Boundedness for a Class of Generalized Commutators of Fractional Hardy Operators with a Rough Kernel

Abstract

The authors consider the generalized commutator of fractional Hardy operator with a rough kernel as follows: ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)=1/(|𝑥|^𝑛−𝛽)∫_{|𝑦|<|𝑥|}(Ω(𝑥−𝑦)/|𝑥−𝑦|^𝑚−1)𝑅_𝑚(𝐴;𝑥,𝑦)𝑓(𝑦)𝑑𝑦, where Ω∈𝐿^𝑟(𝕊^𝑛−1), 0≤𝛽<𝑛, and 𝑅_𝑚(𝐴;𝑥,𝑦)=𝐴(𝑥)−∑_|𝛾|<𝑚(1/𝛾!)𝐷^𝛾𝐴(𝑦)(𝑥−𝑦)^𝛾 with 𝑚∈𝑍⁺. The authors prove that ℋ^𝑚_Ω,𝐴,𝛽 is bounded on Herz type space and 𝜆-Central Morrey space with 𝑚≥1, which is an open problem for 𝑚>2.

1. Introduction

It is well known that the C-Z singular integrals and their commutators have been studied a lot by many mathematicians; please see [1] or [2] for more details. For the generalizations of the commutators of singular integrals, Cohen [3] studied the following generalized commutator 𝑇²_𝐴 which is defined by 𝑇²_𝐴𝑓(𝑥)=_ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^𝑛+1𝐴(𝑥)𝑦𝑦𝑓𝑦−𝐴−∇𝐴𝑥−𝑦𝑑𝑦, (1) where Ω∈𝐿¹(𝕊^𝑛−1) is homogeneous of degree zero and satisfies the moment condition _𝕊𝑛−1Ω(𝑥)𝑥^𝛾𝑑𝜎(𝑥)=0 (2) with |𝛾|=1. Cohen [3] proved that if Ω∈Lip₁(𝕊^𝑛−1) and ∇𝐴∈BMO, then 𝑇²_𝐴 is bounded on 𝐿^𝑝(ℝ^𝑛) with 1<𝑝<∞. Later, Cohen and Gosselin [4] considered another type of generalized commutator as follows: 𝑇^𝑚_𝐴𝑓(𝑥)=_ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^{𝑛+𝑚−1}𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦, (3) where 𝑅_𝑚(𝐴;𝑥,𝑦)(𝑚∈𝑍⁺) is defined by 𝑅_𝑚(𝐴;𝑥,𝑦)=𝐴(𝑥)−∑_|𝛾|<𝑚(1/𝛾!)𝐷^𝛾𝐴(𝑦)(𝑥−𝑦)^𝛾, the mth remainder of Taylor series of the function 𝐴 at 𝑦 about 𝑥, and Ω satisfies the following moment condition: _𝕊𝑛−1Ω(𝑥)𝑥^𝛾𝑑𝜎(𝑥)=0, (4) with |𝛾|=𝑚−1. Obviously, if we choose 𝑚=1, 𝑇^𝑚_𝐴 becomes [𝐴,𝑇], the commutator of 𝑇 generalized by 𝐴 and 𝑇. Furthermore, 𝑇^𝑚_𝐴 becomes 𝑇²_𝐴 if we choose 𝑚=2.

Cohen and Gosselin proved that if 𝑚≥2,Ω∈Lip₁(𝕊^𝑛−1), and the function 𝐴 has derivatives of order 𝑚−1 in BMO(ℝ^𝑛), then the operator 𝑇^𝑚_𝐴 is bounded on 𝐿^𝑝(ℝ^𝑛) for 1<𝑝<∞. Later, 𝑇^𝑚_𝐴 was studied by many mathematicians; please see [5, 6] or [7] for more details. Recently, Wang and Zhang [8] gave a new proof of Wu’s theorem in [9] by using the 𝑊^1,𝑝 estimate for the elliptic equation of divergence form with partially BMO coefficients and the 𝐿^𝑝 boundedness of the Cohen-Gosselin type generalized commutators proved by Yan in [6]. Furthermore, the method used in [8] is much simpler than that in [9]. Recently, Yu and Tao [7] proved that 𝑇^𝑚_𝐴 is bounded on 𝜆-Central Morrey space.

Let 𝑓 be a nonnegative integral on ℝ⁺: then the Hardy operator is defined by (𝑥)=1𝐻𝑓𝑥^𝑥₀𝑓(𝑡)𝑑𝑡,𝑥≠0. (5) In 1920, Hardy [10] proved the following inequality: 𝐻𝑓_{𝐿𝑝(ℝ+)}≤𝑝𝑓𝑝−1^{𝐿𝑝(ℝ+)}, (6) where 1<𝑝<∞ and the constant 𝑝/(𝑝−1) is the best possible.

In 2007, Fu et al. [11] introduced the 𝑛-dimensional fractional type Hardy operator ℋ_𝛽 as follows: ℋ_𝛽𝑓(𝑥)=1|𝑥|^{𝑛−𝛽|𝑡|<|𝑥|}𝑓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑛⧵{0}, (7) where −𝑛<𝛽<𝑛 and 𝑓 is a locally integrable function on ℝ^𝑛.

Obviously, when 𝛽=0, ℋ₀ is just the 𝑛-dimensional Hardy operator ℋ which was proposed by Christ and Grafakos in [12].

In [11], the authors gave the characterization of the CḂMO^𝑞(ℝ^𝑛) by the boundedness of the commutator of the fractional type Hardy operator [ℋ_𝛽,𝑏] on Herz type spaces. Here the CḂMO^𝑞(ℝ^𝑛) space is defined by the following.

Definition 1 (see [13]). Let 1≤𝑞<∞. A function 𝑓∈𝐿^𝑞_loc(ℝ^𝑛) is said to belong to the homogeneous Central BMO space CḂMO^𝑞(ℝ^𝑛) if ‖𝑏‖_Ċ‌BMO^𝑞_(ℝ^𝑛₎∶=sup_𝑟>01|𝐵(0,𝑟)|_𝐵(0,𝑟)|𝑓(𝑥)−𝑓_𝐵|^𝑞𝑑𝑥^1/𝑞<∞, (8) where 𝑓_𝐵=(1/|𝐵(0,𝑟)|)∫_𝐵(0,𝑟)𝑓(𝑥)𝑑𝑥.
From [14], we know that BMO(ℝ^𝑛)⊂CḂMO^𝑞(ℝ^𝑛) for 1≤𝑞<∞.

The CḂMO^𝑞(ℝ^𝑛) space can be regarded as the space of bounded mean oscillation, a local version of BMO(ℝ^𝑛) at the origin. But the famous John-Nirenberg inequality no longer holds in CḂMO^𝑞(ℝ^𝑛).

