# Generalized Carleson Measure Spaces and Their Applications

Authors: Chin-Cheng Lin; Kunchuan Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce the generalized Carleson measure spaces CMOrα,q that extend BMO. Using Frazier and Jawerth's φ-transform and sequence spaces, we show that, for α∈ℝ and 0<p≤1, the duals of homogeneous Triebel-Lizorkin spaces Ḟpα,q for 1<q<∞ and 0<q≤1 are CMO(q′/p)-(q′/q)-α,q′ and CMOr-α+(n/p)-n,∞ (for any r∈ℝ), respectively. As applications, we give the necessary and sufficient conditions for the boundedness of wavelet multipliers and paraproduct operators acting on homogeneous Triebel-Lizorkin spaces.


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## Body

## 1. Introduction

In 1972, Fefferman and Stein [1] proved that the dual of H1 is the BMO space. In 1990, Frazier and Jawerth [2, Theorem  5.13] generalized the above duality to homogeneous Triebel-Lizorkin spaces Ḟpα,q. More precisely, they showed that the dual of Ḟ1α,q is Ḟ∞-α,q′ for α∈ℝ and 0<q<∞, where q′ is the conjugate index of q. Throughout the paper, q′ is interpreted as q′=∞ whenever 0<q≤1, and q′=q/(q-1) for 1<q≤∞. Note that Ḟ10,2=H1 and BMO=Ḟ∞0,2. For α∈ℝ, 0<p<1, and 0<q<∞, it is known (cf. [2–4]) that the dual of Ḟpα,q is Ḟ∞-α+(n/p)-n,∞. Here, we will give another characterization for the duals of Ḟpα,q in terms of the generalized Carleson measure spaces for α∈ℝ, 0<p≤1, and 0<q<∞.We say that a cubeQ⊆ℝn is dyadic if Q=Qjk={x=(x1,x2,…,xn)∈ℝn:2-jki≤xi<2-j(ki+1), i=1,2,…,n} for some j∈ℤ and k=(k1,k2,…,kn)∈ℤn. Denote by ℓ(Q)=2-j the side length of Q and by xQ=2-jk the “left lower corner’’ of Q when Q=Qjk. We use supP and ∑P to express the supremum and summation taken over all dyadic cubes P, respectively. Also, denote the summation taken over all dyadic cubes Q contained in P by ∑Q⊆P. For any dyadic cubes P and Q, either P and Q are nonoverlapping or one contains the other. For any function f defined on ℝn, j∈ℤ, and dyadic cube Q=Qjk, set(1.1)fQ(x)=|Q|-1/2f(x-xQl(Q))=2jn/2f(2jx-k),fj(x)=2jnf(2jx),f̃(x)=f(-x)¯.
It is clear that g̃j*f(xQ)=|Q|-1/2〈f,gQ〉, where 〈f,g〉 denotes the paring in the usual sense for g in a Fréchet space X and f in the dual of X.Choose a fixed functionφ in Schwartz class 𝒮=𝒮(ℝn), the collection of rapidly decreasing C∞ functions on ℝn, satisfying(1.2)supp⁡(φ̂)⊆{ξ:12≤|ξ|≤2},|φ̂(ξ)|≥c>0if35≤|ξ|≤53.
For α∈ℝ and 0<p,q≤+∞, we say that f belongs to the homogeneous Triebel-Lizorkin space Ḟpα,q if f∈𝒮′/𝒫, the tempered distributions modulo polynomials, satisfies(1.3)‖f‖Ḟpα,q:={‖{∑k∈Z(2kα|φk*f|)q}1/q‖Lp<∞for0<p<∞,sup⁡P{|P|-1∫P∑k=-log⁡2l(P)∞(2kα|φk*f(x)|)qdx}1/q<∞forp=∞.
When 0<p<∞ and q=∞, the above ℓq-norm is modified to be the supremum norm as usual, and Ḟ∞α,∞ is defined to be Ḃ∞α,∞, which is(1.4)‖f‖Ḟ∞α,∞:=sup⁡k∈Zsup⁡x∈Ql(Q)=2-k2kα|φk*f(x)|≈sup⁡Q|Q|-(α/n)-(1/2)|〈f,φQ〉|<∞.We now introduce a new spaceCMOrα,q as follows.Definition 1.1.
Letφ∈𝒮 satisfy (1.2). For α,r∈ℝ and 0<q≤∞, the generalized  Carleson measure spacesCMOrα,q is the collection of all f∈𝒮′/𝒫 satisfying ∥f∥CMOrα,q<∞, where
(1.5)‖f‖CMOrα,q:={sup⁡P{|P|-r∫P∑Q⊆P(|Q|-(α/n)-(1/2)|〈f,φQ〉|χQ(x))qdx}1/q,0<q<∞,sup⁡Psup⁡Q⊆P|Q|-(α/n)-(1/2)|〈f,φQ〉|=sup⁡Q|Q|-(α/n)-(1/2)|〈f,φQ〉|,q=∞,
and χQ denotes the characteristic function of Q.Remark 1.2.
By definition, we immediately haveCMOrα,∞=Ḟ∞α,∞ for α,r∈ℝ, and it is easy to check CMOrα,q={0} for r<0 and 0<q<∞. Note that the zero element in CMOrα,q means the class of polynomials. Also note that CMO0α,q=Ḟqα,q with equivalent norms for α∈ℝ and 0<q<∞. It follows from Proposition 3.3 that CMO1α,q=Ḟ∞α,q for α∈ℝ and 0<q<∞. In particular, CMO10,2=BMO, and hence the spaces CMOrα,q generalize BMO.Remark 1.3.
For a dyadic cubeP, denote by  kP=-log2ℓ(P); that is, kP is the integer so that  ℓ(P)=2-kP. In [5, 6], Yang and Yuan introduced the so-called “unified and generalized" Triebel-Lizorkin-type spaces Ḟp,qα,τ with four parameters by
(1.6)‖f‖Ḟp,qα,τ:=sup⁡P|P|-τ{∫P[∑k≥kP(2kα|φk*f(x)|)q]p/qdx}1/p<∞,
for α,τ∈ℝ, p∈(0,∞), q∈(0,∞], and f∈𝒮′/𝒫. Note that in [5] the space Ḟp,qα,τ was defined for τ∈[0,∞), p∈(1,∞), and q∈(1,∞]. It follows from [6, Theorem  3.1] that
(1.7)‖f‖Ḟp,qα,τ≈sup⁡P|P|-τ{∫P[∑Q⊂P(|Q|(-α/n)-(1/2)|〈f,φQ〉|χQ(x))q]p/qdx}1/p.It is clear thatCMOrα,q=Ḟq,qα,r/q for 0<q<∞, and hence CMOrα,q “looks like" a special case of Ḟp,qα,τ. In fact, it was proved in [7, 8] that the space Ḟp,qα,τ is the “same" as the space CMOτq+1-q/pα,q.The definition ofCMOrα,q is independent of the choice of φ∈𝒮 satisfying (1.2). To show that, we need the following Plancherel-Pôlya inequalities.Theorem 1.4 (Plancherel-Pôlya inequality for0<q<∞).
Letφ,ϕ∈𝒮 satisfy  (1.2). For α,r∈ℝ and 0<q<∞,  if f∈𝒮′/𝒫 satisfies
(1.8)sup⁡P{|P|-r∑k=-log⁡2l(P)∞∑Q⊆Pl(Q)=2-k(2kαsup⁡u∈Q|φ̃k*f(u)|)q|Q|}1/q<∞,
then
(1.9)sup⁡P{|P|-r∑k=-log⁡2l(P)∞∑Q⊆Pl(Q)=2-k(2kαsup⁡u∈Q|ϕ̃k*f(u)|)q|Q|}1/q≈sup⁡P{|P|-r∑k=-log⁡2l(P)∞∑Q⊆Pl(Q)=2-k(2kαinf⁡u∈Q|φ̃k*f(u)|)q|Q|}1/q.Theorem 1.5 (Plancherel-Pôlya inequality forq=∞).
