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

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))<∞(doublingproperty). 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.

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 on ℝ^𝑑 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 on ℝ^𝑛 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, 𝜇(𝐵(𝑥,2𝑟))≤𝐶₀𝜇(𝐵(𝑥,𝑟))doublingproperty. (1.1)

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, 𝜇((𝐵(𝑥,𝜆𝑟))≤𝐶𝜆^𝑛𝜇(𝐵(𝑥,𝑟)), (1.2) 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 constants 𝑎₀, 𝐶₀∈(1,∞) such that for all 𝑥∈𝒳 and 0<𝑟<diam(𝒳)/𝑎₀, 𝐶₀𝜇(𝐵(𝑥,𝑟))≤𝜇𝐵𝑥,𝑎₀𝑟 (1.3) that is, some “reverse” doubling condition holds.
Clearly, any Ahlfors 𝑛-regular metric measure space (𝒳,𝑑,𝜇) (which means that there exists some 𝑛>0 such that 𝜇(𝐵(𝑥,𝑟))∼𝑟^𝑛 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 that 𝜇(𝒳)=∞,𝜇({𝑥})=0 for all 𝑥∈𝒳. For any 𝑥,𝑦∈𝒳 and 𝛿>0, set 𝑉_𝛿(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 kernel 𝐻(𝑠,𝑥,𝑦) does in Nagel-Stein's theory [11].

Definition 1.3 (see [10, 12]). A sequence {𝑆_𝑘}_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, 𝑆_𝑘(𝑥,𝑦), the kernel of 𝑆_𝑘 satisfies the following conditions: 𝑆_𝑘𝑥,𝑦=0if𝜌𝑥,𝑦≥𝐶₁2^−𝑘,𝑆_𝑘≲1𝑥,𝑦𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦,𝑆_𝑘𝑥,𝑦−𝑆_𝑘𝑥^′,𝑦≲2^𝑘𝜌𝑥,𝑥^′1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦 (1.4) for 𝜌(𝑥,𝑥^′)≤max{𝐶₁,1}2^−𝑘, 𝑆_𝑘𝑥,𝑦−𝑆_𝑘𝑥,𝑦^′≲2^𝑘𝜌𝑦,𝑦^′1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦 (1.5) for 𝜌(𝑦,𝑦^′)≤max{𝐶₁,1}2^−𝑘, 𝑆_𝑘𝑥,𝑦−𝑆_𝑘𝑥,𝑦^′−𝑆_𝑘𝑥^′,𝑦−𝑆_𝑘𝑥^′,𝑦^′≲2^2𝑘𝜌𝑥,𝑥^′𝜌𝑦,𝑦^′1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦 (1.6) for 𝜌(𝑥,𝑥^′)≤max{𝐶₁,1}2^−𝑘 and 𝜌(𝑦,𝑦^′)≤max{𝐶₁,1}2^−𝑘, _𝒳𝑆_𝑘𝑥,𝑦𝑑𝜇𝑦=_𝒳𝑆_𝑘𝑥,𝑦𝑑𝜇(𝑥)=1. (1.7)
The space of test functions plays a key role in this paper; see [10].

Definition 1.4. Fix two exponents 0<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: 𝑓(𝑥1)≤𝐶𝑉_𝑟𝑥₀+𝑉𝑥,𝑥₀𝑟^𝛾𝑟+𝑑𝑥,𝑥₀^𝛾,𝑓(𝑥)−𝑓𝑥^′𝑑≤𝐶𝑥,𝑥^′𝑟+𝑑𝑥,𝑥₀^𝛽1𝑉_𝑟𝑥₀+𝑉𝑥,𝑥₀𝑟^𝛾𝑟+𝑑𝑥,𝑥₀^𝛾 (1.8)(1.9) for 𝑑(𝑥,𝑥^′)≤(1/2)(𝑟+𝑑(𝑥,𝑥₀)).
If 𝑓 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 𝑓_{ℳ(𝑥0,𝑟,𝛽,𝛾)}=inf{𝐶≥0∶(1.8)and(1.9)hold}. (1.10)
We denote by ℳ(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 given 𝜖∈(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 {𝑓_𝑗}_𝑗∈ℕ⊂ℳ(𝜖,𝜖) such that ‖𝑓−𝑓_𝑗‖_{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)}=lim_𝑗→∞‖𝑓_𝑗‖_{ℳ(𝛽,𝛾)} for the above chosen {𝑓_𝑗}_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), ℒ𝑓≤𝐶𝑓_{ℳ(𝛽,𝛾)}. (1.11) 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. Let 𝒳 be a space of homogeneous type. Then there exist a collection {𝑄^𝑘_𝛼⊂𝒳∶𝑘∈ℤ₊,𝛼∈𝐼_𝑘} of open subsets, where 𝐼_𝑘 is some (possible finite) index set, and constants 𝛿∈(0,1) and 𝐶₅,𝐶₆>0 such that(i)𝜇(𝒳⧵∪_𝛼𝑄^𝑘_𝛼)=0 for each fixed 𝑘 and 𝑄^𝑘_𝛼∩𝑄^𝑘_𝛽=∅ if 𝛼≠𝛽;(ii)for any 𝛼,𝛽,𝑘,𝑙 with 𝑙≥𝑘, either 𝑄^𝑙_𝛽⊂𝑄^𝑘_𝛼 or 𝑄^𝑙_𝛽∩𝑄^𝑘_𝛼=∅;(iii)for each (𝑘,𝛼) and each 𝑙<𝑘 there is a unique 𝛽 such that 𝑄^𝑘_𝛼⊂𝑄^𝑙_𝛽;(iv)diam(𝑄^𝑘_𝛼)≤𝐶₂𝛿^𝑘;(v)each 𝑄^𝑘_𝛼 contains some ball 𝐵(𝑧^𝑘_𝛼,𝐶₃𝛿^𝑘), where 𝑧^𝑘_𝛼∈𝒳.

In fact, we can think of 𝑄^𝑘_𝛼 as being a dyadic cube with a diameter roughly 𝛿^𝑘 and centered at 𝑧^𝑘_𝛼. 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∈𝐼_𝑘, we will denote by 𝑄_𝜏^𝑘,𝜈,𝜈=1,…,𝑁(𝑘,𝜏,𝑀), the set of all cubes 𝑄_𝜏^𝑘+𝑀_′⊂𝑄^𝑘_𝜏, where 𝑀 is a fixed large positive integer.

