).
s+ǫ s+ǫ
1/q
0
1
P
Q
0 1 1/p
Figure 2. Contradiction argument using interpolation.
As the end of this paper, we discuss the optimality of Corollary 1.1. We recall the
counterexample given in [9]. For 1 < p,q < ∞, there exist a function which shows
T : Mp,q (Rn)6→ Mp,q(Rn−1).
1/q′ 0
Since Mq,q = Wq,q, we also have T : Wq,q (Rn) 6→ Wq,q(Rn−1). Hence, Corollary
1/q′ 0
1.1 is sharp for p = q,1 < p,q < ∞ (refer to FIGURE 1). We now claim that it
is also sharp for all 1 < p,q < ∞. Contrary to our claim, suppose s = 1/q′ implies
TWp,q(Rn)= Wp,q(Rn−1). Then,byinterpolationwiththeestimateforapointQ(p ,q )
s 1 1
with s = 1/q′, one would obtain an improvement for thesegment connecting P(p,q) and
1
Q(p ,q ) (refer to FIGURE 2), which is not possible.
1 1
Acknowledgement
ThisworkwassupportedbyJSPS,through”ProgramtoDisseminateTenureTracking
System”. The second author was also partially supported by JSPS, through Grant-in-
Aid for Young Scientists (B) (No. 15K20919).
10 JAYSONCUNANANANDYOHEITSUTSUI
References
[1] A.B´enyi,K.Gr¨ochenig,K.Okoudjou,andL.Rogers.UnimodularFouriermultipliersformodulation
spaces. J. Funct. Anal., 246(2):366–384, 2007.
[2] E. Cordero and F. Nicola. Remarks on Fourier multipliers and applications to wave equation. J.
Math. Anal. Appl., 353:583–591, 2009.
[3] J. Cunanan, M. Sugimoto, and M. Kobayashi. Inclusion relations between Lp-Sobolev and Wiener
amalgam spaces. J. Funct. Anal.,419(2):738–747, 2014.
[4] L. Evans. Partial differential equations. Providence, R.I.: American Mathematical Society, 1998.
[5] H. G. Feichtinger. Banach convolution algebras of wiener’s type. In Proc. Conf. ”Function, Series,
Operators”, Budapest, Colloq.Math.Soc.J´anosBolyai,pages509–524.North-Holland,Amsterdam,
1980.
[6] H. G. Feichtinger. Banach spaces of distributions of wiener type and interpolation. In Functional
Analysis and Approximation, volume60, pages 153–165. Birkh¨auser Basel, 1981.
[7] H.G.Feichtinger.Modulationspacesonlocallycompactabeliangroups.Technicalreport,University
of Vienna, 1983.
[8] H.G.Feichtinger.Generalized amalgams, withapplicationstoFouriertransform. Canad. J. Math.,
42(3):395–409, 1990.
[9] H. G. Feichtinger, C. Huang, and B. Wang. Trace operators for modulation, α-modulation and
besov spaces. Applied and Computational Harmonic Analysis, 30(1):110 – 127, 2011.
[10] M. Frazier and B. Jawerth. Decomposition of besov spaces. Indiana Univ. Math. J., 34:777–799,
1985.
[11] K. Gr¨ochenig and C. Heil. Modulation spaces and pseudodifferential operators. Integral Equations
Operator Theory, 34(4):439–457, 1999.
[12] J. Toft. Continuity properties for modulation spaces, with applications to pseudo-differential
calculus-II. Ann. Global Anal. Geom., 26:73–106, 2004.
[13] H.Triebel. Modulation spaces on theeuclidean n-space.Z. Anal. Anwendungen, 2:443–457, 1983.
[14] H.Triebel. Theory of function spaces. Birkh¨auser-Verlag, 1983.
[15] H.Triebel. Theory of function spaces II.Birkh¨auser, 1992.
[16] B. Wang and C. Huang. Frequency-uniform decomposition method for generalized BO, KdV and
NLS equations. J. Differential Equations, 239:213–250, 2007.
Department of Mathematical Sciences, Faculty of Science, Shinshu University, Asahi
3-1-1, Matsumoto, Nagano, 390-8621, Japan
E-mail address: [email protected]
Department of Mathematical Sciences, Faculty of Science, Shinshu University, Asahi
3-1-1, Matsumoto, Nagano, 390-8621, Japan
E-mail address: [email protected]