Title The Haar wavelets and the Haar scaling function in weighted $L^p$ spaces with $A_p^{\dy ,m}$ weights
Author(s) Izuki, Mitsuo
Citation Hokkaido University Preprint Series in Mathematics, 766, 1-25
Issue Date 2006
DOI 10.14943/83916
Doc URL http://hdl.handle.net/2115/69574
Type bulletin (article)
File Information pre766.pdf
Instructions for use
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP The Haar wavelets and the Haar scaling function
p dy,m in weighted L spaces with Ap weights
Mitsuo IZUKI
Department of Mathematics, Hokkaido University
Sapporo 060-0810, Japan
Abstract
dy,m The new class of weights called Ap weights is introduced. We prove that a characterization and an unconditional basis of the weighted Lp space Lp(Rn, w(x)dx) dy,m with w ∈ Ap (1 < p < ∞) are given by the Haar wavelets and the Haar scaling function. As an application of these results, we establish a greedy basis by using the Haar wavelets and the Haar scaling function again. Keywords and Phrases. The Haar wavelets, the Haar scaling function, weighted Lp dy,m space, Ap weight, greedy basis. 2000 Mathematics S ub ject Classi f ication : Primary: 42C40; Secondary: 46B15; 42B35; 42C15.
1 Introduction
The relations between wavelets and weighted Lp spaces Lp(w):= Lp(Rn, w(x)dx) (1 < p < ∞) has been considered in [ABM], [GK], [Ka] and [L]. In particular, H. A. Aimar, A. L. Bernardis and F. J. Mart´ın-Reyes proved that some characterizations of Lp(w) with w ∈ Ap and an unconditional basis for it are given by 1-regular wavelets (e.g., the Meyer wavelets, the Daubechies wavelets etc.) ([ABM]). They also proved that the similar results dy followed for the case of the Haar wavelets replacing Ap weights by Ap weights. On the other hand, in 1994, P. G. Lemarie-Rieusset´ considered for the case of Daubechies’. He proved that a characterization of Lp(w) and an unconditional basis for it were given by
1 2 Mitsuo IZUKI using not only the wavelet but also the scaling function which constructed the wavelet for ∈ loc loc the case of w Ap ([L]). Let us remark that the class of Ap weights was first defined by V. S. Rychkov in 2001 ([R]). We don’t explain in detail, however, we shall point out that loc he gives some interesting results about Ap weights and weighted function spaces with them. We prove that the similar results to P. G. Lemarie-Rieusset’s´ hold for the case of loc dy,m Haar’s replacing Ap weights by Ap weights, first defined in this article, on the basis of the idea of [ABM]. Let us explain the outline of this article. We describe briefly the basic concept of wavelets associated with an MRA in Section 2. In Section 3, we introduce four classes dy loc dy,m of weights, namely, Ap, Ap , Ap and Ap . Also we give some examples of them re- spectively. Section 4 consists of the main results and the proof of them. We show that a p dy,m characterization and an unconditional basis of the weighted L space with Ap weights are given by the Haar wavelets and the Haar scaling function. Using these results, we es- p dy,m tablish the greedy basis in the weighted L space with Ap weights by the Haar wavelets and the Haar scaling function again in Section 5. So to speak, our weighted wavelet (and scaling function) method is valid for the results of [ABM] and [L], too. We explain them in Section 6. Lastly, we would like to remark on the studies of greedy bases briefly. [CDH], [GH], [KT] and [Ky] give the remarkable results respectively. We shall point out, however, that they consider only for non-weighted cases. This article studies greedy bases in the weighted Lp space with four kinds of weights.
2 Wavelets
, {ψe}2n−1 Definition 2.1 (wavelet set wavelet basis and wavelet). Let e=1 be a sequence of functions belong to L2(dx):= L2(Rn, dx). We define ( ) ( ) jn jn ψe = 2 ψe j − = 2 ψe j − , ··· , j − = , ··· , ∈ Rn j,k(x): 2 2 x k 2 2 x1 k1 2 xn kn (x (x1 xn) ) ≤ ≤ n − ∈ Z = , ··· , ∈ Zn {ψe}2n−1 for each 1 e { 2 } 1, j and k (k1 kn) . The sequence e=1 is called a wavelet set if ψe is an orthonormal basis in L2(dx). Then we say that { } j,k 1≤e≤2n−1, j∈Z,k∈Zn ψe is a wavelet basis in L2(dx) and that any ψe is a wavelet. j,k 1≤e≤2n−1, j∈Z,k∈Zn 2 e 2n−1 Let f ∈ L (dx) and {ψ } = be a wavelet set. We obtain the wavelet expansion of f e 1 ∑ 〈 〉 = , ψe ψe 2 f f j,k j,k in L (dx) 1≤e≤2n−1, j∈Z,k∈Zn and Parseval’s equality
1 ∑ 2 〈 〉 2 ∥ ∥ = , ψe . f L2(dx) f j,k 1≤e≤2n−1, j∈Z,k∈Zn The Haar wavelets and the Haar scaling function in weighted Lp spaces 3
〈 〉 ∫ , ψe = ψe 2 Here f j,k : f (x) j,k(x)dx. By above two equalities, we see that the L -norm of Rn 〈 〉 ffi , ψe ≤ ≤ n − , ∈ Z, ∈ Zn f is characterized by the wavelet coe cients f j,k (1 e 2 1 j k ). We can construct wavelets which have various properties by the proper way of con- structions. The remarkable feature of wavelets, which have a proper smoothness, a proper decay or a compact support, give characterizations and unconditional bases of various function spaces, such as Hardy spaces, Sobolev spaces etc. ([W], [HW], [M], [G], [GM], [D]). Additionally, we shall point out that a sequence of closed subspaces of L2(dx) called MRA gives wavelets.
Definition 2.2 (MRA). An MRA (multiresolution analysis) is a sequence {V j} j∈Z of closed subspaces of L2(dx) such that ⊂ ∈ Z (a) V∪j V j+1 for all j . 2 (b) V j = L (dx). ∩j∈Z (c) V j = {0}. j∈Z − j (d) f ∈ V j holds if and only if f (2 x) ∈ V0 for all j ∈ Z. n (e) f ∈ V0 holds if and only if f (x − k) ∈ V0 for every k ∈ Z . (f) There exists a function ϕ ∈ V0, called a scaling function of {V j} j∈Z, such that the system {ϕ(x − m)} ∈Zn is an orthonormal basis in V . m 0 { } e 2n−1 e Given an MRA {V j} j∈Z, there exists a wavelet set {ψ } = such that ψ e 1 j,k 1≤e≤2n−1,k∈Zn is an orthonormal basis in W j for all j ∈ Z, where W j means the orthogonal complement of V j in V j+1 (cf. [M, Chapter 3] or [W, Chapter 5]). Taking suitable MRAs enables us to obtain wavelets which have various properties. Let us introduce the Haar wavelets and the Haar scaling function which play an important role in this paper.
Definition 2.3 (The Haar wavelet set, the Haar wavelet and the Haar scaling function). n 1 0 Let E := {0, 1} − {(0, ··· , 0)}, ψ := χ , 1 − χ 1 , , ψ := χ[0,1) and [0 2 ) [ 2 1)
∏n e ei n ψ (x):= ψ (xi)(e = (e1, ··· , en) ∈ E, x = (x1, ··· , xn) ∈ R ), i=1 where χF means the characteristic function of a measurable set F. We say that the se- {ψe} ψe quence e∈E is the Haar wavelet set and that any is the Haar wavelet. Additionally we call the function χ[0,1)n the Haar scaling function. On the other hand, we shall also remark that there are well-known examples of smooth wavelets such as the Meyer wavelets which belong to Schwartz class, the Daubechies wavelets which belong to Cr(Rn) for some r ∈ N and have compact support, and so forth. 4 Mitsuo IZUKI
3 Weights
Rn Rn ≥ . . Rn ∈ 1 Rn A weight on is a function w defined on such that w 0 a e and w Lloc( ). Let w be a weight on Rn and 1 < p < ∞ without notices in the following of this paper. We shall explain some notations here in order to introduce four classes of weights.
