The boundedness of commutators of multilinear Marcinkiewicz integral

Nonlinear Analysis 195 (2020) 111727

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Nonlinear Analysis www.elsevier.com/locate/na

The boundedness of commutators of multilinear Marcinkiewicz integral Sha He a , Yiyu Liang b ,∗ a

Department of Basic Science, Beijing International Studies University, Beijing 100024, People’s Republic of China b Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People’s Republic of China

article

info

Article history: Received 11 February 2019 Accepted 4 December 2019 Communicated by Enrico Valdinoci MSC: primary 47B47 secondary 42B20 42B30 42B35 Keywords: Commutator Multilinear Marcinkiewicz integral BMO space Weighted Hardy space Multilinear Calderón–Zygmund operator

abstract Let µ be the multilinear Marcinkiewicz integral, ⃗b ∈ (BMO(Rn ))m and w ⃗ = ⃗ = (p1 , . . . , pm ), pi ∈ [1, ∞), for i ∈ {1, . . . , m}, (w1 , . . . , wm ) ∈ AP⃗ , where P and AP⃗ denotes the multiple weight class introduced by Lerner et al. (2009). In this article, for pi ∈ (1, ∞), we prove that the commutators µ⃗b , generated by the multilinear Marcinkiewicz integral µ and ⃗b, are bounded from Lpw11 (Rn ) × n to Lp (Rn ) if ⃗ b ∈ (BMO(Rn ))m , where 1/p = 1/p1 + · · · + 1/pm · · · × Lpwm ν⃗ m (R ) ∏ m w p/pi . For the endpoint case, we show two kinds of and νw (x) = w (x) i ⃗ i=1 estimates. One is that if w ⃗ ∈ A(1,...,1) and ⃗b ∈ (BMO(Rn ))m , then we obtain a weighted weak L log L type estimate for µ⃗b . On the other hand, it is well known 1 (Rn ) × · · · × H 1 (Rn ) to that the commutator µ⃗b may not be bounded from Hw w 1/m n n m ⃗ Lw (R ) if b ∈ (BMO(R )) . We obtain that when w ∈ A1+min{1/2,γ}/mn (Rn ) ∫ w(x) n m ⊂ (BMO(Rn ))m , the ⃗ satisfying n 1+|x|n dx < ∞ and b ∈ (BMO w (R )) R

1/m

1 (Rn ) × · · · × H 1 (Rn ) to L (Rn ), where commutator µ⃗b is bounded from Hw w w BMOw (Rn ) is a subspace of BMO(Rn ). This result is new even for the linear case. Let T be the multilinear Calderón–Zygmund operator. We also obtain the 1 (Rn ) × · · · × H 1 (Rn ) to L1/m (Rn ) if boundedness of commutator [⃗b, T ] from Hw w w ⃗b ∈ (BMOw (Rn ))m . © 2019 Elsevier Ltd. All rights reserved.

1. Introduction In 1990, Torchinsky and Wang [33] introduced the commutator µb of Marcinkiewicz integral µ and b ∈ BMO(Rn ) as follows, for f ∈ L1loc (Rn ) and x ∈ Rn , ⎧ ⏐ ⏐2 ⎫1/2 ⎨∫ ∞ ⏐∫ ⏐ dt ⎬ Ω (x − y) ⏐ ⏐ µb (f )(x) := [b(x) − b(y)]f (y) dy , ⏐ ⏐ 3 ⎩ 0 ⏐ B(x,t) |x − y|n−1 ⏐ t ⎭ ∗ Corresponding author. E-mail addresses: [email protected] (S. He), [email protected] (Y. Liang).

https://doi.org/10.1016/j.na.2019.111727 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

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S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

