Pointwise Multipliers on Spaces of Homogeneous Type in the Sense of Coifman and Weiss

Abstract

By applying the remarkable orthonormal basis constructed recently by Ausher and Hytönen on spaces of homogeneous type in the sense of Coifman and Weiss, pointwise multipliers of inhomogeneous Besov and Triebel-Lizorkin spaces are obtained. We make no additional assumptions on the quasi-metric or the doubling measure. Hence, the results of this paper extend earlier related results to a more general setting.

1. Introduction

The main purpose of this paper is to provide pointwise multipliers of inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss. By a pointwise multiplier from a function space 𝒜 into another function space ℬ, we meant that a function defines a bounded linear mapping from 𝒜 into ℬ by pointwise multiplication. Pointwise multipliers arise in many different areas of mathematical analysis and have many applications; for example, coefficients of differential operators and symbols of more general pseudodifferential operators may be considered as pointwise multipliers. For the theory of pointwise multipliers acting on several function spaces such as Sobolev, Besov, and Triebel-Lizorkin spaces on ℝ^𝑛 we refer to [1]. See also [2–5] for more details.

It was well known that the Fourier transform is a crucial tool to study pointwise multipliers on ℝ^𝑛. However, it was not clear how to generalize pointwise multipliers on ℝ^𝑛 to spaces of homogeneous type introduced by Coifman and Weiss [6] because the Fourier transform is no longer available on spaces of homogenous type. To be more precise, let us first recall briefly these spaces. A quasi-metric 𝑑 on a set 𝑋 is a function 𝑑: 𝑋×𝑋→[0,∞) satisfying that (i) 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦; (ii) 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋; (iii) there exists a constant 𝐴₀∈[1,∞) such that for all 𝑥,𝑦 and 𝑧∈𝑋, 𝑑𝑥,𝑦≤𝐴₀𝑑(𝑥,𝑧)+𝑑𝑧,𝑦. (1) Any quasi-metric defines a topology, for which the balls 𝐵(𝑥,𝑟)=𝑦∈𝑋∶𝑑𝑦,𝑥<𝑟 (2) for all 𝑥∈𝑋 and all 𝑟>0 form a basis. We say that (𝑋,𝑑,𝜇) is a space of homogeneous type in sense of Coifman and Weiss if 𝑑 is a quasi-metric and 𝜇 is a nonnegative Borel regular measure on 𝑋 satisfying the doubling condition; that is, for all 𝑥∈𝑋, 𝑟>0, then 0<𝜇(𝐵(𝑥,𝑟))<∞ and 𝜇(𝐵(𝑥,2𝑟))≤𝐶𝜇(𝐵(𝑥,𝑟)), (3) where 𝜇 is assumed to be defined on a 𝜎-algebra which contains all Borel sets and all balls 𝐵(𝑥,𝑟) and the constant 0<𝐶<∞ is independent of 𝑥∈𝑋 and 𝑟>0. Spaces of homogenous type in the sense of Coifman and Weiss have many applications in analysis. For example, Coifman and Weiss introduced atomic Hardy space 𝐻^𝑝_𝑎𝑡 for 𝑝∈(0,1] in [6, 7] that proved that if 𝑇 is a Calderón-Zygmund singular integral operator and is bounded on 𝐿², then 𝑇 extends a bounded operator from 𝐻^𝑝 to 𝐿^𝑝 for suitable 𝑝≤1. However, note that the quasi-metric, in contrast to a metric, may not be Hölder regular and quasi-metric balls may not be open. For this reason, in many applications, the additional assumptions on the quasi-metric 𝑑 and the measure 𝜇 are required. For instance, in order to provide the maximal function characterization of the Hardy spaces 𝐻^𝑝_𝑎𝑡 on spaces of homogenous type, Macías and Segovia in [8] showed that the quasi-metric 𝑑 can be replaced by another quasi-metric 𝑑 such that the topologies induced on 𝑋 by 𝑑 and 𝑑 coincide, and, moreover, 𝑑 has the following Hölder regularity: there exist positive constant 𝐶 and 0<𝜃<1 such that for all 0<𝑟<∞ and all 𝑥,𝑥^′,𝑦∈𝑋𝑑𝑥,𝑦−𝑑𝑥^′𝑑,𝑦≤𝐶𝑥,𝑥^′𝜃𝑑𝑥,𝑦+𝑑𝑥^′,𝑦^1−𝜃. (4) Furthermore, if balls 𝐵(𝑥,𝑟) are defined by 𝑑, that is, 𝐵(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑦,𝑥)<𝑟}, then 𝜇(𝐵(𝑥,𝑟))∼𝑟. (5) Macías and Segovia provided the maximal function characterization of the Hardy spaces 𝐻^𝑝(𝑋) for (1+𝜃)⁻¹<𝑝≤1, on spaces of homogeneous type (𝑋,𝑑,𝜇) with the regularity condition (4) on 𝑑 and property (5) on the measure 𝜇.

A fundamental result for these spaces (𝑋,𝑑,𝜇) is the 𝑇(𝑏) theorem of David-Journé-Semmes [9], where 𝑑 and 𝜇 satisfy (4) and (5), respectively. The crucial tool in the proof of the 𝑇(𝑏) theorem is the existence of a suitable approximation to the identity. The construction of such an approximation to the identity is due to Coifman. We would like to point out that for Coifman’s construction the additional assumptions (4) on 𝑑 and (5) on 𝜇 are crucial. Later, based on the conditions in (4) and (5), the Calderón reproducing formula, test function spaces and distributions, the Littlewood-Paley theory, and function spaces on (𝑋,𝑑,𝜇) were developed in [10–12].

In [13], Nagel and Stein developed the product 𝐿^𝑝 (1<𝑝<∞) theory in the setting of the Carnot-Carathéodory spaces formed by vector fields satisfying Hörmander’s finite rank condition. The particular Carnot-Carathéodory spaces studied in [13] are spaces of homogeneous type with a smooth quasi-metric 𝑑 and a measure 𝜇 satisfying the conditions 𝜇(𝐵(𝑥,𝑠𝑟))∼𝑠^𝑚+2𝜇(𝐵(𝑥,𝑟)) (6) for 𝑠≥1 and 𝜇(𝐵(𝑥,𝑠𝑟))∼𝑠⁴𝜇(𝐵(𝑥,𝑟)) (7) for 𝑠≤1.

These conditions on the measure are weaker than property in (5) but are still stronger than the original doubling condition in (3).

Recently, pointwise multiplier theorems of Besov and Triebel-lizorkin spaces were obtained by the first author on spaces of homogeneous type with the additional assumptions (1.3) and (1.4) in [14] and with the conditions (1.3) and (1.5) in [15, 16].

