Instructions for use
T itle W avelet bases in the weighted B esov and T riebel-L izorkin spaces with A ^loc_ p-weights
A uthor(s ) Izuki,Mitsuo; S awano,Y oshihiro; T achizawa,K azuya
C itation Hokkaido University Preprint S eries in Mathematics, 854: 1-22
Is s ue D ate 2007-05-16
D O I 10.14943/84004
D oc UR L http://hdl.handle.net/2115/69663
T ype bulletin (article)
Wavelet bases in the weighted Besov and Triebel-Lizorkin
spaces with
A
locp-weights
Mitsuo Izuki, Yoshihiro Sawano and Kazuya Tachizawa
May 16, 2007
Mitsuo Izuki, Kazuya Tachizawa
Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10 Nishi 8, Kita-ku, Sapporo, Hokkaido 060-0810, JAPAN.
E-mail: [email protected], [email protected]
Yoshihiro Sawano
Department of Mathematics and Information Sciences, Tokyo Metropolitan University Minami-Ohsawa 1-1, Hachioji-shi Tokyo 192-0397
E-mail: [email protected]
Abstract
The aim of this paper is to obtain the wavelet expansion in the Besov spaces and the Triebel-Lizorkin spaces coming with Aloc
p -weights. After characterizing
these spaces in terms of wavelet, we shall obtain unconditional bases and greedy bases.
Keywords Besov space, Triebel-Lizorkin space, wavelet, unconditional basis, greedy basis.
2000 Mathematics Subject Classification Primary 42B35 ; Secondary 41A17 ; 42C40 ; 42C15 ; 46B15.
1
Introduction
space, where w is an Ap-local weight defined by Rychkov [21]. As is well-known, the
class Ap was defined [19] in connection with the Hardy-Littlewood maximal operator
given by
M f(x) := sup
x∈Q
mQ(|f|) = sup x∈Q
1
|Q|
Z
Q
|f(y)|dy, f ∈Lloc1
whereQruns over all cubes whose edges are parallel to the coordinate axis andmQ(g)
denotes the average of g over a cube Q. Below we mean a compact cube whose edges are parallel to the coordinate axis simply by “cube ”. We also mean by “weight ” a non-negative measurable function which is locally integrable onRnand does not vanish dx-almost everywhere in Rn. A weight w is said to be an A1-weight, if there exists a constantc >0 such thatM w(x)≤c w(x) fordx-almost everywherex∈Rn. Letp >1. A weightw is said to be anAp-weight if
sup
Q
mQ(w)·mQ(w−
1
p−1)p−1 <∞,
where Q runs over all cubes. We refer to [8] for more information of this class of weights.
The present paper deals with its local version. To formulate the local version of Ap-class, first we introduce the following maximal operator due to Rychkov [21].
Mlocf(x) := sup
x∈Q ℓ(Q)≤1
mQ(|f|).
Here Q runs over all cubes whose edges are parallel to the coordinate axis and which have sidelength less than 1. Motivated by the definition due to Muckenhoupt, Rychkov defined the class of the weights as follows :
Alocp :={w : w is a weight withAploc(w)<∞},1≤p <∞,
where
Aloc1 (w) := esssupx∈Rn
Mlocw(x) w(x)
Alocp (w) := sup
ℓ(Q)≤1
mQ(w)·mQ(w−
1
p−1)p−1, p >1. We also define Aloc
∞ := S
1≤p<∞Alocp as a set. This class of weights enjoys properties
analogous toApsuch as the openness property and the (local) reverse H¨older inequality.
In [21] Rychkov defined the weighted Besov-spacesBp,qs,w and the weighted
Triebel-Lizorkin spacesFp,qs,w. We remark that for some smaller class of weights these function
spaces are investigated in [22].
The aim of the present paper is to characterize these weighted function spaces in terms of wavelets. We always place ourselves in the setting of function spaces coming with anAloc∞-weightw. This type of approach can be found in the textbook [26] when w ≡ 1. In [26] we can find more about wavelet characterization for the unweighted case. In particular [5, 6, 20, 23, 25] are pioneering works of this attempt withw ≡1. If w ∈ Ap, then Deng, Xu and Yan obtained some characterization of 1-dimensional
dotted function spaces ˙Fs
p,q(R, w) with|s| restricted to very small values [4]. A minor
modification shows that our characterization is applicable to the dotted function spaces ˙
Asp,q(Rn, w) withw∈Ap even if we do not assume that|s|is small.
Finally we describe the organization of this paper. In Section 2 we recall the definition of the function spaces Bp,qs,w and Fp,qs,w with w ∈ Aloc∞. As well as we define
the function spaces, we make a brief review of the maximal operator, the notion of bases in Banach spaces. In Section 3 we give an auxiliary result which is interesting of its own right, where we identify the dual space of the function spaces. This result will serve to define hf, τi, where f is a function in the weighted Besov spaces or the weighted Triebel-Lizorkin spaces and τ is a wavelet or a scaling function. Section 4 devotes to the wavelet characterizations. Finally in Section 5 we obtain a greedy basis, which enjoys a very nice property. Greedy bases, which was introduced in [14], begin to be studied recently. For example Garrig´os, Hern´andez and Martell investigated greedy bases in Orlicz spaces in [7]. In [9, 10, 12] we investigated ones in Lebesgue, weighted Sobolev spaces and Herz spaces. The present paper will deal with those in weighted Besov and Triebel-Lizorkin spaces.
2
Preliminaries
Function spaces As,wp,q with w∈Alocp
Let us describe precisely the function spaces As,wp,q. Throughout this paper, unless
additionally stated, we assume thatw is anAloc∞-weight. We keep to the following notations.
