Wavelets, Sobolev Multipliers, and Application to Schrödinger Type Operators with Nonsmooth Potentials

Abstract

We employ Meyer wavelets to characterize multiplier space 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) without using capacity. Further, we introduce logarithmic Morrey spaces 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) to establish the inclusion relation between Morrey spaces and multiplier spaces. By fractal skills, we construct a counterexample to show that the scope of the index 𝜏 of 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) is sharp. As an application, we consider a Schrödinger type operator with potentials in 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛).

1. Introduction

As useful tools, multipliers on the spaces of differential functions are applied to the study of various problems in harmonic analysis and differential equations. A function 𝑔 is called a multiplier from a Sobolev space 𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛) to another Sobolev space 𝐻^𝑡,𝑝(ℝ^𝑛) if for every function 𝑓∈𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛), the product 𝑓𝑔∈𝐻^𝑡,𝑝(ℝ^𝑛). We denote by 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) the class of all such functions 𝑔. In the famous book [1], Maz'ya and Shaposhnikova gave many characterizations of different kinds of multiplier spaces. For different indices 𝑟, 𝑡, and 𝑝, the multiplier spaces can adapt to the needs of different problems. Further, for different indices 𝑟, 𝑡, and 𝑝, they need different skills to deal with different multiplier spaces. In this paper, we consider multiplier spaces 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) defined as follows.

Definition 1 (see [1]). Given 𝑡≥0, 𝑟>0 and 1<𝑝<𝑛/(𝑟+𝑡), the multiplier space 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) is defined as the set of all the functions 𝑓(𝑥) such that 𝑓_{𝑋𝑡𝑟,𝑝(ℝ𝑛)}∶=sup_{𝑔𝐻𝑛)𝑡+𝑟,𝑝(ℝ≤1}𝑓𝑔_{𝐻𝑡,𝑝(ℝ𝑛)}<∞. (1)

For a compact set 𝑒⊂ℝ^𝑛, the capacity cap(𝑒,𝐻^𝑡,𝑝) is defined by cap𝑒,𝐻^𝑡,𝑝=inf‖𝑢‖^𝑝_{𝐻𝑡,𝑝(ℝ𝑛)}∶𝑢∈𝒮,𝑢≥1on𝑒, (2) where 𝒮 denotes the Schwartz class of rapidly decreasing smooth functions on ℝ^𝑛.

Lemma 2 (see [1]). Given 𝑟>0 and 𝑡≥0. (i)For 1<𝑝<𝑛/(𝑟+𝑡), 𝑓∈𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) if and only if sup_{𝑒⊂ℝ𝑛}(−Δ)^𝑡/2𝑓_{𝐿𝑝(𝑒)}𝑐𝑎𝑝𝑒,𝐻^{𝑡+𝑟,𝑝1/𝑝}+𝑓_{𝐿𝑝(𝑒)}𝑐𝑎𝑝𝑒,𝐻^{𝑟,𝑝1/𝑝}<∞. (3)(ii)For 1<𝑝<𝑛/𝑟 and any cube 𝑄 with a length less than 1, the capacity 𝑐𝑎𝑝(𝑄,𝐻^𝑟,𝑝) is less than 𝐶|𝑄|^{1−𝑝𝑟/𝑛}.

Our motivation is based on the following consideration. For complicated compact sets, it is very difficult to compute the capacity. The main aim of this paper is to establish a relation between 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) and the Morrey spaces 𝑀^𝑡_𝑟,𝑝(ℝ^𝑛), 𝑡≥0, and 𝑟>0 by wavelets. See also Liang et al. [2, 3], Triebel [4], and Yuan et al. [5] for further information on wavelet characterization of Morrey spaces. The special case 𝑡=0 has been studied by Yang [6] and Yang and Zhou [7]. We point out that the result for 𝑡>0 is not a simple generalization. For 𝑡=0, the Sobolev space 𝐻^0,𝑝(ℝ^𝑛) becomes Lebesgue space 𝐿^𝑝(ℝ^𝑛). It is well known that 𝑓_𝐿𝑝∼𝑀𝑓_𝐿𝑝,1<𝑝<∞, (4) where 𝑀 is the Hardy-Littlewood maximal operator. Yang and Zhou [7] used this equivalence to characterize 𝑋⁰_𝑟,𝑝(ℝ^𝑛). See also Yang [6]. However, (4) does not hold for 𝐻^𝑡,𝑝(ℝ^𝑛),𝑡>0.

For the case 𝑡>0, it is necessary to make some progress in technique. The difficulty is to deal with the impact of the maximal operator 𝑀 on the frequencies when we split a product of two functions. To overcome this difficulty, we introduce an almost local operator 𝑇^𝑡. See Definition 12. Let 𝑟>0, 𝑡≥0 with 𝑡+𝑟<1<𝑝<𝑛/(𝑟+𝑡). In Theorem 21, we characterize 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) by Meyer wavelets without using capacity. Also our method can be applied to study the relation between multiplier spaces and Morrey spaces.

Lemma 2 implies that 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛)⊂𝑀^𝑡_𝑟,𝑝(ℝ^𝑛). For the converse imbedding, Fefferman [8] established the following relation: 𝑀^𝑡_𝑟,𝑞ℝ^𝑛⊂𝑋^𝑡_𝑟,𝑝ℝ^𝑛,𝑞>𝑝>1. (5)

Let 𝑓∈𝑀^𝑡_𝑟,𝑝(ℝ^𝑛) and 𝑔∈𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛). From the counterexample in Theorem 33, we can see that the product 𝑓𝑔 may produce a logarithmic type blowup on the fractal sets with Hausdorff dimension 𝑛−𝑝(𝑟+𝑡). To eliminate this defect, we introduce a logarithmic type Morrey space 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) and prove that, for 𝜏>1/𝑝^′, 𝑀^𝑡,𝜏_𝑟,𝑝ℝ^𝑛⊂𝑋^𝑡_𝑟,𝑝ℝ^𝑛, (6) where 𝑟>0, 𝑡≥0, and 1<𝑝<𝑛/(𝑟+𝑡). See Section 4.1. In (6), the scope of 𝜏 is (1/𝑝^′,∞), where 1/𝑝+1/𝑝^′=1. In Section 4.2, our counterexample implies that, for 0<𝜏≤1/𝑝^′, there exists some function 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), but 𝑓∉𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). See Section 4.2 for the details. Theorems 24 and 33 illustrate the difference between Morrey spaces and multiplier spaces.

Remark 3. For the case 𝑡=0, some similar counterexamples have been obtained. Lemarié-Rieusset [9] gave a counterexample to show that 𝑋⁰_𝑟,2(ℝ^𝑛)≠𝑀⁰_𝑟,2(ℝ^𝑛), where 𝑛−2𝑟 is an integer. Recently, Lemarié-Rieusset [10] and Yang and Zhou [7] constructed some counterexamples for 𝑡=0 and 1<𝑝<𝑛/𝑟. The counterexamples there are independent of the wavelet characterization of 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). Our counterexample depends on the wavelet characterization of multiplier space, Theorem 24, and fractal skills. See Theorem 33.

As an application, we apply our results obtained in Section 4 to the Schrödinger operator 𝐿=𝐼−Δ+𝑉, where 𝑉 is the potential function. Maz'ya and Verbitsky [11] considered the multipliers from 𝐻^1,2(ℝ^𝑛) to 𝐻^−1,2(ℝ^𝑛). For a Schrödinger operator 𝐿=𝐼−Δ+𝑉, they got many sufficient and necessary conditions such that 𝑉 is a multiplier from 𝐻^1,2(ℝ^𝑛) to 𝐻^−1,2(ℝ^𝑛). For more information, we refer the reader to Jiang et al. [12], Lemarié-Rieusset [9], Maz'ya and Shaposhnikova [1], Maz'ya and Verbitsky [11, 13], Yang and Yang [14], Yang et al. [15–17], Yang and Zhou [7], and the references therein.

Given 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑟+𝑡), and 𝜏>1/𝑝^′, we consider the following equation: 𝐼+(−Δ)^𝑟/2+𝑉𝑓=𝑔, (7) where 𝑔∈𝐻^𝑡,𝑝(ℝ^𝑛) and 𝑉∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛). If 𝑉 is a function of Hölder class, one usual method to deal with (7) is the boundedness of Calderón-Zygmund operators. As a function in 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), 𝑉 may not be a 𝐿^∞ function. In Section 5, by Theorem 32, we prove that if 𝑉∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), (7) has an unique solution in the Sobolev space 𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛).

The rest of this paper is organized as follows. In Section 2, we state some notations and known results which will be used throughout this paper. In Section 3, we give a wavelet characterization of 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). In Section 4, we introduce a class of logarithmic Morrey spaces 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) such that 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛)⊂𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). Further, we construct a counterexample to prove the sharpness of the scope of the index 𝜏. In the last section, we consider an application to PDE problem.

Notations. 𝑈≈𝑉 represents that there is a constant 𝑐>0 such that 𝑐⁻¹𝑉≤𝑈≤𝑐𝑉 whose right inequality is also written as 𝑈≲𝑉. Similarly, if 𝑉≥𝑐𝑈, we denote 𝑉≳𝑈.

2. Some Preliminaries

In this section, we state some notations, knowledge, and preliminary lemmas which will be used in the sequel. Firstly, we recall some background knowledge of wavelets and multiresolution analysis.

For any 𝑗∈ℕ and 𝑘=(𝑘₁,𝑘₂,…,𝑘_𝑛)∈ℤ^𝑛, let 𝑄_𝑗,𝑘=∏^𝑛_𝑠=1[2^−𝑗𝑘_𝑠,2^−𝑗(𝑘_𝑠+1)] and denote by Ω the set of all dyadic cubes 𝑄_𝑗,𝑘. For arbitrary set 𝑄, we denote by 𝑄 the 2^𝑀+2-multiple of 𝑄. Finally, let 𝜒, 𝜒 be the characteristic functions of the unit cube 𝑄₀ and 𝑄₀, respectively.

We will adopt real-valued tensor product wavelets to study the multiplier spaces in this paper. Let {𝑉¹_𝑗,𝑗∈ℤ} be an orthogonal multiresolution in 𝐿²(ℝ) with the scaling function Φ⁰(𝑥). Denote by 𝑊¹_𝑗 the orthogonal complement space of 𝑉¹_𝑗 in 𝑉¹_𝑗+1; that is, 𝑊¹_𝑗=𝑉¹_𝑗+1⊖𝑉¹_𝑗. Let {Φ¹(𝑥−𝑘),𝑘∈ℤ} be an orthogonal basis in 𝑊¹₀. For 𝜀=(𝜀₁,…,𝜀_𝑛)∈{0,1}^𝑛, denote Φ^𝜀(𝑥)=∏^𝑛_𝑖=1Φ^𝜀𝑖(𝑥_𝑖).

In the proof, we use only Meyer wavelets and regular Daubechies wavelets. We say a Daubechies wavelet is regular if it has sufficient vanishing moment until order 𝑚 and Φ^𝜀(𝑥)∈𝐶^𝑚₀([−2^𝑀,2^𝑀]^𝑛), where the regularity exponent 𝑚 is large enough and 𝑀 is determined by 𝑚; see [18, 19] for more details. For any 𝜀∈{0,1}^𝑛, 𝑗∈ℕ, and 𝑘∈ℤ^𝑛, we denote Φ^𝜀_𝑗,𝑘(𝑥)=2^𝑗𝑛/2Φ^𝜀(2^𝑗𝑥−𝑘). For 𝑗∈ℤ and 𝑘∈ℤ^𝑛, let 𝜒(2^𝑗𝑥−𝑘) and 𝜒(2^𝑗𝑥−𝑘) be the characteristic functions on 𝑄_𝑗,𝑘 and 𝑄_𝑗,𝑘, respectively. For simplicity, we denote by 𝜒_𝑗,𝑘 and 𝜒_𝑗,𝑘 for short.

In addition we define Λ_𝑛=𝜀,𝑗,𝑘∶𝜀∈{0,1}^𝑛,𝑗∈ℕ,𝑘∈ℤ^𝑛,𝜀≠0,if𝑗>0. (8) For fixed tempered distribution 𝑓, if we use wavelets which are sufficiently regular, then we can define 𝑓^𝜀_𝑗,𝑘=⟨𝑓,Φ^𝜀_𝑗,𝑘⟩. And the wavelet representation 𝑓=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘 holds in the sense of distribution.

Let {𝑉_𝑗,𝑗∈ℤ^𝑛} be the orthogonal multiresolution in 𝐿²(ℝ^𝑛) with the scaling function Φ⁰(𝑥). Denote by 𝑊_𝑗 the orthogonal complement space of 𝑉_𝑗 in 𝑉_𝑗+1; that is, 𝑊_𝑗=𝑉_𝑗+1⊖𝑉_𝑗. Denote by 𝑃_𝑗 and 𝑄_𝑗 the projection operators from 𝐿²(ℝ^𝑛) to 𝑉_𝑗 and 𝑊_𝑗, respectively. Dobynski got a decomposition of the product of two functions 𝑓 and 𝑔, which is similar to Bony's paraproduct (see [20]). Denote Λ_𝑛=𝜀,𝜀^′,𝑗,𝑘,𝑘^′,𝜀,𝜀^′∈{0,1}^𝑛⧵{0},𝑗≥0,𝑘,𝑘^′∈ℤ^𝑛,(𝜀,𝑘)≠𝜀^′,𝑘^′. (9) By the projection operators 𝑃_𝑗 and 𝑄_𝑗, we divide the product 𝑓𝑔 into the following terms: 𝑓(𝑥)𝑔(𝑥)=𝑃₀𝑓𝑃₀𝑔+_𝑗≥0𝑃_𝑗𝑓𝑄_𝑗𝑔+_𝑗≥0𝑄_𝑗𝑓𝑃_𝑗𝑔+Λ_𝑛𝑓^𝜀_𝑗,𝑘𝑔^𝜀′_𝑗,𝑘′Φ^𝜀_𝑗,𝑘(𝑥)Φ^𝜀′_𝑗,𝑘′(𝑥)+_{Λ𝑛,𝜀≠0}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)². (10) To facilitate our use, we make a modification to (10) and use special wavelets for different cases. Let 𝑁 be a positive integer. We decompose the product 𝑓𝑔 as 𝑓𝑔=^∞_𝑗=1𝑃_𝑗+1𝑓𝑃_𝑗+1𝑔−𝑃_𝑗𝑓𝑃_𝑗𝑔+𝑃₀𝑓𝑃₀𝑔=^∞_𝑗=𝑁𝑄_𝑗𝑓𝑄_𝑗𝑔+𝑃_𝑗𝑓𝑄_𝑗𝑔+𝑄_𝑗𝑓𝑃_𝑗𝑔+𝑃_𝑁𝑓𝑃_𝑁𝑔, (11) and the term ∑^∞_𝑗=𝑁𝑄_𝑗(𝑓)𝑃_𝑗(𝑔) can be decomposed as ^∞_𝑗=𝑁𝑄_𝑗𝑓𝑃_𝑗𝑔=^∞_𝑗=𝑁𝑄_𝑗𝑓^𝑁_𝑡=1𝑄_𝑗−𝑡𝑔+^∞_𝑗=𝑁𝑄_𝑗𝑓𝑃_𝑗−𝑁𝑔=^∞𝑁_𝑗=0𝑡=1𝑄_𝑗+𝑡𝑓𝑄_𝑗𝑔+^∞_𝑗=0𝑄_𝑗+𝑁𝑓𝑃_𝑗𝑔. (12)

In 1970s, Triebel introduced Triebel-Lizorkin spaces 𝐹_𝑝^𝑟,𝑞(ℝ^𝑛) ([21]). Many function spaces can be seen as the special cases for 𝐹_𝑝^𝑟,𝑞(ℝ^𝑛). For example, 𝐹₁^𝑟,2(ℝ^𝑛) is the fractional Hardy space. For 1<𝑝<∞, 𝐹_𝑝^𝑟,2(ℝ^𝑛) are the Sobolev spaces 𝐻^𝑟,𝑝(ℝ^𝑛). For 𝑝=∞, 𝐹_∞^−𝑟,2(ℝ^𝑛) is the space BMO^𝑟(ℝ^𝑛) defined as BMO^𝑟ℝ^𝑛∶=(𝐼−Δ)^−𝑟/2BMO_locℝ^𝑛, (13) where 𝐼 and Δ denote the unit operator and the Laplace operator, respectively. Here BMO_loc(ℝ^𝑛) denotes the set of all measurable functions 𝑓 with sup_{(𝑄)1/2≤𝑙≤1}𝑓_𝑄+sup_{(𝑄)0<𝑙<1/2}|𝑄|⁻¹_𝑄𝑓(𝑥)−𝑓_𝑄𝑑𝑥<∞, (14) where 𝑓_𝑄 denotes the mean value of 𝑓 on 𝑄. See also Section 3.1 of [1].

For 1≤𝑝<∞ and 𝑟∈ℝ, it is well known that (𝐹_𝑝^𝑟,2(ℝ^𝑛))^′=𝐹_𝑝^−𝑟,2_′(ℝ^𝑛). The following lemma gives a characterization of 𝐹_𝑝^𝑟,2(ℝ^𝑛) via Meyer wavelets and regular Daubechies wavelets. For the proof, we refer the reader to Chapters 5 and 6 of Meyer [18]. See also Yuan et al. [22] and Yang [23].

