Endpoint Estimates for Generalized Commutators of Hardy Operators on 𝐻¹ Space

Abstract

We study the 𝐻¹-boundedness of the generalized commutators of Hardy operator with a homogeneous 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 𝑅_𝑚(𝐴;𝑥,𝑦)=𝐴(𝑥)−∑_|𝛼|<𝑚(1/𝛼!)𝐷^𝛼𝐴(𝑦)(𝑥−𝑦)^𝛼 with 𝑚∈𝑍⁺, 0≤𝛽<𝑛 and Ω∈Lip₁(𝑆^𝑛−1). We prove that, when 𝑚≥1, ℋ^𝑚_Ω,𝐴,𝛽 is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)} unless ℋ^𝑚_Ω,𝐴,𝛽≡0. Finally, we prove that ℋ^𝑚_Ω,𝐴,𝛽 is bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽),∞} with 𝑚≥1.

1. Introduction

Let 𝑓∈𝐿^𝑝(ℝ⁺) with 1<𝑝<∞; the classical Hardy operator is defined by (𝑥)=1𝐻𝑓𝑥^𝑥₀𝑓(𝑡)𝑑𝑡,𝑥≠0. (1)

A famous result proved by Hardy [1] can be stated as follows; 𝐻𝑓_{𝐿𝑝(ℝ+)}≤𝑝𝑓𝑝−1^{𝐿𝑝(ℝ+)}. (2)

Hardy [1] also pointed out the fact that the constant 𝑝/(𝑝−1) in (2) is the best possible. Later, Hardy operator was studied by many mathematicians; please see [2, 3] for more details.

In 1995, Christ and Grafakos [4] studied the following 𝑛-dimensional Hardy operator; (𝑥)=1ℋ𝑓𝜈_𝑛|𝑥|^𝑛_{|𝑦|<|𝑥|}𝑓𝑦𝑑𝑦,𝑥∈ℝ^𝑛⧵{0}, (3) where 𝜈_𝑛 is the volume of the unit ball in ℝ^𝑛, and they proved the following inequality: ℋ𝑓_𝐿𝑝≤𝑝𝑓𝑝−1^𝐿𝑝,1<𝑝<∞. (4)

Furthermore, Christ and Grafakos [4] also showed that the constant 𝑝/(𝑝−1) in (3) is the best possible.

In 2007, Fu et al. [5] considered the following commutator of fractional Hardy operator: ℋ^𝑏_𝛽𝑓(𝑥)=𝑏(𝑥)ℋ_𝛽𝑓(𝑥)−ℋ_𝛽𝑓𝑏(𝑥), (5) where ℋ_𝛽𝑓(𝑥) is defined by ℋ_𝛽𝑓(𝑥)=1|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}𝑓𝑦𝑑𝑦 (6) with −𝑛<𝛽<𝑛. When 𝛽=0, we simply denote ℋ^𝑏₀ by ℋ_𝑏 and ℋ₀ is just the 𝑛-dimensional Hardy operator proposed by Christ and Grafakos in [4] (without considering the constant 𝜈_𝑛).

In 2011, Fu et al. [6] studied the following 𝑛-dimensional fractional Hardy operator with a homogeneous kernel: ℋ_Ω,𝛽𝑓(𝑥)=1|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}Ω𝑥−𝑦𝑓𝑦𝑑𝑦, (7) where Ω∈𝐿^𝑟(𝑆^𝑛−1). Fu et al. [6] proved that ℋ^𝑏_Ω,𝛽 is bounded on Herz type space and 𝜆-central Morrey space. Here ℋ^𝑏_Ω,𝛽 is just the commutator of fractional Hardy operator with a homogeneous kernel.

Recently, Zhao et al. [7] gave a counterexample to show that ℋ_𝑏 is not bounded from 𝐻¹ to 𝐿¹, and they proved that ℋ_𝑏 is bounded from 𝐻¹ to weak 𝐿¹ space where 𝐻¹ denotes the Hardy space.

On the other hand, in 1982, Cohen [8] studied the following generalized commutator: 𝑇²_𝐴𝑓(𝑥)=_ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^𝑛+1×𝐴(𝑥)−𝐴𝑦−∇𝐴𝑦𝑥−𝑦𝑓𝑦𝑑𝑦, (8) where Ω∈𝐿¹(𝑆^𝑛−1) is homogeneous of degree zero and satisfies the moment condition _𝑆𝑛−1Ω(𝑥)𝑥^𝛼𝑑𝜎(𝑥)=0 (9) for |𝛼|=1. Cohen [8] 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 [9] considered another type of generalized commutator 𝑇^𝑚_𝐴(𝑓) as follows: 𝑇^𝑚_𝐴𝑓(𝑥)=_ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^{𝑛+𝑚−1}𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦, (10) where 𝑅_𝑚(𝐴;𝑥,𝑦)(𝑚∈𝑍⁺) is defined by 𝑅_𝑚(𝐴;𝑥,𝑦)=𝐴(𝑥)−∑_|𝛼|<𝑚(1/𝛼!)𝐷^𝛼𝐴(𝑦)(𝑥−𝑦)^𝛼, the 𝑚th remainder of Taylor series of the function 𝐴 at 𝑥 about 𝑦, and Ω satisfies the following moment conditions: _𝑆𝑛−1Ω(𝑥)𝑥^𝛼𝑑𝜎(𝑥)=0 (11) for |𝛼|=𝑚−1. Obviously, if we choose 𝑚=1, 𝑇^𝑚_𝐴 becomes [𝐴,𝑇], the commutator of 𝑇 generalized by 𝐴 and 𝑇.

Cohen and Gosselin [9] proved that 𝑇^𝑚_𝐴 is bounded on 𝐿^𝑝(ℝ^𝑛) for 1<𝑝<∞ if Ω∈Lip₁(𝑆^𝑛−1) and the function 𝐴 has derivatives of order 𝑚−1 in BMO(ℝ^𝑛). Later, 𝑇^𝑚_𝐴 was studied by many mathematicians; for example, see [10, 11] or [12] for more details. Particularly in [11], Lu and Wu studied the endpoint estimates of 𝑇^𝑚_𝐴 on 𝐻¹ space.

It should be pointed out that the generalized commutators of some operators play an important role in the study of partial differential equation. Recently, 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 [12], Wang and Zhang [13] gave a new proof of Wu’s theorem in [14]. Here we would like to point out that the method used in [13] is much simpler than that in [14].

In this paper, we will consider the following generalized commutator of fractional Hardy operator with a homogeneous kernel: ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)=1|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑓𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑑𝑦, (12) where 𝑅_𝑚(𝐴;𝑥,𝑦)=𝐴(𝑥)−∑_|𝛼|<𝑚(1/𝛼!)𝐷^𝛼𝐴(𝑦)(𝑥−𝑦)^𝛼,0≤𝛽<𝑛, and Ω∈Lip₁(𝑆^𝑛−1).