1. We setN0 ={0,1, . . .} and N={1,2, . . .}.
2. B(R) :={x∈Rn : |x|< R}.
3. Letf be a measurable function. Then define
kf : Lwpk:=
µZ
Rn
|f(x)|pw(x)dx ¶1
p
for 1< p <∞.
{fj}j∈N0, we define
k{fj}j∈N0 : lq(Lwp)k:= X
j∈N0
kfj : Lwpkq
1
q
k{fj}j∈N0 : Lwp(lq)k:= ° ° ° ° ° ° °
X
j∈N0
|fj|q
1
q
: Lwp ° ° ° ° ° ° ° .
5. Se denotes the set of all smooth functionsφ satisfying
qN(φ) := sup x∈Rn
X
|α|≤N
eN|x||∂αφ(x)|<∞
for allN ∈N0. TopologizeSe with{qN}N∈N0. Se′ denotes the topological dual of
Se.
Definition 2.1. Let L∈N0∪ {−1} ands∈R. Take a sequence {φj}j∈N0 ofCc∞(Rn)
-functions satisfying the following conditions.
1. L≥[s].
2. supp(φ0)⊂B(1).
3. φ1(x) =φ0(x)−2nφ0(2x).
4. φj(x) = 2(j−1)nφ1(2j−1x) for j ∈N.
5.
Z
Rn
xβφ1(x)dx= 0for all β ∈N0n with |β| ≤L. If L=−1, then this condition
means no condition.
6.
Z
Rn
φ0(x)dx6= 0.
This condition forφ0 is referred to asMs-condition in [21].
Using the sequence {φj}j∈N0 above, Rychkov defined the function space As,wp,q as
follows :
Definition 2.2. Suppose that the parameters p, q, s satisfy
1≤p <∞,1≤q≤ ∞, s∈R.
Assume thatL appearing in Definition 2.1 is greater than max(−1,[s]). Then define
kf : Bp,qs,wk:=k{2jsφj∗f}j∈N0 : lq(Lwp)k
kf : Fp,qs,wk:=k{2jsφj∗f}j∈N0 : Lwp(lq)k
Rychkov actually defined As,wp,q with 0 < p <∞,0 < q≤ ∞, s∈ Rand w ∈Aloc∞.
However, in the present paper we take up As,wp,q with the parameter indicated in the
definition above.
Rychkov proved the following theorem for w∈Aloc
∞.
Theorem 2.3. [21] Suppose that the parameters p, q, s satisfy
1≤p <∞,1≤q≤ ∞, s∈R
and w∈Aloc∞.
1. The sequence of {φj}j∈N0 in Definition 2.1does exist.
2. Different admissible choice of {φj}j∈N0 in Definition 2.1 will yield equivalent
norms, provided φ0 satisfies the Ms-condition.
As for the weighted function spaces, we remark hw
p was investigated by Bui [1],
which was equivalent toFp,0,w2 [21].
We also remark that the following atomic decomposition for these function spaces is obtained. To formulate the result, we need some notations. Let
Qν,m = n
Y
j=1
· mj
2ν ,
mj + 1
2ν
¸ .
Denote by χ(ν,mp) the p-normalized indicator of this cube : χ(ν,mp) (x) := 2
nν p χ
Qν,m(x). Given a doubly indexed sequenceλ={λνm}ν∈N0,m∈Zn, define
kλ : bwp,qk:=
° ° ° ° ° °
( X
m∈Zn
λνmχ(ν,mp)
)
ν∈N0
: lq(Lwp)
° ° ° ° ° °
kλ : fp,qw k:=
° ° ° ° ° °
( X
m∈Zn
λνmχ(ν,mp)
)
ν∈N0
: Lwp(lq)
° ° ° ° ° °.
awp,q denotes eitherbwp,q orfp,qw .
Proposition 2.4. [11] Suppose that the parameters p, q, s and an integer L satisfy
1≤p <∞,1≤q≤ ∞, s∈R, L≥ −1.
Assume in addition thatw∈Aloc
∞. Then there exist constantsc0, c1, c2 >0andcα, α∈
N0n with the following property.
1. Let f ∈As,wp,q. Then f admits the following decomposition.
f =
∞ X
ν=0
X
m∈Zn
where the coefficientλ={λνm}ν∈N0, m∈Zn and{aνm}ν∈N0, m∈Zn satisfy conditions (1)–(4) given below.
kλ : awp,qk ≤c0kf : As,wp,qk (1)
supp(aνm)⊂c1Qν,m (2)
aνm ∈C∞,k∂αaνm : L∞k ≤cα2−ν
“
s−n p ”
+ν|α|
for all α∈N0n (3)
Z
Rn
xβaνm(x)dx= 0 for allν ∈N, m∈Zn, β∈N0n with |β| ≤L. (4)
If L=−1, then (4) means no condition.
2. Assume that the coefficient λ = {λνm}ν∈N0, m∈Zn and {aνm}ν∈N0, m∈Zn satisfy
conditions (2), (4),
kλ : as,wp,qk<∞ (5)
and
aνm∈C(1+[s])+,k∂αaνm : L∞k ≤cα2−ν
“
s−n p ”
+ν|α|
. (6)
for allα∈N0 with|α| ≤(1 + [s])+. Then
f =
∞ X
ν=0
X
m∈Zn
λνmaνm
converges in S′
e and satisfies the norm estimate
kf : As,wp,qk ≤c2kλ : awp,qk.
Local maximal operators Now we collect some auxiliary results concerning the local maximal operator, which is also due to Rychkov [21].
Definition 2.5. Let r >0 and η >0. Then define
Mloc(η)f(x) := sup
x∈Q ℓ(Q)≤r
mQ(|f|η)
1
η.
Define also Mlocf(x) :=Mloc(1)f(x).
It can happen that the precise values of r appearing implicitly in Mloc(η)f(x) are different in every occurrence.
In this paper we use the following boundedness of this maximal operator.
