Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

Abstract

Let 𝐿 be the infinitesimal generator of an analytic semigroup on 𝐿²(ℝ^𝑛) with Gaussian kernel bounds, and let 𝐿^−𝛼/2 be the fractional integrals of 𝐿 for 0<𝛼<𝑛. For any locally integrable function 𝑏, the commutators associated with 𝐿^−𝛼/2 are defined by [𝑏,𝐿^−𝛼/2](𝑓)(𝑥)=𝑏(𝑥)𝐿^−𝛼/2(𝑓)(𝑥)−𝐿^−𝛼/2(𝑏𝑓)(𝑥). When 𝑏∈BMO(𝜔) (weighted BMO space) or 𝑏∈BMO, the authors obtain the necessary and sufficient conditions for the boundedness of [𝑏,𝐿^−𝛼/2] on weighted Morrey spaces, respectively.

1. Introduction and Main Results

Morrey [1] introduced the classical Morrey spaces to investigate the local behavior of solutions to second order elliptic partial differential equations. Chiarenza and Frasca [2] established the boundedness of the Hardy-Littlewood maximal operator, the fractional operator, and a singular integral operator on the Morrey spaces. On the other hand, Coifman and Fefferman [3] and Muckenhoupt [4] studied the boundedness of these operators on weighted 𝐿^𝑝 spaces. Motivated by these works, Komori and Shirai [5] introduced the following weighted Morrey space and investigated the boundedness of classical operators in harmonic analysis, that is, the Hardy-Littlewood maximal operator, a Calderón-Zygmund operator, the fractional integral operator, and so forth.

Let 1≤𝑝<∞ and 0≤𝑘<1. Then for two weights 𝜇 and 𝜈, the weighted Morrey space is defined by 𝐿^𝑝,𝑘𝜇,𝜈=𝑓∈𝐿^𝑝_loc𝜇∶𝑓_{𝐿𝑝,𝑘(𝜇,𝜈)}<∞, (1) where 𝑓_{𝐿𝑝,𝑘(𝜇,𝜈)}=sup_𝑄1𝜈(𝑄)^𝑘𝑄𝑓(𝑥)^𝑝𝜇(𝑥)𝑑𝑥^1/𝑝, (2) and the supremum is taken over all balls 𝑄 in ℝ^𝑛.

If 𝜇=𝜈, then we have the classical Morrey space 𝐿^𝑝,𝑘(𝜇) with measure 𝜇. When 𝑘=0, then 𝐿^𝑝,𝑘(𝜇,𝜈)=𝐿^𝑝(𝜇) is the Lebesgue space with measure 𝜇.

Suppose that 𝐿 is a linear operator on 𝐿²(ℝ^𝑛) which generates an analytic semigroup 𝑒^−𝑡𝐿 with a kernel 𝑝_𝑡(𝑥,𝑦) satisfying a Gaussian upper bound, that is, 𝑝_𝑡≤𝐶𝑥,𝑦𝑡^𝑛/2𝑒^{−𝑐(|𝑥−𝑦|2/𝑡)} (3) for 𝑥,𝑦∈ℝ^𝑛 and all 𝑡>0.

For 0<𝛼<𝑛, the fractional integral 𝐿^−𝛼/2 of the operator 𝐿 is defined by 𝐿^−𝛼/2𝑓(𝑥)=1Γ(𝛼/2)^∞₀𝑒^−𝑡𝐿𝑓𝑑𝑡𝑡^−𝛼/2+1(𝑥). (4) Note that if 𝐿=−Δ is the Laplacian on ℝ^𝑛, then 𝐿^−𝛼/2 is the classical fractional integral 𝐼_𝛼 which plays important roles in many fields. It is well known that 𝐼_𝛼 is bounded from 𝐿^𝑝(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛) for all 𝑝>1, 1/𝑞=1/𝑝−𝛼/𝑛>0 and is also of weak type (1,𝑛/(𝑛−𝛼)).

Let 1≤𝑝<∞ and 𝜔 be a weight function. A locally integrable function 𝑏 is said to be in BMO_𝑝(𝜔) if ‖𝑏‖_{BMO𝑝(𝜔)}=sup_𝑄1𝜔(𝑄)_𝑄𝑏(𝑥)−𝑏_𝑄^𝑝𝜔(𝑥)^1−𝑝𝑑𝑥^1/𝑝≤𝐶<∞, (5) where 𝑏_𝑄=(1/|𝑄|)∫_𝑄𝑏(𝑦)𝑑𝑦 and the supremum is taken over all balls 𝑄∈ℝ^𝑛.

Let 𝜔∈𝐴₁; García-Cuerva [6] proved that the spaces BMO_𝑝(𝜔) coincide, and the norms of ‖⋅‖_{BMO𝑝(𝜔)} are equivalent with respect to different values provided that 1≤𝑝<∞.

Let 𝑏 be a locally integrable function on ℝ^𝑛; we consider the commutator [𝑏,𝐿^−𝛼/2] defined by 𝑏,𝐿^−𝛼/2𝑓(𝑥)=𝑏(𝑥)𝐿^−𝛼/2𝑓(𝑥)−𝐿^−𝛼/2𝑏𝑓(𝑥). (6)

Chanillo [7] proved that the commutator [𝑏,𝐼_𝛼] of the multiplication operator by 𝑏∈BMO is bounded on 𝐿^𝑝 for 1<𝑝<∞.

Duong and Yan [8] proved that [𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝 to 𝐿^𝑞, where 𝑏∈BMO,   1<𝑝<𝑛/𝛼, 1/𝑞=1/𝑝−𝛼/𝑛, 0<𝛼<𝑛.

