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

Authors: Guoen Hu; Haibo Lin; Dachun Yang
Journal: Abstract and Applied Analysis
 (2008-02-03)

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## Abstract

WeightedLp 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.


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## Body

## 1. Introduction

We  will be working on a space of homogeneous type. Let𝒳 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,(1.1)μ(B(x,2r))≤Cμ(B(x,r)).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,(1.2)μ(B(x,λr))≤Cλnμ(B(x,r)).Moreover, there also exist
constants C>0 and N∈[0,n] such that for
all x, y∈𝒳 and r>0,(1.3)μ(B(y,r))≤C(1+d(x,y)r)Nμ(B(x,r)).We remark that although all balls defined byd 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 d˜ such that(i)
there exists a
constantC≥1 such that for
all x,y∈𝒳, C−1d˜(x,y)≤d(x,y)≤Cd˜(x,y);(ii)
there exist
constantsC>0 and γ∈(0,1] such that for
all x,x′,y∈𝒳,(1.4)|d˜(x,y)−d˜(x′,y)|≤C(d˜(x,x′))γ(d˜(x,y)+d˜(x′,y))1−γ.
The balls corresponding to d˜ are open in the
topology induced by d˜. 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,(1.5)|d(x,y)−d(x′,y)|≤C(d(x,x′))γ(d(x,y)+d(x′,y))1−γ,and that the balls B(x,r) for all x∈𝒳 and r>0 are open.Now letk be a positive
integer and b∈BMO(𝒳), define the kth-order
commutator Mb,k of the
Hardy-Littlewood maximal operator with b by(1.6)Mb,kf(x)=sup⁡B϶x1μ(B)∫B|b(x)−b(y)|k|f(y)|dμ(y)for all x∈𝒳. For the case that (𝒳,d,μ) is the
Euclidean space, García-Cuerva et al. [3] proved that Mb,k is bounded on Lp(ℝn) for any p∈(1,∞), and Alphonse [4] proved that Mb,1 enjoys a
weak-type L(log⁡L) estimate, that
is, there exists a positive constant C, depending on ∥b∥BMO(ℝn), such that for all suitable functions f,(1.7)|{x∈ℝn:Mb,1f(x)>λ}|≤C∫ℝn|f(x)|λlog⁡(e+|f(x)|λ)dx.Li et al. [5] established a weighted estimate
with any general weight for Mb,1 in ℝn. As it was shown in [3–5] for the setting of Euclidean spaces, the operator Mb,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 Mb,k in spaces of
homogeneous type. To state our results, we first give some notation.LetE be a measurable
set with μ(E)<∞. For any fixed p∈[1,∞), δ>0, and suitable
function f, set(1.8)∥f∥Lp(log⁡L)δ,E=inf⁡{λ>0:1μ(E)∫E(|f(x)|λ)plog⁡δ(e+|f(x)|λ)dμ(x)≤1}.The maximal operator MLp(log⁡L)δ is defined by MLp(log⁡L)δf(x)=sup⁡B϶x∥f∥Lp(log⁡L)δ,B, where the
supremum is taken over all balls containing x. In the following, we denote ML1(log⁡L)δ by ML(log⁡L)δ for simplicity,
and denote by Lb∞(𝒳) the set of
bounded functions with bounded support.With the notation above, we now formulate our main
results as follows.Theorem 1.1.
Letk 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∈Lb∞(𝒳),(1.9)∫𝒳(Mb,kf(x))pw(x)dμ(x)≤C∥b∥BMO(𝒳)kp∫𝒳|f(x)|pML(log⁡L)kp+δw(x)dμ(x).Theorem 1.2.
Letk be a positive
integer, b∈BMO(𝒳), and δ>0. There exists a positive constant C=Ck,∥b∥BMO(𝒳) such that for
all nonnegative weights w, f∈Lb∞(μ) and λ>0,(1.10)w({x∈𝒳:Mb,kf(x)>λ})≤C∫𝒳|f(x)|λlog⁡k(e+|f(x)|λ)ML(log⁡L)k˜+δw(x)dμ(x),where, and in the
following, k˜=k if k is even and k˜=k+1 if k is odd.As a corollary of Theorem1.2, 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,(1.11)Tf(x)=∫𝒳K(x,y)f(y)dμ(y),where K is a locally
integrable function on 𝒳×𝒳∖{(x,y):x=y} and satisfies
that for all x≠y,(1.12)|K(x,y)|≤Cμ(B(x,d(x,y))),and that for all x,y,y′∈𝒳 with d(x,y)≥2d(y,y′),(1.13)|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))τwith positive constants C and τ≤1. For any ϵ>0, suitable function f and x∈𝒳, define the truncated operator Tϵ by(1.14)Tϵf(x)=∫d(x,y)>ϵK(x,y)f(y)dμ(y).Let b∈BMO(𝒳) and let k be a positive
integer. Define the commutator Tϵ;b,k by(1.15)Tϵ;b,0f(x)=Tϵf(x),Tϵ;b,kf(x)=b(x)Tϵ;b,k−1f(x)−Tϵ;b,k−1(bf)(x)for all x∈𝒳 and f∈Lb∞(𝒳). The maximal operator associated with the commutator Tb,k is defined
by(1.16)Tb,k*f(x)=sup⁡ϵ>0|Tϵ;b,kf(x)|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∈Lb∞(𝒳) and all
nonnegative weights w,(1.17)∫𝒳|Tb,k∗f(x)|pw(x)dμ(x)≤C∥b∥BMO(μ)kp∫𝒳|f(x)|pML(log⁡L)(k+1)p+δw(x)dμ(x).In [5], it was proved that in ℝn, Tb,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(ℝn), such that(1.18)w({x∈ℝn:Tb,1∗f(x)>λ})≤C∫ℝn|f(x)|λlog⁡(e+|f(x)|λ)ML(log⁡)2+δw(x)dx.Using Theorem1.2, we will prove the following
result.Theorem 1.3.