A natural question arises: whether pointwise multipliers still hold on spaces of homogeneous type in the sense of Coifman and Weiss with only the original quasi-metric and a doubling measure?

Very recently, Auscher and Hytönen constructed an orthonormal basis with Hölder regularity and exponential decay on spaces of homogeneous type [17]. This result is remarkable since there are no additional assumptions other than those defining spaces of homogeneous type in the sense of Coifman and Weiss. Motivated by Auscher and Hytönen’s orthonormal basis on spaces of homogeneous type, the purpose of the current paper is to answer the above question. More precisely, in this paper, we will provide pointwise multipliers on spaces of homogeneous type in the sense of Coifman and Weiss with the original quasi-metric 𝑑 and doubling measure 𝜇.

The main tool used in this paper is the orthonormal basis constructed by Auscher and Hytönen [17]. We now briefly recall the orthonormal basis constructed in [17] and inhomogeneous Besov and Triebel-Lizorkin spaces obtained in [18] on spaces of homogeneous type in the sense of Coifman and Weiss.

The orthonormal basis of 𝐿²(𝑋) constructed by Auscher and Hytönen [17] is given by the following.

Theorem 1 (see [17] Theorem 7.1). Let (𝑋,𝑑,𝜇) be a space of homogeneous type in the sense of Coifman and Weiss. There exists an orthonormal basis 𝜓^𝑘_𝛼, 𝑘∈ℤ, 𝑦^𝑘_𝛼∈𝒴^𝑘, of 𝐿²(𝑋), having exponential decay 𝜓^𝑘_𝛼(𝑥≤𝐶)𝜇𝐵𝑦^𝑘_𝛼,𝛿^𝑘𝑑𝑦exp−𝜈^𝑘𝛼,𝑥𝛿^𝑘𝑎, (8) Hölder-regularity 𝜓^𝑘_𝛼(𝑥)−𝜓^𝑘_𝛼𝑦≤𝐶𝜇𝐵𝑦^𝑘_𝛼,𝛿^𝑘𝑑𝑥,𝑦𝛿^𝑘𝜂𝑑𝑦exp−𝜈^𝑘𝛼,𝑥𝛿^𝑘𝑎 (9) for some 𝜂∈(0,1) and for 𝑑(𝑥,𝑦)≤𝛿^𝑘, and the cancellation property _𝑋𝜓^𝑘_𝛼(𝑥)𝑑𝜇(𝑥)=0,𝑘∈ℤ,𝑦^𝑘_𝛼∈𝒴^𝑘. (10) Moreover, 𝑓(𝑥)=_{𝑘∈ℤ𝛼∈𝒴𝑘}𝑓,𝜓^𝑘_𝛼𝜓^𝑘_𝛼(𝑥) (11) in the sense of 𝐿²(𝑋).

Here 𝑎=(1+2log2^𝐴0)⁻¹,𝛿 is a fixed small parameter, say 𝛿≤10⁻³𝐴₀⁻¹⁰, and 𝜈>0 and 𝐶<∞ are constants independent of 𝑘, 𝛼, 𝑥, and 𝑦^𝑘_𝛼; see [17] for more details. In what follows, we also refer to the functions 𝜓^𝑘_𝛼 as wavelets.

To develop function spaces such as the Hardy, Besov and Triebel-Lizorkin spaces, the key point is to introduce test function and distributions spaces. For this purpose, the following definitions were introduced in [18, 19].

Definition 2. For fixed 𝑥₀∈𝑋, 𝑟>0, 𝛾,𝛽∈(0,𝜂], where 𝜂 is given in Theorem 1. A function 𝑓 is said to be a test function of type (𝑥₀,𝑟,𝛽,𝛾) centered at 𝑥₀∈𝑋 with width 𝑟 if 𝑓 satisfies the following decay and Hölder regularity properties.(i)For all 𝑥∈𝑋, 𝑓(𝑥1)≤𝐶𝑉_𝑟𝑥₀+𝑉𝑥,𝑥₀𝑟𝑟+𝑑𝑥,𝑥₀^𝛾. (12)(ii)For all 𝑥,𝑦∈𝑋 with 𝑑(𝑥,𝑦)≤(1/2𝐴₀)(𝑟+𝑑(𝑥,𝑥₀)), 𝑓(𝑥)𝑦𝑑−𝑓≤𝐶𝑥,𝑦𝑟+𝑑𝑥,𝑥₀^𝛽1𝑉_𝑟𝑥₀+𝑉𝑥,𝑥₀𝑟𝑟+𝑑𝑥,𝑥₀^𝛾. (13)If 𝑓 is a test function of type (𝑥₀,𝑟,𝛽,𝛾) centered at 𝑥₀∈𝑋 with width 𝑟>0, we write 𝑓∈𝒢(𝑥₀,𝑟,𝛽,𝛾). The norm of 𝑓 on 𝒢(𝑥₀,𝑟,𝛽,𝛾) is defined by 𝑓_{𝒢(𝑥0,𝑟,𝛽,𝛾)}=inf{𝐶>0∶(i)and(ii)hold}. (14)

We denote by 𝒢(𝛽,𝛾) the class of all 𝑓∈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 𝜆∈(0,𝜂], let 𝒢(𝛽,𝛾) be the completion of the space 𝒢(𝜆,𝜆) in 𝒢(𝛽,𝛾) with 0<𝛽, 𝛾≤𝜆. Obviously, 𝒢(𝜆,𝜆)=𝒢(𝜆,𝜆). Moreover, 𝑓∈𝒢(𝛽,𝛾) if and only if 𝑓∈𝒢(𝛽,𝛾) with 0<𝛽, 𝛾≤𝜆 and there exists {𝑓_𝑗}_𝑗∈ℕ⊂𝒢(𝜆,𝜆) such that ‖𝑓−𝑓_𝑗‖_{𝒢(𝛽,𝛾)}→0 as 𝑗→∞. If 𝑓∈𝒢(𝛽,𝛾), we define ‖𝑓‖_{𝒢(𝛽,𝛾)}=‖𝑓‖_{𝒢(𝛽,𝛾)}. Obviously, 𝒢(𝛽,𝛾) is a Banach space and we also have ‖𝑓‖_{𝒢(𝛽,𝛾)}=lim_𝑗→∞‖𝑓_𝑗‖_{𝒢(𝛽,𝛾)} for the above chosen {𝑓_𝑗}_𝑗∈ℕ.