Letφ,ϕ∈𝒮 satisfy  (1.2). For α,r∈ℝ, if  f∈𝒮′/𝒫 satisfies
(1.10)sup⁡Q(|Q|-(α/n)-rsup⁡u∈Q|φ̃kQ*f(u)|)<∞,
then
(1.11)sup⁡Q(|Q|-(α/n)-rsup⁡u∈Q|ϕ̃kQ*f(u)|)≈sup⁡Q(|Q|-(α/n)-rinf⁡u∈Q|φ̃kQ*f(u)|).Remark 1.6.
Letφ,ϕ∈𝒮 satisfy  (1.2). Denote by CMOrα,q(φ) the collection of all f∈𝒮′/𝒫 satisfying ∥f∥CMOrα,q(φ)<∞ defined in Definition 1.1 with respect to φ. Then, by Theorem 1.4,
(1.12)‖f‖CMOrα,q(ϕ)≤sup⁡P{|P|-r∑k=-log⁡2l(P)∞∑Q⊆Pl(Q)=2-k(2kαsup⁡u∈Q|ϕ̃k*f(u)|)q|Q|}1/q≤Csup⁡P{|P|-r∑k=-log⁡2l(P)∞∑Q⊆Pl(Q)=2-k(2kαinf⁡u∈Q|φ̃k*f(u)|)q|Q|}1/q≤C‖f‖CMOrα,q(φ)for0<q<∞.
Similarly, ∥f∥CMOrα,q(φ)≤C∥f∥CMOrα,q(ϕ) by interchanging the roles of φ and ϕ. Hence, the definition of CMOrα,q(φ) is independent of the choice of φ and, for short, denoted by CMOrα,q. Also, Theorem 1.5 shows that CMOrα,∞ is independent of the choice of φ satisfying (1.2) in the same argument.Remark 1.7.
The classical Plancherel-Pôlya inequality [9] concludes that if {xk} is an appropriate set of points in ℝn, for example, lattice points, where the length of the mesh is sufficiently small, then
(1.13)(∑k=1∞|f(xk)|p)1/p≈‖f‖p
for all 0<p≤∞ with a modification if p=∞.Using the Calderón reproducing formula (either continuous or discrete version), several authors obtain the variant Plancherel-Pôlya inequalities [10–13]. These inequalities give characterizations of the Besov spaces and the Triebel-Lizorkin spaces. Moreover, using these inequalities, one can show that the Littlewood-Paley g-function and Lusin area S-function are equivalent in Lp-norm.Define a linear mapSφ from 𝒮′/𝒫 into the family of complex sequences by(1.14)Sφ(f)={〈f,φQ〉}Q.
Let 𝒮0 denote the family of f∈𝒮 satisfying ∫xkf(x)dx=0 for all k∈(ℕ∪{0})n. For g∈CMOp-α,q′, define a linear functional Lg by (1.15)Lg(f)=〈Sψ(g),Sφ(f)〉=∑Q〈g,ψQ〉〈f,φQ〉forf∈S0.We now state our first main result as follows.Theorem 1.8 (duality forḞpα,q).
Suppose thatα∈ℝ,  0<p≤1, and 0<q<∞.
(a)
For1<q<∞, the dual of Ḟpα,q is CMO(q′/p)-(q′/q)-α,q′ in the following sense.
(i)
Forg∈CMO(q′/p)-(q′/q)-α,q′, the linear functional Lg given by (1.15), defined initially on 𝒮0, extends to a continuous linear functional on Ḟpα,q with ∥Lg∥≤C∥g∥CMO(q′/p)-(q′/q)-α,q′.(ii)
Conversely, every continuous linear functionalL on Ḟpα,q satisfies L=Lg for some g∈CMO(q′/p)-(q′/q)-α,q′ with ∥g∥CMO(q′/p)-(q′/q)-α,q′≤C∥L∥.(b)
For0<q≤1, the dual of Ḟpα,q is CMOr-α+(n/p)-n,∞(anyr∈ℝ) in the following sense.
(i)
Forg∈CMOr-α+(n/p)-n,∞, the linear functional Lg given by (1.15), defined initially on 𝒮0, extends to a continuous linear functional on Ḟpα,q with ∥Lg∥≤C∥g∥CMOr-α+(n/p)-n,∞.(ii)
Conversely, every continuous linear functionalL on Ḟpα,q satisfies L=Lg for some g∈CMOr-α+(n/p)-n,∞ with ∥g∥CMOr-α+(n/p)-n,∞≤C∥L∥.Remark 1.9.
For0<p<1 and 0<q≤1, it follows immediately from [2, 3] (Verbitsky [4] corrected a gap of the proof) and definition that (Ḟpα,q)′=Ḟ∞-α+(n/p)-n,∞=CMOr-α+(n/p)-n,∞ (any r∈ℝ). Theorem 1.8 (b) shows a different approach to the duality and includes the case of p=1.
Forp=1<q<∞, we have CMOq′-(q′/q)-α,q′=(Ḟ1α,q)′=Ḟ∞-α,q′. For 0<p<1<q<∞, CMO(q′/p)-(q′/q)-α,q′=(Ḟpα,q)′=Ḟ∞-α+(n/p)-n,∞, and hence CMO(q′/p)-(q′/q)-α,q′=CMOr-α+(n/p)-n,∞. That is, each CMO(q/p)-(q/q′)α,q coincides with CMOrα+(n/p)-n,∞ for α,r∈ℝ and 0<p<1<q<∞.Remark 1.10.
In Remark1.2 we are aware that CMOrα,q generalize BMO by the viewpoint of spaces directly. Choosing α=0 and q=2 in Theorem 1.8, we immediately have (Hp)′=(Ḟp0,2)′=CMO(2/p)-10,2 for 0<p≤1. In particular, BMO=CMO10,2. Once again, we obtain that CMOrα,q generalize BMO by the viewpoint of duality. It was also proved in [14] that the dual of the multiparameter product Hardy space is the generalized multiparameter Carleson measure space (cf. [14] for more details).Remark 1.11.
Forα,r∈ℝ, in order to make each index works, we defined CMOrα,∞ to be supP|P|-rsupQ⊆P|Q|-(α/n)-(1/2)|〈f,φQ〉| in our earlier version and in [7]. In such a situation, for 0<p,q≤1, the dual of Ḟpα,q would be CMO(1/p)-1-α,∞. In this paper, however, we follow the referee’s suggestion and adopt a more “natural’’ definition of CMOrα,∞ in Definition 1.1, that is, the limit of CMOrα,q as q→∞. The sequence space crα,∞ given in Definition 2.1 has a similar story as well.As applications, we first recall the Haar multipliers introduced in [15, 16]. Given a sequence t={tI}I, where the I’s  are dyadic intervals in ℝ, a Haar multiplier on L2(ℝ) is a linear operator of the form(1.16)Htf(x):=∑ItI〈f,hI〉hI(x),f∈L2(R),
where hI are the Haar functions corresponding to I.Using Meyer’s wavelets, we may generalize the above Haar multiplier toℝn and obtain a necessary and sufficient condition for the boundedness on Triebel-Lizorkin spaces. Let {ψi} for i∈E:={1,2,…2n-1} be Meyer’s wavelets (cf. [17], [18, pages  71–109]). Then, {ψQi}, where i∈E and Q′s are dyadic cubes in ℝn, is a frame for Ḟpα,q for α∈ℝ and 0<p,q≤∞; that is, ∥f∥Ḟpα,q≈∑i∈E∥{〈f,ψQi〉}Q∥ḟpα,q for f∈Ḟpα,q. For t={tQ}Q, define a wavelet multiplierT̃t on ℝn by(1.17)T̃t(f)=∑i∈E∑Q|Q|-1/2tQ〈f,ψQi〉ψQi
for f∈𝒮′/𝒫 such that the above summation is well defined.Theorem 1.12.