As the Hardy operator is controlled by the Hardy-Littlewood maximal function, we have ℋ^𝑚_Ω,𝐴,𝛽≤𝐶𝑀_Ω,𝐴,𝛽𝑓(𝑥), (13) where 𝑀_Ω,𝐴,𝛽𝑓(𝑥) is defined by 𝑀_Ω,𝐴,𝛽𝑓(𝑥)=sup_𝑟>01𝑟^𝑛−𝛽_{|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦. (14)

By a simple computation or from [15, pp. 221-222], we have 𝑀_Ω,𝐴,𝛽𝑓(𝑥)=sup_𝑟>01𝑟^𝑛−𝛽_{|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦≤𝐶sup_𝑟>01𝑟^𝑛−𝛽_{(𝑟/2)<|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦, (15) and thus ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)≤𝐶sup_𝑟>01𝑟^𝑛−𝛽_{|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦≤𝐶sup_𝑟>01𝑟^𝑛−𝛽_{(𝑟/2)<|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦≤𝐶sup_𝑟>0𝑟^{−(𝑛−𝛽+𝑚−1)}_{(𝑟/2)<|𝑥−𝑦|<𝑟}Ω×𝑅𝑥−𝑦_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦. (16)

So we have ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)≤𝐶𝑀^𝐴_Ω,𝛽𝑓(𝑥), (17) where 𝑀^𝐴_Ω,𝛽𝑓(𝑥)=sup_𝑟>0𝑟^{−(𝑛−𝛽+𝑚−1)}×_{(𝑟/2)<|𝑥−𝑦|<𝑟}Ω𝑥−𝑦𝑅_𝑚𝐴;𝑥,𝑦𝑓𝑦𝑑𝑦. (18)

From [15, p. 222], we have the following lemma.

Lemma 1 (see [15]). Let 0<𝛽<𝑛 and Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1). If 1<𝑝,𝑞<∞ with 1/𝑝−1/𝑞=𝛽/𝑛, and 𝐴 has derivatives of order 𝑚−1 in BMO(ℝ^𝑛), then 𝑀^𝐴_Ω,𝛽𝑓_𝐿𝑞≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_𝐵𝑀𝑂𝑓_𝐿𝑝, (19) where the constant 𝐶 is independent of 𝑓 and 𝐴.

By checking [15, p. 222] carefully, we deduce that (19) still holds if we take 𝛽=0. So we have the following proposition.

Proposition 2. Let 0≤𝛽<𝑛, 1/𝑝−1/𝑞=𝛽/𝑛 with 1<𝑝,𝑞<∞. If Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1) and 𝐴 has derivatives of order 𝑚−1 in BMO(ℝ^𝑛), then ℋ^𝑚_Ω,𝐴,𝛽𝑓_𝐿𝑞≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_𝐵𝑀𝑂𝑓_𝐿𝑝, (20) where the constant 𝐶 is independent of 𝑓 and 𝐴.

Definition 3 (see [16]). One says a function 𝑎(𝑥) is an 𝐻¹ atom if 𝑎 satisfies the following conditions: (i)supp(𝑎)⊂𝐵𝑥₀,(,𝑟ii)‖𝑎‖_𝐿^∞≤𝐵𝑥₀,𝑟⁻¹,(iii)𝑎(𝑥)𝑑𝑥=0. (21)

It is well known that, if a function 𝑓 belongs to 𝐻¹, then it can be written as 𝑓=∑^∞_𝑖=1𝜆_𝑖𝑎_𝑖 where each 𝑎_𝑖 is an 𝐻¹ atom. Moreover, one has 𝑓_𝐻1∼inf^+∞_𝑖=−∞𝜆_𝑖, (22) where the infimum is taken over all decompositions of 𝑓.

Definition 4 (see [17]). A function 𝑓 is said to belong to BMO(ℝ^𝑛) if the following sharp maximal function is bounded: 𝑓^♯(𝑥)=sup_𝐵1|𝐵|_𝐵𝑓𝑦−𝑓_𝐵𝑑𝑦<∞, (23) where the supreme is taken over all balls 𝐵⊂ℝ^𝑛 and 𝑓_𝐵=(1/|𝐵|)∫_𝐵𝑓(𝑥)𝑑𝑥 and ‖𝑓‖_BMO=‖𝑓^♯‖_𝐿∞.

Proposition 5 (see [17]). Let 1<𝑝<∞ and 𝑓∈𝐵𝑀𝑂(ℝ^𝑛); then one has(a)‖𝑓‖_𝐵𝑀𝑂∼sup_𝐵((1/|𝐵|)∫_𝐵|𝑓(𝑥)−𝑓_𝐵|^𝑝𝑑𝑥)^1/𝑝;(b)‖𝑓‖_𝐵𝑀𝑂∼sup_𝐵inf_𝑎∈𝑅(1/|𝐵|)∫_𝐵|𝑓(𝑥)−𝑎|𝑑𝑥.
Obviously, when 𝑚=1, ℋ^𝑚_Ω,𝐴,𝛽 can be written as ℋ¹_Ω,𝐴,𝛽𝑓(𝑥)∶=ℋ_Ω,𝐴,𝛽𝑓(𝑥)=𝐴(𝑥)ℋ_Ω,𝛽𝑓(𝑥)−ℋ_Ω,𝛽(𝑓𝐴)(𝑥), just the commutator of fractional Hardy operator with a homogeneous kernel.

For the case 𝑚=2,𝛽=0, and Ω≡1, Lu and Zhao [18] proved that ℋ²_Ω,𝐴,0 is bounded on Herz type spaces and Morrey-Herz type spaces.

In this paper, we would like to show that ℋ^𝑚_Ω,𝐴,𝛽 is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)} for all 𝑚∈𝑍⁺. Furthermore, we will prove that ℋ^𝑚_Ω,𝐴,𝛽 is bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽),∞}, where 𝐿^{𝑛/(𝑛−𝛽),∞} denotes the weak 𝐿^{𝑛/(𝑛−𝛽)} space. Some ideas of this paper come from Zhao et al. [7].

In this chapter, we would like to show that, if 𝐴∈BMO(ℝ^𝑛), ℋ_Ω,𝐴,𝛽(0≤𝛽<𝑛) is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)}.