Notation 3.1 (a) We define a dyadic cube Q j,k by [ ) [ ) − j − j − j − j n Q j,k := 2 k1, 2 (k1 + 1) ×· · ·× 2 kn, 2 (kn + 1) ( j ∈ Z , k = (k1, ··· , kn) ∈ Z ) . ∫ (b) We write w(E):= w(x)dx for a measurable set E ⊂ Rn. And |E| means the E Lebesgue measure of E.
Definition 3.2 (four classes of weights). (a) We define the class of weights Ap which consists of all weights w satisfying ( ) ∫ p−1 1 1 − 1 = p−1 < ∞, Ap(w): sup | |w (Q) | | w(y) dy Q:cube Q Q Q
and say that w ∈ Ap is an Ap weight. dy (b) We define the class of weights Ap which consists of all weights w satisfying ( ) ∫ p−1 1 1 1 dy − − A (w):= sup w Q , w(y) p 1 dy < ∞, p j k Q j,k Q j,k Q j,k Q j,k
dy dy and say that w ∈ Ap is an Ap weight. loc (c) We define the class of weights Ap which consists of all weights w satisfying ( ) ∫ p−1 1 1 − 1 loc = ( ) p−1 < ∞, Ap (w): sup | |w Q | | w(y) dy |Q|≤1 Q Q Q
∈ loc loc and say that w Ap is an Ap weight. dy,m (d) Let m ∈ Z. We define the class of weights Ap which consists of all weights w satisfying ( ) ∫ p−1 dy,m 1 1 − 1 p−1 Ap (w):= sup w Q j,k w(y) dy < ∞, −mn |Q j,k|≤2 Q j,k Q j,k Q j,k
dy,m dy,m and say that w ∈ Ap is an Ap weight. The Haar wavelets and the Haar scaling function in weighted Lp spaces 5
loc The class of Ap weights is defined by V. S. Rychkov. This class is independent of the upper bound for the cube size used in its definitions. Namely, we can replace |Q| ≤ 1 by |Q| ≤ r in Definition 3.2 (c) for any 0 < r < ∞ ([R]). Thus, we obtain the following inclusion relations between above four classes of weights: dy,m−1 dy,m • Ap ⊂ Ap (m ∈ Z). • ⊂ dy, loc. Ap Ap Ap • dy, loc ⊂ dy,m ∈ Z . Ap Ap Ap (m ) However, notice that the signs of equality are not satisfied for each inclusion relations, i.e., strictly speaking, above all three ” ⊂ ” mean ” $ ”. These facts become clear by giving some examples in the following Example 3.3:
Example 3.3 We consider only in the case of n = 1 for convenience. α (a) Let −1 < α < p − 1. Then, we have |x| ∈ Ap (cf. [To]). (b) Let −1 < α, β < p − 1 (α , β). We define a function w0 as follows: | |α > x (x 0) w (x):= 0 (x = 0) 0 |x|β (x < 0)
dy Then, it follows that w0 ∈ Ap − Ap. ∈ R − { } r|x| ∈ loc − (c) Let r 0 . Then, we obtain e Ap Ap. ∑ ∈ Z > ∈ R−{ } rks χ ∈ dy,m − dy,m−1 (d) Let m , s 1 and r 0 . Then it follows that e Qm,k (x) Ap Ap . k∈Z
4 The Haar wavelets and the Haar scaling function in weighted Lp spaces
To state the main result Theorem 4.2, we shall introduce some notations.
Notation 4.1 ∏n n (a) We denote ϕk(x):= ϕ0,k(x) = ϕ(x − k) and Qk := Q0,k = [ki, ki + 1) for each k ∈ Z . i=1 jn χ = 2 χ ∈ Z ∈ Zn (b) We write j,k : 2 Q j,k for each j and k . (c) N means the set of natural numbers. And we denote Z+ := N ∪ {0}. (d) p′ means the conjugate exponent of p, that is, p′ satisfies p−1 + p′−1 = 1.
e Theorem 4.2 (Main Result). Let {ψ }e∈E be the Haar wavelet set, ϕ := χ[0.1)n (i.e., the Haar scaling function), µ be a positive Borel measure on Rn, finite on compact sets, m ∈ Z and 1 < p < ∞. Then, the following three conditions are equivalent: (I1) µ is absolutely continuous with regard to the Lebesgue measure. And there exists a 6 Mitsuo IZUKI
dy,m w ∈ Ap such that dµ(x) = w(x)dx. (I2) We define
1 1 ∑ p ∑ 2 〈 〉 p 〈 〉 2 e M ,µ, ( f ):= f, ϕ , ϕ , + f, ψ χ , . p m m k m k Lp(dµ) j,k j k ∈Zn ∈ , ≥ , ∈Zn k e E j m k Lp(dµ) < ≤ < ∞ ∥ ∥ ≤ Then, there exist two constants 0 c C independent of f such that c f Lp(dµ) M ≤ ∥ ∥ ∈ p µ = p Rn, µ M · p,µ,m( f ) C f Lp(dµ) for all f L (d ): L ( ( d ()x)). Namely p,µ,m( ) defines p n the equivalent norm to ∥ · ∥Lp(dµ) on L (dµ). And µ Q j,k > 0 for all j ≥ m and k ∈ Z . e p (I3) The sequence {ϕ , } ∈Zn ∪ {ψ } ∈ , ≥ , ∈Zn forms an unconditional basis for L (dµ). {( ) } m{ k k } j,k e E j m k ∗ e ∗ p ∗ p ∗ And ϕm,k ∈Zn , (ψ ) ⊂ L (dµ) . Here L (dµ) means the dual space k j,k e∈E,∫j≥m,k∈Zn ( ) p ∗ p of L (dµ) and ϕm,k ( f ):= f (x)ϕm,k(x)dx ( f ∈ L (dµ)). Rn It is known that there are several equivalent definitions of an unconditional basis in a Banach space ([KS], [LT]). We shall remark that we adopt the definition of an uncondi- tional basis by [W, Chapter 7] in this paper.
Definition 4.3 (unconditional convergence, unconditional basis). Let A be a countable index set, {xm}m∈A be a sequence of elements belong to a Banach space X and {x˜k}k∈A be a sequence of elements belong to X∗, where X∗ is the dual space of X. ∑ ∑∞ (a) We say that the series xm is unconditionally convergent in X if the series xσ(i) m∈A i=1 converges in X for all σ : N → A, a 1 to 1 and onto map. (b) We call {xm, x˜m}m∈A an unconditional basis in X if the following three conditions are satisfied: (i) {xm, x˜m}m∈A is a biorthogonal system, i.e., x˜k(xm) = δm,k. Here δm,k means Kronecker’s delta, that is, δm,m = 1 and δm,k = 0 if m , k. X (ii) span{xm}m∈A = X, where span{xm}m∈A means the set of finite linear combinations of elements belong to {xm}m∈A. ∑
(iii) There exists a constant 0 < C < ∞ such that x˜ (x)x ≤ C∥x∥ for every x ∈ X m m X m∈B X and every finite subset B ⊂ A.