Suppose thatα,β∈ℝ,  0<p≤1 and 0<q<∞. Then,
(a)
for1<q<∞, T̃t is bounded from Ḟpα,q into Ḟ1α+β,1 if and only if t∈c(q′/p)-(q′/q)β,q′;(b)
for0<q≤1 and r∈ℝ, T̃t is bounded from Ḟpα,q into Ḟ1α+β,1 if and only if t∈crβ+(n/p)-n,∞, where crα,q is given in Definition 2.1.We consider another application. Letφ and ψ in 𝒮 satisfy (1.2) and (3.1). Choose a function Φ∈𝒮 supported on [0,1]n and ∫Φ=1. For α∈ℝ and g∈Ḟ∞α,∞, define the paraproduct operatorΠg by(1.18)Πg(f)=∑Q〈g,φQ〉|Q|-1/2〈f,ΦQ〉ψQ.
Thus, the adjoint operator Πg* is(1.19)Πg*(f)=∑Q〈g,φQ〉|Q|-1/2〈f,ψQ〉ΦQ.
Then, Πg1=g and Πg*1=0 since 〈1,ΦQ〉=|Q|1/2 and 〈1,ψQ〉=0. Also, if g∈Ḟ∞0,∞, then both Πg and Πg* are singular integral operators satisfying the weak boundedness property. Moreover, Πg is a Calderón-Zygmund operator (i.e., Πg is bounded on L2(ℝn)) if and only if g∈Ḟ∞0,2 by David-Journé’s T1 theorem [19] (also see [12, Theorems 5.4 and 5.8]). The authors showed a more general type of paraproduct operators in [12, page 688], which were derived from the discrete Calderón reproducing formula.Theorem 1.13.
Suppose thatβ∈ℝ,  0<r≤1 and 0<p≤r<q<r/(1-r). (i)
Forα<0, Πg is bounded from Ḟpα,q into Ḟrα+β,r if and only if g∈CMOr(q-p)/p(q-r)β,qr/(q-r).(ii)
Ifα∈ℝ with α+β>0 and g∈CMOr(q-p)/p(q-r)β,qr/(q-r), then Πg* is bounded from Ḟpα,q into Ḟrα+β,r.Remark 1.14.
Whenr=1, 0<p≤1<q<∞, and β∈ℝ, Theorem 1.13 says that Πg is bounded from Ḟpα,q into Ḟ1α+β,1 if and only if g∈CMO(q′/p)-(q′/q)β,q′ for α<0, and Πg* is bounded from Ḟpα,q into Ḟ1α+β,1 for α>-β provided g∈CMO(q′/p)-(q′/q)β,q′. In 1995, Youssfi [20] showed that, for β∈ℝ, 1<p<∞, 1≤q≤2, and g∈Ḟ∞β,∞, Πg is bounded from Ḟp0,q into Ḟpβ,p if and only if g∈Ḟ∞β,p. The special case of Theorem 1.13(i), p=r, generalizes Youssfi’s result to 0<p≤1. More precisely, for α<0, β∈ℝ, 0<p≤1, and p<q<p/(1-p), Πg is bounded from Ḟpα,q to Ḟpα+β,p if and only if g∈CMO1β,pq/(q-p)=Ḟ∞β,pq/(q-p).The paper is organized as follows. In Section2, we introduce the discrete version of the generalized Carleson measure spaces crα,q and show that the duals of sequence Triebel-Lizorkin spaces ḟpα,q for 1<q<∞ and 0<q≤1 are c(q′/p)-(q′/q)-α,q′ and cr-α+(n/p)-n,∞ (for any r∈ℝ), respectively. In Section 3, we prove the duals of homogeneous Triebel-Lizorkin spaces Ḟpα,q for 1<q<∞ and 0<q≤1 to be the generalized Carleson measure spaces CMO(q′/p)-(q′/q)-α,q′ and CMOr-α+(n/p)-n,∞ (for any r∈ℝ), respectively. In Section 4, we prove the Plancherel-Pôlya inequalities that give us the independence of the choice of φ for the definition of the generalized Carleson measure spaces. In the last section, we show the boundedness of wavelet multipliers and paraproduct operators. Throughout, we use C to denote a universal constant that does not depend on the main variables but may differ from line to line. Also, Q and P always mean the dyadic cubes in ℝn, and, for r>0, we denote by rQ the cube concentric with Q whose each edge is r times as long.


## 2. Sequence Spaces

In this section, we introduce sequence spacescrα,q and then characterize the duals of ḟpα,q by means of crα,q. Let us recall the definition of these sequence spaces ḟpα,q defined in [2]. For α∈ℝ and 0<p,q≤∞, the space ḟpα,q consists all such sequences s={sQ}Q satisfying(2.1)‖s‖ḟpα,q:={‖{∑Q(|Q|-(α/n)-(1/2)|sQ|χQ)q}1/q‖Lp<∞if0<p<∞,sup⁡P{|P|-1∫P∑Q⊆P(|Q|(-α/n)-(1/2)|sQ|χQ(x))qdx}1/q<∞ifp=∞.
As before, the previous ℓq-norm is modified to the supremum norm for 0<p<∞ and q=∞. For p=q=∞, we adopt the norm(2.2)‖s‖ḟ∞α,∞:=sup⁡Q|Q|-(α/n)-(1/2)|sQ|.
Note that ∥s∥ḟ∞α,qq is equivalent to the Carleson norm of the measure(2.3)∑Q(|Q|-(α/n)-(1/2)|sQ|)q|Q|δ(xQ,l(Q)),
where δ(x,t) is the point mass at (x,t)∈ℝ+n+1. See [2] for the details.To study the duals ofḟpα,q, we introduce a discrete version of the generalized   Carleson measure spacescrα,q.Definition 2.1.
Forα,r∈ℝ and 0<q≤∞, the space crα,q is the collection of all sequences t={tQ}Q satisfying ∥t∥crα,q<∞, where
(2.4)‖t‖crα,q:={sup⁡P⁡|P|-r∫P∑Q⊆P(|Q|-(α/n)-(1/2)|tQ|χQ(x))qdxfor0<q<∞,sup⁡P⁡sup⁡Q⊆P⁡|Q|-(α/n)-(1/2)|tQ|=sup⁡Q|Q|-(α/n)-(1/2)|tQ|forq=∞.
It is obvious that(2.5)‖t‖crα,q=sup⁡P{|P|-r∑Q⊆P(|Q|-(α/n)-(1/2)+(1/q)|tQ|)q}1/qfor0<q<∞
and ∥t∥crα,∞=∥t∥ḟ∞α,∞ for α,r∈ℝ. Using embedding theorem, Frazier and Jawerth [2, equation (5.14) and Theorem 5.9] obtained that, for α∈ℝ and 0<q<∞, the dual of ḟpα,q is ḟ∞-α+(n/p)-n,∞ when 0<p<1, and the dual of ḟ1α,q is ḟ∞-α,q′. Note that crα,q={0} for r<0 and 0<q<∞. Here we give the dual relationship between sequence spaces ḟpα,q and crα,q.Theorem 2.2 (duality forḟpα,q).
Suppose thatα∈ℝ,  0<p≤1, and 0<q<∞.
(a)
For1<q<∞, the dual of ḟpα,q is c(q′/p)-(q′/q)-α,q′ in the following sense.
(i)
Fort={tQ}Q∈c(q′/p)-(q′/q)-α,q′, the linear functional ℓt on ḟpα,q given by ℓt(s)=∑QsQtQ is continuous with ∥ℓt∥≤C∥t∥c(q′/p)-(q′/q)-α,q′ for s={sQ}Q∈ḟpα,q.(ii)
Conversely, every continuous linear functionalℓ on ḟpα,q satisfies ℓ=ℓt for some t∈c(q′/p)-(q′/q)-α,q′ with ∥t∥c(q′/p)-(q′/q)-α,q′≤C∥ℓ∥.(b)
For0<q≤1, the dual of ḟpα,q is cr-α+(n/p)-n,∞(anyr∈ℝ) in the following sense.
(i)
Fort={tQ}Q∈cr-α+(n/p)-n,∞, the linear functional ℓt on ḟpα,q given by ℓt(s)=∑QsQtQ is continuous with ∥ℓt∥≤C∥t∥cr-α+(n/p)-n,∞ for s={sQ}Q∈ḟpα,q.(ii)
Conversely, every continuous linear functionalℓ on ḟpα,q satisfies ℓ=ℓt for some t∈cr-α+(n/p)-n,∞ with ∥t∥cr-α+(n/p)-n,∞≤C∥ℓ∥.Remark 2.3.
