Commutators of the Hardy-Littlewood Maximal Operator with BMO Symbols on Spaces of Homogeneous Type

Abstract

Weighted 𝐿^𝑝 for 𝑝∈(1,∞) and weak-type endpoint estimates with general weights are established for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. As an application, a weighted weak-type endpoint estimate is proved for maximal operators associated with commutators of singular integral operators with BMO symbols on spaces of homogeneous type. All results with no weight on spaces of homogeneous type are also new.

1. Introduction

We will be working on a space of homogeneous type. Let 𝒳 be a set endowed with a positive Borel regular measure 𝜇 and a symmetric quasimetric 𝑑 satisfying that there exists a constant 𝜅≥1 such that for all 𝑥, 𝑦, 𝑧∈𝒳, 𝑑(𝑥,𝑦)≤𝜅[𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦)]. The triple (𝒳,𝑑,𝜇) is said to be a space of homogeneous type in the sense of Coifman and Weiss [1] if 𝜇 satisfies the following doubling condition: there exists a constant 𝐶≥1 such that for all 𝑥∈𝒳 and 𝑟>0,𝜇(𝐵(𝑥,2𝑟))≤𝐶𝜇(𝐵(𝑥,𝑟)). (1.1)It is easy to see that the above doubling property implies the following strong homogeneity: there exist positive constants 𝐶 and 𝑛 such that for all 𝜆≥1, 𝑟>0, and 𝑥∈𝒳,𝜇(𝐵(𝑥,𝜆𝑟))≤𝐶𝜆^𝑛𝜇(𝐵(𝑥,𝑟)). (1.2)Moreover, there also exist constants 𝐶>0 and 𝑁∈[0,𝑛] such that for all 𝑥, 𝑦∈𝒳 and 𝑟>0,𝜇(𝐵(𝑦,𝑟))≤𝐶(1+𝑑(𝑥,𝑦)𝑟)^𝑁𝜇(𝐵(𝑥,𝑟)). (1.3)

We remark that although all balls defined by 𝑑 satisfy the axioms of complete system of neighborhoods in 𝒳, and therefore induce a (separated) topology in 𝒳, the balls 𝐵(𝑥,𝑟) for 𝑥∈𝒳 and 𝑟>0 need not be open with respect to this topology. However, by a remarkable result of Macías and Segovia in [2], we know that there exists another quasimetric ^0𝑥0𝑑^02𝑑𝑐 such that(i) there exists a constant 𝐶≥1 such that for all 𝑥,𝑦∈𝒳, 𝐶𝑑^{−10𝑥002𝑑𝑐}(𝑥,𝑦)≤𝑑(𝑥,𝑦)≤𝐶𝑑^{0𝑥002𝑑𝑐}(𝑥,𝑦);(ii) there exist constants 𝐶>0 and 𝛾∈(0,1] such that for all 𝑥,𝑥^′,𝑦∈𝒳,|𝑑^{0𝑥002𝑑𝑐}(𝑥,𝑦)−𝑑^{0𝑥002𝑑𝑐}(𝑥^′,𝑦)|≤𝐶(𝑑^{0𝑥002𝑑𝑐}(𝑥,𝑥^′))^𝛾(𝑑^{0𝑥002𝑑𝑐}(𝑥,𝑦)+𝑑^{0𝑥002𝑑𝑐}(𝑥^′,𝑦))^1−𝛾. (1.4) The balls corresponding to ^0𝑥0𝑑^02𝑑𝑐 are open in the topology induced by ^0𝑥0𝑑^02𝑑𝑐. Thus, throughout this paper, we always assume that there exist constants 𝐶>0 and 𝛾∈(0,1] such that for all 𝑥,𝑥^′,𝑦∈𝒳,|𝑑(𝑥,𝑦)−𝑑(𝑥^′,𝑦)|≤𝐶(𝑑(𝑥,𝑥^′))^𝛾(𝑑(𝑥,𝑦)+𝑑(𝑥^′,𝑦))^1−𝛾, (1.5)and that the balls 𝐵(𝑥,𝑟) for all 𝑥∈𝒳 and 𝑟>0 are open.

Now let 𝑘 be a positive integer and 𝑏∈BMO(𝒳), define the 𝑘th-order commutator 𝑀_𝑏,𝑘 of the Hardy-Littlewood maximal operator with 𝑏 by𝑀_𝑏,𝑘𝑓(𝑥)=sup_𝐵϶𝑥1𝜇(𝐵)_𝐵|𝑏(𝑥)−𝑏(𝑦)|^𝑘|𝑓(𝑦)|𝑑𝜇(𝑦) (1.6)for all 𝑥∈𝒳. For the case that (𝒳,𝑑,𝜇) is the Euclidean space, García-Cuerva et al. [3] proved that 𝑀_𝑏,𝑘 is bounded on 𝐿^𝑝(ℝ^𝑛) for any 𝑝∈(1,∞), and Alphonse [4] proved that 𝑀_𝑏,1 enjoys a weak-type 𝐿(log𝐿) estimate, that is, there exists a positive constant 𝐶, depending on ‖𝑏‖_BMO(ℝ^𝑛₎, such that for all suitable functions 𝑓,|{𝑥∈ℝ^𝑛∶𝑀_𝑏,1𝑓(𝑥)>𝜆}|≤𝐶_ℝ𝑛|𝑓(𝑥)|𝜆log(e+|𝑓(𝑥)|𝜆)𝑑𝑥. (1.7)Li et al. [5] established a weighted estimate with any general weight for 𝑀_𝑏,1 in ℝ^𝑛. As it was shown in [3–5] for the setting of Euclidean spaces, the operator 𝑀_𝑏,𝑘 plays an important role in the study of commutators of singular integral operators with BMO symbols. In this paper, we establish weighted estimates with general weights for 𝑀_𝑏,𝑘 in spaces of homogeneous type. To state our results, we first give some notation.

Let 𝐸 be a measurable set with 𝜇(𝐸)<∞. For any fixed 𝑝∈[1,∞), 𝛿>0, and suitable function 𝑓, set‖𝑓‖_{𝐿𝑝(log𝐿)}^𝛿_,𝐸1=inf{𝜆>0∶𝜇(𝐸)_𝐸(|𝑓(𝑥)|𝜆)^𝑝log^𝛿(e+|𝑓(𝑥)|𝜆)𝑑𝜇(𝑥)≤1}. (1.8)The maximal operator 𝑀_{𝐿𝑝(log𝐿)}^𝛿 is defined by 𝑀_{𝐿𝑝(log𝐿)}^𝛿𝑓(𝑥)=sup_𝐵϶𝑥‖𝑓‖_{𝐿𝑝(log𝐿)}^𝛿_,𝐵, where the supremum is taken over all balls containing 𝑥. In the following, we denote 𝑀_{𝐿1(log𝐿)}^𝛿 by 𝑀_{𝐿(log𝐿)}^𝛿 for simplicity, and denote by 𝐿^∞_𝑏(𝒳) the set of bounded functions with bounded support.

With the notation above, we now formulate our main results as follows. Theorem 1.1. Let 𝑘 be a positive integer, 𝑝∈(1,∞), and 𝑏∈BMO(𝒳). Then for any 𝛿>0, there exists a positive constant 𝐶, depending only on 𝑝, 𝑘, and 𝛿, such that for all nonnegative weights 𝑤 and 𝑓∈𝐿^∞_𝑏(𝒳),_𝒳(𝑀_𝑏,𝑘𝑓(𝑥))^𝑝𝑤(𝑥)𝑑𝜇(𝑥)≤𝐶‖𝑏‖^𝑘𝑝_{BMO(𝒳)𝒳}|𝑓(𝑥)|^𝑝𝑀_{𝐿(log𝐿)}^{𝑘𝑝+𝛿}𝑤(𝑥)𝑑𝜇(𝑥). (1.9)Theorem 1.2. Let 𝑘 be a positive integer, 𝑏∈BMO(𝒳), and 𝛿>0. There exists a positive constant 𝐶=𝐶_{𝑘,‖𝑏‖BMO(𝒳)} such that for all nonnegative weights 𝑤, 𝑓∈𝐿^∞_𝑏(𝜇) and 𝜆>0,𝑤({𝑥∈𝒳∶𝑀_𝑏,𝑘𝑓(𝑥)>𝜆})≤𝐶_𝒳|𝑓(𝑥)|𝜆log^𝑘(e+|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)𝑘0𝑥002𝑑𝑐+𝛿}𝑤(𝑥)𝑑𝜇(𝑥), (1.10)where, and in the following, ^0𝑥0𝑘^02𝑑𝑐=𝑘 if 𝑘 is even and ^0𝑥0𝑘^02𝑑𝑐=𝑘+1 if 𝑘 is odd.

As a corollary of Theorem 1.2, we establish a weighted endpoint estimate for the maximal commutator of singular integral operators with BMO(𝒳) symbols. Let 𝑇 be a Calderón-Zygmund operator, that is, 𝑇 is a linear 𝐿²(𝒳)-bounded operator and satisfies that for all 𝑓∈𝐿²(𝒳) with bounded support and almost all 𝑥∉supp𝑓,𝑇𝑓(𝑥)=_𝒳𝐾(𝑥,𝑦)𝑓(𝑦)𝑑𝜇(𝑦), (1.11)where 𝐾 is a locally integrable function on 𝒳×𝒳⧵{(𝑥,𝑦)∶𝑥=𝑦} and satisfies that for all 𝑥≠𝑦,𝐶|𝐾(𝑥,𝑦)|≤𝜇(𝐵(𝑥,𝑑(𝑥,𝑦))), (1.12)and that for all 𝑥,𝑦,𝑦^′∈𝒳 with 𝑑(𝑥,𝑦)≥2𝑑(𝑦,𝑦^′),|𝐾(𝑥,𝑦)−𝐾(𝑥,𝑦^′)|+|𝐾(𝑦,𝑥)−𝐾(𝑦^′,𝑥)|≤𝐶(𝑑(𝑦,𝑦^′))^𝜏𝜇(𝐵(𝑥,𝑑(𝑥,𝑦)))(𝑑(𝑥,𝑦))^𝜏 (1.13)with positive constants 𝐶 and 𝜏≤1. For any 𝜖>0, suitable function 𝑓 and 𝑥∈𝒳, define the truncated operator 𝑇_𝜖 by𝑇_𝜖𝑓(𝑥)=_{𝑑(𝑥,𝑦)>𝜖}𝐾(𝑥,𝑦)𝑓(𝑦)𝑑𝜇(𝑦). (1.14)Let 𝑏∈BMO(𝒳) and let 𝑘 be a positive integer. Define the commutator 𝑇_{𝜖;𝑏,𝑘} by𝑇_𝜖;𝑏,0𝑓(𝑥)=𝑇_𝜖𝑓(𝑥),𝑇_{𝜖;𝑏,𝑘}𝑓(𝑥)=𝑏(𝑥)𝑇_{𝜖;𝑏,𝑘−1}𝑓(𝑥)−𝑇_{𝜖;𝑏,𝑘−1}(𝑏𝑓)(𝑥) (1.15)for all 𝑥∈𝒳 and 𝑓∈𝐿^∞_𝑏(𝒳). The maximal operator associated with the commutator 𝑇_𝑏,𝑘 is defined by𝑇^∗_𝑏,𝑘𝑓(𝑥)=sup_𝜖>0|𝑇_{𝜖;𝑏,𝑘}𝑓(𝑥)| (1.16)for all 𝑥∈𝒳. In [6], it was proved that if 𝑇 is a Calderón-Zygmund operator, then for any 𝑝∈(1,∞), there exists a positive constant 𝐶 such that for all 𝑓∈𝐿^∞_𝑏(𝒳) and all nonnegative weights 𝑤,_𝒳|𝑇^∗_𝑏,𝑘𝑓(𝑥)|^𝑝𝑤(𝑥)𝑑𝜇(𝑥)≤𝐶‖𝑏‖^𝑘𝑝_{BMO(𝜇)𝒳}|𝑓(𝑥)|^𝑝𝑀_{𝐿(log𝐿)}^{(𝑘+1)𝑝+𝛿}𝑤(𝑥)𝑑𝜇(𝑥). (1.17)In [5], it was proved that in ℝ^𝑛, 𝑇^∗_𝑏,1 enjoys the following weighted weak-type endpoint estimate: for any 𝛿>0, there exists a positive constant 𝐶, depending on 𝑛, 𝛿, and ‖𝑏‖_BMO(ℝ^𝑛₎, such that𝑤({𝑥∈ℝ^𝑛∶𝑇^∗_𝑏,1𝑓(𝑥)>𝜆})≤𝐶_ℝ𝑛|𝑓(𝑥)|𝜆log(e+|𝑓(𝑥)|𝜆)𝑀_𝐿(log)^2+𝛿𝑤(𝑥)𝑑𝑥. (1.18)