To show this, let 𝐴(𝑥)=𝜒_(4,∞)(𝑥)∈BMO, Ω≡1, and 𝑓₀(𝑥)=𝜒_(0,4)(𝑥)−𝜒_(−4,0)(𝑥); then for 𝑥>8 and 𝑛=1, we have ℋ_Ω,𝐴,𝛽𝑓₀(𝑥=1)𝑥^1−𝛽4₀()=41−0×1𝑑𝑦𝑥^1−𝛽, (24) and then _𝑅1ℋ_Ω,𝐴,𝛽𝑓₀(𝑥)^1/(1−𝛽)𝑑𝑥≥^∞₈4^1/(1−𝛽)𝑥𝑑𝑥=∞, (25) which indicates that ℋ_Ω,𝐴,𝛽 is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)}.

2. Endpoint Estimates for ℋ^𝑚_Ω,𝐴,𝛽 from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)}

In Section 1, we know that, when 𝑚=1, ℋ¹_Ω,𝐴,𝛽 is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)}. In this section, we will prove that, when 𝑚≥2, ℋ^𝑚_Ω,𝐴,𝛽 is also not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)} unless ℋ^𝑚_Ω,𝐴,𝛽≡0. We have the following conclusions.

Theorem 6. Let 𝑚≥2,0≤𝛽<𝑛, and Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1). Assume that 𝐴 has derivatives of order 𝑚−1 in 𝐵𝑀𝑂(ℝ^𝑛); then the following two statements are equivalent;(i)ℋ^𝑚_Ω,𝐴,𝛽 maps 𝐻¹(ℝ^𝑛) continuously into 𝐿^{𝑛/(𝑛−𝛽)};(ii)for any 𝐻¹ atom supported on certain ball 𝐵 and 𝑢∈3𝐵⧵2𝐵, there is _{(4𝐵)𝑐|𝛼|=𝑚−1}1𝐾𝛼!^𝛼()1𝑥,𝑢|𝑥|^{𝑛−𝛽𝐵}𝐷^𝛼𝐴𝑦𝑎𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)}×𝑑𝑥≤𝐶, (26)where 𝐾_𝛼(𝑥,𝑢)=Ω(𝑥−𝑢)(𝑥−𝑢)^𝛼/|𝑥−𝑢|^𝑚−1 with |𝛼|=𝑚−1.

In order to prove Theorem 6, we need the following lemma.

Lemma 7 (see [9]). 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×_|𝛼|=𝑚1𝑄𝑥,𝑦𝑄_(𝑥,𝑦)𝐷^𝛼𝑏(𝑧)^𝑞𝑑𝑧^1/𝑞, (27) where 𝑄(𝑥,𝑦) is the cube centered at 𝑥 and having diameter 5√𝑛|𝑥−𝑦|.