Now we are interested in the following generalized commutator of Hardy operator: ℋ^𝑚_𝐴𝑓(𝑥)=1|𝑥|^𝑛_{|𝑦|<|𝑥|}1𝑥−𝑦^𝑚−1𝑓𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦, (9) where 𝑅_𝑚(𝐴;𝑥,𝑦)=𝐴(𝑥)−∑_|𝛾|<𝑚(1/𝛾!)𝐷^𝛾𝐴(𝑦)(𝑥−𝑦)^𝛾 and 𝑚∈𝑍⁺.

In 2010, Lu and Zhao [15] proved that when 𝑚=2, ℋ²_𝐴 is bounded on Herz type space and Morrey-Herz type space. Later, Gao and Yu [16] proved that ℋ²_𝐴 is bounded on 𝜆-Central Morrey spaces. However, we would like to point out that the method used in [15, 16] cannot apply to the case when 𝑚>2. An interesting question is whether the boundedness of ℋ^𝑚_𝐴 on Herz type space or 𝜆-Central Morrey space still holds with 𝑚>2. In this paper, we will use a different method to answer this question. Furthermore, we will consider the generalized commutator of fractional Hardy operator with a rough kernel as follows: ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)=1|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑓𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦, (10) where 𝑚∈𝑍⁺, 0≤𝛽<𝑛, and Ω∈𝐿^𝑟(𝕊^𝑛−1).

In [17], we prove that ℋ^𝑚_Ω,𝐴,𝛽 is bounded from 𝐿^𝑝 to 𝐿^𝑞 with 1/𝑝−1/𝑞=𝛽/𝑛. Furthermore, we study the endpoint estimates of ℋ^𝑚_Ω,𝐴,𝛽 on 𝐻¹ spaces with Ω∈Lip₁(𝕊^𝑛−1) in [17]. In this paper, we will prove that ℋ^𝑚_Ω,𝐴,𝛽 is bounded on Herz type space and 𝜆-Central Morrey space when Ω∈𝐿^𝑟(𝕊^𝑛−1).

2. Boundedness of ℋ^𝑚_Ω,𝐴,𝛽 on Herz Type Spaces

In this section, we will give the boundedness of ℋ^𝑚_Ω,𝐴,𝛽 on Herz type spaces. First we introduce some notations that will be used throughout this paper.

Let 𝐵_𝑘={𝑥∈ℝ^𝑛∶|𝑥|≤2^𝑘}, 𝐶_𝑘=𝐵_𝑘⧵𝐵_𝑘−1, and 𝜒_𝑘=𝜒_𝐶𝑘 for 𝑘∈𝑍: here 𝜒_𝐶𝑘 is the characteristic function of the set 𝐶_𝑘.

Definition 2 (see [18]). Let 𝛼∈𝑅,0<𝑝,𝑞≤∞. Then the homogeneous Herz type space 𝐾̇_𝑞^𝛼,𝑝(ℝ^𝑛) is defined by 𝐾̇_𝑞^𝛼,𝑝ℝ^𝑛=𝑓∈𝐿^𝑞_locℝ^𝑛⧵{0}∶𝑓_̇𝐾𝑞^𝛼,𝑝_(ℝ𝑛)<∞, (11) where ‖𝑓‖_̇𝐾𝑞^𝛼,𝑝_(ℝ𝑛) is defined as 𝑓_̇𝐾𝑞^𝛼,𝑝_(ℝ𝑛)=^∞_𝑘=−∞2^𝑘𝛼𝑝𝑓𝜒_𝑘^𝑝_{𝐿𝑞(ℝ𝑛)}^1/𝑝 (12) with the usual modifications made when 𝑝=∞ or 𝑞=∞.
Now we show our main results in this section.

Theorem 3. Suppose 𝑚≥2, Ω∈𝐿^𝑟(𝕊^𝑛−1) with 1<𝑟<∞, and 𝐴 has derivatives of order 𝑚−1 in 𝐶𝐵̇𝑀𝑂^𝑝2 with 𝑛<𝑝₂<∞. Let 0<𝑠≤𝑝<∞,1<𝑞,𝑝₁<∞,1/𝑞=1/𝑝₁+1/𝑝₂−𝛽/𝑛 with 0≤𝛽<𝑛. If  1/𝑟^′−1/𝑞−𝛽/𝑛>0,𝑟>𝑝^′₁, 𝛼₂=𝛼₁+𝑛/𝑝₂ and 𝛼₂ satisfies the following condition: 𝛼₂+𝑛/𝑞−𝑛/𝑟^′−1/𝑟−𝑛/𝑝₂+𝛽<0, (13) then there exists a constant 𝐶, such that ℋ^𝑚_Ω,𝐴,𝛽𝑓_{̇𝐾𝛼1𝑞,𝑠}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_{̇𝐶𝐵𝑀𝑂}^𝑝₂𝑓_̇𝐾^𝛼_2𝑝1^,𝑝. (14)

For 𝑚=1, ℋ¹_Ω,𝐴,𝛽 is just the commutator of Hardy operator; that is, ℋ¹_Ω,𝐴,𝛽=ℋ^𝐴_Ω,𝛽𝑓(𝑥)=𝐴(𝑥)ℋ_Ω,𝛽𝑓(𝑥)−ℋ_Ω,𝛽(𝐴𝑓)(𝑥). We have the following theorem of ℋ^𝐴_Ω,𝛽 on Herz type space.

Theorem 4. Suppose Ω∈𝐿^𝑟(𝕊^𝑛−1) with 1<𝑟<∞ and 𝐴∈𝐶𝐵̇𝑀𝑂^𝑝2. Let 0<𝑠≤𝑝<∞,1<𝑞,𝑝₁,𝑝₂<∞,1/𝑞=1/𝑝₁+1/𝑝₂−𝛽/𝑛 with 0≤𝛽<𝑛. If 1/𝑟^′−1/𝑞−𝛽/𝑛>0,𝑟>𝑝^′₁,  𝛼₂=𝛼₁+𝑛/𝑝₂, and 𝛼₂ satisfies (13), then there exists a constant 𝐶, such that ℋ^𝐴_Ω,𝛽𝑓_{̇𝐾𝛼1𝑞,𝑠}≤𝐶‖𝐴‖_{̇𝐶𝐵𝑀𝑂}^𝑝₂𝑓_̇𝐾^𝛼_2𝑝1^,𝑝. (15)

Remark 5. Comparing Theorems 3 and 4, we find that the restrictions on 𝛼₁ and 𝛼₂ are more rigid in Theorem 4 than in Theorem 3, which indicates that ℋ^𝑚_Ω,𝐴,𝛽 with 𝑚≥2 has better properties than the commutators.

In order to prove Theorems 3 and 4, we need the following lemmas.