Mo and Lu [9] proved that the multilinear commutator generated by ^→𝑏 and 𝐿^−𝛼/2 is bounded from 𝐿^𝑝 to 𝐿^𝑞, where 1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛,0<𝛼<1, ^→𝑏=(𝑏₁,…,𝑏_𝑚),𝑏_𝑖∈BMO, for 𝑖=1,…,𝑚.

Lu et al. [10] proved that [𝑏,𝐼_𝛼] is bounded from 𝐿^𝑝 to 𝐿^𝑞 if and only if 𝑏∈BMO.

Wang [11] proved that [𝑏,𝐼_𝛼] is bounded from 𝐿^𝑝,𝑘(𝜔) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^{1−(1−𝛼/𝑛)𝑞},𝜔), where 𝑏∈BMO(𝜔), 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛,0<𝑘<𝑝/𝑞, and 𝜔^𝑞/𝑝∈𝐴₁.

Inspired by the above results, we study the boundedness properties of the commutator [𝑏,𝐿^−𝛼/2] on weighted Morrey spaces in this work. The main theorems are stated as follows.

Theorem 1. Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛,0≤𝑘<𝑝/𝑞,𝜔^𝑞/𝑝∈𝐴₁, and 𝑟_𝜔>((1−𝑘)/(𝑝/𝑞−𝑘)), where 𝑟_𝜔 denotes the critical index of 𝜔 for the reverse Hölder condition. Then the following conditions are equivalent. (a)𝑏∈𝐵𝑀𝑂(𝜔). (b)[𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝,𝑘(𝜔) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^{1−(1−𝛼/𝑛)𝑞},𝜔).

In particular, when 𝑘=0 in Theorem 1, we get the following.

Corollary 2. Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛,𝜔^𝑞/𝑝∈𝐴₁, and 𝑟_𝜔>𝑞/𝑝, where 𝑟_𝜔 denotes the critical index of 𝜔 for the reverse Hölder condition. Then the following conditions are equivalent. (a)𝑏∈𝐵𝑀𝑂(𝜔). (b)[𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝(𝜔) to 𝐿^𝑞(𝜔^{1−(1−𝛼/𝑛)𝑞}).
Furthermore, if 𝐿=−Δ is the Laplacian, then the following conditions are equivalent. (a^′)𝑏∈𝐵𝑀𝑂(𝜔). (b^′)[𝑏,𝐼_𝛼] is bounded from 𝐿^𝑝(𝜔) to 𝐿^𝑞(𝜔^{1−(1−𝛼/𝑛)𝑞}).

Theorem 3. Let 0<𝛼<𝑛,0≤𝑘<𝑝/𝑞,1/𝑞=1/𝑝−𝛼/𝑛, and 1<𝑟,𝑠<∞ such that 1<𝑟𝑠<𝑝<𝑛/𝛼, 𝜔^𝑟𝑠∈𝐴_{𝑝/𝑟𝑠,𝑞/𝑟𝑠}. Then the following conditions are equivalent. (a)𝑏∈𝐵𝑀𝑂. (b)[𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞).

In particular, when 𝑘=0 in Theorem 3, we obtain the following.

Corollary 4. Let 0<𝛼<𝑛, 1/𝑞=1/𝑝−𝛼/𝑛, and 1<𝑟, 𝑠<∞ such that 1<𝑟𝑠<𝑝<𝑛/𝛼, 𝜔^𝑟𝑠∈𝐴_{𝑝/𝑟𝑠,𝑞/𝑟𝑠}. Then the following conditions are equivalent. (a)𝑏∈𝐵𝑀𝑂. (b)[𝑏,𝐿^−𝛼/2] is bounded from 𝐿^𝑝(𝜔^𝑝) to 𝐿^𝑞(𝜔^𝑞).
Furthermore, if 𝐿=−Δ is the Laplacian, then the following conditions are equivalent. (a^′)𝑏∈𝐵𝑀𝑂. (b^′)[𝑏,𝐼_𝛼] is bounded from 𝐿^𝑝(𝜔^𝑝) to 𝐿^𝑞(𝜔^𝑞).

Remark 5. It is easy to see that our results extend the results in [7, 8, 10, 11] significantly.

2. Prerequisite Material

Let us first recall some definitions.

Definition 6. The Hardy-Littlewood maximal operator 𝑀 is defined by 𝑀𝑓(𝑥)=sup_𝑥∈𝑄1|𝑄|_𝑄𝑓𝑦𝑑𝑦. (7) Let 𝜔 be a weight. The weighted maximal operator 𝑀_𝜔 is defined by 𝑀_𝜔𝑓(𝑥)=sup_𝑥∈𝑄1𝜔(𝑄)_𝑄𝑓𝑦𝜔𝑦𝑑𝑦. (8) A variant of this maximal operator will become the main tool in our scheme; for 1<𝑟<∞, 𝑀_𝑟,𝜔𝑓(𝑥)=𝑀_𝜔𝑓^𝑟1/𝑟(𝑥). (9) For 0<𝛼<𝑛,𝑟≥1, the fractional maximal operator 𝑀_𝛼,𝑟 is defined by 𝑀_𝛼,𝑟𝑓(𝑥)=sup_𝑥∈𝑄1|𝑄|^{1−𝛼𝑟/𝑛𝑄}𝑓𝑦^𝑟𝑑𝑦^1/𝑟, (10) and the fractional weighted maximal operator 𝑀_{𝛼,𝑟,𝜔} is defined by 𝑀_{𝛼,𝑟,𝜔}𝑓(𝑥)=sup_𝑥∈𝑄1𝜔(𝑄)^{1−𝛼𝑟/𝑛𝑄}𝑓𝑦^𝑟𝜔𝑦𝑑𝑦^1/𝑟. (11) For any 𝑓∈𝐿^𝑝(ℝ^𝑛),𝑝≥1, the sharp maximal function 𝑀^♯_𝐿𝑓 associated with the generalized approximations to the identity {𝑒^−𝑡𝐿}_𝑡>0 is given by 𝑀^♯_𝐿𝑓(𝑥)=sup_𝑥∈𝑄1|𝑄|_𝑄𝑓𝑦−𝑒^{−𝑡𝑄𝐿}𝑓𝑦𝑑𝑦, (12) where 𝑡_𝑄=𝑟²_𝑄 and 𝑟_𝑄 is the radius of the ball 𝑄.
In the above definitions, the supremum is taken over all balls 𝑄 containing 𝑥.