Remark 4.4 Let A be a countable index set and {xm, x˜m}m∈A be an unconditional basis in a Banach space X. ∑ (a) ([W, Theorem 7.7 (i)]). The series x˜m(x)xm converges unconditionally in X to x for m∈A every x ∈ X. ∗ (b) ([W, Remark 7.2]). We see that the functionals {x˜k}k∈A ⊂ X are determined by the The Haar wavelets and the Haar scaling function in weighted Lp spaces 7
vectors {xm}m∈A ⊂ X from two conditions (i) and (ii) in Definition 4.3 (b). Thus we often say that {xm}m∈A is an unconditional basis in X. We will prove the equivalence of three conditions (I1), (I2) and (I3) in Theorem 4.2 in the rest of this section. Before starting the proof, let us remark the following facts: mn m n • ϕm,k(x) = 2 2 ϕk(2 x) for all m ∈ Z and k ∈ Z . • Rn ∈ dy,m −m ∈ dy,0 ∈ Z 〈Let w be a weight〉 on . Then,〈w Ap if and〉 only〈 if w(2〉 x) Ap for all m . mn mn • 2 m· , ϕ = 〈 , ϕ 〉 2 m· , ψe = , ψe ∈ ∈ Z 2 f (2 ) m,k f k and 2 f (2 ) j,k f j−m,k for every e E, m , k ∈ Zn and j ≥ m. Thus we see that we have only to prove Theorem 4.2 for the case of m = 0. We will give the proof referring to the statements of [ABM] and partly using the results of [V].
4.1 Proof of (I1) ⇒ (I2) ( ) ( ) First of all, we assume (I1). It is clear that µ Q j,k = w Q j,k > 0 for all j ∈ Z+ and k ∈ Zn.
1 ∑ p p dy,0 1 〈 , ϕ 〉 ∥ϕ ∥ ≤ p ∥ ∥ ∈ p Lemma 4.5 It follows that f k k Lp(w) Ap (w) f Lp(w) for all f L (w). k∈Zn
Proof of Lemma 4.5 By straightforward calculations, Holder’s¨ inequality and |Qk| = 1 for all k ∈ Zn, we have ∑ ∑
〈 , ϕ 〉 ∥ϕ ∥ p = |〈 , ϕ 〉|p · ∥ϕ ∥p f k k Lp(w) f k k Lp(w) k∈Zn k∈Zn ∑ ∫ p ∫ 1 − 1 = p · ϕ p · |ϕ |p f (x)w(x) k(x)w(x) dx k(x) w(x)dx ∈Zn Qk Qk k ( ) ∑ ∫ ∫ p−1 1 p − − ≤ | f (x)| w(x)dx · w(x) p 1 dx · w (Qk) ∈Zn Qk Qk k ∑ ∫ dy,0 p ≤ Ap (w) | f (x)| w(x)dx k∈Zn Qk = dy,0 ∥ ∥p . Ap (w) f Lp(w)
Lemma 4.6 There exists a constant 0 < C1 < ∞ independent of f such that for all ∈ p f L (w), 1 ∑ 2 〈 〉 2 , ψe χ ≤ ∥ ∥ . f j,k j,k C1 f Lp(w) ∈ , ∈Z , ∈Zn e E j + k Lp(w) 8 Mitsuo IZUKI
Proof of Lemma 4.6 We shall follow the lines of [M] using Khintchine’s inequality. ′ , . Ω {− , }Λ Proposition 4.7 (Khintchine s inequality cf [Z]). Let be{ the product set 1 1} and dµ(ω) be the Bernoulli probability measure on Ω for ω = {ω(λ)}λ∈Λ : ω(λ) = ±1 ∈ Ω, 1 obtained by taking the product of the measures on each factor which give a mass of 2 to each of the points −1 and 1. Then, for all 1 < p < ∞, there exist two constants 0 < Cp,1 ≤ Cp,2 < ∞ such that for all {α(λ)}λ∈Λ ⊂ C,
1 ∫ p 1 1 ∑ 2 ∑ p ∑ 2 2 2 C , |α(λ)| ≤ α(λ)ω(λ) dµ(ω) ≤ C , |α(λ)| . p 1 p 2 λ∈Λ Ω λ∈Λ λ∈Λ ∑ 〈 〉 { } e e e e Now we define Tε f (x):= ε , f, ψ , ψ , (x) for each ε = ε , ∈ j k j k j k j k e∈E, j∈Z+,k∈Zn { { } e∈E, j∈Z+,k∈Z}n 〈 〉 e e e e e Ω := ε = ε , : ε , = ±1 . Denoting b , := f, ψ , ψ , , we can express ∑j k e∈E, j∈Z+,k∈Zn j k j k j k j k = εe e < Tε f (x) j,kb j,k(x). By Khintchine’s inequality, there exists a constant 0 e∈E, j∈Z+,k∈Zn a0 < ∞ such that p ∫ ∑ 2 ∑ p e 2 ≤ εe e ν ε . . ∈ Rn a0 b j,k(x) j,kb j,k(x) d ( ) a e x Ω e∈E, j∈Z+,k∈Zn e∈E, j∈Z+,k∈Zn for all f ∈ Lp(w), where dν(ε) denotes the Bernoulli probability measure on Ω. Since ψe 2 = χ 2 j,k j,k , we have
1 p ∫ p ∑ 〈 〉 2 ∑ 2 2 , ψe χ = e 2 f j,k j,k b j,k(x) w(x)dx Rn e∈E, j∈Z+,k∈Zn e∈E, j∈Z+,k∈Zn Lp(w) ∫ ∫ ∑ p ≤ −1 εe e ν ε a0 j,kb j,k(x) d ( ) w(x)dx n R Ω ∈ , ∈Z , ∈Zn ∫ {∫ e E j + k } −1 p = a0 |Tε f (x)| w(x)dx dν(ε) ∫Ω Rn = −1 ∥ ∥p ν ε . a0 Tε f Lp(w) d ( ) Ω ( ) ∏n · χ ⊂ , + ∈ Zn On the other hand, supp Tε f Ql [li li 1] for any l . Besides there exists i=1 n n an unique l(x) ∈ Z such that x ∈ Ql(x) for all x ∈ R . Hence we have
p ∑ ∑ ( ) ( ) p ( ) p p |Tε f (x)| = Tε f · χ (x) = Tε f · χ (x) ≤ Tε f · χ (x) . Ql Ql(x) Ql l∈Zn l∈Zn The Haar wavelets and the Haar scaling function in weighted Lp spaces 9
Thus we obtain ∫ ∑ ( ) p p ∥ ε ∥ ≤ ε · χ T f Lp(w) T f Ql (x) w(x)dx Rn ∫l∈Zn ∑ ( ) = · χ p Tε f Ql (x) w(x)dx ∑l∈Zn Ql ( ) p = ε · χ . T f Ql Lp(w) l∈Zn
n Now we define wl (l ∈ Z ) to fulfill the following two conditions: ∏n • = τ , ··· , τ ∈ , + ν ∈ Z ∈ ν, ν + wl(x) w( l1 (x1) ln (xn)) if x [li li 2), where for and t [ 2), i=1 { t (t ∈ [ν, ν + 1)) τν(t):= 2(ν + 1) − t (t ∈ [ν + 1, ν + 2))
n n • Each wl are 2Z -periodic functions on R . , Then it follows that Ady(w ) ≤ 3npAdy 0(w) for all l ∈ Zn ([R, Proof of Lemma 1.1] or p l p ( ) ε · χ [L, Proof of Proposition 2 (ii)]). To estimate T f Ql Lp(w) we apply the following theorem:
dy Theorem 4.8 ([V, Theorem 4.1]). Let v ∈ Ap . Then, there exist two constants 0 < b0 ≤ dy b1 < ∞ depended on only Ap (v) and p such that
