# Pointwise Multipliers of Triebel-Lizorkin Spaces on Carnot-Carathéodory Spaces

Authors: Yanchang Han; Fang Wang
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

Let(𝒳,d,μ) be a Carnot-Carathéodory space, namely, 𝒳 is a smooth manifold, d is a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. μ is a nonnegative Borel regular measure on 𝒳 satisfying that there exists constant C0∈[1,∞) such that for all x∈𝒳 and 0<r< diam 𝒳, μ(B(x,2r)):=μ({y∈𝒳:d(x,y)<2r})≤C0μ(B(x,r))<∞(doubling property). Using the discrete Calderón reproducing formula and the Plancherel-Pôlya characterization of the inhomogeneous Triebel-Lizorkin spaces developed in Han et al., in press and Han et al., 2008, pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces are obtained.


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## Body

## 1. Introduction

The multiplier theory of function spaces has been studied for a long time, and a lot of results have been obtained. As we know, the multiplier theory is one of the important parts in the studies of the Gleason problem, function space properties, and general operator theory. The pointwise multipliers onℝd are studied as a part of the researches of function spaces in several monographs, [1–8]. Pointwise multipliers have been found many important applications in partial differential equations.However, it was not clear how to generalize the pointwise multipliers onℝn to spaces of homogeneous type introduced by Coifman and Weiss (see [9]) because the Fourier transform is no longer available. The main purpose of this paper is to establish pointwise multipliers on inhomogeneous Triebel-Lizorkin spaces in the setting of Carnot-Carathéodory spaces. To be more precisely, we first recall some necessary definitions. In this paper, we always assume that (𝒳,d) is a metric space with a regular Borel measure μ such that all balls defined by d have finite and positive measures. In what follows, set diam(𝒳)≡sup{d(x,y):x,y∈𝒳}, and for any x∈𝒳 and r>0, set B(x,r)≡{y∈𝒳:d(x,y)<r}.Definition 1.1 (see [10]).
Let(𝒳,d) be a metric space with a Borel regular measure μ such that all balls defined by d have finite and positive measures. The triple (𝒳,d,μ) is called a space of homogeneous type if there exists a constant C0∈[1,∞) such that for all x∈𝒳 and r>0,
(1.1)μ(B(x,2r))≤C0μ(B(x,r))(doublingproperty).Remark 1.2.
We point out that the doubling condition (1.1) implies that there exist positive constants C and n such that for all x∈𝒳 and λ≥1,
(1.2)μ((B(x,λr))≤Cλnμ(B(x,r)),
where C is independent of x and r. Denote by n the homogeneous “dimension” of 𝒳 as in [10].
A space of homogeneous type is called an RD space, if there exist constantsa0, C~0∈(1,∞) such that for all x∈𝒳 and 0<r<diam(𝒳)/a0,
(1.3)C~0μ(B(x,r))≤μ(B(x,a0r))
that is, some “reverse” doubling condition holds.
Clearly, any Ahlforsn-regular metric measure space (𝒳,d,μ) (which means that there exists some n>0 such that μ(B(x,r))~rn for x∈𝒳 and 0<r<diam(𝒳)/2) is a (n,n)-space (see [10]), also is an RD space and space of homogeneous type in the sense of Coifman in [10]. In other words, μ satisfies the doubling condition which is weaker than Ahlfors n-regular metric measure space and RD-spaces.
Another typical such a space is Carnot-Carathéodory space. One example with unbounded total measure studied in [11] is that 𝒳 arises as the boundary of an unbounded model polynomial domain in ℂ2. Let Ω={(z,w)∈ℂ2:Im(w)>P(z)}, where P is a real, subharmonic, non-harmonic polynomial of degree m. Then 𝒳=∂Ω can be identified with ℂ×ℝ={(z,t):z∈ℂ,t∈ℝ}. The basic (0,1) Levi vector field is then Z-=∂/∂z--i(∂P/∂z-)(∂/∂t), and we write Z-=𝕏1+i𝕏2. The real vector fields {𝕏1,𝕏2} and their commutators of order ≤m span the tangent space to 𝒳 at each point. See [10, 12] for more details and references therein.
We will also suppose thatμ(𝒳)=∞,μ({x})=0 for all x∈𝒳. For any x,y∈𝒳 and δ>0, set Vδ(x)≡μ(B(x,δ)) and V(x,y)≡μ(B(x,d(x,y))). It follows from (1.1) that V(x,y)~V(y,x). The following notion of approximations of the identity on RD spaces was first introduced in [10]. Let ℤ+=ℕ∪{0}.
We begin with recalling the definition of an approximation to the identity, which plays the same role as the heat kernelH(s,x,y) does in Nagel-Stein's theory [11].Definition 1.3 (see [10, 12]).
A sequence{Sk}k∈ℤ+ of operators is said to be an approximation to the identity (in short, ATI) if there exists constant C1>0 such that for all k∈ℤ+ and all x,x',y, and y'∈𝒳, Sk(x,y), the kernel of Sk satisfies the following conditions:
(1.4)Sk(x,y)=0ifρ(x,y)≥C12-k,|Sk(x,y)|≲1V2-k(x)+V2-k(y),|Sk(x,y)-Sk(x′,y)|≲2kρ(x,x′)1V2-k(x)+V2-k(y)
for ρ(x,x′)≤max{C1,1}2-k,
(1.5)|Sk(x,y)-Sk(x,y′)|≲2kρ(y,y′)1V2-k(x)+V2-k(y)
for ρ(y,y′)≤max{C1,1}2-k,
(1.6)|[Sk(x,y)-Sk(x,y′)]-[Sk(x′,y)-Sk(x′,y′)]|≲22kρ(x,x′)ρ(y,y′)1V2-k(x)+V2-k(y)
for ρ(x,x′)≤max{C1,1}2-k and ρ(y,y′)≤max{C1,1}2-k,
(1.7)∫𝒳‍Sk(x,y)dμ(y)=∫𝒳‍Sk(x,y)dμ(x)=1.