Proof of Theorem 6. Suppose that 𝑎(𝑥) is an 𝐻¹ atom supported on 𝐵(𝑥₀,𝑟) and satisfies (21). Now we take 𝑎(𝑥)=𝑎(𝑥+𝑥₀); then 𝑎 is also an 𝐻¹ atom and satisfies i^′supp(𝑎)⊂𝐵(0,𝑟),ii^′‖𝑎‖_𝐿^∞≤|𝐵(0,𝑟)|⁻¹,iii^′_ℝ𝑛𝑎(𝑥)𝑑𝑥=0. (28)
Thus by the main results in [19] and the atomic decomposition of the space 𝐻¹(ℝ^𝑛), it suffices to show that, for any 𝐻¹ atom 𝑎, we have ‖ℋ^𝑚_Ω,𝐴,𝛽𝑎‖_{𝐿𝑛/(𝑛−𝛽)}≤𝐶.
Let 𝐵=𝐵(0,𝑟) and 𝐴(𝑥)=𝐴(𝑥)−∑_{|𝛼|=𝑚−1}(1/𝛼!)𝑚_𝐵(𝐷^𝛼𝐴)𝑥^𝛼; then 𝑅_𝑚(𝐴;𝑥,𝑦)=𝑅_𝑚(𝐴;𝑥,𝑦). For each 𝐻¹ atom 𝑎, we split each ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥) as ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)=𝜒_4𝐵(𝑥)ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)+𝜒_(4𝐵)^𝑐(𝑥)ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)∶=𝜇₁(𝑥)+𝜇₂(𝑥). (29)
For 𝜇₁(𝑥), taking 𝑛/(𝑛−𝛽)<𝑞<∞ and 𝑝 so that 1/𝑝−1/𝑞=𝛽/𝑛, it follows from Proposition 2 that 𝜇_{1𝐿𝑛/(𝑛−𝛽)}≤|4𝐵|^{(𝑛−𝛽)/𝑛−1/𝑞}ℋ^𝑚_Ω,𝐴,𝛽𝑎_𝐿𝑞≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO|𝐵|^1−1/𝑝‖𝑎‖_𝐿^𝑝≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (30)
For 𝜇₂(𝑥), as 𝑥∈(4𝐵)^𝑐 and |𝑦|<|𝑥|, we can deduce {𝑦∶|𝑦|<|𝑥|}∩{𝑦∶𝑦∈𝐵(0,𝑟)}={𝑦∶𝑦∈𝐵(0,𝑟)}; thus we have 𝜇₂(𝑥)=𝜒_(4𝐵)^𝑐1|𝑥|^𝑛−𝛽×_𝐵(0,𝑟)Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚𝐴;𝑥,𝑦𝑎𝑦𝑑𝑦. (31)
Next we denote 𝐵(0,𝑟)=𝐵 and 𝐵(0,𝑘𝑟)=𝑘𝐵; then by the vanishing condition of 𝑎, we decompose 𝜇₂ as follows: 𝜇₂(𝑥)=𝜒_(4𝐵)^𝑐(𝑥)1|𝑥|^{𝑛−𝛽𝐵}Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚−1−Ω()𝐴;𝑥,𝑦𝑥−𝑢|𝑥−𝑢|^𝑚−1𝑅_𝑚−1×𝑎𝑦𝑑𝑦−𝜒_(4𝐵)^𝑐(𝑥)1|𝑥|^𝑛−𝛽×_{|𝛼|=𝑚−1}1𝛼!_𝐵𝐾_𝛼𝑥,𝑦−𝐾_𝛼(𝑥,𝑢)𝐷^𝛼𝐴𝑦𝑎𝑦𝑑𝑦−𝜒_(4𝐵)^𝑐(𝑥)1|𝑥|^𝑛−𝛽×_{|𝛼|=𝑚−1}1𝛼!_𝐵𝐾_𝛼(𝑥,𝑢)𝐷^𝛼𝐴𝑦𝑎𝑦𝑑𝑦=𝜇₂₁(𝑥,𝑢)−𝜇₂₂(𝑥,𝑢)−𝜇₂₃(𝑥,𝑢), (32) where 𝑢∈3𝐵⧵2𝐵.
For 𝜇₂₁(𝑥,𝑢), as 𝑢∈3𝐵⧵2𝐵, 𝑦∈𝐵, and 𝑥∈(4𝐵)^𝑐, we have 𝑥−𝑦∼|𝑥−𝑢|∼|𝑥|. (33)
Thus we have Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚−1−Ω()𝐴;𝑥,𝑦𝑥−𝑢|𝑥−𝑢|^𝑚−1𝑅_𝑚−1≤Ω𝐴;𝑥,𝑢𝑥−𝑦𝑥−𝑦^𝑚−1𝑅_𝑚−1−Ω𝐴;𝑥,𝑦𝑥−𝑦|𝑥−𝑢|^𝑚−1𝑅_𝑚−1+Ω𝐴;𝑥,𝑦𝑥−𝑦|𝑥−𝑢|^𝑚−1𝑅_𝑚−1−Ω()𝐴;𝑥,𝑦𝑥−𝑢|𝑥−𝑢|^𝑚−1𝑅_𝑚−1+Ω()𝐴;𝑥,𝑦𝑥−𝑢|𝑥−𝑢|^𝑚−1𝑅_𝑚−1𝐴;𝑥,𝑦−𝑅_𝑚−1𝐴;𝑥,𝑢∶=𝐼+𝐼𝐼+𝐼𝐼𝐼. (34)
For the term 𝐼, by Lemma 7, we have Ω𝐼≤𝐶𝑥−𝑦𝑥−𝑦^𝑚𝑦−𝑢𝑅_𝑚−1𝐴;𝑥,𝑦≤𝐶𝑦−𝑢𝑥−𝑦^𝑚𝑥−𝑦^𝑚−1𝑘_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO≤𝐶𝑘2^−𝑘_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (35)
For the term 𝐼𝐼, by the similar estimates of 𝐼, we obtain 𝐼𝐼≤𝐶𝑘2^−𝑘_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (36)
For the term 𝐼𝐼𝐼, by the following formula (see [20]): 𝑅_𝑚𝐴;𝑥,𝑦−𝑅_𝑚(𝐴;𝑥,𝑧)=_|𝛼|<𝑚1𝑅𝛽!^{𝑚−|𝛼|}𝐷^𝛼𝐴;𝑧,𝑦(𝑥−𝑧)^𝛼, (37) and then together with Lemma 7, we have 𝐼𝐼𝐼≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO^𝑚−2_𝑖=0𝑥−𝑦^{−(𝑚−1)+𝑙}𝑦−𝑢^{𝑚−1−𝑖}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO𝑘2^−𝑘. (38)
Thus for 𝜇₂₁(𝑥,𝑢), by the size condition of 𝑎, we get the following estimates: 𝜇₂₁(⋅,𝑢)_{𝐿𝑛/(𝑛−𝛽)}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO×^∞_𝑘=2𝑘2^−𝑘_{2𝑘+1𝐵⧵2𝑘𝐵}1|𝑥|^𝑛−𝛽×_𝐵𝑎𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO^∞_𝑘=2𝑘2^−𝑘≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (39)
For 𝜇₂₂(𝑥,𝑢), since Ω∈Lip₁(𝑆^𝑛−1), we have the following estimates of |𝐾_𝛼(𝑥−𝑦)−𝐾_𝛼(𝑥−𝑢)|: 𝐾_𝛼𝑥−𝑦−𝐾_𝛼(≤Ω𝑥−𝑢)𝑥−𝑦𝑥−𝑦^𝛼𝑥−𝑦^𝑚−1−Ω𝑥−𝑦(𝑥−𝑢)^𝛼𝑥−𝑦^𝑚−1+Ω𝑥−𝑦(𝑥−𝑢)^𝛼𝑥−𝑦^𝑚−1−Ω(𝑥−𝑢)(𝑥−𝑢)^𝛼𝑥−𝑦^𝑚−1+Ω(𝑥−𝑢)(𝑥−𝑢)^𝛼𝑥−𝑦^𝑚−1−Ω(𝑥−𝑢)(𝑥−𝑢)^𝛼|𝑥−𝑢|^𝑚−1≤𝐶𝑦−𝑢.𝑥−𝑦 (40)
Thus by the size condition of 𝑎 and Lemma 7, we have 𝜇₂₂(⋅,𝑢)_{𝐿𝑛/(𝑛−𝛽)}≤𝐶_|𝛼|^∞_{=𝑚−1𝑘=22𝑘+1𝐵⧵2𝑘𝐵}×_𝐵𝑦−𝑢𝑥−𝑦|𝑥|^𝑛−𝛽𝑎𝑦×𝐷^𝛼𝐴𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO×^∞_𝑘=22^𝑘𝐵^{(𝛽−𝑛)/𝑛}|𝐵|⁻¹|𝐵|2^𝑘𝐵^{(𝑛−𝛽)/𝑛}2^−𝑘≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (41)
Now we can deduce that ‖ℋ^𝑚_Ω,𝐴,𝛽𝑎‖_{𝐿𝑛/(𝑛−𝛽)}≤𝐶 is equivalent to ‖𝜇₂₃(⋅,𝑢)‖_{𝐿𝑛/(𝑛−𝛽)}≤𝐶. By the vanishing condition of 𝑎, we can easily get _{(4𝐵)𝑐|𝛼|=𝑚−1}1𝐾𝛼!^𝛼()1𝑥,𝑢|𝑥|^{𝑛−𝛽𝐵}𝐷^𝛼𝐴𝑦𝑎𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)}×𝑑𝑥≤𝐶. (42)
Consequently, we have finished the proof of Theorem 6.

Next we would like to show that ℋ^𝑚_Ω,𝐴,𝛽 is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)} unless ℋ^𝑚_Ω,𝐴,𝛽≡0. We have the following theorem.

Theorem 8. Let 𝑚≥2,0≤𝛽<𝑛, and Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1), and assume that 𝐴 has derivatives of order 𝑚−1 in 𝐵𝑀𝑂(ℝ^𝑛). Then the following two statements are equivalent:(i)ℋ^𝑚_Ω,𝐴,𝛽 maps 𝐻¹ continuously into 𝐿^{𝑛/(𝑛−𝛽)},(ii)𝐴 is a polynomial of degree no more than 𝑚−1 or Ω≡0.

Remark 9. From Theorem 8, we can draw the conclusion that, when 𝑚≥2, ℋ^𝑚_Ω,𝐴,𝛽 is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)} unless ℋ^𝑚_Ω,𝐴,𝛽≡0.