LetT 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∈Lb∞(𝒳) and nonnegative
weights w,(1.19)∫{x∈𝒳:Tb,k∗f(x)>λ}w(x)dμ(x)≤C∫𝒳|f(x)|λlog⁡k(e+|f(x)|λ)ML(log⁡L)k+1+δw(x)dμ(x).We mention that Theorems1.1, 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 byC 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, χE 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 mE(f) the mean value
of f over E, that is, mE(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 by(1.20)M#f(x)=sup⁡B϶x1μ(B)∫B|f(y)−mB(f)|dμ(y),where the supremum is taken over
all balls B containing x. For any fixed q∈(0,1), the sharp maximal function Mq#f of the function f is defined by Mq#f=(M#(|f|q))1/q.A generalization of Hölder's inequality will be used
in the proofs of our theorems. For any measurable setE with μ(E)<∞, positive integer l, and suitable function f, set(1.21)∥f∥exp⁡L1/l,E=inf⁡{λ>0:1μ(E)∫Eexp⁡(|f(x)|λ)1/ldμ(x)≤2}.Then the following generalization
of Hölder's inequality:(1.22)1μ(E)∫E|f(x)h(x)|dμ(x)≤C∥f∥L(log⁡L)l,E∥h∥exp⁡L1/l,Eholds for any suitable functions f and h; see [7] for details.


## 2. Proof of Theorem1.1

To prove
Theorem1.1, 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 Ms(f)=[M(|f|s)]1/s.Lemma 2.1 (see [8]).
There exists a positive constantC such that for
all weights w and all
nonnegative functions f satisfying μ({x∈𝒳:f(x)>λ})<∞ for all λ>0, then(i)
ifμ(𝒳)=∞,(2.1)∫𝒳f(x)w(x)dμ(x)≤C∫𝒳M#f(x)Mw(x)dμ(x);(ii)
ifμ(𝒳)<∞,(2.2)∫𝒳f(x)w(x)dμ(x)≤C∫𝒳M#f(x)Mw(x)dμ(x)+Cw(𝒳)m𝒳(f).Lemma 2.2.
For anyq∈(0,1), there exists a positive constant C such that for
all f∈Lp(𝒳) with p∈[1,∞) and all x∈𝒳, Mq#(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.
Letp∈(1,∞) and let k be a positive
integer.(a)
There exists a
positive constantC, depending only on k and p, such that for all f∈Lb∞(𝒳) and all weights w,(2.3)∫𝒳[ML(log⁡L)kf(x)]pw(x)dμ(x)≤C∫𝒳|f(x)|pMw(x)dμ(x).(b)
For anyδ1>0, there exists a positive constant C, depending only on k, p, and δ1, such that for all f∈Lb∞(𝒳) and all weights w,(2.4)∫𝒳(Mkf(x))p[ML(log⁡L)kp′−1+δ1w(x)]1−pdμ(x)≤C∫𝒳|f(x)|p[w(x)]1−pdμ(x).For Euclidean spaces, Lemma2.3(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, ML(log⁡L)kw≈Mk+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 anyx∈𝒳 and r>0, set Vr(x)=μ(B(x,r)).Lemma 2.4.
Letγ be as in
(1.5). Then
there exists an approximation of the identity {Sk}k∈ℤ of order γ with bounded
support on 𝒳. Namely, {Sk}k∈ℤ is a sequence
of bounded linear integral operators on L2(𝒳), and there exist constants C0, C˜>0 such that for
all k∈ℤ and all x, x′, y, and y′∈𝒳,Sk(x,y), the integral
kernel of Sk is a measurable
function from 𝒳×𝒳 into ℂ satisfying
(i)
Sk(x,y)=0 if d(x,y)≥C˜2−k and 0≤Sk(x,y)≤C0(1/(V2−k(x)+V2−k(y)));(ii)
Sk(x,y)=Sk(y,x) for all x,y∈𝒳;(iii)
|Sk(x,y)−Sk(x′,y)|≤C02kγd(x,x′)γ(1/(V2−k(x)+V2−k(y))) for d(x,x′)≤max⁡{C˜/κ, 1/κ}21−k;(iv)
C0V2−k(x)Sk(x,x)>1 for all x∈𝒳 and k∈ℤ;(v)
∫𝒳Sk(x,y)dμ(y)=1=∫𝒳Sk(x,y)dμ(x).For anyϵ>0 and x,y∈𝒳, let(2.5)Sϵ(x,y)≡Sk(x,y)χ(2−k−1,2−k](ϵ).Obviously, Sϵ 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˜/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,(2.6)CVϵ(x)Sϵ(x,y)>1.For a positive integer k and a function b∈BMO(𝒳), let M˜b,k be the operator
defined by(2.7)M˜b,kf(x)=sup⁡ϵ>0M˜ϵ;b,kf(x)for all f∈Lb∞(𝒳) and x∈𝒳, where for ϵ>0,(2.8)M˜ϵ;b,kf(x)=∫𝒳Sϵ(x,y)|b(x)−b(y)|k|f(y)|dμ(y).If k=0, we denote M˜b,k and M˜ϵ;b,k simply by M˜ and M˜ϵ, respectively. From (i) of Lemma 2.4 together with
(1.1), it
follows that Sϵ(x,y)≲1/Vϵ(x)≲1/V2C˜ϵ(x). Notice that if d(x,y)≥2C˜ϵ, then Sϵ(x,y)=0. Thus,(2.9)M˜b,kf(x)=sup⁡ϵ>0M˜ϵ;b,kf(x)≲sup⁡ϵ>01V2C˜ϵ(x)∫B(x,2C˜ϵ)|b(x)−b(y)|k|f(y)|dμ(y)≲Mb,kf(x).On the other hand, for each
fixed ϵ>0, by (2.6) and VC′ϵ(x)∼Vϵ(x), we have(2.10)1VC′ϵ(x)∫B(x,C′ϵ)|b(x)−b(y)|k|f(y)|dμ(y)≲∫𝒳Sϵ(x,y)|b(x)−b(y)|k|f(y)|dμ(y)≲M˜b,kf(x).By the definition of Mb,k, we further obtain Mb,kf(x)≲M˜b,kf(x). Thus, there exists some constant C≥1 such that for
all x∈𝒳 and f∈Lb∞(𝒳),(2.11)C−1M˜b,kf(x)≤Mb,kf(x)≤CM˜b,kf(x).For the sharp function estimate
of M˜b,k, we have the following estimate.