Lemma 4. Given |𝑟|<𝑚, let {Φ^𝜀_𝑗,𝑘} be Meyer wavelets or 𝑚-regular Daubechies wavelets. Then,(i)for 1≤𝑝<∞, 𝑔(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)∈𝐹_𝑝^𝑟,2ℝ^𝑛⟺_{𝜀,𝑗,𝑘∈Λ𝑛}2^{2𝑗(𝑟+𝑛/2)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(⋅)^1/2_𝐿𝑝⟺<∞_{𝜀,𝑗,𝑘∈Λ𝑛}2^{2𝑗(𝑟+𝑛/2)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(⋅)^1/2_𝐿𝑝<∞, (15)(ii)𝑔(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)∈𝐹_∞^𝑟,2(ℝ^𝑛) if and only if there exists 𝑝∈(1,∞) such that for 𝑄∈Ω, _{𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{2𝑗(𝑟+𝑛/2)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(⋅)^1/2_𝐿𝑝≤𝐶|𝑄|^1/𝑝. (16)

The wavelet characterizations of function spaces have been studied by many authors. In [18, Chapters 5 and 6], Meyer established wavelet characterizations for many function spaces, for example, Hardy space, BMO spaces, Besov spaces, and Bloch space. Yang et al. [24] used wavelets to characterize Lorentz type Triebel-Lizorkin spaces and Lorentz type Besov spaces. For the wavelet characterization of Besov type Morrey spaces and Triebel-Lizorkin type Morrey spaces, We refer to Yang and Yuan [25, 26], Yang [23], and Yuan et al. [22].

Morrey spaces were introduced by Morrey in 1938 and played an important role in the research of partial differential equations. Xiao [27] established a relation between the homogeneous Morrey space ℒ^{2,𝑛−2𝛼}(ℝ^𝑛) and 𝑄-type space 𝑄_𝛼(ℝ^𝑛) by heat semigroup and the fractional integrals. In the recent 20 years, 𝑄-type spaces are studied extensively. See Essèn et al. [28], Dafni and Xiao [29, 30], Peng and Yang [31], Wu and Xie [32], Yang [23], and Yuan et al. [22].

Define 𝑓_𝑡,𝑄 as 𝑓_𝑡,𝑄=1|𝑄|_𝑄(𝐼−Δ)^𝑡/2𝑓(𝑥)𝑑𝑥. (17) The Morrey spaces 𝑀^𝑡_𝑟,𝑝(ℝ^𝑛) are defined as follows.

Definition 5. Given 1≤𝑝<∞ and 𝑟,𝑡≥0, the Morrey space 𝑀^𝑡_𝑟,𝑝(ℝ^𝑛) is defined as the set of all measurable functions 𝑓 such that sup_|𝑄|=1|𝑓_𝑡,𝑄|≤𝐶 and (𝐼−▵)_𝑄^𝑡/2𝑓(𝑥)−𝑓_𝑡,𝑄^𝑝𝑑𝑥≤𝐶|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛}, (18) where 𝑄 is any cube in ℝ^𝑛 with |𝑄|≤1.

Morrey spaces 𝑀^𝑡_𝑟,𝑝(ℝ^𝑛) can be also characterized by wavelets. We state it as the following theorem and refer to Yuan et al. [22] for the proof.

Theorem 6. Given 𝑡∈ℝ, 1<𝑝<∞ and 0≤𝑝(𝑟+𝑡)<𝑛, 𝑓(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)∈𝑀^𝑡_𝑟,𝑝ℝ^𝑛 (19) if and only if for any 𝑄∈Ω with |𝑄|≤1_{𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥≤𝐶|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛}. (20)

The following lemma can be obtained by Lemmas 2 and 4 immediately.

Lemma 7. Given 𝑟>0, 𝑡≥0 and 1<𝑝<𝑛/(𝑟+𝑡), then 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛)⊂𝑀^𝑡_𝑟,𝑝(ℝ^𝑛).

Now we give two lemmas about BMO^𝑟(ℝ^𝑛).

Lemma 8. Given 𝑟>0, 𝑡≥0 and 1<𝑝<𝑛/(𝑟+𝑡), then 𝑀^𝑡_𝑟,𝑝(ℝ^𝑛)⊂𝐵𝑀𝑂^𝑟(ℝ^𝑛).

Proof. For any dyadic cube 𝑄, we have _{𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗𝑛−2𝑗𝑟}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲|𝑄|𝑑𝑥_{𝑝(𝑟+𝑡)/𝑛𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲|𝑄|.𝑑𝑥 (21)

For any 𝜀∈{0,1}^𝑛, 𝑗∈ℕ and 𝑘∈ℤ^𝑛, we denote 𝑓^𝜀_𝑗,𝑘=⟨𝑓,Φ^𝜀_𝑗,𝑘⟩. We can get the following result.

Lemma 9. Suppose that 𝑟>0 and 𝑓∈𝐵𝑀𝑂^𝑟(ℝ^𝑛). The wavelet coefficients of 𝑓 satisfy 𝑓^𝜀_𝑗,𝑘≲2^{(𝑟−𝑛/2)𝑗},∀𝜀∈1{0,1}^𝑛,𝑗∈ℕ,𝑘∈ℤ^𝑛. (22)

Proof. Take 𝑗∈ℕ and 𝑘∈ℤ^𝑛. We consider two cases 𝜀∈𝐸_𝑛 and 𝜀=0 separately.(i) For 𝜀∈𝐸_𝑛, by Lemma 4, we get 2^{𝑗𝑛−2𝑗𝑟}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥≲2^−𝑗𝑛. (23) It is easy to see that |𝑓^𝜀_𝑗,𝑘|≲2^{𝑗(𝑟−𝑛/2)}. (ii) For 𝜀=0, 𝑓⁰_𝑗,𝑘=_{𝜀′,𝑗′,𝑘′∈Λ𝑛}𝑓^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′},Φ⁰_𝑗,𝑘=_{𝜀′,𝑗′,𝑘′∈Λ𝑛∶𝑗′<𝑗}𝑓^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′},Φ⁰_𝑗,𝑘. (24) Because |∑_{(𝜀′,𝑗′,𝑘′)∈Λ𝑛∶𝑗′<𝑗}𝑓^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′}(𝑥)|≲2^𝑟𝑗, we have 𝑓⁰_𝑗,𝑘≲2^𝑟𝑗,Φ⁰_𝑗,𝑘(𝑥)≲2^{𝑗(𝑟−𝑛/2)}. (25)

Let Ψ¹ and Ψ² be two functions such that Ψ^𝑖∈𝐶^𝜇₀−2^𝑀+1,2^𝑀+1𝑛𝑥,𝑖=1,2,_𝛼Ψ^𝑖(𝑥)𝑑𝑥=0,where|𝛼|≤𝜇,𝑖=1,2. (26) Write 𝑎_{𝑗,𝑘,𝑗′,𝑘′}=⟨Ψ¹_𝑗,𝑘,Ψ²_{𝑗′,𝑘′}⟩. The following lemma can be found in Chapter 8 of Meyer [18] or Chapter 6 of Yang [23].

Lemma 10 (see [18, Chapter 8, Lemma 1]). Let |𝜇|≤𝑚. For |𝑠|<𝜇, the coefficients 𝑎_{𝑗,𝑘,𝑗′,𝑘′} satisfy the following condition: 𝑎_{𝑗,𝑘,𝑗′,𝑘′}≲2^{−|𝑗−𝑗′|(𝑛/2+𝑠)}2^−𝑗+2^−𝑗′2^−𝑗+2^−𝑗′+𝑘2^−𝑗−𝑘^′2^{−𝑗′𝑛+𝑠}. (27)

By Lemma 4, the boundedness of Calderón-Zygmund operators on 𝐻^𝑟,𝑝(ℝ^𝑛) is equivalent to the following lemma. We refer the reader to [18, 23, 33] for the proof.

Lemma 11. Suppose that 𝑠>|𝑟| and 𝑔(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)∈𝐻^𝑟,𝑝(ℝ^𝑛). Let 𝑔^𝜀_𝑗,𝑘=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑗′,𝑘′}𝑔^𝜀_{𝑗′,𝑘′}. If the coefficients 𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑗′,𝑘′} satisfy (27), then _{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑟)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑟)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥. (28)

We say that 𝑇 is a local operator if there exists some constant 𝐶>1 such that, for all 𝑥∈ℝ^𝑛 and 𝑟>0, 𝑇 maps a distribution with the support 𝐵(𝑥,𝑟) to another distribution supported on the ball 𝐵(𝑥,𝐶𝑟). If 𝑡/2 is not a nonnegative integer, the operator (𝐼−Δ)^𝑡/2 is not a local operator. Now we use wavelets to construct some special fractional differential operators 𝑇^𝑡, which are almost local operators and will be used in the proof of our main result.

Definition 12. For 𝑡≥0 and ℎ(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}ℎ^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥), we call that 𝑇^𝑡 is an operator associated to the kernel 𝐾^𝑡(𝑥,𝑦)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}2^−𝑗𝑡Φ^𝜀_𝑗,𝑘(𝑥)Φ^𝜀_𝑗,𝑘(𝑦) if 𝑇^𝑡ℎ(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}2^−𝑗𝑡ℎ^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥). (29)

It is easy to prove that 𝑇⁰ is the identity operator and ‖𝑇^𝑡ℎ‖_𝐿^𝑝=‖ℎ‖_𝐻^−𝑡,𝑝 for 1<𝑝<∞. Furthermore, we have the following.

Lemma 13. Suppose 𝑡≥0. For any 𝑄_𝑗,𝑘∈Ω and 𝑥∈𝑄_𝑗,𝑘, 2^{𝑗(𝑛/2−𝑡)}ℎ⁰_𝑗,𝑘≤𝐶𝑀𝑇^𝑡ℎ(𝑥), (30) where 𝑀 is the Hardy-Littlewood maximal operator.

Proof. If 𝑡=0, the proof was given by Meyer [18]. Now we consider the case 𝑡>0. It is easy to verify that 𝑇^−𝑡Φ⁰(𝑥)=𝐾^−𝑡𝑥,𝑦Φ⁰𝑦𝑑𝑦≤𝐶(1+|𝑥|)^{−𝑛−𝑡}. (31) By the fact that 𝑡>0, we have 2^{𝑗(𝑛/2−𝑡)}ℎ⁰_𝑗,𝑘=2^{𝑗(𝑛/2−𝑡)}𝑇^𝑡ℎ,𝑇^−𝑡Φ⁰_𝑗,𝑘=2^𝑗𝑛/2𝑇^𝑡ℎ,𝑇^−𝑡Φ⁰_𝑗,𝑘. (32) Hence, we can get 2^{𝑗(𝑛/2−𝑡)}ℎ⁰_𝑗,𝑘=2^𝑗𝑛/2𝑇^𝑡ℎ,𝑇^−𝑡Φ⁰_𝑗,𝑘≲2^𝑗𝑛/2𝑇^𝑡ℎ(𝑥)2^𝑗𝑛/2𝑑𝑥1+2^𝑗𝑥−𝑘^𝑛+𝑡≲2^𝑗𝑛_{|2𝑗𝑥−𝑘|≤1}𝑇^𝑡ℎ(𝑥+)𝑑𝑥_{∞𝑙=12𝑙−1≤|2𝑗𝑥−𝑘|≤2𝑙}𝑇^𝑡ℎ(𝑥)𝑑𝑥1+2^𝑗𝑥−𝑘^𝑛+𝑡≲2^𝑗𝑛2^−𝑗𝑛𝑀𝑇^𝑡ℎ(𝑥)+^∞_𝑙=12^−𝑙𝑡𝑀𝑇^𝑡ℎ(𝑥)2^−𝑗𝑛≲𝑀𝑇^𝑡ℎ(𝑥). (33) This completes the proof of Lemma 13.

In the rest of this section, we give a decomposition of Sobolev spaces associated with combination atoms. For |𝑟|<𝑚 and 𝑔(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥), denote 𝑆_𝑟𝑔(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(2𝑟+𝑛)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2, (34) and, for 𝑟=0, denote also 𝑆𝑔(𝑥)=𝑆₀𝑔(𝑥).

Definition 14. Let 𝑟∈ℝ and 𝜆>0. For arbitrary measurable set 𝐸 with finite measure, we say that 𝑔(𝑥) is a (𝑟,𝜆,𝐸)-combination atom if supp(𝑆_𝑟𝑔)⊂𝐸 and 𝑆_𝑟𝑔(𝑥)≤𝜆. If 𝐸 is a dyadic cube, then we say 𝑔(𝑥) is a (𝑟,𝜆,𝐸)-atom.

In the next theorem, we give a combination atom decomposition of Sobolev spaces. This result is a generalization of that of Yang [34].

Theorem 15. If 1<𝑝<∞, |𝑟|<𝑚 and ‖𝑔‖_{𝐻𝑟,𝑝}≤1, there exists a series of (𝑟,2^𝑣,𝐸_𝑣)-combination atoms 𝑔_𝑣(𝑥) such that ∑_𝑣∈ℕ2^𝑝𝑣|𝐸_𝑣|≤𝐶.

Proof. Denote 𝑆_𝑟𝑔(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(2𝑟+𝑛)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (35) For 𝑣≥1, let 𝐸_𝑣={𝑥∶𝑆_𝑟𝑔(𝑥)>2^𝑣}. By wavelet characterization of Sobolev spaces, we have ∑_𝑣∈ℕ2^𝑝𝑣|𝐸_𝑣|≲1. Let 𝐸_𝑣=⋃_𝑙𝑄^𝑣,𝑙, where 𝑄^𝑣,𝑙 are disjoint maximal dyadic cubes with |𝑄^𝑣,𝑙|≤1. Let 𝔉_𝑣,𝑙 be the set of dyadic cubes contained in 𝑄^𝑣,𝑙 but not in 𝐸_𝑣+1, 𝔉_𝑣=⋃_𝑙𝔉_𝑣,𝑙 and 𝔉₀=Ω⧵⋃_𝑣≥1𝔉_𝑣. Let 𝐸₀={𝑥∈𝑄,𝑄∈𝔉₀}, and we can write also 𝐸₀=⋃_𝑙𝑄^0,𝑙, where 𝑄^0,𝑙 are disjoint maximal dyadic cubes in Ω. The related set 𝔉_0,𝑙 is defined as 𝔉_0,𝑙={𝑄⊂𝑄^0,𝑙and𝑄∈𝔉₀}.
For any 𝑣≥0, we write 𝑔_𝑣,𝑙(𝑥)=_{𝑄𝑗,𝑘∈𝔉𝑣,𝑙}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥),𝑔_𝑣(𝑥)=_{𝑄𝑗,𝑘∈𝔉𝑣}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥). (36) Then 𝑔_𝑣(𝑥) is a desired combination atom. This completes the proof.

3. Wavelet Characterization of the Multiplier Spaces

In this section, we use Meyer wavelets to characterize 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). Let Φ be a function in 𝐶^∞₀(𝐵(0,1)) satisfying Φ(𝑥)≥0 and ∫Φ(𝑥)𝑑𝑥=1. For any 𝑔∈𝐻^𝑡,𝑝(ℝ^𝑛), define 𝑔^Φ_𝑗,𝑘=𝑔,2^𝑗𝑛Φ2^𝑗⋅−𝑘. (37) The function space 𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛) is defined as follows.

Definition 16. Given 𝑟>0, 𝑡≥0 and 𝑟+𝑡<1<𝑝<𝑛/(𝑟+𝑡), we say 𝑓∈𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛) if 𝑓(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥) and _{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑡)}𝑔^Φ_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥≲𝑔_{𝐻𝑟+𝑡,𝑝}, (38) where 𝑔∈𝐻^{𝑟+𝑡,𝑝}(ℝ^𝑛) and ‖𝑔‖_{𝐻𝑟+𝑡,𝑝}≤1.

Now we give a wavelet characterization of 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). Let Φ⁰ and Φ^𝜀,𝜀∈𝐸_𝑛, be the scaling function and wavelet functions, respectively. For (𝜀,𝑗,𝑘), (𝜀^′,𝑗^′,𝑘^′), (𝜀^′′,𝑗^′,𝑘^′)∈Λ_𝑛 and 𝑙∈ℤ^𝑛, let 𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}=Φ⁰_{𝑗,𝑘+𝑙}Φ^𝜀_𝑗,𝑘,Φ^𝜀′_{𝑗′,𝑘′},𝑎^{𝜀,𝜀,𝜀′′,0}_{𝑗,𝑘,0,𝑗′,𝑘′}=Φ^𝜀_𝑗,𝑘²−2^𝑗𝑛Φ2^𝑗⋅−𝑘,Φ^𝜀′′_{𝑗′,𝑘′}. (39) Furthermore, for 0≤𝑠≤𝑁, 𝜀^′∈𝐸_𝑛, 𝑙∈ℤ^𝑛, and 𝑠+|𝜀−𝜀^′|+|𝑙|≠0, let 𝑎^{𝜀,𝜀′,𝜀′′,𝑠}_{𝑗,𝑘,𝑙,𝑗′,𝑘′}=Φ^𝜀′_𝑗,𝑘Φ^𝜀_𝑗+𝑠,2^𝑠_𝑘+𝑙,Φ^𝜀′′_{𝑗′,𝑘′}. (40) By the same method of [18, Chapter 8, Lemma 1], we could prove the following lemma.