Forα∈ℝ and 0<q<∞, sequence spaces c1α,q=ḟ∞α,q and crα,∞=ḟ∞α,∞ (for any r∈ℝ) by definitions. Theorem 2.2 shows that (ḟ1α,q)′=ḟ∞-α,q′, which gives a different but simpler proof of Frazier-Jawerth’s result for the duality of ḟ1α,q (cf. [2, Theorem 5.9]).Proof of Theorem2.2.
Fors={sQ}Q∈ḟpα,q and t={tQ}Q∈cr-α,q′, set s̃={s̃Q}Q and t̃={t̃Q}Q to be
(2.6)s̃Q=|Q|-α/nsQ,t̃Q=|Q|α/ntQ.
Then, ℓt̃(s̃)=ℓt(s). Also,
(2.7)‖s̃‖ḟp0,q=‖s‖ḟpα,q,‖t̃‖cr0,q′=‖t‖cr-α,q′.
Without loss of generality, we may assume that α=0.
We first consider the case1<q<∞. Let t∈c(q′/p)-(q′/q)0,q′ and define a linear functional ℓt on ḟp0,q by
(2.8)lt(s)=∑QsQtQfors∈ḟp0,q.
For s={sQ}Q∈ḟp0,q, let
(2.9)Vq(x):=(∑Q(|Q|-1/2|sQ|χQ(x))q)1/q.
For k∈ℤ, let
(2.10)Ωk:={x∈Rn:2k<Vq(x)≤2k+1},Ω̃k:={x∈Rn:MχΩk(x)>12},Bk:={dyadicQ:|Q∩Ωj|>|Q|2,|Q∩Ωj+1|≤|Q|2forsomej≥k},
where M is the Hardy-Littlewood maximal function. Then, for each dyadic cube Q, there exists exactly a k∈ℤ such that Q∈Bk. For every Q∈Bk, let Q̃ denote the maximal dyadic cube in Bk containing Q. Then all of such Q̃’s are pairwise disjoint. Thus, by Hölder’s inequality for q and the inequality (a+b)p≤ap+bp for 0<p≤1,
(2.11)|∑QsQtQ|≤∑k∈Z∑Q̃∈Bk∑Q⊆Q̃Q∈Bk(|Q|-(1/2)+(1/q)|sQ|)(|Q|(1/2)-(1/q)|tQ|)≤{∑k∈Z∑Q̃∈Bk(∑Q⊆Q̃Q∈Bk(|Q|-(1/2)+(1/q)|sQ|)q)p/q(∑Q⊆Q̃Q∈Bk(|Q|-(1/2)+(1/q′)|tQ|)q′)p/q′}1/p≤‖t‖c(q′/p)-(q′/q)0,q′{∑k∈Z∑Q̃∈Bk(∑Q⊆Q̃Q∈Bk(|Q|-(1/2)+(1/q)|sQ|)q)p/q|Q̃|1-(p/q)}1/p.
Since Q̃∈Bk implies Q̃⊆Ω̃k, the disjointness of Q̃’s and Hölder’s inequality yield
(2.12)|∑QsQtQ|≤‖t‖c(q′/p)-(q′/q)0,q′{∑k∈Z|Ω̃k|1-(p/q)(∑Q∈Bk(|Q|-(1/2)+(1/q)|sQ|)q)p/q1/p.
We claim that ∑Q∈Bk(|Q|-(1/2)+(1/q)|sQ|)q≤C2kq|Ω̃k| for k∈ℤ and 0<q<∞. Assume the claim for the moment. The weak (1,1) boundedness of M gives |Ω̃k|≤C|Ωk|, and hence
(2.13)|∑QsQtQ|≤C‖t‖c(q′/p)-(q′/q)0,q′(∑k∈Z|Ω̃k|1-(p/q)(2kq|Ω̃k|)p/q)1/p≤C‖t‖c(q′/p)-(q′/q)0,q′(∑k∈Z2kp|Ωk|)1/p≤C‖t‖c(q′/p)-(q′/q)0,q′‖Vq‖Lp=C‖t‖c(q′/p)-(q′/q)0,q′‖s‖ḟp0,q.
To prove the claim, we note that, for k∈ℤ and 0<q<∞,
(2.14)2q(k+1)|Ω̃k|≥∫Ω̃k∖⋃j=k+1∞Ωj(Vq(x))qdx=∫Ω̃k∖⋃j=k+1∞Ωj∑Q(|Q|-1/2|sQ|χQ(x))qdx≥∑Q∈Bk(|Q|-1/2|sQ|)q|(Ω̃k∖Ωj)∩Q|forsomej≥k+1,
which implies
(2.15)2q(k+1)|Ω̃k|≥12∑Q∈Bk(|Q|-(1/2)+(1/q)|sQ|)q.
For0<q≤1, with a modification, we have
(2.16)|∑QsQtQ|≤∑k∈Z∑Q̃∈Bk∑Q⊆Q̃Q∈Bk(|Q|1/2|sQ|)(|Q|-(1/p)+(1/2)|tQ|)(|Q||Q̃|)(1/p)-1|Q̃|(1/p)-1≤‖t‖cr(n/p)-n,∞{∑k∈Z∑Q̃∈Bk(∑Q⊆Q̃Q∈Bk|Q|1/2|sQ|)p|Q̃|1-p}1/p≤C‖t‖cr(n/p)-n,∞(∑k∈Z|Ω̃k|1-p(2k|Ω̃k|)p)1/p≤C‖t‖cr(n/p)-n,∞‖Vq‖Lp≤C‖t‖cr(n/p)-n,∞‖s‖ḟp0,q.
On the other hand, suppose thatℓ is a continuous linear functional on ḟp0,q. For each dyadic cube P, write eP={(eP)Q}Q to be the sequence defined by
(2.17)(eP)Q={1ifQ=P,0ifQ≠P.
Let tP=ℓ(eP) and t={tP}P. Then, for s={sQ}Q∈ḟp0,q,
(2.18)l(s)=∑QsQtQ=lt(s).
Fix a dyadic cube P. For 1<q<∞, let X be the sequence space consisting of s={sQ}Q⊆P, and define a counting measure on dyadic cubes Q⊆P by dσ(Q)=|Q|/|P|(q′/p)-(q′/q). Then,
(2.19)(1|P|(q′/p)-(q′/q)∑Q⊆P(|Q|-(1/2)+(1/q′)|tQ|)q′)1/q′=sup⁡‖s‖lq(X,dσ)≤1|1|P|(q′/p)-(q′/q)∑Q⊆P|Q|sQ|Q|-1/2|tQ||≤‖l‖sup⁡‖s‖lq(X,dσ)≤1‖{sQ|Q|1/2|P|(q′/p)-(q′/q)}Q⊆P‖ḟp0,q.
Note that
(2.20)‖{sQ|Q|1/2|P|(q′/p)-(q′/q)}Q⊆P‖ḟp0,q≤1|P|(q′/p)-(q′/q){(∑Q⊆P|Q||sQ|q)p/q⋅|P|1-(p/q)}1/p≤C‖s‖lq(X,dσ).
Thus,
(2.21)(1|P|(q′/p)-(q′/q)∑Q⊆P(|Q|(-1/2)+(1/q′)|tQ|)q′)1/q′≤C‖l‖,
and hence t∈c(q′/p)-(q′/q)0,q′. For 0<q≤1, consider eP defined before. Then, ∥eP∥ḟp0,q=|P|-(1/2)+(1/p) and
(2.22)(|P|(1/2)-(1/p)|tP|)‖eP‖ḟp0,q=|tP|=|l(eP)|≤‖l‖‖eP‖ḟp0,q.
Hence, ∥t∥cr(n/p)-n,∞=supP|P|(1/2)-(1/p)|tP|≤∥ℓ∥. This completes the proof.