Lemma 2.5.
Letk 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∈Lb∞(𝒳) and all x∈𝒳,(2.12)Mq#(M˜b,kf)(x)≤C∑j=0k−1∥b∥BMO(𝒳)k−jMs(M˜b,jf)(x)+C∥b∥BMO(𝒳)kML(log⁡L)kf(x).Proof.
By (i), (ii), and (iii) of Lemma2.4, we obtain that for all x,y∈𝒳,(2.13)|Sϵ(x,y)|≲1μ(B(x,d(x,y))),and that for all ϵ>0 and all x,y,y′∈𝒳 with d(x,y)≥2κd(y,y′),(2.14)|Sϵ(x,y)−Sϵ(x,y′)|+|Sϵ(y,x)−Sϵ(y′,x)|≲1μ(B(x,d(x,y)))(d(y,y′)d(x,y))γ.
To verify (2.12), by homogeneity, we may assume that ∥b∥BMO(𝒳)=1. For all f∈Lb∞(𝒳), x∈𝒳, and balls B containing x, it suffices to prove that(2.15)inf⁡c∈ℂ(1μ(B)∫B|M˜b,kf(y)−c|qdμ(y))1/q≲∑j=0k−1Ms(M˜b,jf)(x)+ML(log⁡L)kf(x).We consider the following three
cases. Case 1 (μ(𝒳∖C1B)=0).
Where and in what followsC1=κ(4κ+1). In this case, we have that for all x∈𝒳,(2.16)M˜b,kf(x)≲∑j=0k−1|mB(b)−b(x)|k−jM˜b,jf(x)+M˜((b−mB(b))kf)(x).The Kolmogorov inequality (see
[15, page 102]), along
with the fact that M (and so M˜) is bounded
from L1(𝒳) to L1,∞(𝒳) and the
inequality (1.22) gives us that(2.17){1μ(B)∫B[M˜((b−mB(b))kf)(y)]qdμ(y)}1/q≲1μ(B)∫C1B|b(y)−mB(b)|k|f(y)|dμ(y)≲1μ(B)∫C1B|b(y)−mC1B(b)|k|f(y)|dμ(y)+|mC1B(b)−mB(b)|1μ(B)∫C1B|f(y)|dμ(y)≲ML(log⁡L)kf(x),where the last inequality
follows from the John-Nirenberg inequality, which states that for any ball
Q,(2.18)∥|mQ(b)−b|k∥exp⁡L1/k,Q≲∥b∥BMO(𝒳)k.On the other hand, if 0<q<s<1, an application of Hölder's inequality implies
that(2.19){1μ(B)∫B[M˜b,kf(x)]qdμ(x)}1/q≲∑j=0k−1Ms(M˜b,jf)(x)+{1μ(B)∫B[M˜((b−mB(b))kf)(y)]qdμ(y)}1/q.We then get (2.15).Case 2 (μ(𝒳∖C1B)≠0 and μ(C1B∖B)>0).