Lemma 17. There exist sufficient big integers 𝑁, 𝑁₁, and 𝑁₂ such that min{𝑁,𝑁₁,𝑁₂}>8𝑛+8𝑚 and the following estimates hold. (i)If (𝜀,𝑗,𝑘), (𝜀^′,𝑗^′,𝑘^′)∈Λ_𝑛, 𝑙∈ℤ^𝑛, and 𝑗≥𝑗^′, then 𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}≲(1+|𝑙|)^−𝑁12^{𝑛𝑗′/2+𝑗′−𝑗}1+𝑘^′−2^{𝑗′−𝑗}𝑘^−𝑁2. (41)(ii)If (𝜀,𝑗,𝑘), (𝜀^′′,𝑗^′,𝑘^′)∈Λ_𝑛, 0≤𝑠≤𝑁, 𝜀^′∈𝐸_𝑛, 𝑙∈ℤ^𝑛, and 𝑗≥𝑗^′, then 𝑎^{𝜀,𝜀′,𝜀′′,𝑠}_{𝑗,𝑘,𝑙,𝑗′,𝑘′}≲(1+|𝑙|)^−𝑁12^{𝑛𝑗′/2+𝑗′−𝑗}1+𝑘^′−2^{𝑗′−𝑗}𝑘^−𝑁2. (42)(iii)If (𝜀,𝑗,𝑘), (𝜀^′,𝑗^′,𝑘^′)∈Λ_𝑛, 𝑙∈ℤ^𝑛, and 𝑗<𝑗^′, then 𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}≲(1+|𝑙|)^−𝑁12^{−𝑛𝑗′/2+𝑛𝑗+𝑁(𝑗−𝑗′)}1+𝑘−2^{𝑗−𝑗′}𝑘^′−𝑁2. (43)(iv)If (𝜀,𝑗,𝑘), (𝜀^′′,𝑗^′,𝑘^′)∈Λ_𝑛, 0≤𝑠≤𝑁, 𝜀^′∈𝐸_𝑛, 𝑙∈ℤ^𝑛, and 𝑗<𝑗^′, then 𝑎^{𝜀,𝜀′,𝜀′′,𝑠}_{𝑗,𝑘,𝑙,𝑗′,𝑘′}≲(1+|𝑙|)^−𝑁12^{−𝑛𝑗′/2+𝑛𝑗+𝑁(𝑗−𝑗′)}1+𝑘−2^{𝑗−𝑗′}𝑘^′−𝑁2. (44)

3.1. Decomposition of Products via Multilinear Analysis

At first, we give a wavelet decomposition of the product of 𝑓ℎ. Let Φ⁰ and {Φ^𝜀,𝜀∈𝐸_𝑛} be the scaling function and wavelet functions of Meyer wavelets, respectively. There exists an integer 𝑁≥3 such that 𝑥^𝛼Φ⁰(𝑥)Φ^𝜀2^𝑁𝑥−𝑘𝑑𝑥=0forany𝑘∈ℤ^𝑛,𝛼∈ℕ^𝑛,𝜀∈𝐸_𝑛. (45) Denote Λ_𝜀,𝑛=𝑠,𝜀^′,𝑙,0≤𝑠≤𝑁,𝜀^′∈𝐸_𝑛,𝑙∈ℤ^𝑛,|𝑙|≤2^{(𝑀+2+𝑠)𝑛},andif𝑠=0,(0,𝜀,0)≠0,𝜀^′,ℤ,0_𝑛𝑁=𝑙=𝑙₁,…,𝑙_𝑛∈ℤ^𝑛,0≤𝑙_𝑖≤2^𝑁−1,𝑖=1,…,𝑛. (46) For 𝜀∈𝐸_𝑛, 𝑙∈ℤ^𝑛, |𝑙|≤2^(𝑀+2)𝑛, and (𝜀^′,𝑙)∈Λ_𝜀,𝑛, we denote 𝑇_1,𝜀(𝑥)=_{𝑗∈ℕ,𝑘∈ℤ𝑛𝑙∈ℤ𝑛}𝑓⁰_{𝑗,𝑘+𝑙}ℎ^𝜀_𝑗,𝑘Φ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥),𝑇_{2,0,𝜀,𝜀}(𝑥)=_{𝑗∈ℤ,𝑘∈ℤ𝑛𝑙∈ℤ𝑛,𝑙≠0}𝑓^𝜀_{𝑗,𝑘+𝑙}ℎ^𝜀_𝑗,𝑘Φ^𝜀_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥)+_{𝑗∈ℕ,𝑘∈ℤ𝑛}𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)²−2^𝑗𝑛Φ2^𝑗,𝑇𝑥−𝑘_3,𝜀(𝑥)=_{𝑗∈ℕ,𝑘∈ℤ𝑛𝑙∈ℤ𝑛𝑁𝑙′∈ℤ𝑛}𝑓^𝜀_{𝑗+𝑁,2𝑁𝑘+𝑙}ℎ⁰_{𝑗,𝑘+𝑙′}×Φ⁰_{𝑗,𝑘+𝑙′}(𝑥)Φ^𝜀_{𝑗+𝑁,2𝑁𝑘+𝑙}(𝑥),𝑇_4,𝜀(𝑥)=_{𝑗∈ℕ,𝑘∈ℤ𝑛}𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘2^𝑗𝑛Φ2^𝑗,𝑇𝑥−𝑘₅(𝑥)=_{𝑘∈ℤ𝑛𝑙∈ℤ𝑛}𝑓⁰_0,𝑘ℎ⁰_0,𝑘+𝑙Φ⁰_𝑗,𝑘(𝑥)Φ⁰_0,𝑘+𝑙(𝑥). (47) For 𝜀,𝜀^′∈𝐸_𝑛, 0<𝑠≤𝑁 or 𝑠=0, and 𝜀≠𝜀^′, denote 𝑇_{2,𝑠,𝜀,𝜀′}(𝑥)=_{𝑗∈ℕ,𝑘∈ℤ𝑛𝑙∈ℤ𝑛}𝑓^𝜀′_{𝑗+𝑠,2𝑠𝑘+𝑙}ℎ^𝜀_𝑗,𝑘Φ^𝜀′_𝑗+𝑠,2^𝑠_𝑘+𝑙(𝑥)Φ^𝜀_𝑗,𝑘(𝑥). (48) By (11) and (12), we can decompose 𝑓ℎ as follows: 𝑓(𝑥)ℎ(𝑥)=_{𝜀∈𝐸𝑛}𝑇_1,𝜀(𝑥)+_{0≤𝑠≤𝑁,𝜀,𝜀′∈𝐸𝑛}𝑇_{2,𝑠,𝜀,𝜀′}(𝑥)+_{𝜀∈𝐸𝑛}𝑇_3,𝜀(𝑥)+_{𝜀∈𝐸𝑛}𝑇_4,𝜀(𝑥)+𝑇₅(𝑥)∶=⁵_𝑖=1𝑇_𝑖(𝑥). (49) If 𝑔∈𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛), write 𝑔(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥). For 𝜀,𝜀^′∈𝐸_𝑛, and 0≤𝑠≤𝑁, we define 𝑇_1,𝜀=𝑇_1,𝜀(𝑥)𝑔(𝑥)𝑇𝑑𝑥,_{2,𝑠,𝜀,𝜀′}=𝑇_{2,𝑠,𝜀,𝜀′}(𝑥)𝑔(𝑥)𝑑𝑥. (50) Let 𝑇_{1,1,𝜀,𝑙}=_{𝑗∈ℕ,𝑘∈ℤ𝑛𝑗≥𝑗′≥0,𝜀′,𝑘′∈ℤ𝑛}𝑓⁰_{𝑗,𝑘+𝑙}ℎ^𝜀_𝑗,𝑘𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}𝑔^𝜀′_{𝑗′,𝑘′},𝑇_{1,2,𝜀,𝑙}=_{𝑗∈ℕ,𝑘∈ℤ𝑛𝑗<𝑗′,𝜀′,𝑘′∈ℤ𝑛}𝑓⁰_{𝑗,𝑘+𝑙}ℎ^𝜀_𝑗,𝑘𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}𝑔^𝜀′_{𝑗′,𝑘′},𝑇_{2,1,𝑠,𝜀,𝜀′,𝑙}=_{𝑗∈ℕ,𝑘∈ℤ𝑛𝑗≥𝑗′>0,𝜀′′,𝑘′∈ℤ𝑛}𝑓^𝜀′_{𝑗+𝑠,2𝑠𝑘+𝑙}ℎ^𝜀_𝑗,𝑘𝑎^{𝜀,𝜀′,𝜀′′,𝑠}_{𝑗,𝑘,𝑙,𝑗′,𝑘′}𝑔^𝜀′′_{𝑗′,𝑘′},𝑇_{2,2,𝑠,𝜀,𝜀′,𝑙}=_{𝑗∈ℕ,𝑘∈ℤ𝑛𝑗<𝑗′,𝜀′′,𝑘}𝑓^𝜀′_{𝑗+𝑠,2𝑠𝑘+𝑙}ℎ^𝜀_𝑗,𝑘𝑎^{𝜀,𝜀′,𝜀′′,𝑠}_{𝑗,𝑘,𝑙,𝑗′,𝑘′∈ℤ𝑛}𝑔^𝜀′′_{𝑗′,𝑘′}. (51) It is easy to see that 𝑇_1,𝜀≲_{𝑙∈ℤ𝑛}𝑇_{1,1,𝜀,𝑙}+𝑇_{1,2,𝜀,𝑙},𝑇_{2,𝑠,𝜀,𝜀′}≲_{𝑙∈ℤ𝑛}𝑇_{2,1,𝑠,𝜀,𝜀′,𝑙}+𝑇_{2,2,𝑠,𝜀,𝜀′,𝑙}. (52)

3.2. Several Technical Lemmas

Now we estimate the quantities 𝑇_{1,1,𝜀,𝑙}, 𝑇_{1,2,𝜀,𝑙}, 𝑇_{2,1,𝑠,𝜀,𝜀′,𝑙}, and 𝑇_{2,2,𝑠,𝜀,𝜀′,𝑙} separately. Let 𝑆_𝑡𝑔_𝑗′(𝑥)=_{𝜀′,𝑘′}2^{𝑗′(𝑛/2+𝑡)}𝑔^𝜀′_{𝑗′,𝑘′}𝜒2^𝑗′𝑥−𝑘^′. (53) For fixed 𝑥, there is only one 𝑘^′ such that 𝜒(2^𝑗′𝑥−𝑘^′)≠0 and the number of 𝜀^′ is finite. Then 𝑆_𝑡𝑔_𝑗′(𝑥)=_{𝜀′,𝑘′}2^{2𝑗′(𝑛/2+𝑡)}𝑔^𝜀′_{𝑗′,𝑘′}²𝜒2^𝑗′𝑥−𝑘^′1/2. (54)

Let 𝑀 be the Hardy-Littlewood maximal function. Then, if 𝑗<𝑗^′ and 𝑥∈𝑄_𝑗,𝑘, we have _𝑘′1+𝑘−2^{𝑗−𝑗′}𝑘^′−𝑁22^{𝑗′(−𝑛/2+𝑡)}𝑔^𝜀′_{𝑗′,𝑘′}≲2^−𝑗𝑛𝑀𝑆_𝑡𝑔_𝑗′(𝑥). (55)

Lemma 18. Given 𝑟>0, 𝑡≥0, and 𝑡+𝑟<1<𝑝<𝑛/(𝑡+𝑟). If 𝑓∈𝐵𝑀𝑂^𝑟(ℝ^𝑛), then (i)for 𝑔∈𝐻^{𝑡+𝑟,𝑝} and ℎ∈𝐻^{−𝑡,𝑝′}, 𝑇_1,𝜀+𝑇_{2,𝑠,𝜀,𝜀′}≲𝑔_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}, (56)(ii)for 𝑔∈𝐻^{𝑡+𝑟,𝑝} and ℎ∈𝐻^{−𝑡,𝑝′}, 𝑇₅,𝑔≲𝑔_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (57)

Proof. (1) For 𝑗^′≤𝑗, we have 𝑇_{1,1,𝜀,𝑙}=_{𝑗,𝑘𝑗≥𝑗′,𝜀′,𝑘′}𝑓⁰_{𝑗,𝑘+𝑙}ℎ^𝜀_𝑗,𝑘𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}𝑔^𝜀′_{𝑗′,𝑘′}. (58)
By Lemma 17, we know 𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}≲(1+|𝑙|)^−𝑁11+𝑘^′−2^{𝑗′−𝑗}𝑘^−𝑁22^{𝑛𝑗′/2+𝑗′−𝑗}. (59)
By Lemma 9, 𝑓∈BMO^𝑟(ℝ^𝑛) implies |𝑓^𝜀_𝑗,𝑘|≤𝐶2^{(𝑟−𝑛/2)𝑗}. Now we can get _{𝑙∈ℤ𝑛}𝑇_{1,1,𝜀,𝑙}≲_{𝑙∈ℤ𝑛𝑘,𝑗≥𝑗′}2^{3𝑛𝑗′/2+(𝑗′−𝑗)+(𝑟−𝑛/2)𝑗}_{𝑘′𝜀′}𝑔^𝜀′_{𝑗′,𝑘′}^21/2×ℎ^𝜀_𝑗,𝑘𝜒2^𝑗′𝑥−𝑘^′(1+|𝑙|)^𝑁11+𝑘^′−2^{𝑗′−𝑗}𝑘^𝑁2≲𝑑𝑥_{𝑗≥𝑗′}2^{3𝑛𝑗′/2+(𝑗′−𝑗)+(𝑟−𝑛/2)𝑗}×_{𝑘′𝜀′}2^{2𝑗′(𝑛/2+𝑡+𝑟)}𝑔^𝜀′_{𝑗′,𝑘′}^21/22^{−𝑗′(𝑡+𝑟+𝑛/2)}×_𝑘ℎ^𝜀_𝑗,𝑘𝜒2^𝑗′𝑥−𝑘^′1+𝑘^′−2^{𝑗′−𝑗}𝑘^𝑁22^{𝑗(−𝑛/2−𝑡)}2^{𝑗(𝑛/2+𝑡)}≲𝑑𝑥_{𝑗≥𝑗′}2^{3𝑛𝑗′/2−𝑗′(𝑡+𝑟+𝑛/2)+(𝑗′−𝑗)+(𝑟−𝑛/2)𝑗+𝑗(𝑛/2+𝑡)}×2^{−𝑗′𝑛}𝑆_𝑡+𝑟𝑔_𝑗′(𝑥)𝑀𝑆_−𝑡ℎ_𝑗(𝑥)𝑑𝑥. (60)
Because 0<𝑡+𝑟<1, _{𝑙∈ℤ𝑛}𝑇_{1,1,𝜀,𝑙}≲_{𝑗≥𝑗′}2^{(𝑗′−𝑗)+(𝑟+𝑡)𝑗−𝑗′(𝑟+𝑡)}×𝑆_𝑡+𝑟𝑔_𝑗′(𝑥)𝑀𝑆_−𝑡ℎ_𝑗(𝑥)≲𝑑𝑥_{𝑗≥𝑗′}2^{(𝑗′−𝑗)(1−𝑟−𝑡)}𝑔_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}≲𝑔_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (61)
Now we estimate the term 𝑇_{2,1,𝑠,𝜀,𝜀′,𝑙}(𝑥)=_{𝑗,𝑘𝑗≥𝑗′>0}𝑓^𝜀′_𝑗,𝑘ℎ^𝜀_𝑗+𝑠,2^𝑠_𝑘+𝑙𝑎^{𝜀,𝜀′,𝜀′′,𝑠}_{𝑗,𝑘,𝑙,𝑗′,𝑘′}𝑔^𝜀′′_{𝑗′,𝑘′}. (62)
Because 𝑓∈BMO^𝑟(ℝ^𝑛), by Lemma 9, we have |𝑓^𝜀′_{𝑗,𝑘+𝑙}|≲2^{(𝑟−𝑛/2)𝑗}. By Lemma 17, 𝑎^{𝜀,𝜀′,𝜀′′,𝑠}_{𝑗,𝑘,𝑙,𝑗′,𝑘′}≲(1+|𝑙|)^−𝑁11+𝑘^′−2^{𝑗′−𝑗}𝑘^−𝑁22^{𝑛𝑗′/2+(𝑗′−𝑗)}. (63)
We can get, similarly, _{𝑙∈ℤ𝑛𝑁}𝑇_{2,1,𝑠,𝜀,𝜀′,𝑙}(𝑥)≲_{𝑙∈ℤ𝑛𝑁𝑗,𝑘𝑗≥𝑗′>0}2^{(𝑟−𝑛/2)𝑗}2^{𝑛𝑗′/2+𝑗′−𝑗}×ℎ^𝜀_𝑗+𝑠,2^𝑠_𝑘+𝑙𝑔^𝜀′′_{𝑗′,𝑘′}(1+|𝑙|)^𝑁11+𝑘^′−2^{𝑗′−𝑗}𝑘^𝑁2≲_{𝑙∈ℤ𝑛𝑁𝑗≥𝑗′>0}2^{3𝑛𝑗′/2+(𝑗′−𝑗)+(𝑟−𝑛/2)𝑗}×_{𝜀′′,𝑘′}𝑔^𝜀′_{𝑗′,𝑘′}ℎ^𝜀_𝑗+𝑠,2^𝑠_𝑘+𝑙𝜒2^𝑗′𝑥−𝑘^′(1+|𝑙|)^𝑁11+𝑘^′−2^{𝑗′−𝑗}𝑘^𝑁2𝑑𝑥≲_{𝑗≥𝑗′>0}2^{(𝑗′−𝑗)(1−𝑡−𝑟)}𝑆_𝑡+𝑟𝑔_𝑗′(𝑥)𝑀𝑆_−𝑡ℎ_𝑗+𝑠(𝑥)≲𝑆𝑑𝑥_𝑡+𝑟𝑔_{𝑗′𝐿𝑝}𝑀𝑆_−𝑡ℎ_{𝑗+𝑠𝐿𝑝′}≲𝑔_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (64)
(2) If 𝑗^′>𝑗, the estimates of 𝑇_{1,2,𝜀,𝑙} and 𝑇_{2,2,𝑠,𝜀,𝜀′,𝑙} are easier than those of 𝑇_{1,1,𝜀,𝑙} and 𝑇_{2,1,𝑠,𝜀,𝜀′,𝑙}. For example, we estimate the term 𝑇_{1,2,𝜀,𝑙}=_{𝑗,𝑘0≤𝑗<𝑗′,𝜀′,𝑘′}𝑓⁰_{𝑗,𝑘+𝑙}ℎ^𝜀_𝑗,𝑘𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}𝑔^𝜀′_{𝑗′,𝑘′}. (65)
Because 𝑓∈BMO^𝑟(ℝ^𝑛), by Lemma 9, |𝑓⁰_{𝑗,𝑘+𝑙}|≲2^{(𝑟−𝑛/2)𝑗}. By Lemma 17, 𝑎^𝜀,𝜀′_{𝑗,𝑘,𝑙,𝑗′,𝑘′}≲(1+|𝑙|)^−𝑁11+𝑘−2^{𝑗−𝑗′}𝑘^′−𝑁22^{−𝑛𝑗′/2+𝑛𝑗+𝑁(𝑗−𝑗′)}. (66)
Because 2^𝑛𝑗∫𝜒_𝑗,𝑘(𝑥)𝑑𝑥=1, we can obtain _{𝑙∈ℤ𝑛}𝑇_{1,2,𝜀,𝑙}≲_{𝑙∈ℤ𝑛𝑗,𝑘0≤𝑗<𝑗′,𝜀′,𝑘′}2^{(𝑟−𝑛/2)𝑗}2^{−𝑛𝑗′/2+𝑛𝑗+𝑁(𝑗−𝑗′)}2^𝑛𝑗×ℎ^𝜀_𝑗,𝑘𝑔^𝜀′_{𝑗′,𝑘′}𝜒_𝑗,𝑘(𝑥)(1+|𝑙|)^𝑁11+𝑘−2^{𝑗−𝑗′}𝑘^′𝑁2𝑑𝑥≲_{𝜀,𝑗,𝑘}2^{(𝑟−𝑛/2)}2^{−𝑛𝑗′/2+𝑛𝑗+𝑁(𝑗−𝑗′)}2^𝑛𝑗2^−𝑛𝑗2^{−𝑗′(−𝑛/2+𝑡+𝑟)}×𝑀𝑆_𝑡+𝑟𝑔_𝑗′(𝑥)ℎ^𝜀_𝑗,𝑘𝜒_𝑗,𝑘(𝑥)𝑑𝑥≲_{0≤𝑗<𝑗′}2^{(𝑟−𝑛/2)𝑗}2^{−(𝑛𝑗′/2)+𝑛𝑗+𝑁(𝑗−𝑗′)}2^{−𝑗(𝑛/2−𝑡)}×2^{−𝑗′(−𝑛/2+𝑡+𝑟)}𝑀𝑆_𝑡+𝑟𝑔_𝑗′(𝑥)𝑆_−𝑡ℎ_𝑗(𝑥)≲𝑔𝑑𝑥_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (67)
The estimate for 𝑇_{2,2,𝑠,𝜀,𝜀′,𝑙}(𝑥) can be obtained similarly. By the same methods used in (1) and (2), we can get the estimate of the term 𝑇₅. We omit the details.

