PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 183–208 DOI: 10.2298/PIM1614183L

WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF PARAMETERIZED LITTLEWOOD–PALEY OPERATORS

AND AREA INTEGRALS

Yan Lin and Xiao Xuan

Abstract. We establish the boundedness for commutators of parameterized Littlewood–Paley operators and area integrals on weighted Lebesgue spaces L^{p}(ω) when 1 < p < ∞, where the kernel satisfies certain logarithmic type Lipschitz condition. Moreover, the weighted endpoint estimates when p = 1 are also obtained.

1. Introduction

Suppose that S^{n−1} is the unit sphere of ℝ^{n}(n ⩾ 2) equipped with normalized Lebesgue measure. Let Ω be a homogeneous function of degree zero and

(1.1)  ∫_{S^{n−1}} Ω(x′) dσ(x′) = 0.

The parameterized area integral μ_{Ω,S}^{ρ} and parameterized Littlewood–Paley operator μ_{λ}^{∗,ρ} are defined by

μ_{Ω,S}^{ρ} (f)(x) = ( ∫∫_{Γ(x)} | \frac{1}{t^{ρ}} ∫_{|y−z|<t} \frac{Ω(y − z)}{|y − z|^{n−ρ}} f(z) dz|^{2} \frac{dy dt}{t^{n+1}}) ^{\frac{1}{2}} ,

μ_{λ}^{∗,ρ} (f)(x) = ( ∫∫_{ℝ_{+}^{n+1}} ( \frac{t}{t + |x − y|} ) ^{λn} | \frac{1}{t^{ρ}} ∫_{|y−z|<t} \frac{Ω(y − z)}{|y − z|^{n−ρ}} f(z) dz|^{2} \frac{dy dt}{t^{n+1}}) ^{\frac{1}{2}} ,

respectively, where ρ > 0, λ > 1 and Γ(x) = {(t, y) ∈ ℝ_{+}^{n+1} : |x − y| < t}.

2010 Mathematics Subject Classification: Primary 42B20; Secondary 42B25. Key words and phrases: parameterized Littlewood-Paley operator, parameterized area inte-

gral, commutator, BMO.

Partially supported by the National Natural Science Foundation of China (11171345), Beijing Higher Education Young Elite Teacher Project (YETP0946), and the Fundamental Research Funds for the Central Universities (2009QS16).

Communicated by Michael Ruzhansky.

Define the Hilbert spaces as follows.

H_{1} = \{ h : ∥h∥_{H_{1}} = ( ∫_{0}^{∞} ∫_{|y|<1} |h(t, y)|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}} < ∞\} ,

H_{2} = \{ h : ∥h∥_{H_{2}} = ( ∫_{0}^{∞} ∫_{ℝ^{n}} ( \frac{1}{1 + |y|} ) ^{λn} |h(t, y)|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}} < ∞\} ,

where λ > 1. Then

μ_{Ω,S}^{ρ} (f)(x) = ∥ϕ_{t,y}(f)(x)∥_{H_{1}} , μ_{λ}^{∗,ρ} (f)(x) = ∥ϕ_{t,y}(f)(x)∥_{H_{2}} ,

where ϕ(x) = \frac{Ω(x)}{|x|^{n−ρ}} χ_{\{x:|x|<1\}} , and ϕ_{t,y}(f)(x) = ∫ t^{−n} ϕ(\frac{x−z}{t} − y)f(z) dz.

The commutators of μ_{Ω,S}^{ρ} and μ_{λ}^{∗,ρ} are defined by

μ_{Ω,S}^{ρ,b} (f)(x) = ( ∫∫_{Γ(x)} | \frac{1}{t^{ρ}} ∫_{|y−z|<t} \frac{Ω(y − z)}{|y − z|^{n−ρ}} (b(x) − b(z))f(z) dz|^{2} \frac{dy dt}{t^{n+1}} ) ^{\frac{1}{2}} ,

μ_{λ,b}^{∗,ρ}(f)(x) = ( ∫∫_{ℝ_{+}^{n+1}} ( \frac{t}{t + |x − y|} ) ^{λn}

× | \frac{1}{t^{ρ}} ∫_{|y−z|<t} \frac{Ω(y − z)}{|y − z|^{n−ρ}} (b(x) − b(z))f(z) dz|^{2} \frac{dy dt}{t^{n+1}}) ^{\frac{1}{2}} ,

respectively.

Denote f_{Q} = \frac{1}{|Q|} ∫_{Q} f(y) dy. For δ > 0, we define

M_{δ}(f) = [M(|f|^{δ})]^{\frac{1}{δ}} , M_{δ}^{♯}(f) = [M^{♯}(|f|^{δ})]^{\frac{1}{δ}} ,

where M is the Hardy–Littlewood maximal operator and M^{♯} is the Fefferman–Stein sharp function. The corresponding dyadic maximal operators are denoted by M_{δ}^{Δ} and M_{δ}^{♯,Δ} , respectively.

A function A : [0, ∞) → [0, ∞) is said to be a Young function if it is continuous, convex and increasing satisfying A(0) = 0 and A(t) → ∞ as t → ∞. The complementary Young function Ā(t) of the Young function A(t) is defined by

Ā(s) = sup_{0⩽t<∞} [st − A(t)], 0 ⩽ s < ∞.

As an example, Φ_{m}(t) = t(1 + log^{+} t)^{m}, 1 ⩽ m < ∞, is a Young function with its complementary Φ̄_{m} (t) ≈ e^{t^{1/m}} . If A is a Young function, then the Luxembury norm of f on a cube Q ⊂ ℝ^{n} is defined by

∥f∥_{A,Q} = inf \{ λ > 0 : \frac{1}{|Q|} ∫_{Q} A( \frac{|f(y)|}{λ} ) dy ⩽ 1\} .

If A(t) = Φ_{1}(t), we denote

∥f∥_{L log L,Q} = ∥f∥_{Φ_{1} ,Q} , ∥f∥_{expL,Q} = ∥f∥_{Φ̄_{1} ,Q}, M_{L log L} f(x) = sup_{Q∋x} ∥f∥_{L log L,Q} .

For the Luxembury norm, there is the following generalized Hölder inequality.

\frac{1}{|Q|} ∫_{Q} |f(y)g(y)|dy ⩽ ∥f∥_{A,Q} ∥g∥_{Ā,Q}.

Let us recall the definition of A_{p} weight class. A locally integrable nonnegative function ω is said to belong to A_{p}(1 < p < ∞), if there is a constant C > 0 such that

sup_{Q⊂ℝ^{n}} ( \frac{1}{|Q|} ∫_{Q} ω(x) dx) ( \frac{1}{|Q|} ∫_{Q} ω(x)^{−1/(p−1)} dx) ^{p−1} ⩽ C,

where Q denotes a cube in ℝ^{n}. The smallest constant C such that the above inequality holds is called the A_{p} constant of ω and denoted by [ω]_{A_{p}} . A weight ω is said to be in the class A_{1} if there is a positive constant C such that Mω(x) ⩽ Cω(x), a.e. x ∈ ℝ^{n}. We denote by [ω]_{A_{1}} the infimum of all these C. A weight ω is in the class A_{∞} if there are positive constants C, ϵ such that

\frac{ω(E)}{ω(Q)} ⩽ C( \frac{|E|}{|Q|}) ^{ϵ},

for all cubes Q and all measurable sets E ⊂ Q. We denote by [ω]_{A_{∞}} the infimum of all these C.

Inspired by Hörmander’s work [7] on the parameterized Marcinkiewicz integral, the parameterized Littlewood–Paley g_{λ}^{∗} function μ_{λ}^{∗,ρ} and parameterized area integral μ_{Ω,S}^{ρ} were discussed by Sakamoto and Yabuta [12] in 1999. In [12], the authors studied the L^{p}(1 < p < ∞) boundedness with kernel satisfying the Lip_{α} condition. In 2002, Ding, Lu and Yabuta [2] proved the L^{p}(2 ⩽ p < ∞) boundedness of μ_{λ}^{∗,ρ} and μ_{Ω,S}^{ρ} with kernel satisfying a weaker Llog^{+} L(S^{n−1} ) condition.

Torchinsky and Wang [14] considered the weighted L^{p} boundedness of μ_{λ}^{∗,ρ} and μ_{Ω,S}^{ρ} with kernel satisfying the Lip_{α} condition. In 1999, Ding, Fan and Pan [1] improved Torchinsky and Wang’s result in [14] and gave the weighted L^{p} boundedness of μ_{λ}^{∗,ρ} and μ_{Ω,S}^{ρ} when Ω ∈ L^{q}(S^{n−1} )(q > 1). In 2002, Duoandikoetxea and Seijo [5] studied the weighed L^{p} boundedness of μ_{λ}^{∗,ρ} and μ_{Ω,S}^{ρ} with rough kernel. In 2004, Xue [15] proved the weighed L^{p} boundedness of μ_{λ}^{∗,ρ} and μ_{Ω,S}^{ρ} with Ω satisfying the L^{2}-Dini condition.

Lee and Rim [8], in 2004, established the logarithmic type Lipschitz condition

(1.2)  |Ω(y_{1}) − Ω(y_{2})| ⩽ \frac{C}{( log \frac{2}{|y_{2} −y_{1} |} )^{α}} ,

for any y_{1}, y_{2} ∈ S^{n−1} , where α > 1, and proved the type (L^{∞} , BMO) and (L^{p}, L^{p}) boundedness of the Marcinkiewicz integral with kernel satisfying the condition (1.2). In 2012, Lin, Liu and Gao [9] gave the endpoint estimate of μ_{λ}^{∗,ρ}.

Theorem 1.1. [9] Let n ⩾ 2, Ω ∈ L^{2}(S^{n−1} ) be a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > \frac{3}{2}, then for ρ > n/2, λ > 2 , there exists a constant C > 0, such that for all β > 0 and f ∈ L^{1}(ℝ^{n}),

|{x ∈ ℝ^{n} : |μ_{λ}^{∗,ρ} (f)(x) | > β }| ⩽ C∥f∥_{L^{1}} /β.