Proof of Theorem 8. It is clear that (ii) ⇒ (i) is obvious. We only need to prove (i) ⇒ (ii).
Let 𝑎 be an 𝐻¹ atom supported on the ball 𝐵=𝐵(0,𝑟), and denote 𝐶_𝛼=(1/𝛼!)∫_𝐵𝐷^𝛼𝐴(𝑦)𝑎(𝑦)𝑑𝑦 with |𝛼|=𝑚−1. By Theorem 6, for any 𝑢∈3𝐵⧵2𝐵 and 𝑁>8 with 𝑁∈𝑍⁺, we have 𝐶≥_(4𝐵)𝑐1|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}𝐶_𝛼Ω(𝑥−𝑢)(𝑥−𝑢)^𝛼|𝑥−𝑢|^{𝑚−1𝑛/(𝑛−𝛽)}𝑑𝑥≥𝐶_1(4𝐵)𝑐1|𝑥−𝑢|^𝑛−𝛽_{|𝛼|=𝑚−1}𝐶_𝛼Ω(𝑥−𝑢)(𝑥−𝑢)^𝛼|𝑥−𝑢|^{𝑚−1𝑛/(𝑛−𝛽)}𝑑𝑥≥𝐶_{18𝑟≤|𝑥|≤𝑁𝑟}1|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}𝐶_𝛼Ω(𝑥)𝑥^𝛼|𝑥|^{𝑚−1𝑛/(𝑛−𝛽)}𝑑𝑥=𝐶₁𝑁log8_𝑆^𝑛−1_{|𝛼|=𝑚−1}𝐶_𝛼Ω𝑥^′𝑥^{′𝛼𝑛/(𝑛−𝛽)}𝑑𝜎𝑥^′. (43)
Let 𝑁→+∞, we know log(𝑁/8)→+∞. Thus we have _{𝑆𝑛−1|𝛼|=𝑚−1}𝐶_𝛼Ω𝑥^′𝑥^{′𝛼𝑛/(𝑛−𝛽)}𝑑𝜎𝑥^′=0. (44)
From (44) we can deduce _{|𝛼|=𝑚−1}𝐶_𝛼Ω𝑥^′𝑥^′𝛼=0. (45)
If Ω≡0, (45) is obviously true. Otherwise, we can easily obtain 𝐶_𝛼=1𝛼!_𝐵𝐷^𝛼𝐴𝑦𝑎𝑦𝑑𝑦=0. (46)
Since 𝑎 is arbitrary, 𝐷^𝛼𝐴 must be a constant for each 𝛼 with |𝛼|=𝑚−1. So we can deduce that 𝐴 is a polynomial with degree no more than 𝑚−1.
Consequently, we have finished the proof of Theorem 8.

3. Boundedness of ℋ^𝑚_Ω,𝐴,𝛽 from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽),∞}

In Section 2, we prove that, when 𝑚≥2, ℋ^𝑚_Ω,𝐴,𝛽 is not bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽)} unless ℋ^𝑚_Ω,𝐴,𝛽≡0. In this section, we will prove that ℋ^𝑚_Ω,𝐴,𝛽 is bounded from 𝐻¹ to 𝐿^{𝑛/(𝑛−𝛽),∞} with 𝑚≥1. Here 𝐿^{𝑛/(𝑛−𝛽),∞} is defined by 𝑓_{𝐿𝑛/(𝑛−𝛽),∞}=sup_𝜆>0𝜆{𝑥∈ℝ^𝑛∶𝑓(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}<∞. (47)

Our results can be stated as follows.

Theorem 10. Suppose that 𝑚≥2,0≤𝛽<𝑛, and Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1). If 𝐴 has derivatives of order 𝑚−1 in BMO(ℝ^𝑛), then there exists a constant 𝐶 independent of 𝑓 and 𝐴, such that 𝑥∈ℝ^𝑛∶ℋ^𝑚_Ω,𝐴,𝛽𝑓(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_𝐵𝑀𝑂𝑓_𝐻1𝜆 (48) for any 𝜆>0.

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

Theorem 11. Let 0≤𝛽<𝑛, 𝐴∈𝐵𝑀𝑂(ℝ^𝑛), and Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1); then there exists a constant 𝐶 independent of 𝑓 and 𝐴, such that 𝑥∈ℝ^𝑛∶ℋ¹_Ω,𝐴,𝛽𝑓(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_𝐵𝑀𝑂𝑓_𝐻1𝜆 (49) for any 𝜆>0.

Before the proof of Theorems 10 and 11, we need the following lemma.

Lemma 12. Let ℋ^𝛼_𝛽 be defined by ℋ^𝛼_𝛽ℎ(𝑥)=1|𝑥|^{𝑛−𝛽|𝑦|<|𝑥|}ℎ𝑦𝑥−𝑦^𝛼Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑑𝑦, (50) where 0≤𝛽<𝑛,Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1), and |𝛼|=𝑚−1; then ℋ^𝛼_𝛽 is bounded from 𝐿¹ to 𝐿^{𝑛/(𝑛−𝛽),∞}.

Proof. For any 𝜆>0, we have 𝜆𝑥∈ℝ^𝑛∶ℋ^𝛼_𝛽ℎ(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}=𝜆𝑥∈ℝ^𝑛∶1|𝑥|^𝑛−𝛽∫_{|𝑦|<|𝑥|}ℎ𝑦𝑥−𝑦^𝛼×Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑑𝑦>𝜆^{(𝑛−𝛽)/𝑛}≤𝜆𝑥∈ℝ^𝑛∶1|𝑥|^𝑛−𝛽∫_{|𝑦|<|𝑥|}ℎ𝑦1𝑥−𝑦^{𝑚−1−|𝛼|}𝑑𝑦>𝜆^{(𝑛−𝛽)/𝑛}≤𝜆𝑥∈ℝ^𝑛∶1|𝑥|^𝑛−𝛽∫_{|𝑦|<|𝑥|}ℎ𝑦𝑑𝑦>𝜆^{(𝑛−𝛽)/𝑛}≤𝜆𝑥∈ℝ^𝑛∶|𝑥|<‖ℎ‖_𝐿^{1/(𝑛−𝛽)}₁𝜆^{1/(𝑛−𝛽)(𝑛−𝛽)/𝑛}=𝜆∫_𝑆𝑛−1∫^{(‖ℎ‖𝐿11/(𝑛−𝛽))/(𝜆1/(𝑛−𝛽))}₀𝑟^𝑛−1𝑑𝑟𝑑𝜎𝑥^{′(𝑛−𝛽)/𝑛}‖ℎ‖≤𝐶𝜆^{𝐿𝑛/(𝑛−𝛽)1}𝜆^{𝑛/(𝑛−𝛽)(𝑛−𝛽)/𝑛}=𝐶‖ℎ‖_𝐿¹. (51)

Proposition 13. By the proof of Lemma 12 with minor changes, one can draw the conclusion that ℋ_Ω,𝛽 is bounded from 𝐿¹ to 𝐿^{𝑛/(𝑛−𝛽),∞} with Ω∈𝐿𝑖𝑝₁(𝑆^𝑛−1).