Now we consider the term 𝑇_3,𝜀. We have the following claim.

Lemma 19. Given 𝑟>0, 𝑡≥0, and 𝑡+𝑟<1<𝑝<𝑛/(𝑡+𝑟), if 𝑓∈𝐵𝑀𝑂^𝑟(ℝ^𝑛), then 𝑇_3,𝜀,𝑔≲𝑔_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (68)

Proof. In fact, for 𝑙∈ℤ^𝑛_𝑁, (𝜀,𝑗,𝑘)∈Λ_𝑛 and 𝑙^′∈ℤ^𝑛, let 𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}=2^{−𝑛𝑗/2}Φ⁰_{𝑗,𝑘+𝑙′}Φ^𝜀_{𝑗+𝑁,2𝑁𝑘+𝑙},𝑔. (69) We have 𝑇_3,𝜀=,𝑔_{𝜀,𝑗,𝑘∈Λ𝑛𝑙∈ℤ𝑛𝑁,𝑙′∈ℤ𝑛}2^𝑗𝑛/2ℎ⁰_{𝑗,𝑘+𝑙′}𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}𝑓^𝜀_{𝑗+𝑁,2𝑁𝑘+𝑙}≲_{𝑙∈ℤ𝑛𝑁𝜀,𝑗,𝑘∈Λ𝑛𝑙′∈ℤ𝑛}2^𝑗𝑛/2ℎ⁰_{𝑗,𝑘+𝑙′}𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}×𝑓^𝜀_{𝑗+𝑁,2𝑁𝑘+𝑙}. (70) Because (𝐹₁^𝑟,2(ℝ^𝑛))^′=BMO^𝑟(ℝ^𝑛), by Lemmas 9 and 13, we have 𝑇_3,𝜀,𝑔≲_{𝑙∈ℤ𝑛𝑁}𝑓_{BMO𝑟𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑟+𝑡)}×_{𝑙′∈ℤ𝑛}2^{𝑗(𝑛/2−𝑡)}ℎ⁰_{𝑗,𝑘+𝑙′}𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}²×𝜒_𝑗,𝑘(𝑥)^1/2≲𝑓𝑑𝑥_{BMO𝑟𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑟+2𝑡)}×_{𝑙′∈ℤ𝑛}1+𝑙^′𝑁+𝑛𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}²×𝜒_𝑗,𝑘(𝑥)^1/2𝑀𝑇^𝑡ℎ(𝑥)𝑑𝑥. (71) Because 𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}=2^{−𝑛𝑗/2}Φ⁰_{𝑗,𝑘+𝑙′}Φ^𝜀_{𝑗+𝑁,2𝑁𝑘+𝑙′(𝜀′,𝑗′,𝑘′)∈Λ𝑛}𝑔^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′}=2^{−𝑛𝑗/2}Φ⁰_{𝑗,𝑘+𝑙′}Φ^𝜀_{𝑗+𝑁,2𝑁𝑘+𝑙′}×_{𝜀′,𝑗′,𝑘′∈Λ𝑛,|𝑗′−𝑗−𝑁|≤2}𝑔^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′}, (72) we can get _{𝜀,𝑗,𝑘∈Λ𝑛}2^{(𝑛+2𝑡+2𝑡)𝑗}×_{𝑙′∈ℤ𝑛}1+𝑙^′𝑁+𝑛𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}²𝜒_𝑗,𝑘(𝑥)^1/2≲_{𝑗′−𝑗−𝑁≤2}𝑀𝑆_𝑡+𝑟𝑔_𝑗′(𝑥). (73) Hence, we have _{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑟+𝑡)}×_{𝑙′∈ℤ𝑛}2^{𝑗(𝑛/2−𝑡)}ℎ⁰_{𝑗,𝑘+𝑙′}𝑔^𝜀,𝑙′_{𝑗+𝑁,2𝑁𝑘+𝑙}²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥_{𝑗≥0𝑗′−𝑗−𝑁≤2}𝑀𝑆_𝑡+𝑟𝑔_𝑗′(𝑥)^2𝑝/2𝑑𝑥≲𝑔_{𝐻𝑡+𝑟,𝑝}. (74) Finally, we obtain, by Hölder's inequality, 𝑇_3,𝜀,𝑔≲𝑓_BMO𝑟𝑔_{𝐻𝑡+𝑟,𝑝}𝑀𝑇^𝑡ℎ_𝑝′≲𝑓_BMO𝑟𝑔_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (75) This completes the proof of Lemma 19.

In order to deal with the term 𝑇_4,𝜀(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘2^𝑗𝑛Φ2^𝑗𝑥−𝑘, (76) we need the following estimate.

Lemma 20. Given 𝑟>0, 𝑡≥0 with 𝑡+𝑟<1<𝑝<𝑛/(𝑟+𝑡), 𝑓∈𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛) if and only if 𝑇_4,𝜀,𝑔≲𝑔_{𝐻𝑟+𝑡,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (77)

Proof. In fact, for 𝑔∈𝐻^{𝑟+𝑡,𝑝}(ℝ^𝑛), 𝑇_4,𝜀,𝑔=_{𝜀,𝑗,𝑘∈Λ𝑛}𝑓^𝜀_𝑗,𝑘𝑔^Φ_𝑗,𝑘ℎ^𝜀_𝑗,𝑘. (78) By Lemma 4, using Hölder's inequality, we have 𝑇_4,𝜀≲,𝑔_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^𝑗𝑛𝑔^Φ_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)≲𝑑𝑥_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^{𝑗(𝑛+2𝑡)}𝑔^Φ_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2×_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^{𝑗(𝑛−2𝑡)}ℎ^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲𝑑𝑥_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^{𝑗(𝑛+2𝑡)}𝑔^Φ_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²×𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥^1/𝑝×_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^{𝑗(𝑛−2𝑡)}ℎ^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝′/2𝑑𝑥^1/𝑝′≲𝑓_𝑆^Φ,𝑡_𝑟,𝑝‖ℎ‖_{𝐻′−𝑡,𝑝}. (79) Because 𝑓∈𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛), we can see that |⟨𝑇_4,𝜀,𝑔⟩|≲‖𝑔‖_{𝐻𝑡+𝑟,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}.
Conversely, let 𝜏^𝜀_𝑗,𝑘=2^𝑗𝑡𝑓^𝜀_𝑗,𝑘𝑔^Φ_𝑗,𝑘,𝜏(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}𝜏^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥). (80) Denote ℎ=|𝜏|^𝑝−2𝜏. For ℎ(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}ℎ^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥), we write ℎ_𝑡(𝑥) as ℎ_𝑡(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^𝑗𝑡ℎ^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)∶=_{𝜀,𝑗,𝑘∈Λ𝑛}ℎ_𝑡^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥). (81) It is easy to see that ℎ∈𝐿^𝑝′(ℝ^𝑛) is equivalent to ℎ_𝑡∈𝐻^{−𝑡,𝑝′}(ℝ^𝑛).
By the wavelet characterization of 𝐻^𝑡,𝑝(ℝ^𝑛), we get _{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^{𝑗(𝑛+2𝑡)}𝑔^Φ_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2=𝑑𝑥_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^𝑗𝑛𝜏^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2=|𝜏|𝑑𝑥_𝑝𝑑𝑥=𝜏ℎ𝑑𝑥=_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑡+𝑛/2)}𝑓^𝜀_𝑗,𝑘𝑔^Φ_𝑗,𝑘ℎ^𝜀_𝑗,𝑘=_{𝜀,𝑗,𝑘∈Λ𝑛}2^𝑗𝑛/2𝑓^𝜀_𝑗,𝑘𝑔^Φ_𝑗,𝑘ℎ_𝑡^𝜀_𝑗,𝑘. (82) Further, we can deduce that _{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥0}2^{𝑗(𝑛+2𝑡)}𝑔^Φ_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2=𝑇𝑑𝑥_4,𝜀≲𝑔,𝑔_{𝐻𝑡+𝑟,𝑝}ℎ_{𝑡𝐻′−𝑡,𝑝}≲𝑔_{𝐻𝑡+𝑟,𝑝}‖𝜏‖^𝑝/𝑝′_𝐿𝑝. (83) Hence ‖𝜏‖_𝐿^𝑝<∞ and 𝑓∈𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛). This completes the proof of this lemma.

3.3. Wavelet Characterization of Sobolev Multipliers

Theorem 21. Let 𝑡≥0, 𝑟>0 with 𝑡+𝑟<1<𝑝<𝑛/(𝑡+𝑟). Then, 𝑋^𝑡_𝑟,𝑝ℝ^𝑛=𝑀^𝑡_𝑟,𝑝ℝ^𝑛∩𝑆^Φ,𝑡_𝑟,𝑝ℝ^𝑛. (84)

Proof. We first assume 𝑓∈𝑀^𝑡_𝑟,𝑝(ℝ^𝑛)∩𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛). Let 𝑔∈𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛) and ℎ∈𝐻^{−𝑡,𝑝′}(ℝ^𝑛). By the decomposition given in Section 3.1, we have 𝑓(𝑥)ℎ(𝑥)=_{𝜀∈𝐸𝑛}𝑇_1,𝜀(𝑥)+_{0≤𝑠≤𝑁,𝜀,𝜀′∈𝐸𝑛}𝑇_{2,𝑠,𝜀,𝜀′}(𝑥)+_{𝜀∈𝐸𝑛}𝑇_3,𝜀(𝑥)+_{𝜀∈𝐸𝑛}𝑇_4,𝜀(𝑥)+𝑇₅(𝑥). (85) Notice that 𝑓∈𝑀^𝑡_𝑟,𝑝(ℝ^𝑛) implies that 𝑓∈BMO^𝑟(ℝ^𝑛). By Lemmas 18–20, we obtain that, for 𝑓∈𝑀^𝑡_𝑟,𝑝(ℝ^𝑛)∩𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛), 𝑓ℎ∈𝐻^{−𝑡−𝑟,𝑝′}(ℝ^𝑛) and 𝑓ℎ_{𝐻′−𝑡−𝑟,𝑝}≲𝑓_{𝑀𝑡𝑟,𝑝∩𝑆}^Φ,𝑡_𝑟,𝑝‖ℎ‖_{𝐻′−𝑡,𝑝}, (86) which gives 𝑓_{𝑋𝑡𝑟,𝑝}=sup_{𝑔𝐻𝑟+𝑡,𝑝≤1}sup_{‖ℎ‖𝐻′−𝑡,𝑝}𝑓𝑔,ℎ=sup_{𝑔𝐻𝑟+𝑡,𝑝≤1}sup_{‖ℎ‖𝐻′−𝑡,𝑝}𝑓ℎ,𝑔≤sup_{𝑔𝐻𝑟+𝑡,𝑝≤1}sup_{‖ℎ‖𝐻′−𝑡,𝑝}𝑓ℎ_{𝐻′−𝑡−𝑟,𝑝}𝑔_{𝐻𝑡+𝑟,𝑝}≤𝑓_{𝑀𝑡𝑟,𝑝∩𝑆}^Φ,𝑡_𝑟,𝑝. (87) Hence, we can get 𝑀^𝑡_𝑟,𝑝(ℝ^𝑛)∩𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛)⊂𝑋^𝑡_𝑟,𝑝(ℝ^𝑛).
Conversely, assume that 𝑓∈𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). By Lemma 7, we have 𝑓∈𝑀^𝑡_𝑟,𝑝(ℝ^𝑛)⊂BMO^𝑟(ℝ^𝑛). We apply Lemmas 18 and 19 again to deduce 𝑇_4,𝜀,𝑔≤𝑇_1,𝜀+𝑇_{2,𝑠,𝜀,𝜀′}+𝑇_3,𝜀,𝑔+𝑇₅≤𝑔,𝑔_{𝐻𝑟+𝑡,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}. (88) Then, Lemma 20 implies that 𝑓∈𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛). This completes the proof of this theorem.

Now we characterize 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) by the fractional integration. Let Ψ(𝑥)≥0, Ψ(𝑥)∈𝐶^∞₀(𝐵(0,2)) with Ψ(𝑥)=1 on 𝐵(0,1). It is well known that 𝑔∈𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛) if and only if there exists 𝑔∈𝐿^𝑝(ℝ^𝑛) such that 𝑔(𝑥)=𝐽_𝑡+𝑟𝑔(𝑥)=𝑔𝑦Ψ𝑥−𝑦𝑥−𝑦^{𝑡+𝑟−𝑛}𝑑𝑦. (89) For 𝑔∈𝐿^𝑝(ℝ^𝑛), let 𝑔_𝑗,𝑘=2^−𝑗+𝑦−2^−𝑗𝑘^{𝑡+𝑟−𝑛}𝑔𝑦𝑑𝑦. (90)

Definition 22. Given 𝑟>0, 𝑡≥0 and 1<𝑝<𝑛/(𝑟+𝑡), then 𝑓(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)∈𝑄^𝑡_𝑟,𝑝(ℝ^𝑛) if and only if _{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑡)}𝑔_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥≲1, (91) where 𝑔∈𝐿^𝑝(ℝ^𝑛) and 𝑔(𝑥)≥0.

