# Some Function Spaces via Orthonormal Bases on Spaces of Homogeneous Type

Authors: Chuang Chen; Ji Li; Fanghui Liao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Let(X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure μ satisfies only the doubling property. Adapting the recently developed randomized dyadic structures of X and applying orthonormal bases of L2(X) constructed recently by Auscher and Hytönen, we develop the Besov and Triebel-Lizorkin spaces on such a general setting. In this paper, we establish the wavelet characterizations and provide the dualities for these spaces. The results in this paper extend earlier related results with additional assumptions  on the quasi-metric d and the measure μ to the full generality of the theory of these function spaces.


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## Body

## 1. Introduction

The aim of this paper is to introduce the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss without any additional assumption on the quasi-metric and the doubling measure. The main tool used in this paper is the remarkable orthonormal basis constructed recently by Auscher and Hytönen [1]. It is well known that function spaces play an important role in both classical and modern analysis, ordinary and partial differential equations, and approximation theory, and so forth. Since the seventies of last century, classical theory of the Besov spaces [2] as well as the Triebel-Lizorkin spaces [3, 4] has been developed rapidly from Euclidean spaces to spaces of homogeneous type in the sense of Coifman and Weiss with some additional assumptions on the quasi-metric and the doubling measure. See, for example, [5, 6] and references therein.Let us recall briefly spaces of homogeneous type introduced by Coifman and Weiss [7]. A function d:X×X→[0,∞) is called a quasi-metric on a set X if d satisfies the following: (1) d(x,y)=d(y,x) for all x,y∈X; (2) d(x,y)=0 if and only if x=y; and (3) the quasi-triangle inequality holds: there is a constant A0≥1 such that
(1)d(x,y)≤A0dx,z+dz,y
for all x, y and z∈X. In addition, the quasi-metric ball B(x,r) with center x∈X and radius r>0 is defined by B(x,r)∶={y∈X:d(x,y)<r}. We say that a nonzero measure μ satisfies the doubling condition if there is a constant Cμ>0 such that, for all x∈X and all r>0,
(2)μBx,2r≤CμμBx,r<∞.
We point out that the doubling condition (2) implies that there exist positive constants Q (theupper dimension of μ) and C such that, for all x∈X and λ≥1,
(3)μBx,λr≤CλQμBx,r.A space(X,d,μ) is said to be the space of homogeneous type in the sense of Coifman and Weiss [7] if d is a quasi-metric on X and μ satisfies the doubling condition. Such spaces have many applications in the theory of singular integrals and function spaces [8]. Unfortunately, for some applications, additional assumptions were imposed on these general spaces due to the facts that the original quasi-metric may have no (Hölder) regularity and quasi-metric balls, even Borel sets, may not be open.A recent work on the Besov spaces and the Triebel-Lizorkin spaces on spaces(X,d,μ) of homogeneous type in the sense of Coifman and Weiss was developed by Han et al. [6]. More precisely, in [6], in order to apply Coifman’s construction for the approximation to the identity, they need that the quasi-metric d satisfies the Hölder regularity and the doubling measure μ is required to satisfy the reversed doubling condition; that is, there are constants κ∈(0,Q] and c∈(0,1] such that
(4)cλκμBx,r≤μBx,λr
for all x∈X, 0<r<sup⁡x,y∈X⁡d(x,y)/2, and 1≤λ<sup⁡x,y∈Xd(x,y)/2r. This assumption ensures that
(5)∑k∈Z:δk≥r‍1μBx,δk≤CμBx,r,
which is the key to develop the theory of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. The main tool used in [6] is the so-called frame developed by Han [9] (see also [10]) because the Fourier transform is missing in general spaces of homogeneous type and wavelets were not available at that time. However, things seem to be changed after the quite recent work of Auscher and Hytönen [1], where remarkable orthonormal bases (wavelets) were constructed by using spline functions with the original quasi-metric and the doubling measure. These orthonormal bases open the door for developing function spaces on these general settings. See the very recent work in [11] for the theory of product Hp, CMOp, VMO and duality on such general spaces. Motivated by the work in [11], in this paper, we develop the theory of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss without any additional assumption on the quasi-metric and the doubling measure. Therefore, results in this paper extend earlier related results with additional assumptions on the quasi-metric d and the measure μ to the full generality of the theory of the Besov and Triebel-Lizorkin spaces.This paper is organized as follows. In Section2 we give some notations and preliminaries, including a useful almost orthogonality estimate (see Lemma 7). Wavelet characterizations and dualities of the inhomogeneous Besov and Triebel-Lizorkin spaces are given in Section 3. We introduce homogeneous Besov and Triebel-Lizorkin spaces and discuss relationships between homogeneous and inhomogeneous Besov and Triebel-Lizorkin spaces in Section 4.


## 2. Preliminaries and Notations

Throughout this paper,(X,d,μ) denotes a space of homogeneous type with quasi-triangle constant A0 and μ(X)=∞. By Vr(x) we denote the measure of B(x,r), the ball centered at x with radius r>0, and by V(x,y) we denote the measure of B(x,d(x,y)), the ball centered at x with radius d(x,y)>0. In addition, we use the notation a≲b to mean that there is a constant C>0 such that a≤Cb and the notation a~b to mean that a≲b≲a. The implicit constants, C, are meant to be independent of other relevant quantities.We first recall the orthonormal basis ofL2(X) constructed by Auscher and Hytönen [1]. Let sαk be spline functions constructed in [1, Section 3] and let Vk be the closed linear span in L2(X) of {sαk}α∈Xk. By [1, Theorem 5.1] we observe that
(6)Vk⊂Vk+1,⋂k∈Z‍Vk=0,⋃k∈Z‍Vk¯=L2X.