The space of test functions plays a key role in this paper; see [10].Definition 1.4.
Fix two exponents0<β≤1 and γ>0. A function f defined on 𝒳 is said to be a test function of type (β,γ) centered at x0∈𝒳 with width r>0 if there exists a nonnegative constant C such that f satisfies the following conditions:
(1.8)|f(x)|≤C1(Vr(x0)+V(x,x0))rγ(r+d(x,x0))γ,(1.9)|f(x)-f(x′)|≤C(d(x,x′)r+d(x,x0))β1(Vr(x0)+V(x,x0))rγ(r+d(x,x0))γ
for d(x,x′)≤(1/2)(r+d(x,x0)).
Iff is a test function of type (β,γ) centered at x0 with width r>0, we write f∈ℳ(x0,r,β,γ), and the norm of f in ℳ(x0,r,β,γ) is defined by
(1.10)∥f∥ℳ(x0,r,β,γ)=inf{C≥0:(1.8)and(1.9)hold}.
We denote byℳ(β,γ) the class of all f∈ℳ(x0,1,β,γ). It is easy to see that ℳ(x1,r,β,γ)=ℳ(β,γ) with the equivalent norms for all x1∈𝒳 and r>0. Furthermore, it is also easy to check that ℳ(β,γ) is a Banach space with respect to the norm in ℳ(β,γ).
In what follows, for givenϵ∈(0,1], we let ℳ~(β,γ) be the completion of the space ℳ(ϵ,ϵ) in ℳ(β,γ) when 0<β,γ≤ϵ. Obviously ℳ~(ϵ,ϵ)=ℳ(ϵ,ϵ). Moreover, f∈ℳ~(β,γ) if and only if f∈ℳ(β,γ) when 0<β,γ≤ϵ and there exists {fj}j∈ℕ⊂ℳ(ϵ,ϵ) such that ∥f-fj∥ℳ(β,γ)→0 as j→∞. If f∈ℳ~(β,γ), we then define ∥f∥ℳ~(β,γ)=∥f∥ℳ(β,γ). Obviously ℳ~(β,γ) is a Banach space, and we also have ∥f∥ℳ~(β,γ)=limj→∞∥fj∥ℳ(β,γ) for the above chosen {fj}j∈ℕ.
We denote by(ℳ~(β,γ))′ the dual space of ℳ~(β,γ) consisting of all linear functionals ℒ from ℳ~(β,γ) to ℂ with the property that there exists a constant C such that for all f∈ℳ~(β,γ),
(1.11)|ℒ(f)|≤C∥f∥ℳ~(β,γ).
We denote by 〈h,f〉 the natural pairing of elements h∈(ℳ~(β,γ))′ and f∈ℳ~(β,γ). Since ℳ~(x1,r,β,γ)=ℳ~(β,γ) with the equivalent norms for all x1∈𝒳 and r>0. Thus, for all h∈(ℳ~(β,γ))′,〈h,f〉 is well defined for all f∈ℳ~(x0,r,β,γ) with x0∈𝒳 and r>0.The following constructions, which provide an analogue of the grid of Euclidean dyadic cubes on spaces of homogeneous type, were given by Christ in [13].Lemma 1.5.
Let𝒳 be a space of homogeneous type. Then there exist a collection {Qαk⊂𝒳:k∈ℤ+,α∈Ik} of open subsets, where Ik is some (possible finite) index set, and constants δ∈(0,1) and C5,C6>0 such that(i)
μ(𝒳∖∪αQαk)=0 for each fixed k and Qαk∩Qβk=∅ if α≠β;(ii)
for anyα,β,k,l with l≥k, either Qβl⊂Qαk or Qβl∩Qαk=∅;(iii)
for each(k,α) and each l<k there is a unique β such that Qαk⊂Qβl;(iv)
diam(Qαk)≤C2δk;(v)
eachQαk contains some ball B(zαk,C3δk), where zαk∈𝒳.In fact, we can think ofQαk as being a dyadic cube with a diameter roughly δk and centered at zαk. In what follows, we always suppose δ=1/2. See [14] for how to remove this restriction. Also, in the following, for k∈ℤ+,τ∈Ik, we will denote by Qτk,ν,ν=1,…,N(k,τ,M), the set of all cubes Qτ'k+M⊂Qτk, where M is a fixed large positive integer.Now, we can introduce the inhomogeneous Triebel-Lizorkin spacesFpα,q(𝒳) via the approximation in Definition 1.3. Note that the Triebel-Lizorkin spaces have been already investigated for decades in the study of partial differential equations, interpolation theory, and approximation theory.Definition 1.6.
Suppose that{Sk}k∈ℤ+ is an ATI and let D0=S0, and Dk=Sk-Sk-1 for k∈ℕ. Let M be a fixed large positive integer, Qτ0,ν be as above. Suppose that -1<s<1.
Theinhomogeneous Triebel-Lizorkin spaceFpα,q(𝒳) for max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞ is the collection of f∈(ℳ~(β,γ))', for some β and γ satisfying
(1.12)max(0,s,-s+n(1min{p,1}-1))<β<1,nmin{p,1}-n<γ<1
such that
(1.13)∥f∥Fps,q(𝒳)={∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[mQτ0,ν(|D0(f)|)]p}1/p+∥{∑k=1∞‍[2ks|Dk(f)|]q}1/q∥Lp(𝒳)<∞,
where mQτ0,ν(D0(f)) are averages of D0(f) over Qτ0,ν.
The restrictions (1.12) guarantee that the definitions of the inhomogeneous Triebel-Lizorkin spaceFps,q(𝒳) for max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞ are independent of the choices of β and γ satisfying these conditions and Fps,q(𝒳)⊂(ℳ~(β,γ))′,ℳ~(β,γ)⊂ℳ(β,γ)⊂Fps,q(𝒳) in [10].
