A Note on Weighted Besov-Type and Triebel-Lizorkin-Type Spaces

Abstract

Let 𝑠,𝜏∈ℝ,𝑞∈(0,∞], and ω be in the class 𝐴_∞ of Muckenhoupt. We introduce the weighted Besov-type spaces 𝐵̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) and weighted Triebel-Lizorkin-type spaces 𝐹̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) for 𝑝∈(0,∞) and then establish the φ-transform characterizations of these new spaces in the sense of Frazier and Jawerth.

1. Introduction

Function spaces have been a central topic in modern analysis and are now of increasing applications in areas such as harmonic analysis and partial differential equations.

Since Besov spaces and Triebel-Lizorkin spaces were introduced in [1–3], these spaces became the focus of many scholars. Bui studied weighted Besov and Triebel-Lizorkin spaces in [4, 5]. In 2010, Yuan et al. gave a unified treatment of Morrey, Campanato, Besov, Lizorkin, and Triebel spaces in [6]. A series of research results on these topics can be found in [6–15]. They develop a theory of spaces of Besov-Triebel-Lizorkin type built on Morrey spaces. Based on their work, this note will generalize these spaces to the weighted cases.

Let 𝒮(ℝ^𝑛) be the space of all Schwartz functions on ℝ^𝑛. Let 𝜑 and 𝜓 be functions on ℝ^𝑛 satisfying 𝜑,𝜓∈𝒮ℝ^𝑛,supp𝜑,𝜓⊂𝜉∈ℝ^𝑛∶12≤𝜉,𝜑(𝜉,𝜓(𝜉3≤2))≥𝐶>0if5≤𝜉≤53_𝑗∈ℤ,𝜑2^𝑗𝜉𝜓2^𝑗𝜉=1if𝜉≠0, (1)(2)(3)(4) where 𝑓(𝜉)≡∫_ℝ𝑛𝑓(𝑥)𝑒^{−𝑖𝑥⋅𝜉}𝑑𝑥. Throughout the paper, for all 𝑗∈ℤ and 𝑥∈ℝ^𝑛, we put 𝜑_𝑗(𝑥)≡2^𝑗𝑛𝜑(2^𝑗𝑥).

As in [16], we set 𝒮_∞ℝ^𝑛≡𝜑∈𝒮ℝ^𝑛∶_ℝ𝑛𝜑(𝑥)𝑥^𝛾({0𝑑𝑥=0forallmulti-indices𝛾∈ℕ∪})^𝑛. (5) Let 𝒮^′_∞(ℝ^𝑛) be the topological dual of 𝒮_∞(ℝ^𝑛), namely, the set of all continuous linear functionals on 𝒮_∞(ℝ^𝑛). Endowed 𝒮^′_∞(ℝ^𝑛) with the weak ∗-topology, then 𝒮^′_∞(ℝ^𝑛) is complete; see [17].

Let 𝐴_∞=∪_𝑝≥1𝐴_𝑝. If 𝜔∈𝐴_∞, then there exist independent positive constants 𝐶_𝑖 and 𝛼_𝑖, 𝑖=1,2, such that 𝐶₁|𝐸||𝑄|^𝛼1≤𝜔(𝐸)𝜔(𝑄)≤𝐶₂|𝐸||𝑄|^𝛼2 (6) for each cube 𝑄 and each subcube 𝐸⊃𝑄.

Before stating our theorems on the weighted Triebel-Lizorkin-type spaces, we first give the definition of these spaces.

Definition 1. Let 𝜏,𝑠∈ℝ, 𝑞∈(0,∞], 𝜔∈𝐴_∞ and let 𝜑 be a Schwartz function satisfying (1) through (3).(i) Let 𝑝∈(0,∞]. The weighted Besov-type space 𝐵̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) is defined to be the set of all 𝑓∈𝒮^′_∞(ℝ^𝑛) such that ‖𝑓‖_̇𝐵^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}<∞, where 𝑓_̇𝐵^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≡sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃}2^𝑗𝑠𝜑_𝑗∗𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞} (7) with suitable modifications made when 𝑝=∞ or 𝑞=∞.(ii) Let 𝑝∈(0,∞). The weighted Triebel-Lizorkin-type space 𝐹̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) is defined to be the set of all 𝑓∈𝒮^′_∞(ℝ^𝑛) such that ‖𝑓‖_̇𝐹^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}<∞, where 𝑓_̇𝐹^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≡sup_𝑃dyadic1|𝑃|^𝜏×_𝑃^∞_{𝑗=𝑗𝑃}2^𝑗𝑠𝜑_𝑗∗𝑓(𝑥)^{𝑞𝑝/𝑞}𝜔(𝑥)𝑑𝑥^1/𝑝 (8) with suitable modifications made when 𝑞=∞, where 𝑙(𝑃) is the side length of dyadic cube 𝑃, 𝑗_𝑃≡−log₂𝑙(𝑃), and the supremum is taken over all dyadic cubes 𝑃.

Obviously, letting 𝜏=0, the above spaces are weighted Besov space 𝐵̇^𝑠_{𝑝,𝑞,𝜔}(ℝ^𝑛) and weighted Triebel-Lizorkin space 𝐹̇^𝑠_{𝑝,𝑞,𝜔}(ℝ^𝑛) in [5, 18]. If 𝑝=𝑞∈(0,∞), then 𝐵̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛)=𝐹̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛). For simplicity, in what follows, we use 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) to denote either 𝐵̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) or 𝐹̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛). If 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) means 𝐹̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), then the case 𝑝=∞ is excluded. Below are the main results of the paper.