Proposition 2.6. Suppose that 1 < p < ∞ and 1 < q ≤ ∞. Assume that w ∈Aloc
p .
Then there exists a constantc >0 such that
k{Mlocfj}j∈N0 : Lp(lq)k ≤ck{fj}j∈N0 : Lp(lq)k
Unconditional bases
Finally to formulate our results, let us recall the definition of unconditional bases. LetX be a Banach space and A a countable index set in this section.
We recall the definition of several kinds of bases. The first one is unconditional basis. It is known that there are several equivalent definitions of unconditional basis in Banach spaces [13, 17].
Definition 2.7. Let {xm}m∈A be a sequence of elements in X. The series
X
m∈A
xm is
said to converge unconditionally in X, if
∞ X
i=1
xσ(i) converges for every bijection from N
toA.
Next we introduce several kinds of bases, some of which are defined by Konyagin and Temlyakov [14]. Now we follow closely to [14].
Definition 2.8. {xk}∞k=1 ⊂ X is said to be a Schauder basis if there exists a unique
sequence{ck(x)}∞k=1 ⊂Csuch thatx=
∞ X
k=1
ck(x)xkinXfor allx∈X. Furthermore, if
the convergence above is always unconditional, then the basis is said to be unconditional.
Now we recall the definition of greedy basis and democratic basis.
Definition 2.9. Let {xk}∞k=1 be a Schauder basis in X normalized as kxkkX = 1 for
allk∈N.
1. {xk}∞k=1 is called greedy forX if there exists a constantC >0such that for every x∈X there exists a permutationρ of N which satisfies
¯
¯cρ(1)(x)¯¯≥¯¯cρ(2)(x)¯¯≥. . .≥¯¯cρ(N)(x)¯¯≥. . .
and °
° ° ° °x−
N
X
k=1
cρ(k)(x)xρ(k) ° ° ° ° °
X
≤C inf
y∈ΣN
kx−ykX,
for everyN ∈N, where ΣN :=
( X
ν∈Λ
ανxν : αν ∈C, ♯Λ≤N,Λ⊂N
)
.
2. {xk}∞k=1 is called democratic for X if there exists a constant C >0 independent
of P andQ such that
° ° ° ° °
X
k∈P
xk
° ° ° ° °
X
≤C
° ° ° ° ° °
X
k∈Q
xk
° ° ° ° ° °
X
We remark that “unconditional ” and “democratic ” are independent notions. In-deed, in [14, Section 3] Konyagin and Temlyakov gave some examples of bases, which are not democratic but unconditional, or which are not unconditional but democratic. However, if we assume that the basis is unconditional. Then the above two notions agree. That is, we have the following.
Proposition 2.10. [14, Section 1] Let {xk}∞k=1 be a Schauder basis in X such that
kxkkX = 1for allk∈N. Then{xk}∞k=1 is a greedy basis if and only if it is unconditional
and democratic.
3
Duality
In this section we prove an auxiliary result which is interesting of its own right.
Lemma 3.1. Suppose that θ > 0, 1 ≤ p < ∞,1 ≤ q ≤ ∞ and w ∈ Aloc∞. Then we have
° ° ° ° ° °
( ∞ X
l=0
2−θ|l−j||fl|
)
j∈N0
: Lwp(lq)
° ° ° ° °
°≤ck{fj}j∈
N : Lwp(lq)k °
° ° ° ° °
( ∞ X
l=0
2−θ|l−j||fl|
)
j∈N0
: lq(Lwp)
° ° ° ° °
°≤ck{fj}j∈
N : lq(Lwp)k
for all sequences of measurable functions{fj}j∈N0.
Proof. This lemma is proved easily by H¨older’s inequality and we omit the proof.
Theorem 3.2 (Duality As,wp,q-A−s,w
−p−11
p′,q′ ). Let w be an Alocp -weight. Suppose that the
parametersp, q, s satisfy
1< p <∞,1≤q <∞, s∈R.
Set v:=w−p−11. Then we have the following.
1. For all g∈A−p′s,v,q′, the mapping
f ∈ Se7→ hg, fi ∈C
extends to a continuous linear functional on As,wp,q. Speaking precisely, we have
|hg, fi| ≤ckg : A−p′s,v,q′k · kf : A
s,w p,qk
2. Conversely any continuous functional Φ can be realized by someg∈A−p′s,v,q′
satis-fying
kg : A−p′s,v,q′k ≤ckΦ : A
s,w p,qk,
where c depends only on the parameters p, q, s and the Aloc
p -constant of w.
Proof. We concentrate on theF-scale, the counterpart for theB-scale being the same. Letf ∈ Se. By the local reproducing formula [21, Theorem 1.6], we have
f =
∞ X
j=0
ψj ∗φj∗f,
where the convergence takes place in Se. Therefore, we obtain
hg, fi=
∞ X
j=0
hg, ψj ∗φj∗fi=
∞ X
j=0
Z
Rn ˇ
ψj∗g(x)·φj∗f(x)dx,
where ˇψj(x) :=ψj(−x). A repeated application of the H¨older inequality gives us
° ° ° ° ° °
∞ X
j=0
Z
Rn ˇ
ψj∗g(x)·φj∗f(x)dx
° ° ° ° ° °
≤ k{2−jsψˇj ∗g}∞j=0 : Lvp′(lq′)k · k{2jsφj ∗f}j∞=0 : Lwp(lq)k ≤ckg : Fp−′,qs,v′ k · kf : F
s,w p,q k,
proving1. Here we have used
k{2−jsψˇj∗g}∞j=0 : Lvp′(lq′)k ≤ckg : Fp−′,qs,v′k
to obtain the second inequality.
Let us prove 2. To do this, we pick a continuous functional Φ on Fp,qs,w arbitrarily.