In 2013, the authors in [10] discussed the operators μ_{λ}^{∗,ρ} and μ_{Ω,S}^{ρ} with kernel satisfying (1.2) on weak Hardy spaces. Recently the authors in [11] gave the weighted L^{p} boundedness of μ_{Ω,S}^{ρ} and μ_{λ}^{∗,ρ} with kernel satisfying (1.2).

Theorem 1.2. [11] Let Ω ∈ L^{2}(S^{n−1} ) be a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > \frac{3}{2}. Suppose ω ∈ A_{p}, then for ρ > n/2, λ > 2 and 1 < p < ∞,

∥μ_{Ω,S}^{ρ} (f)∥_{p,ω} ⩽ C∥f∥_{p,ω} , ∥μ_{λ}^{∗,ρ} (f)∥_{p,ω} ⩽ C∥f∥_{p,ω} .

On the other hand, the boundedness of the commutator has also received increasing attentions. Torchinsky and Wang [14] in 1990 proved that if b ∈ BMO, then the commutator of the Marcinkiewicz integral [b, μ_{Ω} ] is bounded on weighted spaces L^{p}(ω) for 1 < p < ∞ and ω ∈ A_{p}. In 2002, Ding, Lu and Yabuta [2] gave the weighted L^{p} boundedness of the higher order commutator μ_{Ω,b}^{m} for rough Marcinkiewicz integral. In 2004, Ding, Lu and Zhang [3] gave the endpoint weighted esimates for the higher order commutator μ_{Ω,b}^{m} . In 2007, Ding and Xue [4] gave the weighted boundedness and the weak L logL estimates for the commutators of parameterized Littlewood–Paley operators and area integrals with kernel satisfying the L^{2}-Dini condition.

Inspired by the above results, in this paper, we will focus on the weighted L^{p} (1 < p < ∞) boundedness and the weighted endoint estimates (p = 1) for the commutators μ_{Ω,S}^{ρ,b} and μ_{λ,b}^{∗,ρ}, where the kernel satisfies the logarithmic type Lipschitz condition (1.2).

2. Main results

Now, we state our main results as follows.

Theorem 2.1. Suppose that ρ > n/2, λ > 2, Ω ∈ L^{2}(S^{n−1} ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > \frac{3}{2}. If b ∈ BMO, then for 1 < p < ∞ and ω ∈ A_{p}, there exists a constant C > 0 such that for any f ∈ L^{p}(ω),

∥μ_{Ω,S}^{ρ,b} (f)∥_{p,ω} ⩽ C∥f∥_{p,ω} , ∥μ_{λ,b}^{∗,ρ}(f)∥_{p,ω} ⩽ C∥f∥_{p,ω} .
Theorem 2.2. Suppose that ρ > n/2, λ > 2, Ω ∈ L^{2}(S^{n−1} ) is a homogeneous

function of degree zero satisfying (1.1) and (1.2) with α > \frac{5}{2}. If b ∈ BMO and ω ∈ A_{1}, then there exists a constant C > 0 such that for any β > 0 and each smooth function f with compact support, the following inequalities hold

ω(\{x ∈ ℝ^{n} : μ_{Ω,S}^{ρ,b} (f)(x) > β\}) ⩽ C ∫_{ℝ^{n}} \frac{|f(x)|}{β} ( 1 + log^{+} \frac{|f(x)|}{β} ) ω(x) dx,

ω(\{x ∈ ℝ^{n} : μ_{λ,b}^{∗,ρ}(f)(x) > β\}) ⩽ C ∫_{ℝ^{n}} \frac{|f(x)|}{β} ( 1 + log^{+} \frac{|f(x)|}{β} ) ω(x) dx.

WEIGHTED BOUNDEDNESS FOR COMMUTATORS... 187

3. Some lemmas

In order to prove the main results, we need the following necessary lemmas. Lemma 3.1. [16] Suppose f ∈ BMO. There exist constants C_{1}, C_{2} > 0,

depending only on the dimension n, such that for 0 < C < \frac{C_{2}}{∥f∥_{∗}} , every cube Q in ℝ^{n}, we have

∫_{Q} e^{C|f(x)−f_{Q} |} dx ⩽ C_{1}C( \frac{C_{2}}{∥f∥_{∗}} − C) ^{−1} |Q|.

Lemma 3.2. Let 1 < p < ∞ and λ′ > 0. Then, when b(x) ∈ BMO with ∥b∥_{∗} < min\{\frac{C_{2}}{λ′} , \frac{C_{2} (p−1)}{λ′} \}, where C_{2} is the constant in Lemma 3.1, we have e^{λ′b(x)} ∈ A_{p}.

Proof. We have

( \frac{1}{|Q|} ∫_{Q} e^{λ′b(x)} dx) ( \frac{1}{|Q|} ∫_{Q} ( e^{λ′b(x)} )^{− \frac{1}{p−1}} dx) ^{p−1}
= ( \frac{1}{|Q|} ∫_{Q} e^{λ′(b(x)−b_{Q} )} dx) ( \frac{1}{|Q|} ∫_{Q} ( e^{−λ′(b(x)−b_{Q} )} ) ^{\frac{1}{p−1}} dx) ^{p−1}
⩽( \frac{1}{|Q|} ∫_{Q} e^{λ′|b(x)−b_{Q} |} dx) ( \frac{1}{|Q|} ∫_{Q} e^{\frac{λ′}{p−1} |b(x)−b_{Q} |} dx) ^{p−1} := I_{Q}.

Let λ_{0} = \frac{λ′}{p−1} . If 1 < p < 2, then λ_{0} > λ′. By taking C = λ_{0} in Lemma 3.1, we have

I_{Q} ⩽ ( \frac{1}{|Q|} ∫_{Q} e^{λ_{0} |b(x)−b_{Q} |} dx) ( \frac{1}{|Q|} ∫_{Q} e^{λ_{0} |b(x)−b_{Q} |} dx) ^{p−1}

= ( \frac{1}{|Q|} ∫_{Q} e^{λ_{0} |b(x)−b_{Q} |} dx) ^{p} ⩽ ( \frac{C_{1}λ_{0}}{\frac{C_{2}}{∥b∥_{∗}} − λ_{0}} ) ^{p}.

If p ⩾ 2, then λ_{0} ⩽ λ′. By taking C = λ′ in Lemma 3.1, we have

I_{Q} ⩽ ( \frac{1}{|Q|} ∫_{Q} e^{λ′|b(x)−b_{Q} |} dx) ( \frac{1}{|Q|} ∫_{Q} e^{λ′|b(x)−b_{Q} |} dx) ^{p−1}

= ( \frac{1}{|Q|} ∫_{Q} e^{λ′|b(x)−b_{Q} |} dx) ^{p} ⩽ ( \frac{C_{1}λ′}{\frac{C_{2}}{∥b∥_{∗}} − λ′} ) ^{p}.

The two cases imply the desired result. □ Remark 3.1. It follows from Lemma 3.2 that if 1 < p < ∞, λ′ > 0 and

a(x), b(x) ∈ BMO with ∥a∥_{∗} ⩽ ∥b∥_{∗} < min\{ \frac{C_{2}}{λ′} , \frac{C_{2} (p−1)}{λ′} \}, then e^{λ′a(x)} ∈ A_{p} and the A_{p} constant of e^{λ′a(x)} satisfies

[e^{λ′a(x)} ]_{A_{p}} ⩽ \{ \begin{matrix}( _{\frac{C_{1} λ_{0}}{\frac{C_{2}}{∥b∥∗} −λ_{0}}} ) ^{p}, & 1 < p < 2, \\ ( \frac{C_{1} λ′}{\frac{C_{2}}{∥b∥∗} −λ′} ) ^{p}, & 2 ⩽ p < ∞. \end{matrix}

Lemma 3.3. Suppose b ∈ BMO. There is a positive constant C such that for all ball B ⊂ ℝ^{n},

∥b − b_{B}∥_{expL,B} ⩽ C∥b∥_{∗}.
Proof. When C > \frac{1}{C_{2}} , by Lemma 3.1, we have

\frac{1}{|B|} ∫_{B} e^{\frac{|b(x)−b_{B} |}{C∥b∥∗}} dx ⩽ C_{1} \frac{1}{C∥b∥_{∗}}( \frac{C_{2}}{∥b∥_{∗}} − \frac{1}{C∥b∥_{∗}}) ^{−1} = \frac{C_{1}}{C_{2}C − 1} .

Taking C ⩾ \frac{C_{1} +1}{C_{2}} , we have \frac{C_{1}}{C_{2} C−1} ⩽ 1, then \frac{1}{|B|} ∫_{B} e^{\frac{|b(x)−b_{B} |}{C∥b∥∗}} dx ⩽ 1. By the definition of ∥b − b_{B}∥_{expL,B} , there is ∥b − b_{B}∥_{expL,B} ⩽ C∥b∥_{∗}. □

Lemma 3.4. [6] Let 0 < r < l < ∞. For each function f, define

∥f∥_{W L^{l}} = sup_{t>0}t|\{x : |f(x)| > t\}|^{\frac{1}{l}} , N_{l,r} (f) = sup_{E} \frac{∥fχ_{E}∥_{r}}{∥χ_{E}∥_{s}} , \frac{1}{s} = \frac{1}{r} − \frac{1}{l},

where the supremum is taken over all the measurable sets E with 0 < |E| < ∞. Then

∥f∥_{W L^{l}} ⩽ N_{l,r} (f) ⩽ ( \frac{l}{l − r} ) ^{\frac{1}{r}} ∥f∥_{W L^{l}} .

Lemma 3.5. [3] (1) Let M^{Δ} , M^{♯,Δ} be the dyadic Hardy–Littlewood maximal operator and the dyadic sharp function, respectively. If ω ∈ A_{∞} , then there exists a positive dimensional constant C for which the following good-λ inequality holds. For all λ, ε > 0,

ω({y ∈ ℝ^{n} : M^{Δ} f(y) > λ, M^{♯,Δ} f(y) ⩽ ελ})
⩽ C[ω]_{A_{∞}} εω({y ∈ ℝ^{n} : M^{Δ} f(y) > λ/2}).