Further on, by writing Wk as the orthogonal (in L2(X)) complement of Vk in Vk+1, Auscher and Hytönen pointed out the following result.Theorem 1 (Theorems 6.1 and 7.1, [1]).
Leta∶=(1+2log⁡2A0)-1. There exists an orthonormal basis {ϕαk}α∈Xk of  Vk satisfying the following exponential decay:
(7)ϕαkx≤CVδkxαkexp⁡-νδ-kdx,xαka
and the Hölder regularity of order η:
(8)ϕαkx-ϕαky≤CVδkxαkdx,yδkη×exp⁡-νδ-kdx,xαka
whenever d(x,y)≤δk for some η∈(0,1]. Here δ>0 is a fixed parameter small enough, say δ<1/1000A0-10, and C, ν are two positive constants independent of k, α, x, and xαk. Clearly, {ϕαk}α∈Xk,k∈Z is an orthonormal basis of L2(X).
Also, there exists an orthonormal basis{ψαk}α∈Yk of Wk satisfying the following exponential decay:
(9)ψαkx≤CVδkyαkexp⁡-νδ-kdx,yαka,
the Hölder regularity of order η:
(10)ψαkx-ψαky≤CVδkyαkdx,yδkη×exp⁡-νδ-kdx,yαka
whenever d(x,y)≤δk for some η∈(0,1], and the cancellation property:
(11)∫X‍ψαkxdμx=0,k∈Z,α∈Yk,
where δ is given above and C, ν are two positive constants independent of k, α, x, and yαk. Clearly, {ψαk}α∈Yk,k∈Z is an orthonormal basis of L2(X). In what follows, we also refer to the functions ψαk as wavelets.In order to introduce the (inhomogeneous) Besov and Triebel-Lizorkin spaces, we need the following concepts of test functions and distributions. Refer to [12] for details and also see [11].Definition 2.
For fixedx0∈X, r>0, β∈(0,η], where η is given in Theorem 1, and γ>0. A function f is said to be a test function of type (x0,r,β,γ) centered at x0 with width r>0 if f satisfies the following decay and Hölder regularity properties:(i)
fx≤C1/Vr(x0)+Vx,x0r/r+dx,x0γ for all x∈X,(ii)
fx-fy≤C(d(x,y)/r+d(x,x0))β1/Vr(x0)+V(x,x0)(r/r+d(x,x0))γ for all x, y∈X such that d(x,y)≤1/2A0(r+d(x,x0)).Iff is a test function of type (x0,r,β,γ), we write f∈G(x0,r,β,γ). The norm of f on G(x0,r,β,γ) is defined by
(12)fG(x0,r,β,γ)∶=inf⁡{C>0:iandiihold}.We denoteG(β,γ)∶=G(x0,1,β,γ). It is easy to check that G(x1,r,β,γ)=G(β,γ) with equivalent norms for any fixed x1∈X and r>0. Furthermore, it is also easy to see that G(β,γ) is a Banach space with respect to the norm on G(β,γ). For given ϵ∈(0,η], let G∘(β,γ) be the completion of the space G(ϵ,ϵ) in G(β,γ) with 0<β, γ≤ϵ. Obviously, G∘(ϵ,ϵ)=G(ϵ,ϵ). In addition, we say that f∈G∘(β,γ) if f∈G(β,γ) and there is a sequence {fn}⊂G(ϵ,ϵ) such that fn-fG(β,γ)→0 as n→∞. For given f∈G∘(β,γ), we define fG∘(β,γ)∶=fG(β,γ). Obviously, G∘(β,γ) is a Banach space with respect to the norm ·G∘(β,γ). As usual, we denote by (G∘(β,γ))′ the dual space of G∘(β,γ) consisting of all continuous linear functionals on G∘(β,γ).Han et al. [11] showed that the wavelets constructed by Auscher and Hytönen [1] are test functions defined above.Lemma 3 (see [11], Theorem 3.3).
Letψαk be given in Theorem 1 with the Hölder regularity of η. Then ψαk/Vδk(xαk)∈G(xαk,δk,η,ϵ) for each ϵ>0.By analogous arguments, we can obtain the next result.Proposition 4.
Letϕαk be given in Theorem 1 with Hölder regularity of η. Then ϕαk/Vδk(xαk)∈G(xαk,δk,η,ϵ) for each ϵ>0.LetPk and Qk be orthogonal projections onto Vk and Wk, respectively. We next give some estimates on the kernels of operators Pk and Qk.Lemma 5.