We denote by (𝒢(𝛽,𝛾))^′ the dual space of 𝒢(𝛽,𝛾) consisting of all linear functional £ from 𝒢(𝛽,𝛾) to ℂ with the property that there exists a constant 𝐶, for all 𝑓∈𝒢(𝛽,𝛾), £𝑓≤𝐶𝑓_{𝒢(𝛽,𝛾)}. (15)

We denote by ⟨𝑓,ℎ⟩ the natural pairing of elements ℎ∈(𝒢(𝛽,𝛾))^′ and 𝑓∈𝒢(𝛽,𝛾). Since 𝒢(𝑥₁,𝑟,𝛽,𝛾)=𝒢(𝛽,𝛾) with the equivalent norms for all 𝑥₁∈𝑋 and 𝑟>0. Thus, for all ℎ∈(𝒢(𝛽,𝛾))^′, ⟨𝑓,ℎ⟩ is well defined for all 𝑓∈𝒢(𝑥₀,𝑟,𝛽,𝛾) with 𝑥₀∈𝑋 and 𝑟>0.

We now give definitions of inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss. Denote 𝑃₀(𝑥,𝑦)=∑_{𝑘≤−1,𝛼}𝜓^𝑘_𝛼(𝑥)𝜓^𝑘_𝛼(𝑦) and 𝑄_𝑘(𝑥,𝑦)=∑_𝛼𝜓^𝑘_𝛼(𝑥)𝜓^𝑘_𝛼(𝑦). Let 𝑃₀(𝑓)(𝑥)=∫𝑃₀(𝑥,𝑦)𝑓(𝑦)𝑑𝜇(𝑦) and 𝑄_𝑘(𝑓)(𝑥)=∫𝑄_𝑘(𝑥,𝑦)𝑓(𝑦)𝑑𝜇(𝑦).

Definition 3 (see [18]). Let 1<𝑝, 𝑞<∞ and |𝑠|<𝜂. The inhomogeneous Besov space 𝐵_𝑝^𝑠,𝑞(𝑋) is defined by 𝐵_𝑝^𝑠,𝑞(𝑋)=𝑓∈𝒢𝛽,𝛾^′∶𝑓_𝐵𝑝^𝑠,𝑞<∞, (16) where 𝑓_𝐵𝑝^𝑠,𝑞=𝑃₀𝑓_𝑝+_𝑘≥0𝛿^−𝑘𝑠𝑄_𝑘𝑓_𝑝^𝑞1/𝑞. (17)
The inhomogeneous Triebel-Lizorkin space 𝐹_𝑝^𝑠,𝑞(𝑋) is defined by 𝐹_𝑝^𝑠,𝑞(𝑋)=𝑓∈𝒢𝛽,𝛾^′∶𝑓_𝐹𝑝^𝑠,𝑞<∞, (18) where 𝑓_𝐹𝑝^𝑠,𝑞=𝑃₀𝑓_𝑝+_𝑘≥0𝛿^−𝑘𝑠𝑄_𝑘𝑓^𝑞1/𝑞_𝑝. (19)
We would like to point out that on ℝ^𝑛, 𝐻^𝑠_𝑝(ℝ^𝑛)=𝐹_𝑝^𝑠,2(ℝ^𝑛) are the Bessel-potential spaces (Lebesgue spaces, Liouville spaces). If 𝑚=0,1,2,… and 1<𝑝<∞, then 𝑊^𝑚_𝑝(ℝ^𝑛)=𝐻^𝑚_𝑝(ℝ^𝑛)=𝐹_𝑝^𝑚,2(ℝ^𝑛) are the usual Sobolev spaces. If 0<𝑠,1<𝑝<∞ and 1≤𝑞≤∞, then 𝐵_𝑝^𝑠,𝑞(ℝ^𝑛) coincides with the classical Besov spaces (Lipschitz spaces 𝐴_𝑝^𝑠,𝑞(ℝ^𝑛)).

In this paper, we will consider the following.

Definition 4. Suppose that 𝑔 is a given function on 𝑋. Then 𝑔 is called a pointwise multiplier for 𝐵_𝑝^𝑠,𝑞(𝑋) if 𝑓→𝑔𝑓 admits a bounded linear mapping from 𝐵_𝑝^𝑠,𝑞(𝑋) into itself. Similarly, 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 results in this paper are as follows.

Theorem 5. Let |𝑠|<𝜂, 1<𝑝, 𝑞<∞; then 𝑔∈𝒞^𝛼 is a multiplier for 𝐵_𝑝^𝑠,𝑞 with 𝛼>|𝑠|. Moreover, there exists a positive constant 𝐶 such that 𝑔𝑓_𝐵𝑝^𝑠,𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐵𝑝^𝑠,𝑞 (20) for all 𝑔∈𝒞^𝛼 and all 𝑓∈𝐵_𝑝^𝑠,𝑞.

Theorem 6. Let |𝑠|<𝜂, 1<𝑝, 𝑞<∞; then 𝑔∈𝒞^𝛼 is a multiplier for 𝐹_𝑝^𝑠,𝑞 with 𝛼>|𝑠|. Moreover, there exists a positive constant 𝐶 such that 𝑔𝑓_𝐹𝑝^𝑠,𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐹𝑝^𝑠,𝑞 (21) for all 𝑔∈𝒞^𝛼 and all 𝑓∈𝐹_𝑝^𝑠,𝑞.

Here, the Hölder space 𝒞^𝛼(𝑋) is defined as the collection of 𝑓 such that 𝑓_𝒞𝛼=𝑓_∞+sup_𝑥≠𝑦𝑓(𝑥)−𝑓𝑦𝑑𝑥,𝑦^𝛼<∞. (22)

We remark that Theorems 5 and 6 were proved in [4] on ℝ^𝑛 based on Fourier transform. As mentioned before, the Fourier transform on spaces of homogeneous type is not available and hence the idea used in [4] does not work for this more general setting. A recent work on pointwise multipliers of Besov and Triebel-Lizorkin spaces on Carnot-Carathéodory spaces was developed in [14–16]. However, all results in those papers require the additional assumptions on both the quasi-metric 𝑑 and the measure 𝜇. Therefore, results in the present paper extend all results given in [4, 14–16].

Throughout this paper, we use 𝐶 to denote positive constants, whose value may change from one occurrence to the next. For the measure of ball 𝐵(𝑥,𝑟)=∶{𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝑟}, we sometimes use the abbreviations 𝑉_𝑟(𝑥)∶=𝜇(𝐵(𝑥,𝑟)),𝑉𝑥,𝑦∶=𝑉𝑥,𝑑𝑥,𝑦. (23)

A brief description of the contents of this paper is as follows. In Section 2 we prove Theorem 5. The proof of Theorem 6 will be given in Section 3.

2. Proof of Theorem 5

Let 𝑃₀ and 𝑄_𝑘 be orthogonal projections onto 𝑉₀ and 𝑊_𝑘 with 𝑘∈ℕ, respectively. The next lemma gives some estimates on kernels of operators 𝑃₀ and 𝑄_𝑘.

