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=𝑗𝑃𝑃}^𝑗_{𝑖=𝑗𝑃}2^{((𝑖−𝑗)𝜀)/2}𝑀_{𝜔𝑙(𝑅)=2−𝑖}𝑟_𝑅𝜒_𝑅(𝑥)^𝑝×𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≲sup_𝑃dyadic1|𝑃|^𝜏×^∞_{𝑖=𝑗𝑃3𝑃𝑙(𝑅)=2−𝑖}𝑡_𝑅𝜒_𝑅^𝑝(𝑥)𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎, (43) where the last inequality follows from Minkowski’s inequality if 𝑞>𝑝 or (29) if 𝑞≤𝑝.
To estimate 𝐽₂, we notice that if 𝑅∩(3𝑃)=∅, then 𝑅⊂𝑃+𝑘𝑙(𝑃) and (𝑃+𝑘𝑙(𝑃))∩(3𝑃)=∅ for some 𝑘∈ℤ^𝑛 with |𝑘|≥2 and 1+𝑙(𝑅)⁻¹|𝑥_𝑄−𝑥_𝑅|∼|𝑘|𝑙(𝑃)/𝑙(𝑅). Therefore, by Hölder’s inequality, 𝐽₂≲sup_𝑃dyadic1|𝑃|^{𝜏+1+𝜀/𝑛}×^∞_{𝑗=𝑗𝑃}2^{−𝑗𝑞𝜀/2}_{𝑃𝑙(𝑄)=2−𝑗}^𝑗_{𝑖=𝑗𝑃}2^{−𝑖(𝑛+𝜀)/2}_{𝑘∈ℤ𝑛|𝑘|≥2}|𝑘|^{−𝑛−𝜀}×_{𝑙(𝑅)=2−𝑖(𝑃)𝑅⊂𝑃+𝑘𝑙}𝑡_𝑅𝜒_𝑄(𝑥)^𝑝×𝜔(𝑥)𝑑𝑥^{𝑞/𝑝1/𝑞}≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎sup_𝑃dyadic1|𝑃|^𝜀/𝑛×^∞_{𝑗=𝑗𝑃}2^{−𝑗𝑞𝜀/2𝑗}_{𝑖=𝑗𝑃}2^{−𝑖𝜀/2}_{𝑘∈ℤ𝑛|𝑘|≥2}|𝑘|^{−𝑛−𝜀𝑞1/𝑞}≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (44) Hence, 𝐴₀𝑡_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (45) The similar computations to 𝐼₁ yield 𝐴₁𝑡_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≲‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (46)
For Triebel-Lizorkin-type spaces 𝑓̇^𝑠,𝜏_{𝑝,𝑞,𝜔}(ℝ^𝑛), we also have 𝐴₀𝑡_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≲sup_𝑃dyadic1|𝑃|^𝜏×_{𝑃𝑄⊂𝑃{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+𝜀𝑞𝑝/𝑞}×𝜔(𝑥)𝑑𝑥^1/𝑝+sup_𝑃dyadic1|𝑃|^𝜏×_{𝑃𝑄⊂𝑃{(𝑅)(𝑃𝑅∶𝑙>𝑙)}}⋯^{𝑞𝑝/𝑞}𝜔(𝑥)𝑑𝑥^1/𝑝≡𝐼₁̃+𝐼₂̃. (47) Similarly to the estimate for 𝐼₂, applying the fact that |𝑡_𝑅|≤|𝑅|^1/2+𝜏𝜔(𝑅)^−1/𝑝‖𝑡‖_̇𝑏^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)} and 0≤𝜏<(𝛼₁/𝑝)+(𝜀/2𝑛), we obtain 𝐼₂̃≲‖𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎sup_𝑃dyadic1|𝑃|^𝜏×_𝑃^∞_{𝑗=𝑗𝑃𝑙(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+𝜀𝑞𝑝/𝑞}×𝜔(𝑥)𝑑𝑥^1/𝑝≲‖𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (48)
To estimate 𝐼₁̃, notice that 𝐼₁̃≲sup_𝑃dyadic1|𝑃|^𝜏×_𝑃^∞_{𝑗=𝑗𝑃𝑙(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+𝜀𝑞𝑝/𝑞}×𝜔(𝑥)𝑑𝑥^1/𝑝+sup_𝑃dyadic1|𝑃|^𝜏×_𝑃^∞_{𝑗=𝑗𝑃𝑙(𝑄)=2−𝑗}×^𝑗_{𝑖=𝑗𝑃}2^{(𝑖−𝑗)(𝑛+𝜀)/2}×_{𝑙(𝑅)=2−𝑖}|𝑢_𝑅|𝜒_𝑄(𝑥)1+𝑙(𝑅)⁻¹𝑥_𝑄−𝑥_𝑅^{𝑛+𝜀𝑞𝑝/𝑞}×𝜔(𝑥)𝑑𝑥^1/𝑝≡𝐽₁̃+𝐽₂̃. (49) Similarly to the estimate of 𝐽₁, by [18, Remark A.3], Hölder’s inequality and the Fefferman-Stein weighted vector-valued inequality, we obtain 