Let 𝑓(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥) and ℎ(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}ℎ^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥). Define 𝑇_Φ𝑓,ℎ(𝑥)=_{𝑗∈ℕ𝑘∈ℤ𝑛}2^𝑗𝑛𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘Φ2^𝑗𝑥−𝑘. (92) Similar to Lemma 20, we can get the following.

Lemma 23. Given 𝑡≥0, 𝑟>0 and 𝑡+𝑟<1<𝑝<𝑛/(𝑡+𝑟), then 𝑓∈𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛) if and only if 𝑇_Φ𝑓,ℎ,𝑔≲𝑔_{𝐻𝑟+𝑡,𝑝}‖ℎ‖_{𝐻′−𝑡,𝑝}, (93) where 𝑔∈𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛) and ℎ∈𝐻^{−𝑡,𝑝′}(ℝ^𝑛).

Now we give another characterization of 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛).

Theorem 24. Given 𝑡≥0, 𝑟>0 and 𝑡+𝑟<1<𝑝<𝑛/(𝑡+𝑟), then 𝑋^𝑡_𝑟,𝑝ℝ^𝑛=𝑀^𝑡_𝑟,𝑝ℝ^𝑛∩𝑄^𝑡_𝑟,𝑝ℝ^𝑛. (94)

Proof. By modifying the coefficients such that 𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘=|𝑓^𝜀_𝑗,𝑘||ℎ^𝜀_𝑗,𝑘|, we could suppose that 𝑔(𝑦)≥0. By Theorem 21 (ii) and Lemma 23, we know that 𝑓∈𝑆^Φ,𝑡_𝑟,𝑝(ℝ^𝑛) is equivalent to _{𝜀,𝑗,𝑘∈Λ𝑛}𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘2^𝑗𝑛Φ×2^𝑗Ψ𝑔𝑦𝑥−𝑘𝑥−𝑦𝑥−𝑦^{𝑛−(𝑡+𝑟)}≲𝑔𝑑𝑦𝑑𝑥_𝐿𝑝‖ℎ‖_{𝐻′−𝑡,𝑝},∀𝑔≥0. (95)
By a simple calculation, we get 2^𝑗𝑛Φ2^𝑗𝑥−𝑘Ψ𝑥−𝑦𝑥−𝑦^{𝑡+𝑟−𝑛}≃2𝑑𝑥_−𝑗+𝑦−2^−𝑗𝑘^{(𝑡+𝑟−𝑛)}. (96)
So (95) is equivalent to the following inequality: _{𝜀,𝑗,𝑘∈Λ𝑛}𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘2^−𝑗+𝑦−2^−𝑗𝑘^{𝑡+𝑟−𝑛}𝑔𝑦≲𝑔𝑑𝑦_𝐿𝑝‖ℎ‖_{𝐻′−𝑡,𝑝},∀𝑔≥0. (97)
That is to say _{𝜀,𝑗,𝑘∈Λ𝑛}2^𝑗𝑛𝑓^𝜀_𝑗,𝑘ℎ^𝜀_𝑗,𝑘𝜒_𝑗,𝑘×(𝑥)𝑔𝑦2^−𝑗+𝑦−2^−𝑗𝑘^{𝑛−(𝑡+𝑟)}≲𝑔𝑑𝑥𝑑𝑦_𝐿𝑝‖ℎ‖_{𝐻′−𝑡,𝑝},∀𝑔≥0. (98)
Let 𝑆_−𝑡ℎ(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛−2𝑡)}ℎ^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (99)
We have ‖𝑆_−𝑡ℎ‖_𝑝′≃𝑒𝑞‖ℎ‖_{𝐻′−𝑡,𝑝}. Then, the inequality (98) is equivalent to _{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝑔_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑔𝑑𝑥_𝑝𝐿𝑝,∀𝑔≥0. (100)
The above inequality is equivalent to 𝑓∈𝑄^𝑡_𝑟,𝑝(ℝ^𝑛).

4. Logarithmic Morrey Spaces and Multipliers

4.1. A Logarithmic Condition

Lemma 7 implies that 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛)⊂𝑀^𝑡_𝑟,𝑝(ℝ^𝑛). In this section, we consider the reverse inclusion relation. At first we introduce a logarithmic Morrey spaces.

Definition 25. Fix 1<𝑝<𝑛/(𝑟+𝑡) and 𝜏≥0. We say 𝑓(𝑥)∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) if sup_|𝑄|=1|𝑓_𝑡,𝑄|≤𝐶 and (𝐼−Δ)_𝑄^𝑡/2𝑓(𝑥)−𝑓_𝑡,𝑄^𝑝𝑑𝑥≲1−log₂|𝑄|^−𝑝𝜏|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛}, (101) for any cube 𝑄 with |𝑄|≤1.

Similar to Theorem 6, we have the following wavelet characterization of 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛).

Theorem 26. Given 𝑡,𝜏≥0, 𝑟>0, 1<𝑝<𝑛/(𝑟+𝑡), 𝑓(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥) belongs to 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) if and only if _{𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥1−log₂|𝑄|^−𝑝𝜏|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛}, (102) where 𝑄∈Ω with |𝑄|≤1.

Proof. Similar to that of Theorem 6, the proof of this theorem can be obtained by the characterization of Triebel-Lizorkin spaces. See Lemma 4. We omit the detail.

In [8], Fefferman established the following relation: 𝑀^𝑡_𝑟,𝑞ℝ^𝑛⊂𝑋^𝑡_𝑟,𝑝ℝ^𝑛⊂𝑀^𝑡_𝑟,𝑝ℝ^𝑛, (103) where 𝑞>𝑝>1. In this section, we use wavelet characterization to give a logarithmic type inclusion. Let 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑟+𝑡), and 𝜏>1/𝑝^′. In Theorem 32, we prove that 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) is a subspace of 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛). Hence, 𝑀^𝑡,𝜏_𝑟,𝑝ℝ^𝑛⊂𝑋^𝑡_𝑟,𝑝ℝ^𝑛⊂𝑀^𝑡_𝑟,𝑝ℝ^𝑛=𝑀^𝑡,0_𝑟,𝑝ℝ^𝑛. (104)

Lemma 27. Given 𝜏>0, 𝑟>0, 𝑡≥0, and 1<𝑝<𝑛/(𝑟+𝑡), if 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), then 𝑓^𝜀_𝑗,𝑘≲2^{𝑗(𝑟−𝑛/2)}1+𝑗^−𝜏. (105)

Proof. Because 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), then for 𝑄∈Ω, _{𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥1−log₂|𝑄|^−𝑝𝜏|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛}. (106)
We have _{𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛−2𝑟)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲|𝑄|𝑑𝑥_{𝑝(𝑟+𝑡)/𝑛𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲|𝑄|𝑑𝑥_{𝑝(𝑟+𝑡)/𝑛}|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛}1−log₂|𝑄|^−𝜏𝑝≲|𝑄|1−log₂|𝑄|^−𝜏𝑝. (107)
For 𝜀∈𝐸_𝑛, 𝑗∈ℕ, and 𝑘∈ℤ^𝑛, take 𝑄=𝑄_𝑗,𝑘. By Lemma 4, we get 2^{𝑗𝑛−2𝑗𝑟}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥≲2^−𝑗𝑛1−log₂2^{−𝑗𝑛−𝜏𝑝}≲2^−𝑗𝑛1+𝑗^−𝜏𝑝, (108) which gives 𝑓^𝜀_𝑗,𝑘≲2^{𝑗(𝑟−𝑛/2)}1+𝑗^−𝜏. (109)
When 𝜀=0, 𝑓⁰_𝑗,𝑘=_{𝜀′,𝑗′,𝑘′∈Λ𝑛}𝑓^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′},Φ⁰_𝑗,𝑘=_{𝜀′,𝑗′,𝑘′∈Λ𝑛∶𝑗′<𝑗}𝑓^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′},Φ⁰_𝑗,𝑘. (110)
Since _{𝜀′,𝑗′,𝑘′∈Λ𝑛∶𝑗′<𝑗}𝑓^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′}(𝑥)≲2^{(𝑟−𝑛/2)𝑗}1+𝑗^−𝜏, (111)
we have 𝑓⁰_𝑗,𝑘≲2^𝑟𝑗1+𝑗^−𝜏,Φ⁰_𝑗,𝑘≲2^{(𝑟−𝑛/2)𝑗}1+𝑗^−𝜏. (112)

Let 𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑛), 𝛽_𝑖∈ℕ₊, where ℕ₊ denotes the set of all positive integers. Denote by 𝑓_𝛽(𝑥) the derivative (𝜕/𝜕𝑥)^𝛽𝑓. We have the following two lemmas.

Lemma 28. Given 𝜏>0, 𝑟>0, 𝑡≥0, and 1<𝑝<𝑛/(𝑟+𝑡), if 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), the derivative 𝑓_𝛽∈𝑀^{𝑡−|𝛽|,𝜏}_{𝑟+|𝛽|,𝑝}(ℝ^𝑛), where |𝛽|=∑^𝑛_𝑖=1𝛽_𝑖.

Proof. If 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) and 𝑓(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥), by Theorem 26, we have _{𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥1−log₂|𝑄|^−𝑝𝜏|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛}. (113) Denote by 𝑓^𝜀,𝛽_𝑗,𝑘 the wavelet coefficients of 𝑓_𝛽(𝑥). We can get 𝑓^𝛽,𝜀_𝑗,𝑘=2^𝑗|𝛽|𝑓^𝜀_𝑗,𝑘 and _{𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡−2|𝛽|)}𝑓^𝛽,𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥_{𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥1−log₂|𝑄|^−𝑝𝜏|𝑄|^{1−𝑝[(𝑟+|𝛽|)+(𝑡−|𝛽|)]/𝑛}. (114) This implies that 𝑓_𝛽(𝑥)∈𝑀^{𝑡−|𝛽|,𝜏}_{𝑟+|𝛽|,𝑝}(ℝ^𝑛).

To get the sufficient condition for multiplier spaces, we need to consider carefully the relationship of different dyadic cubes relative to combination atoms. Because of this reason, we use always the same Daubechies wavelets in the rest of this section. For a cube 𝑄 and a measurable set 𝐸, if |𝑄⋂𝐸|=|𝑄|, we say that 𝑄 is a cube in the set 𝐸.

For (𝑚,2^𝑢,𝐸_𝑢)-combination atoms 𝑔_𝑢(𝑥) defined in Definition 14, the measure of 𝐸_𝑢 is finite. Hence, the number of biggest dyadic cubes in 𝐸_𝑢 is finite. We denote the number of biggest dyadic cubes in 𝐸_𝑢 by 𝑖₁. Denote by 𝐹_𝑢,1 the set {𝑖∈ℕ,𝑖=1,…,𝑖₁}. If 𝑖∈𝐹_𝑢,1, we denote such cube by 𝑄_𝑢,1,𝑖. The volume of 𝑄_𝑢,1,𝑖 is denoted by 2^{−𝑛𝑗𝑢,1}; that is, |𝑄_𝑢,1,𝑖|=2^{−𝑛𝑗𝑢,1}. Denote 𝐸_𝑢,1=𝐸_𝑢⧵(⋃_{𝑖∈𝐹𝑢,1}𝑄_𝑢,1,𝑖).

The measure of 𝐸_𝑢,1 is finite. We denote the number of biggest dyadic cubes in 𝐸_𝑢,1 by 𝑖₂. Denote by 𝐹_𝑢,2 the set {𝑖∈ℕ,𝑖=1,…,𝑖₂}. If 𝑖∈𝐹_𝑢,2, we denote such cube by 𝑄_𝑢,2,𝑖. The volume of 𝑄_𝑢,2,𝑖 is denoted by 2^{−𝑛𝑗𝑢,2}; that is, |𝑄_𝑢,2,𝑖|=2^{−𝑛𝑗𝑢,2}. Denote 𝐸_𝑢,2=𝐸_𝑢,1⧵(⋃_{𝑖∈𝐹𝑢,2}𝑄_𝑢,2,𝑖).

We continue this process until there exists some 𝑠 such that 𝐸_𝑢,𝑠+1 is empty. For 𝑠^′≥𝑠+1, we denote 𝑖_𝑠′=0, and 𝐹_𝑢,𝑠′ and 𝐸_𝑢,𝑠′ are empty sets. Otherwise we continue until infinity. Then 𝐸_𝑢=⋃_{𝑠≥1,𝑖∈𝐹𝑢,𝑠}𝑄_{𝑢,𝑠,𝑖} and 𝑔_𝑢(𝑥)=∑_{𝑠≥1,𝑖∈𝐹𝑢,𝑠}𝑔_{𝑢,𝑠,𝑖}(𝑥), where 𝑔_{𝑢,𝑠,𝑖}(𝑥)=∑_{𝑄𝑗,𝑘⊂𝑄𝑢,𝑠,𝑖}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥).

To compute the norm of 𝑓(𝑥)𝑔_𝑢(𝑥), we need to find out a special set of dyadic cubes denoted by {𝑄_{𝑢,𝑠,𝑖,𝑘}}_{(𝑠,𝑖,𝑘)∈𝐻𝑢} such that supp𝑔_𝑢⊂⋃_{(𝑠,𝑖,𝑘)∈𝐻𝑢}𝑄_{𝑢,𝑠,𝑖,𝑘}. 𝑔_𝑢(𝑥) is nearly 𝐿^∞ function on 𝑄_{𝑢,𝑠,𝑖,𝑘} and satisfies the estimate of Lemma 29. We divide such process into the following three steps.

Step 1. For all 𝑠≥1, if  𝑖∈𝐹_𝑢,𝑠 and 𝑘∈ℤ^𝑛 with |𝑘|<√𝑛2^(𝑀+3)𝑛, we denote (𝑖,𝑘)∈𝐺_𝑢,𝑠. Denote 𝐹_𝑢={(𝑠,𝑖,𝑘),𝑠≥1,(𝑖,𝑘)∈𝐺_𝑢,𝑠}. For (𝑠,𝑖,𝑘)∈𝐹_𝑢, we denote 𝑄_{𝑢,𝑠,𝑖,𝑘}=2^{−𝑗𝑢,𝑠}𝑘+𝑄_{𝑢,𝑠,𝑖}. For 𝑠≥1, we denote 𝐸⁰_𝑢,𝑠=⋃_{(𝑖,𝑘)∈𝐺𝑢,𝑠}𝑄_{𝑢,𝑠,𝑖,𝑘}. In the next step, we choose a special subcover to the support of 𝑔_𝑢(𝑥).

Step 2. We define now 𝐻_𝑢,𝑠(𝑠≥1), 𝐸¹_𝑢,𝑠(𝑠≥1) and 𝐺¹_𝑢,𝑠(𝑠≥2).

For 𝑠=1, denote 𝐻_𝑢,1=𝐺_𝑢,1 and 𝐸¹_𝑢,𝑠=𝐸⁰_𝑢,𝑠. For 𝑠=2, denote (𝑖,𝑘)∈𝐺¹_𝑢,2, if there exists 0≤𝑗≤𝑗_𝑢,1, 𝑙∈ℤ^𝑛 such that 𝑄_𝑗,𝑙⊂⋃_{|𝑘−𝑘′√|≤𝑛2(𝑀+2)𝑛}𝑄_{𝑢,2,𝑖,𝑘′} and ⟨𝑔_𝑢,Φ⁰_𝑗,𝑙(𝑥)⟩≠0. We know that ⋃_{(𝑖,𝑘)∈𝐺}¹_𝑢,2𝑄_{𝑢,2,𝑖,𝑘}⊂𝐸¹_𝑢,1. Denote 𝐻_𝑢,2=𝐺_𝑢,2⧵𝐺¹_𝑢,2 and 𝐸¹_𝑢,2=⋃_{(𝑖,𝑘)∈𝐻𝑢,2}𝑄_{𝑢,2,𝑖,𝑘}.

For 𝑠=3, denote (𝑖,𝑘)∈𝐺¹_𝑢,3, if there exists 0≤𝑗≤𝑗_𝑢,2, 𝑙∈ℤ^𝑛 such that 𝑄_𝑗,𝑙⊂⋃_{|𝑘−𝑘′√|≤𝑛2(𝑀+2)𝑛}𝑄_{𝑢,2,𝑖,𝑘′} and ⟨𝑔_𝑢(𝑥),Φ⁰_{𝑗𝑢,𝑠,𝑙}(𝑥)⟩≠0. We know that ⋃_{(𝑖,𝑘)∈𝐺}¹_𝑢,3𝑄_{𝑢,3,𝑖,𝑘}⊂⋃_1≤𝑠≤2𝐸¹_𝑢,𝑠. Denote 𝐻_𝑢,3=𝐺_𝑢,3⧵𝐺¹_𝑢,3 and 𝐸¹_𝑢,3=⋃_{(𝑖,𝑘)∈𝐻𝑢,3}𝑄_{𝑢,3,𝑖,𝑘}.

We continue this process until infinity. For 𝑠≥2, maybe, a part of 𝐺¹_𝑢,𝑠, 𝐻_𝑢,𝑠, and 𝐸¹_𝑢,𝑠 is empty set.