LetY be the closure of the image of the mapping
f ∈Fp,qs,w7→ {2jsφj∗f}∞j=0∈Lwp(lq).
Then we define a functional Ψ onY uniquely so that it satisfies
Ψ({2jsφj∗f}∞j=0) = Φ(f) for all f ∈Fp,qs,w. Observe that
|Ψ({2jsφj ∗f}∞j=0)|=|Φ(f)| ≤ck2jsφj∗f : Lwp(lq)k.
Therefore Ψ extends to a continuous linear functional on Y. By the Hahn-Banach extension theorem, Ψ extends even to a continuous linear functional onLw
sake of simplicity let us denote it by Ψ again. By the dualityLw
p(lq)-Lvp′(lq′), which is
well-known, there exists{gj}∞j=0∈Lvp′(lq′) such that
k{gj}∞j=0 : Lvp′(lq′)k=kΨ : Lwp(lq)∗k and Ψ({fj}∞j=0) =
∞ X
j=0
Z
Rn
fj(x)gj(x)dx
for all {fj}∞j=0 ∈ Lwp(lq). In particular, putting the observations above together, we
obtain
Φ(f) = Ψ({2jsφj ∗f}∞j=0) =
∞ X
j=0
Z
Rn
2jsφj∗f(x)gj(x)dx.
In view of the observation above, we are to define
g:=
∞ X
j=0
2jsφˇj∗gj. (7)
In order that g makes sense at least in Se′, we shall claim that the sum converges at least inS′
e and then we shall establish
kg : Fp−′,qs,v′ k ≤ckΦ : F
s,w
p,q ∗k. (8)
Once we prove that (7) converges inSe′ and that inequality (8) holds, then the theorem will have been proved.
First, let us verify that the sum converges. To do this we take ϕ∈ Se. It is easy
to show that each summand belongs toSe′. Therefore, we may assume that g0= 0. In view of this assumption let us assumej≥1 below.
h2jsφˇj∗gj, ϕi= 2js
Z
Rn
ϕ∗φj(x)gj(x)dx.
Observe that
ϕ∗φj(x) =
Z
Rn
ϕ(x−y)− X
|β|≤L
(−y)β
β! ∂
βϕ(x)
φj(y)dy
by virtue of the moment condition ofφj. Using the Taylor expansion, we see
eN|x| sup
y∈B(2−j+1)
¯ ¯ ¯ ¯ ¯
¯ϕ(x−y)− X
|β|≤L
(−y)β
β! ∂
βϕ(x)
¯ ¯ ¯ ¯ ¯ ¯
≤c2−j(L+1)eN|x| sup
|β|≤L+1
sup
y∈B(2−j+1)
|∂βϕ(x−y)|
≤c2−j(L+1)qN(ϕ),
for all N ≫L. Hence, we obtain
Adding this overj ∈N, we obtain
g=
∞ X
j=0
2jsφˇj∗gj
converges at least inS′
e.
Having checked that the sum converges, let us turn to the proof that the sum belongs to Fp−′,qs,v′ . To do this, we observe that
|φj∗φˇk(x)| ≤c2jn−(L+1)(k−j)χB(2−j+2)(x), whenever j≤k. Indeed, a simple calculus shows
supp(φj ∗φˇk)⊂supp(φj) + supp( ˇφk)⊂B(2−j+1) +B(2−k+1)⊂B(2−j+2).
To see that theL∞-norm is less thanc2jn−(L+1)(k−j), by dilation we may assumej= 1. We remark that the case when j = 0 can be incorporated readily. Then in this case, going through the same argument as before, we obtain the desired upper bound of φj∗φˇk. That is, we use the Taylor expansion : LettingL=Lφ1, we obtain
φj ∗φˇk(x) =
Z
Rn
φ1(x−y)
φˇk(y)− X
|β|≤L
∂βφˇk(0) β! y
β
dy.
Since the mean value theorem gives us
¯ ¯ ¯ ¯ ¯ ¯
ˇ φk(y)−
X
|β|≤L
∂βφˇk(0)
β! y
β
¯ ¯ ¯ ¯ ¯ ¯≤c2
−(L+1)k,
the desired estimate |φj ∗φˇk(x)| ≤ c2jn−(L+1)(k−j) holds for j, k ≥ 0. Inserting the
estimate, we finally obtain
|φj∗g(x)| ≤c
∞ X
l=0
2ls−(L+1)|j−l|Mlocgl(x).
Therefore, we obtain
k{2−jsφj∗g}∞j=0 : Lvp′(lq′)k ≤c
° ° ° ° ° °
( ∞ X
l=0
2(l−j)s−(L+1)|j−l|Mlocgl
)∞
j=0
: Lvp′(lq′)
° ° ° ° ° °
≤c°°{Mlocgj}j∞=0 : Lvp′(lq′)
° °
≤ckΨ : (Fp,qs,w)∗k
by virtue of Proposition 2.6 and Lemma 3.1. Therefore, we concludeg∈Fp−′,qs,v′ realizes
Ψ.
Corollary 3.3. Suppose that the parameters p, q, s satisfy
1< p <∞,1≤q≤ ∞, s∈R.
Assume in addition thatw∈Aloc
p and setv :=w
− 1
p−1. Then forf ∈As,w
p,q, one has the
following norm equivalence.
c−1kf : As,wp,qk ≤supn|hf, gi| : g∈ Se,kg : A−p′s,v,q′k ≤1
o
≤ckf : A−p,qs,vk.
4
Wavelet characterizations of
A
s,wp,qNotation . Let j∈Z andk∈Zn.
1. Given a functionψ defined onRn, we define
ψj,k(x) := 2jn/2ψ(2jx−k).
2. In order to describe wavelets conveniently, we define the index setE by
E :={1,2, . . . ,2n−1}.