Lemma 7 (see [17, 18]). Let 𝜂 be the Hölder regularity, 𝑘∈ℕ and 𝜖>0. Suppose that 𝑃₀(𝑥,𝑦) and 𝑄_𝑘(𝑥,𝑦) are kernels of 𝑃₀ and 𝑄_𝑘, respectively. Then there exists a constant 𝐶 such that(i)𝑃₀1𝑥,𝑦≤𝐶𝑉₁(𝑥)1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^𝜖, (24)(ii) for 𝑑(𝑦,𝑦^′)≤(1/2𝐴₀)(1+𝑑(𝑥,𝑦)), 𝑃₀𝑥,𝑦−𝑃₀𝑥,𝑦^′𝑑≤𝐶𝑦,𝑦^′1+𝑑𝑥,𝑦^𝜂1𝑉₁(𝑥)1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^𝜖, (25)(iii)_𝑋𝑃₀𝑥,𝑦𝑑𝜇(𝑥)=1, (26)(iv)𝑄_𝑘1𝑥,𝑦≤𝐶𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖, (27)(v)for 𝑑(𝑦,𝑦^′)≤(1/2𝐴₀)(𝛿^𝑘+𝑑(𝑥,𝑦)), 𝑄_𝑘𝑥,𝑦−𝑄_𝑘𝑥,𝑦^′𝑑≤𝐶𝑦,𝑦^′𝛿^𝑘+𝑑𝑥,𝑦^𝜂1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖, (28)(vi)_𝑋𝑄_𝑘𝑥,𝑦𝑑𝜇(𝑥)=0. (29)Note that (ii), (iii), (v), and (vi) still hold with 𝑥 and 𝑦 interchanged.

The key tool used in this paper, as mentioned, is the following version of the wavelet expansion.

Lemma 8 (see [18]). Let 𝑃₀ and {𝑄_𝑘}_𝑘∈ℤ+ be the same as in Definition 3. Then 𝑓=𝑃²₀𝑓+_𝑘≥0𝑄²_𝑘𝑓 (30) holds in 𝐿^𝑝 with 1<𝑝<∞.

For our purpose, we need the following lemmas.

Lemma 9 (see [18]). Let 0<𝛽, 𝛾<𝜂. Then (30) still holds in 𝒢(𝛽,𝛾) and (𝒢(𝛽,𝛾))^′.

Lemma 10. If 𝑓∈𝒢(𝛽,𝛾) with |𝑠|<𝛽<𝜂, 0<𝛾<𝜂, then 𝑓∈𝐵_𝑝^𝑠,𝑞 and 𝑓∈𝐹_𝑝^𝑠,𝑞 with |𝑠|<𝜂, 1<𝑝<∞, 1<𝑞<∞.

Proof. Suppose that 𝑓∈𝒢(𝛽,𝛾) with |𝑠|<𝛽<𝜂, 0<𝛾<𝜂. We claim that _𝑋𝑃₀𝑓(𝑥)(𝑥𝑓𝑥,𝑦𝑑𝜇)≤𝐶_{𝒢(𝛽,𝛾)}1𝑉₁𝑥₀+𝑉𝑦,𝑥₀11+𝑑𝑥₀,𝑦^𝛾 (31) and for 𝑘∈ℕ_𝑋𝑄_𝑘𝑥,𝑦𝑓(𝑥)𝑑𝜇(𝑥)≲𝛿^𝑘𝛽𝑓_{𝒢(𝛽,𝛾)}1𝑉₁𝑥₀+𝑉𝑦,𝑥₀11+𝑑𝑥₀,𝑦^𝛾. (32)
We first verify (31). By the size condition of 𝑃₀ and definition of test functions, we have _𝑋𝑃₀𝑓(𝑥)(𝑥≤𝑥,𝑦𝑑𝜇)_𝑋𝐶𝑓_{𝒢(𝛽,𝛾)}𝑉₁(𝑥)1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^𝛾1𝑉₁𝑥₀+𝑉𝑥,𝑥₀×11+𝑑𝑥,𝑥₀^𝛾(𝑥)𝑓𝑑𝜇≤𝐶_{𝒢(𝛽,𝛾)}1𝑉₁𝑥₀+𝑉𝑦,𝑥₀11+𝑑𝑥₀,𝑦^𝛾. (33)
To estimate (32), by the cancellation condition on 𝑄_𝑘(𝑥,𝑦), we have _𝑋𝑄_𝑘𝑓(𝑥)(𝑥=𝑥,𝑦𝑑𝜇)_𝑋𝑄_𝑘𝑥,𝑦𝑓(𝑥)−𝑓𝑦𝑑𝜇(𝑥)≤_𝑊1𝑄_𝑘𝑓(𝑥)𝑦(𝑥)+𝑥,𝑦−𝑓𝑑𝜇_𝑊2𝑄_𝑘𝑓(𝑥(𝑥)+𝑥,𝑦)𝑑𝜇_𝑊2𝑄_𝑘𝑥,𝑦𝑓𝑦𝑑𝜇(𝑥)∶=𝑅₁+𝑅₂+𝑅₃, (34) where 𝑊₁={𝑥∶𝑑(𝑥,𝑦)≤(1/2𝐴₀)(1+𝑑(𝑥₀,𝑦))} and 𝑊₂=𝑋⧵𝑊₁.
For 𝑅₁, for any 𝜖>0, we have 𝑅₁≤𝐶𝑓_{𝒢(𝛽,𝛾)}×_𝑊11𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖𝑑𝑥,𝑦1+𝑑𝑥₀,𝑦^𝛽×1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑥₀,𝑦^𝛾(𝑥)𝑓𝑑𝜇≤𝐶_{𝒢(𝛽,𝛾)}𝛿^𝛽𝑘1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑥₀,𝑦^𝛾, (35) where 𝜖>𝛽>0.
For 𝑅₂, 𝑑(𝑥,𝑦)≥(1/2𝐴₀)(1+𝑑(𝑥₀,𝑦)) implies 𝑉(𝑥,𝑦)≳𝑉₁(𝑥₀)+𝑉(𝑦,𝑥₀); then 𝑅₂≤𝐶𝑓_{𝒢(𝛽,𝛾)𝑊2}1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖×1𝑉₁𝑥₀+𝑉𝑥₀1,𝑥1+𝑑𝑥₀,𝑥^𝛾(𝑥)𝑓𝑑𝜇≤𝐶_{𝒢(𝛽,𝛾)}𝛿^𝛽𝑘1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑥₀,𝑦^𝛾, (36) where 𝜖>𝛽>0 and 𝜖>𝛾>0.
To estimate 𝑅₃, since 𝑑(𝑥,𝑦)/(𝑑(𝑦,𝑥₀)+1)≳1, then 𝑅₃≤𝐶𝑓_{𝒢(𝛽,𝛾)}×_𝑊21𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖𝑑𝑥,𝑦1+𝑑𝑥₀,𝑦^𝛽×1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑥₀,𝑦^𝛾(𝑥)𝑓𝑑𝜇≤𝐶_{𝒢(𝛽,𝛾)}𝛿^𝛽𝑘1𝑉₁𝑥₀+𝑉𝑥₀1,𝑦1+𝑑𝑥₀,𝑦^𝛾, (37) where 𝜖>𝛽>0. The claim is concluded.
We now return to the proof of Lemma 10 and only prove that 𝒢(𝛽,𝛾)⊂𝐵_𝑝^𝑠,𝑞 since the proof of 𝒢(𝛽,𝛾)⊂𝐹_𝑝^𝑠,𝑞 is similar. By applying (31) and (32), it follows that 𝑓_𝐵𝑝^𝑠,𝑞=𝑃₀𝑓_𝐿𝑝+^∞_𝑘=0𝛿^−𝑘𝑠𝑄_𝑘𝑓_𝐿𝑝^𝑞1/𝑞≤𝐶𝑓_{𝒢(𝛽,𝛾)}+𝐶𝑓_{𝒢(𝛽,𝛾)}^∞_𝑘=0𝛿^{𝑘(𝛽−𝑠)𝑞1/𝑞}≤𝐶𝑓_{𝒢(𝛽,𝛾)}, (38) where 𝛽>𝑠. Thus 𝒢(𝛽,𝛾)⊂𝐵_𝑝^𝑠,𝑞.