The kernelPk(x,y) is symmetric in x and y, and, for any ϵ>0, Pk(x,y) satisfies that(i)
for allx, y∈X,
(13)Pkx,y≤C1Vδkx+Vx,yδkδk+dx,yϵ,(ii)
ford(y,y′)≤1/2A0(δk+d(x,y)),
(14)Pkx,y-Pkx,y′≤Cdy,y′δk+dx,yη1Vδkx+Vx,y×δkδk+dx,yϵ,(iii)
ford(x,x′)≤1/2A0(δk+d(x,y)) and d(y,y′)≤1/2A0(δk+d(x,y′)),
(15)Pkx,y-Pkx′,y-Pkx,y′-Pkx′,y′≤Cdx,x′δk+dx,yηdy,y′δk+dx,yη×1Vδkx+Vx,yδkδk+dx,yϵ.Proof.
Note that, for anyx, x0∈X and r>0 with r≤d(x,x0), from the doubling property on the measure μ it follows that
(16)Vx,x0≤Cdx,x0rQVrx0,(17)1Vrx0≤C1V(x,x0)+Vr(x0)dx,x0rQ.
Also, note that the kernel Pk(x,y) is symmetric in x and y due to [1, Lemma 10.1]. Thus, it suffices to prove (13) for the cases that δk>d(x,y) and δk≤d(x,y). Again, by [1, Lemma 10.1] we have
(18)Pkx,y≤C1Vδkxδkδk+dx,yΓ
for all Γ>0, where the constant C is independent of k. If δk>d(x,y), then V(x,y)<Vδk(x) and hence
(19)1Vδkxδkδk+dx,yΓ≤C1V(x,y)+Vδk(x)δkδk+dx,yΓ.
On the other hand, if δk<d(x,y), then (17) implies that
(20)1Vδk(x)δkδk+dx,yΓ≤C1V(x,y)+Vδk(x)δkδk+dx,yΓ-Q.
By letting ϵ∶=Γ-Q, we obtain (13), immediately.
The proof of (14) is based on the estimates
(21)Pkx,y-Pkx,y′≤Cdy,y′δkη×exp⁡-νδ-kdx,ysVδkxVδky+exp⁡-νδ-kdx,y′sVδkxVδky′
given in [1, Lemma 10.1]. Indeed, since d(y,y′)≤1/2A0(δk+d(x,y)) implies that δk+d(x,y′)≥1/2A0(δk+d(x,y)) and that Vδk(x)+V(x,y′)~Vδk(x)+V(x,y), it follows from (17) and (21) that
(22)II≤Cdy,y′δkη1Vδk(x)+V(x,y)×δkδk+dx,yΓ≤Cdy,y′δk+dx,yη1Vδk(x)+V(x,y)×δkδk+dx,yΓ-η.
Applying ϵ∶=Γ-η yields (14), immediately. The proof of (15) is similar to that of (13) and (14), and we omit it here.Lemma 6 (see [11], Lemma 3.6).
The kernelQk(x,y) is symmetric in x and y and, for any ϵ>0, satisfies that(i)
for allx, y∈X,
(23)Qkx,y≤C1Vδk(x)+Vx,yδkδk+dx,yϵ,(ii)
ford(y,y′)≤1/2A0(δk+d(x,y)),
(24)Qkx,y-Qkx,y′≤Cdy,y′δk+dx,yη×1Vδk(x)+V(x,y)δkδk+dx,yϵ,(iii)
ford(x,x′)≤1/2A0(δk+d(x,y)) and d(y,y′)≤1/2A0(δk+d(x,y))(25)Qkx,y-Qkx′,y-Qkx,y′-Qkx′,y′≤Cdx,x′δk+dx,yηdy,y′δk+dx,yη×1Vδk(x)+V(x,y)δkδk+dx,yϵ.Let{φβl}l∈Z,β∈Yl be another orthonormal basis of L2(X) given by the algorithm in Theorem 1, and let El be the projection onto Wl with respect to the basis {φβl}l∈Z,β∈Yl. Finally, we give the almost orthogonality estimate of the kernel of operator QkEl to end this section. The proof is similar to that in [6, Lemma 3.1], and we omit it here.Lemma 7.
Letϵ>0, and let η>0 be the order of the Hölder regularity given in (8). There exists a constant C=Cϵ>0, dependent only on ϵ, such that, for all σ∈(0,η],
(26)QkElx,y≤Cδk-lσ1Vδk∧lx+Vx,y×δk∧lδk∧l+dx,yϵ,(ii) for all τ∈(0,1),
(27)QkElx,y-QkElx,y′≤Cδk-lστdy,y′δk∧l+dx,yη(1-τ)×1Vδ(k∧l)(x)+V(x,y)δ(k∧l)δk∧l+dx,yϵ
whenever d(y,y′)<1/(2A0)2d(x,y). The same estimate holds with x and y interchanged.


## 3. Inhomogeneous Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

In this section, we introduce the inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and give wavelet characterizations of corresponding spaces.We would like to point out that the wavelet expansion (Theorem9) is the key to develop the theory of the Besov and Triebel-Lizorkin spaces. As defined previously, by Pk and Qk we denote the orthogonal projections onto Vk and Wk, respectively. These two projections provide us with regular Littlewood-Paley decompositions for spaces of homogeneous type.Theorem 8 (Theorem 10.2, [1]).
Letf∈L2(X). Then
(28)f=∑k∈Z‍Qk(f)=∑k∈Z‍Qk2(f),
which gives homogeneous Littlewood-Paley decomposition, while
(29)f=Pl(f)+∑k≥l‍Qk(f)=Pl2(f)+∑k≥l‍Qk2(f)
provides inhomogeneous Littlewood-Paley decomposition.Furthermore, following the proof of Theorem  3.4 in [11], we can show that such wavelet expansions also hold in the space of distributions.Theorem 9.