Proof of Theorem 10. Before giving the proof of Theorem 10, we introduce some notations that are very useful in this section.
For multi-indices 𝛼=(𝛼₁,…,𝛼_𝑛),𝛽=(𝛽₁,…,𝛽_𝑛), we denote 𝛼−𝛽=(𝛼₁−𝛽₁,…,𝛼_𝑛−𝛽_𝑛). Furthermore, 𝛽<𝛼 means that for each 𝑖, we have 𝛽_𝑖<𝛼_𝑖. Finally, we denote 𝐶^𝛽_𝛼=∏^𝑚_𝑗=1𝐶^𝛽𝑗_𝛼𝑗.
From [19] and by the atomic decomposition of 𝐻¹, it suffices to show 𝑥∈ℝ^𝑛∶ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}≤𝐶𝜆_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO (52) for any 𝐻¹ atom 𝑎, where 𝑎 is defined in Section 2. First we have the following decomposition: 𝑥∈ℝ^𝑛∶ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}≤𝑥∈𝐵(0,2𝑟)∶ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}+𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟)∶ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}∶=𝐿₁+𝐿₂. (53)
For the term 𝐿₁, choosing 𝐵(0,𝑟)=𝐵 and 𝐵(0,𝑘𝑟)=𝑘𝐵 with 𝑘∈𝑍⁺, then by Proposition 2, Hölder inequality, and the size condition of 𝑎, we have 𝜆𝑥∈𝐵(0,2𝑟)∶ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)>𝜆^{(𝑛−𝛽)/𝑛}≤𝐶_{𝐵(0,2𝑟)}ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶|2𝐵|^{(𝑛−𝛽)/𝑛−1/𝑞}_{𝐵(0,2𝑟)}ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)^𝑞𝑑𝑥^1/𝑞≤𝐶|2𝐵|^{(𝑛−𝛽)/𝑛−1/𝑞}_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO‖𝑎‖_𝐿^𝑝≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO, (54) where 1/𝑝−1/𝑞=𝛽/𝑛 and 𝑝>1.
For the term 𝐿₂, as 𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟), we have 𝑦∶𝑦<|𝑥|∩𝑦∶𝑦<𝑟=𝑦∶𝑦<𝑟. (55)
Then for a fixed 𝐵=𝐵(0,𝑟), we set 𝐴_𝐵(𝑥)=𝐴(𝑥)−∑_{|𝛼|=𝑚−1}(1/𝛼!)(𝐷^𝛼𝐴)_𝐵𝑥^𝛼. It is easy to check 𝑅_𝑚(𝐴;𝑥,𝑦)=𝑅_𝑚(𝐴_𝐵;𝑥,𝑦).
Thus by the vanishing condition of 𝑎 and the fact supp(𝑎)⊂𝐵(0,𝑟), we can decompose ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥) as follows: ℋ^𝑚_Ω,𝐴,𝛽𝑎(𝑥)=1|𝑥|^{𝑛−𝛽ℝ𝑛}𝑅_𝑚𝐴_𝐵Ω;𝑥,𝑦𝑥−𝑦𝑥−𝑦^𝑚−1−𝑅_𝑚−1𝐴_𝐵;𝑥,0−_{|𝛼|=𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵(𝑥)𝑥^𝛼×Ω(𝑥)|𝑥|^𝑚−1𝑎𝑦1𝑑𝑦=−|𝑥|^{𝑛−𝛽ℝ𝑛}𝑅_𝑚−1𝐴_𝐵;𝑥,0−_{|𝛼|=𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵(𝑥)𝑥^𝛼×Ω(𝑥)|𝑥|^𝑚−1−Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦1𝑑𝑦+|𝑥|^𝑛−𝛽×_{ℝ𝑛|𝛼|<𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵𝑦𝑥−𝑦^𝛼−_{|𝛼|<𝑚−1}𝐷^𝛼𝐴_𝐵(0)𝑥^𝛼Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦1𝑑𝑦+|𝑥|^𝑛−𝛽×_{ℝ𝑛|𝛼|=𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵𝑦𝑥^𝛼−_{|𝛼|=𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵(𝑥)𝑥^𝛼Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦+1𝑑𝑦|𝑥|^{𝑛−𝛽ℝ𝑛}_{|𝛼|=𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵𝑦×_𝛾<𝛼𝐶^𝛾_𝛼𝑥^𝛾−𝑦^𝛼−𝛾Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦𝑑𝑦∶=𝐼₁+𝐼₂+𝐼₃+𝐼₄. (56)
Here we can simply denote each 𝐼_𝑖 by 𝐼_𝑖=1|𝑥|^{𝑛−𝛽ℝ𝑛}𝐾_𝑖𝐴_𝐵;𝑥,𝑦𝑎𝑦𝑑𝑦. (57)
For 𝐾₁, by the fact that Ω∈Lip₁(𝑆^𝑛−1) and |𝑥−𝑦|∼|𝑥|, we have Ω(𝑥)|𝑥|^𝑚−1−Ω𝑥−𝑦𝑥−𝑦^𝑚−1≤Ω(𝑥)|𝑥|^𝑚−1−Ω𝑥−𝑦|𝑥|^𝑚−1+Ω𝑥−𝑦|𝑥|^𝑚−1−Ω𝑥−𝑦𝑥−𝑦^𝑚−1=Ω(𝑥/|𝑥|)|𝑥|/𝑥−𝑦|𝑥|^𝑚−1−Ω𝑥−𝑦/𝑥−𝑦|𝑥|^𝑚−1+Ω𝑥−𝑦|𝑥|^𝑚−1−Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑦≤𝐶|𝑥|^𝑚. (58)
As |𝑥|>𝑟 for 𝑥∈ℝ^𝑛⧵2𝐵, Lemma 7 in this paper and Lemma  2.2 in [8] tell us that 𝐾₁𝐴_𝐵|𝑥|;𝑥,𝑦≤𝐶^𝑚−1_{|𝛼|=𝑚−1}1𝑄𝑄𝐷^𝛼𝐴(𝑧)−𝐷^𝛼𝐴_𝐵^𝑞𝑑𝑧^𝑞+_{|𝛼|=𝑚−1}𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵𝑦|𝑥|^𝑚𝑦≤𝐶|𝑥|_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO𝐶₁|𝑥|+log𝑟+_{|𝛼|=𝑚−1}𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵, (59) where 𝐶₁ is a constant only depending on 𝑛, and 𝑄 is a cube centered at 𝑥 and having diameter 5√𝑛|𝑥|.