Definition 7. A weight function 𝜔 is said to be in the Muckenhoupt class 𝐴_𝑝 with 1<𝑝<∞ if, for every ball 𝑄 in ℝ^𝑛, there exists a positive constant 𝐶 which is independent of 𝑄 such that 1|𝑄|_𝑄𝜔(𝑥)1𝑑𝑥|𝑄|_𝑄𝜔(𝑥)^{−1/(𝑝−1)}𝑑𝑥^𝑝−1≤𝐶. (13) When 𝑝=1,𝜔∈𝐴₁, if 1|𝑄|_𝑄𝜔(𝑥)𝑑𝑥≤𝐶essinf_𝑥∈𝑄𝜔(𝑥). (14)
When 𝑝=∞,𝜔∈𝐴_∞, if there exist positive constants 𝛿 and 𝐶 such that, given a ball 𝑄 and a measurable subset 𝐸 of 𝑄, 𝜔(𝐸)𝜔(𝑄)|𝐸|≤𝐶|𝑄|^𝛿. (15)

Definition 8. A weight function 𝜔 belongs to 𝐴_𝑝,𝑞 for 1<𝑝<𝑞<∞ if, for every ball 𝑄 in ℝ^𝑛, there exists a positive constant 𝐶 which is independent of 𝑄 such that 1|𝑄|_𝑄𝜔(𝑥)^𝑞𝑑𝑥^1/𝑞1|𝑄|_𝑄𝜔(𝑥)^−𝑝′𝑑𝑥^1/𝑝′≤𝐶, (16) where 𝑝^′ denotes the conjugate exponent of 𝑝>1, that is, 1/𝑝+1/𝑝^′=1.

Definition 9. A weight function 𝜔 belongs to the reverse Hölder class RH_𝑟 if there exist two constants 𝑟>1 and 𝐶>0 such that the reverse Hölder inequality 1|𝑄|_𝑄𝜔(𝑥)^𝑟𝑑𝑥^1/𝑟1≤𝐶|𝑄|_𝑄𝜔(𝑥)𝑑𝑥 (17) holds for every ball 𝑄 in ℝ^𝑛.

It is well known that if 𝜔∈𝐴_𝑝 with 1≤𝑝<∞, then there exists 𝑟>1 such that 𝜔∈RH_𝑟. It follows from Hölder's inequality that 𝜔∈RH_𝑟 implies 𝜔∈RH_𝑠 for all 1<𝑠<𝑟. Moreover, if 𝜔∈RH_𝑟,𝑟>1, then we have 𝜔∈RH_𝑟+𝜖 for some 𝜖>0. We thus write 𝑟_𝑤=sup{𝑟>1∶𝜔∈RH_𝑟} to denote the critical index of 𝜔 for the reverse Hölder condition.

We will make use of the following lemmas. We first provide a weighted version of the local good 𝜆 inequality for 𝑀^♯_𝐿 which allows us to obtain an analog of the classical Fefferman-Stein (see [3, 12]) estimate on weighted Morrey spaces.

Lemma 10 (see [13]). Assume that the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (3). Take 𝜆>0,𝑓∈𝐿¹₀(ℝ^𝑛), and a ball 𝑄₀ such that there exists 𝑥₀∈𝑄₀ with 𝑀𝑓(𝑥₀)≤𝜆. Then, for every 𝜔∈𝐴_∞,0<𝜂<1, one can find 𝛾>0 (independent of 𝜆,𝑄₀,𝑓,𝑥₀) and constant 𝐶_𝜔,𝑟>0 (which only depend on 𝜔), such that 𝜔𝑥∈𝑄₀∶𝑀𝑓(𝑥)>𝐴𝜆,𝑀^♯_𝐿𝑓(𝑥)≤𝛾𝜆≤𝐶_𝜔𝜂^𝑟𝜔𝑄₀, (18) where 𝐴>1 is a fixed constant which depends only on 𝑛.

As a consequence, by using the standard arguments, we have the following estimates.