Let0<β, γ<η. Then wavelet expansion (29) holds in G∘(β′,γ′) with β′∈(0,β) and γ′∈(0,γ) and hence holds in (G∘(β′,γ′))′ with β′∈(β,η) and γ′∈(γ,η) as well.Proof.
SinceG(β,γ)⊂L2(X) and (29) holds in the sense of L2(X), (29) holds for such test function f in the (a.e.) pointwise sense (for some subsequence with respect to the convergence).
By analogous arguments given in [11, Theorem 3.4], we can show the convergence of ∑k≥1‍Qk(f) and P1(f) in G(β′,γ′) by using the size and smooth conditions of kernels Qk(x,y) and P1(x,y), respectively. Therefore,
(30)f=P1(f)+∑k≥1‍Qk(f)
in the sense of G(β′,γ′). The second equality of (29) holds for the same reason.We now introduce the inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.Definition 10.
Let1<p, q≤∞ and s<η. The inhomogeneous Besov space Bps,q(X) is defined by
(31)Bps,qX∶=f∈G∘β,γ′:fBps,q<∞,
where
(32)fBps,q∶=P1(f)p+∑k≥1‍δ-ksQkfpq1/q.
For1<p<∞ and 1<q≤∞, the inhomogeneous Triebel-Lizorkin space Fps,q(X) is defined by
(33)Fps,q(X)∶=f∈G∘β,γ′:fFps,q<∞,
where
(34)fFps,q∶=P1(f)p+∑k≥1‍δ-ksQkfq1/qp.Remark 11.
Observe that(35)fBps,q~P1fpq+∑k≥1‍δ-ksQkfpq1/q,hhhhhhhhhhhhhhhhhhhhhhhhhhhf∈Bps,q,
as well as
(36)fFps,q~P1fq+∑k≥1‍δ-ksQkfq1/qp,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhif∈Fps,q.The next result shows that the Besov and Triebel-Lizorkin norms given in Definition10 are both independent of choice of the orthonormal basis {ψαk}α∈Yk,k∈Z of L2(X) and hence, they are independent of the choice of operators P1 and Qk with k≥1. Thus, the Besov and Triebel-Lizorkin spaces given above are both well defined. To see this, let {ϕαk}α∈Yk,k∈Z be another orthonormal basis of L2(X) given in Theorem 1, and let E1 and Dk be projections onto V1 and Wk, respectively, with respect to the basis {ϕαk}α∈Yk,k∈Z.Proposition 12.
Lets<η, and let f∈(G∘(β,γ))′. For all 1<p, q≤∞, then
(37)P1(f)p+∑k≥1‍δ-ksQkfpq1/q~E1(f)p+∑l≥1‍δ-lsDlfpq1/q,
and for all 1<p<∞ and 1<q≤∞(38)P1(f)p+∑k≥1‍δ-ksQkfq1/qp~E1(f)p+∑l≥1‍δ-lsDlfq1/qp.Proof.
Clearly, the equivalence ofP1fp and E1(f)p is a direct consequence of the size condition (13) and the wavelet expansion (29). We next show that
(39)∑k≥1‍δ-ksQkfpq1/q≲∑l≥1‍δ-lsDlfpq1/q.
Indeed, by the wavelet expansion (29) and almost orthogonal estimate (26), we have
(40)Qkfx≤∑l≥1‍∫X‍QkDl(x,y)Dl(f)(y)dμ(y)≤C∑l≥1‍δk-lη∫X‍1Vδ(k∧l)+V(x,y)δk∧lδk∧l+dx,yϵhhhhhhhhhhhhhh×Dlfydμ(y)≤C∑l≥1‍δk-lηMDlfx,
where M denotes the Hardy-Littlewood maximal operator. Thus,
(41)∑k≥1‍δ-ksQkfxpq1/q≤C∑l≥1‍δ-lsDlfpq1/q
due to the fact that s<η. This proves (39), and hence, we obtain (37) symmetrically.
We now prove (38). Clearly, by symmetry, it suffices to show that
(42)∑k≥1‍δ-ksQkfq1/qp≲∑l≥1‍δ-lsDlfq1/qp.
To this end, from (40) it follows that
(43)∑k≥1‍δ-ksQkfxq≤C∑k≥1‍∑l≥1‍δl-ksδk-lηδ-lsMDlfxq≤C∑k≥1‍∑l≥1‍δ(l-k)sδ|k-l|ηδ-lsMDlfxq≤C∑l≥1‍δ-lsMDlfxq
due to the fact that ∑l≥1‍δ(l-k)sδk-lη<∞ and ∑k≥1‍δ(l-k)sδk-lη<∞ for s<η. Thus, applying the Fefferman-Stein vector-valued maximal inequality, we obtain (42), immediately. The proof is complete.Let{ψαk}α∈Yk be an orthonormal basis of  Wk and let {ϕα1}α∈X1 be an orthonormal basis of V1 as shown in Theorem 1. Applying the wavelet expansion, given in Theorem 9,
(44)f=∑α∈X1‍ϕα1,fϕα1+∑k≥1‍∑α∈Yk‍ψαk,fψαk
we now give the wavelet characterizations of the inhomogeneous Besov and Triebel-Lizorkin spaces.Theorem 13.