Lemma 6 (see [19]). Let 1<𝑝₁,𝑝₂<∞,0≤𝛽<𝑛, and 𝛽/𝑛=1/𝑝₁−1/𝑝₂. If Ω∈𝐿^𝑟(𝕊^𝑛−1) with 𝑟>𝑝^′₁, then there exists a constant 𝐶 independent of 𝑓, such that ℋ_Ω,𝛽𝑓_𝐿𝑝2≤𝐶𝑓_𝐿𝑝1, (16) where ℋ_Ω,𝛽 is defined by ℋ_Ω,𝛽𝑓(𝑥)=1|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}Ω𝑥−𝑦𝑓𝑦𝑑𝑦. (17) By checking [19] carefully, one can draw the conclusion that if one replaces ℋ_Ω,𝛽𝑓(𝑥) by ℋ_|Ω|,𝛽|𝑓|(𝑥), then (16) still holds.

Lemma 7. Let 𝑚≥1,1<𝑝₁<𝑝₂<∞ and 0≤𝛽<𝑛. If 𝐴 has derivatives of order 𝑚−1 in 𝐿^𝑟(ℝ^𝑛) with 1/𝑝₂=1/𝑝₁+1/𝑟−𝛽/𝑛 and 𝑟>𝑝^′₁, then one has ℋ^𝑚_Ω,𝐴,𝛽𝑓_𝐿𝑝2≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_𝐿𝑟𝑓_𝐿𝑝1, (18) where the constant 𝐶 is independent of 𝑓 and 𝐴.

Proof. From [20, p. 241], we have the following estimates: 𝑅_𝑚𝐴;𝑥,𝑦𝑥−𝑦^𝑚−1≤𝑅_𝑚−1𝐴;𝑥,𝑦𝑥−𝑦^𝑚−1+𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴𝑦≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^∗(𝑥)+𝐷^𝛾𝐴^∗𝑦, (19) where 𝑚≥1 and (𝑓)^∗ is the Hardy-Littlewood maximal function of 𝑓.
Thus we obtain ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥=1)|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1𝑅_𝑚1𝐴;𝑥,𝑦𝑑𝑦≤𝐶|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}𝑓𝑦Ω×𝑥−𝑦_{𝛾=𝑚−1}𝐷^𝛾𝐴^∗(𝑥)+𝐷^𝛾𝐴^∗𝑦𝑑𝑦≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^∗(𝑥)ℋ_|Ω|,𝛽𝑓(𝑥)+ℋ_|Ω|,𝛽𝐷^𝛾𝐴^∗𝑓(𝑥). (20) By the above estimates, we can get _ℝ𝑛ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞≤𝐶_{𝛾=𝑚−1ℝ𝑛}𝐷^𝛾𝐴^∗(𝑥)ℋ_|Ω|,𝛽𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞+_ℝ𝑛ℋ_|Ω|,𝛽𝐷^𝛾𝐴^∗𝑓(𝑥)^𝑞1/𝑞≤𝐶_{𝛾=𝑚−1}(𝐼+𝐼𝐼). (21)
For the term 𝐼, let 1/𝑞=1/𝑟+1/𝑙=1/𝑟+1/𝑝−𝛽/𝑛; then by the Hölder inequality, Lemma 6, and the boundedness of Hardy-Littlewood maximal function on 𝐿^𝑝 spaces, we obtain 𝐼≤_ℝ𝑛𝐷^𝛾𝐴^∗(𝑥)^𝑟𝑑𝑥^1/𝑟_ℝ𝑛ℋ_|Ω|,𝛽𝑓(𝑥)^𝑙𝑑𝑥^1/𝑙≤𝐶𝐷^𝛾𝐴^∗_𝐿𝑟𝑓_𝐿𝑝≤𝐶𝐷^𝛾𝐴_𝐿𝑟𝑓_𝐿𝑝. (22)
For the term 𝐼𝐼, let 1/𝑞=1/𝑡−𝛽/𝑛=1/𝑟+1/𝑝−𝛽/𝑛; then by the Hölder inequality and Lemma 6, we have 𝐼𝐼≤𝐶_ℝ𝑛𝐷^𝛾𝐴^∗(𝑥)𝑓(𝑥)^𝑡𝑑𝑥^1/𝑡≤𝐶𝐷^𝛾𝐴^∗_𝐿𝑟𝑓_𝐿𝑝≤𝐶𝐷^𝛾𝐴_𝐿𝑟𝑓_𝐿𝑝. (23) Combining the estimates of 𝐼 and 𝐼𝐼, we finish the proof of Lemma 7.

Lemma 8 (see [4]). Let 𝑏 be a function on ℝ^𝑛 with 𝑚th order derivatives in 𝐿^𝑞_loc(ℝ^𝑛) for some 𝑞>𝑛. Then 𝑅_𝑚𝑏;𝑥,𝑦≤𝐶_𝑚,𝑛𝑥−𝑦^𝑚×_|𝛾|=𝑚1𝑄𝑥,𝑦𝑄_(𝑥,𝑦)𝐷^𝛾𝑏(𝑧)^𝑞𝑑𝑧^1/𝑞, (24) where 𝑄(𝑥,𝑦) is the cube centered at 𝑥 having diameter 5√𝑛|𝑥−𝑦|.

Lemma 9 (see [5]). Suppose that 𝑓∈𝐶𝐵̇𝑀𝑂^𝑞(ℝ^𝑛), 1≤𝑞<∞, and 𝑟₁,𝑟₂>0; then 1𝐵0,𝑟_{1𝐵(0,𝑟1)}𝑓(𝑥)−𝑓_{𝐵(0,𝑟2)}^𝑞𝑑𝑥^1/𝑞𝑟≤𝐶1+log¹𝑟₂𝑓_{𝐶̇𝐵𝑀𝑂}^𝑞. (25)