Lemma 11. Let 𝑔∈𝒞^𝛼 and 𝑘,𝑙∈ℕ. For 𝜖>𝜂, 𝑃₀𝑔𝑃₀𝑥,𝑦≤𝐶𝑔_𝒞𝛼1𝑉₁(𝑥)1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^𝜖;𝑄_𝑘𝑔𝑄_𝑙𝑥,𝑦≤𝐶𝑔_𝒞𝛼𝛿^{|𝑘−𝑙|(𝛼∧𝜂)}×1𝑉_𝛿^{(𝑘∧𝑙)}(𝑥)𝛿+𝑉𝑥,𝑦𝛿^{(𝑘∧𝑙)}+𝑑^{(𝑘∧𝑙)}𝑥,𝑦^𝜖;𝑃₀𝑔𝑄_𝑙𝑔𝑥,𝑦≤𝐶_𝒞𝛼𝛿^{𝑙(𝛼∧𝜂)}1𝑉₁(𝑥)1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^𝜖,𝑄_𝑘𝑔𝑃₀𝑔𝑥,𝑦≤𝐶_𝒞𝛼𝛿^{𝑘(𝛼∧𝜂)}1𝑉₁(𝑥)1+𝑉𝑥,𝑦1+𝑑𝑥,𝑦^𝜖, (39)(40)(41)(42) where 𝑎∧𝑏 denotes the minimum of 𝑎 and 𝑏.

Proof. We first consider (39). By the size conditions of 𝑃₀ and the definition of Hölder space 𝒞^𝛼, we have 𝑃₀𝑔𝑃₀𝑥,𝑦≤𝐶𝑔_𝒞𝛼×_𝑋1𝑉₁(𝑥)()1+𝑉𝑥,𝑧1+𝑑(𝑧,𝑥)^𝜖×1𝑉₁𝑦1+𝑉𝑦,𝑧1+𝑑𝑧,𝑦^𝜖(𝑧)1𝑑𝜇≤𝐶𝑉₁𝑦1+𝑉𝑦,𝑥1+𝑑𝑥,𝑦^𝜖. (43)
For (40), we only consider that the case for 𝑙≥𝑘≥1 and the proof for 𝑘≥𝑙≥1 are similar. In fact, if 𝑙≥𝑘≥1, we have 𝑄_𝑘𝑔𝑄_𝑙=𝑥,𝑦_𝑋𝑄_𝑘(𝑥,𝑧)𝑔(𝑧)−𝑄_𝑘𝑥,𝑦𝑔𝑦𝑄_𝑙(𝑧≤𝑧,𝑦𝑑𝜇)_𝑋𝑄_𝑘(𝑥,𝑧)−𝑄_𝑘𝑥,𝑦𝑔(𝑧)𝑄_𝑙(𝑧)+𝑧,𝑦𝑑𝜇_𝑋𝑄_𝑘𝑥,𝑦𝑔(𝑧)−𝑔𝑦𝑄_𝑙𝑧,𝑦𝑑𝜇(𝑧)=∶𝐿₁+𝐿₂. (44)
We estimate 𝐿₁ by further splitting it into 𝐿₁=_𝑊1𝑄_𝑘(𝑥,𝑧)−𝑄_𝑘𝑥,𝑦𝑔(𝑧)𝑄_𝑙𝑧,𝑦𝑑𝜇(𝑧)+_𝑊2𝑄_𝑘(𝑥,𝑧)𝑔(𝑧)𝑄_𝑙(𝑧)+𝑧,𝑦𝑑𝜇_𝑊2𝑄_𝑘𝑥,𝑦𝑔(𝑧)𝑄_𝑙𝑧,𝑦𝑑𝜇(𝑧)=∶𝐿₁₁+𝐿₁₂+𝐿₁₃, (45) where 𝑊₁={𝑧∈𝑋∶𝑑(𝑧,𝑦)≤(1/2𝐴₀)(𝛿^𝑘+𝑑(𝑥,𝑦))} and 𝑊₂=𝑋⧵𝑊₁.
For 𝐿₁₁, for any 𝜖>0, we have 𝐿₁₁≤𝐶𝑔_𝒞𝛼×_𝑊1𝑑𝑧,𝑦𝛿^𝑘+𝑑𝑥,𝑦^𝜂1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖×1𝑉_𝛿^𝑙𝑦𝛿+𝑉𝑦,𝑧𝛿^𝑙+𝑑^𝑙𝑧,𝑦^𝜖(𝑧)𝑔𝑑𝜇≤𝐶_𝒞𝛼𝛿^{(𝑙−𝑘)𝜂}1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖×_𝑋1𝑉_𝛿^𝑙𝑦𝛿+𝑉𝑦,𝑧𝛿^𝑙+𝑑^𝑙𝑧,𝑦^𝜖−𝜂(𝑧)𝑔𝑑𝜇≤𝐶_𝒞𝛼𝛿^{(𝑙−𝑘)𝜂}1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖, (46) where 𝜖>𝜂.
For 𝐿₁₂, note that (1/2𝐴₀)(𝛿^𝑘+𝑑(𝑥,𝑦))≤𝑑(𝑧,𝑦) implies that 𝑉_𝛿^𝑘(𝑦)+𝑉(𝑥,𝑦)≲𝑉(𝑧,𝑦); then we have 𝐿₁₂≤𝐶𝑔_𝒞𝛼1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑙𝑥,𝑦^𝜖×_𝑋1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑦,𝑥𝛿^𝑘+𝑑^𝑘(𝑧,𝑥)^𝜖(𝑧)𝑔𝑑𝜇≤𝐶_𝒞𝛼𝛿^{(𝑙−𝑘)𝜂}1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖, (47) where 𝜖>𝜂.
For 𝐿₁₃, we have 𝐿₁₃≤𝐶𝑔_𝒞𝛼1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖×_𝑊21𝑉_𝛿^𝑙𝑦𝛿+𝑉𝑦,𝑧𝛿^𝑙+𝑑^𝑙𝑧,𝑦^𝜖𝑑𝜇(𝑧). (48) Denoting 𝑡=(1/2𝐴₀)(𝛿^𝑘+𝑑(𝑥,𝑦)), then _𝑊21𝑉_𝛿^𝑙𝑦𝛿+𝑉𝑦,𝑧𝛿^𝑙+𝑑^𝑙𝑧,𝑦^𝜖(𝑧)=𝑑𝜇_{∞𝑗=02𝑗𝑡<𝑑(𝑧,𝑦)≤2𝑗+1𝑡}1𝑉_𝛿^𝑙𝑦×𝛿+𝑉𝑦,𝑧_𝑙𝛿^𝑙+𝑑𝑧,𝑦^𝜖(𝑧)≤𝑑𝜇_∞𝑗=0𝛿^𝑙2^𝑗𝑡^𝜖1𝑉₂^𝑗_𝑡𝑦_{𝑑(𝑧,𝑦)≤2}^𝑗+1_𝑡𝑑𝜇(𝑧)≤𝐶^∞_𝑗=0𝛿^𝑙𝛿^𝑘𝜖𝑉₂^𝑗+1_𝑡𝑦𝑉₂^𝑗_𝑡𝑦≤𝐶𝛿^{(𝑙−𝑘)𝜖}. (49) Thus 𝐿₁₃≤𝐶𝑔_𝒞𝛼𝛿^{(𝑙−𝑘)𝜂}1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖, (50) where 𝜖>𝜂 and we obtain 𝐿₁≤𝐶𝑔_𝒞𝛼𝛿^{(𝑙−𝑘)𝜂}1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖. (51)
We now return to verify 𝐿₂. By the size condition of 𝑄_𝑘, 𝑄_𝑙 and the definition of 𝒞^𝛼, we have 𝐿₂≤𝐶𝑔_𝒞𝛼1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖×_𝑋𝑑𝑧,𝑦^𝛼1𝑉_𝛿^𝑙𝑦𝛿+𝑉𝑧,𝑦𝛿^𝑙+𝑑^𝑙𝑧,𝑦^𝜖(𝑥)𝑔𝑑𝜇≤𝐶_𝒞𝛼𝛿^𝑙𝛼1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖, (52) where 𝜖>𝛼. Then 𝐿₁+𝐿₂≤𝐶𝑔_𝒞𝛼𝛿^{(𝑙−𝑘)(𝛼∧𝜂)}1𝑉_𝛿^𝑘(𝑥)𝛿+𝑉𝑥,𝑦𝛿^𝑘+𝑑^𝑘𝑥,𝑦^𝜖, (53) where 𝑙≥𝑘.
The proofs of (41) and (42) are similar to the proof of (40) and we omit the details.