In this case, decomposef into f=fχC1B+fχ𝒳∖C1B≡f1+f2, recalling that χE denotes the
characteristic function of the set E. Let y0 be a point in B such that(2.20)CB≡sup⁡ϵ>0∫𝒳Sϵ(y0,z)|mB(b)−b(z)|k|f2(z)|dz<∞.With the aid of the formula(2.21)(mB(b)−b(z))k=(b(y)−b(z))k+∑j=0k−1Ckj(b(y)−b(z))j(mB(b)−b(y))k−j,where Ckj is the constant
from Newton's formula, we have(2.22)||b(y)−b(z)|k|f(z)|−|mB(b)−b(z)|k|f2(z)||≲∑j=0k−1|b(y)−b(z)|j|mB(b)−b(y)|k−j|f(z)|+|mB(b)−b(z)|k|f1(z)|.Thus for any y∈B,(2.23)|M˜b,kf(y)−CB|=|sup⁡ϵ>0|M˜ϵ;b,kf(y)|−sup⁡ϵ>0|M˜ϵ((mB(b)−b)kf2)(y0)||≤sup⁡ϵ>0|M˜ϵ;b,kf(y)−M˜ϵ((mB(b)−b)kf2)(y0)|≲M˜((mB(b)−b)kf1)(y)+∑j=0k−1|mB(b)−b(y)|k−jM˜b,jf(y)+sup⁡ϵ>0|M˜ϵ((mB(b)−b)kf2)(y)−M˜ϵ((mB(b)−b)kf2)(y0)|≡I(y)+II(y)+III(y).As in Case 1, we have that(2.24){1μ(B)∫B|I(y)|qdμ(y)}1/q≲ML(log⁡L)kf(x),{1μ(B)∫B|II(y)|qdμ(y)}1/q≲∑j=0k−1Ms(M˜b,jf)(x). As for III(y), by (2.14) and (1.22), it is easy to get(2.25)III(y)≤sup⁡ϵ>0∫𝒳|Sϵ(y,z)−Sϵ(y0,z)||mB(b)−b(z)|k|f2(z)|dμ(z)≲∫𝒳∖C1B(d(y,y0))γμ(B(y,d(y,z)))(d(y,z))γ|mB(b)−b(z)|k|f(z)|dz≲∑l=1∞2−lγ1μ(2lC1B)∫2lC1B|m2lC1B(b)−b(z)|k|f(z)|dμ(z)+∑l=1∞2−lγ|m2lC1B(b)−mB(b)|k1μ(2lC1B)∫2lC1B|f(z)|dμ(z)≲∑l=1∞2−lγ∥|m2lC1B(b)−b|k∥exp⁡L1/k,2lC1B∥f∥L(log⁡L)k,2lC1B+∑l=1∞2−lγ|m2lC1B(b)−mB(b)|kMf(x)≲ML(log⁡L)kf(x),where the last inequality
follows from (2.18) and |m2lQ(b)−mQ(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 takeB′ such that B⊂B′, μ(B′)=μ(B), and μ(C1B′∖B′)>0. We then have that(2.26)inf⁡c∈ℂ1μ(B)∫B|M˜b,kf(x)−c|qdμ(x)≤inf⁡c∈ℂ1μ(B′)∫B′|M˜b,kf(x)−c|qdμ(x).With the ball B replaced by B′ in Case 2,
we also obtain the result that for any y∈B′,(2.27)sup⁡ϵ>0∫𝒳|Sϵ(y,z)−Sϵ(y0,z)||mB′(b)−b(z)|k|f2(z)|dz≤CML(log⁡L)kf(x),which completes the proof of
Lemma 2.5.Lemma 2.6.
Letα,β∈[0,∞). There exists a positive constant C, depending only on α and β, such that for all weights w,(2.28)ML(log⁡L)α(ML(log⁡L)βw)(x)≤CML(log⁡L)α+β+1w(x).For Euclidean spaces, a generalization of Lemma2.6 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,(2.29)μ({x∈𝒳:ML(log⁡L)αf(x)>λ})≲∫𝒳|f(x)|λlog⁡α(e+|f(x)|λ)dμ(x).Using this, Lemma 2.6 can be proved by
applying the ideas used in [16]. For details, see [6, Lemma 7].Proof  of Theorem1.1.
We assume again that∥b∥BMO(𝒳)=1. At first, we claim that when μ(𝒳)=∞, for all λ>0 and f∈Lb∞(𝒳), μ({x∈𝒳:M˜b,kf(x)>λ})<∞. In fact, for
any f∈Lb∞(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), (2.30)Mf(x)≲∥f∥L1(𝒳)μ(B(x,d(x,x0))). It then follows that for p∈(1,∞),(2.31)μ({x∈𝒳∖B(x0,3R):|b(x)−mB(x0,R)(b)|kMf(x)>λ})≤λ−p∫𝒳∖B(x0,3R)|b(x)−mB(x0,R)(b)|kp(Mf(x))pdμ(x)≤λ−p∥f∥L1(𝒳)p∫𝒳∖B(x0,3R)|b(x)−mB(x0,d(x,x0))(b)|kp{μ(B(x,d(x,x0)))}pdμ(x)<∞.This, together with the estimate
that(2.32)μ({x∈𝒳:M((b−mB(x0,R)(b))kf)(x)>λ})≲λ−p∥|b−mB(x0,R)(b)|kf∥Lp(𝒳)p<∞,leads to our claim.
By (2.11), to prove Theorem 1.1, it suffices to prove that
for all weights w,(2.33)∫𝒳(M˜b,kf(x))pw(x)dμ(x)≲∫𝒳|f(x)|pML(log⁡L)kp+δw(x)dμ(x).We proceed our proof by an
inductive argument on k. When k=0, (2.33) is implied by the fact that Mw(x)≤ML(log⁡L)δw(x) for all x∈𝒳 and the
following known inequality:(2.34)∫𝒳[M˜f(x)]pw(x)dμ(x)≲∫𝒳|f(x)|pMw(x)dμ(x).See [18, pages 150-151],  for a proof
of the last inequality when 𝒳=ℝn. 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 M˜b,l  (0≤l≤k−1) can extend to a
bounded operator on Lp(𝒳) for p∈(1,∞) and so for any λ>0 and σ∈(0,1),(2.35)μ({x∈𝒳:M((M˜b,lf)σ)(x)>λ})<∞.