1 ∑ 2 〈 〉 2 e ∥ ∥ p ≤ , ψ χ ≤ ∥ ∥ p b0 g L (v) g j,k A j,k j,k b1 g L (v) ∈ , ∈Z, ∈Zn e E j k Lp(dx)
( ) 1 p p 1 for all g ∈ L (v), where A j,k := v(Q j,k) . |Q j,k| Remark 4.9 (a) A. Volberg gives the above result for the case of v ∈ Ap and one-variable. We can dy extend it to the case of v ∈ Ap and several-variables by the exactly same arguments in [V]. (b) Applying Theorem 4.8 and Khintchine’s inequality gives another proof of [ABM, Theorem 6 (H2) ⇒ (H3)] (Theorem 6.3 in this paper). ( ) dy,0 By Theorem 4.8, there exist two constants 0 < bi = bi Ap (w), p < ∞ (i = 0, 1) such that ( ) ( ) · χ = · χ Tε f Ql p Tε f Ql p L (w) L (wl) 10 Mitsuo IZUKI
1 ∑ 2 〈 ( ) 〉 2 −1 e ≤ ε · χ , ψ , χ , b0 T f Ql j,k A j k j k e∈E, j∈Z,k∈Zn Lp(dx) 1 ∑ 2 〈 〉 2 = −1 · χ , ψe χ b0 f Ql , A j,k j,k j k e∈E, j∈Z+,k∈Zn Lp(dx) 1 2 ∑ 〈 〉 2 −1 e ≤ · χ , ψ , χ , b0 f Ql j,k A j k j k ∈ , ∈Z, ∈Zn e E j k Lp(dx)
≤ −1 · χ b0 b1 f Ql Lp(w ) l = −1 · χ b0 b1 f Ql Lp(w)
( ) 1 p n 1 for all l ∈ Z , where A j,k := w(Q j,k) . Therefore we obtain |Q j,k|
1 p 2 ∫ ∑ 〈 〉 2 ∑ e −1 −p p p f, ψ χ , ≤ a b b f · χ dν(ε) j,k j k 0 0 1 Ql Lp(w) n Ω n e∈E, j∈Z+,k∈Z p l∈Z L (w) ∫ ∑
= −1 −p p ν ε · · χ p a0 b0 b1 d ( ) f Ql p Ω L (w) l∈Zn = −1 −p p ∥ ∥p . a0 b0 b1 f Lp(w)
Lemma 4.10 There exists a constant 0 < C2 < ∞ independent of f such that for all f ∈ Lp(w),
1 1 ∑ p ∑ 〈 〉 2 2 ∥ ∥ ≤ 〈 , ϕ 〉 ∥ϕ ∥ p + , ψe χ C2 f p f k k p f , j,k L (w) L (w) j k k∈Zn e∈E, j∈Z+,k∈Zn ( ) Lp(w) dy,0 1 ≤ p + ∥ ∥ . Ap (w) C1 f Lp(w) Proof of Lemma 4.10 The right hand side inequality follows clearly from Lemma 4.5 and Lemma 4.6. So we have only to prove the left hand side inequality. By the duality, we have ∥ ∥ = |〈 , 〉| ∈ 2 ∥ ∥ ( ) ≤ f Lp(w) sup f g : g L (dx) with g ′ − 1 1 Lp w p−1 for all f ∈ Lp(w) ∩ L2(dx). On the other hand, f can be expanded as follows: ∑ ∑ 〈 〉 = 〈 , ϕ 〉 ϕ + , ψe ψe 2 f f k k f j,k j,k in L (dx) k∈Zn e∈E, j∈Z+,k∈Zn The Haar wavelets and the Haar scaling function in weighted Lp spaces 11
{ϕ } ∪ {ψe } 2 because k k∈Zn j,k e∈E, j∈Z+,k∈Zn forms an orthonormal basis for L (dx). Thus, for all ∈ 2 ∥ ∥ ( ) ≤ g L (dx) with g ′ − 1 1, we have Lp w p−1
∑ ∑ 〈 〉〈 〉 |〈 f, g〉| ≤ 〈 f, ϕ 〉〈ϕ , g〉 + f, ψe ψe , g . k k j,k j,k k∈Zn e∈E, j∈Z+,k∈Zn ∫ 2 To begin with, we consider the first sum in the right hand side. By ϕk(x) dx = 1 and Rn Holder’s¨ inequality, we get
∑ ∑ ∫ 2 〈 f, ϕk〉〈ϕk, g〉 = 〈 f, ϕk〉〈g, ϕk〉 · ϕk(x) dx Rn k∈Zn k∈Zn ∫ ∑ 1 − 1 p p = 〈 f, ϕk〉 ϕk(x)w(x) · 〈g, ϕk〉 ϕk(x)w(x) dx Rn k∈Zn ∑ ∑ 1 − 1 p p ≤ 〈 f, ϕk〉 ϕkw · 〈g, ϕk〉 ϕkw ′ k∈Zn Lp(dx) k∈Zn Lp (dx) 1 1 ′ ∑ p ∑ ′ p 1 p − 1 p ≤ 〈 , ϕ 〉 ϕ p · 〈 , ϕ 〉 ϕ p f k kw g k kw ′ Lp(dx) Lp (dx) k∈Zn k∈Zn 1 ′ ′ 1 p p ∑ p ∑ = 〈 , ϕ 〉 ∥ϕ ∥ p · 〈 , ϕ 〉 ∥ϕ ∥ ( ) . f k k Lp(w) g k k ′ − 1 Lp w p−1 k∈Zn k∈Zn
− 1 dy,0 dy,0 p−1 ∈ ∈ Now we have w Ap′ because of w Ap . Therefore by the proof of Lemma 4.5, we obtain
1 ′ ′ p p ∑ ( ) 1 ( ) 1 ′ ′ ( ) dy,0 − 1 p ( ) dy,0 − 1 p 〈 , ϕ 〉 ∥ϕ ∥ ≤ p−1 · ∥ ∥ ≤ p−1 . g k k ′ − 1 A ′ w g ′ − 1 A ′ w Lp w p−1 p Lp w p−1 p k∈Zn Thus we have the estimation
1 ∑ ( ) 1 ∑ p ′ dy,0 − 1 p p p−1 〈 f, ϕ 〉〈ϕ , g〉 ≤ A ′ w 〈 f, ϕ 〉 ∥ϕ ∥ p . k k p k k L (w) k∈Zn k∈Zn Next we consider the second sum. By Schwarz’s inequality and Holder’s¨ inequality, it follows that
∑ 〈 〉〈 〉 f, ψe ψe , g j,k j,k e∈E, j∈Z+,k∈Zn 12 Mitsuo IZUKI
∑ ∫ 〈 〉〈 〉 = , ψe , ψe · χ 2 f j,k g j,k j,k(x) dx Rn e∈E, j∈Z+,k∈Zn ∫ ∑ 〈 〉 〈 〉 = , ψe χ · , ψe χ f j,k j,k(x) g j,k j,k(x) dx Rn e∈E, j∈Z+,k∈Zn ∫ 1 1 ∑ 2 ∑ 2 〈 〉 2 〈 〉 2 ≤ , ψe χ · , ψe χ f j,k j,k(x) g j,k j,k(x) dx Rn ∈ , ∈Z , ∈Zn ∈ , ∈Z , ∈Zn e E j + k e E j + k 1 1 ∑ 〈 〉 2 ∑ 〈 〉 2 2 1 2 − 1 ≤ , ψe χ p · , ψe χ p f , j,k w g , j,k w j k j k e∈E, j∈Z+,k∈Zn e∈E, j∈Z+,k∈Zn ′ Lp(dx ) Lp (dx) 1 1 ∑ 2 ∑ 2 〈 〉 2 〈 〉 2 e e = , ψ χ , · , ψ χ , . f j,k j k g j,k j k ( ) ∈ , ∈Z , ∈Zn ∈ , ∈Z , ∈Zn ′ − 1 e E j + k Lp(w) e E j + k Lp w p−1
− 1 dy,0 dy,0 p−1 ∈ ∈ Note that w Ap′ because of w Ap . Hence by Lemma 4.6, there exists a constant 0 < a1 < ∞ independent of f and g such that
1 ∑ 2 〈 〉 2 , ψe χ ≤ ∥ ∥ ( ) ≤ . g j,k j,k a1 g ′ − 1 a1 ( ) Lp w p−1 ∈ , ∈Z , ∈Zn ′ − 1 e E j + k Lp w p−1
1 ∑ 2 〈 〉〈 〉 ∑ 〈 〉 2 e e e Thus we obtain f, ψ ψ , g ≤ a f, ψ χ , . Con- j,k j,k 1 j,k j k ∈ , ∈Z , ∈Zn ∈ , ∈Z , ∈Zn e E j + k e E j + k Lp(w) sequently it follows that { } ( ) 1 ′ dy,0 − 1 p ∥ ∥ ≤ p−1 , f Lp(w) max Ap′ w a1 1 1 ∑ p ∑ 2 〈 〉 2 p e × 〈 f, ϕ 〉 ∥ϕ ∥ p + f, ψ χ , . k k L (w) j,k j k ∈Zn ∈ , ∈Z , ∈Zn k e E j + k Lp(w) Following the density of Lp(w) ∩ L2(dx) in Lp(w), the above inequality is satisfied for all f ∈ Lp(w).