Step 3. Let 𝐻_𝑢={(𝑠,𝑖,𝑘),𝑠≥1,(𝑖,𝑘)∈𝐻_𝑢,𝑠}. It is easy to see that the support of 𝑔_𝑢(𝑥) is contained in ⋃_𝑠≥1𝐸¹_𝑢,𝑠. For a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom 𝑔_𝑢(𝑥) and 𝑔⁰_𝑗,𝑘=⟨𝑔_𝑢,Φ⁰_𝑗,𝑘⟩, we have the following estimate.

Lemma 29. Given 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑡+𝑟) and 𝑠≥1, for (𝑖,𝑚^′)∈𝐻_𝑢,𝑠 and 𝑄_𝑗,𝑘⊂𝑄_{𝑢,𝑠,𝑖,𝑚′}, we have |𝑔⁰_𝑗,𝑘|≲2^{𝑢−𝑛𝑗/2}2^{−(𝑡+𝑟)𝑗𝑢,𝑠}.

Proof. By the definition of 𝐻_𝑢,𝑠, we have 𝑔⁰_𝑗,𝑘=_{𝜀′,𝑗′,𝑘′∈Λ𝑛}𝑔^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′},Φ⁰_𝑗,𝑘=_{𝜀′,𝑗′,𝑘′∈Λ𝑛∶𝑗′≥𝑗𝑢,𝑠}𝑔^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′},Φ⁰_𝑗,𝑘. (115)
Because 𝑔_𝑢 is a (𝑚,2^𝑢,𝐸_𝑢)-combination atom, 𝑆_𝑡+𝑟(𝑔_𝑢)(𝑥)≤𝐶2^𝑢. Hence, for every (𝜀^′,𝑗^′,𝑘^′), |𝑔^𝜀′_{𝑗′,𝑘′}|≲2^𝑢2^{−𝑗′(𝑟+𝑡+𝑛/2)}. We can obtain 𝑔⁰_𝑗,𝑘=_{𝜀′,𝑗′,𝑘′∈Λ𝑛∶𝑗′≥𝑗𝑢,𝑠}𝑔^𝜀′_{𝑗′,𝑘′}Φ^𝜀′_{𝑗′,𝑘′},Φ⁰_𝑗,𝑘≲2^𝑢_{𝑗′≥𝑗𝑢,𝑠}2^{−𝑗′(𝑟+𝑡)}2^{−𝑗𝑛′/2}2^{𝑗′𝑛/2}2^𝑗𝑛/2×Φ^𝜀′2^𝑗′𝑥−𝑘^′Φ⁰2^𝑗𝑥−𝑘𝑑𝑥≲2^𝑢2^{−𝑗𝑛/2}2^{−𝑗𝑢,𝑠(𝑟+𝑡)}. (116)

Theorem 30. Suppose that 𝜏>1/𝑝^′, 𝑡≥0, 𝑟>0, and 1<𝑝<𝑛/(𝑟+𝑡). If 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) and 𝑔_𝑢 is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, then, for 𝑠≥1, (𝑖,𝑚^′)∈𝐻_𝑢,𝑠, and 𝑄=𝑄_{𝑢,𝑠,𝑖,𝑚′}, we have 𝑓𝑔_{𝑢𝐻𝑡,𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝. (117)

Proof. First, for 𝑡≥0, we prove ‖𝑓𝑔_𝑢‖_{𝐿𝑝(𝑄)}≤𝐶𝑗^−𝜏_𝑢,𝑠2^𝑢|𝑄|^1/𝑝. Let 𝑓(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥) and 𝑔_𝑢(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥). Denote by Λ^′_𝑛 the set 𝜀,𝜀^′,𝑗,𝑘,𝑙,𝜀,𝜀^′∈𝐸_𝑛,(𝜀,𝑘)≠𝜀^′,𝑘+𝑙,𝑗∈ℤ,𝑘,𝑙∈ℤ^𝑛,|𝑙|≤√𝑛2^(𝑀+2)𝑛. (118) For any dyadic cube 𝑄⊂𝐸_𝑢, by the formulas (11) and (12), we decompose 𝑓𝑔_𝑢 as 𝑇₁(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗=𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓⁰_{𝑗,𝑘+𝑙}𝑔⁰_𝑗,𝑘Φ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ⁰_𝑗,𝑘(𝑥),𝑇₂(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓⁰_{𝑗,𝑘+𝑙}𝑔^𝜀_𝑗,𝑘Φ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥),𝑇₃(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓^𝜀_{𝑗,𝑘+𝑙}𝑔⁰_𝑗,𝑘Φ^𝜀_{𝑗,𝑘+𝑙}(𝑥)Φ⁰_𝑗,𝑘(𝑥),𝑇₄(𝑥)=_{𝜀,𝜀′,𝑗,𝑘,𝑙∈Λ′𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓^𝜀′_{𝑗,𝑘+𝑙}×𝑔^𝜀_𝑗,𝑘Φ^𝜀′_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥),𝑇₅(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)²−2^𝑛𝑗/2Φ⁰_𝑗,𝑘(𝑥),𝑇₆(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘2^𝑛𝑗/2Φ⁰_𝑗,𝑘(𝑥). (119) The rest of the proof is divided into three steps.

Step 1. For 𝑡≥0, we estimate ‖𝑇_𝑖‖_{𝐿𝑝(𝑄)},𝑖=1,2,…,6. By Lemma 4, we obtain 𝑓=_{𝜀,𝑗,𝑘∈Λ𝑛}𝑓^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)∈𝐿^𝑝⟺_{𝜀,𝑗,𝑘∈Λ𝑛}2^𝑗𝑛𝑓^𝜀_𝑗,𝑘²𝜒2^𝑗⋅−𝑘^1/2_𝐿𝑝. (120) Let 𝑆₀𝑓(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}2^𝑗𝑛𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (121) We have ‖𝑓‖_𝐿𝑝≃‖𝑆₀(𝑓)‖_𝐿𝑝.

(1) Because 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝, by Lemma 27, 𝑓⁰_{𝑗𝑢,𝑠,𝑘}≲1+𝑗_𝑢,𝑠^−𝜏2^{(𝑟−𝑛/2)𝑗𝑢,𝑠}. (122) By Lemma 29, we have |𝑔⁰_{𝑗𝑢,𝑠,𝑘}|≲2^{𝑢−(𝑟+𝑡+𝑛/2)𝑗𝑢,𝑠}. Hence we can get 𝑆₀𝑇₁(𝑥)=_{𝑄𝑗,𝑘=𝑄𝑗𝑢,𝑠,𝑘}2^{2𝑗𝑢,𝑠𝑛}𝑓⁰_{𝑗𝑢,𝑠,𝑘}²𝑔⁰_{𝑗𝑢,𝑠,𝑘}²𝜒2^{𝑗𝑢,𝑠}𝑥−𝑘^1/2≲_{𝑄𝑗,𝑘=𝑄𝑗𝑢,𝑠,𝑘}2^{2𝑗𝑢,𝑠𝑛}1+𝑗_𝑢,𝑠^−2𝜏2^{2(𝑟−𝑛/2)𝑗𝑢,𝑠}×2^{2𝑢−2(𝑟+𝑡+𝑛/2)𝑗𝑢,𝑠}2^{2𝑗𝑢,𝑠(𝑛/2+𝑡)}𝜒2^{𝑗𝑢,𝑠}𝑥−𝑘^1/2. (123) Because 𝑗_𝑢,𝑠≥0, we have 𝑆₀𝑇₁(𝑥)≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢_{𝑄𝑗,𝑘=𝑄𝑗𝑢,𝑠,𝑘}𝜒_𝑗,𝑘(𝑥)^1/2,𝑇_{1𝐿𝑝(𝑄)}=𝑆₀𝑇_{1𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝. (124)

(2) Now we estimate 𝑇₂(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓⁰_{𝑗,𝑘+𝑙}𝑔^𝜀_𝑗,𝑘Φ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥). (125) Because 𝑗≥𝑗_𝑢,𝑠≥0, we have 𝑓⁰_𝑗,𝑘≲1+𝑗_𝑢,𝑠^−𝜏2^{(𝑟−𝑛/2)𝑗}≤1+𝑗_𝑢,𝑠^−𝜏2^{(𝑟+𝑡−𝑛/2)𝑗}. (126) Let Λ^𝑄_𝑛,2=𝜀,𝑗,𝑘∈Λ_𝑛∣𝑗≥𝑗_𝑢,𝑠,∀|𝑙|≤√𝑛2^(𝑀+2)𝑛,suppΦ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥)∩𝑄≠0. (127) Then, we can get 𝑆₀𝑇₂(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,2}𝑓⁰_{𝑗,𝑘+𝑙}²𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,2}2^2𝑗𝑛1+𝑗^−2𝜏×2^{2𝑗(𝑟+𝑡−𝑛/2)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲1+𝑗_𝑢,𝑠^−𝜏𝑆_𝑡+𝑟𝑔_𝑢(𝑥). (128) Because 𝑔_𝑢(𝑥) is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, 𝑆₀𝑇_{2𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏𝑆_𝑡+𝑟𝑔_{𝑢𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝. (129)

(3) Since 𝑔_𝑢(𝑥) is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, for 𝑠≥1, (𝑖,𝑚^′)∈𝐻_𝑢,𝑠, 𝑗≥0, and 𝑄=𝑄_{𝑢,𝑠,𝑖,𝑚′}, 𝑔⁰_𝑗,𝑘≲2^{𝑢−𝑛𝑗/2}2^{−(𝑟+𝑡)𝑗}=2^{𝑢−𝑛𝑗/2}|𝑄|^{(𝑟+𝑡)/𝑛}. (130) Let Λ^𝑄_𝑛,3=𝜀,𝑗,𝑘∈Λ_𝑛∣𝑗≥𝑗_𝑢,𝑠,∀|𝑙|≤√𝑛2^(𝑀+2)𝑛,suppΦ⁰_𝑗,𝑘Φ^𝜀_{𝑗,𝑘+𝑙}∩𝑄≠0. (131) We have, by 𝑗_𝑢,𝑠≥0, 𝑆₀𝑇₃(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,3}2^𝑗𝑛𝑔⁰_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑋)^1/2≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,3}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (132) By the fact that 𝑗≥𝑗_𝑢,𝑠 and 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), we get 𝑆₀𝑇_{3𝐿𝑝(𝑄)}≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}1−log₂|𝑄|^−𝜏|𝑄|^{1/𝑝−(𝑟+𝑡)/𝑛}≲2^𝑢1+𝑗_𝑢,𝑠^−𝜏|𝑄|^1/𝑝. (133)

(4) Now we estimate the term 𝑇₄(𝑥). Let Λ^𝑄_𝑛,4=𝜀,𝑗,𝑘∈Λ_𝑛∣𝑗≥𝑗_𝑢,𝑠,∀𝜀,𝜀^′,𝑗,𝑘,𝑙∈Λ^′_𝑛,suppΦ^𝜀′_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥)∩𝑄≠0. (134) Because 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), we have 𝑓^𝜀_𝑗,𝑘≲1+𝑗^−𝜏2^{𝑗(𝑟−𝑛/2)}≲1+𝑗^−𝜏2^{𝑗(𝑟+𝑡−𝑛/2)},𝑆₀𝑇₄(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,4}2^𝑗𝑛1+𝑗^−2𝜏×2^{𝑗(2𝑟+2𝑡−𝑛)}2^𝑗𝑛𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲1+𝑗_𝑢,𝑠^−𝜏𝑆_𝑡+𝑟𝑔_𝑢(𝑥). (135)(136) Then we can get, by the fact that 𝑔_𝑢 is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, 𝑆₀𝑇_{4𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏𝑆_𝑡+𝑟𝑔_{𝑢𝐿𝑝(𝑄)}≲2^𝑢1+𝑗_𝑢,𝑠^−𝜏|𝑄|^1/𝑝. (137)

(5) Now we estimate the term 𝑇₅(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)²−2^𝑛𝑗/2Φ⁰_𝑗,𝑘(𝑥). (138) Because the function (Φ^𝜀_𝑗,𝑘)²−2^𝑛𝑗/2Φ⁰_𝑗,𝑘 plays the role as that of Φ^𝜀_𝑗,𝑘(𝑥), we have 𝑆₀𝑇_{5𝐿𝑝(𝑄)}≲2^𝑢1+𝑗_𝑢,𝑠^−𝜏|𝑄|^1/𝑝. (139)

(6) To estimate the term 𝑇₆(𝑥), we take ℎ∈𝐿^𝑝′(𝑄). Let Λ^𝑄_𝑛,6=𝜀,𝑗,𝑘∈Λ_𝑛∣𝑗≥𝑗_𝑢,𝑠,suppΦ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥)∩𝑄≠0. (140) By the orthogonality of the wavelet function, we have 𝑇₆,ℎ=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘2^𝑗𝑛/2Φ⁰_𝑗,𝑘,ℎ=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘2^𝑛𝑗/2ℎ⁰_𝑗,𝑘. (141) Then, 𝑇₆,ℎ≲_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^𝑛𝑗𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘2^𝑛𝑗/2ℎ⁰_𝑗,𝑘𝜒_𝑗,𝑘(𝑥)≲𝑑𝑥_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^𝑛𝑗𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘𝜒_𝑗,𝑘(𝑥)𝑀(ℎ)(𝑥)𝑑𝑥. (142) By Hölder's inequality and 𝑗≥𝑗_𝑢,𝑠≥0, it can be deduced that _{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^𝑛𝑗𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘𝜒_𝑗,𝑘(𝑥)≲_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛−2𝑡−2𝑟)𝑗}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2×_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛+2𝑡+2𝑟)𝑗}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛−2𝑡−2𝑟)𝑗}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (143) Finally, we can get 𝑇₆,ℎ≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛−2𝑡−2𝑟)𝑗}×𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2_{𝐿𝑝(𝑄)}‖ℎ‖_{𝐿𝑝′(𝑄)}≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}1−log₂𝑄‖^−𝜏|𝑄|^{1/𝑝−(𝑟+𝑡)/𝑛}≲2^𝑢|𝑄|^1/𝑝1+𝑗_𝑢,𝑠^−𝜏. (144) Because ℎ takes over all functions in 𝐿^𝑝′(𝑄), it is obvious that 𝑆₀(𝑇₆)_{𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝. (145)

Step 2. Assume that 0≤𝑡<1. Let 𝑔(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥) and 𝑆_𝑡𝑔(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(2𝑡+𝑛)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (146)

We need to prove that 𝑆_𝑡𝑓𝑔_{𝑢𝐿𝑝(𝑄)}≲2^𝑢1+𝑗_𝑢,𝑠^−𝜏|𝑄|^1/𝑝. (147)

The index sets Λ^𝑄_𝑛,𝑖,𝑖=1,2,…,6, are the same as Step 1.

