Weighted Boundedness for Commutators of Parameterized Littlewood

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 Lp (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 Rn (n > 2) equipped with normalized Lebesgue measure. Let Ω be a homogeneous function of degree zero and Z (1.1) Ω(x′ )dσ(x′ ) = 0. S n−1

The parameterized area integral µρΩ,S and parameterized Littlewood–Paley operator µ∗,ρ λ are defined by Z 2 Z Z 1 1 dy dt 2 Ω(y − z) µρΩ,S (f )(x) = f (z) dz , ρ tn+1 n−ρ Γ(x) t |y−z| 1 and Γ(x) = {(t, y) ∈ Rn+1 : |x − y| < t}. +

2010 Mathematics Subject Classification: Primary 42B20; Secondary 42B25. Key words and phrases: parameterized Littlewood-Paley operator, parameterized area integral, 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. 183

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LIN AND XUAN

Define the Hilbert spaces as follows. Z ∞Z H1 = h : khkH1 = 0

H2 =

Z h : khkH2 =

0

where λ > 1. Then

∞

Z

Rn

2 dy dt

|y| 32 , then for ρ > n/2, λ > 2 , there exists a constant C > 0, such that for all β > 0 and f ∈ L1 (Rn ), |{x ∈ Rn : |µ∗,ρ λ (f )(x)| > β}| 6 Ckf kL1 /β.

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ρ In 2013, the authors in [10] discussed the operators µ∗,ρ λ and µΩ,S with kernel satisfying (1.2) on weak Hardy spaces. Recently the authors in [11] gave the weighted Lp boundedness of µρΩ,S and µ∗,ρ λ with kernel satisfying (1.2).

Theorem 1.2. [11] Let Ω ∈ L2 (S n−1 ) be a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > 23 . Suppose ω ∈ Ap , then for ρ > n/2, λ > 2 and 1 < p < ∞, kµρΩ,S (f )kp,ω 6 Ckf kp,ω ,

kµ∗,ρ λ (f )kp,ω 6 Ckf kp,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 ∈ BM O, then the commutator of the Marcinkiewicz integral [b, µΩ ] is bounded on weighted spaces Lp (ω) for 1 < p < ∞ and ω ∈ Ap . In 2002, Ding, Lu and Yabuta [2] gave the weighted Lp boundedness of the higher order commutator µm Ω,b for rough Marcinkiewicz integral. In 2004, Ding, Lu and Zhang [3] gave the endpoint weighted esimates for the higher order commutator µm Ω,b . In 2007, Ding and Xue [4] gave the weighted boundedness and the weak L log L estimates for the commutators of parameterized Littlewood–Paley operators and area integrals with kernel satisfying the L2 -Dini condition. Inspired by the above results, in this paper, we will focus on the weighted Lp (1 < p < ∞) boundedness and the weighted endoint estimates (p = 1) for ∗,ρ the commutators µρ,b Ω,S and µλ,b , 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 ∈ L2 (S n−1 ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > 23 . If b ∈ BM O, then for 1 < p < ∞ and ω ∈ Ap , there exists a constant C > 0 such that for any f ∈ Lp (ω), kµρ,b Ω,S (f )kp,ω 6 Ckf kp,ω ,

kµ∗,ρ λ,b (f )kp,ω 6 Ckf kp,ω .

Theorem 2.2. Suppose that ρ > n/2, λ > 2, Ω ∈ L2 (S n−1 ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > 25 . If b ∈ BM O and ω ∈ A1 , 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 Z |f (x)| ρ,b + |f (x)| n ω({x ∈ R : µΩ,S (f )(x) > β}) 6 C 1 + log ω(x) dx, β β Rn n

ω({x ∈ R :

µ∗,ρ λ,b (f )(x)