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∈Lb∞(𝒳),(2.36)∫𝒳(M˜b,kf(x))qh(x)dμ(x)≲∫𝒳(ML(log⁡L)kf(x))qMkh(x)dμ(x).We first consider the case that μ(𝒳)=∞. Choose r1,…,rk−1, rk such that 0<q=r0<r1<⋯<rk−1<rk<1. By Lemma 2.5, we obtain that for any 1≤m≤k−1 and any weight h,(2.37)∫𝒳(Mrj#(M˜b,mf)(x))qh(x)dμ(x)≲∑l=0m−1∫𝒳(Mrj+1(M˜b,lf)(x))qh(x)dμ(x)+∫𝒳(ML(log⁡L)mf(x))qh(x)dμ(x).Therefore, applying Lemmas 2.1 and 2.2, and the
estimate (2.35),
we have(2.38)∫𝒳(Mrj+1(M˜b,lf)(x))qh(x)dμ(x)=∫𝒳(M((M˜b,lf)rj+1)(x))q/rj+1h(x)dμ(x)≲∫𝒳(Mq/rj+1#(M((M˜b,lf)rj+1)(x)))q/rj+1Mh(x)dμ(x)≲∫𝒳(M#((M˜b,lf)rj+1(x)))q/rj+1Mh(x)dμ(x)=∫𝒳(Mrj+1#(M˜b,lf)(x))qMh(x)dμ(x),which leads to(2.39)∫𝒳(Mrj#(M˜b,mf)(x))qh(x)dμ(x)≲∑l=0m−1∫𝒳(Mrj+1#(M˜b,lf)(x))qMh(x)dμ(x)+∫𝒳(ML(log⁡L)mf(x))qh(x)dμ(x).Repeating the argument above k−1 times, we then
have that for all weights h,(2.40)∫𝒳(Mq#(M˜b,kf)(x))qh(x)dμ(x)≲∑j=0k−1∫𝒳(Mr1#(M˜b,jf)(x))qMh(x)dμ(x)+∫𝒳(ML(log⁡L)kf(x))qh(x)dμ(x)≲∑j=0k−2∫𝒳(Mr2#(M˜b,jf)(x))qM2h(x)dμ(x)+∑j=0k−1∫𝒳(ML(log⁡L)jf(x))qMh(x)dμ(x)+∫𝒳(ML(log⁡L)kf(x))qh(x)dμ(x)≲∑j=01∫𝒳(Mrk−1#(M˜b,jf)(x))qMk−1h(x)dμ(x)+∑j=0k∫𝒳(ML(log⁡L)jf(x))qMk−1h(x)dμ(x).On the other hand, notice that
for all x∈𝒳, Mrk−1#(M˜b,0f)(x)≲M2f(x), and that, by
(2.12) and the
fact that M2f(x)∼ML(log⁡L)f(x) for all x∈𝒳 (see [12, (4.11)]), we then have
that for all x∈𝒳, Mrk−1#(M˜b,1f)(x)≲Mrk(M˜b,0f)(x)+ML(log⁡L)f(x)≲M2f(x).
From these
inequalities, it then follows that(2.41)∫𝒳(Mq#(M˜b,kf)(x))qh(x)dμ(x)≲∫𝒳(M2f(x))qMk−1h(x)dμ(x)+∑j=0k∫𝒳(ML(log⁡L)jf(x))qMk−1h(x)dμ(x)≲∫𝒳(ML(log⁡L)kf(x))qMk−1h(x)dμ(x),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,(2.42)M˜b,lf(x)≲∑j=0l|b(x)−m𝒳(b)|jM˜((m𝒳(b)−b)l−jf)(x).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),(2.43)1μ(𝒳)∫𝒳|b(x)−m𝒳(b)|rj(M˜((m𝒳(b)−b)l−jf)(x))rdμ(x)≲{1μ(𝒳)∫𝒳[M˜((m𝒳(b)−b)l−j)f(x)]tdμ(x)}r/t≲{1μ(𝒳)∫𝒳|f(x)||m𝒳(b)−b(x)|l−jdμ(x)}r≲(∥f∥L(log⁡L)l−j,𝒳)r.Combining the above estimates,
we obtain(2.44)m𝒳((M˜b,lf)r)≲(inf⁡x∈𝒳ML(log⁡L)lf(x))r.Let q, r1,r2,…,rk be as in the
case of μ(𝒳)=∞. Another application of Kolmogorov inequality and the
fact that M is bounded from L1(𝒳) to L1,∞(𝒳) leads to(2.45){m𝒳(M(M˜b,lf)rj)q/rj}rj/q≲m𝒳((M˜b,lf)rj)≲(inf⁡x∈𝒳ML(log⁡L)lf(x))rj.As in the case of μ(𝒳)=∞, by Lemmas 2.1, 2.2, and 2.5, we have that for any q∈(0,1),(2.46)∫𝒳(M˜b,kf)qh(x)dμ(x)≲∫𝒳(Mq#(M˜b,kf)(x))qMh(x)dμ(x)+h(𝒳)m𝒳((M˜b,kf)q)≲∑j=0k−1∫𝒳(Mr1#(M˜b,jf)(x))qM2h(x)dμ(x)+∫𝒳(ML(log⁡L)kf(x))qMh(x)dμ(x)+∑j=0k−1(Mh)(𝒳)m𝒳(M(M˜b,jf)r1)q/r1+h(𝒳)m𝒳((M˜b,kf)q)≲∫𝒳(ML(log⁡L)kf(x))qMkh(x)dμ(x).Combining the two cases yields
(2.36).
For any fixedp∈(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(2.47)∥(M˜b,kf)q∥Lp/q(w)=sup⁡∥h∥L(p/q)′(w1−(p/q)′)≤1|∫𝒳[M˜b,kf(x)]qh(x)dμ(x)|≲sup⁡∥h∥L(p/q)′(w1−(p/q)′)≤1∫𝒳[ML(log⁡L)kf(x)]qMkh(x)dμ(x)≲sup⁡∥h∥L(p/q)′(w1−(p/q)′)≤1{∫𝒳[ML(log⁡L)kf(x)]pML(log⁡L)kp/q−1+δ1w(x)dμ(x)}q/p×{∫𝒳[Mkh(x)](p/q)′[ML(log⁡L)kp/q−1+δ1w(x)]1−(p/q)′dμ(x)}1/(p/q)′≲{∫𝒳[ML(log⁡L)kf(x)]pML(log⁡L)kp/q−1+δ1w(x)dμ(x)}q/p≲{∫𝒳|f(x)|pML(log⁡L)kp+δw(x)dμ(x)}q/p,where in the last inequality we
have used Lemma  2.6. This completes the proof of Theorem 1.1.