Lets<η and let ϕα1, ψαk be that given in Theorem 1 with k≥1. For 1<p, q≤∞, then(45)fBps,q~∑α∈X1‍μQα11/p-1/2ϕα1,fpq/p+∑k≥1‍δ-ksq∑α∈Yk‍μQαk1/p-1/2ψαk,fpq/p1/q,
and for 1<p<∞, 1<p≤∞, then(46)fFps,q~+∑k≥1‍δ-ksq∑α∈Yk‍μQαk-1/2ψαk,fχQαkq1/q∑α∈Yk‍μQαk-12ψαk,fχQαkq∑α∈Xk‍μQαk-1/2ϕα1,fχQα1qhhhhhi+∑k≥1‍δ-ksq∑α∈Yk‍μQαk-1/2ψαk,fχQαkq1/qp.Proof.
Observe that, by analogous arguments in the proof of (26), we have
(47)P1ϕα1μQα1x≤C1Vδxα1+Vx,xα1δδ+dx,xα1ϵ,QkψαkμQαjx≤Cδj-kηVδj∧kxαj+Vx,xαjδj∧kδj∧k+dx,xαjϵ.
In order to prove (45), by Remark 11, it suffices to show that
(48)P1fpq+∑k≥1‍δ-ksQkfpq1/q~∑α∈Yk‍μQαk1/p-1/2ψαk,fpq/p∑α∈X1‍μQα11/p-1/2ϕα1,fpq/phhhhh+∑k≥1‍δ-ksq∑α∈Yk‍μQαk1/p-1/2ψαk,fpq/p1/q.
By the wavelet expansion (44), we have
(49)P1fx≤∑α∈X1‍μQα1ϕα1μQα1,fhhhhhhh×P1ϕα1μQα1x≤C∑α∈X1‍∫Qα1‍1Vδ(xα1)+V(x,xα1)hhhhhhhhhhhh×δδ+dx,xα1ϵhhhhhhhhhhhh×ϕα1μQα1,fdz≤C∑α∈X1‍∫Qα1‍1Vδ(z)+V(x,z)δδ+dx,zϵhhhhhhhhhhhh×ϕα1μQα1,fdz=C∑α∈X1‍∫X‍1Vδ(z)+V(x,z)δδ+dx,zϵhhhhhhhhhhh×ϕα1μQα1,fχQα1zdz≤CM∑α∈X1‍ϕα1μQα1,fχQα1(x),
so that(50)P1(f)p≤C∑α∈X1‍μQα11/p-1/2ϕα1,fp1/p.
By analogous arguments, we also have
(51)Qkfx≤∑α∈Yk‍μQαkψαkμQαk,fhhhhhhh×QkψαkμQαkx≤CM∑α∈Yk‍ψαkμQαk,fχQαkx,
and hence,
(52)∑k≥1‍δ-ksQkfpq≤C∑k≥1‍δ-ksqM∑α∈Yk‍ψαkμQαk,fχQαkpq≤C∑k≥1‍δ-ksq∑α∈Yk‍ψαkμQαk,fχQαkpq≤C∑k≥1‍δ-ksq∑α∈Yk‍μQαk1/p-1/2ψαk,fpχQαkq/p.
This implies that
(53)P1(f)pq+∑k≥1‍δ-ksQkfpq1/q≤∑α∈Yk‍μQαk1/p-1/2ψαk,fpq/p∑α∈X1‍μQα11/p-1/2ϕα1,fpq/p∫X‍P1fyP1ϕα1μQα1ydyphhhhh+∑k≥1‍δ-ksq∫X‍P1fyP1ϕα1μQα1ydyp∑α∈Yk‍μQαk1/p-1/2ψαk,fpq/p1/q.
On the other hand, by using the wavelet expansion and the size condition, we have(54)∑α∈X1‍μQα11/p-1/2ϕα1,fp≤∑α∈X1‍μQα1hhhhhhh×∫X‍P1fyP1ϕα1μQα1ydyp≤CP1(f)pp
as well as(55)∑α∈Yi‍ψαj,fμQαj1/p-1/2p≤∑α∈Yi‍μQαj∑k≥1‍∫X‍∫XψαjxμQαjQk(x,y)dμ(x)‍Qkfydμyp≤C∫X‍∑k≥1‍δj-kη∫X1Vδ(j∧k)(z)+V(y,z)δj∧kδj∧k+dy,zϵQkfydμypdμ(z)≤C∑k≥1‍δj-kηMQkfpp≤C∑k≥1‍δj-kηpQk(f)pp,and hence, by using the Hölder inequality, we have further that
(56)∑α∈Yj‍μQαj1/p-1/2ψαj,fpq/p∑α∈X1‍μQα11/p-1/2ϕα1,fpq/p+∑j≥1‍δ-jsq∑α∈Yj‍μQαj1/p-1/2ψαj,fpq/p1/q≤C+∑j≥1‍∑k≥1‍δk-jsδj-kηδ-ksQkfpqP1(f)pqhhhhhhi+∑j≥1‍∑k≥1‍δk-jsδj-kηδ-ksQkfpq1/q≤CP1fpq+∑k≥1‍δ-ksQkfpq1/q.
This completes the proof of (48).