(2) If φ : (0, ∞) → (0, ∞) is a doubling function and δ > 0, then there exists a positive constant C such that

sup_{λ>0}φ(λ)ω({y ∈ ℝ^{n} :M_{δ}^{Δ} f(y) > λ}) ⩽ C[ω]_{A_{∞}} sup_{λ>0}φ(λ)ω({y ∈ ℝ^{n} :M_{δ}^{♯,Δ} f(y) > λ})

for all functions f such that the left side is finite.

Lemma 3.6. [3] There exists a constant C > 0 such that for any weight ω and all β > 0,

ω(\{y : M^{m+1} f(y) > β\}) ⩽ C ∫_{ℝ^{n}} \frac{|f(y)|}{β} ( 1 + log^{+} \frac{|f(y)|}{β} ) ^{m}Mω(y) dy

for every locally integrable function f.

Lemma 3.7. As for |y − z| ⩾ 4r, there is

∫_{|y−z|}^{∞} \frac{(log \frac{t}{r})^{4+2ϵ}}{t^{2ρ−n+1}} dt ⩽ C \frac{[log(\frac{|y−z|}{r} )]^{4+2ϵ}}{(|y − z|)^{2ρ−n}} ,

where r > 0, 0 < ϵ < ρ − \frac{n}{2} and ρ > \frac{n}{2} .

The proof of this lemma is similar to that of Lemma 2.1.2 in [15], so we omit the details.

Lemma 3.8. [11] There exists a constant C > 0 such that for any z ∈ (8B^{∗})^{c}, |y − z| ⩾ 6r,

| \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(ω − x_{0} + y − z)}{|ω − x_{0} + y − z|^{n−ρ}}  | ⩽ \frac{C(1 + |Ω(y − z)|)}{|y − z|^{n−ρ} ( log \frac{|y−z|}{r} )^{α}} ,

where Ω ∈ L^{2}(S^{n−1} ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > \frac{3}{2}; B is a ball with center at x̄ and radius r_{0}; B^{∗} is a ball with center at x̄ and radius r = 2r_{0} and x_{0}, ω ∈ B.

Lemma 3.9. Let b ∈ BMO, 0 < δ < l < 1. Suppose that ρ > n/2, λ > 2, Ω ∈ L^{2}(S^{n−1} ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > \frac{5}{2}. Then for any smooth function with compact support f, there exists a positive constant 0 < C = C_{δ} such that

M_{δ}^{♯,Δ} (μ_{λ,b}^{∗,ρ}(f))(x) ⩽ C∥b∥_{∗}(M_{l}^{Δ} (μ_{λ}^{∗,ρ} (f))(x) + M^{2}(f)(x)).

Proof. Given x ∈ ℝ^{n}, let Q = Q(x̄, r_{1}) be a dyadic cube centered at x̄ with half side length r_{1} and x ∈ Q. Let B be a ball centered at x̄ and with radius r_{0} = \sqrt{n}r_{1}, and B^{∗} = B(x̄, r) with r = 2r_{0}. Decompose f = fχ_{8B^{∗}} +f(1−χ_{8B^{∗}} ) := f_{1} + f_{2}. Take C_{Q} = \frac{1}{|Q|} ∫_{Q} μ_{λ}^{∗,ρ} [(b − b_{B^{∗}} )f_{2}](u) du; then

|μ_{λ,b}^{∗,ρ}(f)(u) − C_{Q}|

⩽ |b(u) − b_{B^{∗}} |μ_{λ}^{∗,ρ} (f)(u) + μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{1})(u) + |μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2})(u) − C_{Q}|,

and

( \frac{1}{|Q|} ∫_{Q} ||μ_{λ,b}^{∗,ρ}(f)(u)|^{δ} − |C_{Q}|^{δ}|du) ^{\frac{1}{δ}}

⩽ C_{δ}( \frac{1}{|Q|} ∫_{Q} |(b(u) − b_{B^{∗}} )μ_{λ}^{∗,ρ} (f)(u)|^{δ}du) ^{\frac{1}{δ}}

+ C_{δ}( \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{1})(u)|^{δ}du) ^{\frac{1}{δ}}

+ C_{δ}( \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2})(u) − C_{Q}|^{δ}du) ^{\frac{1}{δ}}

:= C_{δ}(I_{1} + I_{2} + I_{3}).

As for I_{1}, we can choose 1 < r < min\{\frac{l}{δ}, \frac{1}{1−δ} \}, then by Hölder’s inequality,

I_{1} ⩽ ( \frac{1}{|Q|} ∫_{Q} |b(u) − b_{B^{∗}} |^{δr′}du) ^{\frac{1}{δr′}} ( \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} (f)(u)|^{δr}du) ^{\frac{1}{δr}}

⩽ C∥b∥_{∗}( \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} (f)(u)|^{l}du) ^{\frac{1}{l}} ⩽ C∥b∥_{∗}M_{l}^{Δ} (μ_{λ}^{∗,ρ} (f))(x).

190 LIN AND XUAN

As for I_{2}, applying Lemma 3.4 with \frac{1}{s} = \frac{1}{δ} − 1, Theorem 1.1, Hölder’s generalized inequality and Lemma 3.3, we have

I_{2} ⩽ ( \frac{1}{|Q|}) ^{\frac{1}{δ}} ( \frac{1}{1 − δ} ) ^{\frac{1}{δ}} ∥μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{1})∥_{W L^{1}} ∥χ_{Q}∥_{s}

⩽ \frac{C}{|8B^{∗}|} ∫_{8B^{∗}} |b(u) − b_{B^{∗}} ||f(u)|du (3.1)

⩽ ∥b − b_{B^{∗}} ∥_{expL,8B^{∗}} ∥f∥_{L log L,8B^{∗}}
⩽ C∥b∥_{∗}M_{L log L} (f)(x).

Note that M^{2}(f)(x) ≈ M_{L log L} (f)(x), we have I_{2} ⩽ C∥b∥_{∗}M^{2}(f)(x).

Finally, let us estimate I_{3}. Since f ∈ L^{p}, and μ_{λ}^{∗,ρ} is L^{p} bounded for 1 < p < ∞ by Theorem 1.2, then

∫_{Q} |μ_{λ}^{∗,ρ} (f_{2})(u)|du ⩽ |Q|^{\frac{1}{p′}} ( ∫_{Q} |μ_{λ}^{∗,ρ} (f_{2})(u)|^{p}du) ^{\frac{1}{p}} ⩽ C|Q|^{\frac{1}{p′}} ( ∫_{ℝ^{n}} |f(u)|^{p}du) ^{\frac{1}{p}} . This fact shows that μ_{λ}^{∗,ρ} (f_{2})(u) < ∞ a.e. on Q, so except a subset E with measure zero, for all u ∈ Q ∖ E, μ_{λ}^{∗,ρ} (f_{2})(u) < ∞. Thus,

I_{3} ⩽ \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2})(u) − (μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2}))_{Q}|du

⩽ \frac{1}{|Q|^{2}} ∫_{Q∖E} ∫_{Q∖E} |μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2})(u) − μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2})(v)|dvdu.

Next we will prove the following fact. For any x_{0}, w ∈ Q ∖ E,

J = |μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2})(x_{0}) − μ_{λ}^{∗,ρ} ((b − b_{B^{∗}} )f_{2})(w)|

⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz + Cr^{ε/2} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε/2}} dz

+ Cr^{ρ− \frac{n}{2}} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} dz (3.2)

+ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{r} )^{2+ε}} dz

:= T_{1} + T_{2} + T_{3} + T_{4}. write

J = |∥ϕ_{t,y}((b − b_{B^{∗}} )f_{2})(x_{0})∥_{H_{2}} − ∥ϕ_{t,y}((b − b_{B^{∗}} )f_{2})(w)∥_{H_{2}} |

⩽ ∥ϕ_{t,y}((b − b_{B^{∗}} )f_{2})(x_{0}) − ϕ_{t,y}((b(z) − b_{B^{∗}} )f_{2})(w)∥_{H_{2}}

⩽ ( ∫_{0}^{∞} ∫_{|y|<1} ( \frac{1}{1 + |y|} )^{λn} | ∫ t^{−n} ( ϕ( \frac{x_{0} − z}{t} − y) − ϕ( \frac{w − z}{t} − y) )

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|y|⩾1} ( \frac{1}{1 + |y|} ) ^{λn} | ∫ t^{−n} ( ϕ( \frac{x_{0} − z}{t} − y) − ϕ( \frac{w − z}{t} − y) )

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}} := J_{1} + J_{2}.

Since ( \frac{1}{1+|y|} )^{λn} ⩽ 1, there is

J_{1} ⩽ ( ∫_{0}^{∞} ∫_{|y|<1} | ∫\begin{matrix}| \frac{x_{0} −z}{t} −y|<1 \\ | \frac{ω−z}{t} −y|⩾1 \end{matrix} t^{−n} ϕ( \frac{x_{0} − z}{t} − y) (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|y|<1} | ∫\begin{matrix}| \frac{x_{0} −z}{t} −y|⩾1 \\ | \frac{ω−z}{t} −y|<1 \end{matrix} t^{−n} ϕ( \frac{ω − z}{t} − y)

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|y|<1} | ∫\begin{matrix}| \frac{x_{0} −z}{t} −y|<1 \\ | \frac{ω−z}{t} −y|<1 \end{matrix} t^{−n} ( ϕ( \frac{x_{0} − z}{t} − y) − ϕ( \frac{ω − z}{t} − y) )

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}} .

Use the transform y → \frac{x_{0} −y′}{t} (we still use y instead y′), then

J_{1} ⩽ ( ∫_{0}^{∞} ∫_{|x_{0} −y|<t} | ∫\begin{matrix}|y−z|<t \\ |ω−x_{0} +y−z|⩾t \end{matrix}  \frac{Ω(y − z)}{|y − z|^{n−ρ}} (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|x_{0} −y|<t} | ∫\begin{matrix}|y−z|⩾t \\ |ω−x_{0} +y−z|<t \end{matrix} \frac{Ω(ω − x_{0} + y − z)}{|ω − x_{0} + y − z|^{n−ρ}}

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|x_{0} −y|<t} | ∫\begin{matrix}|y−z|<t \\ |ω−x_{0} +y−z|<t \end{matrix} ( \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(ω − x_{0} + y − z)}{|ω − x_{0} + y − z|^{n−ρ}} )

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}}

:= J_{1.1} + J_{1.2} + J_{1.3}.