Now, we can introduce the inhomogeneous Triebel-Lizorkin spaces 𝐹_𝑝^𝛼,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 {𝑆_𝑘}_𝑘∈ℤ+ is an ATI and let 𝐷₀=𝑆₀, and 𝐷_𝑘=𝑆_𝑘−𝑆_𝑘−1 for 𝑘∈ℕ. Let 𝑀 be a fixed large positive integer, 𝑄_𝜏^0,𝜈 be as above. Suppose that −1<𝑠<1.
The inhomogeneous Triebel-Lizorkin space 𝐹_𝑝^𝛼,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 1max0,𝑠,−𝑠+𝑛𝑛min𝑝,1−1<𝛽<1,min𝑝,1−𝑛<𝛾<1 (1.12) such that 𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)=_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝑚_𝑄𝜏^0,𝜈𝐷₀𝑓^𝑝1/𝑝+^∞_𝑘=12^𝑘𝑠𝐷_𝑘𝑓^𝑞1/𝑞_{𝐿𝑝(𝒳)}<∞, (1.13) where 𝑚_𝑄𝜏^0,𝜈(𝐷₀(𝑓)) are averages of 𝐷₀(𝑓) over 𝑄_𝜏^0,𝜈.
The restrictions (1.12) guarantee that the definitions of the inhomogeneous Triebel-Lizorkin space 𝐹_𝑝^𝑠,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 𝐹_𝑝^𝑠,𝑞(𝒳)⊂(ℳ(𝛽,𝛾))^′,ℳ(𝛽,𝛾)⊂ℳ(𝛽,𝛾)⊂𝐹_𝑝^𝑠,𝑞(𝒳) in [10].
The classical scale of inhomogeneous Triebel-Lizorkin spaces contains many well-known function spaces. For example, if 𝛼>0,𝑝=𝑞=∞, one recovers the Hölder-Zygmund spaces 𝒞^𝛼(𝒳), that is, 𝐹_∞^𝛼,∞(𝒳)=𝒞^𝛼(𝒳),𝛼>0. (1.14) The space 𝒞^𝛼(𝒳) is defined as the collection of 𝑓 such that 𝑓_{𝒞𝛼(𝒳)}=𝑓_∞+sup_𝑥≠𝑦𝑓(𝑥)−𝑓𝑦𝑑𝑥,𝑦^𝛼<∞. (1.15)
If −1<𝑠<1 and 1<𝑝<∞, then 𝐻^𝑠_𝑝(𝒳)=𝐹_𝑝^𝑠,2(𝒳) are the Bessel potential spaces (Lebesgue spaces, Liouville spaces). If 𝑚=0,1,2,… and 1<𝑝<∞, then 𝑊^𝑚_𝑝(𝒳)=𝐻^𝑚_𝑝(𝒳)=𝐹_𝑝^𝑚,2(𝒳) are the usual Sobolev spaces. If 𝑛/(𝑛+1)<𝑝≤1, then 𝐹_𝑝^0,2(𝒳)=ℎ^𝑝(𝒳) are the inhomogeneous Hardy spaces, which are closely related to the Hardy spaces 𝐻^𝑝(𝒳) in [15, 16] (more precisely: the homogeneous spaces 𝐹̇_𝑝^0,2(𝒳) coincide with the usual Hardy spaces 𝐻_𝑝(𝒳)). We will use here the notation 𝐹_𝑝^𝑠,2(𝒳)=𝐻^𝑠_𝑝(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 {𝐷_𝑘}_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 𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)∼_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝑚_𝑄𝜏^0,𝜈𝐷₀𝑓^𝑝1/𝑝+^∞_{𝑘=1𝜏∈𝐼𝑘}^{𝑁(𝜏,𝑘,𝑀)}_𝜈=12^𝑘𝑠inf_{𝑧∈𝑄𝜏𝑘,𝜈}𝐷_𝑘𝑓(𝑧)𝜒_{𝑄𝜏𝑘,𝜈}(⋅)^𝑞1/𝑞_{𝐿𝑝(𝒳)}∼_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝑚_𝑄𝜏^0,𝜈𝐷₀𝑓^𝑝1/𝑝+^∞_{𝑘=1𝜏∈𝐼𝑘}^{𝑁(𝜏,𝑘,𝑀)}_𝜈=12^𝑘𝑠sup_{𝑧∈𝑄𝜏𝑘,𝜈}𝐷_𝑘𝑓(𝑧)𝜒_{𝑄𝜏𝑘,𝜈}(⋅)^𝑞1/𝑞_{𝐿𝑝(𝒳)}. (1.16)

We now introduce the following definition of the pointwise multiplier.

Definition 1.8. Suppose that 𝑔 is a given function on 𝒳. Then 𝑔 is called a pointwise multiplier for 𝐹_𝑝^𝑠,𝑞(𝒳) if 𝑓→𝑔𝑓 admits a bounded linear mapping from 𝐹_𝑝^𝑠,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∈𝒞^𝛼(𝒳) with 0<𝛼<1 is a multiplier for 𝐹_𝑝^𝑠,𝑞(𝒳). In other words, 𝑓→𝑔𝑓 yields a bounded linear mapping from 𝐹_𝑝^𝑠,𝑞(𝒳) into itself and there is a positive constant 𝐶 such that 𝑔𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)≤𝐶𝑔_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳) (1.17) holds for all 𝑔∈𝒞^𝛼(𝒳) and 𝑓∈𝐹_𝑝^𝑠,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 𝐹_𝑝^𝑠,𝑞(ℝ^𝑛), where 0<𝑝<∞,0<𝑞≤∞,𝑠∈ℝ have been studied in great detail in [4, 5, 8].

Theorem 1.9 was proved in [8] for pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces on ℝ^𝑛 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 by 𝐶 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 Theorem 1.9