The classical scale of inhomogeneous Triebel-Lizorkin spaces contains many well-known function spaces. For example, ifα>0,p=q=∞, one recovers the Hölder-Zygmund spaces 𝒞α(𝒳), that is,
(1.14)F∞α,∞(𝒳)=𝒞α(𝒳),α>0.
The space 𝒞α(𝒳) is defined as the collection of f such that
(1.15)∥f∥𝒞α(𝒳)=∥f∥∞+supx≠y|f(x)-f(y)|d(x,y)α<∞.
If-1<s<1 and 1<p<∞, then Hps(𝒳)=Fps,2(𝒳) are the Bessel potential spaces (Lebesgue spaces, Liouville spaces). If m=0,1,2,… and 1<p<∞, then Wpm(𝒳)=Hpm(𝒳)=Fpm,2(𝒳) are the usual Sobolev spaces. If n/(n+1)<p≤1, then Fp0,2(𝒳)=hp(𝒳) are the inhomogeneous Hardy spaces, which are closely related to the Hardy spaces Hp(𝒳) in [15, 16] (more precisely: the homogeneous spaces F˙p0,2(𝒳) coincide with the usual Hardy spaces Hp(𝒳)). We will use here the notation Fps,2(𝒳)=Hps(𝒳),s∈(-1,1),n/(n+1)<p≤∞. Then these spaces will be denoted as the (inhomogeneous) Hardy-Sobolev spaces, which include the above Lebesgue-Sobolev spaces for 1<p<∞.
The inhomogeneous Triebel-Lizorkin spaces have the following Plancherel-Pôlya characterizations in [10], which will be one of the the basic tools to prove the main results of this paper.Lemma 1.7.
Let{Dk}k∈ℤ+ be as in Definition 1.6, -1<s<1. Then if max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞, for all f∈(ℳ~(β,γ))' with β,γ satisfying (1.12), one has
(1.16)∥f∥Fps,q(𝒳)~{∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[mQτ0,ν(|D0(f)|)]p}1/p+∥{∑k=1∞‍∑τ∈Ik‍∑ν=1N(τ,k,M)‍[2ksinfz∈Qτk,ν|Dk(f)(z)|χQτk,ν(·)]q}1/q∥Lp(𝒳)~{∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[mQτ0,ν(|D0(f)|)]p}1/p+∥{∑k=1∞‍∑τ∈Ik‍∑ν=1N(τ,k,M)‍[2kssupz∈Qτk,ν|Dk(f)(z)|χQτk,ν(·)]q}1/q∥Lp(𝒳).We now introduce the following definition of the pointwise multiplier.Definition 1.8.
Suppose thatg is a given function on 𝒳. Then g is called a pointwise multiplier for Fps,q(𝒳) if f→gf admits a bounded linear mapping from Fps,q(𝒳) into itself.The main result in this paper is the following.Theorem 1.9.
Let-1<s<1, max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞ and α>max(s,n/min{p,q,1}-n-s), then g∈𝒞α(𝒳) with 0<α<1 is a multiplier for Fps,q(𝒳). In other words, f→gf yields a bounded linear mapping from Fps,q(𝒳) into itself and there is a positive constant C such that
(1.17)∥gf∥Fps,q(𝒳)≤C∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)
holds for all g∈𝒞α(𝒳) and f∈Fps,q(𝒳).We would like to point out that the study of pointwise multipliers is one of important problems in the theory of function spaces. It has attracted a lot of attention in the decades since starting with [7]. Pointwise multipliers in general spaces Fps,q(ℝn), where 0<p<∞,0<q≤∞,s∈ℝ have been studied in great detail in [4, 5, 8].Theorem1.9 was proved in [8] for pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces on ℝn based on the Fourier transform. In the present setting, however, we do not have the Fourier transform at our disposal. Since the Fourier transform on Carnot-Carathéodory spaces is not available and hence the idea used in [8] does not work for this more general setting. A new idea to prove Theorem 1.9 is to use the discrete Calderón reproducing formula, which was developed in [10]. Therefore, this scheme easily extends to geometrical settings where the Fourier transform does not exist. The Fourier transform is missing but a version of pointwise multiplier is still present.Throughout, we also denote byC a positive constant independent of main parameters involved, which may vary at different occurrences. Constants with subscripts do not change through the whole paper. We use f≲g and f≳g to denote f≤Cg and f≥Cg, respectively. If f≲g≲f, we then write f~g. For any a,b∈ℝ, set a∧b≐min{a,b},a∨b≐max{a,b}. If p>1, set 1/p+1/p'=1.


## 2.  Proof of Theorem1.9

In this section, we will prove Theorem1.9. Since there is no the Fourier transforms on spaces of homogeneous type, the proof of Theorem 1.9 is quite different from the proof of Theorem 2.8.2 in [8]. The key new ingredient in the proof of Theorem 1.9 is to apply the following discrete Calderón reproducing formulae established in [10, 12]. This formula can be stated as follows.Lemma 2.1.