For every 𝑓∈𝐿^𝑝,𝑘(𝜇,𝜈), with 1<𝑝<∞. if 𝜇,𝜈∈𝐴_∞,1<𝑝<∞,0≤𝑘<1, then 𝑓_{𝐿𝑝,𝑘(𝜇,𝜈)}≤𝑀𝑓_{𝐿𝑝,𝑘(𝜇,𝜈)}≤𝐶𝑀^♯_𝐿𝑓_{𝐿𝑝,𝑘(𝜇,𝜈)}. (19) In particular, when 𝜇=𝜈=𝜔 and 𝜔∈𝐴_∞, we have 𝑓_{𝐿𝑝,𝑘(𝜔)}≤𝑀𝑓_{𝐿𝑝,𝑘(𝜔)}≤𝐶𝑀^♯_𝐿𝑓_{𝐿𝑝,𝑘(𝜔)}. (20)

Lemma 11 (see [11]). Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼, 1/𝑞=1/𝑝−𝛼/𝑛, and 𝜔^𝑞/𝑝∈𝐴₁. Then if 0<𝑘<𝑝/𝑞 and 𝑟_𝜔>(1−𝑘)/(𝑝/𝑞−𝑘), one has 𝑀_𝛼,1𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶𝑓_{𝐿𝑝,𝑘(𝜔)}. (21) The same conclusion still holds for 𝐼_𝛼.

Lemma 12 (see [11]). Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛, and 𝜔^𝑞/𝑝∈𝐴₁. Then if  0<𝑘<𝑝/𝑞,1<𝑟<𝑝, and 𝑟_𝜔>(1−𝑘)/(𝑝/𝑞−𝑘), one has 𝑀_𝑟,𝜔𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}. (22)

Lemma 13 (see [11]). Consider 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛, 0<𝑘<𝑝/𝑞, and 𝜔∈𝐴_∞. For any 1<𝑟<𝑝, one has 𝑀_{𝛼,𝑟,𝜔}𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔)}≤𝐶𝑓_{𝐿𝑝,𝑘(𝜔)}. (23)

Remark 14. By checking the proof of Lemmas 11–13, we know that the three lemmas above still hold when 𝑘=0.

Lemma 15. Let 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛, and 𝜔^𝑞/𝑝∈𝐴₁. Then if 0≤𝑘<𝑝/𝑞 and 𝑟_𝜔>(1−𝑘)/(𝑝/𝑞−𝑘), one has 𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶𝑓_{𝐿𝑝,𝑘(𝜔)}. (24)

Proof. Since the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (3), it is easy to check that 𝐿^−𝛼/2(𝑓)(𝑥)≤𝐶𝐼_𝛼(|𝑓|)(𝑥) for all 𝑥∈ℝ^𝑛. Using the boundedness property of 𝐼_𝛼 on weighted Morrey space (see Lemma 11), we have 𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐼_𝛼𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}≤𝐶𝑓_{𝐿𝑝,𝑘(𝜔)}, (25) where 1<𝑝<𝑛/𝛼 and 1/𝑞=1/𝑝−𝛼/𝑛.

Remark 16. Since 𝐼_𝛼 is of weak type (1,𝑛/(𝑛−𝛼)). from the proof of Lemma 15, we can get that 𝐿^−𝛼/2 is also of weak type (1,𝑛/(𝑛−𝛼)).

Lemma 17 (see [8, 14]). Assume that the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (3). Then for 0<𝛼<𝑛, the difference operator 𝐿^−𝛼/2−𝑒^−𝑡𝐿𝐿^−𝛼/2 has an associated kernel 𝐾_𝛼,𝑡(𝑥,𝑦) which satisfies 𝐾_𝛼,𝑡≤𝐶𝑥,𝑦𝑥−𝑦^𝑛−𝛼𝑡𝑥−𝑦², (26) for some positive constant 𝐶.

Lemma 18. Assume that the semigroup 𝑒^−𝑡𝐿 has a kernel 𝑝_𝑡(𝑥,𝑦) which satisfies the upper bound (3), and let 𝑏∈𝐵𝑀𝑂(𝜔),𝜔∈𝐴₁. Then, for every function 𝑓∈𝐿^𝑝(ℝ^𝑛),𝑝>1, and for all 𝑥∈ℝ^𝑛, one has sup_𝑥∈𝑄1|𝑄|_𝑄𝑒^{−𝑡𝑄𝐿}𝑏𝑦−𝑏_𝑄𝑓𝑦‖𝑏‖𝑑𝑦≤𝐶_{𝐵𝑀𝑂(𝜔)}𝜔(𝑥)𝑀_𝑟,𝜔𝑓(𝑥), (27) where 𝑡_𝑄=𝑟²_𝑄,𝑟_𝑄 being the radius of 𝑄.