## 3. Proof of the Main Theorem

Let us recall theφ-transform identity given by Frazier and Jawerth [2]. Choose a function φ∈𝒮 satisfying (1.2). Then there exists a function ψ∈𝒮 satisfying the same conditions as φ such that ∑j∈ℤφ̂¯(2-jξ)ψ̂(2-jξ)=1 for ξ≠0. The φ-transform identity is given by(3.1)f=∑Q〈f,φQ〉ψQ,
where the identity holds in the sense of 𝒮′/𝒫, 𝒮0, and Ḟpα,q-norm.Define a linear mapSφ from 𝒮′/𝒫 into the family of complex sequences by(3.2)Sφ(f)={〈f,φQ〉}Q,
and another linear map Tψ from the family of complex sequences into 𝒮′/𝒫 by(3.3)Tψ({sQ}Q)=∑QsQψQ.
Then, Tψ∘Sφ|Ḟpα,q is the identity on Ḟpα,q by [2, Theorem 2.2].Proposition 3.1.
Suppose thatα∈ℝ and, 0<p,q≤+∞, and φ,ψ in 𝒮 satisfy (1.2) and (3.1). The linear operators Sφ:Ḟpα,q↦ḟpα,q and Tψ:ḟpα,q↦Ḟpα,q defined by (3.2) and (3.3), respectively, are bounded. Furthermore, Tψ∘Sφ is the identity on Ḟpα,q. In particular, ∥f∥Ḟpα,q≈∥Sφ(f)∥ḟpα,q and Ḟpα,q can be identified with a complemented subspace of ḟpα,q.Figures1 and 2 illustrate the relationship among Ḟpα,q, ḟpα,q, CMOrα,q, and crα,q.Figure 1
Diagram for spaces and maps for1<q<∞.Figure 2
Diagram for spaces and maps for0<q≤1.One recalls the almost diagonality given by Frazier and Jawerth [2]. For α∈ℝ and 0<p,q≤∞, let J=n/(min{1,p,q}). One says that a matrix A={aQP}Q,P is (α,p,q)-almost diagonal if there exists ε>0 such that(3.4)sup⁡Q,P|aQP|wQP(ε)<+∞,
where(3.5)wQP(ε)=(l(Q)l(P))α(1+|xQ-xP|max⁡(l(P),l(Q)))-J-ε⋅min⁡{(l(Q)l(P))(n+ε)/2,(l(P)l(Q))((n+ε)/2)+J-n}.Lemma 3.2.
Forα,r∈ℝ and 0<q<∞, an (α+nr,q,q)-almost diagonal matrix is bounded on crα,q. Furthermore, when r≥0, an (α+nr,∞,∞)-almost diagonal matrix is bounded on crα,∞.We postpone the proof of Lemma3.2 until the end of Section 4.Letα,r∈ℝ. For q=∞, we have crα,∞=ḟ∞α,∞ and CMOrα,∞=Ḟ∞α,∞. Thus, Sφ:CMOrα,∞↦crα,∞ and Tψ:crα,∞↦CMOrα,∞ are bounded by Proposition 3.1. For 0<q<∞ and f∈CMOrα,q, let s={sQ}Q=Sφ(f). Then, the φ-transform identity (3.1) shows that f=∑QsQψQ and ∥f∥CMOrα,q=∥Sφ(f)∥crα,q=∥s∥crα,q. In particular, ∥f∥CMO1α,q=∥Sφ(f)∥c1α,q=∥Sφ(f)∥ḟ∞α,q≈∥f∥Ḟ∞α,q. Furthermore, for s∈crα,q,(3.6)‖Tψ(s)‖CMOrα,q=‖∑PsPψP‖CMOrα,q=‖{〈∑PsPψP,φQ〉}Q‖crα,q=‖As‖crα,q,
where A:={〈ψP,φQ〉}Q,P is (α+nr,q,q)-almost diagonal (cf. [2, Lemma 3.6]) and hence A is bounded on crα,q by Lemma 3.2. Therefore, Sφ is bounded from CMOrα,q to crα,q and Tψ is bounded from crα,q to CMOrα,q.We summarize thatTψ∘Sφ|CMOrα,q is also the identity on CMOrα,q.Proposition 3.3.
For(α,r,q)∈ℝ×ℝ×(0,∞) or  (α,r,q)∈ℝ×ℝ×{∞},  the linear operators Sφ:CMOrα,q↦crα,q and  Tψ:crα,q↦CMOrα,q are bounded. Furthermore, Tψ∘Sφ is the identity on CMOrα,q and ∥f∥CMOrα,q=∥Sφf∥crα,q.  In particular, ∥f∥CMO1α,q=∥Sφ(f)∥c1α,q=∥Sφ(f)∥ḟ∞α,q≈∥f∥Ḟ∞α,q for α∈ℝ and 0<q<∞,  and ∥f∥CMOrα,∞=∥Sφ(f)∥crα,∞=∥Sφ(f)∥ḟ∞α,∞≈∥f∥Ḟ∞α,∞ for α,r∈ℝ.Theorem1.8 can be proved as a consequence of Propositions 3.1–3.3 and a duality result between two sequence spaces.Proof of Theorem1.8.
First let us consider the case for1<q<∞. Let g∈CMO(q′/p)-(q′/q)-α,q′. Then, by Proposition 3.3, ∥g∥CMO(q′/p)-(q′/q)-α,q′=∥Sψ(g)∥c(q′/p)-(q′/q)-α,q′. It follows from Theorem 2.2 that ℓSψ(g) is a continuous linear functional on ḟpα,q and ∥ℓSψ(g)∥≈∥Sψ(g)∥c(q′/p)-(q′/q)-α,q′. Hence, for f∈𝒮0,
(3.7)|Lg(f)|≤C‖Sψ(g)‖c(q′/p)-(q′/q)-α,q′‖Sφ(f)‖ḟpα,q≤C‖g‖CMO(q′/p)-(q′/q)-α,q′‖f‖Ḟpα,q.
Since 𝒮0 is dense in Ḟpα,q, the functional Lg can be extended to a continuous linear functional on Ḟpα,q satisfying ∥Lg∥≤C∥g∥CMO(q′/p)-(q′/q)-α,q′.
Conversely, letL∈(Ḟpα,q)′, and set ℓ=L∘Tψ on ḟpα,q. By Proposition 3.1, ℓ∈(ḟpα,q)′. Thus, by Theorem 2.2, there exists t={tQ}Q∈c(q′/p)-(q′/q)-α,q′ such that
(3.8)l({sQ}Q)=∑QsQtQfor{sQ}Q∈ḟpα,q,
and ∥t∥c(q′/p)-(q′/q)-α,q′≈∥ℓ∥≤C∥L∥. For f∈Ḟpα,q, we have
(3.9)l∘Sφ(f)=L∘Tψ∘Sφ(f)=L(f).
So, for f∈𝒮0 and letting g=Tψ(t)=∑QtQψQ,
(3.10)L(f)=l∘Sφ(f)=∑Q〈f,φQ〉tQ=〈t,Sφ(f)〉.
It follows from [2, equations (2.7)-(2.8)] that 〈g,f〉=〈Sψ(g),Sφ(f)〉 and 〈t,Sφ(f)〉=〈Tψ(t),f〉 for f∈𝒮0 and g∈𝒮′/𝒫. This shows that L(f)=〈Tψ(t),f〉=Lg(f) for f∈𝒮0. Proposition 3.3 and Theorem 2.2 give
(3.11)‖g‖CMO(q′/p)-(q′/q)-α,q′≤C‖t‖c(q′/p)-(q′/q)-α,q′≤C‖L‖.
A similar argument gives the desired result for0<q≤1 with a slight modification, and hence the proof is finished.Remark 3.4.
As pointed out by one of the referees, Yang and Yuan [8, Theorem 1] show that if τ>1/p and 0<p,q<∞, then Ḟp,qα,τ=Ḟ∞α+nτ-(n/p),∞, where the definition of Ḟp,qα,τ is given in Remark 1.3. Thus, for 0<p<1 and 1<q<∞,
(3.12)(Ḟpα,q)′=Ḟ∞-α+(n/p)-n,∞=Ḟq′,q′-α,(1/p)-(1/q)=CMO(q′/p)-(q′/q)-α,q′,
which demonstrates a different approach to the duality.