> β}) 6 C

Z

Rn

|f (x)| + |f (x)| 1 + log ω(x) dx. β β

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 ∈ BM O. There exist constants C1 , C2 > 0, depending only on the dimension n, such that for 0 < C < kfCk2∗ , every cube Q in Rn , we have −1 Z C2 C|f (x)−fQ | e dx 6 C1 C −C |Q|. kf k∗ Q Lemma 3.2. Let 1 < p < ∞ and λ′ > 0. Then, when b(x) ∈ BM O with kbk∗ < ′ min{ Cλ′2 , C2 (p−1) }, where C2 is the constant in Lemma 3.1, we have eλ b(x) ∈ Ap . λ′ Proof. We have p−1 Z Z 1 − p−1 ′ ′ 1 1 eλ b(x) dx eλ b(x) dx |Q| Q |Q| Q p−1 Z Z 1 p−1 ′ ′ 1 1 eλ (b(x)−bQ ) dx e−λ (b(x)−bQ ) dx = |Q| Q |Q| Q p−1 Z Z λ′ 1 1 |b(x)−bQ | λ′ |b(x)−bQ | p−1 6 e dx e dx := IQ . |Q| Q |Q| Q Let λ0 = we have

λ′ p−1 .

If 1 < p < 2, then λ0 > λ′ . By taking C = λ0 in Lemma 3.1,

p−1 Z 1 eλ0 |b(x)−bQ | dx eλ0 |b(x)−bQ | dx |Q| Q Q !p p Z 1 C1 λ0 λ0 |b(x)−bQ | . = e dx 6 C2 |Q| Q kbk∗ − λ0

IQ 6

1 |Q|

Z

If p > 2, then λ0 6 λ′ . By taking C = λ′ in Lemma 3.1, we have p−1 Z Z 1 1 λ′ |b(x)−bQ | λ′ |b(x)−bQ | IQ 6 e dx e dx |Q| Q |Q| Q !p p Z 1 C1 λ′ λ′ |b(x)−bQ | = . e dx 6 C2 ′ |Q| Q kbk − λ ∗

The two cases imply the desired result.

Remark 3.1. It follows from Lemma 3.2 if 1 < p < ∞, λ′ > 0 and C2that ′ C2 (p−1) a(x), b(x) ∈ BM O with kak∗ 6 kbk∗ < min λ′ , λ′ , then eλ a(x) ∈ Ap and ′ the Ap constant of eλ a(x) satisfies  p  C1 λ0  , 1 < p < 2,  C2 −λ h ′ i 0 p eλ a(x) 6 kbk∗ ′  Ap   CC21 λ ′ , 2 6 p < ∞. kbk∗

−λ

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Lemma 3.3. Suppose b ∈ BM O. There is a positive constant C such that for all ball B ⊂ Rn , kb − bB kexpL,B 6 Ckbk∗ .

Proof. When C > C12 , by Lemma 3.1, we have −1 Z |b(x)−b | B 1 1 C2 1 C1 e Ckbk∗ dx 6 C1 − = . |B| B Ckbk∗ kbk∗ Ckbk∗ C2 C − 1 R |b(x)−bB | 1 1 +1 1 Taking C > CC , we have C2CC−1 6 1, then |B| e Ckbk∗ dx 6 1. By the B 2 definition of kb − bB kexpL,B , there is kb − bB kexpL,B 6 Ckbk∗ . Lemma 3.4. [6] Let 0 < r < l < ∞. For each function f , define 1

kf kW Ll = sup t|{x : |f (x)| > t}| l , Nl,r (f ) = sup t>0

E

kf χE kr 1 1 1 , = − , kχE ks s r l

where the supremum is taken over all the measurable sets E with 0 < |E| < ∞. Then l r1 kf kW Ll . kf kW Ll 6 Nl,r (f ) 6 l−r

Lemma 3.5. [3] (1) Let M ∆ , M ♯,∆ 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 ∈ Rn : M ∆ f (y) > λ, M ♯,∆ f (y) 6 ελ}) 6 C[ω]A∞ εω({y ∈ Rn : M ∆ 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 ϕ(λ)ω({y ∈ Rn : Mδ∆ f (y) > λ}) 6 C[ω]A∞ sup ϕ(λ)ω({y ∈ Rn : Mδ♯,∆ f (y) > λ}) λ>0

λ>0

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, m Z |f (y)| + |f (y)| m+1 ω({y : M f (y) > β}) 6 C 1 + log M ω(y) dy β β Rn for every locally integrable function f . Lemma 3.7. As for |y − z| > 4r, there is 4+2ǫ Z ∞ log( |y−z| (log rt )4+2ǫ r ) dt 6 C , 2ρ−n+1 (|y − z|)2ρ−n |y−z| t

where r > 0, 0 < ǫ < ρ −

n 2

and ρ >

n 2.