Proof of Theorem 3. To prove Theorem 3, first we split each 𝑓 as 𝑓(𝑥)=^+∞_𝑖=−∞𝑓(𝑥)𝜒_𝑖(𝑥)=^+∞_𝑖=−∞𝑓_𝑖(𝑥); (26) then we have ℋ^𝑚_Ω,𝐴,𝛽𝑓𝜒_𝑘^𝑞_𝐿𝑞=_{𝐶𝑘𝐵(0,|𝑥|)}𝑓𝑦Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞×|𝑥|^{(𝛽−𝑛)𝑞}≤𝑑𝑥_{𝐶𝑘𝑘𝑖=−∞𝐶𝑖}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞×|𝑥|^{(𝛽−𝑛)𝑞}𝑑𝑥≤𝐶_𝐶𝑘^𝑘−3_{𝑖=−∞𝐶𝑖}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞×|𝑥|^{(𝛽−𝑛)𝑞}𝑑𝑥+𝐶_𝐶𝑘^𝑘_{𝑖=𝑘−2𝐶𝑖}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞×|𝑥|^{(𝛽−𝑛)𝑞}𝐼𝑑𝑥=𝐶₁+𝐼₂. (27)
For the term 𝐼₁, we denote 𝐴_𝑘(𝑥)=𝐴(𝑥)−∑_{|𝛾|=𝑚−1}(1/𝛾!)𝑚_𝐵𝑘(𝐷^𝛾𝐴)𝑥^𝛾; then it is easy to check 𝑅_𝑚(𝐴;𝑥,𝑦)=𝑅_𝑚(𝐴_𝑘;𝑥,𝑦). By the fact that 𝑥∈𝐶_𝑘, 𝑦∈𝐶_𝑖 with 𝑖≤𝑘−3, we have |𝑥−𝑦|∼|𝑥|∼2^𝑘. As 𝑝₂>𝑛, then by Lemmas 8 and 9, we obtain 𝑅_𝑚𝐴_𝑘≤𝑅;𝑥,𝑦_𝑚−1𝐴_𝑘;𝑥,𝑦+_{𝛾=𝑚−1}1𝐷𝛾!^𝛾𝐴_𝑘(𝑥)𝑥−𝑦^𝑚−1≤𝐶𝑥−𝑦^𝑚−1_{𝛾=𝑚−1}1𝑄×𝑄𝑥,𝑦_(𝑥,𝑦)𝐷^𝛾𝐴_𝑘(𝑧)^𝑝2𝑑𝑧^1/𝑝2+𝐷^𝛾𝐴_𝑘𝑦≤𝐶𝑥−𝑦^𝑚−1_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}+𝐷^𝛾𝐴_𝑘𝑦. (28) As |𝑥−𝑦|∼|𝑥|∼2^𝑘 and 1−1/𝑝₁−1/𝑝₂−1/𝑟=1/𝑟^′−1/𝑞−𝛽/𝑛>0, then by the Hölder inequality, we have 𝐼₁≤𝐶_𝐶𝑘^𝑘−3_{𝑖=−∞𝐶𝑖}Ω𝑓𝑦×𝑥−𝑦_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}+𝐷^𝛾𝐴_𝑘𝑦𝑑𝑦^𝑞×|𝑥|^{(𝛽−𝑛)𝑞}𝑑𝑥≤𝐶2^{−𝑘(𝑛−𝛽)𝑞}_𝐶𝑘^𝑘−3_{𝑖=−∞𝐶𝑖}𝑓𝑦Ω𝑥−𝑦×_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}+𝐷^𝛾𝐴_𝑘𝑦𝑑𝑦^𝑞𝑑𝑥≤𝐶2^{−𝑘(𝑛−𝛽)𝑞}_𝐶𝑘^𝑘−3_{𝑖=−∞𝐶𝑖}𝑓𝑦^𝑝1𝑑𝑦^𝑞/𝑝1×_𝐶𝑖Ω𝑥−𝑦^𝑟𝑑𝑦^𝑞/𝑟×_{𝛾=𝑚−1𝐶𝑖}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}+𝐷^𝛾𝐴_𝑘𝑦^𝑝2𝑑𝑦^𝑞/𝑝2×𝐶_𝑖^{𝑞(1−1/𝑝1−1/𝑝2−1/𝑟)}𝑑𝑥. (29) As _𝐶𝑖𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}+𝐷^𝛾𝐴_𝑘𝑦^𝑝2𝑑𝑦^1/𝑝2≤_𝐶𝑖𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}+𝐷^𝛾𝐴_𝑘𝑦−𝑚_𝐵𝑘𝐷^𝛾𝐴𝑦^𝑝2𝑑𝑦^1/𝑝2≤𝐶𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}𝐶_𝑖|𝑘−𝑖_𝐶𝑖|,Ω𝑥−𝑦^𝑟𝑑𝑦≤^|𝑥|+2𝑖_{|𝑥|−2𝑖𝕊𝑛−1}Ω𝑦^′𝑟𝑑𝜎𝑦^′𝑟^𝑛−1𝑑𝑟≤𝐶2^{𝑘𝑛−𝑘+𝑖}, (30) we obtain the following estimates: 𝐼₁≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝2}|𝑘−𝑖|^𝑞2^{−𝑘𝑛𝑞+𝑘𝛽𝑞}×2^{𝑖𝑛(1−1/𝑝1−1/𝑝2−1/𝑟)𝑞}×2^{((𝑘(𝑛−1)+𝑖)/𝑟)𝑞+(𝑖𝑛/𝑝2)𝑞}×_𝐶𝑘^𝑘−3_𝑖=−∞𝑓_𝑖^𝑞_𝐿𝑝1𝑑𝑥≤𝐶^𝑘−3_𝑖=−∞|𝑘−𝑖|2^{𝑖𝑛(1−1/𝑝1−1/𝑝2−1/𝑟)}×2^{−𝑘𝑛+𝑘𝛽+(𝑘(𝑛−1)+𝑖)/𝑟+𝑖𝑛/𝑝2+𝑘𝑛/𝑞}×𝑓_{𝑖𝐿𝑝1|𝛾|=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}^𝑞. (31)
For the term 𝐼₂, we choose 𝜙∈𝐶^∞₀(ℝ^𝑛) satisfying supp𝜙⊂𝐵(0,4) as well as 𝜙≡1 in 𝐵(0,2) and we set 𝐿=max{‖𝐷^𝛾𝜙‖_𝐿∞,|𝛾|≤𝑚−1}. Let 𝑦₀∈𝐵_𝑘+4 and 𝐴^𝜙_𝑘(𝑥)=𝑅_𝑚−1(𝐴_𝑘;𝑥,𝑦₀)𝜙(2^−𝑘𝑥). Then it is easy to see that 𝑅_𝑚(𝐴;𝑥,𝑦)=𝑅_𝑚(𝐴^𝜙_𝑘;𝑥,𝑦)=𝑅_𝑚(𝐴_𝑘;𝑥,𝑦) for 𝑥∈𝐵_𝑘 and 𝑦∈𝐵_𝑖 with 𝑘−2≤𝑖≤𝑘. Thus we get ℋ^𝑚_Ω,𝐴,𝛽𝑓_𝑖(𝑥)=ℋ^𝑚_{Ω,𝐴𝜙𝑘,𝛽}𝑓_𝑖(𝑥)=ℋ^𝑚_{Ω,𝐴𝑘,𝛽}𝑓_𝑖(𝑥). (32) Thus by Lemma 7, we have 𝐼₂≤𝐶^𝑘_{𝑖=𝑘−2𝐶𝑘}1|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}Ω𝑥−𝑦𝑓_𝑖𝑦𝑥−𝑦^𝑚−1×𝑅_𝑚𝐴^𝜙_𝑘;𝑥,𝑦𝑑𝑦^𝑞𝑑𝑥≤𝐶^𝑘_{𝑖=𝑘−2}ℋ^𝑚_{Ω,𝐴𝜙𝑘,𝛽}𝑓_𝑖^𝑞_𝐿𝑞≤𝐶^𝑘_{𝑖=𝑘−2}𝑓_𝑖^𝑞_{𝐿𝑝1|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝜙_𝑘^𝑞_𝐿𝑝2. (33) From [5, p.80], we have the following estimates: 𝐷^𝛾𝐴^𝜙_{𝑘𝐿𝑝2}≤𝐶2^{𝑘𝑛/𝑝2}_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}, (34) where 𝐶 is dependent on 𝐿. Thus we get 