## 3. Proof of Theorem1.2

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>m𝒳(f)(m𝒳(f)=0 if μ(𝒳)=∞), there exist a sequence of pairwise disjoint balls {Bj}j≥1 and a constant C4≥1 such
that(3.1)mC4Bj(f)≤λ<mBj(f)and mB(f)≤λ for every ball B centered at x∈𝒳∖(∪jC4Bj).Lemma 3.2.
Letd and l be two
nonnegative integers. Then for all t1,t2≥0,(3.2)t1t2dlog⁡(e+t1t2d)≤C(t1log⁡d+l(e+t1)+exp⁡t2).Proof.
We may assume thatd≥1, otherwise the conclusion holds obviously. Set Φ1(t)=tlog⁡l(e+t), Φ2(t)=tlog⁡l+d(e+t), and Φ3(t)=exp⁡(t1/d). Let j=1,2,3. Denote by Φj−1 the inverse of Φj, that is, Φj−1(t)=inf⁡{s>0:Φj(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(t1t2d)≲Φ2(t1)+Φ3(t2d). Our desired
conclusion then follows directly.Proof  of Theorem1.2.
With the notationM˜b,k as in (2.7), by (2.11), it suffices to
prove that for ∥b∥BMO(𝒳)=1 and all f∈Lb∞(𝒳) and λ>0, (3.3)w({x∈𝒳:M˜b,kf(x)>λ})≲∫𝒳|f(x)|λlog⁡k(e+|f(x)|λ)ML(log⁡L)k˜+δw(x)dμ(x), where k˜=k when k is even and k˜=k+1 when k is odd.
Recall that for allf∈Lb∞(𝒳),(3.4)w({x∈𝒳:M˜f(x)>λ})≲1λ∫𝒳|f(x)|Mw(x)dμ(x).(See [18, page 151] for a proof when 𝒳=ℝn. The same idea also works for 𝒳.) By Hölder's inequality, it follows that for all x∈𝒳,(3.5)M˜b,kf(x)≤(M˜b,k+1f(x))k/(k+1)(M˜f(x))1/(k+1)and so when k is
odd,(3.6)w({x∈𝒳:M˜b,kf(x)>λ})≤w({x∈𝒳:M˜b,k+1f(x)>λ})+w({x∈𝒳:M˜f(x)>λ})≲w({x∈𝒳:M˜b,k+1f(x)>λ})+1λ∫𝒳|f(x)|Mw(x)dμ(x).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)≤ML(log⁡L)δw(x) for all x∈𝒳 and (3.4). Now let k be a positive
integer. We may assume that ML(log⁡L)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=Cl,δ such that for
all λ>0,(3.7)w({x∈𝒳:M˜b,lf(x)>λ})≲∫𝒳|f(x)|λlog⁡l(e+|f(x)|λ)ML(log⁡L)l˜+δw(x)dμ(x),where and in what follows, l˜=l when l is even and l˜=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 {Bj}j≥1 with pairwise
disjoint interiors. As in the proof of Lemma 2.10 in [17], set V1=(C4B1)∖(∪n≥2Bn) and Vj=(C4Bj)∖[∪n=1j−1Vn∪∪l≥j+1Bl], it then follows that Bj⊂Vj⊂(C4Bj) and ∪jVj=∪j(C4Bj). Define the functions g and h, respectively, by g≡|f|χ𝒳∖∪jVj+∑jmVj(|f|)χVj and h≡∑jhj with hj≡(|f|−mVj(|f|))χVj. Recall that μ is regular and
the set of continuous function is dense in Lp(𝒳) for any p∈[1,∞). Lemma 3.1 implies that for any fixed j,(3.8)C6−1λ≤1μ(Vj)∫Vj|f(y)|dμ(y)≤C6λwith C6>1 a constant
independent of f and j, which together with the Lebesgue differentiation
theorem and Lemma 3.1 again yields that(3.9)∥g∥L∞(𝒳)≤C6λ.Let Ω=∪j(C7Bj) with C7=C1C4. The doubling property of μ and (3.1) now state
that(3.10)w(Ω)≲∑jw(C7Bj)μ(C7Bj)μ(Bj)≲1λ∑jinf⁡x∈BjMw(x)∫Bj|f(y)|dμ(y)≲1λ∫𝒳|f(y)|Mw(y)dμ(y).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∈C4Bj,(3.11)ML(log⁡L)γ(wχ𝒳∖Ω)(x)≲inf⁡y∈C4BjML(log⁡L)γ(wχ𝒳∖Ω)(y).Thus,(3.12)|mVj(f)|∫VjML(log⁡L)k+δ(wχ𝒳∖Ω)(x)dμ(x)≲λμ(Vj)inf⁡y∈BjML(log⁡L)k+δ(wχ𝒳∖Ω)(y)≲λμ(Bj)inf⁡y∈BjML(log⁡L)k+δ(wχ𝒳∖Ω)(y)≲∫Bj|f(x)|ML(log⁡L)k+δ(wχ𝒳∖Ω)(x)dμ(x).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 that(3.13)w({x∈𝒳∖Ω:M˜b,kg(x)>λ/2})≲λ−p0∫𝒳|g(x)|p0ML(log⁡L)kp0+δ1(wχ𝒳∖Ω)(x)dμ(x)≲λ−1∫𝒳|g(x)|ML(log⁡L)kp0+δ1(wχ𝒳∖Ω)(x)dμ(x)≲λ−1(∫𝒳∖∪jVj|f(x)|ML(log⁡L)k+δ(wχ𝒳∖Ω)(x)dμ(x)+∑j|mVj(f)|∫VjML(log⁡L)k+δ(wχ𝒳∖Ω)(x)dμ(x))≲λ−1∫𝒳|f(x)|ML(log⁡L)k+δw(x)dμ(x).