Using Theorem 1.2, we will prove the following result.Theorem 1.3. Let 𝑇 be a Calderón-Zygmund operator. Then for any 𝑏∈BMO(𝒳), nonnegative integer 𝑘 and 𝛿>0, there exists a positive constant 𝐶, depending on 𝑘, 𝛿, and ‖𝑏‖_BMO(𝒳), such that for all 𝜆>0, 𝑓∈𝐿^∞_𝑏(𝒳) and nonnegative weights 𝑤,∫_{{𝑥∈𝒳∶𝑇∗𝑏,𝑘𝑓(𝑥)>𝜆}}𝑤(𝑥)𝑑𝜇(𝑥)≤𝐶∫_𝒳|𝑓(𝑥)|𝜆log^𝑘(e+|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)}^𝑘+1+𝛿𝑤(𝑥)𝑑𝜇(𝑥). (1.19)

We mention that Theorems 1.1, 1.2, and 1.3 are also new even when 𝑤(𝑥)≡1 for all 𝑥∈𝒳.

We now make some conventions. Throughout the paper, we always denote by 𝐶 a positive constant which is independent of main parameters, but it may vary from line to line. We denote 𝑓≤𝐶𝑔 and 𝑓≥𝐶𝑔 simply by 𝑓≲𝑔 and 𝑓≳𝑔, respectively. If 𝑓≲𝑔≲𝑓, we then write 𝑓∼𝑔. Constant, with subscript such as 𝐶₁, does not change in different occurrences. A weight 𝑤 always means a nonnegative locally integrable function. For a measurable set 𝐸 and a weight 𝑤, 𝜒_𝐸 denotes the characteristic function of 𝐸, 𝑤(𝐸)=∫_𝐸𝑤(𝑥)d𝜇(𝑥). Given 𝜆>0 and a ball 𝐵, 𝜆𝐵 denotes the ball with the same center as 𝐵 and whose radius is 𝜆 times that of 𝐵. For a fixed 𝑝 with 𝑝∈[1,∞),𝑝^′ denotes the dual exponent of 𝑝, namely, 𝑝^′=𝑝/(𝑝−1). For any measurable set 𝐸 and any integrable function 𝑓 on 𝐸, we denote by 𝑚_𝐸(𝑓) the mean value of 𝑓 over 𝐸, that is, 𝑚_𝐸(𝑓)=(1/𝜇(𝐸))∫_𝐸𝑓(𝑥)𝑑𝜇(𝑥). For any locally integrable function 𝑓 and 𝑥∈𝒳, the Fefferman-Stein sharp maximal function 𝑀^#𝑓(𝑥) is defined by𝑀^#𝑓(𝑥)=sup_𝐵϶𝑥1𝜇(𝐵)_𝐵|𝑓(𝑦)−𝑚_𝐵(𝑓)|𝑑𝜇(𝑦), (1.20)where the supremum is taken over all balls 𝐵 containing 𝑥. For any fixed 𝑞∈(0,1), the sharp maximal function 𝑀^#_𝑞𝑓 of the function 𝑓 is defined by 𝑀^#_𝑞𝑓=(𝑀^#(|𝑓|^𝑞))^1/𝑞.

A generalization of Hölder's inequality will be used in the proofs of our theorems. For any measurable set 𝐸 with 𝜇(𝐸)<∞, positive integer 𝑙, and suitable function 𝑓, set‖𝑓‖_exp𝐿^1/𝑙_,𝐸1=inf{𝜆>0∶𝜇(𝐸)_𝐸exp(|𝑓(𝑥)|𝜆)^1/𝑙𝑑𝜇(𝑥)≤2}. (1.21)Then the following generalization of Hölder's inequality:1𝜇(𝐸)_𝐸|𝑓(𝑥)ℎ(𝑥)|𝑑𝜇(𝑥)≤𝐶‖𝑓‖_{𝐿(log𝐿)}^𝑙_,𝐸‖ℎ‖_exp𝐿^1/𝑙_,𝐸 (1.22)holds for any suitable functions 𝑓 and ℎ; see [7] for details.

2. Proof of Theorem 1.1

To prove Theorem 1.1, we need some technical lemmas. In what follows, we denote by 𝑀 the Hardy-Littlewood maximal function. Moreover, for any 𝑠>0 and suitable function 𝑓, we set 𝑀_𝑠(𝑓)=[𝑀(|𝑓|^𝑠)]^1/𝑠.Lemma 2.1 (see [8]). There exists a positive constant 𝐶 such that for all weights 𝑤 and all nonnegative functions 𝑓 satisfying 𝜇({𝑥∈𝒳∶𝑓(𝑥)>𝜆})<∞ for all 𝜆>0, then(i) if 𝜇(𝒳)=∞,_𝒳𝑓(𝑥)𝑤(𝑥)𝑑𝜇(𝑥)≤𝐶_𝒳𝑀^#𝑓(𝑥)𝑀𝑤(𝑥)𝑑𝜇(𝑥); (2.1)(ii) if 𝜇(𝒳)<∞,_𝒳𝑓(𝑥)𝑤(𝑥)𝑑𝜇(𝑥)≤𝐶_𝒳𝑀^#𝑓(𝑥)𝑀𝑤(𝑥)𝑑𝜇(𝑥)+𝐶𝑤(𝒳)𝑚_𝒳(𝑓). (2.2)Lemma 2.2. For any 𝑞∈(0,1), there exists a positive constant 𝐶 such that for all 𝑓∈𝐿^𝑝(𝒳) with 𝑝∈[1,∞) and all 𝑥∈𝒳, 𝑀^#_𝑞(𝑀𝑓)(𝑥)≤𝐶𝑀^#𝑓(𝑥).

For the case that (𝒳,𝑑,𝜇) is the Euclidean space, this lemma was proved in [9]. For spaces of homogeneous type, the proof is similar to the case of Euclidean spaces; see [6].Lemma 2.3. Let 𝑝∈(1,∞) and let 𝑘 be a positive integer.(a) There exists a positive constant 𝐶, depending only on 𝑘 and 𝑝, such that for all 𝑓∈𝐿^∞_𝑏(𝒳) and all weights 𝑤,_𝒳[𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥)]^𝑝𝑤(𝑥)𝑑𝜇(𝑥)≤𝐶_𝒳|𝑓(𝑥)|^𝑝𝑀𝑤(𝑥)𝑑𝜇(𝑥). (2.3)(b) For any 𝛿₁>0, there exists a positive constant 𝐶, depending only on 𝑘, 𝑝, and 𝛿₁, such that for all 𝑓∈𝐿^∞_𝑏(𝒳) and all weights 𝑤,_𝒳(𝑀^𝑘𝑓(𝑥))^𝑝[𝑀_{𝐿(log𝐿)′1𝑘𝑝−1+𝛿}𝑤(𝑥)]^1−𝑝𝑑𝜇(𝑥)≤𝐶_𝒳|𝑓(𝑥)|^𝑝[𝑤(𝑥)]^1−𝑝𝑑𝜇(𝑥). (2.4)

For Euclidean spaces, Lemma 2.3(a) is just Corollary 1.8 in [10] and Lemma 2.3(b) is included in the proof of Theorem 2 in [11] together with (4.11) in [12]. For spaces of homogeneous type, Lemma 2.3(a) is a simple corollary of Theorem 1.4 in [13]. On the other hand, by Theorem 1.4 in [13], and the estimate that for all weights 𝑤, 𝑀_{𝐿(log𝐿)}^𝑘𝑤≈𝑀^𝑘+1𝑤 (see [12]), we can prove Lemma 2.3(b) by the ideas used in [11, page 751]. For details, see [6].

By a similar argument that was used in the proof of Theorem 2.1 in [14], we can verify the existence of the following approximation of the identity of order 𝛾 with bounded support on 𝒳. We omit the details here.

For any 𝑥∈𝒳 and 𝑟>0, set 𝑉_𝑟(𝑥)=𝜇(𝐵(𝑥,𝑟)).Lemma 2.4. Let 𝛾 be as in (1.5). Then there exists an approximation of the identity {𝑆_𝑘}_𝑘∈ℤ of order 𝛾 with bounded support on 𝒳. Namely, {𝑆_𝑘}_𝑘∈ℤ is a sequence of bounded linear integral operators on 𝐿²(𝒳), and there exist constants 𝐶₀, ^0𝑥0𝐶^02𝑑𝑐>0 such that for all 𝑘∈ℤ and all 𝑥, 𝑥^′, 𝑦, and 𝑦^′∈𝒳,𝑆_𝑘(𝑥,𝑦), the integral kernel of 𝑆_𝑘 is a measurable function from 𝒳×𝒳 into ℂ satisfying
(i)𝑆_𝑘(𝑥,𝑦)=0 if 𝑑(𝑥,𝑦)≥𝐶^{0𝑥002𝑑𝑐}2^−𝑘 and 0≤𝑆_𝑘(𝑥,𝑦)≤𝐶₀(1/(𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘(𝑦)));(ii)𝑆_𝑘(𝑥,𝑦)=𝑆_𝑘(𝑦,𝑥) for all 𝑥,𝑦∈𝒳;(iii)|𝑆_𝑘(𝑥,𝑦)−𝑆_𝑘(𝑥^′,𝑦)|≤𝐶₀2^𝑘𝛾𝑑(𝑥,𝑥^′)^𝛾(1/(𝑉₂^−𝑘(𝑥)+𝑉₂^−𝑘(𝑦))) for 𝑑(𝑥,𝑥^′)≤max{𝐶^{0𝑥002𝑑𝑐}/𝜅,1/𝜅}2^1−𝑘;(iv)𝐶₀𝑉₂^−𝑘(𝑥)𝑆_𝑘(𝑥,𝑥)>1 for all 𝑥∈𝒳 and 𝑘∈ℤ;(v)∫_𝒳𝑆_𝑘(𝑥,𝑦)𝑑𝜇(𝑦)=1=∫_𝒳𝑆_𝑘(𝑥,𝑦)𝑑𝜇(𝑥).