Suppose that{Sk}k∈ℤ+ is an approximation to the identity as in Definition 1.3. Set Dk=Sk-Sk-1 for k∈ℕ and D0=S0. Then for any fixed M∈ℕ large enough, there exists a family of functions {D~k(x,y)}k∈ℤ+ and {D~~k(x,y)}k∈ℤ+ such that for any fixed yτk,ν∈Qτk,ν,k∈ℕ,τ∈Ik and ν∈{1,…,N(k,τ,M)} and all f∈(M~(β,γ))′ with 0<β,γ<ϵ and x∈𝒳(2.1)f(x)=∑τ∈I0‍∑ν=1N(0,τ,M)‍∫Qτ0,ν‍D~0(x,y)dμ(y)mQτ0,ν(D0(f))+∑k∈ℕ‍∑τ∈Ik‍∑ν=1N(k,τ,M)‍μ(Qτk,ν)D~k(x,yτk,ν)Dk(f)(yτk,ν)=∑τ∈I0‍∑ν=1N(0,τ,M)‍∫Qτ0,ν‍D0(x,y)dμ(y)mQτ0,ν(D~~0(f))+∑k∈ℕ‍∑τ∈Ik‍∑ν=1N(k,τ,M)‍μ(Qτk,ν)Dk(x,yτk,ν)D~~k(f)(yτk,ν),
where the series converges in the norm of Fps,q(𝒳) with max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞,-1<s<1, and M~(β′,γ′) for f∈M~(β,γ) with β′<β and γ′<γ, and (M~(β′,γ′))′ for f∈(M~(β,γ))′ with ϵ>β′>β and ϵ>γ′>γ. Moreover, D~k(x,y) and D~~k(x,y), the kernels of D~k and D~~k, satisfy the similar estimates but with x and y interchanged in (2.3): for 0<ϵ<1,
(2.2)|D~k(x,y)|≤C1V2-k(x)+V2-k(y)+V(x,y)2-kϵ(2-k+d(x,y))ϵ,(2.3)|D~k(x,y)-D~k(x′,y)|≤C(d(x,x′)2-k+d(x,y))ϵ1V2-k(x)+V2-k(y)+V(x,y)2-kϵ(2-k+d(x,y))ϵ,
for d(x,x')≤(2-k+d(x,y))/2,
(2.4)∫𝒳‍D~k(x,y)dμ(y)=∫𝒳‍D~k(x,y)dμ(x)=0,
when k∈ℕ;∫𝒳‍D~k(x,y)dμ(y)=∫𝒳‍D~k(x,y)dμ(x)=1 when k=0.
To prove Theorem1.9, we first show the following lemma.Lemma 2.2.
Let{Sk(x,y)}k∈ℤ+ and {Gk(x,y)}k∈ℤ+ be two approximations to the identity as in Lemma 2.1 above and Dk=Sk-Sk-1,Ek=Gk-Gk-1 for k∈ℕ and D0=S0,E0=G0. For any given ϵ∈(0,1) and g∈𝒞α(𝒳) with 0<α<ϵ, then
(2.5)|EkgD~k′(x,y)|≲∥g∥𝒞α(𝒳)2-|k-k′|α1V2-(k∧k′)(x)+V2-(k∧k′)(y)+V(x,y)2-(k∧k′)ϵ(2-(k∧k′)+d(x,y))ϵ,
where k,k'∈ℤ+.Proof.
We first show the inequality (2.5) of the case k=k'=0 above. In fact, in this case, it follows that
(2.6)|E0gD~0(x,y)|≤|∥g∥𝒞α(𝒳)∫‍E0(x,z)D~0(z,y)dμ(z)|≲∥g∥𝒞α(𝒳)1V1(x)+V1(y)+V(x,y)1(1+d(x,y))ϵ.
We next consider the casek'≥k for k>0 and k'≥0 or k≥0 and k'>0. We write
(2.7)|EkgD~k′(x,y)|=|∫‍[Ek(x,z)g(z)-Ek(x,y)g(y)]D~k′(z,y)dμ(z)|≤∫‍|Ek(x,z)-Ek(x,y)||g(z)||D~k′(z,y)|dμ(z)+∫‍|Ek(x,y)||g(z)-g(y)||D~k′(z,y)|dμ(z):=E+F.
ForE, we can obtain
(2.8)E≲∥g∥𝒞α(𝒳)2-(k′-k)ϵ1V2-(k∧k′)(x)+V2-(k∧k′)(y)+V(x,y)2-kϵ(2-k+d(x,y))ϵ,
where k,k'∈ℤ+.
We may rewrite this integral as(2.9)F≲∥g∥𝒞α(𝒳)1V2-k(x)+V2-k(y)+V(x,y)2-kϵ(2-k+d(x,y))ϵ×[∫W1‍d(z,y)α1V2-k′(z)+V2-k′(y)+V(z,y)2-k′ϵ(2-k′+d(z,y))ϵdμ(z)+∫W2‍d(z,y)α1V2-k′(z)+V2-k′(y)+V(z,y)2-k′ϵ(2-k′+d(z,y))ϵdμ(z)]:=∥g∥𝒞α(𝒳)1V2-k(x)+V2-k(y)+V(x,y)2-kϵ(2-k+d(x,y))ϵ[F1+F2],
where W1={z:d(z,y)≥2-k'} and W2={z:d(z,y)<2-k'}.
Observe further, a simple argument yields the same estimate fork,k'∈ℤ+ as follows:
(2.10)F1≲∑i=-k′∞‍∫{z:2i≤d(z,y)<2i+1}‍d(z,y)α1V2-k′(z)+V2-k′(y)+V(z,y)2-k′ϵ(2-k′+d(z,y))ϵdμ(z)≲∑i=-k'∞‍μ(B(y,2i+1))μ(B(y,2i))2(i+1)α2-k'ϵ2-iϵ≲2-k'α≤2-(k′-k)α,
where we used 0<α<ϵ.
ForF2, we have
(2.11)F2≲2-k'α≤2-(k′-k)α.
Combining the estimate forE,F, we conclude that if k'≥k, then (2.5) holds. The case k≥k' is similar to above. This finishes the proof of Lemma 2.2.Now we show the following technical version of Theorem1.9.Proposition 2.3.
For anyg∈𝒞α(𝒳),f∈ℳ~(β,γ) with β and γ satisfying (1.12), then
(2.12)∥fg∥Fps,q(𝒳)≤C∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳),
where -1<s<1, max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞ and α>max(s,n/min{p,q,1}-n-s).Proof.