4.2 Proof of (I2) ⇒ (I3) In this subsection, we gives the proof of (I2) ⇒ (I3) of Theorem 4.2. First of all, we show ϕ ∗ , ψe ∗ ∈ p µ ∗ ∈ ∈ Z ∈ Zn that ( k) ( j,k) L (d ) for all e E, j + and k . Following (I2), there exists The Haar wavelets and the Haar scaling function in weighted Lp spaces 13
< < ∞ M ≤ ∥ ∥ ∈ p µ a constant 0 C3 such that p,µ,0( f ) C3 f Lp(dµ) for all f L (d ). Besides we see that 0 < µ (Qk) , µ(Q j,k) < ∞. Then we have
| ϕ ∗ | = 〈 , ϕ 〉 ∥ϕ ∥ · ∥ϕ ∥−1 ( k) ( f ) f k k Lp(dµ) k Lp(dµ) 1 ∑ p p − 1 ≤ 〈 , ϕ 〉 ∥ϕ ∥ · µ p f K K Lp(dµ) (Qk) K∈Zn − 1 ≤ Mp,µ,0( f ) · µ (Qk) p − 1 ≤ µ p ∥ ∥ . C3 (Qk) f Lp(dµ)
On the other hand, we get
( )∗ 〈 〉 − e e 1 ψ ( f ) = f, ψ χ , · χ , j,k j,k j k Lp(dµ) j k Lp(dµ) ( ) ∫ 1 ( )− 〈 〉 p p ( ) 1 1 jn p = , ψe χ µ · 2 µ f j,k j,k(x) d (x) 2 Q j,k Rn 1 1 · p ∫ 2 p ∑ 〈 〉 2 ( )− 1 λ − jn p ≤ , ψ χ µ · 2 µ f J,K J,K(x) d (x) 2 Q j,k Rn λ∈E,J∈Z+,K∈Zn
1 2 ∑ 〈 〉 2 ( )− 1 λ − jn p = , ψ χ · 2 µ f J,K J,K 2 Q j,k λ∈ , ∈Z , ∈Zn E J + K Lp(dµ) ( )− 1 − jn p ≤ Mp,µ,0( f ) · 2 2 µ Q j,k ( )− 1 − jn p ≤ 2 µ ∥ ∥ . C32 Q j,k f Lp(dµ)
ϕ ∗ , ψe ∗ ∈ p µ ∗ Consequently we have proved ( k) ( j,k) L (d ) . {ϕ } ∪ {ψe } In order to prove that the sequence k k∈Zn j,k e∈E, j∈Z+,k∈Zn forms an unconditional basis for Lp(dµ), we shall check the following two conditions:
< < ∞ , ≤ (I) There exists a constant 0 C4 independent of f , A and B such that TA B f Lp(dµ) p n n C ∥ f ∥ p for all f ∈ L (dµ) and all finite subsets A ⊂ Z and B ⊂ E ×Z+ ×Z , where 4 L (d∑µ) ∑ 〈 〉 = 〈 , ϕ 〉 ϕ + , ψe ψe TA,B f : f k k f j,k j,k. k∈A (e, j,k)∈B Lp(dµ) p µ = {ϕ } ∪ {ψe } (II) L (d ) span k k∈Zn span j,k e∈E, j∈Z+,k∈Zn .
We show the condition (I) first. By the assumption (B), there exists a constant 0 <
< ∞ , ≤ M ,µ, , C5 independent of f , A and B such that C5 TA B f Lp(dµ) p 0(TA B f ). Since 14 Mitsuo IZUKI
{ϕ } ∪ {ψe } 2 k k∈Zn j,k e∈E, j∈Z+,k∈Zn forms an orthonormal system for L (dx), we obtain
1 1 ∑ p ∑ 〈 〉 2 p 2 M = 〈 , ϕ 〉 ∥ϕ ∥ + , ψe χ p,µ,0(TA,B f ) f k k Lp(dµ) f j,k j,k k∈A (e, j,k)∈B Lp(dµ ) 1 1 ∑ p ∑ 〈 〉 2 p 2 ≤ 〈 , ϕ 〉 ∥ϕ ∥ + , ψe χ f k k Lp(dµ) f j,k j,k ∈Zn ∈ , ∈Z , ∈Zn k e E j + k Lp(dµ) = Mp,µ,0( f ) ≤ ∥ ∥ . C3 f Lp(dµ)
−1 , ≤ ∥ ∥ p . Consequently, we have showed TA B f Lp(dµ) C5 C3 f L (dµ) Lastly we check (II). We shall prove that lim Mp,µ,0( f − TA,B f ) = 0, since A↗Zn,B↗E×Z+×Zn p − , ≤ M ,µ, − , ∈ µ C5 f TA B f Lp(dµ) p 0( f TA B f ) for all f L (d ) from the hypothesis (I2). Now we define
1 1 ∑ p ∑ 〈 〉 2 p 2 M = 〈 , ϕ 〉 ∥ϕ ∥ M = , ψe χ . 1( f ): f k k Lp(dµ) and 2( f ): f j,k j,k ∈Zn ∈ , ∈Z , ∈Zn k e E j + k Lp(dµ)
Then we have Mp,µ,0( f − TA,B f ) = M1( f − TA,B f ) + M2( f − TA,B f ). We omit the detail {ϕ } ∪{ψe } because it is obvious that the orthonormalities of the system k k∈Zn j,k e∈E, j∈Z+,k∈Zn with 2 regard to the L -product, the boundednesses of Mi( f − TA,B f )(i = 1, 2) and Lebesgue’s dominated convergence theorem give us the desired result.