𝐼₂≤𝐶^𝑘_{𝑖=𝑘−2}𝑓_𝑖^𝑞_𝐿𝑝12^{𝑘𝑛𝑞/𝑝2}_{|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝2}. (35) As 0<𝑠≤𝑝<∞, we obtain the following estimates: ^+∞_𝑘=−∞2^{𝑘𝛼1𝑠}‖ℋ^𝑚_Ω,𝐴,𝛽𝑓𝜒_𝑘‖^𝑠_𝐿^𝑞1/𝑠≤^+∞_𝑘=−∞2^{𝑘𝛼1𝑝}ℋ^𝑚_Ω,𝐴,𝛽𝑓𝜒_𝑘^𝑝_𝐿𝑞^1/𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞2^{𝑘𝛼1𝑝}×^𝑘−3_𝑖=−∞|𝑘−𝑖|2^{𝑖𝑛(1−1/𝑝1−1/𝑝2−1/𝑟)}𝑓_{𝑖𝐿𝑝1}×2^{−𝑘𝑛+𝑘𝛽+(𝑘(𝑛−1)+𝑖)/𝑟+𝑖𝑛/𝑝2)+𝑘𝑛/𝑞𝑝1/𝑝}+𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞2^{𝑘𝛼1𝑝𝑘}_{𝑖=𝑘−2}𝑓_𝑖^𝑝_𝐿𝑝12^{(𝑘𝑛/𝑝2)𝑝1/𝑝}=𝐶(𝐴+𝐵). (36) For the term 𝐴, we have 𝐴=_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞2^{𝑘𝛼1𝑝}×^𝑘−3_𝑖=−∞|𝑘−𝑖|2^{𝑖𝑛(1−1/𝑝1−1/𝑝2−1/𝑟)}2^{−𝑘𝑛+𝑘𝛽}×2^{(𝑘(𝑛−1)+𝑖)/𝑟+𝑖𝑛/𝑝2+𝑘𝑛/𝑞}‖𝑓_𝑖‖_𝐿^𝑝₁^𝑝1/𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞^𝑘−3_𝑖=−∞|𝑘−𝑖|2^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′+𝛽−1/𝑟−𝑛/𝑝2)}×2^{(𝑘−𝑖)(𝛼1+𝑛/𝑝2)}×2^{𝑖(𝛼1+𝑛/𝑝2)}𝑓_{𝑖𝐿𝑝1}^𝑝1/𝑝. (37) When 0<𝑝≤1, by condition (13), we get 𝐴^𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞^𝑘−3_𝑖=−∞2^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽)}×2^{(𝑘−𝑖)(𝛼1+𝑛/𝑝2)}×|𝑘−𝑖|2^{𝑖(𝛼1+𝑛/𝑝2)}𝑓_{𝑖𝐿𝑝1}^𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞2^{𝑖𝛼2𝑝𝑘−3}_𝑖=−∞|𝑘−𝑖|×2^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽+𝛼2)}×𝑓_{𝑖𝐿𝑝1}^𝑝≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}𝑓^𝑝_{̇𝐾𝛼2𝑝1,𝑝}. (38) When 𝑝≥1, by the Hölder inequality and condition (13), we have 𝐴^𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞2^{𝑘𝛼1𝑝𝑘−3}_𝑖=−∞|𝑘−𝑖|×2^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽)}×2^{𝑘𝑛/𝑝2}𝑓_{𝑖𝐿𝑝1}^𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞^𝑘−3_𝑖=−∞|𝑘−𝑖|2^𝑘𝛼12^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽)}×2^{𝑘𝑛/𝑝2}𝑓_{𝑖𝐿𝑝1}^𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞^𝑘−3_𝑖=−∞|𝑘−𝑖|2^𝑖𝛼22^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽+𝛼2)}×𝑓_{𝑖𝐿𝑝1}^𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}×^+∞_𝑘=−∞^𝑘−3_𝑖=−∞2^{𝑖𝛼2𝑝}𝑓_𝑖^𝑝_𝐿𝑝12^{𝑝(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽+𝛼2)/𝑝}×^𝑘−3_𝑖=−∞|𝑘−𝑖|^𝑝′2^{((𝑘−𝑖)/𝑝′)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽+𝛼2)𝑝′𝑝/𝑝′}≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}×^+∞∞_{𝑖=−∞𝑘=𝑖+3}2^{𝑖𝑝(𝛼1+𝑛/𝑝2)}2^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽+𝛼2)}𝑓_{𝑖𝐿𝑝1}×^𝑘−3_𝑖=−∞|𝑘−𝑖|^𝑝′2^{(𝑘−𝑖)(𝑛/𝑞−𝑛/𝑟′−1/𝑟−𝑛/𝑝2+𝛽+𝛼2)𝑝/𝑝′}≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}^+∞_𝑖=−∞2^{𝑖𝛼2𝑝}𝑓_𝑖^𝑝_𝐿𝑝1≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝑝_{Ċ‌BMO𝑝2}𝑓^𝑝_{̇𝐾𝛼2𝑝1,𝑝}. (39) For the term 𝐵, we have the following estimates: 𝐵≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}^+∞_𝑘=−∞2^{𝑘𝛼1𝑝𝑘}_{𝑖=𝑘−2}2^{𝑘𝑛𝑝/𝑝2}𝑓_𝑖^𝑝_𝐿𝑝1^1/𝑝≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}^+∞_𝑖=−∞2^{𝑖𝛼2𝑝}𝑓_𝑖^𝑝_𝐿𝑝1^1/𝑝≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝2}𝑓_̇𝐾^𝛼_2𝑝1^,𝑝. (40) Combining the estimates of 𝐴 and 𝐵, we finish the proof of Theorem 3.