Let 𝜑 and 𝜓 satisfy (1) through (4). Recall that the 𝜑-transform 𝑆_𝜑 is defined to be the map taking each 𝑓∈𝒮^′_∞(ℝ^𝑛) to the sequence 𝑆_𝜑𝑓={(𝑆_𝜑𝑓)_𝑄}_𝑄, where (𝑆_𝜑𝑓)_𝑄≡⟨𝑓,𝜑_𝑄⟩ for all dyadic cubes 𝑄; the inverse 𝜑-transform 𝑇_𝜓 is defined to be the map taking a sequence 𝑡={𝑡_𝑄}_𝑄 to 𝑇_𝜓𝑡=∑_𝑄𝑡_𝑄𝜓_𝑄; see, for example, [19, 20]. The following theorem is about the 𝜑-transform characterizations of the spaces 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛).

Theorem 2. Let 𝑠∈ℝ, 𝜏∈[0,∞), 𝑝,𝑞∈(0,∞], and 𝜑 and 𝜓 satisfy (1) through (4). Then the operators 𝑆_𝜑∶𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛)→𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) and 𝑇_𝜓∶𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛)→𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) are bounded. Furthermore, 𝑇_𝜓∘𝑆_𝜑 is the identity on 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛).

Next is the result of an 𝜀-almost diagonal operator (defined in Section 3) on 𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛).

Theorem 3. Let 𝜔∈𝐴_∞, 𝑝,𝑞∈(0,∞], 𝜀∈((𝑛(𝛼₂−𝛼₁))/𝑝,∞), 𝑠∈ℝ, and 𝜏∈[0,(𝛼₁/𝑝)+(𝜀/2𝑛)). Then all 𝜀-almost diagonal operators are bounded on 𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛).

Remark 4. Let 𝛼₁=𝛼₂=1, then 𝜀∈(0,∞) and 𝜏∈[0,(1/𝑝)+(𝜀/2𝑛)). Then all 𝜀-almost diagonal operators are bounded on 𝑎̇^𝑠,𝜏_𝑝,𝑞(ℝ^𝑛), which is just the results established by Yuan et al.; see [6, Theorem 3.1] and [10, Theorem 4.1].

This paper is organized as follows. In Section 2, we establish the 𝜑-transform characterizations of the spaces 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛). And the boundedness of almost diagonal operators on 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) is considered in Section 3.

At the end of this section, we make some conventions on notation. Throughout the paper, 𝐶 denotes unspecified positive constants, possibly different at each occurrence; the symbol 𝑋≲𝑌 means that there exists a positive constant 𝐶 such that 𝑋≤𝐶𝑌, and 𝑋∼𝑌 means 𝐶⁻¹𝑌≤𝑋≤𝐶𝑌. For any 𝜑∈𝒮(ℝ^𝑛), we set 𝜑(𝑥)≡𝜑(−𝑥) for all 𝑥∈ℝ^𝑛. For 𝑘=(𝑘₁,…,𝑘_𝑛)∈ℤ^𝑛 and 𝑗∈ℤ, 𝑄_𝑗𝑘 denotes the dyadic cube 𝑄_𝑗𝑘≡{(𝑥₁,…,𝑥_𝑛)∶𝑘_𝑖≤2^𝑗𝑥_𝑖<𝑘_𝑖+1for𝑖=1,…,𝑛} and 𝒬≡{𝑄_𝑗𝑘}_𝑗,𝑘. We denote by 𝑥_𝑄 the lower left-corner 2^−𝑗𝑘 of 𝑄=𝑄_𝑗𝑘. Throughout the paper, when dyadic cube 𝑄 appears as an index, such as ∑_𝑄 and {⋅}_𝑄, it is understood that 𝑄 runs over all dyadic cubes in ℝ^𝑛. For each cube 𝑄, we denote its side length by 𝑙(𝑄) and its center by 𝑐_𝑄, and for 𝑟>0, we denote by 𝑟𝑄 the cube concentric with 𝑄 having the side length 𝑟𝑙(𝑄). Let 𝐸 be a set of ℝ^𝑛. Denote by 𝜒_𝐸 its characteristic function and 𝐸^∘ its interior. Also, set ℕ≡{1,2,…} and ℤ₊≡ℕ∪{0}.

2. 𝜑-Transform Characterizations of 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛)

In this section, we establish the 𝜑-transform characterizations of the spaces 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛). To this end, we introduce their corresponding sequence spaces as follows.

Definition 5. Let 𝜏,𝑠∈ℝ, 𝑞∈(0,∞], and 𝜔∈𝐴_∞.(i) Let 𝑝∈(0,∞]. The sequence space 𝑏̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) is defined to be the set of all sequences {𝑡_𝑄}_𝑄⊂ℂ such that ‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎<∞, where ‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≡sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃}2^𝑗𝑠𝑞_{𝑃𝑙(𝑄)=2−𝑗}𝑡_𝑄𝜒_𝑄(𝑥)^𝑝𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞} (9) and 𝜒_𝑄≡|𝑄|^−1/2𝜒_𝑄.(ii)Let 𝑝∈(0,∞). The sequence space 𝑓̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) is defined to be the set of all sequences {𝑡_𝑄}_𝑄⊂ℂ such that ‖𝑡‖_̇𝑓^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎<∞, where ‖𝑡‖_̇𝑓^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≡sup_𝑃dyadic1|𝑃|^𝜏×_{𝑃𝑄⊂𝑃}|𝑄|^−𝑠/𝑛𝑡_𝑄𝜒_𝑄(𝑥)^{𝑞𝑝/𝑞}𝜔(𝑥)𝑑𝑥^1/𝑝. (10)

Obviously, we have ‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎=sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑙(𝑄)=2−𝑗𝑄⊂𝑃}|𝑄|^{−𝑠/𝑛−1/2}𝜔(𝑄)^1/𝑝𝑡_𝑄^{𝑝𝑞/𝑝1/𝑞}. (11)

In a similar manner to consider 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), we use 𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) to denote either 𝑏̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) or 𝑓̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛). If 𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) means 𝑓̇^𝑠,𝜏_𝑝,𝑞(ℝ^𝑛), then the case 𝑝=∞ is excluded.