In this section, we will prove Theorem 1.9. 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 {𝑆_𝑘}_𝑘∈ℤ+ is an approximation to the identity as in Definition 1.3. Set 𝐷_𝑘=𝑆_𝑘−𝑆_𝑘−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 𝑦_𝜏^𝑘,𝜈∈𝑄_𝜏^𝑘,𝜈,𝑘∈ℕ,𝜏∈𝐼_𝑘 and 𝜈∈{1,…,𝑁(𝑘,𝜏,𝑀)} and all 𝑓∈(𝑀(𝛽,𝛾))^′ with 0<𝛽,𝛾<𝜖 and 𝑥∈𝒳𝑓(𝑥)=_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_{𝜈=1𝑄𝜏0,𝜈}𝐷₀𝑥,𝑦𝑑𝜇𝑦𝑚_𝑄𝜏^0,𝜈𝐷₀𝑓+_{𝑘∈ℕ𝜏∈𝐼𝑘}^{𝑁(𝑘,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^𝑘,𝜈𝐷_𝑘𝑥,𝑦_𝜏^𝑘,𝜈𝐷_𝑘𝑓𝑦_𝜏^𝑘,𝜈=_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_{𝜈=1𝑄𝜏0,𝜈}𝐷₀𝑥,𝑦𝑑𝜇𝑦𝑚_𝑄𝜏^0,𝜈𝐷₀𝑓+_{𝑘∈ℕ𝜏∈𝐼𝑘}^{𝑁(𝑘,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^𝑘,𝜈𝐷_𝑘𝑥,𝑦_𝜏^𝑘,𝜈𝐷_𝑘𝑓𝑦_𝜏^𝑘,𝜈, (2.1) where the series converges in the norm of 𝐹_𝑝^𝑠,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, 𝐷_𝑘1𝑥,𝑦≤𝐶𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦2+𝑉𝑥,𝑦2^−𝑘+𝑑^−𝑘𝜖𝑥,𝑦^𝜖,𝐷_𝑘𝑥,𝑦−𝐷_𝑘𝑥^′𝑑,𝑦≤𝐶𝑥,𝑥^′2^−𝑘+𝑑𝑥,𝑦^𝜖1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦2+𝑉𝑥,𝑦2^−𝑘+𝑑^−𝑘𝜖𝑥,𝑦^𝜖, (2.2)(2.3) for 𝑑(𝑥,𝑥^′)≤(2^−𝑘+𝑑(𝑥,𝑦))/2, _𝒳𝐷_𝑘𝑥,𝑦𝑑𝜇𝑦=_𝒳𝐷_𝑘𝑥,𝑦𝑑𝜇(𝑥)=0, (2.4) when 𝑘∈ℕ;∫_𝒳𝐷_𝑘(𝑥,𝑦)𝑑𝜇(𝑦)=∫_𝒳𝐷_𝑘(𝑥,𝑦)𝑑𝜇(𝑥)=1 when 𝑘=0.
To prove Theorem 1.9, we first show the following lemma.

Lemma 2.2. Let {𝑆_𝑘(𝑥,𝑦)}_𝑘∈ℤ+ and {𝐺_𝑘(𝑥,𝑦)}_𝑘∈ℤ+ be two approximations to the identity as in Lemma 2.1 above and 𝐷_𝑘=𝑆_𝑘−𝑆_𝑘−1,𝐸_𝑘=𝐺_𝑘−𝐺_𝑘−1 for 𝑘∈ℕ and 𝐷₀=𝑆₀,𝐸₀=𝐺₀. For any given 𝜖∈(0,1) and 𝑔∈𝒞^𝛼(𝒳) with 0<𝛼<𝜖, then 𝐸_𝑘𝑔𝐷_𝑘′𝑥,𝑦≲𝑔_{𝒞𝛼(𝒳)}2^{−|𝑘−𝑘′|𝛼}1𝑉_{2′)−(𝑘∧𝑘}(𝑥)+𝑉_{2′)−(𝑘∧𝑘}𝑦2+𝑉𝑥,𝑦2^{−(𝑘∧𝑘′)}+𝑑^{−(𝑘∧𝑘′)𝜖}𝑥,𝑦^𝜖, (2.5) 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 𝐸₀𝑔𝐷₀𝑥,𝑦≤𝑔_{𝒞𝛼(𝒳)}𝐸₀(𝑥,𝑧)𝐷₀(𝑧≲𝑔𝑧,𝑦𝑑𝜇)_{𝒞𝛼(𝒳)}1𝑉₁(𝑥)+𝑉₁𝑦1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^𝜖. (2.6)
We next consider the case 𝑘^′≥𝑘 for 𝑘>0 and 𝑘^′≥0 or 𝑘≥0 and 𝑘^′>0. We write 𝐸_𝑘𝑔𝐷_𝑘′=𝐸𝑥,𝑦_𝑘(𝑥,𝑧)𝑔(𝑧)−𝐸_𝑘𝑥,𝑦𝑔𝑦𝐷_𝑘′(𝑧≤𝐸𝑧,𝑦𝑑𝜇)_𝑘(𝑥,𝑧)−𝐸_𝑘𝑥,𝑦𝑔(𝑧)𝐷_𝑘′(𝑧)+𝐸𝑧,𝑦𝑑𝜇_𝑘𝑥,𝑦𝑔(𝑧)−𝑔𝑦𝐷_𝑘′𝑧,𝑦𝑑𝜇(𝑧)∶=𝐸+𝐹. (2.7)
For 𝐸, we can obtain 𝐸≲𝑔_{𝒞𝛼(𝒳)}2^{−(𝑘′−𝑘)𝜖}1𝑉_{2′)−(𝑘∧𝑘}(𝑥)+𝑉_{2′)−(𝑘∧𝑘}𝑦2+𝑉𝑥,𝑦2^−𝑘+𝑑^−𝑘𝜖𝑥,𝑦^𝜖, (2.8) where 𝑘,𝑘^′∈ℤ₊.
We may rewrite this integral as 𝐹≲𝑔_{𝒞𝛼(𝒳)}1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦2+𝑉𝑥,𝑦2^−𝑘+𝑑^−𝑘𝜖𝑥,𝑦^𝜖×_𝑊1𝑑𝑧,𝑦^𝛼1𝑉_2′−𝑘(𝑧)+𝑉_2′−𝑘𝑦2+𝑉𝑧,𝑦2^−𝑘′+𝑑^{−𝑘′𝜖}𝑧,𝑦^𝜖(𝑧)+𝑑𝜇_𝑊2𝑑𝑧,𝑦^𝛼1𝑉_2′−𝑘(𝑧)+𝑉_2′−𝑘𝑦2+𝑉𝑧,𝑦2^−𝑘′+𝑑^{−𝑘′𝜖}𝑧,𝑦^𝜖(𝑧)𝑔𝑑𝜇∶=_{𝒞𝛼(𝒳)}1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦2+𝑉𝑥,𝑦2^−𝑘+𝑑^−𝑘𝜖𝑥,𝑦_𝜖𝐹₁+𝐹₂, (2.9) 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 for 𝑘,𝑘^′∈ℤ₊ as follows: 𝐹₁≲^∞_{𝑖=−𝑘′{𝑧∶2𝑖≤𝑑(𝑧,𝑦)<2𝑖+1}}𝑑𝑧,𝑦^𝛼1𝑉_2′−𝑘(𝑧)+𝑉_2′−𝑘𝑦2+𝑉𝑧,𝑦2^−𝑘′+𝑑^{−𝑘′𝜖}𝑧,𝑦^𝜖(𝑧)≲𝑑𝜇_{∞𝑖=−𝑘′}𝜇𝐵𝑦,2^𝑖+1𝜇𝐵𝑦,2^𝑖2^(𝑖+1)𝛼2^{−𝑘′𝜖}2^−𝑖𝜖≲2^{−𝑘′𝛼}≤2^{−(𝑘′−𝑘)𝛼}, (2.10) where we used 0<𝛼<𝜖.
For 𝐹₂, we have 𝐹₂≲2^{−𝑘′𝛼}≤2^{−(𝑘′−𝑘)𝛼}. (2.11)
Combining the estimate for 𝐸,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 Theorem 1.9.