Proof. For any 𝑓∈𝐿^𝑝(ℝ^𝑛),1<𝑝<∞ and 𝑥∈𝑄. We have 1|𝑄|_𝑄𝑒^{−𝑡𝑄𝐿}𝑏(⋅)−𝑏_𝑄𝑓𝑦≤1𝑑𝑦|𝑄|_𝑄ℝ^𝑛𝑝_𝑡𝑄𝑦,𝑧𝑏(𝑧)−𝑏_𝑄𝑓(𝑧≤1)𝑑𝑧𝑑𝑦|𝑄|_𝑄2𝑄𝑝_𝑡𝑄𝑦,𝑧𝑏(𝑧)−𝑏_𝑄𝑓(𝑧+1)𝑑𝑧𝑑𝑦|𝑄|_𝑄^∞_𝑘=12^𝑘+1_𝑄⧵2^𝑘_𝑄𝑝_𝑡𝑄×𝑏(𝑧)𝑦,𝑧−𝑏_𝑄𝑓(𝑧)𝑑𝑧𝑑𝑦≐𝐌+𝐍. (28) For any 𝑦∈𝑄 and 𝑧∈2𝑄. We have 𝑝_𝑡𝑄𝑦,𝑧≤𝐶𝑡_𝑄^−𝑛/21≤𝐶||.2𝑄 (29) Thus, 1𝐌≤𝐶|2𝑄|_2𝑄𝑏(𝑧)−𝑏_𝑄𝑓(𝑧1)𝑑𝑧≤𝐶|2𝑄|_2𝑄𝑏(𝑧)−𝑏_𝑄^𝑟′𝜔(𝑧)^1−𝑟′𝑑𝑧^1/𝑟′×_2𝑄𝑓(𝑧)^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝜔(2𝑄)||12𝑄𝜔(2𝑄)_2𝑄𝑓(𝑧)^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)𝑀_𝑟,𝜔𝑓(𝑥). (30) Moreover, for any 𝑦∈𝑄 and 𝑧∈2^𝑘+1𝑄⧵2^𝑘𝑄, we have |𝑦−𝑧|≥2^𝑘−1𝑟_𝑄 and |𝑝_𝑡𝑄|≤𝐶(𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛/|2^𝑘+1𝑄|): 𝐍=1|𝑄|_𝑄^∞_𝑘=12^𝑘+1_𝑄⧵2^𝑘_𝑄𝑝_𝑡𝑄𝑦,𝑧𝑏(𝑧)−𝑏_𝑄𝑓(𝑧)𝑑𝑧𝑑𝑦≤𝐶^∞_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛2^𝑘+1𝑄^2𝑘+1𝑄𝑏(𝑧)−𝑏_𝑄𝑓(𝑧)𝑑𝑧≤𝐶^∞_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛2^𝑘+1𝑄^2𝑘+1𝑄𝑏(𝑧)−𝑏₂^𝑘+1_𝑄𝑓(𝑧)𝑑𝑧+𝐶^∞_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛2^𝑘+1𝑄^2𝑘+1𝑄𝑏₂^𝑘+1_𝑄−𝑏_𝑄𝑓(𝑧)𝑑𝑧≐𝐍₁+𝐍₂. (31) We estimate each term in turn. For 𝐍₁, we apply Hölder's inequalities with exponent 𝑟. Then we have 𝐍₁≤𝐶^∞_𝑘=1𝑒^{−𝑐22(𝑘−1)}2^(𝑘+1)𝑛2^𝑘+1𝑄×_2𝑘+1𝑄𝑏(𝑧)−𝑏₂^𝑘+1_𝑄^𝑟′𝜔(𝑧)^1−𝑟′𝑑𝑧^1/𝑟′×_2𝑘+1𝑄𝑓(𝑧)^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶^∞_𝑘=12^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}‖𝑏‖_BMO(𝜔)𝜔2^𝑘+1𝑄2^𝑘+1𝑄×1𝜔2^𝑘+1𝑄^2𝑘+1𝑄𝑓(𝑧)^𝑟𝜔(𝑧)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)𝑀_𝑟,𝜔𝑓(𝑥). (32)
Since 𝜔∈𝐴₁, then |𝑏₂^𝑘+1𝑄−𝑏_𝑄|≤𝐶𝑘𝜔(𝑥)‖𝑏‖_BMO(𝜔). This fact together with Hölder's inequality implies𝐍₂≤𝐶^∞_𝑘=12^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}𝑘2^𝑘+1𝑄𝜔(𝑥)×‖𝑏‖_{BMO(𝜔)𝑄}𝑓(𝑧)𝑑𝑧≤𝐶^∞_𝑘=1𝑘2^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}𝜔(𝑥)×‖𝑏‖_BMO(𝜔)12^𝑘+1𝑄^2𝑘+1𝑄𝑓(𝑧)^𝑟𝑑𝑧^1/𝑟=𝐶^∞_𝑘=1𝑘2^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}𝜔(𝑥)×‖𝑏‖_BMO(𝜔)𝜔2^𝑘+1𝑄2^𝑘+1𝑄1𝜔2^𝑘+1𝑄^2𝑘+1𝑄𝑓(𝑧)^𝑟𝑑𝑧^1/𝑟≤𝐶^∞_𝑘=1𝑘2^(𝑘+1)𝑛𝑒^{−𝑐22(𝑘−1)}𝜔(𝑥)×‖𝑏‖_BMO(𝜔)1𝜔2^𝑘+1𝑄^2𝑘+1𝑄𝑓(𝑧)^𝑟𝜔(𝑥)𝑑𝑧^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)𝑀_𝑟,𝜔𝑓(𝑥). (33)Then Lemma 18 is proved.

Lemma 19. Let 0<𝛼<𝑛,𝜔∈𝐴₁, and 𝑏∈𝐵𝑀𝑂(𝜔). Then for all 𝑟>1 and for all 𝑥∈ℝ^𝑛, one has 𝑀^♯_𝐿𝑏,𝐿^−𝛼/2𝑓(𝑥)≤𝐶‖𝑏‖_{𝐵𝑀𝑂(𝜔)}𝜔(𝑥)𝑀_𝑟,𝜔𝐿^−𝛼/2𝑓(𝑥)+𝜔(𝑥)^{1−𝛼/𝑛}×𝑀_{𝛼,𝑟,𝜔}𝑓(𝑥)+𝜔(𝑥)𝑀_𝛼,1𝑓(𝑥). (34)