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


189

Lemma 3.8. [11] There exists a constant C > 0 such that for any z ∈ (8B ∗ )c , |y − z| > 6r, Ω(y − z) Ω(ω − x0 + y − z) C(1 + |Ω(y − z)|) α , |y − z|n−ρ − |ω − x0 + y − z|n−ρ 6 |y − z|n−ρ log |y−z| r

where Ω ∈ L2 (S n−1 ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > 32 ; B is a ball with center at x ¯ and radius r0 ; B ∗ is a ball with center at x ¯ and radius r = 2r0 and x0 , ω ∈ B. Lemma 3.9. Let b ∈ BM O, 0 < δ < l < 1. Suppose that ρ > n/2, λ > 2, Ω ∈ L2 (S n−1 ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > 25 . Then for any smooth function with compact support f , there exists a positive constant 0 < C = Cδ such that ∆ ∗,ρ 2 Mδ♯,∆ µ∗,ρ λ,b (f ) (x) 6 Ckbk∗ Ml (µλ (f ))(x) + M (f )(x) .

Proof. Given x ∈ Rn , let Q = Q(¯ x, r1 ) be a dyadic cube centered at x¯ with √ half side length r1 and x ∈ Q. Let B be a ball centered at x ¯ and with radius ∗ +f (1−χ8B ∗ ) := x , r) with r = 2r . Decompose f = f χ r0 = nr1 , and B ∗ = B(¯ 0 8B R ∗,ρ 1 ∗ )f2 ](u) du; then f1 + f2 . Take CQ = |Q| µ [(b − b B Q λ |µ∗,ρ λ,b (f )(u) − CQ |

∗,ρ ∗,ρ 6 |b(u) − bB ∗ |µ∗,ρ λ (f )(u) + µλ ((b − bB ∗ )f1 )(u) + |µλ ((b − bB ∗ )f2 )(u) − CQ |,

and

1 |Q|

Z 1δ ∗,ρ δ δ |µ (f )(u)| − |C | λ,b Q du Q

1δ Z 1 δ |(b(u) − bB ∗ )µ∗,ρ (f )(u)| du λ |Q| Q 1δ Z 1 ∗,ρ δ ∗ + Cδ |µ ((b − bB )f1 )(u)| du |Q| Q λ δ1 Z 1 ∗,ρ δ ∗ + Cδ |µ ((b − bB )f2 )(u) − CQ | du |Q| Q λ

6 Cδ

:= Cδ (I1 + I2 + I3 ).

1 As for I1 , we can choose 1 < r < min{ δl , 1−δ }, then by Hölder’s inequality,

I1 6

1 |Q|

Z

6 Ckbk∗

′

Q

|b(u) − bB ∗ |δr du

1 |Q|

Z

Q

1 δr′

l |µ∗,ρ λ (f )(u)| du

1 |Q|

1l

Z

Q

δr |µ∗,ρ λ (f )(u)| du

1 δr

6 Ckbk∗ Ml∆ (µ∗,ρ λ (f ))(x).

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LIN AND XUAN

As for I2 , applying Lemma 3.4 with 1s = 1δ − 1, Theorem 1.1, Hölder’s generalized inequality and Lemma 3.3, we have 1 1δ 1 δ 1 I2 6 kµ∗,ρ λ ((b − bB ∗ )f1 )kW L1 kχQ ks |Q| 1−δ Z C (3.1) 6 |b(u) − bB ∗ ||f (u)|du |8B ∗ | 8B ∗ 6 kb − bB ∗ kexpL,8B ∗ kf kL log L,8B ∗ 6 Ckbk∗ ML log L (f )(x).