To prove Theorem 2, we need some technical lemmas. The next lemma is a special case of [21, Lemma 2.11].

Lemma 6. Let 𝛿∈ℝ, 𝜔∈𝐴_∞. Then there exist positive constants 𝐿₀ and 𝐶 such that for all 𝑗∈ℤ, _{𝑄∈𝒬,𝑙(𝑄)=2−𝑗}𝜔(𝑄)^𝛿1+𝑥_𝑄^𝑛/max{1,|𝑄|}^𝐿0≤𝐶2^{𝑛(2|𝛿|+1)|𝑗|}. (12)

For all 𝑠∈ℝ, 𝜏∈[0,∞), 𝑝∈(0,∞), and 𝑞∈(0,∞], we see that 𝑏̇^𝑠,𝜏_{𝑝,min{𝑝,𝑞},𝜔}(ℝ^𝑛)⊂𝑓̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛)⊂𝑏̇^𝑠,𝜏_{𝑝,max{𝑝,𝑞},𝜔}(ℝ^𝑛) by Minkowski’s inequality. The following conclusions is easily verified similarly to the proof of lemma  2.7 in [6]. Here we omit the proof.

Lemma 7. Let 𝑠∈ℝ, 𝜏∈[0,∞), 𝑝,𝑞∈(0,∞], and 𝜓∈𝒮_∞(ℝ^𝑛), 𝜔∈𝐴_∞. Then for all 𝑡∈𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), 𝑇_𝜓𝑡≡∑_𝑄𝑡_𝑄𝜓_𝑄 converges in 𝒮^′_∞(ℝ^𝑛); moreover, 𝑇_𝜓∶𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛)→𝒮^′_∞(ℝ^𝑛) is continuous.

For a sequence 𝑡={𝑡_𝑄}_𝑄, 𝑟∈(0,∞], and a fixed 𝜆∈(0,∞), set 𝑡^∗_{𝑟,𝜆𝑄}≡_{{𝑅∶𝑙(𝑅)=𝑙(𝑄)}}𝑡_𝑅^𝑟1+𝑙(𝑅)⁻¹𝑥_𝑅−𝑥_𝑄^𝜆1/𝑟 (13) and 𝑡^∗_𝑟,𝜆≡{(𝑡^∗_𝑟,𝜆)_𝑄}_𝑄. We have the following estimates.

Lemma 8. Suppose 0<𝑎≤𝑟<∞ and 𝜆>𝑛𝑟𝛼₂/𝑎. Let 𝜇,𝑣∈ℤ and 𝜇≤𝑣. Then for each 𝑥∈𝑄 and 𝑙(𝑄)=2^−𝑣, we have _{𝑙(𝑅)=2−𝜇}𝑡_𝑅^𝑟1+𝑙(𝑅)⁻¹𝑥_𝑅−𝑥_𝑄^𝜆1/𝑟≤𝐶𝑀_{𝜔𝑙(𝑅)=2−𝜇}𝑡_𝑅^𝑎𝜒_𝑘(𝑥)^1/𝑎. (14)

Proof. Assume 𝑥_𝑄=0. Let 𝐴₀=𝑅dyadic∶𝑙(𝑅)=2^−𝜇,𝑥_𝑅𝑙(𝑅),𝐴≤1_𝑘=𝑅dyadic∶𝑙(𝑅)=2^−𝜇,2^𝑘−1<𝑥_𝑅𝑙(𝑅)≤2^𝑘 (15) for 𝑘=1,2,3,….
Denote 𝐵⊃∪_{𝑅∈𝐴𝑘}𝑅, then _{𝑅∈𝐴𝑘}𝑡_𝑅^𝑟1+𝑥_𝑅/𝑙(𝑅)^𝜆≤𝐶2^−𝑘𝜆_{𝑅∈𝐴𝑘}𝑡_𝑅^𝑟≤𝐶2^−𝑘𝜆_{𝑅∈𝐴𝑘}𝑡_𝑅^{𝑎𝑟/𝑎}≤𝐶2^−𝑘𝜆𝐵_{𝑅∈𝐴𝑘}𝑡_𝑅^𝑎𝜔(𝑅)𝜒_𝑅𝜔(𝑥)𝑑𝑥^𝑟/𝑎≤𝐶2^−𝑘𝜆|𝐵||𝑅|^{𝑟𝛼2/𝑎}𝑀_{𝜔𝑅∈𝐴𝑘}𝑡_𝑅^𝑎𝜒_𝑅^𝑟/𝑎≤𝐶2^{−𝑘(𝜆−𝑛𝑟𝛼2/𝑎)}𝑀_{𝜔𝑅∈𝐴𝑘}𝑡_𝑅^𝑎𝜒_𝑅^𝑟/𝑎. (16) Summing on 𝑘 and taking the 𝑟th roots yield the result.