We are now ready to prove Theorem 5.

Proof of Theorem 5. We first show Theorem 5 for the special case; that is, if 𝑓∈𝒢(𝛽,𝛾) with |𝑠|<𝛽<𝜂, 0<𝛾<𝜂 and 𝑔∈𝒞^𝛼 with |𝑠|<𝛼, then 𝑓𝑔_𝐵𝑝^𝑠,𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐵𝑝^𝑠,𝑞. (54) To verify (54), we write 𝑓𝑔_𝐵𝑝^𝑠,𝑞=𝑃₀𝑔𝑓_𝐿𝑝+^∞_𝑘=0𝛿^−𝑘𝑠𝑄_𝑘𝑔𝑓_𝐿𝑝^𝑞1/𝑞=∶𝑌₁+𝑌₂. (55) By the wavelet expansion, Hölder’s inequality, and the estimates in (39) and (41), we obtain 𝑌₁≤𝑃₀𝑔𝑃²₀𝑓+^∞_𝑙=0𝑄²_𝑙𝑓_𝐿𝑝≤𝑃₀𝑔𝑃₀𝑃₀𝑓_𝐿𝑝+^∞_𝑙=0𝑃₀𝑔𝑄_𝑙𝑄_𝑙𝑓_𝐿𝑝≤𝐶𝑔_𝒞𝛼𝐸₀𝑓_𝐿𝑝+𝐶𝑔_𝒞𝛼×^∞_𝑙=0𝛿^{𝑙(𝛼∧𝜂+𝑠)}𝛿^−𝑙𝑠𝐷_𝑙(𝑓)_𝐿𝑝≤𝐶𝑔_𝒞𝛼𝐸₀𝑓_𝐿𝑝+𝐶𝑔_𝒞𝛼×^∞_𝑙=0𝛿^−𝑙𝑠𝐷_𝑙𝑓_𝐿𝑝^𝑞1/𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐵𝑝^𝑠,𝑞, (56) where we use the fact that (𝛼∧𝜂)+𝑠>0 and in the third inequality, by (39) and (41), we use the estimates ∫|𝑃₀𝑔𝑃₀(𝑥,𝑦)|𝑑𝜇(𝑥)≤𝐶‖𝑔‖_𝒞^𝛼, ∫|𝑃₀𝑔𝑃₀(𝑥,𝑦)|𝑑𝜇(𝑦)≤𝐶‖𝑔‖_𝒞^𝛼,∫|𝑃₀𝑔𝑄_𝑙(𝑥,𝑦)|𝑑𝜇(𝑥)≤𝐶‖𝑔‖_𝒞^𝛼𝛿^{𝑙(𝛼∧𝜂)} and ∫|𝑃₀𝑔𝑄_𝑙(𝑥,𝑦)|𝑑𝜇(𝑦)≤𝐶‖𝑔‖_𝒞^𝛼𝛿^{𝑙(𝛼∧𝜂)}.
For 𝑌₂, instead of using (40) and (42), we have 𝑌₂≤^∞_𝑘=0𝛿^−𝑘𝑠𝑄_𝑘𝑔𝑃²₀𝑓+^∞_𝑙=0𝑄²_𝑙𝑓_𝐿𝑝^𝑞1/𝑞≤𝐶^∞_𝑘=0𝛿^−𝑘𝑠𝛿^{𝑘(𝛼∧𝜂)}𝑃₀𝑓_𝐿𝑝^𝑞1/𝑞+𝐶^∞_𝑘=0𝛿^−𝑘𝑠𝑄_𝑘𝑔𝑄_𝑙𝑄_𝑙𝑓_𝐿𝑝^𝑞1/𝑞≤𝐶𝑔_𝒞𝛼𝑃₀𝑓_𝐿𝑝+𝐶^∞_𝑘=0^∞_𝑙=0𝛿^−𝑘𝑠𝛿^{|𝑘−𝑙|(𝛼∧𝜂)}𝑔_𝒞𝛼𝑄_𝑙𝑓_𝐿𝑝^𝑞1/𝑞≤𝐶𝑔_𝒞𝛼𝑃₀𝑓_𝐿𝑝+𝐶𝑔_𝒞𝛼^∞∞_𝑙=0𝑘=0𝛿^{(𝑙−𝑘)𝑠}𝛿^{|𝑘−𝑙|(𝛼∧𝜂)}𝛿^−𝑙𝑠𝑄_𝑙𝑓_𝐿𝑝^𝑞1/𝑞≤𝐶𝑔_𝒞𝛼𝑃₀𝑓_𝐿𝑝+𝐶𝑔_𝒞𝛼^∞_𝑙=0𝛿^−𝑙𝑠𝑄_𝑙𝑓_𝐿𝑝^𝑞1/𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐵𝑝^𝑠,𝑞, (57) where |𝑠|<𝜂 and |𝑠|<𝛼. Thus, 𝑌₁+𝑌₂≤𝐶𝑔_𝒞𝛼𝑓_𝐵𝑝^𝑠,𝑞. (58) This completes the proof of (54).