Note that M 2 (f )(x) ≈ ML log L (f )(x), we have I2 6 Ckbk∗ M 2 (f )(x). p Finally, let us estimate I3 . Since f ∈ Lp , and µ∗,ρ λ is L bounded for 1 < p < ∞ by Theorem 1.2, then Z Z p1 Z p1 1 1 ∗,ρ p p′ p′ |µ (f )(u)| du |f (u)|p du . |µ∗,ρ (f )(u)|du 6 |Q| 6 C|Q| 2 2 λ λ Q

Q

Rn

This fact shows that µ∗,ρ λ (f2 )(u) < ∞ a.e. on Q, so except a subset E with measure zero, for all u ∈ Q r E, µ∗,ρ λ (f2 )(u) < ∞. Thus, Z 1 I3 6 |µ∗,ρ ((b − bB ∗ )f2 )(u) − (µ∗,ρ λ ((b − bB ∗ )f2 ))Q |du |Q| Q λ Z Z 1 6 |µ∗,ρ ((b − bB ∗ )f2 )(u) − µ∗,ρ λ ((b − bB ∗ )f2 )(v)|dvdu. |Q|2 QrE QrE λ Next we will prove the following fact. For any x0 , w ∈ Q r E,

∗,ρ J = |µ∗,ρ λ ((b − bB ∗ )f2 )(x0 ) − µλ ((b − bB ∗ )f2 )(w)| Z Z |b(z) − bB ∗ ||f (z)| |b(z) − bB ∗ ||f (z)| ε/2 6 Crε dz + Cr dz n+ε |z − x0 | |z − x0 |n+ε/2 (8B ∗ )c (8B ∗ )c Z |b(z) − bB ∗ ||f (z)| n (3.2) + Crρ− 2 dz n ∗ c |z − x0 | 2 +ρ (8B ) Z |b(z) − bB ∗ ||f (z)| +C dz |z−x0 | 2+ε ∗ c ) (8B ) |z − x0 |n (log r := T1 + T2 + T3 + T4 .

write J = kφt,y ((b − bB ∗ )f2 )(x0 )kH2 − kφt,y ((b − bB ∗ )f2 )(w)kH2

6 kφt,y ((b − bB ∗ )f2 )(x0 ) − φt,y ((b(z) − bB ∗ )f2 )(w)kH2 Z ∞Z 1 λn Z t−n φ x0 − z − y − φ w − z − y 6 t t 0 |y|1 2 12 dy dt × (b(z) − bB ∗ )f2 (z) dz := J1 + J2 . t ∞

Z

Since

J1 6

Z

0

+

λn

1 1+|y|

∞

Z

|y||z−x0 |/2 |y−z|6r |y − z| Z |x0 −y| λn 12 t (log t+|y−x0r|+C(ε)r )4+2ε dt × dy dz. (t + |x0 − y|)λn−2n tn+2ρ+1 |y−z|

(8B ∗ )c

Z

Notice that the function G(s) =

(log s)4+2ε sε

is decreasing when s > e(4+2ε)/ε and

t + |y − x0 | + C(ε)r |y − z| + C(ε)r > > C(ε) = e(4+2ε)/ε , r r then

[log( t+|y−x0r|+C(ε)r )]4+2ε ( t+|y−x0r|+C(ε)r )ε

6

)]4+2ε [log( |y−z|+C(ε)r r ( |y−z|+C(ε)r )ε r

.


201

Since t + |y − x0 | ∼ t + |y − x0 | + C(ε)r and 0 < ε < min{ 21 , (λ−2)n , ρ − n2 , α − 52 }, 2 then Z |x0 −y| λn t (log t+|y−x0r|+C(ε)r )4+2ε dt λn−2n n+2ρ+1 (t + |x − y|) t 0 |y−z| Z ∞ [log( |y−z|+C(ε)r )]4+2ε dt r 6C ε 2ρ−n+1−ε (|y − z| + C(ε)r) t |y−z| 6C