## 4. Proofs of the Plancherel-Pôlya Inequalities

In this section we demonstrate the Plancherel-Pôlya inequalities.Proof of Theorem1.4.
Without loss of generality, we may assume thatα=0. By (3.1), we rewrite ϕ̃j*f(u) as
(4.1)ϕ̃j*f(u)=∑Q〈f,φQ〉∫ϕ̃j(u-x)ψQ(x)dx=∑k∈Z∑Ql(Q)=2-k|Q|〈f,φk(⋅-xQ)〉∫ϕ̃j(u-x)ψk(x-xQ)dx.
Using the inequality [2, page 151, equation (B.5)]
(4.2)|∫ϕ̃j(u-x)ψk(x-xQ)dx|≤C2-K|j-k|2-(j∧k)(2-(j∧k)+|u-xQ|)n+1,
where j∧k=min{j,k} and K>1+nr, we obtain
(4.3)|ϕ̃j*f(u)|≤C∑k∈Z∑Ql(Q)=2-k2-K|j-k||Q|2-(j∧k)(2-(j∧k)+|u-xQ|)n+1|φ̃k*f(xQ)|.
Thus, for ℓ(Q′)=2-j,
(4.4)(sup⁡u∈Q′|ϕ̃j*f(u)|)q≤C(∑k∈Z∑Ql(Q)=2-k2-K|j-k||Q|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1|φ̃k*f(xQ)|)q≤C∑k∈Z∑Ql(Q)=2-k2-K|j-k||Q|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1|φ̃k*f(xQ)|q,
where the last inequality is followed by Hölder’s inequality and
(4.5)∑Ql(Q)=2-k|Q|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1≤C.
Denote TQ by
(4.6)TQ:=inf⁡u∈Q|φ̃k*f(u)|q.
Since xQ can be replaced by any point in Q in the last inequality,
(4.7)(sup⁡u∈Q′|ϕ̃j*f(u)|)q≤C∑k∈Z∑Ql(Q)=2-k2-K|j-k||Q|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1TQ.
Given a dyadic cube P with ℓ(P)=2-k0, the above estimates yield
(4.8)∑j=k0∞∑Q′⊆Pl(Q′)=2-j(sup⁡u∈Q′|ϕ̃j*f(u)|)q|Q′|≤C∑j=k0∞∑Q′⊆Pl(Q′)=2-j∑k∈Z∑Ql(Q)=2-k2-K|j-k||Q′|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1TQ|Q|:=CA1+CA2,
where
(4.9)A1=∑j=k0∞∑Q′⊆Pl(Q′)=2-j∑k≥k0∑Ql(Q)=2-k2-K|j-k||Q′|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1TQ|Q|,A2=∑j=k0∞∑Q′⊆Pl(Q′)=2-j∑k<k0∑Ql(Q)=2-k2-K|j-k||Q′|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1TQ|Q|.
Then, A1 can be further decomposed as
(4.10)A1=∑j=k0∞∑Q′⊆Pl(Q′)=2-j∑k≥k0∑Q⊆3Pl(Q)=2-k2-K|j-k||Q′|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1TQ|Q|+∑j=k0∞∑Q′⊆Pl(Q′)=2-j∑k≥k0∑Q∩3P=∅l(Q)=2-k2-K|j-k||Q′|2-(j∧k)(2-(j∧k)+|xQ′-xQ|)n+1TQ|Q|:=A11+A12.
There are 3n dyadic cubes in 3P with the same side length as P, so
(4.11)∑Q⊆3Pl(Q)≤l(P)TQ|Q|≤3nsup⁡P′⊆3Pl(P′)=l(P)∑Q⊆P′l(Q)≤l(P′)TQ|Q|.
Thus,
(4.12)|P|-rA11≤C|P|-r∑j=k0∞∑Q′⊆Pl(Q′)=2-j∑k≥k0∑Q⊆3Pl(Q)=2-k2-K|j-k||Q′|2-(j∧k)(2-(j∧k)+|xQ′-xP|)n+1TQ|Q|≤Csup⁡P′|P′|-r∑k=-log⁡2l(P′)∞∑Q⊆P′l(Q)=2-kinf⁡u∈Q|φ̃k*f(u)|q|Q|.
Next we decompose the set of dyadic cubes{Q:Q∩3P=∅,ℓ(Q)=ℓ(P)} into {Bi}i∈ℕ according to the distance between each Q and P. Namely, for each i∈ℕ,
(4.13)Bi:={P′:P′∩3P=∅,l(P)=l(P′),2i-k0≤|yP′-yP|<2i-k0+1},
where yQ denotes the center of Q. Then, we obtain
(4.14)|P|-rA12≤C∑i=1∞∑P′∈Bi|P′|-r∑j=k0∞∑Q′⊆Pl(Q′)=2-j∑k≥k0∑Q⊆P′l(Q)=2-k2-K|j-k||Q′|×2-(j∧k)(2-(j∧k)+|xP′-xP|)n+1TQ|Q|.
Since ∑Q′⊆Pℓ(Q′)=2-j|Q′|=|P| for each j≥k0 and |xP′-xP|≈2i-k0 for P′∈Bi, the right-hand side of (4.14) is dominated by
(4.15)C∑i=1∞∑P′∈Bi|P|2-k02(i-k0)(n+1)∑k≥k0(∑j=k0∞2k0-(j∧k)-|k-j|)(|P′|-r∑Q⊆P′l(Q)=2-kTQ|Q|).
There are at most 2(i+2)n cubes in Bi, and hence
(4.16)|P|-rA12≤C{sup⁡P′|P′|-r∑k≥k0∑Q⊆P′l(Q)=2-kTQ|Q|}∑i=1∞|P|2-k02(i-k0)(n+1)2in=Csup⁡P′|P′|-r∑k=-log⁡2l(P′)∞∑Q⊆P′l(Q)=2-kinf⁡u∈Q|φ̃k*f(u)|q|Q|.
To estimateA2, for i∈ℕ and k<k0, set
(4.17)Ei,k:={Q:l(Q)=2-k,xQ∈2iP∖2i-1P}.
Then, |xQ-xP|≈2i-k0 for Q∈Ei,k and
(4.18)A2=∑j=k0∞∑k<k0∑i=1∞∑Q∈Ei,k2-K|j-k||P||Q|-r2-(j∧k)(2-(j∧k)+|xQ-xP|)n+1|Q|-rTQ|Q|.
Since, for Q∈Ei,k,
(4.19)|Q|-rTQ|Q|≤sup⁡P′|P′|-r∑m=-log⁡2l(P′)∞∑Q′⊆P′l(Q′)=2-mTQ′|Q′|
and the number of dyadic cubes contained in Ei,k is at most 2(i+k-k0)n,
(4.20)|P|-rA2≤C{sup⁡P′|P′|-r∑m=-log⁡2l(P′)∞∑Q′⊆P′l(Q′)=2-mTQ′|Q′|}×∑j=k0∞∑k<k0∑i=1∞2(k0-k)nr2K(k-j)2-k(n+1)2(i-k0)(n+1)2(i+k-k0)n=Csup⁡P′|P′|-r∑m=-log⁡2l(P′)∞∑Q′⊆P′l(Q′)=2-minf⁡u∈Q′|φ̃m*f(u)|q|Q′|,
where the condition K>1+nr is used in the last equality. Combining the estimates of A1 and A2, we prove Theorem 1.4.By modifying the proof above, we may easily show Theorem1.5. Detailed verifications are left to the reader.We now return to show Lemma3.2.Proof of Lemma3.2.
Forr<0, crα,q={0}, and hence the result holds. For r=0, c0α,q=ḟqα,q, and so the matrix is bounded by [2, Theorem 3.3]. To complete the proof, it suffices to show the boundedness of (α+nr,q,q)-almost diagonal matrices for the case r>0.