Take 0 < ε < min\{ \frac{1}{2}, ρ − \frac{n}{2} , α − \frac{5}{2}, \frac{(λ−2)}{2} n\} (we always restrict that ε satisfies this in the whole proof of this lemma). As for J_{1.1}, it follows from the Minkowski

192 LIN AND XUAN

inequality that

J_{1.1} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|[( ∫∫\begin{matrix}y∈2B^{∗} \\ |y−z|<t \\ |x_{0} −y|<t \\ |ω−x_{0} +y−z|⩾t \end{matrix}

+ ∫∫\begin{matrix}y∈(2B^{∗})^{c} \\ |y−z|<t \\ |x_{0} −y|<t \\ |ω−x_{0} +y−z|⩾t \end{matrix} ) \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ]^{\frac{1}{2}} dz

⩽ J_{1.1.1} + J_{1.1.2} ,

where

J_{1.1.1} = ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}y∈2B^{∗} \\ |y−z|<t \\ |x_{0} −y|<t \\ |ω−x_{0} +y−z|⩾t \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz,

J_{1.1.2} = ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}y∈(2B^{∗})^{c} \\ |y−z|<t \\ |x_{0} −y|<t \\ |ω−x_{0} +y−z|⩾t \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz.

For J_{1.1.1} , since y ∈ 2B^{∗}, z ∈ (8B^{∗})^{c}, then |y − z| ∼ |x_{0} − z| ∼ |ω − x_{0} + y − z|. We have

J_{1.1.1} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{y∈2B^{∗}} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}}

× ∫_{|y−z|<t⩽|ω−x_{0} +y−z|} \frac{dt}{t^{n+2ρ+1}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{y∈2B^{∗}} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{r}{|y − z|^{n+2ρ+1}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} ( ∫_{|y−z|>6r} \frac{r|Ω(y − z)|^{2}}{|y − z|^{n+1−2ε}} dy) ^{\frac{1}{2}} dz (3.3)

⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz.

For J_{1.1.2} , we have

J_{1.1.2} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

× ( ∫∫\begin{matrix}y∈(2B^{∗})^{c} , |y−z|<t \\ |x_{0} −y|<t, 2|y−z|⩾|z−x_{0} | \\ |ω−x_{0} +y−z|⩾t \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

+ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

WEIGHTED BOUNDEDNESS FOR COMMUTATORS... 193

× ( ∫∫\begin{matrix}y∈(2B^{∗})^{c} , |y−z|<t \\ |x_{0} −y|<t, 2|y−z|<|z−x_{0} | \\ |ω−x_{0} +y−z|⩾t \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

:= J_{1.1.2′} + J_{1.1.2′′} .

First we give the estimate of J_{1.1.2′} .

J_{1.1.2′} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

× ( ∫\begin{matrix}y∈(2B^{∗})^{c} \\ 2|y−z|⩾|z−x_{0} | \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} ∫_{|y−z|}^{|ω−x_{0} +y−z|} \frac{dt}{t^{n+2ρ+1}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

× ( ∫\begin{matrix}y∈(2B^{∗})^{c} \\ 2|y−z|⩾|z−x_{0} | \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{r}{|y − z|^{n+2ρ+1}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} ( ∫_{|y−z|>3r} \frac{r|Ω(y − z)|^{2}}{|y − z|^{n+1−2ε}} dy) ^{\frac{1}{2}} dz.

Similarly to the estimate of (3.3), we have

J_{1.1.2′} ⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz.

Then we give the estimate of J_{1.1.2′′} .

J_{1.1.2′′} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

× ( ∫∫\begin{matrix}y∈(2B^{∗})^{c} , |y−z|<t \\ |x_{0} −y|<t, 2|y−z|<|z−x_{0} | \\ |ω−x_{0} +y−z|⩾t, |y−z|<2r \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

+ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

× ( ∫∫\begin{matrix}y∈(2B^{∗})^{c} , |y−z|<t \\ |x_{0} −y|<t, 2|y−z|<|z−x_{0} | \\ |ω−x_{0} +y−z|⩾t, |y−z|⩾2r \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

:= J_{1.1.2.1′′} + J_{1.1.2.2′′} .

For J_{1.1.2.1′′} , since |y − z| < \frac{|z−x_{0} |}{2} , then |y − x_{0}| > \frac{|z−x_{0} |}{2} and we get

J_{1.1.2.1′′} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

× ( ∫\begin{matrix}|y−z|<2r \\ |y−x_{0} |> \frac{|z−x_{0} |}{2} \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} ∫_{|y−x_{0} |}^{∞} \frac{dt}{t^{n+2ρ+1}} dy) ^{\frac{1}{2}} dz

194 LIN AND XUAN

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} ( ∫_{|y−z|<2r} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} dy) ^{\frac{1}{2}} dz

⩽ Cr^{ρ− \frac{n}{2}} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} dz.

For J_{1.1.2.2′′} , since t > |y − x_{0}| > \frac{|z−x_{0} |}{2} , and |y − z| ∼ |ω − x_{0} + y − z|, we have

J_{1.1.2.2′′} ⩽ ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{(\frac{|z−x_{0} |}{2} )^{n+ε}} ( ∫\begin{matrix}y∈(2B^{∗})^{c} \\ 2|y−z|<|z−x_{0} | \\ |y−z|⩾2r \end{matrix}  \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}}

× ∫_{|y−z|<t⩽|ω−x_{0} +y−z|} \frac{dt}{t^{2ρ−n+1−2ε}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} ( ∫_{|y−z|⩾2r} \frac{r|Ω(y − z)|^{2}}{|y − z|^{n+1−2ε}} dy) ^{\frac{1}{2}} dz.

Similarly to the estimate of (3.3), we have

J_{1.1.2.2′′} ⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz.

Combining the estimates of J_{1.1.1} , J_{1.1.2′} , J_{1.1.2.1′′} and J_{1.1.2.2′′} , we obtain

J_{1.1} ⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz + Cr^{ρ− \frac{n}{2}} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} dz.

Similarly as we deal with J_{1.1}, we can get

J_{1.2} ⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz + Cr^{ρ− \frac{n}{2}} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} dz.

Next we give the estimate of J_{1.3}. Apply the Minkowski inequality to J_{1.3} and divide the region by |y − z| ⩾ 6r and |y − z| < 6r. When |y − z| < 6r, we have y ∈ (2B^{∗})^{c}, so

J_{1.3} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}y∈(2B^{∗})^{c} \\ |y−z|<t \\ |x_{0} −y|<t \\ |y−z|<6r \\ |ω−x_{0} +y−z|<t \end{matrix}  | \frac{Ω(y − z)}{|y − z|^{n−ρ}}

− \frac{Ω(ω − x_{0} + y − z)}{|ω − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz
+ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}|y−z|⩾6r \\ |y−z|<t \\ |x_{0} −y|<t \\ |ω−x_{0} +y−z|<t \end{matrix}  | \frac{Ω(y − z)}{|y − z|^{n−ρ}}
− \frac{Ω(ω − x_{0} + y − z)}{|ω − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz := J_{1.3.1} + J_{1.3.2} .

When z ∈ (8B^{∗})^{c} and |y − z| < 6r, there are |y − x_{0}| ∼ |z − x_{0}| and |ω − x_{0} + y − z| ⩽ |ω − x_{0}| + |y − z| < 8r. Then

J_{1.3.1} ⩽ C ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫\begin{matrix}y∈(2B^{∗})^{c} , |y−z|<6r \\ |x_{0} −y|<t, |ω−x_{0} +y−z|<8r \end{matrix}  ( \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}}

+ \frac{|Ω(ω − x_{0} + y − z)|^{2}}{|ω − x_{0} + y − z|^{2n−2ρ}} ) ∫_{|y−x_{0} |}^{∞} \frac{dt}{t^{n+2ρ+1}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} ( ∫_{|y−z|<6r} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} dy) ^{\frac{1}{2}} dz

+ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}}

× ( ∫_{|ω−x_{0} +y−z|<8r} \frac{|Ω(ω − x_{0} + y − z)|^{2}}{|ω − x_{0} + y − z|^{2n−2ρ}} dy) ^{\frac{1}{2}} dz

⩽ Cr^{ρ− \frac{n}{2}} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{ρ+ \frac{n}{2}}} dz.

Next we estimate J_{1.3.2} . Note that |z−x_{0}| ⩽ |x_{0}−y|+|y−z| < 2t, so t > \frac{|z−x_{0} |}{2} .

J_{1.3.2} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}t>|z−x_{0} |/2, |y−z|<t \\ |y−x_{0} |<t, |y−z|⩾6r \\ |ω−x_{0} +y−z|<t \end{matrix}  | \frac{Ω(y − z)}{|y − z|^{n−ρ}}

− \frac{Ω(ω − x_{0} + y − z)}{|ω − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{(log \frac{t}{r})^{4+2ε} dy dt}{t^{2ρ−n+1} t^{2n} (log \frac{t}{r})^{4+2ε}} ) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{2r} )^{2+ε}} ( ∫_{|y−z|⩾6r} | \frac{Ω(y − z)}{|y − z|^{n−ρ}}

− \frac{Ω(ω − x_{0} + y − z)}{|ω − x_{0} + y − z|^{n−ρ}}  |^{2} ∫_{|y−z|}^{∞} \frac{(log \frac{t}{r})^{4+2ε}}{t^{2ρ−n+1}} dtdy) ^{\frac{1}{2}} dz. (3.4)

By Lemma 3.7 and Lemma 3.8, there is

J_{1.3.2} ⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{2r} )^{2+ε}}

× ( ∫_{|y−z|⩾6r} \frac{(1 + |Ω(y − z)|)^{2}}{|y − z|^{n}(log \frac{|y−z|}{r} )^{2α−4−2ε}} dy) ^{\frac{1}{2}} dz (3.5)

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{r} )^{2+ε}} dz.