Proof of Theorem 4. The proof of Theorem 4 is quite similar and much easier than Theorem 3 and we omit the details here.

3. Boundedness of ℋ^𝑚_Ω,𝐴,𝛽 on 𝜆-Central Morrey Spaces

In [21], Wiener gave a way to describe the behavior of a function at the infinity. Later, Beurling [22] extended Weiner’s idea and introduced a pair of dual Banach spaces, 𝐴^𝑞 and 𝐵^𝑞′, with 1/𝑞+1/𝑞^′=1. In [23], Feichtinger proved that 𝐵^𝑞 can be described as 𝑓_𝐵𝑞=sup_𝑘≥02^{−𝑘𝑛/𝑞}𝑓𝜒_𝑘𝐿𝑞<∞, (41) where 𝜒₀ is the characterization of the unit ball {𝑥∈ℝ^𝑛∶|𝑥|≤1} and 𝜒_𝑘 is defined as in Section 2.

Now by duality, the space 𝐴^𝑞, which is called the Beurling algebra, can be described by 𝑓_𝐴𝑞=^∞_𝑘=02^{𝑘𝑛/𝑞′}𝑓𝜒_𝑘𝐿𝑞<∞. (42)

Later, Chen and Lau [24] as well as García-Cuerva [25] introduced atomic spaces 𝐻𝐴^𝑞(ℝ^𝑛) associated with the Buerling algebra 𝐴^𝑞 and the dual space of 𝐻𝐴^𝑞(ℝ^𝑛) can be described by 𝑓_CBMO𝑞∶=sup_𝑅≥11|𝐵(0,𝑅)|_𝐵(0,𝑅)𝑓(𝑥)−𝑓_𝐵(0,𝑅)^𝑞𝑑𝑥^1/𝑞<∞; (43) here the CBMO^𝑞 can be regarded as the inhomogeneous central BMO spaces.

In 2000, Alvarez et al. [26] introduced the 𝜆-Central bounded mean oscillation space and 𝜆-Central Morrey space, respectively.

Definition 10 (see [26]). Given 𝜆<1/𝑛,1<𝑞<∞, then a function 𝑓∈𝐿^𝑞_loc(ℝ^𝑛) is said to belong to the 𝜆-Central bounded mean oscillation space CḂMO^𝑞,𝜆(ℝ^𝑛) if 𝑓_{Ċ‌BMO𝑞,𝜆}∶=sup_𝑅>01|𝐵(0,𝑅)|^{1+𝜆𝑞𝐵(0,𝑅)}𝑓(𝑥)−𝑓_𝐵(0,𝑅)^𝑞𝑑𝑥^1/𝑞<∞. (44)

Definition 11 (see [26]). Let 𝜆∈ℝ and 1<𝑞<∞. Then the 𝜆-Central Morrey space 𝐸̇^𝑞,𝜆(ℝ^𝑛) is defined by 𝑓_{̇𝐸𝑞,𝜆}=sup_𝑅>01|𝐵(0,𝑅)|^{1+𝜆𝑞𝐵(0,𝑅)}𝑓(𝑥)^𝑞𝑑𝑥^1/𝑞<∞. (45)

From [27], we know that if 1<𝑞₁<𝑞₂<∞, we obtain 𝐸̇^𝑞2,𝜆⊂𝐸̇^𝑞1,𝜆 for 𝜆∈𝑅 and CḂMO^𝑞2,𝜆⊂CḂMO^𝑞1,𝜆 for 𝜆<1/𝑛. Furthermore, when 𝜆<−1/𝑞, 𝐸̇^𝑞,𝜆 reduces to {0} and CḂMO^𝑞,𝜆 reduces to the space of constant functions. When 𝜆=−1/𝑞, CḂMO^𝑞,𝜆 coincides with 𝐿^𝑞(ℝ^𝑛) modulo constant and 𝐸̇^𝑞,𝜆=𝐿^𝑞.

In 2011, Fu et al. [19] proved the boundedness of the commutator of fractional Hardy operator with a rough kernel on 𝜆-Central Morrey space. Later, Fu et al. [28] proved the boundness of the weighted Hardy operator and its commutator on 𝜆-Central Morrey space. In this paper, we will give the boundedness of ℋ^𝑚_Ω,𝐴,𝛽 on 𝜆-Central Morrey space with 𝑚≥1.

Our results can be stated as follows.

Theorem 12. Suppose 𝑚≥2, 𝑛<𝑝₂<∞,1<𝑝₁<∞, 1/𝑞=1/𝑝₁+1/𝑝₂−𝛽/𝑛 with 0≤𝛽<𝑛 and 𝜆=𝜆₁+𝜆₂+𝛽/𝑛. Let Ω∈𝐿^𝑟(𝕊^𝑛−1) with 1/𝑟^′>𝛽/𝑛+1/𝑞, and 𝐴 has derivatives of order 𝑚−1 in 𝐶𝐵𝑀̇𝑂^𝑝2,𝜆2. If 𝑟>𝑝^′₁, 𝜆₁>−1/𝑝₁, and 𝑞𝜆+1>0, then one has ℋ^𝑚_Ω,𝐴,𝛽𝑓_{̇𝐸𝑞,𝜆}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_{̇𝐶𝐵𝑀𝑂}^𝑝₂₂^,𝜆𝑓_{̇𝐸𝑝11,𝜆}, (46) where the constant 𝐶 is independent of 𝑓 and 𝐴.

For the case 𝑚=1, we have the following theorem.

Theorem 13. Suppose 1<𝑝₁,𝑝₂<∞, 1/𝑞=1/𝑝₁+1/𝑝₂−𝛽/𝑛 with 0≤𝛽<𝑛 and 𝜆=𝜆₁+𝜆₂+𝛽/𝑛. Let Ω∈𝐿^𝑟(𝕊^𝑛−1) with 1/𝑟^′>𝛽/𝑛+1/𝑞 and 𝐴∈𝐶𝐵𝑀̇𝑂^𝑝2,𝜆2. If 𝜆₁>−1/𝑝₁,𝑟>𝑝^′₁, and 𝑞𝜆+1>0, then one has ℋ^𝐴_Ω,𝛽𝑓_{̇𝐸𝑞,𝜆}≤𝐶‖𝐴‖_{̇𝐶𝐵𝑀𝑂}^𝑝₂₂^,𝜆𝑓_{̇𝐸𝑝11,𝜆}, (47) where the constant 𝐶 is independent of 𝑓 and 𝐴.

In order to prove Theorems 12 and 13, by a standard argument, we have the following lemma.