Using the Calderón reproducing formula, for anyg∈𝒞α(𝒳),f∈ℳ~(β,γ), We write
(2.13)∥gf∥Fps,q(𝒳)≤{∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[∑τ′∈I0‍∑ν′=1N(0,τ′,M)‍mQτ0,ν(|E0gD~Qτ′0,ν′|)μ(Qτ′0,ν′)|mQτ′0,ν′(D0(f))|]p}1/p+{∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[∑k′=1∞‍∑τ′∈Ik′‍∑ν′=1N(k′,τ′,M)‍μ(Qτ′k′,ν′)mQτ0,ν×(|E0gD~k′(·,yτ′k′,ν′)|)|Dk′(f)(yτ′k′,ν′)|∑k′=1∞‍∑τ′∈Ik′‍∑ν′=1N(k′,τ′,M)‍μ(Qτ′k′,ν′)mQτ0,ν]p{∑τ∈I0‍}}1/p+∥{∑k=1∞‍2ksq[∑τ′∈I0‍∑ν′=1N(0,τ′,M)‍μ(Qτ′0,ν′)|EkgD~Qτ′0,ν′(·)|mQτ′0,ν′(|D0(f)|)]q}1/q∥Lp(𝒳)+∥{∑k=1∞‍2ksq[∑k′=1∞‍∑τ′∈Ik′‍∑ν′=1N(k′,τ′,M)‍μ(Qτ′k′,ν′)|EkgD~k′(·,yτ′k′,ν′)||Dk′(f)(yτ′k′,ν′)|]q}1/q∥Lp(𝒳):=L1+L2+L3+L4.
Applying the Hölder inequality forp>1 and
(2.14)(∑k‍|ak|)p≤∑k‍|ak|p
for all ak∈ℂ and p≤1, for L1, it follows that
(2.15)L1≲∥g∥𝒞α(𝒳){∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)∑τ′∈I0‍∑ν′=1N(0,τ′,M)‍μ(Qτ′0,ν′)|mQτ′0,ν′(D0(f))|p×μ(Qτ'0,ν')p∧1-1infx∈Qτ0,νinfy∈Qτ'0,ν'[1V1(x)+V1(y)+V(x,y)1(1+d(x,y))ϵ]p∧1}1/p≲∥g∥𝒞α(𝒳){∑τ′∈I0‍∑ν′=1N(0,τ′,M)‍μ(Qτ′0,ν′)|mQτ′0,ν′(D0(f))|pμ(Qτ'0,ν')p∧1-1[V1(yτ′0,ν′)]1-p∧1}1/p≲∥g∥𝒞α(𝒳)∥f∥Bps,q(𝒳),
where we used the fact that for any yτ'0,ν'∈Qτ'0,ν',V1(yτ'0,ν')μ(Qτ'0,ν').
ForL2, in fact for all k'∈ℕ,τ'∈Ik' and ν'=1,2,…,N(k',τ',M), and all x∈𝒳,z∈Qτ'k',ν',
(2.16)μ(Qτ′k′,ν′)~V2-k′(z),V1(yτ′k′,ν′)+V(x,yτ′k′,ν′)~V1(z)+V(x,z),(2.17)V1(z)≲2k′nμ(Qτ′k′,ν′),V2i+1(z)≲2inV1(z),1+d(x,yτ′k′,ν′)~1+d(x,z).
From the inequality (2.5), the Hölder inequality for q>1 and (2.14) for q≤1, the Fefferman-Stein vector-valued maximal function inequality in [17] and Lemma 1.7, it follows that
(2.18)L2≲∥g∥𝒞α(𝒳){∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[∑k′=1∞‍2-k′(s+α)∑τ′∈Ik′‍∑ν′=1N(k′,τ′,M)‍μ(Qτ′k′,ν′)2k's|Dk′(f)(yτ′k′,ν′)|×infx∈Qτ0,ν1V1(x)+V1(yτ′k′,ν′)+V(x,yτ′k′,ν′)1(1+d(x,yτ′k′,ν′))ϵ]p}1/p≲{{×1(1+d(xτ0,ν,yτ′k′,ν′))ϵr)1r∥g∥𝒞α(𝒳)]p}∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[{×1(1+d(xτ0,ν,yτ′k′,ν′))ϵr)1r∥g∥𝒞α(𝒳)}∑k'=1∞‍2-k′(s+α)(∑τ'∈Ik'‍∑ν'=1N(k′,τ′,M)‍μ(Qτ'k',ν')r|2k′sDk′(f)(yτ′k′,ν′)|r×1(V1(xτ0,ν)+V1(yτ′k′,ν′)+V(xτ0,ν,yτ′k′,ν′))r×1(1+d(xτ0,ν,yτ′k′,ν′))ϵr)1/r∥g∥𝒞α(𝒳)]p}1/p≲∥g∥𝒞α(𝒳){∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[∑k'=1∞‍2-k′(s+α)(1V1(xτ0,ν)∫d(xτ0,ν,z)<1‍∑τ'∈Ik'‍×∑ν′=1N(k′,τ′,M)‍V1(xτ0,ν)1-rμ(Qτ′k′,ν′)1-r|2k′sDk′(f)(yτ′k′,ν′)|rχQτ′k′,ν′(z)dμ(z)+∑i=0∞‍1V2i+1(xτ0,ν)∫2i≤d(xτ0,ν,z)<2i+1‍V1(z)1-rμ(Qτ'k',ν')1-rV2i+1(z)1-rV1(z)1-r2-iϵr×∑τ'∈Ik'‍∑ν'=1N(k′,τ′,M)‍|2k′sDk′(f)(yτ′k′,ν′)|rχQτ′k′,ν′(z)dμ(z))1/r]p}1/p≲∥g∥𝒞α(𝒳)×∥∑k'=1∞‍2-k′(n+s+α-n/r)[M(∑τ'∈Ik'‍∑ν'=1N(k′,τ′,M)‍2k'sr|Dk′(f)(yτ′k′,ν′)|rχQτ'k',ν')]1/r∥Lp(𝒳)≲∥g∥𝒞α(𝒳)∥{{[M(∑τ′∈Ik′‍∑ν′=1N(k′,τ′,M)‍2k′sr|Dk′(f)(yτ′k′,ν′)|rχQτ′k′,ν′)]qr}∑k'=1∞‍2-k′(n+s+α-n/r)(q∧1)×[M(∑τ'∈Ik'‍∑ν'=1N(k′,τ′,M)‍2k'sr|Dk′(f)(yτ′k′,ν′)|rχQτ'k',ν')]q/r}1/q∥Lp(𝒳)≲∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳),
where we choose r satisfying max(n/(n+ϵ),n/(n+s+α))<r<min{1,p,q}.