Proof. For any given 𝑥∈ℝ^𝑛, fix a ball 𝑄=𝑄(𝑥₀,𝑟_𝑄) which contains 𝑥. We decompose 𝑓=𝑓₁+𝑓₂, where 𝑓₁=𝑓𝜒_2𝑄. Observe that 𝑏,𝐿^−𝛼/2𝑓(𝑥)=𝑏−𝑏_𝑄𝐿^−𝛼/2𝑓−𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₁−𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₂,𝑒^{−𝑡𝑄𝐿}𝑏,𝐿^−𝛼/2𝑓=𝑒^{−𝑡𝑄𝐿}𝑏−𝑏_𝑄𝐿^−𝛼/2𝑓−𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₁−𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₂. (35) Then 1|𝑄|_𝑄𝑏,𝐿^−𝛼/2𝑓𝑦−𝑒^{−𝑡𝑄𝐿}𝑏,𝐿^−𝛼/2𝑓𝑦≤1𝑑𝑦|𝑄|_𝑄𝑏𝑦−𝑏_𝑄𝐿^−𝛼/2𝑓𝑦+1𝑑𝑦|𝑄|_𝑄𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₁𝑦+1𝑑𝑦|𝑄|_𝑄𝑒^{−𝑡𝑄𝐿}𝑏−𝑏_𝑄𝐿^−𝛼/2𝑓𝑦+1𝑑𝑦|𝑄|_𝑄𝑒^{−𝑡𝑄𝐿}𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₁𝑦+1𝑑𝑦|𝑄|_𝑄𝐿^−𝛼/2−𝑒^{−𝑡𝑄𝐿}𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₂𝑦𝑑𝑦≐𝐈+𝐈𝐈+𝐈𝐈𝐈+𝐈𝐕+𝐕. (36) We estimate each term separately.
Since 𝜔∈𝐴₁, then it follows from Hölder's inequality that 1𝐈≤|𝑄|_𝑄𝑏𝑦−𝑏_𝑄𝐿^−𝛼/2𝑓𝑦≤1𝑑𝑦|𝑄|_𝑄𝑏𝑦−𝑏_𝑄^𝑟′𝜔𝑦^1−𝑟′𝑑𝑦^1/𝑟′×_𝑄𝐿^−𝛼/2𝑓𝑦^𝑟𝜔𝑦𝑑𝑦^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑄)|𝑄|1𝜔(𝑄)_𝑄𝐿^−𝛼/2𝑓𝑦^𝑟𝜔𝑦𝑑𝑦^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)𝑀_𝑟,𝜔𝐿^−𝛼/2𝑓(𝑥). (37)
Applying Kolmogorov's inequality (see [15, page 485]), Hölder's inequality, and the continuity of 𝐿^−𝛼/2, we thus have 1𝐈𝐈=|𝑄|_𝑄𝐿^−𝛼/2𝑏−𝑏_𝑄𝑓₁𝑦1𝑑𝑦≤𝐶|𝑄|^{1−𝛼/𝑛}𝐿^−𝛼/2(𝑏−𝑏_𝑄)𝑓_{1𝐿𝑛/(𝑛−𝛼),∞}1≤𝐶|𝑄|^{1−𝛼/𝑛𝑄}𝑏−𝑏_𝑄𝑓₁𝑦1𝑑𝑦≤𝐶|𝑄|^{1−𝛼/𝑛𝑄}𝑏𝑦−𝑏_𝑄^𝑟′𝜔(𝑦)^1−𝑟′𝑑𝑦^1/𝑟′×_𝑄𝑓𝑦^𝑟𝜔(𝑦)𝑑𝑦^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝑤(𝑄)^{1−𝛼/𝑛}|𝑄|^{1−𝛼/𝑛}×1𝑤(𝑄)^{1−𝑟𝛼/𝑛𝑄}𝑓𝑦^𝑟𝜔𝑦𝑑𝑦^1/𝑟≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼,𝑟,𝜔}𝑓(𝑥). (38) By Lemma 18, we have 𝐈𝐈𝐈≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)𝑀_𝑟,𝜔𝐿^−𝛼/2𝑓(𝑥). (39) For 𝐈𝐕, using the estimate obtained in 𝐈𝐈, we get 1𝐈𝐕≤|𝑄|_𝑄2𝑄𝑝_𝑡𝑄𝑦,𝑧𝑏(𝑧)−𝑏_𝑄𝑓(𝑧≤1)𝑑𝑧𝑑𝑦|2𝑄|_2𝑄𝐿^−𝛼/2𝑏(𝑧)−𝑏_𝑄𝑓(𝑧‖𝑏‖)𝑑𝑧≤𝐶_BMO(𝜔)𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼,𝑟,𝜔}𝑓(𝑥). (40)
By virtue of Lemma 17, we have1𝐕≤|𝑄|_𝑄(2𝑄)^𝑐𝐾_{𝛼,𝑡𝑄}𝑦,𝑧𝑏(𝑧)−𝑏_𝑄𝑓(𝑧)𝑑𝑧𝑑𝑦≤𝐶^∞_{𝑘=12𝑘𝑟𝑄≤|𝑥0−𝑧|<2𝑘+1𝑟𝑄}1𝑥₀−𝑧^𝑛−𝛼𝑡_𝑄𝑥₀−𝑧²×𝑏(𝑧)−𝑏_𝑄𝑓(𝑧)𝑑𝑧≤𝐶^∞_𝑘=12^−2𝑘12^𝑘+1𝑄^{1−𝛼/𝑛2𝑘+1𝑄}𝑏(𝑧)−𝑏_𝑄𝑓(𝑧)𝑑𝑧≤𝐶^∞_𝑘=12^−2𝑘12^𝑘+1𝑄^{1−𝛼/𝑛2𝑘+1𝑄}𝑏(𝑧)−𝑏₂^𝑘+1_𝑄𝑓(𝑧)𝑑𝑧+𝐶^∞_𝑘=12^−2𝑘𝑏₂^𝑘+1_𝑄−𝑏_𝑄12^𝑘+1𝑄^{1−𝛼/𝑛2𝑘+1𝑄}𝑓(𝑧)𝑑𝑧≐𝐕𝐈+𝐕𝐈𝐈. (41)
For 𝐕𝐈, applying the same arguments as in 𝐈𝐈, we get 𝐕𝐈≤𝐶‖𝑏‖_BMO^∞_(𝜔)𝑘=12^−2𝑘𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼,𝑟,𝜔}𝑓(𝑥)≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼,𝑟,𝜔}𝑓(𝑥). (42)
Since 𝜔∈𝐴₁, then |𝑏₂^𝑘+1𝑄−𝑏_𝑄|≤𝐶𝑘𝜔(𝑥)‖𝑏‖_BMO(𝜔). Thus, 𝐕𝐈𝐈≤𝐶‖𝑏‖_BMO^∞_(𝜔)𝑘=12^−2𝑘𝑘𝜔(𝑥)𝑀_𝛼,1𝑓(𝑥)≤𝐶‖𝑏‖_BMO(𝜔)𝜔(𝑥)𝑀_𝛼,1𝑓(𝑥). (43) Then 𝐕≤𝐶‖𝑏‖_BMO(𝜔)×𝜔(𝑥)^{1−𝛼/𝑛}𝑀_{𝛼,𝑟,𝜔}𝑓(𝑥)+𝜔(𝑥)𝑀_𝛼,1𝑓(𝑥). (44) Combining the above estimates 𝐈--𝐕, we get (34). The proof of Lemma 19 is complete.