Lemma 9. Let 𝜏=0, then 𝑡^∗_{𝑝∧𝑞,𝜆̇𝑏𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑏^𝑠,0_{𝑝,𝑞,𝜔(ℝ}^𝑛₎,𝑡^∗_{𝑝∧𝑞,𝜆̇𝑓𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑓^𝑠,0_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (17)

Proof. For all 𝑄, we have |𝑡_𝑄|≤(𝑡^∗_{𝑝∧𝑞,𝜆})_𝑄. Then it is obvious that ‖𝑡‖_̇𝑓^𝑠,0_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≤𝐶𝑡^∗_{𝑝∧𝑞,𝜆̇𝑓𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}. (18)
Let 𝜀=−1+(𝜆/𝑛𝛼₂)>0, choose 𝑎=(𝑝∧𝑞)/(1+𝜀/2), then 𝑝/𝑎,𝑞/𝑎>1. Using weighted Fefferman-Stein inequality, we have 𝑡^∗_{𝑝∧𝑞,𝜆̇𝑓𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}=_𝑄|𝑄|^−𝑠/𝑛𝑡^∗_{𝑝∧𝑞,𝜆𝑄}𝜒𝑄^𝑞1/𝑞_𝐿𝑝𝜔≤_𝑄|𝑄|^−𝑠/𝑛_{{(𝑅)(𝑄𝑅∶𝑙=𝑙)}}𝑡_𝑅^𝑝∧𝑞1+𝑙(𝑅)⁻¹𝑥_𝑅−𝑥_𝑄^{𝜆1/𝑝∧𝑞}×𝜒𝑄^𝑞1/𝑞_𝐿𝑝𝜔≤𝐶_𝜈∈ℤ𝑀_{𝜔𝑙(𝑅)=2−𝜈}|𝑅|^−𝑠/𝑛𝑡_𝑅𝜒_𝑅^{𝑎𝑞/𝑎𝑎/𝑞}_𝐿^1/𝑎_{𝜔𝑝/𝑎}≤𝐶_{𝜈∈ℤ𝑙(𝑅)=2−𝜈}|𝑅|^−𝑠/𝑛𝑡_𝑅𝜒_𝑅^𝑞1/𝑞_𝐿𝑝𝜔=‖𝑡‖_̇𝑓^𝑠,0_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (19) Therefore, ‖𝑡^∗_{𝑝∧𝑞,𝜆}‖_{̇𝑓𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑓^𝑠,0_{𝑝,𝑞,𝜔(ℝ𝑛)}. Similarly, we can verify that ‖𝑡^∗_{𝑝∧𝑞,𝜆}‖_{̇𝑏𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑏^𝑠,0_{𝑝,𝑞,𝜔(ℝ𝑛)}.

Lemma 10. Let 𝜔∈𝐴_∞, 𝑠∈ℝ, 𝜏∈[0,∞), 𝑝,𝑞∈(0,∞], and 𝜆∈(𝑛+𝑛𝛼₂,∞). Then there exists a constant 𝐶∈[1,∞) such that for all 𝑡∈𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), ‖𝑡‖_̇𝑎^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≤𝑡^∗_{min𝑝,𝑞,𝜆̇𝑎𝑠,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}≤𝐶‖𝑡‖_̇𝑎^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (20)