Analogously, we can also obtain (46) by using the Fefferman-Stein vector-valued maximal inequality.As an application of wavelet characterizations, we now turn to discuss the dual spaces of the inhomogeneous Besov and Triebel-Lizorkin spaces. For convenience, we rewrite (44) as
(57)f=∑k≥0‍∑α∈Zk‍φαk,fφαk,
where Z0=X1, φα0=ϕα1, Zk=Yk, and φαk=ψαk for k≥1. The following concepts of spaces of sequences were first introduced by Frazier and Jawerth [13].Definition 14.
Letλ∶={λαk:k≥0,α∈Zk}, and let s<η and 1<p, q≤∞. The space bps,q of sequences is defined by bps,q∶={λ:λbps,q<∞} with
(58)λbps,q∶=∑k≥0‍δ-ksq∑α∈Zk‍μQαk1/p-1/2λαkpq/p1/q,
and, for 1<p<∞, the space fps,q is defined by fps,q∶={λ:λfps,q<∞} with
(59)λfps,q∶=∑k≥0‍δ-ksq∑α∈Zk‍μQαk-1/2λαkχQαkq1/qp.Remark 15.
It is easy to see thatf∈Bps,q⇔{〈φαk,f〉}∈bps,q and f∈Fps,q⇔{〈φαk,f〉}∈fps,q due to (57), (45), and (46).The next result shows thatL2 is dense in Bps,q as well as Fps,q.Proposition 16.
Let0<β, γ<η, and s<η. Then L2 is dense in both Bps,q for 1<p, q≤∞ and Fps,q for 1<p<∞ and 1<q≤∞.Proof.
Lets<η′<η. Since G(η′,η′)⊂L2, to show the conclusion, it suffices to show that G(η′,η′) is dense in Bps,q and Fps,q. Let f∈Bps,q. For any fixed M, write
(60)fM∶=P12(f)+∑k=1M‍Qk2(f).
By the estimates (26) and (27), we can show that fM∈G(η′,η′) and fMBps,q≤fBps,q. Further on, we can also show that
(61)lim⁡M→∞∑k>M‍Qk2(f)Bps,q=0.
This is a direct consequence of the wavelet expansion (29) and estimate (26). Indeed,
(62)DlQkf≤Cδk-lη′MQkfx,
where Dl is defined as the same as Ek for l≥0. By wavelet expansion (29), we then have
(63)∑k≥M‍Qk2(f)Bps,q≤C∑k>M‍δ-ksQkfpq1/q.
This implies that fM→f in Bps,q as M→∞, and hence, L2 is dense in Bps,q. By analogous arguments, we can also show that L2 is dense in Fps,q.We are ready to give the dualities of the Besov and Triebel-Lizorkin spaces.Theorem 17.
Let1<p, q<∞ and s<η, and let 1/p+1/p′=1/q+1/q′=1. Then (Bps,q)*=Bp′-s,q′. More precisely, given g∈Bp′-s,q′, then £g(f)=f,g defines a linear functional on Bps,q∩G(η′,η′) with s<η′<η such that
(64)£gf≤CfBps,qgBp′-s,q′,
and this linear functional can be extended to Bps,q with norm at most C∥g∥Bp′-s,q′.
Conversely, if£ is a linear functional on Bps,q, then there exists a unique g∈Bp′-s,q′ such that
(65)£g(f)=f,g
defines a linear functional on Bps,q∩G(η′,η′) with s<η′<η, and £ is the extension of £g with gBp′-s,q′≤C£.Remark 18.
Theorem17 also holds with Bps,q and Bp′-s,q′ replaced by Fps,q and Fp′-s,q′, respectively.Proof of Theorem17.
(i)
Suppose thatg∈Bp′-s,q′ and f∈Bps,q∩G(η′,η′). By Remark 15, we observe that
(66)g=∑k≥0‍∑α∈Zk‍φαk,gφαk
is an element of Bp′-s,q′, and hence, {φαk,g}∈bp′-s,q′. Also,
(67)f=∑k≥0‍∑α∈Zk‍φαkφαk,f
is an element of Bps,q∩G(η′,η′), and hence, {〈φαk,f〉}∈bps,q. Write now
(68)£g(f)∶=∑k≥0‍∑α∈Zk‍φαk,gφαk,f.
By the Hölder inequality and Minkowski inequality, we have
(69)£g(f)≤φαk,gbp′-s,q′φαk,fbps,q≤gBp′-s,q′fBps,q,
and hence, £g is a linear functional defined on Bps,q∩G(η′,η′). Clearly, £g has bounded extension to Bps,q due to the density of Bps,q∩G(η′,η′) in Bps,q. This proves that Bp′-s,q′⊂(Bps,q)*.(ii)
Suppose that£∈(Bps,q)*. We need to prove that there exists a g∈Bp′-s,q′ such that
(70)£g(f)=f,g,f∈Bps,q∩G(η′,η′).Note that
(71)£(f)≤£fBps,q≤£φαk,fbps,q,
and define a linear functional £~ on S={〈φαk,f〉:f∈Bps,q} by
(72)£~φαk,f=£f.
From (71) it follows that
(73)£~φαk,f=£f≤£φαk,fbps,q.