Combining the estimates of J_{1.3.1} and J_{1.3.2} , we obtain

J_{1.3} ⩽ Cr^{ρ− \frac{n}{2}} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} dz + C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{r} )^{2+ε}} dz.

As for J_{2},

J_{2} ⩽ ( ∫_{0}^{∞} ∫_{|y|⩾1} ( \frac{1}{1 + |y|} ) ^{λn} | ∫\begin{matrix}| \frac{x_{0} −z}{t} −y|<1 \\ | \frac{w−z}{t} −y|⩾1 \end{matrix} t^{−n}

× ϕ( \frac{x_{0} − z}{t} − y) (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|y|⩾1} ( \frac{1}{1 + |y|} ) ^{λn} | ∫\begin{matrix}| \frac{x_{0} −z}{t} −y|⩾1 \\ | \frac{w−z}{t} −y|<1 \end{matrix} t^{−n}

× ϕ( \frac{w − z}{t} − y) (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|y|⩾1} ( \frac{1}{1 + |y|} ) ^{λn} | ∫\begin{matrix}| \frac{x_{0} −z}{t} −y|<1 \\ | \frac{w−z}{t} −y|<1 \end{matrix} t^{−n}

× ( ϕ( \frac{x_{0} − z}{t} − y) − ϕ( \frac{w − z}{t} − y) ) (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t}) ^{\frac{1}{2}} .

Using the trasform y → \frac{x_{0} −y′}{t} again (we still use y instead y′ ), we have

J_{2} ⩽ ( ∫_{0}^{∞} ∫_{|x_{0} −y|⩾t} ( \frac{t}{t + |x_{0} − y|} ) ^{λn} | ∫\begin{matrix}|y−z|<t \\ |w−x_{0} +y−z|⩾t \end{matrix}  \frac{Ω(y − z)}{|y − z|^{n−ρ}}

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|x_{0} −y|⩾t} ( \frac{t}{t + |x_{0} − y|} ) ^{λn} | ∫\begin{matrix}|y−z|⩾t \\ |w−x_{0} +y−z|<t \end{matrix} \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}

× (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}}

+ ( ∫_{0}^{∞} ∫_{|x_{0} −y|⩾t} ( \frac{t}{t + |x_{0} − y|} ) ^{λn} | ∫\begin{matrix}|y−z|<t \\ |w−x_{0} +y−z|<t \end{matrix} ( \frac{Ω(y − z)}{|y − z|^{n−ρ}}

− \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}} ) (b(z) − b_{B^{∗}} )f_{2}(z) dz|^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}}

:= J_{2.1} + J_{2.2} + J_{2.3}.

For J_{2.1}, we claim that y ∈ (2B^{∗})^{c}. Otherwise if y ∈ 2B^{∗}, then t ⩽ |x_{0}−y| < 4r. But z ∈ (8B^{∗})^{c} and t > |y − z| > 6r. Thus by the Minkowski inequality, we get

J_{2.1} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{0}^{∞} ∫\begin{matrix}|x_{0} −y|⩾t \\ y∈(2B^{∗})^{c} \\ |y−z|<t \\ |w−x_{0} +y−z|⩾t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

WEIGHTED BOUNDEDNESS FOR COMMUTATORS... 197

× \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz ⩽ J_{2.1.1} + J_{2.1.2} ,

where

J_{2.1.1} = ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{0}^{∞} ∫\begin{matrix}|y−z|<8r , |x_{0} −y|⩾t \\ y∈(2B^{∗})^{c} , |y−z|<t \\ |w−x_{0} +y−z|⩾t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz,

J_{2.1.2} = ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{0}^{∞} ∫\begin{matrix}|y−z|⩾8r, |x_{0} −y|⩾t \\ y∈(2B^{∗})^{c} , |y−z|<t \\ |w−x_{0} +y−z|⩾t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz.

For J_{2.1.1} , since |y −z| < 8r, z ∈ (8B^{∗})^{c} and y ∈ (2B^{∗})^{c}, then |y −x_{0}| ∼ |z −x_{0}|. So

J_{2.1.1} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{0}^{∞} ∫\begin{matrix}|y−z|<8r, |x_{0} −y|⩾t \\ y∈(2B^{∗})^{c} , |y−z|<t \end{matrix} ( \frac{1}{t + |x_{0} − y|} ) ^{2n+2ε}

× ( \frac{t}{t + |x_{0} − y|} ) ^{λn−2n−2ε} t^{2n+2ε} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{0}^{∞} ∫\begin{matrix}|y−z|<8r \\ |x_{0} −y|⩾t \\ y∈(2B^{∗})^{c} \\ |y−z|<t \end{matrix} \frac{1}{|z − x_{0}|^{2n+2ε}}

× \frac{|Ω(y − z)|^{2}}{|y − z|^{n−ε}} \frac{dy dt}{t^{1−ε}}) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε/2}} ( ∫_{|y−z|<8r} \frac{|Ω(y − z)|^{2}}{|z − x_{0}|^{ε}|y − z|^{n−ε}}

× ( ∫_{0}^{|x_{0} −y|} \frac{1}{t^{1−ε}} dt) dy) ^{\frac{1}{2}} dz

⩽ Cr^{ε/2} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε/2}} dz.

For J_{2.1.2} , there is

J_{2.1.2} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{0}^{∞} ∫\begin{matrix}2|y−z|⩾|z−x_{0} | \\ |x_{0} −y|⩾t, y∈(2B^{∗})^{c} \\ |y−z|<t, |y−z|⩾8r \\ |w−x_{0} +y−z|⩾t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

198 LIN AND XUAN

+ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫_{0}^{∞} ∫\begin{matrix}2|y−z|<|z−x_{0} | \\ |x_{0} −y|⩾t, y∈(2B^{∗})^{c} \\ |y−z|<t, |y−z|⩾8r \\ |w−x_{0} +y−z|⩾t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz := J_{2.1.2′} + J_{2.1.2′′} .

As for J_{2.1.2′} , we have

J_{2.1.2′} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫\begin{matrix}y∈(2B^{∗})^{c} \\ |y−z|⩾8r \\ 2|y−z|⩾|z−x_{0} | \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}}

× ∫_{|y−z|}^{|w−x_{0} +y−z|} \frac{dt}{t^{n+2ρ+1}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} ( ∫_{|y−z|⩾8r} \frac{r|Ω(y − z)|^{2}}{|y − z|^{n+1−2ε}} dy) ^{\frac{1}{2}} dz

⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz.

For J_{2.1.2′′} , by |z − x_{0}| > 2|y − z|, we have |y − x_{0}| > |z − x_{0}|/2 and |y − z| ∼ |w − x_{0} + y − z|. Then

J_{2.1.2′′} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}y∈(2B^{∗})^{c} , |y−z|⩾8r \\ |y−x_{0} |⩾|z−x_{0} |/2 \\ |y−z|<t, |w−x_{0} +y−z|  ⩾t \end{matrix} \frac{t^{2n+2ε}}{|x_{0} − y|^{2n+2ε}}

× ( \frac{t}{t + |x_{0} − y|} ) ^{λn−2n−2ε} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} ( ∫\begin{matrix}y∈(2B^{∗})^{c} , |y−z|⩾8r \\ |y−x_{0} |⩾|z−x_{0} |/2 \end{matrix} \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}}

× ( ∫_{|y−z|}^{|w−x_{0} +y−z|} \frac{1}{t^{2ρ−n−2ε+1}} dt) dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} ( ∫_{|y−z|⩾8r} \frac{r|Ω(y − z)|^{2}}{|y − z|^{n+1−2ε}} dy) ^{\frac{1}{2}} dz.

Similarly to the estimate of (3.3), we have J_{2.1.2′′} ⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z)−b_{B^{∗}} ||f(z)|}{|z−x_{0} |^{n+ε}} dz. Combining the estimates of J_{2.1.1} , J_{2.1.2′} and J_{2.1.2′′} , we obtain

J_{2.1} ⩽ Cr^{ε/2} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε/2}} dz + Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz.

Similarly as J_{2.1}, we can obtain

J_{2.2} ⩽ Cr^{ε/2} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε/2}} dz + Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε}} dz.

WEIGHTED BOUNDEDNESS FOR COMMUTATORS... 199

Finally, we deal with the last part J_{2.3}. By the Minskowski inequality,

J_{2.3} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}|y−z|<6r, |x_{0} −y|⩾t \\ |y−z|<t, |w−x_{0} +y−z|<t \end{matrix}  ( \frac{t}{t + |x_{0} − y|} ) ^{λn}
× | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

+ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|(∫∫\begin{matrix}|y−z|⩾6r, |x_{0} −y|⩾t \\ |y−z|<t, |w−x_{0} +y−z|<t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz := J_{2.3.1} + J_{2.3.2} .

For J_{2.3.1} , it follows from |y − z| < 6r and z ∈ (8B^{∗})^{c} that |y − x̄| ⩾ |z − x̄| − |y − z| > 2r. We can get y ∈ (2B^{∗})^{c} and |w − x_{0} + y − z| ⩽ |w − x_{0}| + |y − z| < 8r.

J_{2.3.1} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}y∈(2B^{∗})^{c} , |x_{0} −y|⩾t \\ |y−z|<6r, |y−z|<t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× \frac{|Ω(y − z)|^{2}}{|y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

+ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}y∈(2B^{∗})^{c} , |x_{0} −y|⩾t \\ |w−x_{0} +y−z|<8r \\ |w−x_{0} +y−z|<t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× \frac{|Ω(w − x_{0} + y − z)|^{2}}{|w − x_{0} + y − z|^{2n−2ρ}} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz := J_{2.3.1′} + J_{2.3.1′′} .

Estimate J_{2.3.1′} and J_{2.3.1′′} by the similar method as we deal with J_{2.1.1} . There is

J_{2.3.1} ⩽ Cr^{ε/2} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε/2}} dz.