Proof. Notice that |𝑡_𝑄|≤(𝑡^∗_𝑟,𝜆)_𝑄 holds for all dyadic cubes 𝑄. This observation immediately implies that ‖𝑡‖_̇𝑎^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≤‖𝑡^∗_{𝑝∧𝑞,𝜆}‖_{̇𝑎𝑠,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}, where 𝑝∧𝑞=min{𝑝,𝑞}.
To see the converse, fix a dyadic cube 𝑃. Let 𝑟_𝑄≡𝑡_𝑄 if 𝑄⊂3𝑃 and 𝑟_𝑄≡0 otherwise, and let 𝑢_𝑄≡𝑡_𝑄−𝑟_𝑄. Set 𝑟≡{𝑟_𝑄}_𝑄 and 𝑢≡{𝑢_𝑄}_𝑄. Then for all dyadic cubes 𝑄, we have 𝑡^∗_{𝑝∧𝑞,𝜆𝑄}^𝑝∧𝑞=𝑟^∗_{𝑝∧𝑞,𝜆𝑄}^𝑝∧𝑞+𝑢^∗_{𝑝∧𝑞,𝜆𝑄}^𝑝∧𝑞. (21)
Applying the fact of Lemma 9 that for each sequence 𝑡={𝑡_𝑄}_𝑄, ‖𝑡^∗_{𝑝∧𝑞,𝜆}‖_{̇𝑏𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑏^𝑠,0_{𝑝,𝑞,𝜔(ℝ𝑛)} and ‖𝑡^∗_{𝑝∧𝑞,𝜆}‖_{̇𝑓𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑓^𝑠,0_{𝑝,𝑞,𝜔(ℝ𝑛)}, we then have 𝐼_𝑃≡1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑙(𝑄)=2−𝑗𝑄⊂𝑃}|𝑄|^{−𝑠/𝑛−1/2}𝜔(𝑄)^1/𝑝𝑟^∗_{𝑝∧𝑞,𝜆𝑄}^{𝑝𝑞/𝑝1/𝑞}≤1|𝑃|^𝜏𝑟^∗_{𝑝∧𝑞,𝜆̇𝑏𝑠,0𝑝,𝑞,𝜔(ℝ𝑛)}≲1|𝑃|^𝜏‖𝑟‖_̇𝑏^𝑠,0_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≲‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎ (22) and similarly, 𝐼_𝑃̃≡1|𝑃|^𝜏_{𝑃𝑄⊂𝑃}|𝑄|^{−𝑠/𝑛−1/2}𝑟^∗_{𝑝∧𝑞,𝜆𝑄}𝜒_𝑄(𝑥)^{𝑞𝑝/𝑞}×𝜔(𝑥)𝑑𝑥^1/𝑝≲‖𝑡‖_̇𝑓^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (23)
On the other hand, let 𝑄⊂𝑃 be a dyadic cube with 𝑙(𝑄)=2^−𝑖𝑙(𝑃) for some 𝑖∈ℤ₊. Suppose 𝑄 is any dyadic cube with 𝑙(𝑄)=𝑙(𝑄)=2^−𝑖𝑙(𝑃) and 𝑄⊂𝑃+𝑘𝑙(𝑃)⊆3̸𝑃 for some 𝑘∈ℤ^𝑛, where 𝑃+𝑘𝑙(𝑃)≡{𝑥+𝑘𝑙(𝑃)∶𝑥∈𝑃}. Then |𝑘|≥2 and 1+𝑙(𝑄)⁻¹|𝑥_𝑄−𝑥_𝑄|∼2^𝑖|𝑘|. For 𝜔(𝑄)𝜔𝑄=𝜔(𝑄)𝜔(𝑃)𝜔(𝑃)𝜔((𝑃𝜔((𝑃𝑃+𝑘𝑙))𝑃+𝑘𝑙))𝜔𝑄≤𝐶𝑘^𝑛𝛼22^{𝑖𝑛(𝛼2−𝛼1)}, (24) we have 𝐽_𝑃≡1|𝑃|^𝜏×^∞_{𝑖=0𝑙(𝑄)=2−𝑖𝑙(𝑃)𝑄⊂𝑃}|𝑄|^{−𝑠/𝑛−1/2}𝜔(𝑄)^1/𝑝×𝑢^∗_{𝑝∧𝑞,𝜆𝑄}^{𝑝𝑞/𝑝1/𝑞}≲1|𝑃|^𝜏×^∞_𝑖=02^{𝑖𝑛(𝛼2−𝛼1)𝑞/𝑝−𝑖𝜆𝑞/(𝑝∧𝑞)}×_{𝑘∈ℤ𝑛|𝑘|≥2}|𝑘|^{𝑛𝛼2−𝜆}×_𝑙𝑄^=2−𝑖_𝑙(𝑃)𝑄_{(𝑃)⊂𝑃+𝑘𝑙}𝑄^{−𝑠/𝑛−1/2}𝜔𝑄^1/𝑝×𝑡_𝑄^{𝑝∧𝑞𝑞/(𝑝∧𝑞)1/𝑞}. (25) When 𝑝≤𝑞, by 𝜆>𝑛+𝑛𝛼₂, we have 𝐽_𝑃≲‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎×^∞_𝑖=02^{𝑖𝑛(𝛼2−𝛼1)𝑞/𝑝−𝑖𝜆𝑞/𝑝}_{𝑘∈ℤ𝑛|𝑘|≥2}|𝑘|^{𝑛𝛼2−𝜆𝑞/𝑝1/𝑞}≲‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (26) When 𝑝>𝑞, by Hölder’s inequality and 𝜆>𝑛+𝑛𝛼₂, we obtain 𝐽_𝑃≲1|𝑃|^𝜏×^∞_𝑖=02^{𝑖𝑛(𝛼2−𝛼1)𝑞/𝑝−𝑖𝜆}×_{𝑘∈ℤ𝑛|𝑘|≥2}|𝑘|^{𝑛𝛼2−𝜆}×_𝑙𝑄₌₂^−𝑖_𝑙(𝑃)𝑄_{(𝑃)⊂𝑃+𝑘𝑙}𝑄^{−𝑠/𝑛−1/2}𝜔𝑄^1/𝑝𝑡_𝑄^𝑞1/𝑞≲‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎^∞_𝑖=02^{𝑖𝑛(𝛼2−𝛼1)−𝜆}_{𝑘∈ℤ𝑛|𝑘|≥2}|𝑘|^{𝑛𝛼2−𝜆1/𝑞}≲‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (27) Therefore, by (21), ‖𝑡^∗_{min{𝑝,𝑞},𝜆}‖_{̇𝑏𝑠,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}≲sup_𝑃∈𝒬(𝐼_𝑃+𝐽_𝑃)≲‖𝑡‖_̇𝑏^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎.
To complete the proof, for any 𝑖∈ℤ₊, 𝑘∈ℤ^𝑛₊, and dyadic cube 𝑃, set 𝐴(𝑖,𝑘,𝑃)≡𝑄∈𝒬∶𝑙𝑄=2^−𝑖𝑙(𝑃),𝑄⊂𝑃+𝑘𝑙(𝑃),𝑄∩(3𝑃)=∅. (28) Recall that 1+𝑙(𝑄)⁻¹|𝑥_𝑄−𝑥_𝑄|∼2^𝑖|𝑘| for any 𝑄⊂𝑃 and 𝑄∈𝐴(𝑖,𝑘,𝑃), and that for all 𝑑∈[0,1] and {𝑎_𝑗}_𝑗⊂ℂ, _𝑗𝑎_𝑗^𝑑≤_𝑗𝑎_𝑗^𝑑. (29) Similarly to the proof of Lemma A.2 (see [18, Remark A.3]), we obtain that for all 𝑥∈𝑃 and 𝑎∈(0,𝑝∧𝑞], 𝑄^{∈𝐴(𝑖,𝑘,𝑃)}𝑄^{−𝑠/𝑛−1/2}𝑡_𝑄^𝑝∧𝑞1+𝑙𝑄⁻¹𝑥_𝑄−𝑥_𝑄^𝜆≲2^𝑖|𝑘|^−𝜆×_{𝑃+𝑘𝑙(𝑃)}𝑄^{−𝑠/𝑛−1/2}𝑡_𝑄^𝑎𝜔𝑄𝜒_𝑄𝜔(𝑥)𝑑𝑥^{(𝑝∧𝑞)/𝑎}≲2^{−𝑖(𝜆−𝑛𝛼2(𝑝∧𝑞)/𝑎)}|𝑘|^−𝜆×𝑀_𝜔𝑙𝑄₌₂^−𝑖_𝑙(𝑃)𝑄₍^𝑃₎^{⊂𝑃+𝑘𝑙}𝑄^−𝑠/𝑛𝑡_𝑄𝜒_𝑄^𝑎×(𝑥+𝑘𝑙(𝑃))^{(𝑝∧𝑞)/𝑎}, (30) where herein and in what follows, 𝑀_𝜔 denotes the Hardy-Littlewood maximal function on ℝ^𝑛. Let 𝑎≡(2𝑛(𝑝∧𝑞)𝛼₂)/(𝜆+𝑛𝛼₂). Then 𝑎∈(0,𝑝∧𝑞). Applying Minkowski’s inequality, Fefferman-Stein’s weighted vector-valued inequality, and Hölder’s inequality, we have 𝐽_𝑃̃≡1|𝑃|^𝜏×_{𝑃𝑄⊂𝑃}|𝑄|^{−(𝑠/𝑛)−(1/2)}𝑢^∗_{𝑝∧𝑞,𝜆𝑄}𝜒_𝑄(𝑥)^{𝑞𝑝/𝑞}𝑑𝑥^1/𝑝≲1|𝑃|^𝜏×_𝑃^∞_{𝑖=0𝑘∈ℤ𝑛|𝑘|≥2}2^{−𝑖(𝜆−𝑛𝛼2(𝑝∧𝑞)/𝑎)}|𝑘|^−𝜆×𝑀_𝜔𝑙𝑄₌₂^−𝑖_𝑙(𝑃)𝑄_{(𝑃)⊂𝑃+𝑘𝑙}𝑄^−𝑠/𝑛𝑡_𝑄𝜒_𝑄^𝑎×(𝑥+𝑘𝑙(𝑃))^{(𝑝∧𝑞)/𝑎𝑞/(𝑝∧𝑞)𝑝/𝑞}×𝑑𝑥^1/𝑝≲‖𝑡‖_̇𝑓^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (31) Therefore, by (21) again, ‖𝑡^∗_{min{𝑝,𝑞},𝜆}‖_{̇𝑓𝑠,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}≲sup_𝑃∈𝒬(𝐼_𝑃̃+𝐽_𝑃̃)≲‖𝑡‖_̇𝑓^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎, which completes the proof of Lemma 10.