4.3 Proof of (I3) ⇒ (I1) In the end of this section, we prove (I3) ⇒ (I1) of Theorem 4.2. Since the hypothesis (I3) {ϕ } ∪ {ψe } p µ states that k k∈Zn j,k e∈E, j∈Z+,k∈Zn forms an unconditional basis for L (d ), we have the following Lemma from the arguments in [ABM]:
Lemma 4.11 (cf. [ABM, Proof of Theorem 4 (D1) ⇒ (D2), Step a and Step b]). µ is equivalent to the Lebesgue measure. Namely, µ (E) = 0 if and only if |E| = 0 for all bounded Borel set E ⊂ Rn. ∈ 1 Using Lemma 4.11 and Radon-Nikodym theorem, there exists an unique w Lloc(dx) such that dµ(x) = w(x)dx. dy,0 Finally we prove w ∈ Ap . We define ∑ ∑ 〈 〉 = 〈 , ϕ 〉 ϕ + , ψe ψe ∈ N , S j f : f k k f l,k l,k ( j ) k∈Zn e∈E,0≤l≤ j−1,k∈Zn The Haar wavelets and the Haar scaling function in weighted Lp spaces 15 ∑ 〈 〉 P j f := f, ϕ j,k ϕ j,k ( j ∈ Z). k∈Zn
n 2 Lemma 4.12 It follows that S j f = P j f a.e. R for all j ∈ N and f ∈ L (dx). Proof of Lemma 4.12 We see that for every j ∈ N,
L2(dx) L2(dx) L2(dx) {ϕ } ⊕ {ψe } = {ϕ } , span k k∈Zn span l,k e∈E,0≤l≤ j−1,k∈Zn span j,k k∈Zn
2 2 in the sense of subspaces in L (dx). Hence we have S j f =∫P j f in L (dx) for all f ∈
2 2 n 2 L (dx). Thus we get S j f (x) − P j f (x) = 0 a.e. x ∈ R since S j f (x) − P j f (x) dx = 0. Rn n Namely we have showed S j f (x) = P j f (x) a.e. x ∈ R . { } Lemma 4.13 ([HW, Lemma 2.7 in Chapter 5]). Let X be a Banach space and x j be { } j∈N an unconditional basis for X. Then, there exists an unique sequence f j(x) ⊂ C for ∑ j∈N every x ∈ X such that x = f j(x)x j converges unconditionally in X. Additionally we ∑ j∈N { } define S β(x):= β j f j(x)x j for every β = β j ⊂ C with β j ≤ 1 ( j ∈ N). Then, S β(x) j∈N j∈N { } β converges unconditionally in X, and S β is uniformly bounded on X. Now we define { 1 ( 0 ≤ l ≤ j − 1) β j := l 0 ( l ≥ j) for each j ∈ N. Then we can denote ∑ ∑ 〈 〉 = 〈 , ϕ 〉 ϕ + β j , ψe ψe ∈ p . S j f f k k l f l,k l,k ( f L (w)) k∈Zn e∈E,l∈Z+,k∈Zn p ∈ p Hence, using Lemma 4.13, S j{( f )} converges unconditionally in L (w) for all f L (w) p and j ∈ N. We also obtain that S j is uniformly bounded on L (w). On the other hand, j∈N we have ∑ ∑ 〈 〉 = 〈 , ϕ 〉 ϕ + · , ψe ψe ∈ p . P0 f f k k 0 f l,k l,k ( f L (w)) k∈Zn e∈E,l∈Z+,k∈Zn p p Thus, it follows that P0( f ) converges unconditionally in L (w) for all f ∈ L (w) and that p P0 is bounded on L (w) by using Lemma 4.13 similarly. Consequently{ } we obtain the uniformly weak type (p, p) inequality with respect to w(x)dx for P j . Namely there j∈Z+ exists a constant 0 < C5 < ∞ such that ({ }) 1 · ∈ Rn > p ≤ ∥ ∥ s w x : P j f (x) s C5 f Lp(w) 16 Mitsuo IZUKI
p 2 for all j ∈ Z+, f ∈ L (w) ∩ L (dx) and 0 < s < ∞. We consider a dyadic cube∫ Q with − jn − 1 |Q| = 2 for a fixed j ∈ Z+. We define σε(x):= (w(x) + ε) p−1 and σε(Q):= σε(y)dy Q for every ε > 0. Then we get ∫ ( ) ∑ ( ) σ χ = jn χ χ · σ χ P j ε Q (x) 2 Q j,k (x) Q j,k (y) ε Q (y)dy Rn ∫ k∈Zn jn = 2 σε(y)dy Q −1 = |Q| σε(Q) a.e. x ∈ Q.
p 2 Since σεχI ∈ L (w) ∩ L (dx), we have
({ ( ) }) 1 n p s · w x ∈ R : P σεχ (x) > s ≤ σεχ . j Q C5 Q Lp(w) ({ ( ) }) ({ }) n −1 Now notice that w x ∈ R : P j σεχQ (x) > s ≥ w x ∈ Q : |Q| σε(Q) > s and (∫ ) 1 p p −1 σεχ = |σε | . δ = σε (| | + δ) Q Lp(w) (y) w(y)dy Therefore defining s : (Q) Q for any Q −1 fixed δ > 0, we have |Q| σε(Q) > sδ and ({ }) −1 w(Q) = w x ∈ Q : |Q| σε(Q) > sδ ∫ −p p p ≤ sδ C5 |σε(y)| w(y)dy ∫Q − p p p − − = sδ C5 (w(y) + ε) p 1 w(y)dy ∫Q − 1 p p − − ≤ sδ C5 (w(y) + ε) p 1 dy Q p p 1−p = C5 (|Q| + δ) σε(Q) .
≤ p| |pσ 1−p δ → ε → Hence it follows that w(Q() C5 Q ε)(Q) when 0. Additionally letting 0, ∫ 1−p 1 p p − − we have w(Q) ≤ C5 |Q| w(y) p 1 dy , i.e., we obtain Q ( ) ∫ p−1 1 1 − 1 p−1 ≤ p. | |w(Q) | | w(y) dy C5 Q Q Q
dy,0 Consequently we have proved w ∈ Ap . The Haar wavelets and the Haar scaling function in weighted Lp spaces 17
p dy,m 5 The greedy bases in L (w) with w ∈ Ap
The theory of greedy approximations on several non-weighted function spaces has been studied so far ([CDH], [GH], [KT], [Ky]). In this section, we give the greedy basis of p dy,m L (w) with w ∈ Ap using our result Theorem 4.2.
5.1 Definitions and statement of the result Let us begin with introducing two kinds of bases. { }∞ ∥ ∥ = Let X be a Banach space and xk k=1 be a Schauder basis in X such that xk X 1 for ∑∞ ∈ N { }∞ ⊂ C = all k . Then there exists an unique sequence ck(x) k=1 such that x ck(x)xk k=1 in X for all x ∈ X.
{ }∞ Definition 5.1 (greedy basis). We call xk k=1 a greedy basis for X if there exists a con- stant 0 < C < ∞ such that for every x ∈ X there exists a permutation ρ of N which satisfies
∑N ≥ − ≤ ∥ − ∥ , cρ(N)(x) cρ(N+1)(x) and x cρ(k)(x)xρ(k) C inf x y X y∈ΣN k=1 X ∑ ∈ N Σ = α α ∈ C, ♯Λ ≤ , Λ ⊂ N for every N , where N : i xi : i N . i∈Λ
∞ Definition 5.2 (democratic basis). We say that {xk} = is a democratic basis for X if there k 1 ∑ ∑ exists a constant 0 < D < ∞ independent of P and Q such that x ≤ D x k k ∈ ∈ k P X k Q X for any finite subsets P, Q ⊂ N with the same cardinality ♯P = ♯Q. Theorem 5.3 we describe next is the key to the proof of Theorem 5.5 which is the main result of this section:
, { }∞ Theorem 5.3 ([KT Theorem 1]). xk k=1 is a greedy basis if and only if it is an uncondi- tional and democratic basis.
Remark 5.4 ([KT, Section 3]). S. V. Konyagin and V. N. Temlyakov give some examples of bases, which are not democratic but unconditional, or which are not unconditional but democratic. The following is the conclusion of this section: 18 Mitsuo IZUKI
e Theorem 5.5 Let {ψ }e∈E be the Haar wavelet set, ϕ := χ[0.1)n (i.e., the Haar scaling dy,m function), m ∈ Z, 1 < p < ∞ and w ∈ Ap . We define
ψe ϕm,k − mn ( )− 1 j,k − jn − 1 ϕ = = 2 p ϕ ψee = = 2 p ψe . ˜ m,k : 2 w Qm,k m,k and j,k : 2 w(Q j,k) j,k ϕ , ψe m k Lp(w) , j k Lp(w)
{ϕ } ∪ {ψee } p Then, the sequence ˜ m,k k∈Zn j,k e∈E, j≥m,k∈Zn forms a greedy basis for L (w).
5.2 Two lemmas We prepare the following two Lemmas which we will need in the next subsection.
dy,m Lemma 5.6 Let m ∈ Z, 1 < p < ∞ and w ∈ Ap . Then, w satisfies the dyadic reverse doubling condition, i.e., there exists a constant 1 < d < ∞ independent of I and I′ such that dw(I′) ≤ w(I) for all dyadic cubes I, I′ satisfying I′ $ I.