As for J_{2.3.2} , we have

J_{2.3.2} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}2|y−z|⩾|z−x_{0} | \\ |y−z|⩾6r, |x_{0} −y|⩾t \\ |y−z|<t, |w−x_{0} +y−z|<t \end{matrix}  ( \frac{t}{t + |x_{0} − y|} ) ^{λn}
× | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

+ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}2|y−z|<|z−x_{0} | \\ |y−z|⩾6r, |x_{0} −y|⩾t \\ |y−z|<t, |w−x_{0} +y−z|<t \end{matrix} ( \frac{t}{t + |x_{0} − y|} ) ^{λn}

× | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

:= J_{2.3.2′} + J_{2.3.2′′} .

For J_{2.3.2′} , there is

J_{2.3.2′} ⩽ C ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|

× ( ∫_{|y−z|⩾6r} | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2}

× ( ∫_{max\{|y−z|,|z−x_{0} |/2\}}^{∞} \frac{(log \frac{t}{r})^{4+2ε} dt}{t^{2ρ−n+1} |z − x_{0}|^{2n} (log \frac{|z−x_{0} |}{2r} )^{4+2ε}} ) dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{2r} )^{2+ε}}

× ( ∫_{|y−z|⩾6r} | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2}

× ( ∫_{|y−z|}^{∞} \frac{(log \frac{t}{r})^{4+2ε} dt}{t^{2ρ−n+1}}) dy) ^{\frac{1}{2}} dz.

By the estimate of (3.4), we get J_{2.3.2′} ⩽ C ∫_{(8B^{∗})^{c}} _{\frac{|b(z)−b_{B^{∗}} ||f(z)|}{|z−x_{0} |^{n} (log \frac{|z−x_{0} |}{r} )^{2+ε}}} dz.

For J_{2.3.2′′} , denote C(ε) = e^{(4+2ε)/ε} . Since 2|y − z| < |z − x_{0}|, then |x_{0} − y| ⩾ |z − x_{0}| − |y − z| > \frac{|z−x_{0} |}{2} , thus

J_{2.3.2′′} ⩽ ∫_{(8B^{∗})^{c}} |b(z) − b_{B^{∗}} ||f(z)|( ∫∫\begin{matrix}|y−z|⩾6r, |x_{0} −y|⩾t \\ |x_{0} −y|>|z−x_{0} |/2 \\ |y−z|<t \end{matrix}

× \frac{t^{λn}}{(t + |x_{0} − y|)^{λn−2n+2n}} \frac{(log \frac{t+|y−x_{0} |+C(ε)r}{r} )^{4+2ε}}{(log \frac{t+|y−x_{0} |+C(ε)r}{r} )^{4+2ε}}

× | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{dy dt}{t^{n+2ρ+1}} ) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{2r} )^{2+ε}}

× ( ∫_{|y−z|⩾6r} | \frac{Ω(y − z)}{|y − z|^{n−ρ}} − \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2}

× ( ∫_{|y−z|}^{|x_{0} −y|} \frac{t^{λn} (log \frac{t+|y−x_{0} |+C(ε)r}{r} )^{4+2ε}}{(t + |x_{0} − y|)^{λn−2n}} \frac{dt}{t^{n+2ρ+1}} ) dy) ^{\frac{1}{2}} dz.

Notice that the function G(s) = \frac{(log s)^{4+2ε}}{s^{ε}} is decreasing when s ⩾ e^{(4+2ε)/ε} and

\frac{t + |y − x_{0}| + C(ε)r}{r} ⩾ \frac{|y − z| + C(ε)r}{r} ⩾ C(ε) = e^{(4+2ε)/ε} ,

then

\frac{[log(\frac{t+|y−x_{0} |+C(ε)r}{r} )]^{4+2ε}}{(\frac{t+|y−x_{0} |+C(ε)r}{r} )^{ε}} ⩽ \frac{[log(\frac{|y−z|+C(ε)r}{r} )]^{4+2ε}}{(\frac{|y−z|+C(ε)r}{r} )^{ε}} .

Since t + |y − x_{0}| ∼ t + |y − x_{0}| + C(ε)r and 0 < ε < min\{\frac{1}{2}, \frac{(λ−2)n}{2} , ρ − \frac{n}{2} , α − \frac{5}{2}\}, then

∫_{|y−z|}^{|x_{0} −y|} \frac{t^{λn} (log \frac{t+|y−x_{0} |+C(ε)r}{r} )^{4+2ε}}{(t + |x_{0} − y|)^{λn−2n}} \frac{dt}{t^{n+2ρ+1}}

⩽ C ∫_{|y−z|}^{∞} \frac{[log(\frac{|y−z|+C(ε)r}{r} )]^{4+2ε}}{(|y − z| + C(ε)r)^{ε}} \frac{dt}{t^{2ρ−n+1−ε}}

⩽ C \frac{[log(\frac{|y−z|+C(ε)r}{r} )]^{4+2ε}}{|y − z|^{2ρ−n}} .

Since |y −z| ⩾ 6r, there exists a constant l ⩾ 1 such that |y −z|+C(ε)r ⩽ 2^{l}|y −z|. Hence

J_{2.3.2′′} ⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{2r} )^{2+ε}} ( ∫_{|y−z|⩾6r} | \frac{Ω(y − z)}{|y − z|^{n−ρ}}

− \frac{Ω(w − x_{0} + y − z)}{|w − x_{0} + y − z|^{n−ρ}}  |^{2} \frac{(log \frac{2^{l} |y−z|}{r} )^{4+2ε}}{|y − z|^{2ρ−n}} dy) ^{\frac{1}{2}} dz

⩽ C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{2r} )^{2+ε}}

× ( ∫_{|y−z|⩾6r} \frac{(1 + |Ω(y − z)|)^{2}}{|y − z|^{n}(log \frac{|y−z|}{r} )^{2α−4−2ε}} dy) ^{\frac{1}{2}} dz.

By the estimate of (3.5), we get J_{2.3.2′′} ⩽ C ∫_{(8B^{∗})^{c}} _{\frac{|b(z)−b_{B^{∗}} ||f(z)|}{|z−x_{0} |^{n} (log \frac{|z−x_{0} |}{r} )^{2+ε}}} dz.

Combining the estimates of J_{2.3.1} , J_{2.3.2′} and J_{2.3.2′′} , we obtain

J_{2.3} ⩽ Cr^{ε/2} ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n+ε/2}} dz + C ∫_{(8B^{∗})^{c}} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{r} )^{2+ε}} dz.

Then we complete the proof of (3.2). Next we will show that T_{i} ⩽ C∥b∥_{∗}M^{2}(f)(x), for i = 1, 2, 3, 4.

Denote B_{j} = {z : |z − x̄| < 2^{j} r}, then |b_{B_{j+1}} − b_{B^{∗}} | ⩽ j∥b∥_{∗}. Since M^{2}(f) ≈ M_{L log L} (f), by Lemma 3.3 we get

T_{1} ⩽ Cr^{ε} ∑_{j=3}^{∞} ∫_{2^{j} r⩽|z−x̄|<2^{j+1} r} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x̄|^{n+ε}} dz

⩽ C ∑_{j=3}^{∞} \frac{1}{2^{jε}} ( ∫_{B_{j+1}} \frac{|b(z) − b_{B_{j+1}} ||f(z)|}{(2^{j} r)^{n}} dz + |b_{B_{j+1}} − b_{B^{∗}} |M(f)(x))

⩽ C ∑_{j=3}^{∞} \frac{1}{2^{jε}} (∥b − b_{B_{j+1}} ∥_{expL,B_{j+1}} ∥f∥_{L log L,B_{j+1}} + j∥b∥_{∗}M(f)(x))

⩽ C ∑_{j=3}^{∞} \frac{1}{2^{jε}} (∥b∥_{∗}M_{L log L} (f)(x) + j∥b∥_{∗}M(f)(x)) ⩽ C∥b∥_{∗}M^{2}(f)(x).

Taking ε/2 and ρ − \frac{n}{2} instead of ε in the above inequality respectively, we get T_{2} ⩽ C∥b∥_{∗}M^{2}(f)(x) and T_{3} ⩽ C∥b∥_{∗}M^{2}(f)(x).

Similarly to the way in estimating T_{1}, we obtain

T_{4} ⩽ C ∑_{j=3}^{∞} ∫_{2^{j} r⩽|z−x̄|<2^{j+1} r} \frac{|b(z) − b_{B^{∗}} ||f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x̄|}{r} )^{2+ε}} dz

⩽ C ∑_{j=3}^{∞} \frac{1}{j^{2+ε}} (∥b∥_{∗}M_{L log L} (f)(x) + j∥b∥_{∗}M(f)(x))

⩽ C∥b∥_{∗}M^{2}(f)(x).

So we have J ⩽ C∥b∥_{∗}M^{2}(f)(x) and I_{3} ⩽ C∥b∥_{∗}M^{2}(f)(x). Combining the estimates of I_{1}, I_{2} and I_{3}, we have

M_{δ}^{♯,Δ} (μ_{λ,b}^{∗,ρ}(f))(x) = [M^{♯,Δ} (|μ_{λ,b}^{∗,ρ}(f)|^{δ})(x)]^{\frac{1}{δ}}

⩽ sup_{Q∋x} ( \frac{1}{|Q|} ∫_{Q} ||μ_{λ,b}^{∗,ρ}(f)|^{δ} − |C_{Q}|^{δ}|du) ^{\frac{1}{δ}}

⩽ C∥b∥_{∗}(M_{l}^{Δ} (μ_{λ}^{∗,ρ} (f))(x) + M^{2}(f)(x)),

where the supremum is taken over all dyadic cubes Q with x ∈ Q. □

Lemma 3.10. Suppose that 0 < δ < 1, ρ > n/2, λ > 2, and Ω ∈ L^{2}(S^{n−1} ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > \frac{5}{2}. Then for any smooth function with compact support f, there exists a positive constant 0 < C = C_{δ} such that

M_{δ}^{♯,Δ} (μ_{λ}^{∗,ρ} (f))(x) ⩽ CM(f)(x).