Thus we obtain 𝜆𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟)∶𝐼₁>𝜆^{(𝑛−𝛽)/𝑛}≤∫_{ℝ𝑛⧵𝐵(0,2𝑟)}𝐼₁^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶∫_{ℝ𝑛⧵𝐵(0,2𝑟)}|𝑥|^{(𝛽−𝑛)(𝑛/(𝑛−𝛽))}×∫_ℝ𝑛𝑦|𝑥|_|𝛼|=∑_𝑚−1𝐷^𝛼𝐴_BMO𝐶₁|𝑥|+log𝑟+∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵×𝑎𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶∑^+∞_𝑘=1∫_{2𝑘+1𝐵⧵2𝑘𝐵}|𝑥|^−𝑛|𝐵|^{1/(𝑛−𝛽)}|𝑥|^{𝑛/(𝛽−𝑛)}×∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO𝐶₁|𝑥|+log𝑟+∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶∑^+∞_𝑘=1|𝐵|^{1/(𝑛−𝛽)}2^𝑘𝐵^{(1/(𝛽−𝑛))−1}∫_{2𝑘+1𝐵⧵2𝑘𝐵}×∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO𝐶₁|𝑥|+log𝑟+∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴∑^{𝑛/(𝑛−𝛽)}_BMO^+∞_𝑘=1𝐶₁2+log^𝑘+1𝑟𝑟^{𝑛/(𝑛−𝛽)}×|𝐵|^{1/(𝑛−𝛽)}2^𝑘+1𝐵^{(1/(𝛽−𝑛))−1}2^𝑘+1𝐵^{(𝑛−𝛽)/𝑛}+∑^+∞_𝑘=1|𝐵|^{1/(𝑛−𝛽)}2^𝑘𝐵^{(1/(𝑛−𝛽))−1}∫_{2𝑘+1𝐵⧵2𝑘𝐵}×∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴∑^{𝑛/(𝑛−𝛽)}_BMO^∞_𝑘=1𝑘+𝐶₁^{𝑛/(𝑛−𝛽)}|𝐵|^{1/(𝑛−𝛽)}×2^𝑘+1𝐵^{(1/(𝛽−𝑛))−1}2^𝑘+1𝐵^{(𝑛−𝛽)/𝑛}≤𝐶∑_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (60)
Next we will give the estimates of 𝐼₂. First by a cumbersome but straightforward computation, we have 𝐾₂𝐴_𝐵,𝑥,𝑦≤^𝑚−2_{𝑘=0|𝛼|=𝑘}𝑅_{𝑚−𝑘−1}𝐷^𝛼𝐴_𝐵;0,𝑦|𝑥|^{−𝑚+𝑘+1}. (61)
Also noting the fact that 𝑅_{𝑚−𝑘−1}𝐷^𝛼𝐴_𝐵𝑦;0,𝑦≤𝐶^{𝑚−𝑘−1}×_{|𝛼|=𝑚−1}1𝑄₀𝑄₀𝐷^𝛼𝐴(𝑧)−𝐷^𝛼𝐴_𝐵^𝑞𝑑𝑧^1/𝑞≤𝐶𝑟^{𝑚−𝑘−1}_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO, (62) where 𝑞>𝑛, and 𝑄₀ is a cube centered at 0 and having diameter 5√𝑛|𝑦|.
Thus we obtain 𝐾₂𝐴_𝐵,𝑥,𝑦≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO^𝑚−2_𝑘=0𝑟^{𝑚−𝑘−1}|𝑥|^{−𝑚+𝑘+1}. (63)
So we have the following estimates of 𝐼₂: 𝐼₂1≤𝐶|𝑥|^{𝑛−𝛽ℝ𝑛𝑚−2}_𝑘=0𝑟^{𝑚−𝑘−1}|𝑥|^{−𝑚+𝑘+1}𝑎𝑦𝑑𝑦=𝐶^𝑚−2_𝑘=0𝑟^{𝑚−𝑘−1}|𝑥|^{−𝑚+𝑘+1−𝑛+𝛽}_𝐵𝑎𝑦𝑑𝑦≤𝐶^𝑚−2_𝑘=0𝑟^{𝑚−𝑘−1}|𝑥|^{−𝑚+𝑘+1−𝑛+𝛽}. (64)
Now we get 𝜆𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟)∶𝐼₂>𝜆^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_{BMOℝ𝑛⧵𝐵(0,2𝑟)}𝐼₂^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO×^𝑚−2_𝑘=0𝑟^{(𝑚−𝑘−1)(𝑛/(𝑛−𝛽))}×^+∞_{𝑗=12𝑗+1𝐵⧵2𝑗𝐵}|𝑥|^{(𝑛(1+𝑘−𝑚)/(𝑛−𝛽))−𝑛}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO×^𝑚−2_𝑘=0|𝐵|^{(𝑚−𝑘−1)/(𝑛−𝛽)}×^∞_𝑗=12^𝑗+1𝐵^{((1+𝑘−𝑚)/(𝑛−𝛽))−1}2^𝑗+1𝐵^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO×^𝑚−2_𝑘=0|𝐵|^{((𝑚−𝑘−1)/(𝑛−𝛽))+((1+𝑘−𝑚)/(𝑛−𝛽))−1+1}×^∞_𝑗=12^{𝑗𝑛(((1+𝑘−𝑚)/(𝑛−𝛽))−1+1)(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (65)
For 𝐼₃, by the vanishing condition of 𝑎(𝑦), we can split 𝐼₃ as follows: 𝐼₃=1|𝑥|^{𝑛−𝛽ℝ𝑛}_{|𝛼|=𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵𝑦𝑥^𝛼−_{|𝛼|=𝑚−1}1𝐷𝛼!^𝛼𝐴_𝐵(𝑥)𝑥^𝛼×Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦≤1𝑑𝑦|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝛼!_ℝ^𝑛𝐷^𝛼𝐴𝑦−𝐷^𝛼𝐴_𝐵×𝑥−𝑦^𝛼Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦+1𝑑𝑦|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝛼!_ℝ^𝑛𝐷^𝛼𝐴_𝐵𝑦×_𝛾<𝛼𝐶^𝛾_𝛼𝑥−𝑦^𝛾𝑦^𝛼−𝛾×Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦+1𝑑𝑦|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝛼!_ℝ^𝑛𝑥^𝛼𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵×Ω𝑥−𝑦𝑥−𝑦^𝑚−1𝑎𝑦𝑑𝑦=𝐼₃₁+𝐼₃₂+𝐼₃₃. (66)
For the term 𝐼₃₁, by Lemma 12 and the size condition of 𝑎, we obtain 𝜆𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟)∶𝐼₃₁>𝜆≤_{|𝛼|=𝑚−1}𝐷^𝛼𝐴−𝐷^𝛼𝐴_𝐵𝑎_𝐿1≤_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (67)