Let 𝜑 satisfy (1) through (3). Since 𝜑(𝑥)≡𝜑(−𝑥) also satisfy (1) through (3), we may take 𝜑 in place of 𝜑 in the definition of 𝐵̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛). For any 𝑓∈𝒮^′_∞(ℝ^𝑛) and 𝑄∈𝒬 with 𝑙(𝑄)=2^−𝑗, define the sequence sup(𝑓)≡{sup_𝑄(𝑓)}_𝑄 by setting sup_𝑄(𝑓)≡|𝑄|^1/2sup_𝑦∈𝑄|𝜑_𝑗∗𝑓(𝑦)|, and for any 𝛾∈ℤ₊, the sequence inf_𝛾(𝑓)≡{inf_𝑄,𝛾(𝑓)}_𝑄 by setting inf_𝑄,𝛾(𝑓)≡|𝑄|^1/2max{inf_𝑦∈𝑄|𝜑_𝑗∗𝑓(𝑦)|∶𝑙(𝑄)=2^−𝛾𝑙(𝑄),𝑄⊂𝑄}. Choosing 𝜆>𝑛+𝑛𝛼₂ as in the proof of [6, Lemma 2.9], we have the following estimates.

Lemma 11. Let 𝑠∈ℝ, 𝜏∈[0,∞), 𝑝,𝑞∈(0,∞], and 𝛾∈ℤ₊ be sufficiently large. Then there exists a constant 𝐶∈[1,∞) such that for all 𝑓∈𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), 𝐶⁻¹‖inf_𝛾(𝑓)‖_{̇𝑎𝑠,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}≤‖𝑓‖_̇𝐴^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≤‖sup(𝑓)‖_̇𝑎^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≤𝐶‖inf_𝛾(𝑓)‖_{̇𝑎𝑠,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}.

Proof of Theorem 2. With Lemmas 7, 10, and 11, the argument for Theorem 2 follows from the method pioneered by Frazier and Jawerth (see [18, pages 50-51]); see also the proof of [10, Theorem 3.1]. We omit the details.

From Theorem 2, we immediately deduce the following conclusion.

Corollary 12. With all the notations as in Definition 1, then the spaces 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) are independent of the choice of 𝜑.

3. Almost Diagonal Operators on 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛)

As an application of Theorem 2, we study boundedness of operators in 𝐴̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) by first considering their boundedness in corresponding 𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛). In this section, we show that almost diagonal operators are bounded on 𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) for appropriate indices, which generalize the classical results on 𝑏̇^𝑠,0_{𝑝,𝑞,𝜔}(ℝ^𝑛) and 𝑓̇^𝑠,0_{𝑝,𝑞,𝜔}(ℝ^𝑛); see [16, 18].

Definition 13. Let 𝑠∈ℝ, 𝑝,𝑞∈(0,∞], 𝐽≡𝑛/min{1,𝑝} for the space 𝑏̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), 𝐽≡𝑛/min{1,𝑝,𝑞} for the space 𝑓̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), and 𝜀∈(0,∞). An operator 𝐴 associated with a matrix {𝑎_𝑄𝑃}_𝑄,𝑃, namely, for all sequences 𝑡={𝑡_𝑄}_𝑄⊂ℂ, 𝐴𝑡≡{(𝐴𝑡)_𝑄}_𝑄≡{∑_𝑃𝑎_𝑄𝑃𝑡_𝑃}_𝑄, is called 𝜀-almost diagonal on 𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) if the matrix {𝑎_𝑄𝑃}_𝑄,𝑃 satisfies sup_𝑄,𝑃𝑎_𝑄𝑃𝜔_𝑄𝑃(𝜀)<∞, (32) where 𝜔_𝑄𝑃(𝜀)≡𝑙(𝑄)𝑙(𝑃)^𝑠𝑥1+^𝑄−𝑥_𝑃max(𝑙(𝑃),𝑙(𝑄))^{−𝐽−𝜀}𝑙(𝑄)×min𝑙(𝑃)^{(𝑛+𝜀)/2},𝑙(𝑃)𝑙(𝑄)^{(𝑛+𝜀)/2+𝐽−𝑛}. (33)