3. Proofs of the Main Results

In this section we prove our main results. We start with the proof of Theorem 1.

Proof. (a)⇒(b): Applying Lemmas 10 and 19, we get 𝑏,𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤𝑀^♯_𝐿𝑏,𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤𝐶‖𝑏‖_BMO(𝜔)×𝜔𝑀_𝑟,𝜔𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}+𝜔^{1−𝛼/𝑛}𝑀_{𝛼,𝑟,𝜔}𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}+𝜔𝑀_𝛼,1𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤𝐶‖𝑏‖_BMO(𝜔)×𝑀_𝑟,𝜔𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}+𝑀_{𝛼,𝑟,𝜔}𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔)}+𝑀_𝛼,1𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}. (45)
Since 0≤𝑘<𝑝/𝑞,𝜔^𝑞/𝑝∈𝐴₁, and 𝑟_𝜔>(1−𝑘)/(𝑝/𝑞−𝑘), by making use of Lemmas 11–13, then we obtain 𝑏,𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔1−(1−𝛼/𝑛)𝑞,𝜔)}≤𝐶‖𝑏‖_BMO(𝜔)𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞/𝑝,𝜔)}+𝑓_{𝐿𝑝,𝑘(𝜔)}≤𝐶‖𝑏‖_BMO(𝜔)𝑓_{𝐿𝑝,𝑘(𝜔)}. (46) The last inequality follows from Lemma 15. This completes the proof of (a)⇒(b).
(b)⇒(a): Let 𝐿=−Δ be the Laplacian on ℝ^𝑛; then 𝐿^−𝛼/2 is the classical fractional integral 𝐼_𝛼. Choose 𝑍₀∈ℝ^𝑛 so that |𝑍₀|=3. For 𝑥∈𝑄(𝑍₀,2),   |𝑥|^−𝛼+𝑛 can be written as the absolutely convergent Fourier series |𝑥|^−𝛼+𝑛=∑_{𝑚∈𝑍𝑛}𝑎_𝑚𝑒^{𝑖⟨𝜈𝑚,𝑥⟩} with ∑_𝑚|𝑎_𝑚|<∞ since |𝑥|^−𝛼+𝑛∈𝐶^∞(𝑄(𝑍₀,2)). For any 𝑥₀∈ℝ^𝑛 and 𝜌>0, let 𝑄=𝑄(𝑥₀,𝜌) and 𝑄_𝑍0=𝑄(𝑥₀+𝑍₀𝜌,𝜌),_𝑄𝑏(𝑥)−𝑏_𝑄𝑍0=1𝑑𝑥𝑄_𝑍0^𝑄𝑄_𝑍0𝑏(𝑥)𝑦=1−𝑏𝑑𝑦𝑑𝑥𝜌^𝑛_𝑄𝑠(𝑥)_𝑄𝑍0𝑏(𝑥)−𝑏𝑦𝑥−𝑦^𝛼−𝑛𝑥−𝑦^𝑛−𝛼𝑑𝑦𝑑𝑥, (47)where 𝑠(𝑥)=sgn(∫_𝑄𝑍0(𝑏(𝑥)−𝑏(𝑦))𝑑𝑦). Fix 𝑥∈𝑄 and 𝑦∈𝑄_𝑍0 and we have (𝑦−𝑥)/𝜌∈𝑄(𝑍₀,2); hence, we have𝜌^−𝛼+𝑛𝜌^𝑛_𝑄𝑠(𝑥)_𝑄𝑍0𝑏(𝑥)−𝑏𝑦𝑥−𝑦^𝛼−𝑛𝑥−𝑦𝜌^𝑛−𝛼𝑑𝑦𝑑𝑥=𝜌^−𝛼_{𝑚∈𝑍𝑛}𝑎_𝑚×_𝑄𝑠(𝑥)_𝑄𝑍0𝑏(𝑥)𝑦×−𝑏𝑥−𝑦^𝛼−𝑛𝑒^{𝑖⟨𝜈𝑚,𝑦/𝜌⟩}𝑑𝑦𝑒^{−𝑖⟨𝜈𝑚,𝑥/𝜌⟩}𝑑𝑥≤𝜌^−𝛼_{𝑚∈𝑍𝑛}𝑎_𝑚𝑄𝑠(𝑥)𝑏,𝐿^−𝛼/2×𝜒_𝑄𝑍0𝑒^{𝑖⟨𝜈𝑚,⋅/𝜌⟩}𝜒_𝑄(𝑥)𝑒^{−𝑖⟨𝜈𝑚,𝑥/𝜌⟩}𝑑𝑥≤𝜌^−𝛼_{𝑚∈𝑍𝑛}𝑎_𝑚𝑏,𝐿^−𝛼/2𝜒_𝑄𝑍0𝑒^{𝑖⟨𝜈𝑚,⋅/𝜌⟩}_{𝐿𝑞,0(𝜔1−(1−(𝛼/𝑛))𝑞,𝜔)}×_𝑄𝜔(𝑥)^{𝑞′[(1−(𝛼/𝑛))−1/𝑞]}𝑑𝑥^1/𝑞′≤𝐶𝜌^−𝛼_{𝑚∈𝑍𝑛}𝑎_𝑚𝜒_{𝑄𝑍0𝐿𝑝,0(𝜔)𝑄}𝜔(𝑥)^{𝑞′(1/𝑞′−𝛼/𝑛)}𝑑𝑥^1/𝑞′≤𝐶𝜔(𝑄)^{1/𝑝+1/𝑞′−𝛼/𝑛}=𝐶𝜔(𝑄). (48)
This implies 𝑏∈BMO(𝜔). Thus Theorem 1 is proved.