By the inequality (2.5), similarly,
(2.19)L3≲∥g∥𝒞α(𝒳){{1(1+d(x,yτ′0,ν′))ϵ(p∧1)}∫𝒳‍∑τ′∈I0‍∑ν′=1N(0,τ′,M)‍μ(Qτ′0,ν′)[mQτ′0,ν′(|D0(f)|)]p1(V1(x)+V1(yτ′0,ν′)+V(x,yτ′0,ν′))p∧11(1+d(x,yτ′0,ν′))ϵ(p∧1)dμ(x)}1/p≲∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳),
where s can be any number in (-ϵ,α)⊂(-1,1).
We now consider the estimate ofL4, Similar to the estimate of L2, using the equality (2.5), the Hölder inequality if q>1 and (2.14) if q≤1, the Fefferman-Stein vector-valued maximal function inequality in [17], we obtain
(2.20)L4≲∥g∥𝒞α(𝒳)∥{{[M(∑τ′∈Ik′‍∑ν′=1N(k′,τ′,M)‍2k′s|Dk′(f)(yτ′k′,ν′)χQτ′k′,ν′|)r]qr}∑k=1∞‍∑k'=1∞‍[2(k-k′)s2-|k′-k|α2-k'n2(k∧k′)n2[k′-(k∧k′)]n/r]q∧1×[M(∑τ'∈Ik'‍∑ν'=1N(k′,τ′,M)‍2k's|Dk′(f)(yτ′k′,ν′)χQτ′k′,ν′|)r]q/r}1/q∥Lp(𝒳)≲∥g∥𝒞α(𝒳)∥{∑k'=1∞‍[M(∑τ'∈Ik'‍∑ν'=1N(k′,τ′,M)‍2k's|Dk′(f)(yτ'k',ν'|rχQτ'k',ν')]q/r}1/q∥Lp(𝒳)≲∥g∥𝒞α(𝒳)∥f∥Fpα,q(𝒳),
where the first and the second inequalities follow from the estimates
(2.21)supk∑k'=0∞‍2(k-k′)s2-|k′-k|α2-k'n2(k∧k′)n2[k′-(k∧k′)]n/r≤C,supk'∑k=0∞‍[2(k-k′)s2-|k′-k|α2-k'n2(k∧k′)n2[k′-(k∧k′)]n/r]q∧1≤C
with max(n/(n+1),n/(n+s+α))<r<min{1,p,q}, s can be any number in (-ϵ,α), which verifies Proposition 2.3.To introduce our definition of pointwise multiplication, the interesting estimate is needed.Lemma 2.4.
Let{Sk(x,y)}k∈ℤ+ be a approximation to the identity as in Lemma 2.1 above and Dk=Sk-Sk-1 for k∈ℕ and D0=S0. For any g∈𝒞α(𝒳) with 0<α<1, h∈ℳ~(β,γ) with β and γ satisfying (1.12). Then
(2.22)|〈Dk(•,y)g,h〉|≤C∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)2-k(β∧α)1V1(x0)+V(x0,y)1(1+d(y,x0))γ,
where k∈ℤ+.Proof.
We first prove inequality (2.22) with k=0. In fact, since D0(z,y)=0 if d(z,y)≥2C1, then
(2.23)|〈D0(•,y)g,h〉|≲∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)∫d(y,z)<2C1‍1V1(z)+V1(y)1V1(x0)+V(x0,z)1(1+d(z,x0))γdμ(z)=∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)[∫d(y,z)<2C1,d(x0,z)≥d(x0,y)/2‍⋯+∫d(y,z)<2C1,d(x0,z)<d(x0,y)/2‍⋯]:=∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)[B1+B2].
ForB1, since V(x0,y)=μ(B(x0,d(x0,y)))≤μ(B(x0,2d(x0,z)))μ(B(x0,d(x0,z)))=V(x0,z), it follows that
(2.24)B1≲1V1(x0)+V(x0,y)1(1+d(y,x0))γ.
ForB2, the fact d(y,z)<2C1,d(x0,z)<d(x0,y)/2 implies that 2C1>d(y,z)>d(x0,y)/2, then
(2.25)B2≲1V1(x0)+V(x0,y)≲1V1(x0)+V(x0,y)1(1+d(y,x0))γ.
That is, (2.22) with k=0 holds.
To prove (2.22) with k∈ℕ, we write
(2.26)|〈Dk(•,y)g,h〉|=|∫𝒳‍Dk(x,y)[g(x)h(x)-g(y)h(y)]dμ(x)|≤∫𝒳‍|Dk(x,y)||g(x)-g(y)||h(x)|dμ(x)+∫𝒳‍|Dk(x,y)||g(y)||h(x)-h(y)|dμ(x):=I+II.
ForI, since Dk(x,y)=0 if d(x,y)≥2C12-k, we obtain
(2.27)I≲∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)2-kα×∫d(x,y)<2C12-k‍1V2-k(x)+V2-k(y)1V1(x0)+V(x0,x)1(1+d(x,x0))γdμ(x)≲∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)2-kα[∫d(x,y)<2C12-k,d(x,y)≤(1/2)(1+d(y,x0))‍⋯+∫2C12-k>d(x,y)>(1/2)(1+d(y,x0))‍⋯]≐∥g∥𝒞α(X)∥h∥ℳ(β,γ)2-kα[I1+I2].