Proof. Let f_{1}, f_{2}, Q and B^{∗} be the same as in the proof of Lemma 3.9. Then applying Lemma 3.4 with \frac{1}{s} = \frac{1}{δ} − 1 and Theorem 1.1, similarly to get (3.1), we have

( \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} (f_{1})(y)|^{δ}dy) ^{\frac{1}{δ}} ⩽ \frac{C}{|8B^{∗}|} ∫_{8B^{∗}} |f(y)|dy ⩽ CM(f)(x).

Since f ∈ L^{p} for 1 < p < ∞, and μ_{λ}^{∗,ρ} is L^{p} bounded, then μ_{λ}^{∗,ρ} (f_{2})(u) < ∞ a.e.

on Q, so except a subset E with measure zero, for all u ∈ Q ∖ E, μ_{λ}^{∗,ρ} (f_{2})(u) < ∞. Next we will prove the following fact. For any x_{0}, w ∈ Q ∖ E,

J = |μ_{λ}^{∗,ρ} (f_{2})(x_{0}) − μ_{λ}^{∗,ρ} (f_{2})(w)| ⩽ CM(f)(x).

In fact, similarly to (3.2), we know that

J ⩽ Cr^{ε} ∫_{(8B^{∗})^{c}} \frac{|f(z)|}{|z − x_{0}|^{n+ε}} dz + Cr^{ε/2} ∫_{(8B^{∗})^{c}} \frac{|f(z)|}{|z − x_{0}|^{n+ε/2}} dz

+ Cr^{ρ− \frac{n}{2}} ∫_{(8B^{∗})^{c}} \frac{|f(z)|}{|z − x_{0}|^{\frac{n}{2} +ρ}} dz + C ∫_{(8B^{∗})^{c}} \frac{|f(z)|}{|z − x_{0}|^{n}(log \frac{|z−x_{0} |}{r} )^{2+ε}} dz

⩽ Cr^{ε} ∑_{j=3}^{∞} ∫_{2^{j} r⩽|z−x̄|<2^{j+1} r} \frac{|f(z)|}{|z − x̄|^{n+ε}} dz

+ Cr^{ε/2} ∑_{j=3}^{∞} ∫_{2^{j} r⩽|z−x̄|<2^{j+1} r} \frac{|f(z)|}{|z − x̄|^{n+ε/2}} dz

+ Cr^{ρ− \frac{n}{2}} ∑_{j=3}^{∞} ∫_{2^{j} r⩽|z−x̄|<2^{j+1} r} \frac{|f(z)|}{|z − x̄|^{\frac{n}{2} +ρ}} dz

+ C ∑_{j=3}^{∞} ∫_{2^{j} r⩽|z−x̄|<2^{j+1} r} \frac{|f(z)|}{|z − x̄|^{n}(log \frac{|z−x̄|}{r} )^{2+ε}} dz

⩽ C ∑_{j=3}^{∞} \frac{1}{2^{jε}} \frac{1}{(2^{j+1} r)^{n}} ∫_{|z−x̄|<2^{j+1} r} |f(z)|dz

+ C ∑_{j=3}^{∞} \frac{1}{2^{jε/2}} \frac{1}{(2^{j+1} r)^{n}} ∫_{|z−x̄|<2^{j+1} r} |f(z)|dz

+ C ∑_{j=3}^{∞} \frac{1}{2^{j(ρ− \frac{n}{2} )}} \frac{1}{(2^{j+1} r)^{n}} ∫_{|z−x̄|<2^{j+1} r} |f(z)|dz

+ C ∑_{j=3}^{∞} \frac{1}{j^{2+ε}} \frac{1}{(2^{j+1} r)^{n}} ∫_{|z−x̄|<2^{j+1} r} |f(z)|dz ⩽ CM(f)(x).

Let C_{Q} = (μ_{λ}^{∗,ρ} (f_{2}))_{Q}. Since |μ_{λ}^{∗,ρ} (f)(u) − C_{Q} | ⩽ |μ_{λ}^{∗,ρ} (f_{1})(u)| + |μ_{λ}^{∗,ρ} (f_{2})(u) −

C_{Q}|, then

M_{δ}^{♯,Δ} ( μ_{λ}^{∗,ρ} (f)) (x)

⩽ C_{δ} sup_{Q∋x} ( \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} (f_{1})(u)|^{δ}du) ^{\frac{1}{δ}}

+ C_{δ} sup_{Q∋x} ( \frac{1}{|Q|} ∫_{Q} |μ_{λ}^{∗,ρ} (f_{2}) − C_{Q} |^{δ}du) ^{\frac{1}{δ}}

⩽ CM(f)(x) + C sup_{Q∋x} \frac{1}{|Q|^{2}} ∫_{Q∖E} ∫_{Q∖E} |μ_{λ}^{∗,ρ} (f_{2})(u) − μ_{λ}^{∗,ρ} (f_{2})(v)|dvdu

⩽ CM(f)(x).

□

For b ∈ BMO, let b_{k}(x) = b(x) if |b(x)| ⩽ k, b_{k}(x) = k if b(x) > k and b_{k}(x) = −k if b(x) < −k for k = 1, 2, 3, · · · . Then b_{k} ∈ L^{∞} and ∥b_{k}∥_{∗} ⩽ ∥b∥_{∗}. The following lemma shows that μ_{λ,b_{k}}^{∗,ρ} (f) can be controlled by the maximal operator.

Lemma 3.11. [4] Suppose that Ω ∈ L^{2}(S^{n−1} ) is a homogeneous function of degree zero. If ρ > n/2, λ > 2, suppf ⊂ B(0, R) and |x| ⩾ 2R, then for any k, there exists a constant C independent of k, f, R and x such that μ_{λ,b_{k}}^{∗,ρ} (f)(x) ⩽ CkMf(x).

Remark 3.2. By checking the proof of Lemma 3.7 in [4], we find out that the condition “Ω ∈ L^{2}(S^{n−1} ) and Ω is a homogeneous function of degree zero” is sufficient to get the desired result.

4. Proof of theorems

First, we give the proof of Theorem 2.1 as follows.

Proof. It is easy to check that μ_{Ω,S}^{ρ,b} ⩽ 2^{λn/2} μ_{λ,b}^{∗,ρ}, so we only give the proof of Theorem 2.1 for μ_{λ,b}^{∗,ρ}.

Since ω ∈ A_{p} , there is an ε > 0 such that ω^{1+ε} ∈ A_{p}, then by Theorem 1.2, we have

(4.1)  ∥μ_{λ}^{∗,ρ} (ϕ)∥_{p,ω^{1+ε}} ⩽ C̄_{1}∥ϕ∥_{p,ω^{1+ε}} , for ϕ ∈ L^{p}(ω^{1+ε} ).

Take λ′ = p(1 + ε)/ε and η = min{C_{2}/λ′, C_{2}(p − 1)/λ′}, where C_{2} is the constant in Lemma 3.1. Without loss of generality we may assume that ∥b∥_{∗} < η. Otherwise we take 0 < δ < η and set b_{0}(x) = δb(x)/∥b∥_{∗}, then ∥b_{0}∥_{∗} = δ < η and μ_{λ,b}^{∗,ρ}(f)(x) = (∥b∥_{∗}/δ)μ_{λ,b_{0}}^{∗,ρ} (f)(x). Therefore, it suffices to consider μ_{λ,b_{0}}^{∗,ρ} (f)(x). By Lemma 3.2, we have e^{p(1+ε)b(x)/ε} ∈ A_{p} for b(x) ∈ BMO with ∥b∥_{∗} < η. Since b(x) ∈ BMO implies that tb(x) ∈ BMO with ∥tb∥_{∗} ⩽ ∥b∥_{∗} for |t| ⩽ 1, by Remark 3.1 we have

(4.2)  e^{p(1+ε)tb(x)/ε} ∈ A_{p}, forb(x) ∈ BMO with |b∥_{∗} < η and|t| ⩽ 1,

where [e^{p(1+ε)tb(x)/ε} ]_{A_{p}} can be dominated by a constant independent of t.

By (4.2) and Theorem 1.2, we know that for any ϕ ∈ L^{p}(e^{p(1+ε)b(x)cosθ/ε} ) and θ ∈ [0, 2π],

(4.3)  ∥μ_{λ}^{∗,ρ} (ϕ)∥_{p,e^{p(1+ε)b(x)cosθ/ε}}  ⩽ C̄_{2}∥ϕ∥_{p,e^{p(1+ε)b(x)cosθ/ε}} ,

where C̄_{2} depends on n, p, b, Ω, but not on θ and ϕ.

Applying the Stein-Weiss interpolation theorem with change of measure in [13] between (4.1) and (4.3), we have for any θ ∈ [0, 2π] and ϕ ∈ L^{p}(ωe^{pb(x)cosθ} ),

(4.4)  ∥μ_{λ}^{∗,ρ} (ϕ)∥_{p,ωe^{pb(x)cosθ}} ⩽ C̄∥ϕ∥_{p,ωe^{pb(x)cosθ}} ,

where C̄ = max{C̄_{1}, C̄_{2}} depending only on n, p, b, ω, Ω, but not on θ and ϕ.

Denote F (y) = e^{y[b(x)−b(z)]} , y ∈ ℂ, then by the analyticity of F (y) on ℂ and the Cauchy integration formula, we have

(4.5) b(x) − b(z) = F′(0) = \frac{1}{2πi} ∫_{|y|=1} \frac{F (y)}{y^{2}}dy = \frac{1}{2π} ∫_{0}^{2π} e^{e^{iθ} (b(x)−b(z))} e^{−iθ} dθ.