For any 𝜖>0 and 𝑥,𝑦∈𝒳, let𝑆_𝜖(𝑥,𝑦)≡𝑆_𝑘(𝑥,𝑦)𝜒₍₂^−𝑘−1_,2−𝑘](𝜖). (2.5)Obviously, 𝑆_𝜖 satisfies (i) through (v) of Lemma 2.4 with 2^−𝑘 replaced by 𝜖. From (iii) and (iv) of Lemma 2.4, it follows that there exist constants 𝐶^′∈(0,min{𝐶^{0𝑥002𝑑𝑐}/𝜅,1/𝜅,(𝐶₀)^−2/𝛾}) and 𝐶>1 such that for all 𝜖>0 and all 𝑥,𝑦∈𝒳 satisfying 𝑑(𝑥,𝑦)<𝐶^′𝜖,𝐶𝑉_𝜖(𝑥)𝑆_𝜖(𝑥,𝑦)>1. (2.6)For a positive integer 𝑘 and a function 𝑏∈BMO(𝒳), let ^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘 be the operator defined by^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘𝑓(𝑥)=sup𝑀_𝜖>0^{0𝑥002𝑑𝑐}_{𝜖;𝑏,𝑘}𝑓(𝑥) (2.7)for all 𝑓∈𝐿^∞_𝑏(𝒳) and 𝑥∈𝒳, where for 𝜖>0,^0𝑥𝑀^002𝑑𝑐_{𝜖;𝑏,𝑘}𝑓(𝑥)=_𝒳𝑆_𝜖(𝑥,𝑦)|𝑏(𝑥)−𝑏(𝑦)|^𝑘|𝑓(𝑦)|𝑑𝜇(𝑦). (2.8)If 𝑘=0, we denote ^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘 and ^0𝑥𝑀^002𝑑𝑐_{𝜖;𝑏,𝑘} simply by ^0𝑥𝑀^002𝑑𝑐 and ^0𝑥𝑀^002𝑑𝑐_𝜖, respectively. From (i) of Lemma 2.4 together with (1.1), it follows that 𝑆_𝜖(𝑥,𝑦)≲1/𝑉_𝜖(𝑥)≲1/𝑉_{20𝑥002𝑑𝑐𝐶𝜖}(𝑥). Notice that if 𝑑(𝑥,𝑦)≥2𝐶^{0𝑥002𝑑𝑐}𝜖, then 𝑆_𝜖(𝑥,𝑦)=0. Thus,^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘𝑓(𝑥)=sup𝑀_𝜖>0^{0𝑥002𝑑𝑐}_{𝜖;𝑏,𝑘}𝑓(𝑥)≲sup_𝜖>01𝑉_{20𝑥002𝑑𝑐𝐶𝜖}(𝑥)_𝐵(𝑥,2^{0𝑥002𝑑𝑐}_𝐶𝜖)|𝑏(𝑥)−𝑏(𝑦)|^𝑘|𝑓(𝑦)|𝑑𝜇(𝑦)≲𝑀_𝑏,𝑘𝑓(𝑥). (2.9)On the other hand, for each fixed 𝜖>0, by (2.6) and 𝑉_𝐶′𝜖(𝑥)∼𝑉_𝜖(𝑥), we have1𝑉_𝐶′𝜖(𝑥)_{𝐵(𝑥,𝐶}^′_𝜖)|𝑏(𝑥)−𝑏(𝑦)|^𝑘|𝑓(𝑦)|𝑑𝜇(𝑦)≲_𝒳𝑆_𝜖(𝑥,𝑦)|𝑏(𝑥)−𝑏(𝑦)|^𝑘≲|𝑓(𝑦)|𝑑𝜇(𝑦)_{0𝑥002𝑑𝑐}𝑀_𝑏,𝑘𝑓(𝑥). (2.10)By the definition of 𝑀_𝑏,𝑘, we further obtain 𝑀_𝑏,𝑘𝑓(𝑥)≲𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥). Thus, there exists some constant 𝐶≥1 such that for all 𝑥∈𝒳 and 𝑓∈𝐿^∞_𝑏(𝒳),𝐶𝑀^{−10𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)≤𝑀_𝑏,𝑘𝑓(𝑥)≤𝐶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥). (2.11)For the sharp function estimate of ^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘, we have the following estimate. Lemma 2.5. Let 𝑘 be a positive integer and 𝑏∈BMO(𝒳). For any 𝑞 and 𝑠 with 0<𝑞<𝑠<1, there exists a positive constant 𝐶 such that for all 𝑓∈𝐿^∞_𝑏(𝒳) and all 𝑥∈𝒳,𝑀^#_𝑞(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)(𝑥)≤𝐶^𝑘−1_𝑗=0‖𝑏‖^𝑘−𝑗_BMO(𝒳)𝑀_𝑠(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)(𝑥)+𝐶‖𝑏‖^𝑘_BMO(𝒳)𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥). (2.12)

Proof. By (i), (ii), and (iii) of Lemma 2.4, we obtain that for all 𝑥,𝑦∈𝒳,|𝑆_𝜖1(𝑥,𝑦)|≲𝜇(𝐵(𝑥,𝑑(𝑥,𝑦))), (2.13)and that for all 𝜖>0 and all 𝑥,𝑦,𝑦^′∈𝒳 with 𝑑(𝑥,𝑦)≥2𝜅𝑑(𝑦,𝑦^′),|𝑆_𝜖(𝑥,𝑦)−𝑆_𝜖(𝑥,𝑦^′)|+|𝑆_𝜖(𝑦,𝑥)−𝑆_𝜖(𝑦^′1,𝑥)|≲(𝜇(𝐵(𝑥,𝑑(𝑥,𝑦)))𝑑(𝑦,𝑦^′))𝑑(𝑥,𝑦)^𝛾. (2.14)
To verify (2.12), by homogeneity, we may assume that ‖𝑏‖_BMO(𝒳)=1. For all 𝑓∈𝐿^∞_𝑏(𝒳), 𝑥∈𝒳, and balls 𝐵 containing 𝑥, it suffices to prove thatinf_𝑐∈ℂ(1𝜇(𝐵)_𝐵|𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑦)−𝑐|^𝑞𝑑𝜇(𝑦))^1/𝑞≲^𝑘−1_𝑗=0𝑀_𝑠(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)(𝑥)+𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥). (2.15)We consider the following three cases. Case 1 (𝜇(𝒳⧵𝐶₁𝐵)=0). Where and in what follows 𝐶₁=𝜅(4𝜅+1). In this case, we have that for all 𝑥∈𝒳,^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘𝑓(𝑥)≲^𝑘−1_𝑗=0|𝑚_𝐵(𝑏)−𝑏(𝑥)|𝑀^{𝑘−𝑗0𝑥002𝑑𝑐}_𝑏,𝑗𝑓(𝑥)+𝑀^{0𝑥002𝑑𝑐}((𝑏−𝑚_𝐵(𝑏))^𝑘𝑓)(𝑥). (2.16)The Kolmogorov inequality (see [15, page 102]), along with the fact that 𝑀 (and so ^0𝑥𝑀^002𝑑𝑐) is bounded from 𝐿¹(𝒳) to 𝐿^1,∞(𝒳) and the inequality (1.22) gives us that{1𝜇(𝐵)_𝐵[𝑀^{0𝑥002𝑑𝑐}((𝑏−𝑚_𝐵(𝑏))^𝑘𝑓)(𝑦)]^𝑞𝑑𝜇(𝑦)}^1/𝑞≲1𝜇(𝐵)_𝐶1𝐵|𝑏(𝑦)−𝑚_𝐵(𝑏)|^𝑘≲1|𝑓(𝑦)|𝑑𝜇(𝑦)𝜇(𝐵)_𝐶1𝐵|𝑏(𝑦)−𝑚_𝐶1𝐵(𝑏)|^𝑘|𝑓(𝑦)|𝑑𝜇(𝑦)+|𝑚_𝐶1𝐵(𝑏)−𝑚_𝐵1(𝑏)|𝜇(𝐵)_𝐶1𝐵|𝑓(𝑦)|𝑑𝜇(𝑦)≲𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥), (2.17)where the last inequality follows from the John-Nirenberg inequality, which states that for any ball 𝑄,‖|𝑚_𝑄(𝑏)−𝑏|^𝑘‖_{exp𝐿1/𝑘,𝑄}≲‖𝑏‖^𝑘_BMO(𝒳). (2.18)On the other hand, if 0<𝑞<𝑠<1, an application of Hölder's inequality implies that{1𝜇(𝐵)_𝐵[𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)]^𝑞𝑑𝜇(𝑥)}^1/𝑞≲^𝑘−1_𝑗=0𝑀_𝑠(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗1𝑓)(𝑥)+{𝜇(𝐵)_𝐵[𝑀^{0𝑥002𝑑𝑐}((𝑏−𝑚_𝐵(𝑏))^𝑘𝑓)(𝑦)]^𝑞𝑑𝜇(𝑦)}^1/𝑞. (2.19)We then get (2.15).Case 2 (𝜇(𝒳⧵𝐶₁𝐵)≠0 and 𝜇(𝐶₁𝐵⧵𝐵)>0). In this case, decompose 𝑓 into 𝑓=𝑓𝜒_𝐶1𝐵+𝑓𝜒_{𝒳⧵𝐶1𝐵}≡𝑓₁+𝑓₂, recalling that 𝜒_𝐸 denotes the characteristic function of the set 𝐸. Let 𝑦₀ be a point in 𝐵 such that𝐶_𝐵≡sup_𝜖>0𝒳𝑆_𝜖(𝑦₀,𝑧)|𝑚_𝐵(𝑏)−𝑏(𝑧)|^𝑘|𝑓₂(𝑧)|𝑑𝑧<∞. (2.20)With the aid of the formula(𝑚_𝐵(𝑏)−𝑏(𝑧))^𝑘=(𝑏(𝑦)−𝑏(𝑧))^𝑘+^𝑘−1_𝑗=0𝐶^𝑗_𝑘(𝑏(𝑦)−𝑏(𝑧))^𝑗(𝑚_𝐵(𝑏)−𝑏(𝑦))^𝑘−𝑗, (2.21)where 𝐶^𝑗_𝑘 is the constant from Newton's formula, we have||𝑏(𝑦)−𝑏(𝑧)|^𝑘|𝑓(𝑧)|−|𝑚_𝐵(𝑏)−𝑏(𝑧)|^𝑘|𝑓₂≲(𝑧)||_{𝑘−1𝑗=0}|𝑏(𝑦)−𝑏(𝑧)|^𝑗|𝑚_𝐵(𝑏)−𝑏(𝑦)|^𝑘−𝑗|𝑓(𝑧)|+|𝑚_𝐵(𝑏)−𝑏(𝑧)|^𝑘|𝑓₁(𝑧)|. (2.22)Thus for any 𝑦∈𝐵,|𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑦)−𝐶_𝐵|=|sup_𝜖>0|𝑀^{0𝑥002𝑑𝑐}_{𝜖;𝑏,𝑘}𝑓(𝑦)|−sup_𝜖>0|𝑀^{0𝑥002𝑑𝑐}_𝜖((𝑚_𝐵(𝑏)−𝑏)^𝑘𝑓₂)(𝑦₀)||≤sup_𝜖>0|𝑀^{0𝑥002𝑑𝑐}_{𝜖;𝑏,𝑘}𝑓(𝑦)−𝑀^{0𝑥002𝑑𝑐}_𝜖((𝑚_𝐵(𝑏)−𝑏)^𝑘𝑓₂)(𝑦₀≲)|_{0𝑥002𝑑𝑐}𝑀((𝑚_𝐵(𝑏)−𝑏)^𝑘𝑓₁)(𝑦)+^𝑘−1_𝑗=0|𝑚_𝐵(𝑏)−𝑏(𝑦)|𝑀^{𝑘−𝑗0𝑥002𝑑𝑐}_𝑏,𝑗𝑓(𝑦)+sup_𝜖>0|𝑀^{0𝑥002𝑑𝑐}_𝜖((𝑚_𝐵(𝑏)−𝑏)^𝑘𝑓₂)(𝑦)−𝑀^{0𝑥002𝑑𝑐}_𝜖((𝑚_𝐵(𝑏)−𝑏)^𝑘𝑓₂)(𝑦₀)|≡I(𝑦)+II(𝑦)+III(𝑦). (2.23)As in Case 1, we have that{1𝜇(𝐵)_𝐵|I(𝑦)|^𝑞𝑑𝜇(𝑦)}^1/𝑞≲𝑀_{𝐿(log𝐿)}^𝑘{1𝑓(𝑥),𝜇(𝐵)_𝐵|II(𝑦)|^𝑞𝑑𝜇(𝑦)}^1/𝑞≲^𝑘−1_𝑗=0𝑀_𝑠(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)(𝑥). (2.24) As for III(𝑦), by (2.14) and (1.22), it is easy to getIII(𝑦)≤sup_𝜖>0𝒳|𝑆_𝜖(𝑦,𝑧)−𝑆_𝜖(𝑦₀,𝑧)||𝑚_𝐵(𝑏)−𝑏(𝑧)|^𝑘|𝑓₂≲(𝑧)|𝑑𝜇(𝑧)_{𝒳⧵𝐶1𝐵}(𝑑(𝑦,𝑦₀))^𝛾𝜇(𝐵(𝑦,𝑑(𝑦,𝑧)))(𝑑(𝑦,𝑧))^𝛾|𝑚_𝐵(𝑏)−𝑏(𝑧)|^𝑘≲|𝑓(𝑧)|𝑑𝑧_∞𝑙=12^−𝑙𝛾1𝜇(2^𝑙𝐶₁𝐵)^2𝑙𝐶₁^𝐵|𝑚₂^𝑙_𝐶1𝐵(𝑏)−𝑏(𝑧)|^𝑘+|𝑓(𝑧)|𝑑𝜇(𝑧)_∞𝑙=12^−𝑙𝛾|𝑚₂^𝑙_𝐶1𝐵(𝑏)−𝑚_𝐵(𝑏)|^𝑘1𝜇(2^𝑙𝐶₁𝐵)^2𝑙𝐶₁^𝐵≲|𝑓(𝑧)|𝑑𝜇(𝑧)_∞𝑙=12^−𝑙𝛾‖|𝑚₂^𝑙_𝐶1𝐵(𝑏)−𝑏|^𝑘‖_{exp𝐿1/𝑘,2𝑙𝐶1𝐵}‖𝑓‖_{𝐿(log𝐿)}^𝑘_{,2𝑙𝐶1𝐵}+^∞_𝑙=12^−𝑙𝛾|𝑚₂^𝑙_𝐶1𝐵(𝑏)−𝑚_𝐵(𝑏)|^𝑘𝑀𝑓(𝑥)≲𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥), (2.25)where the last inequality follows from (2.18) and |𝑚₂^𝑙_𝑄(𝑏)−𝑚_𝑄(𝑏)|≲𝑙. This leads to our desired estimate (2.15).Case 3 (𝜇(𝒳⧵𝐶₁𝐵)≠0 and 𝜇(𝐶₁𝐵⧵𝐵)=0). In this case, we take 𝐵^′ such that 𝐵⊂𝐵^′, 𝜇(𝐵^′)=𝜇(𝐵), and 𝜇(𝐶₁𝐵^′⧵𝐵^′)>0. We then have thatinf_𝑐∈ℂ1𝜇(𝐵)_𝐵|𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)−𝑐|^𝑞𝑑𝜇(𝑥)≤inf_𝑐∈ℂ1𝜇(𝐵^′)^𝐵′|𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)−𝑐|^𝑞𝑑𝜇(𝑥). (2.26)With the ball 𝐵 replaced by 𝐵^′ in Case 2, we also obtain the result that for any 𝑦∈𝐵^′,sup_𝜖>0𝒳|𝑆_𝜖(𝑦,𝑧)−𝑆_𝜖(𝑦₀,𝑧)||𝑚_𝐵′(𝑏)−𝑏(𝑧)|^𝑘|𝑓₂(𝑧)|𝑑𝑧≤𝐶𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥), (2.27)which completes the proof of Lemma 2.5.

Lemma 2.6. Let 𝛼,𝛽∈[0,∞). There exists a positive constant 𝐶, depending only on 𝛼 and 𝛽, such that for all weights 𝑤,𝑀_{𝐿(log𝐿)}^𝛼(𝑀_{𝐿(log𝐿)}^𝛽𝑤)(𝑥)≤𝐶𝑀_{𝐿(log𝐿)}^𝛼+𝛽+1𝑤(𝑥). (2.28)

For Euclidean spaces, a generalization of Lemma 2.6 was proved in [16]. For spaces of homogeneous type, by a standard argument involving a covering lemma in [17, page 138], we have that for any 𝜆>0 and suitable function 𝑓,𝜇({𝑥∈𝒳∶𝑀_{𝐿(log𝐿)}^𝛼𝑓(𝑥)>𝜆})≲_𝒳|𝑓(𝑥)|𝜆log^𝛼(e+|𝑓(𝑥)|𝜆)𝑑𝜇(𝑥). (2.29)Using this, Lemma 2.6 can be proved by applying the ideas used in [16]. For details, see [6, Lemma 7].