Proposition 2.3. For any 𝑔∈𝒞^𝛼(𝒳),𝑓∈ℳ(𝛽,𝛾) with 𝛽 and 𝛾 satisfying (1.12), then 𝑓𝑔_𝐹𝑝^𝑠,𝑞_(𝒳)≤𝐶𝑔_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳), (2.12) 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 any 𝑔∈𝒞^𝛼(𝒳),𝑓∈ℳ(𝛽,𝛾), We write 𝑔𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)≤_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈_{𝜏′∈𝐼0}^{𝑁(0,𝜏′,𝑀)}_𝜈′=1𝑚_𝑄𝜏^0,𝜈𝐸₀𝑔𝐷_{𝑄′𝜏′0,𝜈}𝜇𝑄^0,𝜈′_𝜏′𝑚_{𝑄′𝜏′0,𝜈}𝐷₀𝑓^𝑝1/𝑝+_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈∞_{𝑘′=1𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^{𝑘′,𝜈′}_𝜏′𝑚_𝑄𝜏^0,𝜈×𝐸₀𝑔𝐷_𝑘′⋅,𝑦^{𝑘′,𝜈′}_𝜏′𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑝1/𝑝+^∞_𝑘=12^𝑘𝑠𝑞_{𝜏′∈𝐼0}^{𝑁(0,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^0,𝜈′_𝜏′𝐸_𝑘𝑔𝐷_{𝑄′𝜏′0,𝜈}(⋅)𝑚_{𝑄′𝜏′0,𝜈}𝐷₀𝑓^𝑞1/𝑞_{𝐿𝑝(𝒳)}+^∞_𝑘=12^{𝑘𝑠𝑞∞}_{𝑘′=1𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^{𝑘′,𝜈′}_𝜏′𝐸_𝑘𝑔𝐷_𝑘′⋅,𝑦^{𝑘′,𝜈′}_𝜏′𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑞1/𝑞_{𝐿𝑝(𝒳)}∶=𝐿₁+𝐿₂+𝐿₃+𝐿₄. (2.13)
Applying the Hölder inequality for 𝑝>1 and _𝑘𝑎_𝑘^𝑝≤_𝑘𝑎_𝑘^𝑝 (2.14) for all 𝑎_𝑘∈ℂ and 𝑝≤1, for 𝐿₁, it follows that 𝐿₁≲𝑔_{𝒞𝛼(𝒳)𝜏∈𝐼0}^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈_{𝜏′∈𝐼0}^{𝑁(0,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^0,𝜈′_𝜏′𝑚_{𝑄′𝜏′0,𝜈}𝐷₀𝑓^𝑝×𝜇𝑄^0,𝜈′_𝜏′^𝑝∧1−1inf_{𝑥∈𝑄𝜏0,𝜈}inf_{𝑦∈𝑄′𝜏′0,𝜈}1𝑉₁(𝑥)+𝑉₁𝑦1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^{𝜖𝑝∧11/𝑝}≲𝑔_{𝒞𝛼(𝒳)𝜏′∈𝐼0}^{𝑁(0,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^0,𝜈′_𝜏′𝑚_{𝑄′𝜏′0,𝜈}𝐷₀𝑓^𝑝𝜇𝑄^0,𝜈′_𝜏′^𝑝∧1−1𝑉₁𝑦^0,𝜈′_𝜏′^{1−𝑝∧11/𝑝}≲𝑔_{𝒞𝛼(𝒳)}𝑓_𝐵𝑝^𝑠,𝑞_(𝒳), (2.15) where we used the fact that for any 𝑦^0,𝜈′_𝜏′∈𝑄^0,𝜈′_𝜏′,𝑉₁(𝑦^0,𝜈′_𝜏′)𝜇(𝑄^0,𝜈′_𝜏′).
For 𝐿₂, in fact for all 𝑘^′∈ℕ,𝜏^′∈𝐼_𝑘′ and 𝜈^′=1,2,…,𝑁(𝑘^′,𝜏^′,𝑀), and all 𝑥∈𝒳,𝑧∈𝑄^{𝑘′,𝜈′}_𝜏′, 𝜇𝑄^{𝑘′,𝜈′}_𝜏′∼𝑉_2−𝑘′(𝑧),𝑉₁𝑦^{𝑘′,𝜈′}_𝜏′+𝑉𝑥,𝑦^{𝑘′,𝜈′}_𝜏′∼𝑉₁(𝑧)()𝑉+𝑉𝑥,𝑧,₁(𝑧)≲2^𝑘′𝑛𝜇𝑄^{𝑘′,𝜈′}_𝜏′,𝑉₂^𝑖+1(𝑧)≲2ⁱⁿ𝑉₁(𝑧),1+𝑑𝑥,𝑦^{𝑘′,𝜈′}_𝜏′∼1+𝑑(𝑥,𝑧). (2.16)(2.17) 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 𝐿₂≲𝑔_{𝒞𝛼(𝒳)𝜏∈𝐼0}^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈∞_𝑘′=12^{−𝑘′(𝑠+𝛼)}_{𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^{𝑘′,𝜈′}_𝜏′2^𝑘′𝑠𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′×inf_{𝑥∈𝑄𝜏0,𝜈}1𝑉₁(𝑥)+𝑉₁𝑦^{𝑘′,𝜈′}_𝜏′+𝑉𝑥,𝑦^{𝑘′,𝜈′}_𝜏′11+𝑑𝑥,𝑦^{𝑘′,𝜈′}_𝜏′^{𝜖𝑝1/𝑝}≲_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈∞_𝑘′=12^{−𝑘′(𝑠+𝛼)}_{𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^{𝑘′,𝜈′}_𝜏′^𝑟2^𝑘′𝑠𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑟×1𝑉₁𝑥_𝜏^0,𝜈+𝑉₁𝑦^{𝑘′,𝜈′}_𝜏′+𝑉𝑥_𝜏^0,𝜈,𝑦^{𝑘′,𝜈′}_𝜏′^𝑟×11+𝑑𝑥_𝜏^0,𝜈,𝑦^{𝑘′,𝜈′}_𝜏′^{𝜖𝑟1/𝑟}𝑔_{𝒞𝛼(𝒳)}^𝑝1/𝑝≲𝑔_{𝒞𝛼(𝒳)𝜏∈𝐼0}^