Lemma 14 (see [16]). Suppose 𝑓∈𝐶𝐵𝑀̇𝑂^𝑝,𝜆, 1≤𝑝<∞, 𝜆<1/𝑛, and 𝑟₁,𝑟₂∈𝑅⁺; then 1𝐵0,𝑟₁^{1+𝑝𝜆𝐵(0,𝑟}₁⁾𝑓(𝑥)−𝑓_{𝐵(0,𝑟2)}^𝑝𝑑𝑥^1/𝑝𝑟≤𝐶1+log¹𝑟₂𝑓_{̇𝐶𝐵𝑀𝑂}^𝑝,𝜆. (48)

Proof of Theorem 12. For any 𝑅>0, we denote 𝐵(0,𝑅) by 𝐵 and 𝐵(0,𝑘𝑅) by 𝑘𝐵 for any 𝑘∈𝑍⁺. Thus we have the following estimates: 1|𝐵|_𝐵ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)^𝑞≤1𝑑𝑥|𝐵|_𝐵1|𝑥|^{𝑛−𝛽𝐵(0,|𝑥|)}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1×𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞=1𝑑𝑥|𝐵|⁰_𝑘=−∞2^𝑘_𝐵⧵2^𝑘−1_𝐵1|𝑥|^{𝑘𝑛−𝛽}_𝑖=−∞^{2𝑖𝐵⧵2𝑖−1𝐵}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1×𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞≤1𝑑𝑥|𝐵|⁰_𝑘=−∞2^𝑘_𝐵⧵2^𝑘−1_𝐵1|𝑥|^{𝑛−𝛽𝑘−3}_𝑖=−∞^{2𝑖𝐵⧵2𝑖−1𝐵}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1×𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞+1𝑑𝑥|𝐵|×⁰_{𝑘=−∞2𝑘𝐵⧵2𝑘−1𝐵}1|𝑥|^{𝑘𝑛−𝛽}_{𝑖=𝑘−2}^{2𝑖𝐵⧵2𝑖−1𝐵}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1×𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦^𝑞𝑑𝑥=𝐼+𝐼𝐼. (49)
For the term 𝐼, let 𝐴(𝑥)=𝐴(𝑥)−∑_{|𝛾|=𝑚−1}(1/𝛾!)𝑚₂^𝑘_𝐵(𝐷^𝛾𝐴)𝑥^𝛾. Then it is easy to see 𝑅_𝑚(𝐴;𝑥,𝑦)=𝑅_𝑚(𝐴;𝑥,𝑦). As 𝑝₂>𝑛, then by Lemmas 8 and 9, we have 𝑅_𝑚𝐴;𝑥,𝑦≤𝑅_𝑚−1𝐴;𝑥,𝑦+_{𝛾=𝑚−1}1𝐷𝛾!^𝛾𝐴𝑦𝑥−𝑦^𝑚−1≤𝐶𝑥−𝑦^𝑚−1_{𝛾=𝑚−1}𝑄^𝜆2𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}+𝐷^𝛾𝐴𝑦, (50) where 𝑄(𝑥,𝑦) is the cube centered at 𝑥 and having diameter 5√𝑛|𝑥−𝑦|.
As 𝑥∈2^𝑘𝐵⧵2^𝑘−1𝐵 and 𝑦∈2^𝑖𝐵⧵2^𝑖−1𝐵 with 𝑖≤𝑘−3, we have |𝑥−𝑦|∼|𝑥|∼|2^𝑘𝐵|^1/𝑛 and |𝑥−𝑦|≥𝐶₁|2^𝑖𝐵|^1/𝑛.
Thus by the Hölder inequality and the condition 1−1/𝑝₁−1/𝑝₂−1/𝑟=1/𝑟^′−𝛽/𝑛−1/𝑞>0, we have 𝐼≤𝐶|𝐵|⁻¹×⁰_{𝑘=−∞2𝑘𝐵⧵2𝑘−1𝐵}1|𝑥|^𝑛−𝛽×^𝑘−3_{𝑖=−∞2𝑖𝐵⧵2𝑖−1𝐵}Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑚−1×𝑥−𝑦^𝑚−1×_{|𝛾|=𝑚−1}𝑄^𝜆2𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}+𝐷^𝛾𝐴𝑦𝑑𝑦^𝑞|𝐵|𝑑𝑥≤𝐶⁻¹×⁰_{𝑘=−∞2𝑘𝐵⧵2𝑘−1𝐵}1|𝑥|^𝑛−𝛽×^𝑘−3_{𝑖=−∞2𝑖𝐵⧵2𝑖−1𝐵}Ω𝑥−𝑦𝑓𝑦×_{|𝛾|=𝑚−1}𝑄^𝜆2×𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}+𝐷^𝛾𝐴𝑦𝑑𝑦^𝑞|𝐵|𝑑𝑥≤𝐶⁻¹×⁰_{𝑘=−∞2𝑘𝐵⧵2𝑘−1𝐵}1|𝑥|^{(𝑛−𝛽)𝑞}×^𝑘−3_{𝑖=−∞2𝑖𝐵⧵2𝑖−1𝐵}Ω𝑥−𝑦^𝑟𝑑𝑦^1/𝑟×_{2𝑖𝐵⧵2𝑖−1𝐵𝛾=𝑚−1}𝑥−𝑦^𝜆2𝑛×𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}+𝐷^𝛾𝐴𝑦^𝑝2𝑑𝑦^1/𝑝2×_{2𝑖𝐵⧵2𝑖−1𝐵}𝑓𝑦^𝑝1𝑑𝑦^1/𝑝1×2^𝑖𝐵^{1−1/𝑝1−1/𝑝2−1/𝑟𝑞}𝑑𝑥. (51) Note the following fact: _{2𝑖𝐵⧵2𝑖−1𝐵𝛾=𝑚−1}𝑥−𝑦^𝜆2𝑛𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}+𝐷^𝛾𝐴𝑦^𝑝2𝑑𝑦^1/𝑝2≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}2^𝑘𝐵^𝜆22^𝑖𝐵^1/𝑝2+_2𝑖𝐵𝐷^𝛾𝐴𝑦𝑄−𝑚𝐷^𝛾𝐴^𝑝2𝑑𝑦^1/𝑝2≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}2^𝑖𝐵^1/𝑝22^𝑘𝐵^𝜆2+2^𝑖𝐵^𝜆2, (52) where the last inequality follows from Lemma 14 and the fact |𝑥−𝑦|≥𝐶₁|2^𝑖𝐵|^1/𝑛. Thus by the condition 1+𝑞𝜆>0 and 𝜆₁+1>𝜆₁+1/𝑝₁>0, we have 