Proof of Theorem 1.1. We assume again that ‖𝑏‖_BMO(𝒳)=1. At first, we claim that when 𝜇(𝒳)=∞, for all 𝜆>0 and 𝑓∈𝐿^∞_𝑏(𝒳), 𝜇({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)>𝜆})<∞. In fact, for any 𝑓∈𝐿^∞_𝑏(𝒳), let 𝑅 be large enough such that supp𝑓⊂𝐵(𝑥₀,𝑅) for some 𝑥₀∈𝒳. Notice that for all 𝑥∈𝒳⧵𝐵(𝑥₀,3𝑅), 𝑀𝑓(𝑥)≲‖𝑓‖_𝐿¹_(𝒳)𝜇(𝐵(𝑥,𝑑(𝑥,𝑥₀))). (2.30) It then follows that for 𝑝∈(1,∞),𝜇({𝑥∈𝒳⧵𝐵(𝑥₀,3𝑅)∶|𝑏(𝑥)−𝑚_{𝐵(𝑥0,𝑅)}(𝑏)|^𝑘𝑀𝑓(𝑥)>𝜆})≤𝜆^−𝑝_{𝒳⧵𝐵(𝑥0,3𝑅)}|𝑏(𝑥)−𝑚_{𝐵(𝑥0,𝑅)}(𝑏)|^𝑘𝑝(𝑀𝑓(𝑥))^𝑝𝑑𝜇(𝑥)≤𝜆^−𝑝‖𝑓‖^𝑝_{𝐿1(𝒳)𝒳⧵𝐵(𝑥0,3𝑅)}|𝑏(𝑥)−𝑚_{𝐵(𝑥0,𝑑(𝑥,𝑥0))}(𝑏)|^𝑘𝑝{𝜇(𝐵(𝑥,𝑑(𝑥,𝑥₀)))}^𝑝𝑑𝜇(𝑥)<∞. (2.31)This, together with the estimate that𝜇({𝑥∈𝒳∶𝑀((𝑏−𝑚_{𝐵(𝑥0,𝑅)}(𝑏))^𝑘𝑓)(𝑥)>𝜆})≲𝜆^−𝑝‖|𝑏−𝑚_{𝐵(𝑥0,𝑅)}(𝑏)|^𝑘𝑓‖^𝑝_{𝐿𝑝(𝒳)}<∞, (2.32)leads to our claim.
By (2.11), to prove Theorem 1.1, it suffices to prove that for all weights 𝑤,(𝑀_𝒳^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥))^𝑝𝑤(𝑥)𝑑𝜇(𝑥)≲_𝒳|𝑓(𝑥)|^𝑝𝑀_{𝐿(log𝐿)}^{𝑘𝑝+𝛿}𝑤(𝑥)𝑑𝜇(𝑥). (2.33)We proceed our proof by an inductive argument on 𝑘. When 𝑘=0, (2.33) is implied by the fact that 𝑀𝑤(𝑥)≤𝑀_{𝐿(log𝐿)}^𝛿𝑤(𝑥) for all 𝑥∈𝒳 and the following known inequality:[𝑀_𝒳^{0𝑥002𝑑𝑐}𝑓(𝑥)]^𝑝𝑤(𝑥)𝑑𝜇(𝑥)≲_𝒳|𝑓(𝑥)|^𝑝𝑀𝑤(𝑥)𝑑𝜇(𝑥). (2.34)See [18, pages 150-151], for a proof of the last inequality when 𝒳=ℝ^𝑛. The same ideas also work for 𝒳. Now we assume that 𝑘 is a positive integer and (2.33) holds for any integer 𝑙 with 0≤𝑙≤𝑘−1. Then ^0𝑥𝑀^002𝑑𝑐_𝑏,𝑙 (0≤𝑙≤𝑘−1) can extend to a bounded operator on 𝐿^𝑝(𝒳) for 𝑝∈(1,∞) and so for any 𝜆>0 and 𝜎∈(0,1),𝜇({𝑥∈𝒳∶𝑀((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)^𝜎)(𝑥)>𝜆})<∞. (2.35)
We now prove (2.33). To begin with, we prove that for any given 𝑞∈(0,1) and 𝑘∈ℕ, and for all weights ℎ and all 𝑓∈𝐿^∞_𝑏(𝒳),(𝑀_𝒳^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥))^𝑞ℎ(𝑥)𝑑𝜇(𝑥)≲_𝒳(𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥))^𝑞𝑀^𝑘ℎ(𝑥)𝑑𝜇(𝑥). (2.36)We first consider the case that 𝜇(𝒳)=∞. Choose 𝑟₁,…,𝑟_𝑘−1, 𝑟_𝑘 such that 0<𝑞=𝑟₀<𝑟₁<⋯<𝑟_𝑘−1<𝑟_𝑘<1. By Lemma 2.5, we obtain that for any 1≤𝑚≤𝑘−1 and any weight ℎ,_𝒳(𝑀^#_𝑟𝑗(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑚𝑓)(𝑥))^𝑞≲ℎ(𝑥)𝑑𝜇(𝑥)_{𝑚−1𝑙=0𝒳}(𝑀_𝑟𝑗+1(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)(𝑥))^𝑞ℎ(𝑥)𝑑𝜇(𝑥)+_𝒳(𝑀_{𝐿(log𝐿)}^𝑚𝑓(𝑥))^𝑞ℎ(𝑥)𝑑𝜇(𝑥). (2.37)Therefore, applying Lemmas 2.1 and 2.2, and the estimate (2.35), we have_𝒳(𝑀_𝑟𝑗+1(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)(𝑥))^𝑞ℎ(𝑥)𝑑𝜇(𝑥)=_𝒳(𝑀((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)^𝑟𝑗+1)(𝑥))^{𝑞/𝑟𝑗+1}≲ℎ(𝑥)𝑑𝜇(𝑥)_𝒳(𝑀^#_{𝑞/𝑟𝑗+1}(𝑀((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)^𝑟𝑗+1)(𝑥)))^{𝑞/𝑟𝑗+1}𝑀ℎ(𝑥)𝑑𝜇(𝑥)≲_𝒳(𝑀^#((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)^𝑟𝑗+1(𝑥)))^{𝑞/𝑟𝑗+1}𝑀ℎ(𝑥)𝑑𝜇(𝑥)=_𝒳(𝑀^#_𝑟𝑗+1(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)(𝑥))^𝑞𝑀ℎ(𝑥)𝑑𝜇(𝑥), (2.38)which leads to_𝒳(𝑀^#_𝑟𝑗(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑚𝑓)(𝑥))^𝑞≲ℎ(𝑥)𝑑𝜇(𝑥)_{𝑚−1𝑙=0𝒳}(𝑀^#_𝑟𝑗+1(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)(𝑥))^𝑞𝑀ℎ(𝑥)𝑑𝜇(𝑥)+_𝒳(𝑀_{𝐿(log𝐿)}^𝑚𝑓(𝑥))^𝑞ℎ(𝑥)𝑑𝜇(𝑥). (2.39)Repeating the argument above 𝑘−1 times, we then have that for all weights ℎ,_𝒳(𝑀^#_𝑞(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)(𝑥))^𝑞≲ℎ(𝑥)𝑑𝜇(𝑥)_{𝑘−1𝑗=0𝒳}(𝑀^#_𝑟1(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)(𝑥))^𝑞𝑀ℎ(𝑥)𝑑𝜇(𝑥)+_𝒳(𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥))^𝑞≲ℎ(𝑥)𝑑𝜇(𝑥)_{𝑘−2𝑗=0𝒳}(𝑀^#_𝑟2(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)(𝑥))^𝑞𝑀²ℎ(𝑥)𝑑𝜇(𝑥)+^𝑘−1_𝑗=0𝒳(𝑀_{𝐿(log𝐿)}^𝑗𝑓(𝑥))^𝑞+𝑀ℎ(𝑥)𝑑𝜇(𝑥)_𝒳(𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥))^𝑞≲ℎ(𝑥)𝑑𝜇(𝑥)_1𝑗=0𝒳(𝑀^#_𝑟𝑘−1(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)(𝑥))^𝑞𝑀^𝑘−1ℎ(𝑥)𝑑𝜇(𝑥)+^𝑘_𝑗=0𝒳(𝑀_{𝐿(log𝐿)}^𝑗𝑓(𝑥))^𝑞𝑀^𝑘−1ℎ(𝑥)𝑑𝜇(𝑥). (2.40)On the other hand, notice that for all 𝑥∈𝒳, 𝑀^#_𝑟𝑘−1(𝑀^{0𝑥002𝑑𝑐}_𝑏,0𝑓)(𝑥)≲𝑀²𝑓(𝑥), and that, by (2.12) and the fact that 𝑀²𝑓(𝑥)∼𝑀_{𝐿(log𝐿)}𝑓(𝑥) for all 𝑥∈𝒳 (see [12, (4.11)]), we then have that for all 𝑥∈𝒳, 𝑀^#_𝑟𝑘−1(𝑀^{0𝑥002𝑑𝑐}_𝑏,1𝑓)(𝑥)≲𝑀_𝑟𝑘(𝑀^{0𝑥002𝑑𝑐}_𝑏,0𝑓)(𝑥)+𝑀_{𝐿(log𝐿)}𝑓(𝑥)≲𝑀²𝑓(𝑥). From these inequalities, it then follows that_𝒳(𝑀^#_𝑞(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)(𝑥))^𝑞≲ℎ(𝑥)𝑑𝜇(𝑥)_𝒳(𝑀²𝑓(𝑥))^𝑞𝑀^𝑘−1ℎ(𝑥)𝑑𝜇(𝑥)+^𝑘_𝑗=0𝒳(𝑀_{𝐿(log𝐿)}^𝑗𝑓(𝑥))^𝑞𝑀^𝑘−1≲ℎ(𝑥)𝑑𝜇(𝑥)_𝒳(𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥))^𝑞𝑀^𝑘−1ℎ(𝑥)𝑑𝜇(𝑥), (2.41)which together with (i) of Lemma 2.1 gives (2.36).
We turn our attention to (2.36) for the case of 𝜇(𝒳)<∞. For all 𝑥∈𝒳,^0𝑥𝑀^002𝑑𝑐_𝑏,𝑙𝑓(𝑥)≲^𝑙_𝑗=0|𝑏(𝑥)−𝑚_𝒳(𝑏)|𝑀^{𝑗0𝑥002𝑑𝑐}((𝑚_𝒳(𝑏)−𝑏)^𝑙−𝑗𝑓)(𝑥). (2.42)Moreover, the Kolmogorov inequality, together with Hölder's inequality, the inequalities (1.22), and (2.18), tells us that for any 0≤𝑗≤𝑘, 𝑟∈(0,1), and 𝑡∈(𝑟,1),1𝜇(𝒳)_𝒳|𝑏(𝑥)−𝑚_𝒳(𝑏)|^𝑟𝑗(𝑀^{0𝑥002𝑑𝑐}((𝑚_𝒳(𝑏)−𝑏)^𝑙−𝑗𝑓)(𝑥))^𝑟1𝑑𝜇(𝑥)≲{𝜇(𝒳)_𝒳[𝑀^{0𝑥002𝑑𝑐}((𝑚_𝒳(𝑏)−𝑏)^𝑙−𝑗)𝑓(𝑥)]^𝑡𝑑𝜇(𝑥)}^𝑟/𝑡1≲{𝜇(𝒳)_𝒳|𝑓(𝑥)||𝑚_𝒳(𝑏)−𝑏(𝑥)|^𝑙−𝑗𝑑𝜇(𝑥)}^𝑟≲(‖𝑓‖_{𝐿(log𝐿)}^𝑙−𝑗_,𝒳)^𝑟. (2.43)Combining the above estimates, we obtain𝑚_𝒳((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)^𝑟)≲(inf_𝑥∈𝒳𝑀_{𝐿(log𝐿)}^𝑙𝑓(𝑥))^𝑟. (2.44)Let 𝑞, 𝑟₁,𝑟₂,…,𝑟_𝑘 be as in the case of 𝜇(𝒳)=∞. Another application of Kolmogorov inequality and the fact that 𝑀 is bounded from 𝐿¹(𝒳) to 𝐿^1,∞(𝒳) leads to{𝑚_𝒳(𝑀(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)^𝑟𝑗)^{𝑞/𝑟𝑗}}^{𝑟𝑗/𝑞}≲𝑚_𝒳((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓)^𝑟𝑗)≲(inf_𝑥∈𝒳𝑀_{𝐿(log𝐿)}^𝑙𝑓(𝑥))^𝑟𝑗. (2.45)As in the case of 𝜇(𝒳)=∞, by Lemmas 2.1, 2.2, and 2.5, we have that for any 𝑞∈(0,1),(𝑀_𝒳^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)^𝑞ℎ(𝑥)𝑑𝜇(𝑥)≲_𝒳(𝑀^#_𝑞(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)(𝑥))^𝑞𝑀ℎ(𝑥)𝑑𝜇(𝑥)+ℎ(𝒳)𝑚_𝒳((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)^𝑞)≲^𝑘−1_𝑗=0𝒳(𝑀^#_𝑟1(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)(𝑥))^𝑞𝑀²+ℎ(𝑥)𝑑𝜇(𝑥)_𝒳(𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥))^𝑞+𝑀ℎ(𝑥)𝑑𝜇(𝑥)_{𝑘−1𝑗=0}(𝑀ℎ)(𝒳)𝑚_𝒳(𝑀(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑗𝑓)^𝑟1)^𝑞/𝑟1+ℎ(𝒳)𝑚_𝒳((𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)^𝑞)≲_𝒳(𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥))^𝑞𝑀^𝑘ℎ(𝑥)𝑑𝜇(𝑥). (2.46)Combining the two cases yields (2.36).
For any fixed 𝑝∈(1,∞) and 𝛿>0, choose 𝑞∈(0,1) and 𝛿₁>0 such that 𝑘𝑝/𝑞+𝛿₁<𝑘𝑝+𝛿. This, via a duality argument, (2.36), and Lemma 2.3, leads to‖(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓)^𝑞‖_𝐿^𝑝/𝑞_(𝑤)=sup_{‖ℎ‖𝐿′′)(𝑝/𝑞)(𝑤1−(𝑝/𝑞)≤1}|_𝒳[𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)]^𝑞ℎ(𝑥)𝑑𝜇(𝑥)|≲sup_{‖ℎ‖𝐿′′)(𝑝/𝑞)(𝑤1−(𝑝/𝑞)≤1𝒳}[𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥)]^𝑞𝑀^𝑘ℎ(𝑥)𝑑𝜇(𝑥)≲sup_{‖ℎ‖𝐿′′)(𝑝/𝑞)(𝑤1−(𝑝/𝑞)≤1}{_𝒳[𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥)]^𝑝𝑀_{𝐿(log𝐿)1}^{𝑘𝑝/𝑞−1+𝛿}𝑤(𝑥)𝑑𝜇(𝑥)}^𝑞/𝑝×{_𝒳[𝑀^𝑘ℎ(𝑥)]^{(𝑝/𝑞)′}[𝑀_{𝐿(log𝐿)1}^{𝑘𝑝/𝑞−1+𝛿}𝑤(𝑥)]^{1−(𝑝/𝑞)′}𝑑𝜇(𝑥)}^{1/(𝑝/𝑞)′}≲{_𝒳[𝑀_{𝐿(log𝐿)}^𝑘𝑓(𝑥)]^𝑝𝑀_{𝐿(log𝐿)1}^{𝑘𝑝/𝑞−1+𝛿}𝑤(𝑥)𝑑𝜇(𝑥)}^𝑞/𝑝≲{_𝒳|𝑓(𝑥)|^𝑝𝑀_{𝐿(log𝐿)}^{𝑘𝑝+𝛿}𝑤(𝑥)𝑑𝜇(𝑥)}^𝑞/𝑝, (2.47)where in the last inequality we have used Lemma 2.6. This completes the proof of Theorem 1.1.