We remark that an 𝜀-almost diagonal operator is also an almost diagonal operator introduced by Frazier and Jawerth in [18]. In [18, Section 9], Frazier and Jawerth showed that certain appropriate Calderón-Zygmund operators and certain classes of Fourier multiplier operators correspond to almost diagonal matrices, and hence, the 𝜑-transform simultaneously “almost diagonalizes” these operators. Moreover, Yang and Yuan proved that all almost diagonal operators are bounded on 𝑓̇^𝑠,𝜏_𝑝,𝑞(ℝ^𝑛) and 𝑏̇^𝑠,𝜏_𝑝,𝑞(ℝ^𝑛); see [9, 10]. These results can be generalized into the weighted Besov- and Triebel-Lizorkin-type spaces. We turn to prove it.

Proof of Theorem 3. Let 𝑡={𝑡_𝑄}_𝑄∈𝑎̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) and let 𝐴 be an 𝜀-almost diagonal operator associated with the matrix {𝑎_𝑄𝑅}_𝑄,𝑅 and 𝜀∈(𝑛(𝛼₁−𝛼₂)/𝑝,∞). Without loss of generality, we may assume that 𝑠=0. Indeed, if the conclusion holds for 𝑠=0, let 𝑡_𝑅̃≡𝑙(𝑅)^−𝑠𝑡_𝑅 and let 𝐵 be the operator associated with the matrix {𝑏_𝑄𝑅}_𝑄,𝑅, where 𝑏_𝑄𝑅≡(𝑙(𝑅)/𝑙(𝑄))^𝑠𝑎_𝑄𝑅 for all 𝑄,𝑅∈𝒬. Then we have ‖𝐴𝑡‖_̇𝑎^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎=‖𝐵𝑡‖̃_{̇𝑎0,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}≲‖𝑡‖̃_{̇𝑎0,𝜏𝑝,𝑞,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑎^𝑠,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎, which deduces the desired conclusion.
We now consider the space 𝑏̇^0,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛) in the case min(𝑝,𝑞)>1. For all 𝑄∈𝒬, we write 𝐴≡𝐴₀+𝐴₁ with (𝐴₀𝑡)_𝑄≡∑_{{𝑅∶𝑙(𝑅)≥𝑙(𝑄)}}𝑎_𝑄𝑅𝑡_𝑅 and (𝐴₁𝑡)_𝑄≡∑_{{𝑅∶𝑙(𝑅)<𝑙(𝑄)}}𝑎_𝑄𝑅𝑡_𝑅. By Definition 13, we see that for all 𝑄∈𝒬, 𝐴₀𝑡_𝑄≲_{{𝑅∶𝑙(𝑅)≥𝑙(𝑄)}}𝑙(𝑄)𝑙(𝑅)^{(𝑛+𝜀)/2}×𝑡_𝑅1+𝑙(𝑅)⁻¹𝑥_𝑄−𝑥_𝑅^𝑛+𝜀, (34) and therefore 𝐴₀𝑡_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≲sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗{𝑅∶𝑙(𝑄)≤𝑙(𝑅)≤𝑙(𝑃)}}𝑙(𝑄)𝑙(𝑅)^{(𝑛+𝜀)/2}×𝑡_𝑅𝜒_𝑄(𝑥)1+𝑙(𝑅)⁻¹𝑥_𝑄−𝑥_𝑅^{𝑛+𝜀𝑝}×𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}+sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗{𝑅∶𝑙(𝑅)>𝑙(𝑃)}}⋯^𝑝×𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≡𝐼₁+𝐼₂. (35)
For all 𝑖∈ℤ and 𝑚∈ℕ, set 𝑈_0,𝑖≡{𝑅∈𝒬∶𝑙(𝑅)=2^−𝑖 and |𝑥_𝑄−𝑥_𝑅|<𝑙(𝑅)} and 𝑈_𝑚,𝑖≡{𝑅∈𝒬∶𝑙(𝑅)=2^−𝑖 and 2^𝑚−1𝑙(𝑅)≤|𝑥_𝑄−𝑥_𝑅|<2^𝑚𝑙(𝑅)}. Then we have #𝑈_𝑚,𝑖≲2^𝑚𝑛, where #𝑈_𝑚,𝑖 denotes the cardinality of 𝑈_𝑚,𝑖. Notice that |𝑡_𝑅|≤|𝑅|^1/2+𝜏𝜔(𝑅)^−1/𝑝‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)} for all 𝑅∈𝒬. Since 2^𝑚−1𝑙(𝑅)≤|𝑥_𝑄−𝑥_𝑅|<2^𝑚𝑙(𝑅), 𝑙(𝑄)≤𝑙(𝑅), 𝑥−𝑥_𝑅≤𝑥−𝑥_𝑄+𝑥_𝑄−𝑥_𝑅≤√𝑛𝑙(𝑄)+2^𝑚𝑙(𝑅)≤2^𝑚+√𝑛𝑙(𝑅)≲𝐶2^𝑚𝑙(𝑅)≤𝐶2^{𝑚+𝑗−𝑖}𝑙(𝑄). (36)