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈∞_𝑘′=12^{−𝑘′(𝑠+𝛼)}1𝑉₁𝑥_𝜏^{0,𝜈𝑑(𝑥}_𝜏^{0,𝜈,𝑧)<1}_{𝜏′∈𝐼𝑘′}×^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=1𝑉₁𝑥_𝜏^{0,𝜈1−𝑟}𝜇𝑄^{𝑘′,𝜈′}_𝜏′^1−𝑟2^𝑘′𝑠𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑟𝜒_{𝑄𝑘′′𝜏′,𝜈}(𝑧)(𝑧)+𝑑𝜇_∞𝑖=01𝑉₂^𝑖+1𝑥_𝜏^{0,𝜈2𝑖≤𝑑(𝑥}_𝜏^{0,𝜈,𝑧)<2𝑖+1}𝑉₁(𝑧)^1−𝑟𝜇𝑄^{𝑘′,𝜈′}_𝜏′^1−𝑟𝑉₂^𝑖+1(𝑧)^1−𝑟𝑉₁(𝑧)^1−𝑟2^{−𝑖𝜖𝑟}×_{𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=12^𝑘′𝑠𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑟𝜒_{𝑄𝑘′′𝜏′,𝜈}(𝑧)𝑑𝜇(𝑧)^{1/𝑟𝑝1/𝑝}≲𝑔_{𝒞𝛼(𝒳)}×^∞_𝑘′=12^{−𝑘′(𝑛+𝑠+𝛼−𝑛/𝑟)}𝐌_{𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=12^{𝑘′𝑠𝑟}𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑟𝜒_{𝑄𝑘′′𝜏′,𝜈}^1/𝑟_{𝐿𝑝(𝒳)}≲𝑔_{𝒞𝛼(𝒳)}^∞_𝑘′=12^{−𝑘′(𝑛+𝑠+𝛼−𝑛/𝑟)(𝑞∧1)}×𝐌_{𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=12^{𝑘′𝑠𝑟}𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑟𝜒_{𝑄𝑘′′𝜏′,𝜈}^{𝑞/𝑟1/𝑞}_{𝐿𝑝(𝒳)}≲𝑔_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳), (2.18) where we choose 𝑟 satisfying max(𝑛/(𝑛+𝜖),𝑛/(𝑛+𝑠+𝛼))<𝑟<min{1,𝑝,𝑞}.
By the inequality (2.5), similarly, 𝐿₃≲𝑔_{𝒞𝛼(𝒳)𝒳𝜏′∈𝐼0}^{𝑁(0,𝜏′,𝑀)}_𝜈′=1𝜇𝑄^0,𝜈′_𝜏′𝑚_{𝑄′𝜏′0,𝜈}𝐷₀𝑓^𝑝1𝑉₁(𝑥)+𝑉₁𝑦^0,𝜈′_𝜏′+𝑉𝑥,𝑦^0,𝜈′_𝜏′^𝑝∧111+𝑑𝑥,𝑦^0,𝜈′_𝜏′^{𝜖(𝑝∧1)}𝑑𝜇(𝑥)^1/𝑝≲𝑔_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳), (2.19) where 𝑠 can be any number in (−𝜖,𝛼)⊂(−1,1).
We now consider the estimate of 𝐿₄, 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 𝐿₄≲𝑔_{𝒞𝛼(𝒳)}^∞∞_{𝑘=1𝑘′=1}2^{(𝑘−𝑘′)𝑠}2^{−|𝑘′−𝑘|𝛼}2^{−𝑘′𝑛}2^{(𝑘∧𝑘′)𝑛}2^{[𝑘′−(𝑘∧𝑘′)]𝑛/𝑟𝑞∧1}×𝐌_{𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=12^𝑘′𝑠𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′𝜒_{𝑄𝑘′′𝜏′,𝜈}^{𝑟𝑞/𝑟1/𝑞}_{𝐿𝑝(𝒳)}≲𝑔_{𝒞𝛼(𝒳)}^∞_𝑘′=1𝐌_{𝜏′∈𝐼𝑘′}^{𝑁(𝑘′,𝜏′,𝑀)}_𝜈′=12^𝑘′𝑠𝐷_𝑘′𝑓𝑦^{𝑘′,𝜈′}_𝜏′^𝑟𝜒_{𝑄𝑘′′𝜏′,𝜈}^{𝑞/𝑟1/𝑞}_{𝐿𝑝(𝒳)}≲𝑔_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝛼,𝑞_(𝒳), (2.20) where the first and the second inequalities follow from the estimates sup_𝑘^∞_𝑘′=02^{(𝑘−𝑘′)𝑠}2^{−|𝑘′−𝑘|𝛼}2^{−𝑘′𝑛}2^{(𝑘∧𝑘′)𝑛}2^{[𝑘′−(𝑘∧𝑘′)]𝑛/𝑟}≤𝐶,sup_𝑘′^∞_𝑘=02^{(𝑘−𝑘′)𝑠}2^{−|𝑘′−𝑘|𝛼}2^{−𝑘′𝑛}2^{(𝑘∧𝑘′)𝑛}2^{[𝑘′−(𝑘∧𝑘′)]𝑛/𝑟𝑞∧1}≤𝐶 (2.21) 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 {𝑆_𝑘(𝑥,𝑦)}_𝑘∈ℤ+ be a approximation to the identity as in Lemma 2.1 above and 𝐷_𝑘=𝑆_𝑘−𝑆_𝑘−1 for 𝑘∈ℕ and 𝐷₀=𝑆₀. For any 𝑔∈𝒞^𝛼(𝒳) with 0<𝛼<1, ℎ∈ℳ(𝛽,𝛾) with 𝛽 and 𝛾 satisfying (1.12). Then 𝐷_𝑘•,𝑦𝑔,ℎ≤𝐶𝑔_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)}2^{−𝑘(𝛽∧𝛼)}1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾, (2.22) 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 𝐷₀≲𝑔•,𝑦𝑔,ℎ_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)𝑑(𝑦,𝑧)<2𝐶1}1𝑉₁(𝑧)+𝑉₁𝑦1𝑉₁𝑥₀+𝑉𝑥₀1,𝑧1+𝑑𝑧,𝑥₀^𝛾(𝑧)=𝑔𝑑𝜇_{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. (2.23)