𝐽₁̃≲sup_𝑃dyadic1|𝑃|^𝜏×_𝑃^∞_{𝑗=𝑗𝑃}^𝑗_{𝑖=𝑗𝑃}2^{(𝑖−𝑗)𝜀/2}𝑀_𝐻𝐿×_{𝑙(𝑅)=2−𝑖}𝑟_𝑅𝜒_𝑅(𝑥)^{𝑞𝑝/𝑞}×𝜔(𝑥)𝑑𝑥^1/𝑝≲sup_𝑃dyadic1|𝑃|^𝜏×_3𝑃^∞_{𝑖=𝑗𝑃𝑙(𝑅)=2−𝑖}𝑡_𝑅𝜒_𝑅(𝑥)^{𝑞𝑝/𝑞}𝜔(𝑥)𝑑𝑥^1/𝑝≲‖𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (50) For 𝐽₂̃, similarly to the estimate for 𝐽₂, by Minkowski’s inequality and the Fefferman-Stein weighted vector-valued maximal inequality, we obtain 𝐽₂̃≲_{𝑘∈ℤ𝑛|𝑘|≥2}|𝑘|^{−(𝑛+𝜀)}sup_𝑃dyadic1|𝑃|^{𝜏+𝜀/𝑛}×_𝑃^∞_{𝑗=𝑗𝑃}^𝑗_{𝑖=𝑗𝑃}2^{(𝑖−𝑗)𝜀/2}2^−𝑖𝜀×𝑀_{HL𝑙(𝑅)=2−𝑖𝑅⊂𝑃+𝑘𝑙(𝑃)}|𝑅|^−1/2𝑡_𝑅𝜒_𝑅×(𝑥+𝑘𝑙(𝑃))^{𝑞𝑝/𝑞}×𝜔(𝑥)𝑑𝑥^1/𝑝≲‖𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. (51) Hence ‖𝐴₀𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞(ℝ𝑛)}≲‖𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}.
Some similar estimates to 𝐼₁̃ also yield that ‖𝐴₁𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}≲‖𝑡‖_̇𝑓^0,𝜏_{𝑝,𝑞,𝜔(ℝ𝑛)}. Thus, we obtain the desired conclusion for the case min{𝑝,𝑞}>1.
The case min{𝑝,𝑞}≤1 is a simple consequence of the case min{𝑝,𝑞}>1. Indeed, choosing a 𝛿∈(0,𝑝∧𝑞), then 𝑝/𝛿>1 and 𝑞/𝛿>1. Let 𝐴 be an operator associated with the matrix {𝑎_𝑄𝑃}_𝑄,𝑃≡{|𝑎_𝑄𝑃|^𝛿(𝑙(𝑄)/𝑙(𝑃))^{𝑛/2−𝛿𝑛/2}}_𝑄,𝑃. Then 𝐴 is a 𝜀̃-almost diagonal operator with 𝑠=0 and 𝜀̃=𝛿𝜀.
Define 𝑡̃≡{𝑙(𝑄)^{𝑛/2−𝛿𝑛/2}|𝑡_𝑄|^𝛿}_𝑄. Then ‖𝑡‖̃_̇𝑎^1/𝛿_{0,𝜏𝛿𝑝/𝛿,𝑞/𝛿,𝜔(ℝ𝑛)}=‖𝑡‖_̇𝑎^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎. Since 𝛿<1, we have ‖𝐴𝑡‖_̇𝑎^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≲‖𝐴𝑡‖̃_̇𝑎^1/𝛿_{0,𝜏𝛿𝑝/𝛿,𝑞/𝛿,𝜔(ℝ𝑛)}. Applying the conclusions for min{𝑝,𝑞}>1 yields ‖𝐴𝑡‖_̇𝑎^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎≲‖𝐴𝑡‖̃_̇𝑎^1/𝛿_{0,𝜏𝛿𝑝/𝛿,𝑞/𝛿,𝜔(ℝ𝑛)}≲‖𝑡‖̃_̇𝑎^1/𝛿_{0,𝜏𝛿𝑝/𝛿,𝑞/𝛿,𝜔(ℝ𝑛)}∼‖𝑡‖_̇𝑎^0,𝜏_{𝑝,𝑞,𝜔(ℝ}^𝑛₎, which completes the proof of Theorem 3.

Remark 14. Recall that 𝑏̇^𝑠,0_{𝑝,𝑞,𝜔}(ℝ^𝑛)=𝑏̇^𝑠_{𝑝,𝑞,𝜔}(ℝ^𝑛) and 𝑓̇^𝑠,0_{𝑝,𝑞,𝜔}(ℝ^𝑛)=𝑓̇^𝑠_{𝑝,𝑞,𝜔}(ℝ^𝑛). Thus, if 𝜏=0, by Theorem 3, we see that for all 𝜀∈(0,∞), all 𝜀-almost diagonal operators are bounded on 𝑏̇^𝑠_{𝑝,𝑞,𝜔}(ℝ^𝑛) and 𝑓̇^𝑠_{𝑝,𝑞,𝜔}(ℝ^𝑛), which is just the classical results established by Frazier and Jawerth; see [18, Theorem 3.3] and [16, Theorem 6.20].

Acknowledgments

The author cordially thanks Prof. D. C. Yang and Dr. W. Yuan for their valuable comments. This work was supported by National Natural Science Foundation (Grand no. 11021043) of China and the Fundamental Research Funds for the Central Universities. Also, we thank the referee for his many valuable remarks which greatly improved the presentation of this paper.