By the Hahn-Banach theorem, £~ can be extended to bps,q as a continuous linear functional. In addition, by arguments analogous to that in ([13, Remark 5.11]), we can show that (bps,q)*=bp′-s,q′, and hence, there exists a unique sequence {gk}∈bp′-s,q′ such that ∥£∥bp′-s,q′≤C£ and
(74)£~(ak)=∑k≥0‍akgk,{ak}∈bps,q.
Thus, for each f∈Bps,q∩G(η′,η′), we have
(75)£f=£~φαk,f=∑k≥0‍φαk,fgk=∑k≥0‍φαkgk,f.
Let g∶=∑k≥0‍φαkgk. By the orthogonality of φαk, we have
(76)gbp′-s,q′=φαk,∑l‍glφαlp′q′/p′∑k≥0‍δ-ksq′hhhhh×φαk,∑l‍glφαlp′∑α‍φαk,∑l‍glφαlμQαk1/p-1/2hhhhhhhhhhhh×φαk,∑l≥0‍glφαlp′q′/p′1/q′≤CgkμQαk1/p-1/2p′∑α‍μQαk1/p-1/2q′/p′∑k‍δ-ksq′∑k≥0‍δ-ksq′hhhhhhh×gkμQαk1/p-1/2p′∑α‍μQαk1/p-1/2∑α‍μQαk1/p-1/2hhhhhhhhhhhhh×gkμQαk1/p-1/2p′∑α‍μQαk1/p-1/2q′/p′∑k‍δ-ksq′1/q′≤C£,
and £(f)=£g(f)=g,f≤£fBps,q. Note that Bps,q∩G(η′,η′) is dense in Bps,q. The linear functional £ can be extended to Bps,q. Thus, £g(f)=〈g,f〉 for all f∈Bps,q.The next result gives elementary embedding properties of Besov and Triebel-Lizorkin spaces which can be proved by an approach analogous to that of Proposition  2 in [14, Section 2.3.2] and we omit the details here.Proposition 19.
(i)
Let1<q0≤q1<∞,1<p<∞, and s<η. Then
(77)Bps,q0⊂Bps,q1,Fps,q0⊂Fps,q1.
(ii)
Let1<q0≤q1<∞ and s+θ<η. Then
(78)Bps+θ,q0⊂Bps,q1,Fps+θ,q0⊂Fps,q1.
(iii)
Let1<p<∞, 1<q≤∞, and s<η. Then
(79)Bps,min⁡(p,q)⊂Fps,q⊂Bps,max⁡(p,q).


## 4. Homogeneous Besov and Triebel-Lizorkin Spaces

In this section, we develop the theory of the homogeneous Besov and Triebel-Lizorkin spaces and discuss relationships between homogeneous and inhomogeneous versions.In order to introduce the homogeneous Besov and Triebel-Lizorkin spaces, we recall again the spaces of test functions and distributions on spaces of homogeneous type given in [11]. Let G(x0,r,β,γ) be given in Definition 2. By G0(β,γ) we denote the collection of all test functions in G(β,γ) by ∫X‍f(x)dμ(x)=0. For 0<β,γ<η, let G0(β,γ) be the completion of the space G0(η,η) in the norm of G(β,γ). For f∈G0(β,γ), we define fG0(β,γ)=fG(β,γ).The distribution spaceG0(β,γ)′ is defined to be the set of all linear functionals £ from G0(β,γ) to C with the property that there exists C>0 such that, for all f∈G0(β,γ),
(80)£f≤CfG0β,γ.The homogeneous Besov and Triebel-Lizorkin spaces are defined as follows.Definition 20.
Lets<η and 0<β, γ<η. For 1<p, q<∞, the homogeneous Besov space B˙ps,qX is defined by
(81)B˙ps,q(X)∶=f∈G0β,γ′:fB˙ps,q<∞,
with
(82)fB˙ps,q∶=∑k∈Z‍δ-ksQkfpq1/q,
and for 1<p<∞,1<q≤∞, the homogeneous Triebel-Lizorkin space F˙ps,qX is defined by
(83)F˙ps,q(X)=f∈G0β,γ′:fF˙ps,q<∞,
with
(84)fF˙ps,q=∑k∈Z‍δ-ksQkfq1/qp.The authors in [11] showed that the wavelet expansion (28) holds in G0(β,γ) as well as G0(β,γ)′ with 0<β,γ<θ. The next result shows that Definition 20 is independent of choice of Qk for k∈Z.Proposition 21.
Letf∈G0(β,γ)′ and s<η. For 1<p,q≤∞, then
(85)∑k∈Z‍δ-ksQk(f)pq1/q~∑l∈Z‍δ-lsDl(f)pq1/q,
and for 1<p<∞ and 1<q≤∞, then
(86)∑k∈Z‍δ-ks|Qk(f)|q1/qp~∑l∈Z‍δ-ls|Dl(f)|q1/qp.We next give some relationships between homogeneous and inhomogeneous spaces.Proposition 22.
Let0<β,γ<η, and 0<s<η. For 1<p,q≤∞, then
(87)Bps,q=B˙ps,q∩Lp,
and for 1<p<∞, 1<q≤∞, then
(88)Fps,q=F˙ps,q∩Lp.We only give the proof of Proposition22. Others can be proved by arguments analogous to that of inhomogeneous versions.Proof.