where Ω is homogeneous of degree zero and satisfies Lipschitz and vanish conditions on the unit sphere. They obtained the weighted boundedness of µb , namely, for p ∈ (1, ∞) and w belongs to the Muckenhoupt weight class Ap , there exists a positive constant C independent of f , such that ∥µb (f )∥Lpw (Rn ) ≤ C∥b∥BMO(Rn ) ∥f ∥Lpw (Rn ) . For the endpoint case, for b ∈ BMO(Rn ), the commutator µb may not be bounded from the weighted Lebesgue space L1w (Rn ) to the weighted weak Lebesgue space W L1w (Rn ) or from the weighted Hardy space Hw1 (Rn ) to the weighted Lebesgue space L1w (Rn ). To fix this problem, usually two kinds of ways are proposed. One is replacing L1w (Rn ) or Hw1 (Rn ) by some smaller spaces. Like [12], Ding, Lu and Zhang obtain a weighted weak L log L type estimate for µb with b ∈ BMO(Rn ), where weighted L log L is a weighted Orlicz space smaller than L1w (Rn ). The other one is to find a subspace X of BMO(Rn ) such that, for b ∈ X, µb is bounded from L1w (Rn ) to W L1w (Rn ) or from Hw1 (Rn ) to L1w (Rn ). In this paper, we find the space X (in fact, X is BMOw (Rn ), see Definition 1.8) and show that for b ∈ X, µb is bounded from Hw1 (Rn ) to L1w (Rn ). Thus, we partially fix the problem. From 1970s, the multilinear operators have obtained extensive study, see [1,7,8,20,22] for example. Multilinear Marcinkiewicz integral is a kind of the multilinear Littlewood–Paley operators, and the multilinear Littlewood–Paley operators play an important role in partial differential equations. In 1982, Fabes, Jerison and Kenig [13] studied a collection of multilinear Littlewood–Paley estimates and then applied them to two problems in partial differential equations. The first problem is the estimation of the square root of an elliptic operator in divergence form, and the second is the estimation of solutions to the Cauchy problem for nondivergence form parabolic equations. In 1983, Coifman, Deng and Meyer [5] studied the domain of the square root of certain accretive differential operators, in their proof they also need a multilinear Littlewood– Paley estimate. For more examples of applications of multilinear Marcinkiewicz integrals and multilinear Littlewood–Paley operators, see also [6,11,14,15]. On the other hand, Lerner et al. [23] introduced the multiple weight when they studied the multilinear Calder´ on–Zygmund theory. They proved the commutator [⃗b, T ] of multilinear Calder´on–Zygmund operator ⃗ = (w1 , . . . , wm ) T and ⃗b ∈ (BMO(Rn ))m is bounded from Lpw11 (Rn ) × · · · × Lpwmm (Rn ) to Lpνw⃗ (Rn ) for w satisfies the multiple AP⃗ condition and P⃗ = (p1 , . . . , pm ) ∈ (1, ∞)m , where 1/p = 1/p1 + · · · + 1/pm and ∏m νw⃗ (x) = i=1 wi (x)p/pi . From then on, there are many relevant works on weighted estimates for multilinear Calder´ on–Zygmund operators with similar assumptions on weights, see [9,24,26–28] for example. Later, Xue, Yabuta and Yan [35] obtained the strong weighted boundedness of generalized commutator of multilinear square operators, including the iterated commutator introduced by P´erez et al. [31] and the commutator first studied by P´erez and Gonz´ alez [30]. For the endpoint case, they also obtained the weighted L log L type estimates of these commutators. In this article, when w ⃗ = (w1 , . . . , wm ) satisfies the multiple AP⃗ condition with P⃗ = (p1 , . . . , pm ) and pi ∈ (1, ∞), i ∈ {1, . . . , m}, we prove that the commutators µ⃗b , generated by the multilinear Marcinkiewicz integral µ and ⃗b, is bounded from Lpw11 (Rn ) × · · · × Lpwmm (Rn ) to Lpνw⃗ (Rn ) if ⃗b ∈ (BMO(Rn ))m , where ∏m 1/p = 1/p1 + · · · + 1/pm and νw⃗ (x) = i=1 wi (x)p/pi , see Theorem 1.5. Compared to the linear case proved in [33], we give the proof of the weighted Lp boundedness of µ⃗b in detail and obtain a better sharp maximal function estimate by means of the maximal function ML(log L) (f⃗) introduced by Lerner et al. [23]. For the endpoint case, we show two kinds of estimates. One is that if w ⃗ ∈ A(1,...,1) and ⃗b ∈ (BMO(Rn ))m , then a weighted weak L log L type estimate for µ⃗b is obtained, see Theorem 1.6. On the other hand, we ∫ w(x) n m ⃗ show that when w ∈ A∞ (Rn ) satisfying Rn 1+|x| n dx < ∞ and b ∈ (BMO w (R )) , the commutator µ⃗ b is 1/m

bounded from Hw1 (Rn )×· · ·×Hw1 (Rn ) to Lw (Rn ), where BMOw (Rn ) is a subspace of BMO(Rn ) considered firstly by Bloom [2], see Theorem 1.9. This result is new even for the linear case, namely, µb is bounded from Hw1 (Rn ) to L1w (Rn ) if b ∈ BMOw (Rn ). But we still do not know whether µb is bounded from L1w (Rn ) to W L1w (Rn ) if b ∈ BMOw (Rn ).