To show Theorem 5 for 𝑓∈𝐵_𝑝^𝑠,𝑞, note that if 𝑓∈𝐵_𝑝^𝑠,𝑞, in general, 𝑓 could be a distribution and the multiplication of 𝑔𝑓 is not well defined even for 𝑔∈𝒞^𝛼. For this purpose, we make the following observation: for any 𝑓∈𝐵_𝑝^𝑠,𝑞 with 1<𝑝, 𝑞<∞, |𝑠|<𝜂 and 𝑔∈𝒞^𝛼 with 𝛼>|𝑠|, there exists a sequence {𝑓_𝑛}_𝑛∈ℕ such that 𝑓_𝑗∈𝒢(𝜆,𝜆) with 0<𝜆<𝜂, ‖𝑓_𝑛‖_𝐵𝑝^𝑠,𝑞≲‖𝑓‖_𝐵𝑝^𝑠,𝑞 and lim_𝑛→∞⟨𝑔𝑓_𝑛,ℎ⟩ converges for any ℎ∈𝒢(𝛽,𝛾) with 𝛽,𝛾 satisfying |𝑠|<𝛽<𝜂, 0<𝛾<𝜂. Indeed, for any 𝑓∈𝐵_𝑝^𝑠,𝑞 with 1<𝑝, 𝑞<∞, |𝑠|<𝜂, set 𝑓_𝑛=^𝑛_𝑘=−1𝐵²_𝑘𝑓, (59) where 𝐵₋₁=𝑃₀, 𝐵_𝑘=𝑄_𝑘 for 𝑘∈ℤ₊. By the Proposition 4.4 of [18], 𝑓_𝑛∈𝒢(𝜆,𝜆), and ‖𝑓_𝑛‖_𝐵𝑝^𝑠,𝑞≲‖𝑓‖_𝐵𝑝^𝑠,𝑞. Now we prove that lim_𝑛→∞⟨𝑔𝑓_𝑛,ℎ⟩ converges for any ℎ∈𝒢(𝛽,𝛾) with 𝛽,𝛾 satisfying |𝑠|<𝛽<𝜂, 0<𝛾<𝜂. To do this, for 𝑛,𝑚∈ℕ₊, 𝑚<𝑛, by duality in [18] and the estimate in (54), we have 𝑓_𝑛−𝑓_𝑚,𝑔ℎ≤𝑓_𝑛−𝑓_𝑚𝐵𝑝^𝑠,𝑞𝑔ℎ_{𝐵′−𝑠,𝑞𝑝′}≤𝐶𝑔_𝒞𝛼𝑓_𝑛−𝑓_𝑚𝐵𝑝^𝑠,𝑞‖ℎ‖_{𝐵′−𝑠,𝑞𝑝′}. (60) Note that ‖ℎ‖_{𝐵′−𝑠,𝑞𝑝′}≤𝐶‖ℎ‖_{𝒢(𝛽,𝛾)} and ‖𝑓_𝑛−𝑓_𝑚‖_𝐵𝑝^𝑠,𝑞 tend to zero as 𝑛,𝑚 tend to infinity. This implies that |⟨𝑓_𝑛−𝑓_𝑚,𝑔ℎ⟩|→0 as 𝑛,𝑚→∞ with 1<𝑝, 𝑞<∞, |𝑠|<𝜂.
Now for any 𝑔∈𝒞^𝛼 with |𝑠|<𝛼<𝜂 and 𝑓∈𝐵_𝑝^𝑠,𝑞 with 1<𝑝, 𝑞<∞, |𝑠|<𝜂, by the above observation, lim_𝑛→∞⟨𝑔𝑓_𝑛,ℎ⟩ exists. Therefore, we define 𝑔𝑓,ℎ=lim_𝑛→∞𝑔𝑓_𝑛,ℎ (61) for ℎ∈𝒢(𝛽,𝛾) with 𝛽,𝛾 satisfying |𝑠|<𝛽<𝜂, 0<𝛾<𝜂. It is easy to see that limit is independent of the choice of 𝑓_𝑛. By Fatou’s lemma and (54), we have 𝑔𝑓_𝐵𝑝^𝑠,𝑞≤lim_𝑛→∞inf𝑔𝑓_𝑛𝐵𝑝^𝑠,𝑞≤lim_𝑛→∞inf𝐶𝑔_𝒞𝛼𝑓_𝑛𝐵𝑝^𝑠,𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐵𝑝^𝑠,𝑞, (62) which gives the proof of Theorem 5.

3. Proof of Theorem 6

We first prove the following technical version of Theorem 6.

Lemma 12. For any 𝑔∈𝒞^𝛼, 𝑓∈𝒢(𝛽,𝛾) when 0<𝛽, 𝛾<𝜂, then 𝑓𝑔_𝐹𝑝^𝑠,𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐹𝑝^𝑠,𝑞, (63) where 1<𝑝<∞, 1<𝑞<∞, −𝜂<𝑠<𝜂, and −𝛼<𝑠<𝛼.