We first show thatBps,q⊂B˙ps,q∩Lp. Let f∈Bps,q. By the wavelet expansion (29) and the Minkowski inequality, we have f∈Lp and
(89)fLp≤P1(f)p+∑k=1∞‍Qk(f)p≤P1(f)p+C∑k=1∞‍δ-ksQk(f)pq1/q≤CfBps,q.
Applying the wavelet expansion (29) and the size condition of QkP1(26), we also have f∈B˙ps,q and
(90)fB˙ps,q≤∑k<1‍δ-ksQk(f)pq1/q+∑k≥1‍δ-ksQkfpq1/q≤Cfp+fBps,q.
Thus, f∈B˙ps,q∩Lp and fLp+fB˙ps,q≲fBps,q.
Conversely, we show thatB˙ps,q∩Lp⊂Bps,q. Indeed, let f∈B˙ps,q∩Lp. By the size condition of P1, we have
(91)|P1(f)(x)|≤CM(f)(x),
and hence, P1(f)p≤Cfp. This implies that fBps,q≤Cfp+fB˙ps,q.
By analogous arguments, we can also obtain the second desired equality for the inhomogeneous Triebel-Lizorkin spaces.We now introduce the homogeneous Besov and Triebel-Lizorkin spaces by wavelet coefficients.Definition 23.
Let1<p, q≤∞, and s<η. The homogeneous Besov space B˙p,(2)s,q is defined by
(92)B˙p,2s,q(X)∶=f∈G0β,γ′:fB˙ps,q<∞,
where
(93)fB˙p,(2)s,q∶=∑k∈Z‍δ-ksq∑α∈Yk‍μQαk1/p-1/2ψαk,fpq/p1/q.
For1<p<∞, the homogeneous Triebel-Lizorkin space F˙p,(2)s,q is defined by
(94)F˙p,2s,q(X)=f∈G0β,γ′:fF˙ps,q<∞,
where
(95)fF˙p,(2)s,q:=∑k∈Z‍δ-ksq∑α∈Yk‍μQαk-1/2ψαk,fχQαkq1/qLp.
We are ready to give the wavelet characterizations of the homogeneous Besov and Triebel-Lizorkin spaces.Theorem 24.
Let0<s<η and 1<q≤∞. Then
(96)fB˙ps,q~fB˙p,(2)s,q,1<p≤∞,fF˙ps,q~fF˙p,(2)s,q,1<p<∞.Now we recall the homogenous spaces of sequencesb˙p,s,q and f˙ps,q; refer to [13] for details. Let λ∶={λαk:k∈Z,α∈Yk}. For s<η and 1<p,q≤∞, the space of sequence b˙ps,q is defined by b˙ps,q∶=λ:λb˙ps,q<∞ with
(97)λb˙ps,q∶=∑k∈Z‍δ-ksq∑α∈Yk‍μQαk1/p-1/2λαkpq/p1/q,
and, for 1<p<∞, the space f˙ps,q is defined by f˙ps,q∶=λ:λf˙ps,q<∞ with
(98)λf˙ps,q∶=∑k∈Z‍δ-ksq∑α∈Yk‍μQαk-1/2λαkχQαkq1/qp.Remark 25.
Suppose thatφαk be the same as in Theorem 1 for k∈Z, using Theorem 24 and [11, Corollary 3.5] then, we have f∈B˙ps,q⇔φαk,f∈b˙ps,q and f∈F˙ps,q⇔φαk,f∈f˙ps,q.Proposition 26.
Let0<s<η′<η. Then Gη′,η′ is dense in both B˙ps,q for 1<p and q≤∞ and F˙ps,q for 1<p<∞ and 1<q≤∞.Remark 27.
As the proof of Proposition16, we can also obtain that (28) converges in both the norm of B˙ps,q for 1<p, q≤∞ and the norm of F˙ps,q for 1<p<∞, 1<q≤∞.The homogeneous version of Theorem17 is given as follows.Theorem 28.
Let1<p, q<∞, and s<η, and let 1/p+1/p′=1/q+1/q′=1. Then B˙ps,q*=B˙p′-s,q′. More precisely, given g∈B˙p′-s,q′, then £gf=f,g defines a linear functional on B˙ps,q∩G0η′,η′ with s<η′<η such that
(99)£gf≤CfB˙ps,qgB˙p′-s,q′,
and this linear functional can be extended to B˙ps,q with norm at most CgB˙p′-s,q′.
Conversely, if£ is a linear functional on B˙ps,q, then there exists a unique g∈B˙p′-s,q′ such that
(100)£gf=〈f,g〉
defines a linear functional on B˙ps,q∩G0(η′,η′) with s<η′<η, and £ is the extension of £g with gB˙p′-s,q′≤C£.Remark 29.
Theorem28 also holds with B˙ps,q and B˙p′-s,q′ replaced by F˙ps,q and F˙p′-s,q′, respectively.Finally, we give the following embedding properties in homogeneous Besov and Triebel-Lizorkin spaces to end this section.Proposition 30.
(i)
Let1<q0≤q1≤∞, 1<p<∞, and s<η. Then
(101)B˙ps,q0⊂B˙ps,q1,F˙ps,q0⊂F˙ps,q1.
(ii)
Let1<p<∞, 1<q≤∞ and s<η. Then
(102)B˙ps,min⁡(p,q)⊂F˙ps,q⊂B˙ps,max⁡(p,q).



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