According to wavelet theory, there exists compactly supportedCr-functions{ϕ, ψǫ :
ǫ∈E} such that
Z
Rn
xαψǫ(x)dx= 0 for all ǫ∈E and all α∈N0n with|α| ≤r, (9)
and the sequences
©
ϕ0,k, ψj,kǫ : ǫ∈E, j ∈N0, k∈Znª (10)
form orthonormal bases inL2(Rn) [18, 27]. We often say that the functionϕis a scaling function and eachψǫ is a wavelet in terms of a multiresolution analysis. In particular, we can construct them so thatϕand eachψǫ are real-valued and compactly supported
with suppϕ= suppψǫ = [0,2N −1]n for any positive integer N ≥2. In this case, the constantr is an increasing function ofN [3, 16, 18].
In order to obtain characterizations of function spaces, we sometimes need not only wavelets but also scaling functions [15, 18]. Throughout this paper, we consider a set of functions{ϕ, ψǫ : ǫ∈E}satisfying the following three conditions (A), (B) and (C):
(A) ϕand each ψǫ are compactly supported.
(B) ϕ and each ψǫ belong to Cr(Rn), and each ψǫ satisfies condition (9) for some r∈N0.
Lemma 4.1. Assume the functions ψǫ, ϕ ∈ C(1+[s])+ are compactly supported
func-tions such that
Z
Rn
xβψǫ(x)dx= 0
for allǫ∈E and β∈N0n with |β| ≤[s]. Define
mj,A,B(y) := (1 + 2j|y|)A2|y|B,
where A, B are sufficiently large. Then we have
° ° ° ° °ysup∈Rn
|hf, ϕ(· −x+y)i|
m0,A,B(y)
: Lwp(Rnx)
° ° ° °
°≤ckf : A
s,w p,qk
and
sup
ǫ∈E
° ° ° ° ° ° (
2j(s+n2) sup
y∈Rn
|hf, ψj,ǫ0(· −x+y)i|
mj,A,B(y)
)
j∈N0
: lq(Lwp(Rnx))
° ° ° ° °
°≤ckf : B
s,w
p,qk (11)
sup
ǫ∈E
° ° ° ° ° ° (
2j(s+n2) sup
y∈Rn
|hf, ψj,ǫ0(· −x+y)|
mj,A,B(y)
)
j∈N0
: Lwp(Rnx, lq)
° ° ° ° °
° ≤ckf : F
s,w
p,q k. (12)
for allf ∈ Se.
Proof. The proof is obtained by using the fact that ψǫ
jk has a vanishing moment of
order up toL(≥[s]). The proof is similar to the one in [21, Theorem 2.5] and we omit it.
Definition 4.2. Suppose that the parameters p, q, s and the weightw satisfy
1< p <∞,1≤q ≤ ∞, s∈R, w∈Alocp .
Assume in addition that {ϕ, ψǫ : ǫ ∈ E} satisfy conditions (A), (B) and (C) with
r= max((1 + [s])+,(1 + [−s])+). Define
Bs,wp,q(f)
:= ° ° ° ° ° X
k∈Zn
hf, ϕ0,kiχQ0,k : L
w p ° ° ° ° °+ X
ǫ∈E
° ° ° ° n
2j(s+n2)hf, ψǫ
j,kiχQj,k
o
j∈N0, k∈Zn : b
w p,q ° ° ° °
Fp,qs,w(f)
:= ° ° ° ° ° X
k∈Zn
hf, ϕ0,kiχQ0,k : L
w p ° ° ° ° °+ X
ǫ∈E
° ° ° ° n
2j(s+n2)hf, ψǫ
j,kiχQj,k
o
j∈N0, k∈Zn : f
w p,q ° ° ° °.
for f ∈ Bp,qs,w and f ∈ Fp,qs,w respectively. To simplify our formulation, we denote by
As,wp,q(f) eitherBp,qs,w(f) or Fp,qs,w(f).
Remark 4.3. Since ϕ and each ψǫ are bounded and compactly supported, we see
that each hf, ϕ0,ki and hf, ψj,kǫ i make sense. Indeed, if φis a C([−s]+1)+-function with
compact support, then it is not so hard to see that φ∈A−p,qs,v by virtue of Proposition
2.4. Therefore, we are in the position of using Theorem 3.2 to see that each hf, ϕ0,ki
and hf, ψǫ
j,ki make sense.
Theorem 4.4. Suppose that the parameters p, q, s and the weight w satisfy
1< p <∞,1≤q ≤ ∞, s∈R, w∈Alocp .
Assume in addition that the functions{ϕ, ψǫ : ǫ∈E} satisfy conditions(A), (B) and
(C)withr= max((1 + [s])+,(1 + [−s])+). Then there exists a constantc >0 such that
c−1kf : As,wp,qk ≤ As,wp,q(f)≤ckf : As,wp,qk
for allf ∈As,wp,q.
Proof. We assume that the functions ϕ, ψǫ, ǫ ∈ E are all real-valued. When they are complex-valued, a minor modification of the proof below works. We begin with the proof of the right-inequality. We observe that δ > 0 can be taken so that r = max((1 + [s+δ])+,(1 + [−s+δ])+). By Proposition 2.4 there exists a sequence of functions{fm}∞m=1⊂ Se such that
lim
m→∞fm=f
in the topology ofS′ and that
kfm : As,wp,qk ≤ckf : As,wp,qk. (13)
By Proposition 2.4 we may as well assume that the convergence in (13) takes place in Asp,q−δ. Therefore, by Theorem 3.2 we see that
lim
m→∞ X
k∈Zn
hfm, ϕ0,kiχQ0,k =
X
k∈Zn
hf, ϕ0,kiχQ0,k
lim
m→∞ X
k∈Zn
2j(s+n2)hf
m, ψj,kǫ iχQj,k =
X
k∈Zn
2j(s+n2)hf, ψǫ
j,kiχQj,k
in the sense of the pointwise convergence. By using Fatou’s lemma we obtain
As,wp,q(f)≤lim inf
m→∞ A
s,w
p,q(fm). (14)
Therefore, (13) and (14) justify that we may assume f ∈ Se even when q =∞. With
this in mind, let us prove the right-inequality. Now that
X
k∈Zn
2j(s+n2)|hf, ψǫ
j,ki|χQj,k(x)≤2
j(s+n
2) sup
y∈Rn
|hf, ψǫj,0(· −x+y)|
mj,A,B(y)
.