ForI1, by d(x,y)≤(1/2)(1+d(y,x0)) implies that d(y,x0)≤2(d(x,x0)+1) and
(2.28)1V1(x0)+V(x0,x)≲1V1(x0)+μ(B(x0,d(x0,y)))=1V1(x0)+V(x0,y).
Thus
(2.29)I1≲1V1(x0)+V(x0,y)1(1+d(y,x0))γ∫d(x,y)<2C12-k‍1V2-k(x)+V2-k(y)dμ(x)≲1V1(x0)+V(x0,y)1(1+d(y,x0))γ.
ForI2, by 2C12-k>d(x,y)>(1/2)(1+d(y,x0)) implies that
(2.30)1+d(y,x0)<C,1V2-k(x)+V2-k(y)≲1V(y,x)≲1V1(x0)+V(x0,y).
Thus
(2.31)I2≲1V1(x0)+V(x0,y)1(1+d(x0,y))γ∫𝒳‍1V1(x0)+V(x0,x)1(1+d(x,x0))γdμ(x)≲1V1(x0)+V(x0,y)1(1+d(x0,y))γ.
To obtain the estimate of theII, we write
(2.32)II≲∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ){∫ϖ1‍|Dk(x,y)|1V1(x0)+V(x0,y)d(x,y)β(1+d(y,x0))β+γdμ(x)+∫ϖ2‍|Dk(x,y)|[1V1(x0)+V(x0,x)1(1+d(x,x0))γ+1V1(x0)+V(x0,y)1(1+d(y,x0))γ]dμ(x)}:=∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)[II1+II2+II3],
where ϖ1={x∈𝒳:d(x,y)≤(1/2)(1+d(y,x0))} and ϖ2={x∈𝒳:d(x,y)>(1/2)(1+d(y,x0))}.
ForII1, we have
(2.33)II1≲1V1(x0)+V(x0,y)2-kβ(1+d(x0,y))γ∫ϖ1‍|Dk(x,y)|dμ(x)≲1V1(x0)+V(x0,y)2-kβ(1+d(x0,y))γ.
ForII2, similar to I2, we have
(2.34)|Dk(x,y)|≲1V1(x0)+V(x0,y)2-kβ(1+d(x0,y))γ,
thus
(2.35)II2≲1V1(x0)+V(x0,y)2-kβ(1+d(x0,y))γ∫ϖ2‍1V1(x0)+V(x0,x)1(1+d(x,x0))γdμ(x)≤1V1(x0)+V(x0,y)2-kβ(1+d(x0,y))γ.
ForII3, obviously
(2.36)II3≲2-kβV1(x0)+V(x0,y)1(1+d(y,x0))γ∫𝒳‍|Dk(x,y)|dμ(x)≲2-kβV1(x0)+V(x0,y)1(1+d(y,x0))γ.
Combining the estimate ofI and II, we obtain that for k∈ℕ(2.37)|〈Dk(•,y)g,h〉|≤C∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ)2-k(β∧α)1V1(x0)+V(x0,y)1(1+d(y,x0))γ.
Thus, (2.22) also holds. This finishes the proof of Lemma 2.4.Ifg∈𝒞α(X) and f∈Fps,q(𝒳) for s∈(-1,1), max(n/(n+1),d/(d+ϵ+s))<p<∞, max(n/(n+1),d/(d+ϵ+s))<q≤∞, it is not clear in general what is meant by gf (pointwise multiplication). Our approach is the following.Lemma 2.5.
For anyf∈Fps,q(𝒳) with max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞, -1<s<1, and g∈𝒞α(𝒳) with 1>α>max(s,(n/min{p,q,1})-n-s). There exist a constant C and a sequence {fj}j∈ℕ such that fj∈ℳ~(ϵ,ϵ),∥fj∥Fps,q(𝒳)≤C∥f∥Fps,q(𝒳) and limj→∞〈gfj,h〉 converges for any h∈ℳ~(β,γ) with β and γ satisfying (1.12).Proof.
For anyf∈Fps,q(𝒳) with max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞, -1<s<1, and g∈𝒞α(𝒳) with α>max(s,(n/min{p,q,1})-n-s), denote
(2.38)fj=∑τ=1j‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)mQτ0,ν(D~~0(f))DQτ0,ν(x)+∑k=1j‍∑τ=1j‍∑ν=1N(k,τ,M)‍μ(Qτk,ν)Dk(x,yτk,ν)D~~k(f)(yτk,ν),
where DQτ0,ν(x)=(1/μ(Qτ0,ν))∫Qτ0,ν‍D0(x,y)dμ(y).
It is easy to see thatfj∈ℳ~(ϵ,ϵ). Applying a similar proof as in Proposition 2.3 with g=1 and f=fj gives ∥fj∥Fps,q(𝒳)≤C∥f∥Fps,q(𝒳). We write
(2.39)|〈fj-fm,gh〉|≤|∑τ=m+1j‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)mQτ0,ν(D~~0(f))〈DQτ0,ν,gh〉|+|∑k=m+1j‍∑τ=1j‍∑ν=1N(k,τ,M)‍μ(Qτk,ν)〈Dk(•,yτk,ν),gh〉D~~k(f)(yτk,ν)|+|∑k=1j‍∑τ=m+1j‍∑ν=1N(k,τ,M)‍μ(Qτk,ν)〈Dk(•,yτk,ν),gh〉D~~k(f)(yτk,ν)|:=R+T+Y.
The fact (2.22) implies
(2.40){∑τ=m+1j‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)|DQτ0,ν(gh)|p'}1/p'≲∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ){∑τ≥log2m‍∫{y:2τ<d(x0,y)≤2τ+1}‍[1V1(x0)+V(x0,y)]p′×1(1+d(y,x0))γp′dμ(y)}1/p'≲∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ){∑τ≥log2m∞‍2-τγp'}1/p'=m-γ∥g∥𝒞α(𝒳)∥h∥ℳ(β,γ).