For 𝐵₁, 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 𝐵₁≲1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾. (2.24)
For 𝐵₂, the fact 𝑑(𝑦,𝑧)<2𝐶₁,𝑑(𝑥₀,𝑧)<𝑑(𝑥₀,𝑦)/2 implies that 2𝐶₁>𝑑(𝑦,𝑧)>𝑑(𝑥₀,𝑦)/2, then 𝐵₂≲1𝑉₁𝑥₀+𝑉𝑥₀≲1,𝑦𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾. (2.25) That is, (2.22) with 𝑘=0 holds.
To prove (2.22) with 𝑘∈ℕ, we write 𝐷_𝑘•,𝑦𝑔,ℎ=_𝒳𝐷_𝑘𝑔(𝑥)ℎ(𝑥)𝑦ℎ𝑦(𝑥≤𝑥,𝑦−𝑔𝑑𝜇)_𝒳𝐷_𝑘𝑔(𝑥)𝑦ℎ(𝑥(𝑥)+𝑥,𝑦−𝑔|)|𝑑𝜇_𝒳𝐷_𝑘𝑥,𝑦𝑔𝑦ℎ(𝑥)−ℎ𝑦𝑑𝜇(𝑥)∶=𝐼+𝐼𝐼. (2.26)
For 𝐼, since 𝐷_𝑘(𝑥,𝑦)=0 if 𝑑(𝑥,𝑦)≥2𝐶₁2^−𝑘, we obtain 𝐼≲𝑔_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)}2^−𝑘𝛼×_{𝑑(𝑥,𝑦)<2𝐶12−𝑘}1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦1𝑉₁𝑥₀+𝑉𝑥₀1,𝑥1+𝑑𝑥,𝑥₀^𝛾(𝑥)≲𝑔𝑑𝜇_{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^−𝑘𝛼𝐼₁+𝐼₂. (2.27)
For 𝐼₁, by 𝑑(𝑥,𝑦)≤(1/2)(1+𝑑(𝑦,𝑥₀)) implies that 𝑑(𝑦,𝑥₀)≤2(𝑑(𝑥,𝑥₀)+1) and 1𝑉₁𝑥₀+𝑉𝑥₀≲1,𝑥𝑉₁𝑥₀+𝜇𝐵𝑥₀,𝑑𝑥₀=1,𝑦𝑉₁𝑥₀+𝑉𝑥₀.,𝑦 (2.28) Thus 𝐼₁≲1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾_{𝑑(𝑥,𝑦)<2𝐶12}^−𝑘1𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦(𝑥)≲1𝑑𝜇𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾. (2.29)
For 𝐼₂, by 2𝐶₁2^−𝑘>𝑑(𝑥,𝑦)>(1/2)(1+𝑑(𝑦,𝑥₀)) implies that 1+𝑑𝑦,𝑥₀1<𝐶,𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘𝑦≲1𝑉≲1𝑦,𝑥𝑉₁𝑥₀+𝑉𝑥₀.,𝑦 (2.30) Thus 𝐼₂≲1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑥₀,𝑦^𝛾_𝒳1𝑉₁𝑥₀+𝑉𝑥₀1,𝑥1+𝑑𝑥,𝑥₀^𝛾(𝑥)≲1𝑑𝜇𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑥₀,𝑦^𝛾. (2.31)
To obtain the estimate of the 𝐼𝐼, we write 𝐼𝐼≲𝑔_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)𝜛1}𝐷_𝑘1𝑥,𝑦𝑉₁𝑥₀+𝑉𝑥₀𝑑,𝑦𝑥,𝑦^𝛽1+𝑑𝑦,𝑥₀^𝛽+𝛾(𝑥)+𝑑𝜇_𝜛2𝐷_𝑘1𝑥,𝑦𝑉₁𝑥₀+𝑉𝑥₀1,𝑥1+𝑑𝑥,𝑥₀^𝛾+1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾(𝑥𝑔𝑑𝜇)∶=_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)}𝐼𝐼₁+𝐼𝐼₂+𝐼𝐼₃, (2.32) 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))}.
For 𝐼𝐼₁, we have 𝐼𝐼₁≲1𝑉₁𝑥₀+𝑉𝑥₀2,𝑦1+𝑑𝑥^−𝑘𝛽₀,𝑦^𝛾_𝜛1𝐷_𝑘(𝑥)≲1𝑥,𝑦𝑑𝜇𝑉₁𝑥₀+𝑉𝑥₀2,𝑦1+𝑑𝑥^−𝑘𝛽₀,𝑦^𝛾. (2.33)
For 𝐼𝐼₂, similar to 𝐼₂, we have 𝐷_𝑘≲1𝑥,𝑦𝑉₁𝑥₀+𝑉𝑥₀2,𝑦1+𝑑𝑥^−𝑘𝛽₀,𝑦^𝛾, (2.34) thus 𝐼𝐼₂≲1𝑉₁𝑥₀+𝑉𝑥₀2,𝑦1+𝑑𝑥^−𝑘𝛽₀,𝑦^𝛾_𝜛21𝑉₁𝑥₀+𝑉𝑥₀1,𝑥1+𝑑𝑥,𝑥₀^𝛾(𝑥)≤1𝑑𝜇𝑉₁𝑥₀+𝑉𝑥₀2,𝑦1+𝑑𝑥^−𝑘𝛽₀,𝑦^𝛾. (2.35)
For 𝐼𝐼₃, obviously 𝐼𝐼₃≲2^−𝑘𝛽𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾_𝒳𝐷_𝑘(𝑥)≲2𝑥,𝑦𝑑𝜇_−𝑘𝛽𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾. (2.36)
Combining the estimate of 𝐼 and 𝐼𝐼, we obtain that for 𝑘∈ℕ𝐷_𝑘•,𝑦𝑔,ℎ≤𝐶𝑔_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)}2^{−𝑘(𝛽∧𝛼)}1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑦,𝑥₀^𝛾. (2.37) Thus, (2.22) also holds. This finishes the proof of Lemma 2.4.