Thus, our proof is now reduced to
proving(3.14)w({x∈𝒳∖Ω:M˜b,k∗h(x)>λ2})≲∫𝒳|f(x)|λlog⁡k(e+|f(x)|λ)ML(log⁡L)k+δw(x)dμ(x),where
(3.15)M˜b,k∗h(x)=sup⁡ϵ>0|∫𝒳Sϵ(x,y)|b(x)−b(y)|kh(y)dμ(y)|.
For anyj≥1, let(3.16)M˜∗hj(x)=sup⁡ϵ>0|∫𝒳Sϵ(x,y)hj(y)dμ(y)|.We now prove (3.14). With the aid of
the formula that for all x,y∈𝒳,(3.17)(b(x)−b(y))k=∑l=0kCkl(b(x)−mBj(b))l(mBj(b)−b(y))k−l,since k is even, for x∈𝒳∖Ω, we write(3.18)M˜b,k∗h(x)≲∑j|b(x)−mBj(b)|kM˜∗hj(x)+∑l=0k−1M˜b,l(∑j(b(⋅)−mBj(b))k−lhj)(x)≡G(x)+H(x).Recall that {Vj}j are mutually
disjoint. If we set Φl(t)=tlog⁡l(e+t), our inductive hypothesis (3.7) via (3.11) now tells us
that(3.19)w({x∈𝒳∖Ω:H(x)>λ4})≲∑l=0k−1∑j∫VjΦl(|b(y)−mBj(b)|k−l|f(y)|λ)dμ(y)inf⁡z∈C4BjML(log⁡L)l˜+δ(wχ𝒳∖Ω)(z)+∑l=0k−1∑j∫VjΦl(|b(y)−mBj(b)|k−l|mVj(f)|λ)dμ(y)inf⁡z∈BjML(log⁡L)l˜+δ(χ𝒳∖Ωw)(z).An application of Lemma 3.2 then
gives that(3.20)∫VjΦl(|b(y)−mBj(b)|k−l|f(y)|λ)dμ(y)≲∫VjΦl(|b(y)−mC4Bj(b)|k−l|f(y)|λ)dμ(y)+Φl(|mC4Bj(b)−mBj(b)|k−l)∫VjΦl(|f(y)|λ)dμ(y)≲∫Vjexp⁡(|b(y)−mC4Bj(b)|C2)dμ(x)+∫VjΦk(|f(x)|λ)dμ(x)≲∫VjΦk(|f(x)|λ)dμ(x).For each fixed j, notice that by (3.8) and Lemma
3.2,
(3.21)∫VjΦl(|b(y)−mBj(b)|k−l|mVj(f)|λ)dμ(y)≲μ(Bj).It then follows
that(3.22)w({x∈𝒳∖Ω:H(x)>λ4})≲∑j∫VjΦk(|f(x)|λ)ML(log⁡L)k+δw(x)dμ(x)+∑jμ(Bj)inf⁡y∈BjML(log⁡L)k+δw(y)≲∫𝒳Φk(|f(x)|λ)ML(log⁡L)k+δw(x)dμ(x).
It remains to prove that(3.23)w({x∈𝒳∖Ω:G(x)>λ4})≲λ−1∫𝒳|f(x)|ML(log⁡L)kw(x)dμ(x).For each fixed j, let yj and rj be the center
and radius of Bj, respectively. If x∈𝒳∖Ω, then by the vanishing moment of hj and the
estimate (2.14), we obtain that(3.24)M˜∗hj(x)≤sup⁡ϵ>0∫𝒳|Sϵ(x,y)−Sϵ(x,yj)||hj(y)|dμ(y)≲(rj)γ{d(x,yj)}−γμ(B(yj,d(x,yj)))∫𝒳|hj(y)|dμ(y).This in turn implies
that(3.25)w({x∈𝒳∖Ω:G(x)>λ4})≲1λ∑j(rj)γ∫𝒳∖C7Bj|b(x)−mBj(b)|kμ(B(yj,d(x,yj)))w(x){d(x,yj)}γdμ(x)∫𝒳|hj(y)|dμ(y)≤1λ∑j(rj)γ∑l=1∞∫(2lC7Bj)∖(2l−1C7Bj)|b(x)−mBj(b)|kw(x)μ(B(yj,d(x,yj)))(d(x,yj))γdμ(x)∫𝒳|hj(y)|dμ(y)≲1λ∑j∫𝒳|hj(y)|dμ(y)inf⁡y∈C7BjML(log⁡L)kw(y)≲1λ∫𝒳|f(x)|ML(log⁡L)kw(x)dμ(x),where in the second to the last
inequality, we use the fact that for each fixed j, yj∈Vj and positive
integer l, a standard argument involving the inequalities
(1.22) and
(2.18)
yields(3.26)∫(2lC7Bj)∖(2l−1C7Bj)|b(x)−mBj(b)|kw(x)μ(B(yj,d(x,yj)))(d(x,yj))γdμ(x)≲lk(2lrj)−γinf⁡y∈C7BjML(log⁡L)kw(y);see also the proof of (2.25). We then
complete the proof of Theorem 1.2.


## 4. Proof of Theorem1.3

This section is devoted to the proof of Theorem1.3.Lemma 4.1.