We may assume thatα=0 since the case implies the general case. The proof is similar to the proof of Theorem 1.4. Here, we only outline the proof. First let us consider the case for q>1. Let A={aQP}Q,P be an (nr,q,q)-almost diagonal matrix. Then, for ℓ(Q)=2-k,
(4.21)|(As)Q|≤C∑j∈Z∑l(P)=2-j2(j-k)(nr+((n+ε)/2))(1+2j|xQ-xP|)-n-ε|sP|,(|Q|-1/2|(As)Q|)q≤C∑j∈Z∑l(P)=2-j2(j-k)(nr+(ε/2))(1+2j|xQ-xP|)-n-ε(|P|-1/2|sP|)q
due to Hölder’s inequality. Given a dyadic cube R with ℓ(R)=2-δ,
(4.22)∑k≥δ∑Q⊆Rl(Q)=2-k(|Q|-1/2|(As)Q|)q|Q|≤CI+CII,
where
(4.23)I=∑k≥δ∑Q⊆Rl(Q)=2-k∑j≥δ∑l(P)=2-j2(j-k)(nr+n+(ε/2))(1+2j|xQ-xP|)-n-ε(|P|-1/2|sP|)q|P|,II=∑k≥δ∑Q⊆Rl(Q)=2-k∑j<δ∑l(P)=2-j2(j-k)(nr+n+(ε/2))(1+2j|xQ-xP|)-n-ε(|P|-1/2|sP|)q|P|.
Then, I can be further decomposed as
(4.24)I=∑k≥δ∑Q⊆Rl(Q)=2-k∑j≥δ∑P⊆3Rl(P)=2-j2(j-k)(nr+n+(ε/2))(1+2j|xQ-xP|)-n-ε(|P|-1/2|sP|)q|P|+∑k≥δ∑Q⊆Rl(Q)=2-k∑j≥δ∑P∩3R=∅l(P)=2-j2(j-k)(nr+n+(ε/2))(1+2j|xQ-xP|)-n-ε(|P|-1/2|sP|)q|P|:=I11+I12.
The same argument showed in the proof of Theorem 1.4 for the term A1 gives us
(4.25)|R|-rI≤C‖s‖cr0,qq.
To estimateII, for i∈ℕ and j<δ, let
(4.26)Ei,j:={Q:l(Q)=2-j,xQ∈2iR∖2i-1R}.
Then, using the same argument as Theorem 1.4 for A2, we have
(4.27)|R|-rII≤C‖s‖cr0,qq.
Both estimates for I and II show the desired result for q>1.
Whenq≤1, we modify the previous proof by replacing Hölder’s inequality with q-triangle inequality to get the result.
Whenq=∞ and r≥0, the space crα,∞=ḟ∞α,∞, and hence an (α+nr,∞,∞)-almost diagonal matrix is bounded on crα,∞ by Proposition 5.3.Remark 4.1.
Note thatc1α,q=ḟ∞α,q. By a duality argument and [2, Theorem 3.3 and page 81], one can show that the (α+n,q,q)-almost diagonal matrix is bounded on ḟ∞α,q. When q>1 and r>1, we can prove Lemma 3.2 by duality in Theorem 2.2. Let A={aQP}Q,P be an (nr,q,q)-almost diagonal matrix. Also define the transpose of A by A′={aPQ¯}Q,P. For q>1 and r>1, let p=(q+q′)/(q′r+q). Then, p<1. Since A is (nr,q,q)-almost diagonal, A′ is (0,p,q′)-almost diagonal by a calculation for a different value of ε. Thus, by Theorem 2.2 (a) and Proposition 5.3, A′ is bounded on cr0,q.


## 5. Applications

We define another wavelet multiplier onℝn by using φ-transform identity as follows. Let φ and ψ in 𝒮 satisfy (1.2) and (3.1). For a sequence t={tQ}Q, where the Q′s are dyadic cubes in ℝn, define the wavelet multiplierTt by(5.1)Tt(f)=∑Q|Q|-1/2tQ〈f,φQ〉ψQ
for f∈𝒮′/𝒫 such that the above summation is well defined. Thus, we have the following characterization.Theorem 5.1.
Suppose thatα,β∈ℝ,  0<p≤1, and 0<q<∞. Then,
(a)
for1<q<∞, Tt is bounded from Ḟpα,q into Ḟ1α+β,1 if t∈c(q′/p)-(q′/q)β,q′,(b)
for0<q≤1 and r∈ℝ, Tt is bounded from Ḟpα,q into Ḟ1α+β,1 if t∈crβ+(n/p)-n,∞.Proof.
We show the caseα=0 only, which implies the general case by (2.7). For β∈ℝ, 0<p≤1, and 1<q<∞, let f∈Ḟp0,q and t∈c(q′/p)-(q′/q)β,q′. It follows from Theorem 2.2 and Proposition 3.1 that
(5.2)‖Tt(f)‖Ḟ1β,1≤C‖{|Q|-1/2tQ〈f,φQ〉}Q‖ḟ1β,1=C∑Q(|Q|-β/n|tQ|)|〈f,φQ〉|≤C‖{〈f,φQ〉}Q‖ḟp0,q‖{|Q|-β/ntQ}Q‖c(q′/p)-(q′/q)0,q′≤C‖f‖Ḟp0,q‖t‖c(q′/p)-(q′/q)β,q′.
This shows that Tt is bounded from Ḟp0,q into Ḟ1β,1 and ∥Tt∥≤C∥t∥c(q′/p)-(q′/q)β,q′. A similar argument yields the boundedness of Tt for the case 0<q≤1.In order to prove Theorem1.12, we demonstrate a similar result in sequence spaces first. For a sequence t={tQ}Q, define Dt by(5.3)Dt(s)={|Q|-1/2tQsQ}Qfors={sQ}Qwithfinitelymanynonzeroterms.Theorem 5.2.
Suppose thatα,β∈ℝ,  0<p≤1, and 0<q<∞. Then,
(a)
for1<q<∞, Dt is extendible to be bounded from ḟpα,q into ḟ1α+β,1 if and only if t∈c(q′/p)-(q′/q)β,q′,(b)
for0<q≤1 and r∈ℝ, Dt is extendible to be bounded from ḟpα,q into ḟ1α+β,1 if and only if t∈crβ+(n/p)-n,∞.Proof.
We still assume thatα=0. For β∈ℝ, 0<p≤1, and 1<q<∞, let s={sQ}Q∈ḟp0,q and t={tQ}Q∈c(q′/p)-(q′/q)β,q′. It follows from Theorem 2.2 that
(5.4)‖Dt(s)‖ḟ1β,1=∑Q(|Q|-β/n|tQ|)|sQ|≤C‖s‖ḟp0,q‖{|Q|-β/ntQ}Q‖c(q′/p)-(q′/q)0,q′=C‖s‖ḟp0,q‖t‖c(q′/p)-(q′/q)β,q′.
Conversely, suppose thatDt maps from ḟp0,q into ḟ1β,1 boundedly. For t={tQ}Q, let t̃={|Q|-β/ntQ}Q. Define a linear functional ℓt̃ by
(5.5)lt̃(s)=∑QsQt̃Qfors={sQ}Qwithfinitelymanynonzeroterms.
Then,
(5.6)|lt̃(s)|≤∑Q(|Q|-β/n|tQ|)|sQ|=‖Dt(s)‖ḟ1β,1.
The assumption shows that ℓt̃ is a continuous linear functional on ḟp0,q. Using Theorem 2.2, we have t̃∈c(q′/p)-(q′/q)0,q′, and hence t∈c(q′/p)-(q′/q)β,q′.
For0<q≤1, a similar argument gives the desired result of (b).Proof of Theorem1.12.
The “if’’ part follows from Theorem5.1. To show the “only if’’ part, define T̃ti by
(5.7)T̃ti(f)=∑Q|Q|-1/2tQ〈f,ψQi〉ψQi.
The boundedness of T̃t says that T̃ti is bounded from Ḟpα,q into Ḟ1α+β,1. Clearly,
(5.8)Sψi∘T̃ti∘Tψi(s)=Dt(s)fors∈ḟpα,q.