For the term 𝐼₃₂, by the vanishing condition of 𝑎 and the fact Ω∈Lip₁(𝑆^𝑛−1), we have 𝐼₃₂=1|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝛼!_ℝ^𝑛𝐷^𝛼𝐴_𝐵𝑦−𝐷^𝛼𝐴_𝐵×_𝛾<𝛼𝐶^𝛾_𝛼𝑥−𝑦^𝛾𝑦^𝛼−𝛾×Ω𝑥−𝑦𝑥−𝑦^𝑚−1×𝑎𝑦1𝑑𝑦≤𝐶|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝛼!_ℝ^𝑛𝐷^𝛼𝐴_𝐵𝑦−𝐷^𝛼𝐴_𝐵×_{𝛾<𝑚−1}𝑟^{𝑚−1−|𝛾|}𝑥−𝑦^{1−𝑚+|𝛾|}×𝑎𝑦𝑑𝑦≤𝐶_{𝛾<𝑚−1}1|𝑥|^𝑛−𝛽|𝐵|^{(𝑚−1−|𝛾|)/𝑛}×_{ℝ𝑛|𝛼|=𝑚−1}𝐷^𝛼𝐴_𝐵𝑦−𝐷^𝛼𝐴_𝐵×𝑥−𝑦^{1−𝑚+|𝛾|}𝑎𝑦𝑑𝑦. (68)
Thus we get𝜆𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟)∶𝐼₃₂>𝜆^{(𝑛−𝛽)/𝑛}≤_{ℝ𝑛⧵𝐵(0,2𝑟)}𝐼₃₂^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{𝛾<𝑚−1}|𝐵|^{∞((𝑚−1−|𝛾|)/𝑛)(𝑛/(𝑛−𝛽))}_{𝑘=12𝑘+1𝐵⧵2𝑘𝐵}×1|𝑥|^{𝑛−𝛽ℝ𝑛}_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_𝐵𝑦−𝐷^𝛼𝐴_𝐵×𝑥−𝑦^{1−𝑚+|𝛾|}𝑎𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{𝛾<𝑚−1}|𝐵|^{∞(𝑚−1−|𝛾|)/(𝑛−𝛽)}_𝑘=12^𝑘𝐵^{(1−𝑚+|𝛾|+𝛽−𝑛)/(𝑛−𝛽)}2^𝑘𝐵×_{𝐵|𝛼|=𝑚−1}𝐷^𝛼𝐴_𝐵𝑦−𝐷^𝛼𝐴_𝐵𝑎𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_{BMO𝛾<𝑚−1}|𝐵|^{(𝑚−1−|𝛾|)/(𝑛−𝛽)}|𝐵|^{(1−𝑚+|𝛾|+𝛽−𝑛)/(𝑛−𝛽)}×|𝐵|^∞1−1+1_𝑘=1𝑘2^{𝑘𝑛((1−𝑚+|𝛾|+𝛽−𝑛)/(𝑛−𝛽))(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (69)
For the term 𝐼₃₃, by the vanishing condition of 𝑎, Ω∈Lip₁(𝑆^𝑛−1), and (58), we have 𝐼₃₃=1|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝑥𝛼!^𝛼𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵×_ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^𝑚−1−Ω(𝑥)|𝑥|^𝑚−1𝑎𝑦≤1𝑑𝑦|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝑥𝛼!^𝛼𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵×_ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^𝑚−1−Ω(𝑥)𝑥−𝑦^𝑚−1𝑎𝑦+1𝑑𝑦|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1𝑥𝛼!^𝛼𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵×_ℝ𝑛Ω(𝑥)𝑥−𝑦^𝑚−1−Ω(𝑥)|𝑥|^𝑚−1𝑎𝑦1𝑑𝑦≤𝐶|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1|𝑥|𝛼!^𝑚−1𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵×_𝐵𝑦𝑥−𝑦^𝑚𝑎𝑦𝑑𝑦. (70)
Thus by Lemma  2.2 in [8], we obtain 𝜆𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟)∶𝐼₃₃>𝜆^{(𝑛−𝛽)/𝑛}≤_{ℝ𝑛⧵𝐵(0,2𝑟)}𝐼₃₃^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{ℝ𝑛⧵𝐵(0,2𝑟)}×1|𝑥|^𝑛−𝛽_{|𝛼|=𝑚−1}1|𝑥|𝛼!^𝑚−1𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵×_𝐵𝑦𝑥−𝑦^𝑚𝑎𝑦𝑑𝑦^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶^∞_{𝑘=12𝑘+1⧵2𝑘𝐵}|𝑥|^{(𝛽−𝑛−1)(𝑛/(𝑛−𝛽))}|𝐵|^{1/(𝑛−𝛽)}×_{|𝛼|=𝑚−1}𝐷^𝛼𝐴(𝑥)−𝐷^𝛼𝐴_𝐵^{𝑛/(𝑛−𝛽)}𝑑𝑥^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO×^∞_𝑘=1(𝑘+1)2^𝑘𝐵^{(𝛽−𝑛−1)/(𝑛−𝛽)}2^𝑘𝐵×|𝐵|^{1/(𝑛−𝛽)(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (71)
Finally, we will give the estimates of 𝐼₄. By a similar argument as in the estimates of 𝐼₃₂, we can easily get 𝜆𝑥∈ℝ^𝑛⧵𝐵(0,2𝑟)∶𝐼₄>𝜆^{(𝑛−𝛽)/𝑛}≤𝐶_{|𝛼|=𝑚−1}𝐷^𝛼𝐴_BMO. (72)
Combining the estimates of 𝐼₁,𝐼₂,𝐼₃, and 𝐼₄, we finish the proof of Theorem 10.

Proof of Theorem 11. Theorem 11 was proved in [7] in the case 𝛽=0 and Ω≡1. For the case 0≤𝛽<𝑛 and Ω≡1, we can easily prove Theorem 11 by the proof of Theorem  5.3 in [7] with minor changes. Then for the case 0≤𝛽<𝑛 and Ω∈Lip₁(𝑆^𝑛−1), by the main idea used in the proof of Theorem 10, we can prove Theorem 11 easily and we omit the details here.

Acknowledgments

The authors would like to express their gratitude to the referee for his/her valuable suggestions. This work was partially supported by National Natural Science Foundation of China under Grants nos. 10931001, 11226104, and 11226108, Natural Science Foundation of Jiangxi Province under Grants no. 20114BAB211007, and the Science Foundation of Jiangxi Education Department under Grants no. GJJ13703. This work was also supported by the Key Laboratory of Mathematics and Complex System (Beijing Normal University), Ministry of Education, China.