[log( |y−z|+C(ε)r )]4+2ε r . 2ρ−n |y − z|

Since |y − z| > 6r, there exists a constant l > 1 such that |y − z| + C(ε)r 6 2l |y − z|. Hence Z Z Ω(y − z) |b(z) − bB ∗ ||f (z)| ′′ J2.3.2 6 C |y − z|n−ρ |z−x0 | 2+ε ) (8B ∗ )c |z − x0 |n (log |y−z|>6r 2r 2 12 2l |y−z| Ω(w − x0 + y − z) (log r )4+2ε dy − dz |w − x0 + y − z|n−ρ |y − z|2ρ−n Z |b(z) − bB ∗ ||f (z)| 6C |z−x0 | 2+ε (8B ∗ )c |z − x0 |n (log 2r ) Z 12 (1 + |Ω(y − z)|)2 × dy dz. |y−z| 2α−4−2ε |y−z|>6r |y − z|n (log r ) R |b(z)−bB ∗ ||f (z)| By the estimate of (3.5), we get J2.3.2′′ 6 C (8B ∗ )c |z−x0 | 2+ε dz. n |z−x0 | (log

r

)

Combining the estimates of J2.3.1 , J2.3.2′ and J2.3.2′′ , we obtain Z Z |b(z) − bB ∗ ||f (z)| |b(z) − bB ∗ ||f (z)| ε/2 dz. J2.3 6 Cr dz + C n+ε/2 |z−x0 | 2+ε |z − x | ∗ c ∗ c 0 ) (8B ) (8B ) |z − x0 |n (log r

Then we complete the proof of (3.2). Next we will show that Ti 6 Ckbk∗ M 2 (f )(x), for i = 1, 2, 3, 4. Denote Bj = {z : |z − x ¯| < 2j r}, then |bBj+1 − bB ∗ | 6 jkbk∗ . Since M 2 (f ) ≈ ML log L (f ), by Lemma 3.3 we get ∞ Z X |b(z) − bB ∗ ||f (z)| ε T1 6 Cr dz |z − x ¯|n+ε j j+1 x| 2, and Ω ∈ L2 (S n−1 ) is a homogeneous function of degree zero satisfying (1.1) and (1.2) with α > 52 . Then for any smooth function with compact support f , there exists a positive constant 0 < C = Cδ such that Mδ♯,∆ µ∗,ρ λ (f ) (x) 6 CM (f )(x).

Proof. Let f1 , f2 , Q and B ∗ be the same as in the proof of Lemma 3.9. Then applying Lemma 3.4 with 1s = 1δ − 1 and Theorem 1.1, similarly to get (3.1), we have 1δ Z Z 1 C ∗,ρ δ |µ (f1 )(y)| dy 6 |f (y)|dy 6 CM (f )(x). |Q| Q λ |8B ∗ | 8B ∗

∗,ρ p Since f ∈ Lp for 1 < p < ∞, and µ∗,ρ λ is L bounded, then µλ (f2 )(u) < ∞ a.e. on Q, so except a subset E with measure zero, for all u ∈ Q r E, µ∗,ρ λ (f2 )(u) < ∞. Next we will prove the following fact. For any x0 , w ∈ Q r E, ∗,ρ J = µ∗,ρ λ (f2 )(x0 ) − µλ (f2 )(w) 6 CM (f )(x).

In fact, similarly to (3.2), we know that Z Z |f (z)| |f (z)| ε ε/2 J 6 Cr dz + Cr dz n+ε |z − x | |z − x0 |n+ε/2 0 (8B ∗ )c (8B ∗ )c Z Z n |f (z)| |f (z)| + Crρ− 2 dz + C dz n +ρ |z−x0 | 2+ε (8B ∗ )c |z − x0 | 2 (8B ∗ )c |z − x0 |n (log ) r


6 Crε

∞ Z X

|f (z)| dz |z − x ¯|n+ε

2j r6|z−¯ x| 0. Denote 1 Lδ,b (f ) = sup ω({x ∈ Rn : Mδ∆ (µ∗,ρ λ,b (f ))(x) > t}), t>0 Φ(1/t) then it is easy to see that 1 (4.7) sup ω({x ∈ Rn : µ∗,ρ λ,b (f )(x) > t}) 6 Lδ,b (f ). t>0 Φ(1/t) First we will prove that for any 0 < δ < 1 and r > 0, there is 1 kbk∗ (4.8) Lδ,b (f ) 6 Cδ rLδ,b (f )+CΦ 1/δ sup ω({x ∈ Rn : M 2 (f )(x)) > t}). r t>0 Φ(1/t) By Lemma 3.5(1), we get for any t > 0, ω({x ∈ Rn : Mδ∆ (µ∗,ρ λ,b (f ))(x) > t})