If we assume that f ∈As,wp,q, then we can expandf into a wavelet series :
f = X
k∈Zn
hf, ϕ0kiϕ0.k+
X
ǫ∈E
X
ν∈N X
k∈Zn
hf, ψνkǫ iψǫνk.
Indeed, the right-hand side converges by virtue of the right inequality, which we have just established. To verify that the right-hand side agrees withf, we have only to check by evaluating atg∈ Se. It is not so hard to see
hf, gi= X
k∈Zn
hf, ϕ0kihϕ0,k, gi+
X
ǫ∈E
X
ν∈N X
k∈Zn
hf, ψǫνkihψǫνk, gi.
Therefore, we are in a position of using Proposition 2.4 to see that
kf : As,wp,qk ≤cAs,wp,q(f).
This is the desired result.
5
Wavelet bases in
A
s,wp,q5.1 Unconditional bases in As,wp,q
Theorem 5.1. Let 1 < p < ∞, 1 ≤ q ≤ ∞, s ∈ R, w ∈ Alocp , and the functions
{ϕ, ψǫ : ǫ∈E}satisfy conditions(A), (B)and(C)withr= max((1+[s])
+,(1+[−s])+).
Then the sequence ©
ϕ0,k, ψj,kǫ : ǫ∈E, j ∈N0, k∈Znª
forms an unconditional basis in As,wp,q.
Proof. There are some equivalent definitions of unconditional basis [13, 17, 27]. Now we shall check one of them. It suffices to check the following :
1. There exists a constantC >0 independent off,A and B such that
°
°TA,Bf : As,wp,q
°
°≤C°°f : As,wp,q°° (15)
for allf ∈As,wp,q and all finite subsetsA⊂Znand B ⊂E×N0×Zn, where
TA,Bf :=
X
k∈A
hf, ϕ0,kiϕ0,k+
X
(ǫ,j,k)∈B
f, ψǫj,k®ψj,kǫ .
2. The set span{ϕ0,k}k∈Zn∪span{ψǫ
j,k : ǫ∈E, j ∈N0, k ∈Zn} is dense inA s,w p,q.
We show (15) first. By the orthonormality and Theorem 4.4, we obtain
c−1°°TA,Bf : As,wp,q
°
°≤ As,wp,q(TA,Bf)≤ As,wp,q(f)≤c
°
wherec≥1 is the constant appearing in Theorem 4.4.
Next we check the condition 2. It suffices to prove that for all f ∈As,wp,q,
lim
A→Zn,B→E×N0×ZnA
s,w
p,q(f−TA,Bf) = 0,
sincekf−TA,Bf : As,wp,qk ≤cAs,wp,q(f−TA,Bf) by Theorem 4.4. Now define
A1(f) :=
° ° ° ° ° X
k∈Zn
hf, ϕ0,kiχQ0,k : L
w p ° ° ° ° °
A2(f) :=X
ǫ∈E
° ° ° ° n
2j(s+n2)hf, ψǫ
j,kiχQj,k
o
j∈N0, k∈Zn : a
w p,q ° ° ° °,
whereawp,q:=fp,qw whenAs,wp,q =Fp,qs,w, andawp,q:=bwp,qwhenAs,wp,q =Bp,qs,w. Then we have
As,wp,q(f−TA,Bf) =A1(f−TA,Bf) +A2(f −TA,Bf). We omit the detail because it is
obvious that the orthonormality of the system{ϕ0,k}k∈Zn∪ {ψǫ
j,k : ǫ∈E, j∈N0, k∈
Zn} with regard to theL2-inner product, the boundedness ofAν(f −TA,Bf) for each
ν= 1,2 and Lebesgue’s dominated convergence theorem give us the desired result.
5.2 Greedy bases in Fp,qs,w
Theorem 5.2. Assume the same condition as Theorem 5.1. Define
g ϕ0,k:=
ϕ0,k
kϕ0,k :Fp,qs,wk
and ψgǫ j,k :=
ψǫ j,k
kψǫ j,k :F
s,w p,q k
.
Then the sequence n
g
ϕ0,k,ψgj,kǫ : ǫ∈E, j ∈N0, k∈Zn
o
forms a greedy basis in Fp,qs,w.
Lemma 5.3. Let 1< p <∞ and w ∈Aloc
p . Then there exists a constant 1< d < ∞
such that for all dyadic cubes Q, Q′ satisfyingQ′ (Q and |Q′|,|Q| ≤1,
d w(Q′)≤w(Q).
Proof. The proof of Lemma 5.3 is a local version of [8, p.141] and [24, Proof of Corollary 1.1]. Going through an argument similar to the ones in these literature, we can prove Lemma 5.3.