(1) For the term 𝑇₁, we have 𝑆_𝑡𝑇₁(𝑥)=_{𝑄𝑗,𝑘=𝑄𝑗𝑢,𝑠,𝑘}2^{𝑗𝑢,𝑠(𝑛+2𝑡)}2^{𝑗𝑢,𝑠𝑛}×𝑓⁰_{𝑗𝑢,𝑠,𝑘}²𝑔⁰_{𝑗𝑢,𝑠,𝑘}²𝜒2^{𝑗𝑢,𝑠}𝑥−𝑘^1/2≲_{𝑄𝑗,𝑘=𝑄𝑗𝑢,𝑠,𝑘}2^{𝑗𝑢,𝑠(𝑛+2𝑡)}2^{𝑗𝑢,𝑠𝑛}1+𝑗_𝑢,𝑠^−2𝜏2^{2(𝑟−𝑛/2)𝑗𝑢,𝑠}×2^{2𝑢−2(𝑟+𝑡+𝑛/2)𝑗𝑢,𝑠}2^{2𝑗𝑢,𝑠(𝑛/2+𝑡)}𝜒2^{𝑗𝑢,𝑠}𝑥−𝑘^1/2≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢_{𝑄𝑗,𝑘=𝑄𝑗𝑢,𝑠,𝑘}𝜒_𝑗,𝑘(𝑥)^1/2. (148) Then, we have 𝑇_{1𝐻𝑡,𝑝(𝑄)}=𝑆_𝑡𝑇_{1𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝. (149)

(2) For the term 𝑇₂(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓⁰_{𝑗,𝑘+𝑙}𝑔^𝜀_𝑗,𝑘Φ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥), (150) we have 𝑆_𝑡𝑇₂(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,2}2^{𝑗(𝑛+2𝑡)}2^𝑗𝑛𝑓⁰_{𝑗,𝑘+𝑙}²𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,2}2^{𝑗(𝑛+2𝑡)}2^𝑗𝑛1+𝑗^−2𝜏×2^{2𝑗(𝑟−𝑛/2)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲1+𝑗_𝑢,𝑠^−𝜏𝑆_𝑡+𝑟𝑔_𝑢(𝑥). (151) Because 𝑔_𝑢(𝑥) is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, we have 𝑆_𝑡𝑇_{2𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏𝑆_𝑡+𝑟𝑔_{𝑢𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝. (152)

(3) Because 𝑔_𝑢(𝑥) is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, for 𝑠≥1, (𝑖,𝑚^′)∈𝐻_𝑢,𝑠, and 𝑄=𝑄_{𝑢,𝑠,𝑖,𝑚′}, we can get 𝑔⁰_𝑗,𝑘≲2^{𝑢−𝑛𝑗/2}2^{−(𝑟+𝑡)𝑗}=2^{𝑢−𝑛𝑗/2}|𝑄|^{(𝑟+𝑡)/𝑛},𝑆_𝑡𝑇₃(𝑥)≲2^𝑢_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,3}2^{𝑗(𝑛+2𝑡)}2^−𝑛𝑗2^{−(𝑟+𝑡)𝑗}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,3}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (153) From the fact that 𝑗≥𝑗_𝑢,𝑠 and 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), we deduce that 𝑆_𝑡(𝑇₃)_{𝐿𝑝(𝑄)}≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}−log₂|𝑄|^−𝜏|𝑄|^{1/𝑝−(𝑟+𝑡)/𝑛}≲2^𝑢𝑗^−𝜏_𝑢,𝑠|𝑄|^1/𝑝. (154)

(4) For the term 𝑇₄(𝑥), because 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) and 𝑔_𝑢 is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, we have |𝑓^𝜀_𝑗,𝑘|≲(1+𝑗)^−𝜏2^{𝑗(𝑟−𝑛/2)} and 𝑆_𝑡𝑇₄(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,4}2^{𝑗(𝑛+2𝑡)}1+𝑗^−2𝜏×2^{𝑗(2𝑟−𝑛)}2^𝑗𝑛𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲1+𝑗_𝑢,𝑠^−𝜏𝑆_𝑡+𝑟𝑔_𝑢(𝑥). (155) Then, we can get 𝑆_𝑡𝑇_{4𝐿𝑝(𝑄)}≲(1+𝑗_𝑢,𝑠)^−𝜏𝑆_𝑡+𝑟𝑔_{𝑢𝐿𝑝(𝑄)}≲2^𝑢1+𝑗_𝑢,𝑠^−𝜏|𝑄|^1/𝑝. (156)

(5) Now we estimate the term 𝑇₅(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)²−2^𝑛𝑗/2Φ⁰_𝑗,𝑘(𝑥). (157) Because the function (Φ^𝜀_𝑗,𝑘(𝑥))²−2^𝑛𝑗/2Φ⁰_𝑗,𝑘(𝑥) plays the role as that of Φ^𝜀_𝑗,𝑘(𝑥), we have 𝑆_𝑡𝑇_{5𝐿𝑝(𝑄)}≲2^𝑢1+𝑗_𝑢,𝑠^−𝜏|𝑄|^1/𝑝. (158)

(6) For the term 𝑇₆(𝑥), we take ℎ∈𝐻^{−𝑡,𝑝′}(𝑄). By the orthogonality of the wavelet functions, we have 𝑇₆,ℎ=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘2^𝑗𝑛/2Φ⁰_𝑗,𝑘,ℎ=_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘2^𝑛𝑗/2ℎ⁰_𝑗,𝑘. (159) We can get 𝑇₆,ℎ≲_{𝜀,𝑗,𝑘∈Λ𝑄𝑛}2^𝑛𝑗/2𝑔^𝜀_𝑗,𝑘𝑓^𝜀_𝑗,𝑘2^𝑗𝑛ℎ⁰_𝑗,𝑘𝜒_𝑗,𝑘(𝑥)≲𝑑𝑥_{𝜀,𝑗,𝑘∈Λ𝑄𝑛}2^𝑛𝑗𝑔^𝜀_𝑗,𝑘𝑓^𝜀_𝑗,𝑘𝑀(ℎ)(𝑥)𝜒_𝑗,𝑘(𝑥)𝑑𝑥. (160) By Hölder's inequality, we have _{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^𝑛𝑗𝑓^𝜀_𝑗,𝑘𝑔^𝜀_𝑗,𝑘𝜒_𝑗,𝑘(𝑥)≲_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛−2𝑡−2𝑟)𝑗}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2×_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛+2𝑡+2𝑟)𝑗}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2≲𝑆_𝑡+𝑟𝑔_𝑢(𝑥)_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛−2𝑡−2𝑟)𝑗}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2. (161) Because 𝑗≥𝑗_𝑢,𝑠 implies that 2^{−𝑗(𝑟+𝑡)}≤2^{−𝑗𝑢,𝑠(𝑟+𝑡)}, we can get 𝑇₆,ℎ≲2^𝑢_{𝜀,𝑗,𝑘∈Λ𝑄𝑛,6}2^{(𝑛−2𝑡−2𝑟)𝑗}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2_{𝐿𝑝(𝑄)}×‖ℎ‖_{𝐿𝑝′(𝑄)}≲2^𝑢|𝑄|^{(𝑟+𝑡)/𝑛}1−log₂|𝑄|^−𝜏|𝑄|^{1/𝑝−(𝑟+𝑡)/𝑛}≲2^𝑢|𝑄|^1/𝑝1+𝑗_𝑢,𝑠^−𝜏. (162)

Because ℎ takes over all functions in 𝐻^{−𝑡,𝑝′}(𝑄), we can get 𝑆_𝑡𝑇_{6𝐿𝑝(𝑄)}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝. (163)

This completes the proof for the case 0≤𝑡<1.

Step 3. Now we consider the case 𝑡≥1. In this case, there exists an integer [𝑡] such that [𝑡]≤𝑡<[𝑡]+1. For any 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛)∈ℕ^𝑛 with |𝛼|=∑^𝑛_𝑖=1𝛼_𝑖=[𝑡], the derivative 𝜕^𝛼/𝜕𝑥^𝛼 of the product 𝑓𝑔_𝑢 can be represented as 𝜕^𝛼𝜕𝑥^𝛼𝑓𝑔_𝑢=_𝛾,𝛽𝐶_{𝛼,𝛽,𝛾}𝜕^𝛽𝜕𝑥^𝛽𝑓(𝑥)𝜕^𝛾𝜕𝑥^𝛾𝑔_𝑢(𝑥), (164) where |𝛼|=|𝛽|+|𝛾|. Denote (𝜕^𝛽/𝜕𝑥^𝛽)𝑓(𝑥) and (𝜕^𝛾/𝜕𝑥^𝛾)𝑔_𝑢(𝑥) by 𝑓_𝛽(𝑥) and 𝑔_𝑢,𝛾(𝑥), respectively. Applying the conclusion in Step 1, we only need to prove 𝑓_𝛽𝑔_{𝑢,𝛾𝐻𝑡−[𝑡],𝑝}≲1+𝑗_𝑢,𝑠^−𝜏2^𝑢|𝑄|^1/𝑝, (165) that is, ‖𝑆_{𝑡−[𝑡]}(𝑓_𝛽𝑔_𝑢,𝛾)‖_𝐿𝑝≲2^𝑢(1+𝑗_𝑢,𝑠)^−𝜏|𝑄|^1/𝑝.

If 𝑔_𝑢 is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, then 𝑔_𝑢,𝛾(𝑥)=_{𝑄𝑗,𝑘∈ℱ𝑣,𝑙}𝑔^𝜀_𝑗,𝑘𝜕𝜕𝑥^𝛾2^𝑛𝑗/2Φ^𝜀_𝑗,𝑘(𝑥)=_{𝑄𝑗,𝑘∈ℱ𝑣,𝑙}𝑔^𝜀_𝑗,𝑘2^𝑛𝑗/22^𝑗|𝛾|𝜕^𝛾𝜕𝑥^𝛾Φ^𝜀2^𝑗𝑥−𝑘. (166) Hence, 𝑆_{𝑡−|𝛾|}𝑔_𝑢,𝛾(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑡−2|𝛾|)}𝑔^𝜀_𝑗,𝑘²2^𝑗|𝛾|𝜒_𝑗,𝑘(𝑥)^1/2=_{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑡)}𝑔^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^1/2=𝑆_𝑡𝑔_𝑢(𝑥). (167) On the other hand, if 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), then _{𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡)}𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥1−log₂|𝑄|^−𝑝𝜏|𝑄|^{1−𝑝(𝑟+𝑡)/𝑛} (168) and (𝑓_𝛽)^𝜀_𝑗,𝑘=2^𝑗|𝛽|𝑓^𝜀_𝑗,𝑘. We have _{𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡−2|𝛽|)}𝑓^𝛽,𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2=𝑑𝑥_{𝑄𝜀∈𝐸𝑛,𝑄𝑗,𝑘⊂𝑄}2^{𝑗(𝑛+2𝑡−2|𝛽|)}2^2𝑗|𝛽|𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2≲𝑑𝑥1−log₂|𝑄|^−𝑝𝜏|𝑄|^{1−𝑝(𝑡−|𝛽|+𝑟+|𝛽|)/𝑛}; (169) that is, 𝑓_𝛽(𝑥)∈𝑀^{𝑡−|𝛽|,𝜏}_{𝑟+|𝛽|,𝑝}(ℝ^𝑛).

For any cube 𝑄, the function 𝑓_𝛽(𝑥)=∑_{(𝜀,𝑗,𝑘)∈Λ𝑛}𝑓^𝛽,𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥), and any dyadic cube 𝑄⊂𝑆_𝑢, we divide the product 𝑓_𝛽⋅𝑔_𝑢,𝛾 into the following parts: 𝑇₁^𝛽,𝛾(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗=𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓^𝛽,0_{𝑗,𝑘+𝑙}𝑔^𝛾,0_𝑗,𝑘Φ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ⁰_𝑗,𝑘(𝑥),𝑇₂^𝛽,𝛾(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓^𝛽,0_{𝑗,𝑘+𝑙}𝑔^𝛾,𝜀_𝑗,𝑘Φ⁰_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥),𝑇₃^𝛽,𝛾(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓^𝛽,𝜀_{𝑗,𝑘+𝑙}𝑔^𝛾,0_𝑗,𝑘Φ^𝜀_{𝑗,𝑘+𝑙}(𝑥)Φ⁰_𝑗,𝑘(𝑥),𝑇₄^𝛽,𝛾(𝑥)=_{𝜀,𝜀′,𝑗,𝑘,𝑙∈Λ′𝑛,𝑗≥𝑗𝑢,𝑠|𝑙|≤√𝑛2(𝑀+2)𝑛}𝑓^𝛽,𝜀′_{𝑗,𝑘+𝑙}𝑔^𝛾,𝜀_𝑗,𝑘Φ^𝜀′_{𝑗,𝑘+𝑙}(𝑥)Φ^𝜀_𝑗,𝑘(𝑥),𝑇₅^𝛽,𝛾(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠}𝑓^𝛽,𝜀_𝑗,𝑘𝑔^𝛾,𝜀_𝑗,𝑘Φ^𝜀_𝑗,𝑘(𝑥)²−2^𝑛𝑗/2Φ⁰_𝑗,𝑘(𝑥),𝑇₆^𝛽,𝛾(𝑥)=_{𝜀,𝑗,𝑘∈Λ𝑛,𝑗≥𝑗𝑢,𝑠}𝑓^𝛽,𝜀_𝑗,𝑘𝑔^𝛾,𝜀_𝑗,𝑘2^𝑛𝑗/2Φ⁰_𝑗,𝑘(𝑥). (170) Similar to the method used in the case 0≤𝑡<1, we can complete the proof of the case 𝑡≥1. This completes the proof of this theorem.

By Theorem 30, we can get the following lemma.

Lemma 31. Given 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑟+𝑡), and 𝜏>1/𝑝^′, if 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) and 𝑔_𝑢 is a (𝑡+𝑟,2^𝑢,𝐸_𝑢)-combination atom, then 𝑓𝑔_{𝑢𝐻𝑡,𝑝}≲(1+𝑢)^−𝜏2^𝑢𝐸_𝑢^1/𝑝. (171)

Theorem 32. Given 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑟+𝑡), and 𝜏>1/𝑝^′, then 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛)⊂𝑋^𝑡_𝑟,𝑝(ℝ^𝑛).

Proof. By Lemma 31, we have 𝑓𝑔_{𝐻𝑡,𝑝}≲_𝑢≥0(1+𝑢)^−𝜏2^𝑢𝐸_𝑢^1/𝑝≲_𝑢≥0(1+𝑢)^{−𝑝′𝜏1/𝑝′}_𝑢≥02^𝑝𝑢𝐸_𝑢^1/𝑝≲𝑔_{𝐻𝑡+𝑟,𝑝}. (172)

4.2. The Sharpness for 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛)

In this section, applying our wavelet characterization of multiplier spaces and fractal theory, we prove that the scope of the index of 𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) obtained in Theorem 32 is sharp for 𝑟+𝑡<1. Precisely, by Meyer wavelets, we construct a counterexample to show that Theorem 32 is not true for the case 0≤𝜏≤1/𝑝^′.

Our key idea is to construct a group of sets 𝐶_𝑠 composed by special dyadic cubes and fractal set 𝐻 with Hausdorff dimension 𝑛−(𝑡+𝑟)𝑝. Denote by 𝑆_𝑠 the union ⋃_{𝑄∈𝐶𝑠}𝑄 and 𝐻=⋂_𝑠≥1𝑆_𝑠. By the above dyadic cubes 𝑆_𝑠,𝑠≥0, we construct a special 𝐿^𝑝 function 𝑔(𝑥), which is bounded on 𝑆_𝑠⧵𝑆_𝑠+1 for all 𝑠≥1. The fractional integration 𝐼_𝑟+𝑡𝑔(𝑥) bumps on the fractal set 𝐻. Then we construct a multiplier 𝑓(𝑥) such that its wavelet coefficients are based on these special dyadic cubes 𝐶_𝑠 for all 𝑠≥1. Applying our wavelet characterization of multiplier spaces, we prove that the product of the above multiplier 𝑓(𝑥) and the function 𝐼_𝑟+𝑡𝑔(𝑥) will go out to the desired space 𝐻^𝑡,𝑝(ℝ^𝑛).

Theorem 33. If 0≤𝜏≤1/𝑝^′, there exists 𝑓∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛) such that 𝑓∉𝑋^𝑡_𝑟,𝑝(ℝ^𝑛).