If 𝑔∈𝒞^𝛼(𝑋) and 𝑓∈𝐹_𝑝^𝑠,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 any 𝑓∈𝐹_𝑝^𝑠,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∈𝒞^𝛼(𝒳) with 1>𝛼>max(𝑠,(𝑛/min{𝑝,𝑞,1})−𝑛−𝑠). There exist a constant 𝐶 and a sequence {𝑓_𝑗}_𝑗∈ℕ such that 𝑓_𝑗∈ℳ(𝜖,𝜖),‖𝑓_𝑗‖_𝐹𝑝^𝑠,𝑞_(𝒳)≤𝐶‖𝑓‖_𝐹𝑝^𝑠,𝑞_(𝒳) and lim_𝑗→∞⟨𝑔𝑓_𝑗,ℎ⟩ converges for any ℎ∈ℳ(𝛽,𝛾) with 𝛽 and 𝛾 satisfying (1.12).

Proof. For any 𝑓∈𝐹_𝑝^𝑠,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∈𝒞^𝛼(𝒳) with 𝛼>max(𝑠,(𝑛/min{𝑝,𝑞,1})−𝑛−𝑠), denote 𝑓_𝑗=^𝑗_𝜏=1^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝑚_𝑄𝜏^0,𝜈𝐷₀𝑓𝐷_𝑄𝜏^0,𝜈(𝑥)+^𝑗𝑗_𝑘=1𝜏=1^{𝑁(𝑘,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^𝑘,𝜈𝐷_𝑘𝑥,𝑦_𝜏^𝑘,𝜈𝐷_𝑘𝑓𝑦_𝜏^𝑘,𝜈, (2.38) where 𝐷_𝑄𝜏^0,𝜈(𝑥)=(1/𝜇(𝑄_𝜏^0,𝜈))∫_{𝑄𝜏0,𝜈}𝐷₀(𝑥,𝑦)𝑑𝜇(𝑦).
It is easy to see that 𝑓_𝑗∈ℳ(𝜖,𝜖). Applying a similar proof as in Proposition 2.3 with 𝑔=1 and 𝑓=𝑓_𝑗 gives ‖𝑓_𝑗‖_𝐹𝑝^𝑠,𝑞_(𝒳)≤𝐶‖𝑓‖_𝐹𝑝^𝑠,𝑞_(𝒳). We write 𝑓_𝑗−𝑓_𝑚,𝑔ℎ≤^𝑗_𝜏=𝑚+1^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝑚_𝑄𝜏^0,𝜈𝐷₀𝑓𝐷_𝑄𝜏^0,𝜈+,𝑔ℎ_{𝑗𝑗𝑘=𝑚+1𝜏=1𝑁(𝑘,𝜏,𝑀)𝜈=1}𝜇𝑄_𝜏^𝑘,𝜈𝐷_𝑘•,𝑦_𝜏^𝑘,𝜈,𝑔ℎ𝐷_𝑘𝑓𝑦_𝜏^𝑘,𝜈+^𝑗𝑗_{𝑘=1𝜏=𝑚+1}^{𝑁(𝑘,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^𝑘,𝜈𝐷_𝑘•,𝑦_𝜏^𝑘,𝜈,𝑔ℎ𝐷_𝑘𝑓𝑦_𝜏^𝑘,𝜈∶=𝑅+𝑇+𝑌. (2.39)
The fact (2.22) implies ^𝑗_𝜏=𝑚+1^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝐷_𝑄𝜏^0,𝜈𝑔ℎ^{𝑝′1/𝑝′}≲𝑔_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)𝜏≥log𝑚2{𝑦∶2𝜏<𝑑(𝑥0,𝑦)≤2𝜏+1}}1𝑉₁𝑥₀+𝑉𝑥₀,𝑦^𝑝′×11+𝑑𝑦,𝑥₀^𝛾𝑝′𝑑𝜇𝑦^1/𝑝′≲𝑔_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)}^∞_{𝜏≥log𝑚2}2^{−𝜏𝛾𝑝′1/𝑝′}=𝑚^−𝛾𝑔_{𝒞𝛼(𝒳)}‖ℎ‖_{ℳ(𝛽,𝛾)}. (2.40) Thus 𝑓𝑅≲^{𝐹𝑝𝑠,𝑞(𝒳)𝑗}∑∑_𝜏=𝑚+1^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝐷^∗_{𝑄𝜏0,𝜈}𝑔ℎ^{𝑝′1/𝑝′}𝑓∶𝑝>1_{𝐹𝑝𝑠,𝑞(𝒳)}sup_{𝑦𝜏0,𝜈𝑥,𝑚<𝑑0,𝑦𝜏0,𝜈≤𝑗}𝜇𝑄_𝜏^{0,𝜈1−1/𝑝}𝑉₁𝑥₀+𝑉𝑥₀,𝑦_𝜏^0,𝜈11+𝑑𝑦_𝜏^0,𝜈,𝑥₀^𝛾≲𝑔∶𝑝≤1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}𝑚^−𝛾, (2.41) this finishes the proof of 𝑅→0 as 𝑗,𝑚→∞.
We now consider the estimate of 𝑇. By Hölder inequality if 𝑝,𝑞>1, and (2.14) if 𝑝,𝑞<1 and (2.22), it follows that 𝑓𝑇≲^{𝐹𝑝𝑠,𝑞(𝒳)𝑗}_𝑘=∑_𝑚+12^−𝑘𝑠𝐷^∗_𝑘𝑔ℎ^{𝑞′1/𝑞′}_{𝐿𝑝′(𝒳)}𝑓∶𝑝>1,𝑞>1,_{𝐵𝑝𝑠−𝜖,∞𝑗(𝒳)𝑘=}∑_𝑚+1sup_𝑥∈𝒳2^{𝑘[−𝑠+𝜀]}𝜇𝑄_𝜏^{𝑘,𝜈1−1/𝑝}𝐷^∗_𝑘(𝑥𝑓𝑔ℎ)∶𝑝≤1,𝑞>1,_{𝐹𝑝𝑠,𝑞(𝒳)}sup_{𝑚+1≤𝑘≤𝑗}2^−𝑘𝑠𝐷^∗_𝑘𝑔ℎ_{𝐿𝑝′(𝒳)}𝑓∶𝑝>1,𝑞≤1,_{𝐵𝑝𝑠,1(𝒳)}sup_𝑥∈𝒳sup_{𝑚+1≤𝑘≤𝑗}2^−𝑘𝑠𝜇𝑄_𝜏^{𝑘,𝜈1−1/𝑝}𝐷^∗_𝑘(𝑥≲𝑓𝑔ℎ)∶𝑝≤1,𝑞≤1,_{𝐹𝑝𝑠,𝑞(𝒳)𝑗𝑘=}∑_𝑚+12^−𝑘𝑠𝐷^∗_𝑘𝑔ℎ^{𝑞′1/𝑞′}_{𝐿𝑝′(𝒳)}𝑔,∶𝑝>1,𝑞>1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}2^{−𝑚[𝑠−𝜀+𝑛(1−1/𝑝)+𝛼∧𝛽]}𝑔,∶𝑝≤1,𝑞>1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}2^{−𝑚[𝑠+𝛼∧𝛽]}𝑔,∶𝑝>1,𝑞≤1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}2^{−𝑚[𝑠+𝑛(1−1/𝑝)+𝛼∧𝛽]},∶𝑝≤1,𝑞≤1, (2.42) 