We need another characterization of Fp,qs,w in order to obtain greedy bases in terms
of wavelets. Let us write
f
F1(f) :=
° °
°©hf, ϕ0,ki
°
°ϕ0,k : Fp,qs,w
° °ª
k∈Zn : l
p(Zn)°°
°
f
F2(f) :=
X
ǫ∈E
° ° ° ° n
w(Qj,k)−1/p
°
°ψj,kǫ : Fp,qs,w°°hf, ψj,kǫ iχQj,k
o
j∈N0,k∈Zn : f
Lemma 5.4. Under the same condition as Theorem 5.1 there exists a constant c >0
such that for all f ∈Fp,qs,w,
c−1Ff1(f)≤
° ° ° ° ° X
k∈Zn
hf, ϕ0,kiχ0,k : Lwp
° ° ° °
°≤cFf1(f) (16)
and
c−1Ff2(f)≤X
ǫ∈E
° ° ° ° n
2j(s+n2)hf, ψǫ
j,kiχQj,k
o
j∈N0, k∈Zn : f
w p,q
° ° °
°≤cFf2(f). (17)
Proof of Lemma 5.4. By Theorem 4.4, there exists a constantc≥1 such that for each ǫ∈E,j ∈N0 andk∈Zn,
c−1°°ψj,kǫ : Fp,qs,w°°≤ Fp,qs,w(ψj,kǫ )≤c°°ψj,kǫ : Fp,qs,w°°.
On the other hand, we have Fp,qs,w(ψǫj,k) = 2j(s+n/2)w(Qj,k)1/p. Thus we can easily
obtain the inequality (17).
Next we show the estimate (16). Using Theorem 4.4 again, we get
c−1°°ϕ0,k : Fp,qs,w
°
°≤ Fp,qs,w(ϕ0,k)≤c
°
°ϕ0,k : Fp,qs,w
° °.
Meanwhile we haveFp,qs,w(ϕ0,k) =kχQ0,k : L
w pkand
° ° ° ° ° X
k∈Zn
hf, ϕ0,kiχ0,k : Lwp
° ° ° ° °= ° °
°©hf, ϕ0,kikχQ0,k : L
w pk
ª
k∈Zn : l
p(Zn)°°
°.
These facts imply (16).
Proof of Theorem 5.2. The proof we give here is based on the proof of [2, Lemma 4.1]. In view of Theorem 5.1 and Proposition 2.10, it is enough to prove that the sequence
n g
ϕ0,k,ψgj,kǫ : ǫ∈E, j ∈N0, k ∈Zn
o
is democratic. We see that{ϕ0,k, ψǫj,k : ǫ∈E, j ∈N0, k∈Zn}forms an unconditional basis forFp,qs,w by Theorem 5.1. Thus for all f ∈Fp,qs,w we can write
f = X
k∈Zn
ak(f)ϕg0,k+
X
ǫ∈E
∞ X
j=0
X
k∈Zn
bǫj,k(f)ψgǫ j,k,
whereak(f) :=hf, ϕ0,kikϕ0,k : Fp,qs,wk and bǫj,k(f) :=hf, ψj,kǫ ikψǫj,k : Fp,qs,wk. Combining
Theorem 4.4 and Lemma 5.4, we see that the norm
à X
k∈Zn
|ak(f)|p
!1/p
+X
ǫ∈E
° ° ° ° ° ° ° ∞ X j=0 X
k∈Zn
¯ ¯
¯w(Qj,k)−1/pbǫj,k(f)χQj,k
¯ ¯ ¯q 1/q
is equivalent to kf : Fp,qs,wk. Let us denote ϕfQ := ϕgj,k and ψfQǫ := ψgǫj,k for a dyadic
cube Q = Qj,k. Now we take finite subsets Aν ⊂ {Q0,k : k ∈ Zn} and Λν ⊂ E ×
{Qj,k : j ∈ N0, k ∈ Zn} satisfying ♯A1 +♯Λ1 = ♯A2 +♯Λ2 arbitrarily, and write gν :=
X
I∈Aν
f ϕI+
X
(ǫ,J)∈Λν
f ψǫ
J forν = 1,2. We also denote
Bν :={Qj,k : (ǫ, Qj,k)∈Λν for someǫ∈E}.
Then we see that ♯Bν ≤♯Λν ≤(2n−1)♯Bν for ν= 1,2. Using (18), we obtain
c−1kg1 : Fp,qs,wk
≤(♯A1)1/p+
° ° ° ° ° ° ° X
(ǫ,J)∈Λ1
¯ ¯
¯w(J)−1/pχJ
¯ ¯ ¯q 1/q
: Lwp ° ° ° ° ° ° °
≤(♯A1)1/p+ (2n−1)1/q
Z [
J′∈B1
J′ X
J∈B1
w(J)−q/pχJ(x)
p/q
w(x)dx 1/p . (19)
For eachx ∈ [
J∈B1
J, J1(x) denotes the minimal dyadic cube in B1 with regard to the
inclusion relation that containsx. Then we get
X
J∈B1
w(J)−q/pχJ(x)≤
∞ X
r=0
w(Jr)−q/p, (20)
where J0 := J1(x), Jr is a dyadic cube satisfying Jr−1 ⊂ Jr and 2n|Jr−1| = |Jr| for
everyr ∈N. By Lemma 5.3, there exists a constant 1< d <∞such that for allr ∈N,
w(Jr)≥dw(Jr−1)≥. . .≥drw(J0) =drw(J1(x)).
Thus we have
∞ X
r=0
w(Jr)−q/p≤
∞ X
r=0
(drw(J1(x)))−q/p= (1−d−q/p)−1w(J1(x))−q/p. (21) By virtue of (20) and (21), we obtain
Z [
J′∈B1
J′ X
J∈B1
w(J)−q/pχJ(x)
p/q
w(x)dx
≤
Z [
J′∈B1
J′ ³
c1w(J1(x))−q/p
´p/q
w(x)dx
=cp/q Z
[
J′∈B1
J′w(J1(x))
Now we set ˜J :=
x∈
[
J′∈B1
J′ :J1(x) =J
for eachJ ∈B1. Then, since ˜J ⊂J and [
J′∈B1
J′= [
J∈B1 ˜
J, it follows that
Z [
J′∈B1
J′w(J1(x))
−1w(x)dx=Z
[
J∈B1 ˜
Jw(J1(x))
−1w(x)dx
≤ X
J∈B1
Z
˜
J
w(J1(x))−1w(x)dx
= X
J∈B1
Z
J
w(J)−1w(x)dx
=♯B1. (23)
By virtue of (19)–(23), we have
ckg1 : Fp,qs,wk ≤(♯A1)1/p+ (2n−1)1/qc(♯B1)1/p
≤(♯A1)1/p+ (2n−1)1/qc(♯Λ1)1/p.