Thus
(2.41)R≲{∥f∥Fps,q(𝒳){∑τ=m+1j‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)|DQτ0,ν*(gh)|p'}1/p':p>1∥f∥Fps,q(𝒳)supyτ0,ν,m<d(x0,yτ0,ν)≤jμ(Qτ0,ν)1-1/pV1(x0)+V(x0,yτ0,ν)1(1+d(yτ0,ν,x0))γ:p≤1,≲∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)m-γ,
this finishes the proof of R→0 as j,m→∞.
We now consider the estimate ofT. By Hölder inequality if p,q>1, and (2.14) if p,q<1 and (2.22), it follows that
(2.42)T≲{∥f∥Fps,q(𝒳)∥{∑k=m+1j‍|2-ksDk*(gh)|q'}1/q'∥Lp′(𝒳):p>1,q>1,∥f∥Bps-ϵ,∞(𝒳)∑k=m+1j‍supx∈𝒳2k[-s+ε]μ(Qτk,ν)1-1/p|Dk*(gh)(x)|:p≤1,q>1,∥f∥Fps,q(𝒳)∥supm+1≤k≤j|2-ksDk*(gh)|∥Lp′(𝒳):p>1,q≤1,∥f∥Bps,1(𝒳)supx∈𝒳supm+1≤k≤j2-ksμ(Qτk,ν)1-1/p|Dk*(gh)(x)|:p≤1,q≤1,≲{∥f∥Fps,q(𝒳)∥{∑k=m+1j‍|2-ksDk*(gh)|q'}1/q'∥Lp′(𝒳),:p>1,q>1,∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)2-m[s-ε+n(1-1/p)+α∧β],:p≤1,q>1,∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)2-m[s+α∧β],:p>1,q≤1,∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)2-m[s+n(1-1/p)+α∧β],:p≤1,q≤1,
where let ε be a positive number with s-ε+n(1-1/p)+α∧β>0 when p≤1 and s+α∧β>0 if p>1, and the second inequality can be obtained by using the fact that
(2.43)Fps,q(𝒳)⊂Bps,max(p,q)(𝒳)⊂Bps-ε,∞(𝒳),
when -ϵ<s-ε, see [10, Proposition 5.31].
By the Proposition2.3, then gh∈Fp'-s,q'(𝒳) if s∈(-1,1),p,q>1. Thus by the Calderón reproducing that the series converges in the norm of Fp'-s,q'(𝒳) for gh∈Fp'-s,q'(𝒳) with s∈(-1,1),p,q>1 in Lemma 2.1, which proves T→0 as j,m→∞ when max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞, -1<s<1, and g∈𝒞α(𝒳) with α>max(s,(n/min{p,q,1})-d-s).
ForY, applying the Hölder inequality for p,q>1 and (2.14) for p,q≤1 and (2.22), we also have
(2.44)Y≲{∥f∥Fps,q(𝒳){∫{y∣m<d(x0,y)≤j}‍{∑k=1j‍|2-ksDk*(gh)(y)|q′}p′/q′dμ(y)}1/p':p>1,q>1,∥f∥Bps-ε,∞(𝒳)supk∈ℕsupy∈{y∣m<d(x0,y)≤j}2-k[s-ε]μ(Qτk,ν)1-1/p|Dk*(gh)(y)|:p≤1,q>1,∥f∥Fps,q(𝒳){∫{y∣m<d(x0,y)≤j}‍{sup1≤k≤j|2-ksDk*(gh)(y)|}p′dμ(y)}1/p':p>1,q≤1,∥f∥Bps,1(𝒳)supy∈{y∣m<d(x0,y)≤j}sup1≤k≤j2-ksμ(Qτk,ν)1-1/p|Dk*(gh)(y)|:p≤1,q≤1,≲{<∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)m-γ1/q':p>1,q>1,∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)m-γ{supk∈ℕ2-k[s-ε+n(1-1/p)+(α∧β)]q'}1/q':p≤1,q>1,∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)m-γsupk∈ℕ2-k[s+α∧β]:p>1,q≤1,∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)m-γsupk∈ℕ2-k[s+n(1-1/p)+α∧β]:p≤1,q≤1≲∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳)∥h∥ℳ(β,γ)m-γ,
where let ε be a positive number with s-ε+n(1-1/p)+α∧β>0 when p≤1 and s+α∧β>0 if p>1, and we also used the fact that Fps,q(𝒳)⊂Bps,max(p,q)(𝒳)⊂Bps-ε,∞(𝒳), when -ϵ<s-ε. This finishes the proof of Y→0 as j,m→∞, and hence the proof of Lemma 2.5 is concluded.The above estimate showslimj→∞〈gfj,h〉 exists and the limit is independent of the choice of fj. Therefore, for g∈𝒞α(𝒳),f∈Fps,q(𝒳) with max(n/(n+1),n/(n+1+s))<p<∞ and max(n/(n+1),n/(n+1+s))<q≤∞s∈(-1,1),α>max(s,(n/min{p,q,1})-n-s). We define
(2.45)〈gf,h〉=limj→∞〈gfj,h〉
for any h∈ℳ~(β,γ) with β and γ satisfying (1.12), fj is fundamental sequence defined in Lemma 2.5.We now prove Theorem1.9.Proof of Theorem1.9.
By Proposition2.3 and Lemma 2.5, for any g∈𝒞α(𝒳),f∈Fps,q(𝒳), Fatou's lemma implies that
(2.46)∥gf∥Fps,q(𝒳)≤{∑τ∈I0‍∑ν=1N(0,τ,M)‍μ(Qτ0,ν)[mQτ0,ν(|limj→∞E0(fjg)|)]p}1/p+∥{∑k=1∞‍[2kαlimj→∞|Ek(fjg)|]q}1/q∥Lp(𝒳)≤liminfj→∞∥gfj∥Fps,q(𝒳)≲∥g∥𝒞α(𝒳)∥f∥Fps,q(𝒳).
We complete the proof of Theorem 1.9.



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*Source: 153849-2012-11-11.xml*