It follows from Proposition 3.1 that Dt is bounded from ḟpα,q into ḟ1α+β,1, and hence t∈c(q′/p)-(q′/q)β,q′ for 1<q<∞ and t∈crβ+(n/p)-n,∞ for 0<q≤1 and r∈ℝ by Theorem 5.2.In order to study the boundedness of the paraproduct operators acting on Triebel-Lizorkin spaces, we need more results described as follows.Proposition 5.3 ([2, pages 54 and 81]).
Forα∈ℝ and 0<p,q≤∞,  an (α,p,q)-almost diagonal matrix is bounded on ḟpα,q.Lemma 5.4.
Define a matrix byG={〈ψP,ΦQ〉}Q,P. Then, for α<0 and  0<p,q≤+∞, G is (α,p,q)-almost diagonal and hence is bounded on ḟpα,q.Proof.
Forℓ(P)≤ℓ(Q), since ∫xγψP(x)dx=0 for all γ, by [2, page 150, Lemma  B.1], we have
(5.9)|〈ψP,ΦQ〉|≤C(l(Q)l(P))α(1+|xQ-xP|l(Q))-J-ε(l(P)l(Q))((n+ε)/2)+J-n
for ε>0 and α<J-n+(ε/2), where J=n/min{1,p,q} and C is independent of P and Q. For ℓ(Q)<ℓ(P), by [2, page 152, Lemma B.2], we obtain
(5.10)|〈ψP,ΦQ〉|≤C(1+|xQ-xP|l(P))-J-ε(l(Q)l(P))n/2=C(l(Q)l(P))α(1+|xQ-xP|l(P))-J-ε(l(Q)l(P))(n-2α)/2.
Choosing ε=-2α, we obtain the result.We now can prove Theorem1.13.Proof of Theorem1.13.
To simplify notations, letq0=qr/(q-r) and (1/p0)=(1/p)-(1/q)+(1/q0′). The requirement p≤r<q<r/(1-r) guarantees that  p0≤1≤q0. Now assume that g∈CMO(q0/p0)-(q0/q0′)β,q0 and f∈Ḟpα,q. To prove part (i), by (3.1) we rewrite Πg(f) as
(5.11)Πg(f)=∑Q〈g,φQ〉|Q|-1/2〈∑P〈f,φP〉ψP,ΦQ〉ψQ=∑Q〈g,φQ〉|Q|-1/2(Gs)QψQ,
where s={〈f,φp〉}P. Proposition 3.1 and Theorem 2.2 give
(5.12)‖Πg(f)‖Ḟrα+β,rr≤C‖{|Q|-1/2〈g,φQ〉(Gs)Q}Q‖ḟrα+β,rr=C∑Q(|Q|-(β/n)-(1/2)+(1/2r)|〈g,φQ〉|)r⋅(|Q|-(α/n)-(1/2)+(1/2r)|(Gs)Q|)r≤C‖{(|Q|-(β/n)-(1/2)+(1/2r)|〈g,φQ〉|)r}Q‖c((q/r)′/(p/r))-((q/r)′/(q/r))0,(q/r)′×‖{(|Q|-(α/n)-(1/2)+(1/2r)|(Gs)Q|)r}Q‖ḟp/r0,q/r.
It is clear that
(5.13)‖{(|Q|-(β/n)-(1/2)+(1/2r)|〈g,φQ〉|)r}Q‖c((q/r)′/(p/r))-((q/r)′/(q/r))0,(q/r)′=sup⁡P{|P|-r(q/r)′((1/p)-(1/q))∫P∑Q⊆P(|Q|-(β/n)-(1/2)|〈g,φQ〉|χQ(x))r(q/r)′dx}1/(q/r)′=‖{〈g,φQ〉}Q‖c(q0/p0)-(q0/p0′)β,q0r
and
(5.14)‖{(|Q|-(α/n)-(1/2)+(1/2r)(Gs)Q)r}Q‖ḟp/r0,(q/r)=‖(∑Q(|Q|-(α/n)-(1/2)|(Gs)Q|)qχQ(x))r/q‖Lp/r=‖Gs‖ḟpα,qr.
Hence, by Propositions 3.1 and 3.3, and Lemma 5.4,
(5.15)‖Πg(f)‖Ḟrα+β,r≤C‖{〈g,φQ〉}Q‖c(q0/p0)-(q0/q0′)β,q0‖Gs‖ḟpα,q≤C‖g‖CMO(q0/p0)-(q0/q0′)β,q0‖s‖ḟpα,q≤C‖g‖CMO(q0/p0)-(q0/q0′)β,q0‖f‖Ḟpα,q.
Next suppose thatΠg is bounded from Ḟpα,q into Ḟrα+β,r. Without lost of generality, we may assume that  α=0. A computation yields
(5.16)(|P|-q0((1/p0)+(1/q0)-1)∫P∑Q⊆P(|Q|-(β/n)-(1/2)|〈g,φQ〉|χQ(x))q0dx)1/q0=|P|-(1/p)+(1/q)(∫P∑Q⊆P(|Q|-(β/n)-(1/2)|〈g,φQ〉|χQ(x))qr/(q-r)dx)(q-r)/qr≤C|P|-(1/p)(∫P∑Q⊆P(|Q|-(β/n)-(1/2)|〈g,φQ〉|χQ(x))rdx)1/r.
Fix an integer N>(n/p)-n. Choose a function θ∈𝒮(ℝn) satisfying θ(x)=1 on [0,1]n, θ(x)=0 if x∉3[0,1]n and ∫xγθ(x)dx=0 for all multi-indices γ with |γ|≤N. By the molecular theory [2, page 56], it follows that θ∈Ḟp0,q. For each dyadic cube P, define θP by
(5.17)θP(x)=θ(x-xPl(P)).
Then, 〈θP,ΦQ〉=∫ΦQ(x)dx=|Q|1/2 for all dyadic cubes Q⊆P and ∥θP∥Ḟp0,q=C|P|1/p by the translation invariance and the dilation properties of Ḟp0,q. By Proposition 3.1,
(5.18)‖Πg(θP)‖Ḟrβ,r≈‖{〈g,φQ〉|Q|-1/2〈θP,ΦQ〉}Q‖ḟrβ,r≥(∫P∑Q⊆P(|Q|-(β/n)-(1/2)|〈g,φQ〉|χQ(x))rdx)1/r,
and hence, by the boundedness of Πg,
(5.19)(|P|-q0((1/p0)+(1/q0)-1)∫P∑Q⊆P(|Q|-(β/n)-(1/2)|〈g,φQ〉|χQ(x))q0dx)1/q0≤C.
Taking the supremum on P, we show that  g∈CMO(q0/p0)-(q0/q0′)β,q0.
To prove part (ii), assume thatg∈CMO(q0/p0)-(q0/q0′)β,q0 and f∈Ḟpα,q. Let t={〈g,φQ〉}Q and s={〈f,ψQ〉}Q. By Proposition 3.1,
(5.20)‖Πg*(f)‖Ḟrα+β,r≈‖{∑P|P|-1/2〈g,φP〉〈ΦP,φQ〉〈f,ψP〉}Q‖ḟrα+β,r=‖G̃Dts‖ḟrα+β,r,
where G̃:={〈ΦP,φQ〉}Q,P is the transpose of {〈φP,ΦQ〉}Q,P. Since α+β>0, by Lemma 5.4, G̃ is (α+β,r,r)-almost diagonal and hence is bounded on ḟrα+β,r. Following the same argument as the proof of part (i), we get
(5.21)‖Πg*(f)‖Ḟrα+β,rr≤C‖Dts‖ḟrα+β,rr=C∑Q(|Q|-(β/n)-(1/2)+(1/2r)|〈g,φQ〉|)r⋅(|Q|-(α/n)-(1/2)+(1/2r)|〈f,ψQ〉|)r≤C‖g‖CMO(q0/p0)-(q0/q0′)β,q0r‖f‖ḟpα,qr,
which completes the proof.



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*Source: 879073-2012-06-21.xml*