Hence there exists a constantc2 >0 independent ofg1,A1 and Λ1 such that
kg1 : Fp,qs,wk ≤c (♯A1+♯Λ1)1/p. (24)
On the other hand, applying (18) tof =g2, we have
ckg2 : Fp,qs,wk ≥(♯A2)1/p+
° ° ° ° ° ° ° X
(ǫ,J)∈Λ2
¯ ¯
¯w(J)−1/pχJ
¯ ¯ ¯q 1/q
: Lwp ° ° ° ° ° ° °
≥(♯A2)1/p+
Z [
J′∈B2
J′ X
J∈B2
w(J)−q/pχJ(y)
p/q
w(y)dy 1/p . (25)
For each y∈ [
J∈B2
J,J2(y) denotes the minimal dyadic cube inB2 with regard to the
inclusion relation that containsy. Then we have
X
J∈B2
w(J)−q/pχJ(y)
p/q
≥w(J2(y))−1. (26)
Now going through the same argument as (20)–(21), replacing “B1, −q/pand J1(x) ” by “B2, −1 andJ2(y) ” respectively, we get
X
J∈B2
wherec >0 is a constant independent ofg2,A2 and Λ2. Using (25)–(27), we obtain
ckg2 : Fp,qs,wk ≥(♯A2)1/p+
Z
[
J′∈B2
J′c −1 X
J∈B2
w(J)−1χJ(y)w(y)dy
1/p
= (♯A2)1/p+
c−1 X
J∈B2
w(J)−1
Z
J
w(y)dy
1/p
= (♯A2)1/p+c−1/p(♯B2)1/p
≥(♯A2)1/p+c−1/p(♯Λ2)1/p.
Namely we can take a constantc >0 independent ofg2,A2 and Λ2 so that
ckg2 : Fp,qs,wk ≥(♯A2+♯Λ2)1/p. (28)
Since ♯A1+♯Λ1 =♯A2+♯Λ2, (24) and (28) yield
kg1 : Fp,qs,wk ≤ckg2 : Fp,qs,wk.
Consequently we have proved that the sequence {ϕg0,k,ψgj,kǫ : ǫ∈E, j ∈N0, k ∈Zn} is democratic.
Acknowledgement
Yoshihiro Sawano was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Kazuya Tachizawa was partly supported by Grants-in-Aid for Scientific Research, the Japan Society for the Promotion of Science.
References
[1] H. Q. Bui, Weighted Hardy spaces, Math. Nach.103 (1981), 45–62.
[2] A. Cohen, R. A. DeVore and R. Hochmuth, Restricted nonlinear approximation, Constr. Approx.16(2000), 85–113.
[3] I. Daubechies,Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math.41 (1988), 909–996.
[4] D. Deng, M. Xu and L. Yan,Wavelet characterization of weighted Triebel-Lizorkin spaces, Approx. Theory Appl.18(2002), no. 4, 76–92.
[6] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS-AMS Regional Conf. Ser.79, 1991.
[7] G. Garrig´os, E. Hern´andez and J. M. Martell,Wavelets, Orlicz spaces, and greedy bases, preprint.
[8] J. Garc´ia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland, Amsterdam, 1985.
[9] M. Izuki,The Haar wavelets and the Haar scaling function in weightedLp spaces withAdyp ,m weights, to appear in Hokkaido Math. J.36(2007).
[10] M. Izuki,The characterizations of weighted Sobolev spaces by wavelets and scaling functions, preprint.
[11] M. Izuki and Y. Sawano,Atomic decomposition for the weighted Besov / Triebel-Lizorkin spaces with local class of weights, preprint.
[12] M. Izuki and K. Tachizawa, Wavelet characterizations of weighted Herz spaces, preprint.
[13] K. S. Kashin and A. A. Saakyan, Orthogonal Series, Amer. Math. Soc., Provi-dence, R. I., 1989.
[14] S. V. Konyagin and V. N. Temlyakov, A remark on greedy approximation in Banach spaces, East. J. Approx. 5 (1999), 365–379.
[15] P. G. Lemari´e-Rieusset, Ondelettes et poids de Muckenhoupt, Studia Math.108 (1994), 127–147.
[16] P. G. Lemari´e-Rieusset and G. Malgouyres, Support des fonctions de base dans une analyse multi-r´esolution, C. R. Acad. Sci. Paris 313(1991), 377–380.
[17] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, Berlin, 1977.
[18] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992.
[19] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc.165 (1972), 207–226.
[20] T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytzkij operators and nonlinear partial differential equations, Berlin, de Gruyter, 1996.
[21] V. S. Rychkov, Littlewood-Paley theory and function spaces with Aloc
p weights,
Math. Nachr.224 (2001), 145–180.
[22] T. Schott,Functions spaces with exponential weights I, Math. Nachr,189(1998), 221–242.
[24] K. Tachizawa, On weighted dyadic Carleson’s inequalities, J. Inequal. Appl. 6 (2001), 415–433.
[25] H. Triebel, Fractal and Spectra, Monographs in Math., Vol. 91, Birkh¨auser, Basel/Boston, 1997.
[26] H. Triebel,Theory of function spaces, Monographs in Math., Vol. 78, Birkh¨auser, Basel/Boston, 1983.