By (4.5) and the Minkowski inequality we get

μ_{λ,b}^{∗,ρ}(f)(x) = ( ∫∫_{ℝ_{+}^{n+1}} ( \frac{t}{t + |x − y|} )^{λn} | \frac{1}{2π} ∫_{0}^{2π} \frac{1}{t^{ρ}} ∫_{|y−z|<t} \frac{Ω(y − z)}{|y − z|^{n−ρ}}

× e^{e^{iθ} (b(x)−b(z))} e^{−iθ} f(z) dzdθ|^{2} \frac{dy dt}{t^{n+1}}) ^{\frac{1}{2}}

WEIGHTED BOUNDEDNESS FOR COMMUTATORS... 205

⩽ \frac{1}{2π} ∫_{0}^{2π} ( ∫∫_{ℝ_{+}^{n+1}} ( \frac{t}{t + |x − y|} )^{λn} | \frac{1}{t^{ρ}} ∫_{|y−z|<t} \frac{Ω(y − z)}{|y − z|^{n−ρ}}

× e^{−e^{iθ} b(z)} f(z) dz|^{2} \frac{dy dt}{t^{n+1}}) ^{\frac{1}{2}} |e^{e^{iθ} b(x)} ||e^{−iθ} |dθ

= \frac{1}{2π} ∫_{0}^{2π} μ_{λ}^{∗,ρ} (f^{θ})(x)e^{b(x)cosθ} dθ,

where f^{θ}(z) = f(z)e^{−e^{iθ} b(z)} for θ ∈ [0, 2π]. Then by the Minkowski inequality and (4.4) we get

∥μ_{λ,b}^{∗,ρ}(f)∥_{p,ω} ⩽ ( ∫_{ℝ^{n}} | \frac{1}{2π} ∫_{0}^{2π} μ_{λ}^{∗,ρ} (f^{θ})(x)e^{b(x)cosθ} dθ|^{p}ω(x) dx) ^{\frac{1}{p}}

⩽ \frac{1}{2π} ∫_{0}^{2π} ( ∫_{ℝ^{n}} |μ_{λ}^{∗,ρ} (f^{θ})(x)|^{p}ω(x)e^{pb(x)cosθ} dx) ^{\frac{1}{p}} dθ

⩽ C ∫_{0}^{2π} ( ∫_{ℝ^{n}} |f^{θ}(x)|^{p}ω(x)e^{pb(x)cosθ} dx) ^{\frac{1}{p}} dθ = C∥f∥_{p,ω} .

Thus we complete the proof of Theorem 2.1. □ Now, we turn to the proof of Theorem 2.2. Proof. Since μ_{Ω,S}^{ρ,b} ⩽ 2^{λn/2} μ_{λ,b}^{∗,ρ}, we only give the proof of Theorem 2.2 for

μ_{λ,b}^{∗,ρ}. Let Φ(t) = t(1 + log^{+} t). We first prove the following inequality

sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : μ_{λ,b}^{∗,ρ}(f)(x) > t\})

⩽ C_{∥b∥_{∗}} sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x)) > t\}). (4.6)

If ∥b∥_{∗} = 0, (4.6) holds obviously. As below we may assume ∥b∥_{∗} > 0. Denote

L_{δ,b}(f) = sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M_{δ}^{Δ} (μ_{λ,b}^{∗,ρ}(f))(x) > t\}),

then it is easy to see that

(4.7) sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : μ_{λ,b}^{∗,ρ}(f)(x) > t\}) ⩽ L_{δ,b}(f).

First we will prove that for any 0 < δ < 1 and r > 0, there is

(4.8) L_{δ,b}(f) ⩽ C_{δ}rL_{δ,b}(f)+CΦ( \frac{∥b∥_{∗}}{r^{1/δ}}) sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x)) > t\}).

By Lemma 3.5(1), we get for any t > 0, ω({x ∈ ℝ^{n} : M_{δ}^{Δ} (μ_{λ,b}^{∗,ρ}(f))(x) > t})

⩽ ω({x ∈ ℝ^{n} : M^{Δ} ([μ_{λ,b}^{∗,ρ}(f)]^{δ})(x) > t^{δ}, M^{♯,Δ} ([μ_{λ,b}^{∗,ρ}(f)]^{δ})(x) ⩽ rt^{δ}})
+ ω({x ∈ ℝ^{n} : M^{♯,Δ} ([μ_{λ,b}^{∗,ρ}(f)]^{δ})(x) > rt^{δ}})

⩽ Crω(\{x ∈ ℝ^{n} : M_{δ}^{Δ} (μ_{λ,b}^{∗,ρ}(f))(x) > t/2^{\frac{1}{δ}} \})
+ ω(\{x ∈ ℝ^{n} : M_{δ}^{♯,Δ} (μ_{λ,b}^{∗,ρ}(f))(x) > r^{\frac{1}{δ}} t\}).

Since 0 < δ < 1, we can choose an l satisfying 0 < δ < l < 1. By Lemma 3.9,

ω(\{x ∈ ℝ^{n} : M_{δ}^{♯,Δ} (μ_{λ,b}^{∗,ρ}(f))(x) > r^{\frac{1}{δ}} t\})
⩽ ω(\{x ∈ ℝ^{n} : M_{l}^{Δ} (μ_{λ}^{∗,ρ} (f))(x) > r^{\frac{1}{δ}} t/(2C∥b∥_{∗})\})
+ ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > r^{\frac{1}{δ}} t/(2C∥b∥_{∗})\}).

Note that Φ(ab) ⩽ Φ(a)Φ(b) for a, b ⩾ 0, and Φ is increasing and doubling, so we have

\frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M_{δ}^{Δ} (μ_{λ,b}^{∗,ρ}(f))(x) > t\})

⩽ \frac{Cr}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M_{δ}^{Δ} (μ_{λ,b}^{∗,ρ}(f))(x) > t/2^{\frac{1}{δ}} \})

+ \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M_{l}^{Δ} (μ_{λ}^{∗,ρ} (f))(x) > r^{\frac{1}{δ}} t/(2C∥b∥_{∗})\})

+ \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > r^{\frac{1}{δ}} t/(2C∥b∥_{∗})\})

⩽ C_{δ}rL_{δ,b}(f) + Φ( \frac{2C∥b∥_{∗}}{r^{1/δ}}) sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M_{l}^{Δ} (μ_{λ}^{∗,ρ} (f))(x) > t\})

+ Φ( \frac{2C∥b∥_{∗}}{r^{1/δ}}) sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t\}).

Applying Lemma 3.5(2) and Lemma 3.10, we have

L_{δ,b}(f) ⩽ C_{δ}rL_{δ,b}(f) + CΦ( \frac{∥b∥_{∗}}{r^{1/δ}}) sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t\})

+ CΦ( \frac{∥b∥_{∗}}{r^{1/δ}}) sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M_{l}^{♯,Δ} (μ_{λ}^{∗,ρ} (f))(x) > t\})

⩽ C_{δ}rL_{δ,b}(f) + CΦ( \frac{∥b∥_{∗}}{r^{1/δ}}) sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t\}).

Thus we obtain the result of (4.8). Next we will show that

(4.9)  L_{δ,b}(f) ⩽ C_{∥b∥_{∗}} sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t\}).

For b ∈ BMO, let b_{k} be the same as in Lemma 3.11. Then b_{k} ∈ L^{∞} and ∥b_{k}∥_{∗} ⩽ ∥b∥_{∗}. Since f is smooth with compact support, we may assume suppf ⊂ B(0, R). Then by Lemma 3.11, Theorem 2.1, Lemma 3.6 and tΦ(1/t) ⩾ 1, we get

\frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M_{δ}^{Δ} (μ_{λ,b_{k}}^{∗,ρ} (f))(x) > t\})

⩽ \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M(χ_{B(0,2R)} μ_{λ,b_{k}}^{∗,ρ} (f))(x) > t/2\})

+ \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M(χ_{B^{c} (0,2R)} μ_{λ,b_{k}}^{∗,ρ} (f))(x) > t/2\})

⩽ \frac{C}{tΦ(1/t)} ∫_{B(0,2R)} |μ_{λ,b_{k}}^{∗,ρ} (f)(x)|ω(x) dx

+ \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t/(Ck)\})

⩽ Cω(B(0, 2R))^{\frac{1}{2}} ( ∫_{B(0,2R)} |μ_{λ,b_{k}}^{∗,ρ} (f)(x)|^{2}ω(x) dx) ^{\frac{1}{2}}

+ \frac{C}{Φ(1/t)} ∫_{ℝ^{n}} Φ( \frac{Ck|f(x)|}{t} ) ω(x) dx

⩽ Cω(B(0, 2R))^{\frac{1}{2}} ( ∫_{B(0,R)} |f(x)|^{2}ω(x) dx) ^{\frac{1}{2}} + C_{k} ∫_{B(0,R)} Φ(|f(x)|)ω(x) dx.

So L_{δ,b_{k}} (f) < ∞. Then choose an r > 0 with r < 1/C_{δ}, applying (4.8) for b_{k}, we have

(1 − C_{δ}r)L_{δ,b_{k}} (f) ⩽ C_{δ,r,∥b∥_{∗}} sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t\}).

That is

(4.10)  L_{δ,b_{k}} (f) ⩽ C_{∥b∥_{∗}} sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t\}),

where C is independent of k. Thus we get (4.9) by letting k → ∞ in (4.10). By (4.7) and (4.9), we prove that (4.6) holds.

For β = 1, applying (4.6) and Lemma 3.6, we obtain

ω({x ∈ ℝ^{n} : μ_{λ,b}^{∗,ρ}(f)(x) > 1})

⩽ sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : μ_{λ,b}^{∗,ρ}(f)(x) > t\})

⩽ C_{∥b∥_{∗}} sup_{t>0} \frac{1}{Φ(1/t)} ω(\{x ∈ ℝ^{n} : M^{2}(f)(x) > t\})

⩽ C_{∥b∥_{∗}} sup_{t>0} \frac{1}{Φ(1/t)} ∫_{ℝ^{n}} \frac{|f(x)|}{t} ( 1 + log^{+} \frac{|f(x)|}{t} ) ω(x) dx

⩽ C_{∥b∥_{∗}} ∫_{ℝ^{n}} Φ(|f(x)|)ω(x) dx

= C_{∥b∥_{∗}} ∫_{ℝ^{n}} |f(x)|(1 + log^{+} |f(x)|)ω(x) dx.

Then by homogeneity, we complete the proof of Theorem 2.2. □

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School of Sciences (Received 15 05 2014) China University of Mining and Technology (Revised 07 07 2015) Beijing China

 linyan@cumtb.edu.cn
 xuanxiao1989@126.com