Similarly, to prove Theorem 3, we need the following lemmas.

Lemma 20. Let 0<𝛼<𝑛,1<𝑟,𝑠<∞ such that 𝑟𝑠<𝑝<𝑛/𝛼 and 𝑏∈𝐵𝑀𝑂. Then for all 𝑟>1 and for all 𝑥∈ℝ^𝑛, one has 𝑀^♯_𝐿𝑏,𝐿^−𝛼/2𝑓(𝑥)≤𝐶‖𝑏‖_𝐵𝑀𝑂𝑀_𝑟𝐿^−𝛼/2𝑓(𝑥)+𝑀_{𝛼,𝑟𝑠}𝑓(𝑥), (49) where 𝑀_𝑟(𝑓)(𝑥)=𝑀(|𝑓|^𝑟)^1/𝑟(𝑥).

Proof. The case 0<𝛼<1 was proved by Duong and Yan (see [8] for details). The general case 0<𝛼<𝑛 follows by repeating the same steps as in Lemma 19. Since the main steps and the ideas are almost the same, here we omit the proof.

Lemma 21 (see [5]). If 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛,0<𝑘<𝑝/𝑞, and 𝜔∈𝐴_𝑝,𝑞, then the fractional maximal operator 𝑀_𝛼,1 is bounded from 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞).

Lemma 22 (see [5]). If 0<𝛼<𝑛,1<𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛,0<𝑘<𝑝/𝑞, and 𝜔∈𝐴_𝑝,𝑞, then the fractional maximal operator 𝐼_𝛼 is bounded from 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞).

Lemma 23 (see [5]). If 1<𝑝<∞,0<𝑘<1, and 𝜔∈𝐴_𝑝, then 𝑀 is bounded on 𝐿^𝑝,𝑘(𝜔).

Remark 24. By applying the same argument as in Lemma 15, we know that the conclusion in Lemma 22 still holds for 𝐿^−𝛼/2. We omit the proof here.

Remark 25. By checking the proof of Lemmas 21–23, we know that the three lemmas above still hold when 𝑘=0.

Now we prove Theorem 3.

Proof. (a)⇒(b): Since 𝜔^𝑟𝑠∈𝐴_{𝑝/𝑟𝑠,𝑞/𝑟𝑠}, then we get 𝜔^𝑞∈𝐴_{𝑞/𝑟𝑠} and 𝜔^𝑝∈𝐴_{𝑝/𝑟𝑠}. Applying Lemmas 10 and 20–23, we get𝑏,𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}≤𝑀^♯_𝐿𝑏,𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}≤𝐶‖𝑏‖_BMO𝑀_𝑟𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}+𝑀_{𝛼,𝑟𝑠}𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}≤𝐶‖𝑏‖_BMO𝐿^−𝛼/2𝑓_{𝐿𝑞,𝑘𝑞/𝑝(𝜔𝑞)}+𝑓_{𝐿𝑝,𝑘(𝜔𝑝,𝜔𝑞)}≤𝐶‖𝑏‖_BMO𝑓_{𝐿𝑝,𝑘(𝜔𝑝,𝜔𝑞)}. (50)In the last inequality, we used the fact that 𝐿^−𝛼/2 is bounded from 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞) to 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞) (see Remark 24).
(b)⇒(a): Let 𝐿=−Δ be the Laplacian on ℝ^𝑛; then 𝐿^−𝛼/2 is the classical fractional integral 𝐼_𝛼. Let 𝑘=0 and weight 𝜔≡1, and then 𝐿^𝑝,𝑘(𝜔^𝑝,𝜔^𝑞)=𝐿^𝑝 and 𝐿^{𝑞,𝑘𝑞/𝑝}(𝜔^𝑞)=𝐿^𝑞. From [10] we deduce that the (𝐿^𝑝,𝐿^𝑞) boundedness of [𝑏,𝐼_𝛼] implies 𝑏∈BMO. Thus Theorem 3 is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11226102) and the Doctoral Foundation of Henan Polytechnic University (no. B2012-055).