Proof. We use Meyer wavelets and suppose that 𝜀=(1,1,…,1) and Φ^𝜀(0)>0. First of all, we construct a group of self-similar cubes {𝑄_𝑢} such that the limitation is a set with Hausdorff measure 𝑛−(𝑡+𝑟)𝑝.
We construct two series of integers {𝜏_𝑠}_𝑠≥1 and {𝑣_𝑠}_𝑠≥1 such that 𝑉₁=max4𝑛𝑛−(𝑡+𝑟)𝑝,2^(𝑀+2)𝑛,1≤𝜏_𝑠<4𝑛𝑛−(𝑡+𝑟)𝑝≤𝑣_𝑠≤max5𝑛𝑛−(𝑡+𝑟)𝑝,2^(𝑀+2)𝑛. (173) Denote 𝜎_𝑠=∑_{1≤𝑖≤𝑠}𝜏_𝑖 and 𝑢_𝑠=∑_{1≤𝑖≤𝑠}𝑣_𝑖. We take 𝜏_𝑠 such that there exists 𝐶>0 satisfying that 2^{𝑛𝜎𝑠−(𝑛−(𝑡+𝑟)𝑝)𝑢𝑠}≥2^−𝑛𝑢1,lim_𝑠→+∞2^{𝑛𝜎𝑠−(𝑛−(𝑡+𝑟)𝑝)𝑢𝑠}()=𝐶,𝑛−𝑡+𝑟𝑝=lim_𝑠→+∞𝑛𝜎_𝑠𝑢_𝑠. (174) For 𝜏_𝑠 and 𝑣_𝑠, denote 𝑙_𝑠=(𝑙_𝑠,1,𝑙_𝑠,2,…,𝑙_𝑠,𝑛)∈ℤ^𝑛_𝑠,0 if 𝑙_𝑠∈ℤ^𝑛, and for 𝑖=1,2,…,𝑛, we have 0≤𝑙_𝑠,𝑖<2^𝜏𝑠−1or0≤𝑙_𝑠,𝑖+2^𝜏𝑠−1−2^𝑣𝑠<2^𝜏𝑠−1. (175) Denote ℤ^𝑛₁=ℤ^𝑛_1,0, and, for 𝑠≥2, denote 𝑘_𝑠∈ℤ^𝑛_𝑠, if there exists 𝑘_𝑠−1∈ℤ^𝑛_𝑠−1 and 𝑙_𝑠∈ℤ^𝑛_𝑠,0 such that 𝑘_𝑠=2^𝑣𝑠𝑘_𝑠−1+𝑙_𝑠.
We divide the unit dyadic cube 𝑄₀=[0,1]^𝑛 into 2^𝑛𝑣1 dyadic cubes. Then, we reserve the 2^𝑛𝜏1 dyadic cubes which are near the 2^𝑛 vertex points; that is, we reserve 𝐶₁={𝑄_𝑣1,𝑙1∶𝑙₁∈ℤ^𝑛_1,0} and denote 𝑥∈𝑆₁, if there exists 𝑙₁∈ℤ^𝑛_1,0 such that 𝑥∈𝑄_𝑣1,𝑙1.
For the dyadic cube 𝑄_𝑣1,𝑙1∈𝐶₁, we divide it into 2^𝑛𝑣2 dyadic cubes, and we reserve the 2^𝑛𝜏2 dyadic cubes which are near the 2^𝑛 vertex points; that is, we reserve 𝐶_2,𝑙1=𝑄_𝑢2,2^𝑣_2𝑙1+𝑙2,𝑙₂∈ℤ^𝑛_1,0. (176) For 𝑠=2, denote 𝐶₂=𝑄_𝑢2,𝑘2,𝑘₂=2^𝑣2𝑙₁+𝑙₂,𝑙₁,𝑙₂∈ℤ^𝑛_1,0 (177) and denote 𝑥∈𝑆₂ if there exists 𝑘₂∈ℤ^𝑛₂ such that 𝑥∈𝑄_𝑢2,𝑘2.
We continue this process until infinity, and we get a series of dyadic cubes 𝐶_𝑠 and sets 𝑆_𝑠. We know that |𝑆_𝑠|=2^{𝑛(𝜎𝑠−𝑢𝑠)} and the limitation of {𝑆_𝑠} is a fractal set 𝐻 with Hausdorff dimension 𝑛−(𝑡+𝑟)𝑝.
For 𝑠≥1, let 𝑓_𝑠(𝑥)=_{𝑄𝑢𝑠𝑠,𝑘⊂𝐶𝑠}𝑓^𝜀_{𝑢𝑠,𝑘𝑠}Φ^𝜀_{𝑢𝑠,𝑘𝑠}(𝑥), (178) where 𝑓^𝜀_{𝑢𝑠,𝑘𝑠}=𝑠^−1/𝑝′2^{−𝑡𝑢𝑠}2^{(𝑡+𝑟)𝑢𝑠}. Let 𝑓(𝑥)=∑_𝑠≥1𝑓_𝑠(𝑥). Applying the wavelet characterization of Morrey spaces, 𝑓∈𝑀^{𝑡,1/𝑝′}_𝑟,𝑝(ℝ^𝑛). In fact, for 𝑠,𝑙≥1 and any cube 𝑄 with 2^{−𝑛𝑢𝑠+1}≤|𝑄|<2^{−𝑛𝑢𝑠}, we have 𝑄∩𝑆_𝑠+𝑙⧵𝑆_𝑠+𝑙+1≲2^{−𝑛(𝜎𝑠+𝑙−𝜎𝑠)−𝑛𝑢𝑠}. (179) Hence, we get _{𝑄𝑄𝑢𝑠𝑠,𝑘⊂𝑄}2^{𝑢𝑠(𝑛+2𝑡)}𝑓^𝜀_{𝑢𝑠,𝑘𝑠}²𝜒2^𝑢𝑠𝑥−𝑘_𝑠^𝑝/2≲𝑑𝑥_{𝑄⧵𝑆𝑠+1𝑄𝑢𝑠𝑠,𝑘⊂𝑄}2^{𝑢𝑠(𝑛+2𝑡)}𝑠^−2/𝑝′×2^{−2𝑡𝑢𝑠}2^{2(𝑟+𝑡)𝑢𝑠}𝜒2^𝑢𝑠𝑥−𝑘_𝑠^𝑝/2𝑑𝑥+_{𝑙≥1𝑄∩(𝑆𝑠+𝑙⧵𝑆𝑆+𝑙+1)𝑄𝑢𝑠𝑠,𝑘⊂𝑄}2^{𝑢𝑠(𝑛+2𝑡)}𝑠^−2/𝑝′2^{−2𝑡𝑢𝑠}2^{2(𝑟+𝑡)𝑢𝑠}×𝜒2^𝑢𝑠𝑥−𝑘_𝑠^𝑝/2𝑑𝑥≲s^{−𝑝/𝑝′}2^{(𝑡+𝑟)𝑝𝑢𝑠}2^{−𝑛𝑢𝑠}+_𝑙≥1(𝑠+𝑙)^{−𝑝/𝑝′}2^{(𝑡+𝑟)𝑝𝑢𝑠+𝑙}2^{−𝑛(𝜎𝑠+𝑙−𝜎𝑠)−𝑛𝑢𝑠}≲1−log₂|𝑄|^{−𝑝/𝑝′}|𝑄|^{1−(𝑡+𝑟)𝑝/𝑛}. (180) Let 𝛿 be a sufficient small positive real number. For 𝑥∈[0,1]^𝑛⧵𝑆₁, then 𝑔(𝑥)=1. For 𝑠≥1 and 𝑥∈𝑆_𝑠⧵𝑆_𝑠+1, we take 𝑔(𝑥)=𝑠^−1/𝑝log₂(1+𝑠)^{−(1+𝛿)/𝑝}2^{(𝑡+𝑟)𝑢𝑠}. (181) Then, we have 𝑔∈𝐿^𝑝([0,1]^𝑛). In fact, _[0,1]𝑛𝑔(𝑥)^𝑝=𝑑𝑥_{[0,1]𝑛⧵𝑆1}𝑔^𝑝(𝑥)𝑑𝑥+_{𝑠≥1𝑆𝑠⧵𝑆𝑠+1}𝑔^𝑝(𝑥)𝑑𝑥≲1+_{𝑠≥1𝑆𝑠⧵𝑆𝑠+1}𝑠⁻¹log₂(1+𝑠)^−(1+𝛿)2^{(𝑡+𝑟)𝑝𝑢𝑠}𝑑𝑥≲_𝑠≥1𝑠⁻¹log₂(1+𝑠)^−(1+𝛿)2^{(𝑡+𝑟)𝑝𝑢𝑠}2^{𝑛(𝜎𝑠−𝑢𝑠)}≲1. (182)
Now we estimate the coefficients |𝑔_{𝑢𝑠,𝑘𝑠}|. We divide the estimate into two cases.
(1) For dist(𝑄_{𝑢𝑠,𝑘𝑠},𝑆_𝑠)≤2^−𝑢𝑠, 𝑔_𝑗,𝑘=_𝑆𝑠𝑔𝑦2^−𝑢𝑠+𝑦−2^−𝑢𝑠𝑘_𝑠^{(𝑡+𝑟−𝑛)}𝑑𝑦+_{1≤𝑙≤𝑠−1𝑆𝑙⧵𝑆𝑙+1}𝑙^−1/𝑝log₂(1+𝑙)^{−(1+𝛿)/𝑝}2^{𝑛(𝑢𝑙−𝜎𝑙)/𝑝}×2^−𝑢𝑙+𝑦−2^−𝑢𝑙𝑘_𝑙^{(𝑡+𝑟−𝑛)}𝑑𝑦≳𝑠^−1/𝑝log₂(1+𝑠)^{−(1+𝛿)/𝑝}+_{1≤𝑙≤𝑠−1}𝑠^−1/𝑝log₂(1+𝑠)^{−(1+𝛿)/𝑝}≳𝑠^1/𝑝′log₂(1+𝑠)^{−(1+𝛿)/𝑝}. (183)
(2) For dist(𝑄_{𝑢𝑠,𝑘𝑠},𝑆_𝑠)>2^−𝑢𝑠, there exists 𝑙<𝑠 such that 𝑄_{𝑢𝑠,𝑘𝑠}⊂𝑆_𝑙−1⧵𝑆_𝑙. It is easy to see that 𝑔_𝑗,𝑘 is equivalent to 𝑙^−1/𝑝′[log₂(1+𝑙)]^{−(1+𝛿)/𝑝}.
Finally, we have _{𝜀,𝑗,𝑘∈Λ𝑛}2^{𝑗(𝑛+2𝑡)}𝑔_𝑗,𝑘²𝑓^𝜀_𝑗,𝑘²𝜒_𝑗,𝑘(𝑥)^𝑝/2𝑑𝑥≳_{𝑠≥2𝑄dist𝑢𝑠𝑠,𝑘,𝑆𝑠≤2𝑠−𝑢}2^{𝑢𝑠(𝑛+2𝑡)}𝑔_{𝑢𝑠,𝑘𝑠}²×𝑓^𝜀_{𝑢𝑠,𝑘𝑠}²𝜒2^𝑢𝑠𝑥−𝑘_𝑠^𝑝/2𝑑𝑥≳_{𝑠≥2𝑄dist𝑢𝑠𝑠,𝑘,𝑆𝑠≤2𝑠−𝑢}2^{𝑢𝑠(𝑛+2𝑡)}𝑠^2/𝑝′×log₂(1+𝑠)^{−2(1+𝛿)/𝑝}𝑠^−2/𝑝′×2^{−2𝑡𝑢𝑠+2(𝑡+𝑟)𝑢𝑠}𝜒2^𝑢𝑠𝑥−𝑘_𝑠^𝑝/2𝑑𝑥≳_𝑠≥2log₂(1+𝑠)^−(1+𝛿)2^{(𝑡+𝑟)𝑝𝑢𝑠}2^{𝑛(𝜎𝑠−𝑢𝑠)}≳_𝑠≥2log₂(1+𝑠)^−(1+𝛿)=∞. (184) This completes the proof.

5. An Application to Schrödinger Type Operators

Let 𝐿=𝐼−Δ+𝑉 be a Schrödinger operator. Maz'ya and Verbitsky [11] established many sufficient and necessary conditions such that 𝑉 is a multiplier from 𝐻^1,2(ℝ^𝑛) to 𝐻^−1,2(ℝ^𝑛). Their results can be used to study the Schrödinger operators 𝐿. In Section 4, Theorem 32 gives a relation between Morrey spaces and Sobolev multipliers. In this section, we give an application of the wavelet characterization of 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) to the Schrödinger type operator 𝐼+(−Δ)^𝑟/2+𝑉.

For 𝑉∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), 𝜏>1/𝑝^′, and 𝑔∈𝐻^𝑡,𝑝(ℝ^𝑛), we want to find a solution 𝑓∈𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛) to the equation 𝐼+(−Δ)^𝑟/2+𝑉𝑓(𝑥)=𝑔(𝑥). (185)

Remark 34. Fixed 𝑟>0, 𝑡≥0, and 1<𝑝<𝑛/(𝑟+𝑡). (i)If there exists a 𝛿>0 such that ‖𝑉‖_𝐶^{𝑟+𝑡+𝛿} is sufficiently small, according to the continuity of Calderón-Zygmund operator (𝐼+(−Δ)^𝑟/2)(𝐼+(−Δ)^𝑟/2+𝑉)⁻¹, (185) can be solved easily. But if we consider a nonsmooth potential 𝑉∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), applying the same proof in Lemmas 8 and 9, it is possible that 𝑉 is not a 𝐿^∞ function.(ii)The condition 𝜏>1/𝑝^′ cannot be weaken to 𝜏≥1/𝑝^′. In fact, according to our counterexample in Section 5, if 𝑟+𝑡<1, there exists some 𝑉∈𝑀^{𝑡,1/𝑝′}_𝑟,𝑝(ℝ^𝑛) such that the operator 𝐼+(−Δ)^𝑟/2+𝑉 is not continuous from 𝐻^{𝑡+𝑟,𝑝}(ℝ^𝑛) to 𝐻^𝑡,𝑝(ℝ^𝑛).

Now, we use our sufficient condition of multiplier spaces 𝑋^𝑡_𝑟,𝑝(ℝ^𝑛) to get the solution of (185). We need the following two operators. For 𝑡,𝑟>0, let 𝑇_𝑡,𝑟=𝐼+(−Δ)^𝑡/2,𝐼+(−Δ)^𝑟/2,𝑆_𝑡,𝑟=𝐼+(−Δ)^𝑡/2𝑉𝑇⁻¹_𝑡,𝑟. (186)

Lemma 35. Given 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑟+𝑡), and 𝜏>1/𝑝^′, if 𝑉∈𝑀^𝑡,𝜏_𝑟,𝑝(ℝ^𝑛), the operator 𝑆_𝑡,𝑟 is bounded from 𝐿^𝑝(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛) with the operator norm less than 𝐶_{𝑡,𝑟,𝜏,𝑝}‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝, where 𝐶_{𝑡,𝑟,𝜏,𝑝} denotes a constant associated with 𝑡, 𝑟, 𝜏, and 𝑝.

Proof. By Theorem 32, for any 𝑓∈𝐿^𝑝(ℝ^𝑛), we have 𝑆_𝑡,𝑟𝑓_𝐿𝑝≲𝐼+(−Δ)^𝑡/2𝑉𝑇⁻¹_𝑡,𝑟𝑓_𝐿𝑝≲𝑉𝑇⁻¹_𝑡,𝑟𝑓_{𝐻𝑡,𝑝}≲𝐶_{𝑟,𝑡,𝜏,𝑝}‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝𝑇⁻¹_𝑡,𝑟𝑓_{𝐻𝑡+𝑟,𝑝}≲𝐶_{𝑟,𝑡,𝜏,𝑝}‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝𝑓_𝐿𝑝. (187) Then, 𝑆_𝑡,𝑟 is a bounded operator on 𝐿^𝑝(ℝ^𝑛) with the norm less than 𝐶_{𝑟,𝑡,𝜏,𝑝}‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝.

In the following lemma, we prove that (𝐼+𝑆_𝑡,𝑟) is invertible in 𝐿^𝑝(ℝ^𝑛) and the inverse (𝐼+𝑆_𝑡,𝑟)⁻¹ can be written formally as ∑^∞_𝑛=0(−1)^𝑛𝑆^𝑛_𝑡,𝑟.

Lemma 36. Given 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑟+𝑡), and 𝜏>1/𝑝^′, if ‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝<1/𝐶_{𝑟,𝑡,𝜏,𝑝}, the operator 𝐼+𝑆_𝑡,𝑟 is invertible in 𝐿^𝑝(ℝ^𝑛).

Proof. By Lemma 35, 𝑆_𝑡,𝑟 is bounded on 𝐿^𝑝(ℝ^𝑛). Hence, for any 𝑓∈𝐿^𝑝(ℝ^𝑛), ^∞_𝑛=0(−1)^𝑛𝑆^𝑛_𝑡,𝑟𝑓_𝐿𝑝≲^∞_𝑛=0𝑆^𝑛_𝑡,𝑟𝑓_𝐿𝑝≲^∞_𝑛=0𝐶_{𝑟,𝑡,𝜏,𝑝}‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝^𝑛𝑓_𝐿𝑝. (188)
If ‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝<1/𝐶_{𝑟,𝑡,𝜏,𝑝}, the above series is convergent in 𝐿^𝑝(ℝ^𝑛). Further, 𝐼+𝑆_𝑡,𝑟^∞_𝑛=0(−1)^𝑛𝑆^𝑛_𝑡,𝑟=^∞_𝑛=0(−1)^𝑛𝑆^𝑛_𝑡,𝑟−^∞_𝑛=1(−1)^𝑛𝑆^𝑛_𝑡,𝑟=𝐼. (189) Similarly, we can also get ^∞_𝑛=0(−1)^𝑛𝑆^𝑛_𝑡,𝑟𝐼+𝑆_𝑡,𝑟=𝐼; (190) that is, the operator 𝐼+𝑆_𝑡,𝑟 is invertible in 𝐿^𝑝(ℝ^𝑛).

Theorem 37. Given 𝑟>0, 𝑡≥0, 1<𝑝<𝑛/(𝑟+𝑡), and 𝜏>1/𝑝^′, if ‖𝑉‖_𝑀^𝑡,𝜏_𝑟,𝑝<1/𝐶_{𝑟,𝑡,𝜏,𝑝}, then, for 𝑔∈𝐻^𝑡,𝑝(ℝ^𝑛), there exists a unique solution 𝑓∈𝐻^{𝑟+𝑡,𝑝}(ℝ^𝑛) for (185).

Proof. Because 𝑔∈𝐻^𝑡,𝑝(ℝ^𝑛), we have 𝑔=(𝐼+(−Δ)^𝑡/2)𝑔∈𝐿^𝑝(ℝ^𝑛). By Lemma 36, the operator (𝐼+𝑆_𝑡,𝑟) is invertible in 𝐿^𝑝(ℝ^𝑛). Hence, we can get that there exists a unique solution to the following equation in 𝐿^𝑝(ℝ^𝑛): 𝐼+𝑆_𝑡,𝑟𝑓=𝑔, (191) where 𝑔∈𝐿^𝑝(ℝ^𝑛). Hence, for the above 𝑔∈𝐿^𝑝(ℝ^𝑛), 𝑓=𝐼+𝑆_𝑡,𝑟⁻¹𝑔=𝐼+𝑆_𝑡,𝑟⁻¹𝐼+(−Δ)^𝑡/2𝑔(𝑥)∈𝐿^𝑝ℝ^𝑛 (192) is a solution to (191). Write 𝑓=(𝐼+(−Δ)^𝑟/2)⁻¹(𝐼+(−Δ)^𝑡/2)⁻¹𝑓. Then, 𝑓∈𝐿^𝑝(ℝ^𝑛) is equivalent to 𝑓∈𝐻^{𝑟+𝑡,𝑝}(ℝ^𝑛). It is easy to verify that 𝑓 is a solution to (185). This completes the proof.

Conflict of Interests

The authors declare that they have no conflict of interests in this submitted paper.

Acknowledgments

The research is supported by NSFC nos. 11171203, 11201280, and 11271209; New Teachers’ Fund for Doctor Stations, Ministry of Education 20114402120003; Guangdong Natural Science Foundation S2011040004131; MYRG116(Y1-L3)-FST13-QT; MYRG115(Y1-L4)-FST13-QT; and FDCT 098/2012/A3.