LetT be a
Calderón-Zygmund operator. Then, there exists a positive constant C such that for
all λ>0, f∈Lb∞(𝒳), and weights w,(4.1)∫{x∈𝒳:T∗f(x)>λ}w(x)dμ(x)≤C∫𝒳|f(x)|λML(logL)1+δw(x)dμ(x).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 Theorem1.3.
The argument here is similar to that used in
the proof of Theorem1.2, 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∈Lb∞(𝒳), (4.2)w({x∈𝒳:Tb,l∗f(x)>λ})≲∫𝒳|f(x)|λlog⁡l(e+|f(x)|λ)ML(log⁡L)l+1+δw(x)dμ(x).
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 {Bj}j, {Vj}j, Ω as in the proof
of Theorem 1.2, we decompose f≡g+h, where g≡fχ𝒳∖∪jVj+∑jmVj(f)χVj and h≡∑jhj with hj≡(f−mVj(f))χVj. Applying the estimate
(1.17), and a
similar argument to that used to deal with the term M˜b,kg gives us
that(4.3)w({x∈(𝒳∖Ω):Tb,k∗g(x)>λ2})≲λ−p0∫𝒳|g(x)|p0ML(log⁡L)(k+1)p0+δ1(wχ𝒳∖Ω)(x)dμ(x)≲λ−1∫𝒳|f(x)|ML(log⁡L)k+1+δw(x)dμ(x),where p0∈(1,∞) and δ1>0 such that (k+1)p0+δ1<k+1+δ.
We now turn to the termTb,k∗h. For any x∈𝒳∖Ω and ϵ>0, set(4.4)I1(x,ϵ)={j:∀y∈C4Bj,d(x,y)≤ϵ},I2(x,ϵ)={j:∀y∈C4Bj,d(x,y)>ϵ},I3(x,ϵ)={j:(C4Bj)∩{y∈𝒳:d(x,y)>ϵ}≠∅,(C4Bj)∩{y∈𝒳:d(x,y)≤ϵ}≠∅}. It then  follows
that(4.5)|Tϵ;b,kh(x)|≤|Tϵ;b,k(∑j∈I2(x,ϵ)hj)(x)|+|Tϵ;b,k(∑j∈I3(x,ϵ)hj)(x)|≲|∑j∈I2(x,ϵ)(b(x)−mBj(b))kTϵhj(x)|+∑l=0k−1|Tϵ;b,l(∑j(b(⋅)−mBj(b))k−lhj)(x)|+∑l=0k−1|Tϵ;b,l(∑j∈I3(x,ϵ)(b(⋅)−mBj(b))k−lhj)(x)|+|Tϵ;b,k(∑j∈I3(x,ϵ)hj)(x)|≡Uϵ(x)+Vϵ(x)+Wϵ(x)+Xϵ(x).Notice that for x∈(𝒳∖Ω) and j∈I2(x,ϵ), we have that Tϵhj(x)=Thj(x). By the vanishing moment of hj and the
regularity condition (1.13), we have(4.6)sup⁡ϵ>0Uϵ(x)≤∑j|b(x)−mBj(b)|k∫𝒳(d(y,yj))τμ(B(y,d(x,y)))(d(x,y))τ|hj(y)|dμ(y)and so(4.7)w({x∈(𝒳∖Ω):sup⁡ϵ>0Uϵ(x)>λ6})≲λ−1∑j∫𝒳|hj(y)|(d(y,yj))τ∫𝒳∖C7Bj|b(x)−mBj(b)|kw(x)μ(B(y,d(x,y)))(d(x,y))τdμ(x)dμ(y)≲λ−1∫𝒳|f(x)|ML(log⁡L)kw(x)dμ(x).Our inductive hypothesis (4.2),
via the argument for the term H in the proof of
Theorem 1.2, leads to(4.8)w({x∈(𝒳∖Ω):sup⁡ϵ>0Vϵ(x)>λ4})≲∫𝒳|f(x)|λlog⁡k(e+|f(x)|λ)ML(log⁡L)k+1+δw(x)dμ(x).Notice that for x∈𝒳∖Ω and j∈I3(x,ϵ), we have that C4Bj⊂{B(x,C8ϵ)∖B(x,C9ϵ)}, where C8 and C9 with C8>C9 are two
positive constants. Therefore, for all x∈𝒳∖Ω,(4.9)sup⁡ϵ>0(Wϵ(x)+Xϵ(x))≲∑l=0kMb,l(∑j|b−mBj(b)|k−l|hj|)(x).This, along with Theorem 1.2 and
an argument for the term H in the proof of
Theorem 1.2, leads to (4.10)w({x∈(𝒳∖Ω):sup⁡ϵ>0(Wϵ(x)+Xϵ(x))>λ6})≲∑l=0k∑j∫VjΦl(|b(y)−mBj(b)|k−l|hj(y)|λ)ML(log⁡L)l˜+δ(wχχ∖Ω)(y)dμ(y)≲∫𝒳|f(y)|λlog⁡k(e+|f(y)|λ)ML(log⁡L)k˜+δw(y)dμ(y),where l˜=l when l is even and l˜=l+1 when l is odd.
Combining the estimates for the terms supϵ>0Uϵ, sup⁡ϵ>0Vϵ, and sup⁡ϵ>0(Wϵ+Xϵ) gives us
that(4.11)w({x∈(𝒳∖Ω):Tb,k∗h(x)>λ2})≲∫𝒳|f(y)|λlog⁡k(e+|f(y)|λ)ML(log⁡L)k+1+δw(y)dμ(y),which completes the proof of
Theorem 1.3.



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*Source: 237937-2008-03-12.xml*