3. Proof of Theorem 1.2

We begin with some preliminary lemmas.Lemma 3.1 (see [17]). Let (𝒳,𝑑,𝜇) be a space of homogeneous type and let 𝑓 be a nonnegative integrable function. Then for every 𝜆>𝑚_𝒳(𝑓)(𝑚_𝒳(𝑓)=0 if 𝜇(𝒳)=∞), there exist a sequence of pairwise disjoint balls {𝐵_𝑗}_𝑗≥1 and a constant 𝐶₄≥1 such that𝑚_{𝐶4𝐵𝑗}(𝑓)≤𝜆<𝑚_𝐵𝑗(𝑓) (3.1)and 𝑚_𝐵(𝑓)≤𝜆 for every ball 𝐵 centered at 𝑥∈𝒳⧵(∪_𝑗𝐶₄𝐵_𝑗).Lemma 3.2. Let 𝑑 and 𝑙 be two nonnegative integers. Then for all 𝑡₁,𝑡₂≥0,𝑡₁𝑡^𝑑₂log(e+𝑡₁𝑡^𝑑₂)≤𝐶(𝑡₁log^𝑑+𝑙(e+𝑡₁)+exp𝑡₂). (3.2)

Proof. We may assume that 𝑑≥1, otherwise the conclusion holds obviously. Set Φ₁(𝑡)=𝑡log^𝑙(e+𝑡), Φ₂(𝑡)=𝑡log^𝑙+𝑑(e+𝑡), and Φ₃(𝑡)=exp(𝑡^1/𝑑). Let 𝑗=1,2,3. Denote by Φ_𝑗⁻¹ the inverse of Φ_𝑗, that is, Φ_𝑗⁻¹(𝑡)=inf{𝑠>0∶Φ_𝑗(𝑠)>𝑡}. It is well known that Φ₁⁻¹(𝑡)≈𝑡log^−𝑙(e+𝑡) and Φ₂⁻¹(𝑡)≈𝑡log^{−(𝑑+𝑙)}(e+𝑡) (see [19]). On the other hand, it is easy to verify that Φ₃⁻¹(𝑡)=0 when 𝑡∈[0,1) and Φ₃⁻¹(𝑡)=log^𝑑𝑡 when 𝑡∈[1,∞). Therefore, for all 𝑡∈[0,∞), Φ₂⁻¹(𝑡)Φ₃⁻¹(𝑡)≲Φ₁⁻¹(𝑡). This via [7, Lemma 6, page 63] tells us that Φ₁(𝑡₁𝑡^𝑑₂)≲Φ₂(𝑡₁)+Φ₃(𝑡^𝑑₂). Our desired conclusion then follows directly.