The proof of Lemma 5.6 is found in [GR, Section II. 1, p. 141] or [Ta, Proof of Corollary 1.1]. Additionally, we can get the following Lemma by easy calculations. ( ) 1−p 1 1 1 1 1 Lemma 5.7 Given 0 ≤ a, b < ∞, it follows that 2 p a p + b p ≤ (a + b) p ≤ a p + b p . In particular, the right hand side equality sign holds if and only if a = 0 or b = 0. On the other hand, the left hand side equality sign holds if and only if a = b.
5.3 Proof of Theorem 5.5 The proof of Theorem 5.5 we give here is based on [CDH, the proof of Lemma 4.1]. Following Theorem 4.2 (I3) and Theorem 5.3, it is enough to prove that {ϕ˜ m,k}k∈Zn ∪ {ψee } j,k e∈E, j≥m,k∈Zn is democratic. {ϕ } ∪ {ψe } p We see that m,k k∈Zn j,k e∈E, j≥m,k∈Zn forms an unconditional basis for L (w) by Theorem 4.2 (I3). Thus we can write ∑ ∑ = ϕ + e ψee f am,k( f ) ˜ m,k b j,k( f ) j,k k∈Zn e∈E, j≥m,k∈Zn
〈 〉 〈 〉 ∈ p = , ϕ ϕ e = , ψe ψe for all f L (w), where am,k( f ): f m,k m,k p , b , ( f ): f , , . L (w) j k j k j k Lp(w) Using Theorem 4.2 (I2), we have
1 1 ∑ p ∑ 2 〈 〉 p 〈 〉 2 e c ∥ f ∥ p ≤ f, ϕ , ϕ , + f, ψ χ , L (w) m k m k Lp(w) j,k j k ∈Zn ∈ , ≥ , ∈Zn k e E j m k Lp(w) The Haar wavelets and the Haar scaling function in weighted Lp spaces 19
1 1 ∑ p ∑ 2 ( )− 1 2 p p e = , + , χ am k( f ) w Q j k b j,k( f ) Q j,k ∈Zn ∈ , ≥ , ∈Zn k e E j m k Lp(w) ≤ ∥ ∥ . C f Lp(w) (1) e e Let us denoteϕ ˜ := ϕ˜ , and ψe := ψe for a dyadic cube Q = Q , . Now we take Q { j k Q} j,k { j k } ⊂ ∈ Zn ⊂ ⊂ ≥ , ∈ Zn = , finite subsets Ai Qm,k : k , Ei E and Bi Q j,k : j m∑k (∑i 1 2) ♯ + ♯ × = ♯ + ♯ × = ϕ + ψee satisfying A1 (E1 B1) A2 (E2 B2) arbitrarily, and set g : ˜ I J ∑ ∑ I∈A1 e∈E1,J∈B1 = ϕ + ψee ≤ ♯ ≤ ♯ = n − and h : ˜ I J. Using (1) and 1 E1 E 2 1, we obtain I∈A2 e∈E2,J∈B2
1 ∑ 2 ( ) 1 − 1 2 ∥ ∥ ≤ ♯ p + p χ c g Lp(w) A1 w(J) J e∈E ,J∈B 1 1 Lp(w) 1 p ∫ p ∑ 2 ( ) 1 ( ) 1 − 2 = ♯ p + ♯ 2 · ∪ p χ A1 E1 w(J) J(x) w(x)dx ′ J J∈B1 ′ J ∈B1 1 p ∫ p ∑ 2 ( ) 1 1 ( ) 1 − 2 ≤ ♯ p + n − 2 ♯ p · ∪ p χ . A1 (2 1) E1 w(J) J(x) w(x)dx (2) ′ J J∈B1 ′ J ∈B1 ∪ For each x ∈ J, we write J1(x):= min {J ∈ B1 : x ∈ J}. Then we get
J∈B1
∑ ∑∞ − 2 − 2 w(J) p χJ(x) ≤ w (Jr) p , (3)
J∈B1 r=0 where J0 := J1(x), Jr is a dyadic cube satisfying Jr−1 ⊂ Jr and 2 |Jr−1| = |Jr| for every r ∈ N. By Lemma 5.6, we obtain
r r w (Jr) ≥ dw (Jr−1) ≥ · · · ≥ d w (J0) = d w (J1(x)) for all r ∈ N. Thus we have ∑∞ ∑∞ − 2 r − 2 − 2 w (Jr) p ≤ (d w (J1(x))) p = C0w (J1(x)) p , (4) r=0 r=0 20 Mitsuo IZUKI
( )− − 2 1 where C0 := 1 − d p . Following (3) and (4), we obtain ∫ p ∫ ∑ 2 ( ) p − 2 − 2 2 ∪ p χ ≤ ∪ p ′ w(J) J(x) w(x)dx C0w (J1(x)) w(x)dx J J∈B1 J ′∈ ∈ J B1 J B∫1 p ∪ −1 = C0 2 w (J1(x)) w(x)dx. (5) J
J∈B1 ∪ ˜ = ∈ = ∈ ˜ ⊂ Now we set J : x J : J1(x) J for each J B1. Then, since J J and ∪ ∪ J∈B1 J = J˜, it follows that
J∈B1 J∈B1 ∫ ∫ ∪ −1 ∪ −1 w (J1(x)) w(x)dx = w (J1(x)) w(x)dx J J˜ ∈ ∈ J B1 ∫J B1 = ∪ w (J)−1 w(x)dx J˜ J∈B ∑ ∫1 ≤ w (J)−1 w(x)dx J J∈B1
= ♯B1. (6)
( ) 1 1 1 ( ) 1 ( ) 1 ∥ ∥ ≤ ♯ p + 2 n − 2 ♯ p ♯ p Following (2)-(6), we have c g Lp(w) A1 C0 (2 1) E1 B1 . In addition, using Lemma 5.7, there exists a constant 0 < C1 < ∞ independent of g, A1, E1 and B1 such that
{ } 1 ∥ ∥ ≤ ♯ + ♯ × p . g Lp(w) C1 A1 (E1 B1) (7) ∑ ∑ = ϕ + ψee ≤ ♯ ≤ ♯ = On the other hand, applying (1) to h ˜ I J and using 1 E2 E I∈A2 e∈E2,J∈B2 2n − 1, we have
1 ∑ 2 ( ) 1 − 1 2 ∥ ∥ ≥ ♯ p + p χ C h Lp(w) A2 w(J) J e∈E ,J∈B 2 2 Lp(w) 1 p ∫ p ∑ 2 ( ) 1 ( ) 1 − 2 = ♯ p + ♯ 2 · ∪ p χ A2 E2 w(J) J(y) w(y)dy ′ J J∈B2 ′ J ∈B2 The Haar wavelets and the Haar scaling function in weighted Lp spaces 21
1 p ∫ p ∑ 2 ( ) 1 − 1 ( ) 1 − 2 ≥ ♯ p + n − p ♯ p · ∪ p χ . A2 (2 1) E2 w(J) J(y) w(y)dy (8) ′ J J∈B2 J′∈B ∪ 2 For each y ∈ J, we write J2(y):= min {J ∈ B2 : y ∈ J}. Then we have
J∈B2
p ∑ 2 − 2 − p χ ≥ 1 . w(J) J(y) w (J2(y)) (9) J∈B2 , − 2 , − Now using the same argument as (3)-(4), replacing ” B1 p and J1(x) ” by ” B2 1 and J (y) ” respectively, we get 2 ∑ −1χ ≤ ′ −1 , w(J) J(y) C0w (J2(y)) (10) J∈B2 ′ where C0 is a constant depended on only p and d. Following (8)-(10), we obtain
1 p ∫ ∑ ( ) 1 − 1 ( ) 1 ′ − − p n p ∪ 1 1 C ∥h∥ p ≥ ♯A + (2 − 1) p ♯E · C w(J) χ (y)w(y)dy L (w) 2 2 ′ 0 J J J∈B2 ′ J ∈B2 ∫ 1 ( ) ( ) ∑ p 1 − 1 1 ′ −1 − = ♯ p + n − p ♯ p · 1 A2 (2 1) E2 C0 w(J) w(y)dy J J∈B2 ( ) 1 ′ − 1 − 1 ( ) 1 ( ) 1 = ♯ p + p n − p ♯ p ♯ p . A2 C0 (2 1) E2 B2
By Lemma 5.7, there exists a constant 0 < C2 < ∞ independent of h, A2, E2 and B2 such that { } 1 ∥ ∥ ≥ ♯ + ♯ × p . C2 h Lp(w) A2 (E2 B2) (11) ♯ +♯ × = ♯ +♯ × ∥ ∥ ≤ ∥ ∥ . Following A1 (E1 B1) A2 (E2 B2), (7) and (11), we get g Lp(w) C1C2 h Lp(w) {ϕ } ∪ {ψee } Consequently we have proved that ˜ m,k k∈Zn j,k e∈E, j≥m,k∈Zn is democratic.