𝐼≤𝐶|𝐵|⁻¹×⁰_𝑘=−∞2^𝑘𝐵^{(𝛽/𝑛−1)𝑞}×_{2𝑘𝐵⧵2𝑘−1𝐵}^𝑘−3_𝑖=−∞𝑓_{̇𝐸𝑝11,𝜆}2^𝑖𝐵^{1/𝑝1+𝜆1}2^𝑖𝐵^1/𝑟×_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}2^𝑖𝐵^1/𝑝2×2^𝑘𝐵^𝜆2+2^𝑖𝐵^𝜆2×2^𝑖𝐵^{1−1/𝑝1−1/𝑝2−1/𝑟𝑞}𝑓𝑑𝑥≤𝐶^𝑞_{̇𝐸𝑝11,𝜆𝛾=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝22,𝜆}|𝐵|⁻¹×⁰_𝑘=−∞2^𝑘𝐵^{(𝛽/𝑛−1)𝑞}2^𝑘𝐵2^𝑘𝐵^𝜆2𝑞×^𝑘−3_𝑖=−∞2^𝑖𝐵^{1/𝑝1+𝜆1}×2^𝑖𝐵^1/𝑝22^𝑖𝐵^1/𝑟2^𝑖𝐵^{1−1/𝑝1−1/𝑝2−1/𝑟𝑞}≤𝐶𝑓^𝑞_{̇𝐸𝑝11,𝜆𝛾=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝22,𝜆}×|𝐵|^{−1+𝛽𝑞/𝑛−𝑞+𝜆2𝑞+1+𝜆1𝑞+𝑞}×⁰_𝑘=−∞2^{𝑛𝑘𝛽𝑞/𝑛−𝑘𝑛𝑞+𝑘𝑛+𝑘𝜆2𝑛𝑞+𝑘𝜆1𝑛𝑞+𝑘𝑛𝑞}≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝22,𝜆}𝑓^𝑞_{̇𝐸𝑝11,𝜆}|𝐵|^𝑞𝜆. (53)
To estimate the term 𝐼𝐼, we adopt some basic ideas from the estimates of the term 𝐼₂ in Theorem 3. First, we denote 𝑅_𝑚(𝐴;𝑥,𝑦)=𝑅_𝑚(𝐴^𝜙_𝑘;𝑥,𝑦)=𝑅_𝑚(𝐴_𝑘;𝑥,𝑦) for 𝑥∈2^𝑘𝐵 and 𝑦∈2^𝑖𝐵 with 𝑘−2≤𝑖≤𝑘, where 𝐴^𝜙_𝑘(𝑥)=𝑅_𝑚−1(𝐴_𝑘;,𝑥,𝑦₀)𝜙(|𝑥−𝑦₀|⁻¹𝑥) with 𝑦₀∈2^𝑖𝐵⧵2^𝑖−1𝐵. Here 𝐴_𝑘(𝑥)=𝐴(𝑥)−∑_{|𝛾|=𝑚−1}(1/𝛾!)𝑚₂^𝑘_𝐵(𝐷^𝛾𝐴)𝑥^𝛾 and 𝜙 is defined as in Section 2.
As Ω∈𝐿^𝑟(𝕊^𝑛−1) with 𝑟>𝑝^′₁, by Lemma 7, we get 1𝐼𝐼≤|𝐵|^0𝑘_{𝑘=−∞𝑖=𝑘−2}ℋ^𝑚_{Ω,𝐴𝜙𝑘,𝛽}𝑓_𝑖^𝑞_𝑞≤𝐶|𝐵|^0𝑘_{𝑘=−∞𝑖=𝑘−2}𝑓_𝑖^𝑞_{𝐿𝑝1|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝜙_𝑘^𝑞_𝐿𝑝2, (54) where 𝑓_𝑖 is defined by 𝑓_𝑖(𝑥)=𝑓(𝑥)𝜒_{2𝑖𝐵⧵2𝑖−1𝐵}(𝑥).
For 𝐷^𝛾𝐴^𝜙_𝑘, as |𝑥−𝑦₀|≤𝐶|𝑦₀|, then by Lemmas 8 and 14, we have the following estimates: 𝐷^𝛾𝐴^𝜙_𝑘(𝑥)≤_{𝜇+|𝜈|=𝑚−1}𝐶_𝜇𝜈𝑅_{𝑚−1−|𝜇|}𝐷^𝜇𝐴_𝑘;𝑥,𝑦₀×𝐷^𝜈𝜙𝑥−𝑦₀⁻¹𝑥𝑥−𝑦₀^−|𝜈|≤𝐶_{|𝜇|+|𝜈|=𝑚−1}𝑥−𝑦₀^{𝑚−1−|𝜇|−|𝜈|}×_{𝛾′𝜇=𝑚−1−}1𝑄𝑥,𝑦₀𝑄_(𝑥,𝑦0)𝐷^𝛾′𝐷^𝜇𝐴_𝑘(𝑧)^𝑝2𝑑𝑧^1/𝑝2×𝜒_{|𝑥|≤|𝑥−𝑦0|}(𝑥)≤𝐶_{𝛾=𝑚−1}𝐷^𝛾𝐴_{Ċ‌BMO𝑝22,𝜆}𝑥−𝑦₀^𝜆2𝑛𝜒_{|𝑥|≤𝐶|𝑦0|}(𝑥). (55) Thus we have _{𝛾=𝑚−1}𝐷^𝛾𝐴^𝜙_𝑘^𝑞_𝐿𝑝2≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝22,𝜆}2^𝑖𝐵^𝑞𝜆22^𝑖𝐵^𝑞/𝑝2. (56) By the above estimates, we obtain 𝐶𝐼𝐼≤|𝐵|^0𝑘_{𝑘=−∞𝑖=𝑘−2}1|2^𝑖𝐵|^{1+𝑝1𝜆12𝑖𝐵}𝑓(𝑥)^𝑝1𝑑𝑥^𝑞/𝑝1×2^𝑖𝐵^{𝑞/𝑝1+𝑞𝜆1}×_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝22,𝜆}2^𝑖𝐵^{𝑞𝜆2+𝑞/𝑝2}≤𝐶^0𝑘_{𝑘=−∞𝑖=𝑘−2}2^{𝑖𝑛(1+𝜆𝑞)}×_{𝛾=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝22,𝜆}𝑓^𝑞_{̇𝐸𝑝11,𝜆}|𝐵|^𝑞𝜆≤𝐶_{|𝛾|=𝑚−1}𝐷^𝛾𝐴^𝑞_{Ċ‌BMO𝑝22,𝜆}𝑓^𝑞_{̇𝐸𝑝11,𝜆}|𝐵|^𝑞𝜆. (57) Combining the estimates of 𝐼 and 𝐼𝐼 and by the definition of 𝐸̇^𝑞,𝜆, we finish the proof of Theorem 12.

Proof of Theorem 13. The proof of Theorem 13 is quite similar but much simpler than Theorem 12 and we omit the details here.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant nos. 10931001, 11226104, and 11271175, the Natural Science Foundation of Jiangxi Province under Grant no. 20114BAB211007, and the Science Foundation of Jiangxi Education Department under Grant no. GJJ13703. This work was also supported by the Key Laboratory of Mathematics and Complex System (Beijing Normal University), Ministry of Education, China.