Proof of Theorem 1.2. With the notation ^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘 as in (2.7), by (2.11), it suffices to prove that for ‖𝑏‖_BMO(𝒳)=1 and all 𝑓∈𝐿^∞_𝑏(𝒳) and 𝜆>0, 𝑤({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)>𝜆})≲∫_𝒳|𝑓(𝑥)|𝜆log^𝑘(e+|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)𝑘0𝑥002𝑑𝑐+𝛿}𝑤(𝑥)𝑑𝜇(𝑥), (3.3) where ^0𝑥0𝑘^02𝑑𝑐=𝑘 when 𝑘 is even and ^0𝑥0𝑘^02𝑑𝑐=𝑘+1 when 𝑘 is odd.
Recall that for all 𝑓∈𝐿^∞_𝑏(𝒳),𝑤({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}1𝑓(𝑥)>𝜆})≲𝜆_𝒳|𝑓(𝑥)|𝑀𝑤(𝑥)𝑑𝜇(𝑥). (3.4)(See [18, page 151] for a proof when 𝒳=ℝ^𝑛. The same idea also works for 𝒳.) By Hölder's inequality, it follows that for all 𝑥∈𝒳,^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘𝑓(𝑥)≤(𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘+1𝑓(𝑥))^{𝑘/(𝑘+1)}(𝑀^{0𝑥002𝑑𝑐}𝑓(𝑥))^1/(𝑘+1) (3.5)and so when 𝑘 is odd,𝑤({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑓(𝑥)>𝜆})≤𝑤({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘+1𝑓(𝑥)>𝜆})+𝑤({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}𝑓(𝑥)>𝜆})≲𝑤({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘+11𝑓(𝑥)>𝜆})+𝜆_𝒳|𝑓(𝑥)|𝑀𝑤(𝑥)𝑑𝜇(𝑥). (3.6)Thus, it suffices to prove (3.3) for the case that 𝑘 is even. We employ some ideas from [20], and proceed our proof of (3.3) by an inductive argument. When 𝑘=0, (3.3) is implied by the fact that 𝑀𝑤(𝑥)≤𝑀_{𝐿(log𝐿)}^𝛿𝑤(𝑥) for all 𝑥∈𝒳 and (3.4). Now let 𝑘 be a positive integer. We may assume that 𝑀_{𝐿(log𝐿)}^𝑘+𝛿𝑤 is finite almost everywhere, otherwise there is nothing to be proved. For any fixed 𝛿>0, we assume that for any nonnegative integer 𝑙 with 0≤𝑙≤𝑘−1, there exists a constant 𝐶=𝐶_𝑙,𝛿 such that for all 𝜆>0,𝑤({𝑥∈𝒳∶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑙𝑓(𝑥)>𝜆})≲∫_𝒳|𝑓(𝑥)|𝜆log^𝑙(e+|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)𝑙0𝑥002𝑑𝑐+𝛿}𝑤(𝑥)𝑑𝜇(𝑥), (3.7)where and in what follows, ^0𝑥00𝑙^2𝑑𝑐=𝑙 when 𝑙 is even and ^0𝑥00𝑙^2𝑑𝑐=𝑙+1 when 𝑙 is odd. If 𝜇(𝒳)<∞ and 𝜆≤‖𝑓‖_𝐿¹_(𝒳)[𝜇(𝒳)]⁻¹, the inequality (3.3) is trivial. So it remains to consider the case that 𝜆>‖𝑓‖_𝐿¹_(𝒳)[𝜇(𝒳)]⁻¹. For each fixed bounded function 𝑓 with bounded support and 𝜆>‖𝑓‖_𝐿¹_(𝒳)[𝜇(𝒳)]⁻¹, applying Lemma 3.1 to |𝑓| at level 𝜆, we obtain a sequence of balls {𝐵_𝑗}_𝑗≥1 with pairwise disjoint interiors. As in the proof of Lemma 2.10 in [17], set 𝑉₁=(𝐶₄𝐵₁)⧵(∪_𝑛≥2𝐵_𝑛) and 𝑉_𝑗=(𝐶₄𝐵_𝑗)⧵[∪^𝑗−1_𝑛=1𝑉_𝑛∪∪_{𝑙≥𝑗+1}𝐵_𝑙], it then follows that 𝐵_𝑗⊂𝑉_𝑗⊂(𝐶₄𝐵_𝑗) and ∪_𝑗𝑉_𝑗=∪_𝑗(𝐶₄𝐵_𝑗). Define the functions 𝑔 and ℎ, respectively, by 𝑔≡|𝑓|𝜒_{𝒳⧵∪𝑗𝑉𝑗}+∑_𝑗𝑚_𝑉𝑗(|𝑓|)𝜒_𝑉𝑗 and ℎ≡∑_𝑗ℎ_𝑗 with ℎ_𝑗≡(|𝑓|−𝑚_𝑉𝑗(|𝑓|))𝜒_𝑉𝑗. Recall that 𝜇 is regular and the set of continuous function is dense in 𝐿^𝑝(𝒳) for any 𝑝∈[1,∞). Lemma 3.1 implies that for any fixed 𝑗,𝐶₆⁻¹1𝜆≤𝜇(𝑉_𝑗)_𝑉𝑗|𝑓(𝑦)|𝑑𝜇(𝑦)≤𝐶₆𝜆 (3.8)with 𝐶₆>1 a constant independent of 𝑓 and 𝑗, which together with the Lebesgue differentiation theorem and Lemma 3.1 again yields that‖𝑔‖_𝐿^∞_(𝒳)≤𝐶₆𝜆. (3.9)Let Ω=∪_𝑗(𝐶₇𝐵_𝑗) with 𝐶₇=𝐶₁𝐶₄. The doubling property of 𝜇 and (3.1) now state that𝑤(Ω)≲_𝑗𝑤(𝐶₇𝐵_𝑗)𝜇(𝐶₇𝐵_𝑗)𝜇(𝐵_𝑗1)≲𝜆_𝑗inf_{𝑥∈𝐵𝑗}𝑀𝑤(𝑥)_𝐵𝑗1|𝑓(𝑦)|𝑑𝜇(𝑦)≲𝜆_𝒳|𝑓(𝑦)|𝑀𝑤(𝑦)𝑑𝜇(𝑦). (3.10)Following an argument similar to the case of Euclidean spaces (see [18, page 159]), we have that for any 𝛾≥0, there exists a positive constant 𝐶, depending only on 𝛾, such that for all 𝑥∈𝐶₄𝐵_𝑗,𝑀_{𝐿(log𝐿)}^𝛾(𝑤𝜒_𝒳⧵Ω)(𝑥)≲inf_{𝑦∈𝐶4𝐵𝑗}𝑀_{𝐿(log𝐿)}^𝛾(𝑤𝜒_𝒳⧵Ω)(𝑦). (3.11)Thus,|𝑚_𝑉𝑗(𝑓)|_𝑉𝑗𝑀_{𝐿(log𝐿)}^𝑘+𝛿(𝑤𝜒_𝒳⧵Ω)(𝑥)𝑑𝜇(𝑥)≲𝜆𝜇(𝑉_𝑗)inf_{𝑦∈𝐵𝑗}𝑀_{𝐿(log𝐿)}^𝑘+𝛿(𝑤𝜒_𝒳⧵Ω)(𝑦)≲𝜆𝜇(𝐵_𝑗)inf_{𝑦∈𝐵𝑗}𝑀_{𝐿(log𝐿)}^𝑘+𝛿(𝑤𝜒_𝒳⧵Ω≲)(𝑦)_𝐵𝑗|𝑓(𝑥)|𝑀_{𝐿(log𝐿)}^𝑘+𝛿(𝑤𝜒_𝒳⧵Ω)(𝑥)𝑑𝜇(𝑥). (3.12)For each fixed 𝛿>0, choose 𝑝₀∈(1,∞) and 𝛿₁>0 such that 𝑘𝑝₀+𝛿₁<𝑘+𝛿. From the last estimate, (2.11), Theorem 1.1, and (3.9), it follows that𝑤({𝑥∈𝒳⧵Ω∶𝑀^{0𝑥002𝑑𝑐}_𝑏,𝑘𝑔(𝑥)>𝜆/2})≲𝜆^−𝑝0_𝒳|𝑔(𝑥)|^𝑝0𝑀_{𝐿(log𝐿)01}^{𝑘𝑝+𝛿}(𝑤𝜒_𝒳⧵Ω)(𝑥)𝑑𝜇(𝑥)≲𝜆⁻¹_𝒳|𝑔(𝑥)|𝑀_{𝐿(log𝐿)01}^{𝑘𝑝+𝛿}(𝑤𝜒_𝒳⧵Ω)(𝑥)𝑑𝜇(𝑥)≲𝜆⁻¹(_{𝒳⧵∪𝑗𝑉𝑗}|𝑓(𝑥)|𝑀_{𝐿(log𝐿)}^𝑘+𝛿(𝑤𝜒_𝒳⧵Ω)(𝑥)𝑑𝜇(𝑥)+_𝑗|𝑚_𝑉𝑗(𝑓)|_𝑉𝑗𝑀_{𝐿(log𝐿)}^𝑘+𝛿(𝑤𝜒_𝒳⧵Ω)(𝑥)𝑑𝜇(𝑥))≲𝜆⁻¹_𝒳|𝑓(𝑥)|𝑀_{𝐿(log𝐿)}^𝑘+𝛿𝑤(𝑥)𝑑𝜇(𝑥). (3.13) Thus, our proof is now reduced to proving𝑤({𝑥∈𝒳⧵Ω∶𝑀^{0𝑥002𝑑𝑐∗}_𝑏,𝑘𝜆ℎ(𝑥)>2})≲∫_𝒳|𝑓(𝑥)|𝜆log^𝑘(e+|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)}^𝑘+𝛿𝑤(𝑥)𝑑𝜇(𝑥), (3.14)where ^0𝑥𝑀^{002𝑑𝑐∗}_𝑏,𝑘ℎ(𝑥)=sup_𝜖>0|_𝒳𝑆_𝜖(𝑥,𝑦)|𝑏(𝑥)−𝑏(𝑦)|^𝑘ℎ(𝑦)𝑑𝜇(𝑦)|. (3.15)
For any 𝑗≥1, let^0𝑥𝑀^{002𝑑𝑐∗}ℎ_𝑗(𝑥)=sup_𝜖>0|_𝒳𝑆_𝜖(𝑥,𝑦)ℎ_𝑗(𝑦)𝑑𝜇(𝑦)|. (3.16)We now prove (3.14). With the aid of the formula that for all 𝑥,𝑦∈𝒳,(𝑏(𝑥)−𝑏(𝑦))^𝑘=^𝑘_𝑙=0𝐶^𝑙_𝑘(𝑏(𝑥)−𝑚_𝐵𝑗(𝑏))^𝑙(𝑚_𝐵𝑗(𝑏)−𝑏(𝑦))^𝑘−𝑙, (3.17)since 𝑘 is even, for 𝑥∈𝒳⧵Ω, we write^0𝑥𝑀^{002𝑑𝑐∗}_𝑏,𝑘ℎ(𝑥)≲_𝑗|𝑏(𝑥)−𝑚_𝐵𝑗(𝑏)|𝑀^{𝑘0𝑥002𝑑𝑐∗}ℎ_𝑗(𝑥)+𝑀^𝑘−1_𝑙=0^{0𝑥002𝑑𝑐}_𝑏,𝑙(_𝑗(𝑏(⋅)−𝑚_𝐵𝑗(𝑏))^𝑘−𝑙ℎ_𝑗)(𝑥)≡G(𝑥)+H(𝑥). (3.18)Recall that {𝑉_𝑗}_𝑗 are mutually disjoint. If we set Φ_𝑙(𝑡)=𝑡log^𝑙(e+𝑡), our inductive hypothesis (3.7) via (3.11) now tells us that𝜆𝑤({𝑥∈𝒳⧵Ω∶H(𝑥)>4≲})_{𝑘−1𝑙=0𝑗𝑉𝑗}Φ_𝑙(|𝑏(𝑦)−𝑚_𝐵𝑗(𝑏)|^𝑘−𝑙|𝑓(𝑦)|𝜆)𝑑𝜇(𝑦)inf_{𝑧∈𝐶4𝐵𝑗}𝑀_{𝐿(log𝐿)𝑙0𝑥002𝑑𝑐+𝛿}(𝑤𝜒_𝒳⧵Ω+)(𝑧)_{𝑘−1𝑙=0𝑗𝑉𝑗}Φ_𝑙(|𝑏(𝑦)−𝑚_𝐵𝑗(𝑏)|^𝑘−𝑙|𝑚_𝑉𝑗(𝑓)|𝜆)𝑑𝜇(𝑦)inf_{𝑧∈𝐵𝑗}𝑀_{𝐿(log𝐿)𝑙0𝑥002𝑑𝑐+𝛿}(𝜒_𝒳⧵Ω𝑤)(𝑧). (3.19)An application of Lemma 3.2 then gives that_𝑉𝑗Φ_𝑙(|𝑏(𝑦)−𝑚_𝐵𝑗(𝑏)|^𝑘−𝑙|𝑓(𝑦)|𝜆)𝑑𝜇(𝑦)≲_𝑉𝑗Φ_𝑙(|𝑏(𝑦)−𝑚_{𝐶4𝐵𝑗}(𝑏)|^𝑘−𝑙|𝑓(𝑦)|𝜆)𝑑𝜇(𝑦)+Φ_𝑙(|𝑚_{𝐶4𝐵𝑗}(𝑏)−𝑚_𝐵𝑗(𝑏)|^𝑘−𝑙)_𝑉𝑗Φ_𝑙(|𝑓(𝑦)|𝜆≲)𝑑𝜇(𝑦)_𝑉𝑗exp(|𝑏(𝑦)−𝑚_{𝐶4𝐵𝑗}(𝑏)|𝐶₂)𝑑𝜇(𝑥)+_𝑉𝑗Φ_𝑘(|𝑓(𝑥)|𝜆≲)𝑑𝜇(𝑥)_𝑉𝑗Φ_𝑘(|𝑓(𝑥)|𝜆)𝑑𝜇(𝑥). (3.20)For each fixed 𝑗, notice that by (3.8) and Lemma 3.2, _𝑉𝑗Φ_𝑙(|𝑏(𝑦)−𝑚_𝐵𝑗(𝑏)|^𝑘−𝑙|𝑚_𝑉𝑗(𝑓)|𝜆)𝑑𝜇(𝑦)≲𝜇(𝐵_𝑗). (3.21)It then follows that𝜆𝑤({𝑥∈𝒳⧵Ω∶H(𝑥)>4})≲_𝑗𝑉𝑗Φ_𝑘(|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)}^𝑘+𝛿𝑤(𝑥)𝑑𝜇(𝑥)+_𝑗𝜇(𝐵_𝑗)inf_{𝑦∈𝐵𝑗}𝑀_{𝐿(log𝐿)}^𝑘+𝛿≲𝑤(𝑦)_𝒳Φ_𝑘(|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)}^𝑘+𝛿𝑤(𝑥)𝑑𝜇(𝑥). (3.22)
It remains to prove that𝜆𝑤({𝑥∈𝒳⧵Ω∶G(𝑥)>4})≲𝜆⁻¹_𝒳|𝑓(𝑥)|𝑀_{𝐿(log𝐿)}^𝑘𝑤(𝑥)𝑑𝜇(𝑥). (3.23)For each fixed 𝑗, let 𝑦_𝑗 and 𝑟_𝑗 be the center and radius of 𝐵_𝑗, respectively. If 𝑥∈𝒳⧵Ω, then by the vanishing moment of ℎ_𝑗 and the estimate (2.14), we obtain that^0𝑥𝑀^{002𝑑𝑐∗}ℎ_𝑗(𝑥)≤sup_𝜖>0𝒳|𝑆_𝜖(𝑥,𝑦)−𝑆_𝜖(𝑥,𝑦_𝑗)||ℎ_𝑗(𝑦)|𝑑𝜇(𝑦)≲(𝑟_𝑗)^𝛾{𝑑(𝑥,𝑦_𝑗)}^−𝛾𝜇(𝐵(𝑦_𝑗,𝑑(𝑥,𝑦_𝑗)))_𝒳|ℎ_𝑗(𝑦)|𝑑𝜇(𝑦). (3.24)This in turn implies that𝜆𝑤({𝑥∈𝒳⧵Ω∶G(𝑥)>4≲1})𝜆_𝑗(𝑟_𝑗)^𝛾_{𝒳⧵𝐶7𝐵𝑗}|𝑏(𝑥)−𝑚_𝐵𝑗(𝑏)|^𝑘𝜇(𝐵(𝑦_𝑗,𝑑(𝑥,𝑦_𝑗)))𝑤(𝑥){𝑑(𝑥,𝑦_𝑗)}^𝛾𝑑𝜇(𝑥)_𝒳|ℎ_𝑗≤1(𝑦)|𝑑𝜇(𝑦)𝜆_𝑗(𝑟_𝑗)^𝛾∞_{𝑙=1(2𝑙𝐶7𝐵𝑗)⧵(2𝑙−1𝐶7𝐵𝑗)}|𝑏(𝑥)−𝑚_𝐵𝑗(𝑏)|^𝑘𝑤(𝑥)𝜇(𝐵(𝑦_𝑗,𝑑(𝑥,𝑦_𝑗)))(𝑑(𝑥,𝑦_𝑗))^𝛾𝑑𝜇(𝑥)_𝒳|ℎ_𝑗≲1(𝑦)|𝑑𝜇(𝑦)𝜆_𝑗𝒳|ℎ_𝑗(𝑦)|𝑑𝜇(𝑦)inf_{𝑦∈𝐶7𝐵𝑗}𝑀_{𝐿(log𝐿)}^𝑘1𝑤(𝑦)≲𝜆_𝒳|𝑓(𝑥)|𝑀_{𝐿(log𝐿)}^𝑘𝑤(𝑥)𝑑𝜇(𝑥), (3.25)where in the second to the last inequality, we use the fact that for each fixed 𝑗, 𝑦_𝑗∈𝑉_𝑗 and positive integer 𝑙, a standard argument involving the inequalities (1.22) and (2.18) yields_{(2𝑙𝐶7𝐵𝑗)⧵(2𝑙−1𝐶7𝐵𝑗)}|𝑏(𝑥)−𝑚_𝐵𝑗(𝑏)|^𝑘𝑤(𝑥)𝜇(𝐵(𝑦_𝑗,𝑑(𝑥,𝑦_𝑗)))(𝑑(𝑥,𝑦_𝑗))^𝛾𝑑𝜇(𝑥)≲𝑙^𝑘(2^𝑙𝑟_𝑗)^−𝛾inf_{𝑦∈𝐶7𝐵𝑗}𝑀_{𝐿(log𝐿)}^𝑘𝑤(𝑦); (3.26)see also the proof of (2.25). We then complete the proof of Theorem 1.2.