6 Further results
As we described in Section 1, H. A. Aimar et al. and P. G. Lemarie-Rieusset´ give the important results on characterizations and unconditional bases of weighted Lp spaces (1 < p < ∞) respectively ([ABM], [L]). We will introduce their results and state that we can obtain three conclusions about greedy bases in weighted Lp spaces by applying the same arguments as Subsection 5.5 to them respectively. 22 Mitsuo IZUKI
p dy 6.1 The greedy bases in L (w) with w ∈ Ap or w ∈ Ap Let us begin with the definition of 1-regular functions.
Definition 6.1 (1-regular function). A function f on Rn is 1-regular if for every m ∈ N α −m n there exists a constant 0 < Cm < ∞ such that |∂ f (x)| ≤ Cm(1 + |x|) for all x ∈ R and ∏n α ∑n ∂ i n α for all α = (α , ··· , α ) ∈ Z+ with |α| ≤ 1. Here ∂ := α and |α| := α . 1 n ∂x i i i=1 i i=1 We shall remark that we can construct a wavelet set which consists of 1-regular wavelets for a given MRA which has a 1-regular scaling function (cf. [M, Chapter 3] or [W, Chapter 5]). Now we are ready to describe two results of [ABM]:
Theorem 6.2 ([ABM, Theorem 4]). Let 1 < p < ∞, µ be a positive Borel measure on Rn {ψe}2n−1 ψe , finite on compact sets and e=1 be a wavelet set such that each are 1-regular. Then, the following{ conditions} are equivalent: (D1) The sequence ψe forms an unconditional basis for Lp(dµ), and j,k 1≤e≤2n−1, j∈Z,k∈Zn {( )∗} ψe ⊂ Lp(dµ)∗. j,k 1≤e≤2n−1, j∈Z,k∈Zn (D2) µ is absolutely continuous with regard to the Lebesgue measure. In addition, there exists a w ∈ A such that dµ(x) = w(x)dx. p e n n (D3) ψ , > 0 for all 1 ≤ e ≤ 2 − 1, j ∈ Z and k ∈ Z . Addtionally there exist two j k Lp(dµ) constants 0 < c ≤ C < ∞ independent of f such that for every f ∈ Lp(dµ),
1 2 ∑ 〈 〉 2 ∥ ∥ ≤ , ψe ψe ≤ ∥ ∥ . c f p µ f , , C f p µ L (d ) j k j k L (d ) 1≤e≤2n−1, j∈Z,k∈Zn Lp(dµ) ( ) n (D4) µ Q j,k > 0 for all j ∈ Z and k ∈ Z . And there exist two constants 0 < c ≤ C < ∞ independent of f such that for every f ∈ Lp(dµ),
1 2 ∑ 〈 〉 2 ∥ ∥ ≤ , ψe χ ≤ ∥ ∥ . c f p µ f , j,k C f p µ L (d ) j k L (d ) 1≤e≤2n−1, j∈Z,k∈Zn Lp(dµ) Theorem 6.3 ([ABM, Theorem 6]). Let 1 < p < ∞, µ be a positive Borel measure Rn {ψe} on , finite on compact sets and e∈E be the Haar wavelet set. Then, the following conditions are equivalent:{ } (H1) The sequence ψe forms an unconditional basis for Lp(dµ). Additionally, j,k e∈E, j∈Z,k∈Zn The Haar wavelets and the Haar scaling function in weighted Lp spaces 23
{( )∗} ψe ⊂ Lp(dµ)∗. j,k e∈E, j∈Z,k∈Zn (H2) µ is absolutely continuous with regard to the Lebesgue measure. In addition, there ∈ dy µ = exists( a)w Ap such that d (x) w(x)dx. n (H3) µ Q j,k > 0 for all j ∈ Z and k ∈ Z . Additionally, there exist two constants 0 < c ≤ C < ∞ independent of f such that for every f ∈ Lp(dµ),
1 ∑ 2 〈 〉 2 ∥ ∥ ≤ , ψe χ ≤ ∥ ∥ . c f Lp(dµ) f j,k j,k C f Lp(dµ) ∈ , ∈Z, ∈Zn e E j k Lp(dµ) Applying the same arguments as Subsection 5.5 to above two theorems respectively, we can obtain the following two results about greedy bases in Lp(w). Here let us mention that both of Theorem 6.2 and Theorem 6.3 state that one sequence given by the wavelets, which forms an orthonormal basis in L2(dx), becomes an unconditional basis for Lp(w). Hence we don’t need Lemma 5.7 in order to obtain the following two conclusions.
< < ∞ ∈ {ψe}2n−1 ψe Corollary 6.4 Let 1 p ,{w }Ap and e=1 be a wavelet set such that each are ee p 1-regular. Then, the sequence ψ , forms a greedy basis for L (w). j k 1≤e≤2n−1, j∈Z,k∈Zn
< < ∞ ∈ dy {ψe} Corollary{ 6.5 }Let 1 p , w Ap and e∈E be the Haar wavelet set. Then, the ee p sequence ψ , forms a greedy basis for L (w). j k e∈E, j∈Z,k∈Zn
p ∈ loc 6.2 The greedy bases in L (w) with w Ap P. G. Lemarie-Rieusset´ gives the next result. He proved it in the case of one-variable, however, it is true in the case of several-variables with obvious modifications.
Theorem 6.6 (cf. [L, Proposition 2 (ii)]). Let 1 < p < ∞, w be a weight on Rn, m ∈ Z, ϕ {ψe}2n−1 be the Daubechies scaling function and e=1 be the Daubechies wavelet set. Then, the following conditions are equivalent: {ϕ } ∪{ψe } p (E1) The sequence m,k k∈Zn j,k 1≤e≤2n−1, j≥m,k∈Zn forms an unconditional basis for L (w). (E2) We define
1 2 1 ∑ p ∑ 〈 〉 〈 〉 p 2 M = , ϕ ϕ + , ψe χ . p,w,m( f ): f m,k m,k p f , j,k L (w) j k k∈Zn 1≤e≤2n−1, j≥m,k∈Zn Lp(w) < ≤ < ∞ ∥ ∥ ≤ M ≤ Then, there exist two constants 0 c C such that c f Lp(w) p,w,m( f ) ∥ ∥ ∈ p C f Lp(w) for all f L (w). ∈ loc (E3) w Ap . 24 Mitsuo IZUKI
We obtain the following conclusion by applying the arguments in Subsection 5.5 to Theorem 6.6 directly:
< < ∞ ∈ loc ∈ Z ϕ Corollary 6.7 Let 1 p , w Ap , m , be the Daubechies scaling function and {ψe}2n−1 {ϕ } ∪{ψee } e=1 be the Daubechies wavelet set. Then, the sequence ˜ m,k k∈Zn j,k 1≤e≤2n−1, j≥m,k∈Zn forms a greedy basis for Lp(w).
Acknowledgements. The author has started the study of wavelets when he was at Kyushu University. He is sincerely grateful to Professor Joe Kamimoto and Professor Katsunori Iwasaki for their kind and great encouragements. He also thanks Professor Kazuya Tachizawa and Professor Yasuo Komori for their helpful and useful advices.
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