Let 𝑄=2^{𝑚+𝑗−𝑖}𝑄. If 𝑝≤1, 𝜏<(𝜀/2𝑛)+(𝛼₁/𝑝), 𝜀>(𝑛(𝛼₂−𝛼₁)/𝑝), then 1ℬ≡|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀/2×|𝑅|^𝜏𝜔(𝑅)^−1/𝑝2^{−𝑚(𝑛+𝜀)}𝜒_𝑄(𝑥)^𝑝×𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≤2^{𝑗𝑃𝑛𝜏∞}_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀𝑝/2×𝜔(𝑅)⁻¹|𝑅|^𝜏𝑝2^{−(𝑛+𝜀)𝑚𝑝}×𝜒_𝑄^𝑝(𝑥)𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≤2^{𝑗𝑃𝑛𝜏∞}_{𝑗=𝑗𝑃𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀𝑝/2×𝑙(𝑅)^𝑛𝜏𝑝2^{−(𝑛+𝜀)𝑚𝑝}𝜔(𝑄)𝜔(𝑅)^{𝑞/𝑝1/𝑞}≤2^{𝑗𝑃𝑛𝜏∞}_{𝑗=𝑗𝑃l(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀𝑝/2×𝑙(𝑅)^𝑛𝜏𝑝2^{−(𝑛+𝜀)𝑚𝑝}𝜔𝑄𝜔(𝑅)𝜔(𝑄)𝜔𝑄^{𝑞/𝑝1/𝑞}≤𝐶2^{𝑗𝑃𝑛𝜏∞}_{𝑗=𝑗𝑃}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0}2^{(𝑖𝜀𝑝/2)−𝑖𝑛𝜏𝑝+𝑖𝑛𝛼1}2^{−(𝑗𝜀𝑝/2)−𝑗𝑛𝛼1}×2^{𝑚𝑛(𝛼2−𝛼1)}2^{−𝑚𝜀𝑝)𝑞/𝑝1/𝑞}≤𝐶, (37) here we use 𝜔(𝑄)/𝜔(𝑅)≲2^{𝑚𝑛𝛼2} and 𝜔(𝑄)/𝜔(𝑄)≲2^{−𝑛(𝑚+𝑗−𝑖)𝛼1}.
If 𝑝>1, using Minkowski’s inequality, we have _{𝑃𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀/2×|𝑅|^𝜏𝜔(𝑅)^−1/𝑝2^{−𝑚(𝑛+𝜀)}𝜒_𝑄(𝑥)^𝑝𝜔(𝑥)𝑑𝑥^1/𝑝≤_{𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀/2×𝜔(𝑄)𝜔(𝑅)^1/𝑝|𝑅|^𝜏2^{−𝑚(𝑛+𝜀)}≤_{𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀/2×𝜔𝑄𝜔(𝑅)^1/𝑝𝜔(𝑄)𝜔𝑄^1/𝑝|𝑅|^𝜏2^{−𝑚(𝑛+𝜀)}. (38)
Therefore, ℬ≤𝐶2^{𝑗𝑃𝑛𝜏}×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}2^{𝑖((𝜀/2)+(𝑛𝛼1/𝑝)−𝑛𝜏)}×2^{𝑗(−(𝜀/2)−(𝑛𝛼1/𝑝))}2^{−𝑚(𝑛+𝜀)}2^{𝑛𝑚(𝛼2−𝛼1)/𝑝𝑞1/𝑞}≤𝐶. (39) Thus, by 0≤𝜏<(𝛼₁/𝑝)+(𝜀/2𝑛), 𝐼₂≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^{(𝑛+𝜀)/2}×|𝑅|^1/2+𝜏𝜔(𝑅)^−1/𝑝𝜒_𝑄(𝑥)1+𝑙(𝑅)⁻¹𝑥_𝑄−𝑥_𝑅^{𝑛+𝜀𝑝}×𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗}^{𝑗𝑃−1∞}_{𝑖=−∞𝑚=0𝑅∈𝑈𝑚,𝑖}𝑙(𝑄)𝑙(𝑅)^𝜀/2×|𝑅|^𝜏𝜔(𝑅)^−1/𝑝2^{−𝑚(𝑛+𝜀)}𝜒_𝑄(𝑥)^𝑝×𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞𝜔(ℝ}^𝑛₎. (40)
For 𝐼₁, let 𝑟 and 𝑢 be the same as in the proof of Lemma 10. We see that 𝐼₁≲sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗}^𝑗_{𝑖=𝑗𝑃}2^{(𝑖−𝑗)(𝑛+𝜀)/2}×_{𝑙(𝑅)=2−𝑖}𝑟_𝑅𝜒_𝑄(𝑥)1+𝑙(𝑅)⁻¹𝑥_𝑄−𝑥_𝑅^{𝑛+𝜀𝑝}×𝑑𝑥^{𝑞/𝑝1/𝑞}+sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑗=𝑗𝑃𝑃𝑙(𝑄)=2−𝑗}^𝑗_{𝑖=𝑗𝑃}2^{(𝑖−𝑗)(𝑛+𝜀)/2}×_{𝑙(𝑅)=2−𝑖}𝑢_𝑅𝜒_𝑄(𝑥)1+𝑙(𝑅)⁻¹𝑥_𝑄−𝑥_𝑅^{𝑛+𝜀𝑝}×𝑑𝑥^{𝑞/𝑝1/𝑞}≡𝐽₁+𝐽₂. (41) Applying Lemma 8 with 𝑎=1, for all 𝑥∈𝑄, we have _{𝑙(𝑅)=2−𝑖}𝑟_𝑅1+𝑙(𝑅)⁻¹𝑥_𝑄−𝑥_𝑅^𝑛+𝜀≲𝑀_{𝜔𝑙(𝑅)=2−𝑖}𝑟_𝑅𝜒_𝑅(𝑥). (42) Hence Hölder’s inequality and the 𝐿^𝑝_𝜔(ℝ^𝑛) boundedness for 𝑝∈(1,∞] of the Hardy-Littlewood maximal operator yield 𝐽₁≲sup_𝑃dyadic1|𝑃|^𝜏×^∞_𝑗=