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 𝐹_𝑝^𝑠,𝑞(𝒳)⊂𝐵_𝑝^{𝑠,max(𝑝,𝑞)}(𝒳)⊂𝐵_𝑝^{𝑠−𝜀,∞}(𝒳), (2.43) when −𝜖<𝑠−𝜀, see [10, Proposition 5.31].
By the Proposition 2.3, then 𝑔ℎ∈𝐹^{−𝑠,𝑞′}_𝑝′(𝒳) if 𝑠∈(−1,1),𝑝,𝑞>1. Thus by the Calderón reproducing that the series converges in the norm of 𝐹^{−𝑠,𝑞′}_𝑝′(𝒳) for 𝑔ℎ∈𝐹^{−𝑠,𝑞′}_𝑝′(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∈𝒞^𝛼(𝒳) with 𝛼>max(𝑠,(𝑛/min{𝑝,𝑞,1})−𝑑−𝑠).
For 𝑌, applying the Hölder inequality for 𝑝,𝑞>1 and (2.14) for 𝑝,𝑞≤1 and (2.22), we also have 𝑓𝑌≲^{𝐹𝑝𝑠,𝑞(𝒳)}∫∑_{{𝑦∣𝑚<𝑑(𝑥0,𝑦)≤𝑗}}^𝑗_𝑘=12^−𝑘𝑠𝐷^∗_𝑘𝑔ℎ𝑦^{𝑞′𝑝′/𝑞′}𝑑𝜇𝑦^1/𝑝′𝑓∶𝑝>1,𝑞>1,_{𝐵𝑝𝑠−𝜀,∞(𝒳)}sup_𝑘∈ℕsup_{𝑥𝑦∈{𝑦∣𝑚<𝑑0,𝑦≤𝑗}}2^{−𝑘[𝑠−𝜀]}𝜇𝑄_𝜏^{𝑘,𝜈1−1/𝑝}𝐷^∗_𝑘𝑦𝑓𝑔ℎ∶𝑝≤1,𝑞>1,_{𝐹𝑝𝑠,𝑞(𝒳)}∫_{{𝑦∣𝑚<𝑑(𝑥0,𝑦)≤𝑗}}sup_{1≤𝑘≤𝑗}2^−𝑘𝑠𝐷^∗_𝑘𝑔ℎ𝑦^𝑝′𝑑𝜇𝑦^1/𝑝′𝑓∶𝑝>1,𝑞≤1,_{𝐵𝑝𝑠,1(𝒳)}sup_{𝑥𝑦∈{𝑦∣𝑚<𝑑0,𝑦≤𝑗}}sup_{1≤𝑘≤𝑗}2^−𝑘𝑠𝜇𝑄_𝜏^{𝑘,𝜈1−1/𝑝}𝐷^∗_𝑘𝑦≲<𝑔𝑔ℎ∶𝑝≤1,𝑞≤1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}𝑚^{−𝛾1/𝑞′}𝑔∶𝑝>1,𝑞>1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}𝑚^−𝛾sup_𝑘∈ℕ2^{−𝑘[𝑠−𝜀+𝑛(1−1/𝑝)+(𝛼∧𝛽)]𝑞′1/𝑞′}𝑔∶𝑝≤1,𝑞>1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}𝑚^−𝛾sup_𝑘∈ℕ2^{−𝑘[𝑠+𝛼∧𝛽]}𝑔∶𝑝>1,𝑞≤1,_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}𝑚^−𝛾sup_𝑘∈ℕ2^{−𝑘[𝑠+𝑛(1−1/𝑝)+𝛼∧𝛽]}≲𝑔∶𝑝≤1,𝑞≤1_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)‖ℎ‖_{ℳ(𝛽,𝛾)}𝑚^−𝛾, (2.44) 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 𝐹_𝑝^𝑠,𝑞(𝒳)⊂𝐵_𝑝^{𝑠,max(𝑝,𝑞)}(𝒳)⊂𝐵_𝑝^{𝑠−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 shows lim_𝑗→∞⟨𝑔𝑓_𝑗,ℎ⟩ exists and the limit is independent of the choice of 𝑓_𝑗. Therefore, for 𝑔∈𝒞^𝛼(𝒳),𝑓∈𝐹_𝑝^𝑠,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 𝑔𝑓,ℎ=lim_𝑗→∞𝑔𝑓_𝑗,ℎ (2.45) for any ℎ∈ℳ(𝛽,𝛾) with 𝛽 and 𝛾 satisfying (1.12), 𝑓_𝑗 is fundamental sequence defined in Lemma 2.5.

We now prove Theorem 1.9.

Proof of Theorem 1.9. By Proposition 2.3 and Lemma 2.5, for any 𝑔∈𝒞^𝛼(𝒳),𝑓∈𝐹_𝑝^𝑠,𝑞(𝒳), Fatou's lemma implies that 𝑔𝑓_𝐹𝑝^𝑠,𝑞_(𝒳)≤_𝜏∈𝐼0^{𝑁(0,𝜏,𝑀)}_𝜈=1𝜇𝑄_𝜏^0,𝜈𝑚_𝑄𝜏^0,𝜈lim_𝑗→∞𝐸₀𝑓_𝑗𝑔^𝑝1/𝑝+^∞_𝑘=12^𝑘𝛼lim_𝑗→∞𝐸_𝑘𝑓_𝑗𝑔^𝑞1/𝑞_{𝐿𝑝(𝒳)}≤liminf_𝑗→∞𝑔𝑓_{𝑗𝐹𝑝𝑠,𝑞(𝒳)}≲𝑔_{𝒞𝛼(𝒳)}𝑓_𝐹𝑝^𝑠,𝑞_(𝒳). (2.46) We complete the proof of Theorem 1.9.