δ δ ♯,∆ δ δ 6 ω({x ∈ Rn : M ∆ ([µ∗,ρ ([µ∗,ρ λ,b (f )] )(x) > t , M λ,b (f )] )(x) 6 rt }) δ δ + ω({x ∈ Rn : M ♯,∆ ([µ∗,ρ λ,b (f )] )(x) > rt })

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LIN AND XUAN 1

δ 6 Crω({x ∈ Rn : Mδ∆ (µ∗,ρ λ,b (f ))(x) > t/2 }) 1

δ + ω({x ∈ Rn : Mδ♯,∆ (µ∗,ρ λ,b (f ))(x) > r t}).

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

δ ω({x ∈ Rn : Mδ♯,∆ (µ∗,ρ λ,b (f ))(x) > r t}) 1

δ 6 ω({x ∈ Rn : Ml∆ (µ∗,ρ λ (f ))(x) > r t/(2Ckbk∗ )}) 1

+ ω({x ∈ Rn : M 2 (f )(x) > r δ t/(2Ckbk∗ )}).

Note that Φ(ab) 6 Φ(a)Φ(b) for a, b > 0, and Φ is increasing and doubling, so we have 1 ω({x ∈ Rn : Mδ∆ (µ∗,ρ λ,b (f ))(x) > t}) Φ(1/t) Cr 1 δ 6 ω({x ∈ Rn : Mδ∆ (µ∗,ρ λ,b (f ))(x) > t/2 }) Φ(1/t) 1 1 δ ω({x ∈ Rn : Ml∆ (µ∗,ρ + λ (f ))(x) > r t/(2Ckbk∗ )}) Φ(1/t) 1 1 + ω({x ∈ Rn : M 2 (f )(x) > r δ t/(2Ckbk∗ )}) Φ(1/t) 2Ckbk∗ 1 6 Cδ rLδ,b (f ) + Φ sup ω({x ∈ Rn : Ml∆ (µ∗,ρ λ (f ))(x) > t}) r1/δ t>0 Φ(1/t) 2Ckbk∗ 1 +Φ sup ω({x ∈ Rn : M 2 (f )(x) > t}). r1/δ t>0 Φ(1/t) Applying Lemma 3.5(2) and Lemma 3.10, we have kbk∗ 1 Lδ,b (f ) 6 Cδ rLδ,b (f ) + CΦ 1/δ sup ω({x ∈ Rn : M 2 (f )(x) > t}) r t>0 Φ(1/t) kbk∗ 1 + CΦ 1/δ sup ω({x ∈ Rn : Ml♯,∆ (µ∗,ρ λ (f ))(x) > t}) r t>0 Φ(1/t) kbk∗ 1 6 Cδ rLδ,b (f ) + CΦ 1/δ sup ω({x ∈ Rn : M 2 (f )(x) > t}). r t>0 Φ(1/t) Thus we obtain the result of (4.8). Next we will show that (4.9)

Lδ,b (f ) 6 Ckbk∗ sup t>0

1 ω({x ∈ Rn : M 2 (f )(x) > t}). Φ(1/t)

For b ∈ BM O, let bk be the same as in Lemma 3.11. Then bk ∈ L∞ and kbk k∗ 6 kbk∗. 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 1 ω({x ∈ Rn : Mδ∆ (µ∗,ρ λ,bk (f ))(x) > t}) Φ(1/t) 1 6 ω({x ∈ Rn : M (χB(0,2R) µ∗,ρ λ,bk (f ))(x) > t/2}) Φ(1/t)


207

1 ω({x ∈ Rn : M (χB c (0,2R) µ∗,ρ λ,bk (f ))(x) > t/2}) Φ(1/t) Z C 6 |µ∗,ρ (f )(x)|ω(x) dx tΦ(1/t) B(0,2R) λ,bk 1 ω({x ∈ Rn : M 2 (f )(x) > t/(Ck)}) + Φ(1/t) Z 21 1 2 6 Cω(B(0, 2R)) 2 |µ∗,ρ (f )(x)| ω(x) dx λ,bk +

B(0,2R)

Ck|f (x)| Φ ω(x) dx t Rn Z Z 12 1 2 2 |f (x)| ω(x) dx + Ck 6 Cω(B(0, 2R)) C + Φ(1/t)

Z

B(0,R)

Φ(|f (x)|)ω(x) dx.