Proof. Applying the wavelet expansion, for any 𝑔∈𝒞^𝛼, 𝑓∈𝒢(𝛽,𝛾) when 0<𝛽, 𝛾<𝜂, we have 𝑓𝑔_𝐹𝑝^𝑠,𝑞≤𝑃₀𝑔𝑃²₀𝑓_𝐿𝑝+𝑃₀𝑔^∞_𝑙=0𝑄²_𝑙𝑓_𝐿𝑝+^∞_𝑘=0𝛿^−𝑘𝑠𝑄_𝑘𝑔𝑃²₀𝑓^𝑞1/𝑞_𝐿𝑝+^∞_𝑘=0𝛿^−𝑘𝑠𝑄_𝑘𝑔^∞_𝑙=0𝑄²_𝑙𝑓^𝑞1/𝑞_𝐿𝑝=∶𝑍₁+𝑍₂+𝑍₃+𝑍₄. (64)
The estimate of 𝑍₁ is the same as in the proof of Theorem 5. We only estimate 𝑍₂, 𝑍₃, and 𝑍₄. By applying the inequality (39)–(42), the Hölder inequality and the Fefferman-stein vector-valued maximal function inequality for 1<𝑝<∞, 1<𝑞<∞ in [20], it follows that 𝑍₂≤𝐶𝑔_𝒞𝛼^∞_𝑙=0𝛿^{𝑙((𝛼∧𝜂)+𝑠)}𝛿^−𝑙𝑠𝑀𝑄_𝑙𝑓_𝐿𝑝≤𝐶𝑔_𝒞𝛼^∞_𝑙=0𝛿^−𝑙𝑠𝑄_𝑙𝑓^𝑞1/𝑞_𝐿𝑝≤𝐶𝑔_𝒞𝛼𝑓_𝐹𝑝^𝑠,𝑞;𝑍₃≤𝐶𝑔_𝒞𝛼^∞_𝑘=0𝛿^{𝑘((𝛼∧𝜂)−𝑠)}𝑀𝑃₀𝑓^𝑞1/𝑞_𝐿𝑝≤𝐶𝑔_𝒞𝛼𝑃₀𝑓_𝐿𝑝≤𝐶𝑔_𝒞𝛼𝑓_𝐹𝑝^𝑠,𝑞;𝑍₄≤𝐶𝑔_𝒞𝛼^∞_𝑘=0𝛿^{∞−𝑘𝑠}_𝑙=0𝛿^{−|𝑘−𝑙|(𝜂∧𝛼)}𝑀𝑄_𝑙𝑓^𝑞1/𝑞_𝐿𝑝≤𝐶𝑔_𝒞𝛼^∞_𝑘=0^∞_𝑙=0𝛿^{−(𝑘−𝑙)𝑠}𝛿^{|𝑘−𝑙|(𝜂∧𝛼)𝑞/𝑞′}×^∞_𝑙=0𝛿^{−(𝑘−𝑙)𝑠}𝛿^{|𝑘−𝑙|(𝜂∧𝛼)}𝛿^−𝑙𝑠𝑀𝑄_𝑙𝑓^𝑞1/𝑞_𝐿𝑝≤𝐶𝑔_𝒞𝛼^∞_𝑙=0𝛿^−𝑘𝑠𝑀𝐷_𝑙𝑓^𝑞1/𝑞_𝐿𝑝≤𝐶𝑔_𝒞𝛼𝑓_𝐹𝑝^𝑠,𝑞, (65) where we use the fact that −(𝜂∧𝛼)<𝑠<(𝜂∧𝛼). This verifies Lemma 12.

To show Theorem 6, we also need the following technical lemma.

Lemma 13 (see [18]). For any 𝑓∈𝐹_𝑝^𝑠,𝑞, there exists a sequence {𝑓_𝑛}_𝑛∈ℤ+∈𝒢(𝛽,𝛾) for 0<𝛽,𝛾<𝜂, such that 𝑓_𝑛→𝑓 in 𝐹_𝑝^𝑠,𝑞 with ‖𝑓_𝑛‖_𝐹𝑝^𝑠,𝑞≤𝐶‖𝑓‖_𝐹𝑝^𝑠,𝑞, where 1<𝑝<∞, 1<𝑞<∞, −𝜂<𝑠<𝜂.

Suppose that 𝑓,𝑓_𝑛 are given as in Lemma 13 and 𝑔∈𝒞^𝛼. Note that if |𝑠|<𝛽<𝜂, 0<𝛾<𝜂, then ℎ∈𝒢(𝛽,𝛾)⊂𝐹_𝑝^𝑠,𝑞∩𝐹^{−𝑠,𝑞′}_𝑝′. By duality given in [18] and Lemma 12, it follows that 𝑔𝑓_𝑗−𝑓_𝑘,ℎ≤𝑔𝑓_𝑗−𝑓_{𝑘𝐹𝑝𝑠,𝑞}‖ℎ‖_{𝐹′𝑝′−𝑠,𝑞}≤𝑔_𝒞𝛼𝑓_𝑗−𝑓_{𝑘𝐹𝑝𝑠,𝑞}‖ℎ‖_{𝒢(𝛽,𝛾)}⟶0as𝑗,𝑘→∞. (66)

The above estimate implies that lim_𝑛→∞⟨𝑔𝑓_𝑛,ℎ⟩ exists and the limit is independent of the choice of 𝑓_𝑛. Therefore, for 𝑔∈𝒞^𝛼, 𝑓∈𝐹_𝑝^𝑠,𝑞 we define 𝑔𝑓,ℎ=lim_𝑛→∞𝑔𝑓_𝑛,ℎ, (67) where ℎ∈𝒢(𝛽,𝛾) for 0<𝛽, 𝛾<𝜂 and 𝑓_𝑛 is a sequence defined in Lemma 13.

We now apply Fatou’s lemma and Lemma 13 to show Theorem 6.

Proof of Theorem 6. For any 𝑔∈𝒞^𝛼, 𝑓∈𝐹_𝑝^𝑠,𝑞, applying Fatou’s lemma and Lemma 13 implies that 𝑔𝑓_𝐹𝑝^𝑠,𝑞≤^∞_𝑘=0𝛿^−𝑘𝑠lim_𝑛→∞𝐵_𝑘𝑔𝑓_𝑛^𝑞1/𝑞_𝑝≤liminf_𝑛→∞𝑔𝑓_𝑛𝐹𝑝^𝑠,𝑞≤𝐶liminf_𝑛→∞𝑔_𝒞𝛼𝑓_𝑛𝐹𝑝^𝑠,𝑞≤𝐶𝑔_𝒞𝛼𝑓_𝐹𝑝^𝑠,𝑞, (68) where, as before, 𝐵₋₁=𝑃₀, 𝐵_𝑘=𝑄_𝑘 for 𝑘∈ℤ₊.
The proof of Theorem 6 is concluded.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The second and third authors are supported by NNSF-China (Grant no. 11171345), the Doctoral Fund of Ministry of Education of China (Grant no. 20120023110003). The second author is also supported by China Scholarship Council. The third author is also supported by NNSF-China (Grant no. 51234005).