4. Proof of Theorem 1.3

This section is devoted to the proof of Theorem 1.3.Lemma 4.1. Let 𝑇 be a Calderón-Zygmund operator. Then, there exists a positive constant 𝐶 such that for all 𝜆>0, 𝑓∈𝐿^∞_𝑏(𝒳), and weights 𝑤,_{{𝑥∈𝒳∶𝑇∗𝑓(𝑥)>𝜆}}𝑤(𝑥)𝑑𝜇(𝑥)≤𝐶_𝒳|𝑓(𝑥)|𝜆𝑀_{𝐿(l𝑜𝑔𝐿)}^1+𝛿𝑤(𝑥)𝑑𝜇(𝑥). (4.1)Lemma 4.1 can be proved by a similar but more careful argument as that used in the proof of Theorem 1.2 in [8]. We omit the proof here for brevity.

Proof of Theorem 1.3. The argument here is similar to that used in the proof of Theorem 1.2, and we will only give an outline. Also, we proceed our proof by an inductive argument. By Lemma 4.1, it is obvious that (1.19) is true when 𝑘=0. Now let 𝑘 be a positive integer. For any fixed 𝛿>0, and any nonnegative integer 𝑙 with 0≤𝑙≤𝑘−1, we assume that for all 𝜆>0 and 𝑓∈𝐿^∞_𝑏(𝒳), 𝑤({𝑥∈𝒳∶𝑇^∗_𝑏,𝑙𝑓(𝑥)>𝜆})≲_𝒳|𝑓(𝑥)|𝜆log^𝑙(e+|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)}^𝑙+1+𝛿𝑤(𝑥)𝑑𝜇(𝑥). (4.2) We need only consider the case that 𝜆>‖𝑓‖_𝐿¹_(𝒳)[𝜇(𝒳)]⁻¹. For each fixed bounded function 𝑓 with bounded support and 𝜆>‖𝑓‖_𝐿¹_(𝒳)[𝜇(𝒳)]⁻¹, applying Lemma 3.1 to |𝑓| at level 𝜆, with the same notation {𝐵_𝑗}_𝑗, {𝑉_𝑗}_𝑗, Ω as in the proof of Theorem 1.2, we decompose 𝑓≡𝑔+ℎ, where 𝑔≡𝑓𝜒_{𝒳⧵∪𝑗𝑉𝑗}+∑_𝑗𝑚_𝑉𝑗(𝑓)𝜒_𝑉𝑗 and ℎ≡∑_𝑗ℎ_𝑗 with ℎ_𝑗≡(𝑓−𝑚_𝑉𝑗(𝑓))𝜒_𝑉𝑗. Applying the estimate (1.17), and a similar argument to that used to deal with the term ^0𝑥𝑀^002𝑑𝑐_𝑏,𝑘𝑔 gives us that𝑤({𝑥∈(𝒳⧵Ω)∶𝑇^∗_𝑏,𝑘𝜆𝑔(𝑥)>2})≲𝜆^−𝑝0_𝒳|𝑔(𝑥)|^𝑝0𝑀_{𝐿(log𝐿)01}^{(𝑘+1)𝑝+𝛿}(𝑤𝜒_𝒳⧵Ω)(𝑥)𝑑𝜇(𝑥)≲𝜆⁻¹_𝒳|𝑓(𝑥)|𝑀_{𝐿(log𝐿)}^𝑘+1+𝛿𝑤(𝑥)𝑑𝜇(𝑥), (4.3)where 𝑝₀∈(1,∞) and 𝛿₁>0 such that (𝑘+1)𝑝₀+𝛿₁<𝑘+1+𝛿.
We now turn to the term 𝑇^∗_𝑏,𝑘ℎ. For any 𝑥∈𝒳⧵Ω and 𝜖>0, setI₁(𝑥,𝜖)={𝑗∶∀𝑦∈𝐶₄𝐵_𝑗I,𝑑(𝑥,𝑦)≤𝜖},₂(𝑥,𝜖)={𝑗∶∀𝑦∈𝐶₄𝐵_𝑗I,𝑑(𝑥,𝑦)>𝜖},₃(𝑥,𝜖)={𝑗∶(𝐶₄𝐵_𝑗)∩{𝑦∈𝒳∶𝑑(𝑥,𝑦)>𝜖}≠∅,(𝐶₄𝐵_𝑗)∩{𝑦∈𝒳∶𝑑(𝑥,𝑦)≤𝜖}≠∅}. (4.4) It then follows that|𝑇_{𝜖;𝑏,𝑘}ℎ(𝑥)|≤|𝑇_{𝜖;𝑏,𝑘}(_{𝑗∈I2(𝑥,𝜖)}ℎ_𝑗)(𝑥)|+|𝑇_{𝜖;𝑏,𝑘}(_{𝑗∈I3(𝑥,𝜖)}ℎ_𝑗)(𝑥)|≲|_{𝑗∈I2(𝑥,𝜖)}(𝑏(𝑥)−𝑚_𝐵𝑗(𝑏))^𝑘𝑇_𝜖ℎ_𝑗(𝑥)|+^𝑘−1_𝑙=0|𝑇_{𝜖;𝑏,𝑙}(_𝑗(𝑏(⋅)−𝑚_𝐵𝑗(𝑏))^𝑘−𝑙ℎ_𝑗+)(𝑥)|_{𝑘−1𝑙=0}|𝑇_{𝜖;𝑏,𝑙}(_{𝑗∈I3(𝑥,𝜖)}(𝑏(⋅)−𝑚_𝐵𝑗(𝑏))^𝑘−𝑙ℎ_𝑗)(𝑥)|+|𝑇_{𝜖;𝑏,𝑘}(_{𝑗∈I3(𝑥,𝜖)}ℎ_𝑗)(𝑥)|≡U_𝜖(𝑥)+V_𝜖(𝑥)+W_𝜖(𝑥)+X_𝜖(𝑥). (4.5)Notice that for 𝑥∈(𝒳⧵Ω) and 𝑗∈I₂(𝑥,𝜖), we have that 𝑇_𝜖ℎ_𝑗(𝑥)=𝑇ℎ_𝑗(𝑥). By the vanishing moment of ℎ_𝑗 and the regularity condition (1.13), we havesup_𝜖>0U_𝜖(𝑥)≤_𝑗|𝑏(𝑥)−𝑚_𝐵𝑗(𝑏)|^𝑘_𝒳(𝑑(𝑦,𝑦_𝑗))^𝜏𝜇(𝐵(𝑦,𝑑(𝑥,𝑦)))(𝑑(𝑥,𝑦))^𝜏|ℎ_𝑗(𝑦)|𝑑𝜇(𝑦) (4.6)and so𝑤({𝑥∈(𝒳⧵Ω)∶sup_𝜖>0U_𝜖𝜆(𝑥)>6})≲𝜆⁻¹_𝑗𝒳|ℎ_𝑗(𝑦)|(𝑑(𝑦,𝑦_𝑗))^𝜏_{𝒳⧵𝐶7𝐵𝑗}|𝑏(𝑥)−𝑚_𝐵𝑗(𝑏)|^𝑘𝑤(𝑥)𝜇(𝐵(𝑦,𝑑(𝑥,𝑦)))(𝑑(𝑥,𝑦))^𝜏𝑑𝜇(𝑥)𝑑𝜇(𝑦)≲𝜆⁻¹_𝒳|𝑓(𝑥)|𝑀_{𝐿(log𝐿)}^𝑘𝑤(𝑥)𝑑𝜇(𝑥). (4.7)Our inductive hypothesis (4.2), via the argument for the term H in the proof of Theorem 1.2, leads to𝑤({𝑥∈(𝒳⧵Ω)∶sup_𝜖>0V_𝜖𝜆(𝑥)>4})≲_𝒳|𝑓(𝑥)|𝜆log^𝑘(e+|𝑓(𝑥)|𝜆)𝑀_{𝐿(log𝐿)}^𝑘+1+𝛿𝑤(𝑥)𝑑𝜇(𝑥). (4.8)Notice that for 𝑥∈𝒳⧵Ω and 𝑗∈I₃(𝑥,𝜖), we have that 𝐶₄𝐵_𝑗⊂{𝐵(𝑥,𝐶₈𝜖)⧵𝐵(𝑥,𝐶₉𝜖)}, where 𝐶₈ and 𝐶₉ with 𝐶₈>𝐶₉ are two positive constants. Therefore, for all 𝑥∈𝒳⧵Ω,sup_𝜖>0(W_𝜖(𝑥)+X_𝜖(𝑥))≲^𝑘_𝑙=0𝑀_𝑏,𝑙(_𝑗|𝑏−𝑚_𝐵𝑗(𝑏)|^𝑘−𝑙|ℎ_𝑗|)(𝑥). (4.9)This, along with Theorem 1.2 and an argument for the term H in the proof of Theorem 1.2, leads to 𝑤({𝑥∈(𝒳⧵Ω)∶sup_𝜖>0(W_𝜖(𝑥)+X_𝜖𝜆(𝑥))>6≲})_{𝑘𝑙=0𝑗𝑉𝑗}Φ_𝑙(|𝑏(𝑦)−𝑚_𝐵𝑗(𝑏)|^𝑘−𝑙|ℎ_𝑗(𝑦)|𝜆)𝑀_{𝐿(log𝐿)𝑙0𝑥002𝑑𝑐+𝛿}(𝑤𝜒_𝜒⧵Ω)(𝑦)𝑑𝜇(𝑦)≲_𝒳|𝑓(𝑦)|𝜆log^𝑘(e+|𝑓(𝑦)|𝜆)𝑀_{𝐿(log𝐿)𝑘0𝑥002𝑑𝑐+𝛿}𝑤(𝑦)𝑑𝜇(𝑦), (4.10)where ^0𝑥00𝑙^2𝑑𝑐=𝑙 when 𝑙 is even and ^0𝑥00𝑙^2𝑑𝑐=𝑙+1 when 𝑙 is odd. Combining the estimates for the terms sup_𝜖>0U_𝜖, sup_𝜖>0V_𝜖, and sup_𝜖>0(W_𝜖+X_𝜖) gives us that𝑤({𝑥∈(𝒳⧵Ω)∶𝑇^∗_𝑏,𝑘𝜆ℎ(𝑥)>2})≲_𝒳|𝑓(𝑦)|𝜆log^𝑘(e+|𝑓(𝑦)|𝜆)𝑀_{𝐿(log𝐿)}^𝑘+1+𝛿𝑤(𝑦)𝑑𝜇(𝑦), (4.11)which completes the proof of Theorem 1.3.

Acknowledgments

The authors would like to thank the referees for their many helpful suggestions and corrections, which improve the presentation of this paper. The first author is supported by National Natural Science Foundation of China (Grant no. 10671210), and the third author is supported by National Science Foundation for Distinguished Young Scholars (Grant no. 10425106) and NCET (Grant no. 04-0142) of Ministry of Education of China.