B(0,R)

So Lδ,bk (f ) < ∞. Then choose an r > 0 with r < 1/Cδ , applying (4.8) for bk , we have 1 (1 − Cδ r)Lδ,bk (f ) 6 Cδ,r,kbk∗ sup ω({x ∈ Rn : M 2 (f )(x) > t}). Φ(1/t) t>0 That is (4.10)

Lδ,bk (f ) 6 Ckbk∗ sup t>0

1 ω({x ∈ Rn : M 2 (f )(x) > t}), Φ(1/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 ∈ Rn : µ∗,ρ λ,b (f )(x) > 1}) 1 6 sup ω({x ∈ Rn : µ∗,ρ λ,b (f )(x) > t}) t>0 Φ(1/t) 1 6 Ckbk∗ sup ω({x ∈ Rn : M 2 (f )(x) > t}) t>0 Φ(1/t) Z 1 |f (x)| |f (x)| 6 Ckbk∗ sup 1 + log+ ω(x) dx t t t>0 Φ(1/t) Rn Z 6 Ckbk∗ Φ(|f (x)|)ω(x) dx n ZR = Ckbk∗ |f (x)| 1 + log+ |f (x)| ω(x) dx. Rn

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

References 1. Y. Ding, D. Fan, Y. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, J. Indiana Univ. Math. 48 (1999), 1037–1055. 2. Y. Ding, S. Z. Lu, K. Yabuta, A problem on rough parametric Marcinkiewicz functions, J. Austral. Math. Soc. 72 (2002), 13–21.

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3. Y. Ding, S. Z. Lu, P. Zhang, Weighted week type estimates for commutators of Marcinkiewicz integrals, Sci. China (Set. A) 47 (2004), 83–95. 4. Y. Ding, Q. Y. Xue, Endpoint estimates for commutators of a class of Littlewood–Paley operators, Hokkaido Math. 36 (2007), 245–282. 5. J. Duoandikoetxea, E. Seijo, Weighted inequalities for rough square functions through extrapolation, Studia Math. 149 (2002), 239–252. 6. J. Garcia-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities related topics, Amsterdam North-Holland, (1985), 239–252. 7. L. Hömander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93–140. 8. J. Lee, K. S. Rim, Estimates of Marcinkiewicz integral with bounded homogeneous kernel of degree zero, Integral Equations Oper. Theory 48 (2004), 213–223. 9. Y. Lin, Z. G. Liu, F. L. Gao, Endpoint estimates for parametrized Littlewood–Paley operator, Commun. Math. Anal. 12 (2012), 11–25. 10. Y. Lin, Z. G. Liu, D. L. Mao, Z. K. Sun, Parametrized area integrals on Hardy spaces and weak Hardy spaces, Acta Math. Sinica 29 (2013), 1857–1870. 11. Y. Lin, C. H. Song, X. Xuan, Sharp maximal function estimates for parameterized Littlewood– Paley operator and area integrals, Commun. Math. Anal. 16 (2014), 66–83. 12. M. Sakamoto, K. Yabuta, Boundedness of Marcinkiewicz functions, Studia Math. 135 (1999), 103–142. 13. E. M. Stein, G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159–172. 14. A. Torchinsky, S. Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235–243. 15. Q. Y. Xue, Parametrized Littlewood–Paley operators, Doctor Degree Dissertation, Beijing Normal University, Beijing, 2004. 16. M. Q. Zhou, Harmonic analysis notes, Peking University Press, Beijing, 1999. School of Sciences China University of Mining and Technology Beijing China [email protected] [email protected]

(Received 15 05 2014) (Revised 07 07 2015)