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

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For the multilinear Calder´ on–Zygmund operator T , we also obtain the boundedness of [⃗b, T ] from 1/m 1 n × · · · × Hw (R ) to Lw (Rn ) if ⃗b ∈ (BMOw (Rn ))m , see Theorem 1.13. Note that, for the linear case, Liang, Ky and Yang [25] proved the commutator [b, T ] is bounded from Hw1 (Rn ) to L1w (Rn ) if and only if b ∈ BMOw (Rn ). But there are much differences in studying the linear and the multilinear operators on Hardy space, so the method of [25] cannot be used in our proof. In 2001, Grafakos and Kalton [19] studied the multilinear Calder´ on–Zygmund operators on Hardy space, and obtained they are bounded from p1 n pm n H (R ) × · · · × H (R ) to Lp (Rn ) with 1/p = 1/p1 + · · · + 1/pm . We will make use of the main idea of [19] in the proof of our theorems. To state our main results, we recall some definitions and notations. Hw1 (Rn )

Definition 1.1 ([4]). Let Ω be a function defined on (Rn )m with the following properties: (i) Ω is homogeneous of degree 0, i.e., for any λ > 0 and y = (y1 , . . . , ym ) ∈ (Rn )m , Ω (λy) = Ω (y).

(1.1)

(ii) Ω is Lipschitz continuous on (S n−1 )m , i.e. there exist 0 < γ < 1 and C > 0 such that for any ξ = (ξ1 , . . . , ξm ), η = (η1 , . . . , ηm ) ∈ (Rn )m , γ

|Ω (ξ) − Ω (η)| ≤ C|ξ ′ − η ′ | , where y ′ = (y1 , . . . , ym )′ =

(1.2)

(y1 ,...,ym ) |y1 |+···+|ym | .

(iii) The integration of Ω on (B(0, 1))m vanishes, namely, ∫ Ω (y) dy = 0. m(n−1) m (B(0,1)) |y| For any f⃗ = (f1 , . . . , fm ) ∈ S(Rn ) × · · · × S(Rn ), t > 0, we define Ω (y1 , . . . , ym )χ(B(0,1))m (y1 , . . . , ym ) , (|y1 | + · · · + |ym |)m(n−1) (y ym ) 1 1 ,..., , Kt (y1 , . . . , ym ) = mn K t t t

K(y1 , . . . , ym ) :=

and Gt (f⃗)(y) :=Kt ∗ (f1 ⊗ · · · ⊗ fm )(y) ∫ m ∏ 1 Ω (y − z1 , . . . , y − zm ) = m fi (zi ) dzi , m(n−1) t (B(y,t))m (|y − z1 | + · · · + |y − zm |) i=1 where B(x, t) := { y ∈ Rn : |y − x| ≤ t }. Then the multilinear Marcinkiewicz integral µ is defined by µ(f⃗)(x) :=

(∫

∞

|Gt (f⃗)(x)|2

0

dt t

)1/2 .

Definition 1.2. Given a collection of locally integrable functions ⃗b = (b1 , . . . , bm ). If µ is the multilinear Marcinkiewicz integral, then the multilinear commutators of µ are defined to be, for any f⃗ = (f1 , . . . , fm ) ∈ (S(Rn ))m and x ∈ Rn , m ∑ µ⃗b (f⃗)(x) := µbj (f⃗)(x), j=1

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

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where for j = 1, . . . , m, µbj (f⃗)(x) ⎛ ⎞1/2 ⏐2 ∫ ∞ ⏐⏐∫ m ⏐ ∏ Ω (x − y , . . . , x − y ) dt ⏐ ⏐ 1 m (bj (x) − bj (yj )) := ⎝ fi (yi )dyi ⏐ 2m+1 ⎠ . ⏐ ⏐ t ⏐ (B(x,t))m (|x − y1 | + · · · + |x − ym |)m(n−1) 0 i=1 Recall that a locally integrable function b is said to be in BMO(Rn ) if ∫ 1 ∥b∥BMO(Rn ) := sup |b(x) − bB | dx < ∞, B⊂Rn |B| B where the supremum is taken over all balls B ⊂ Rn . To state our main results, we first recall the definition of the Muckenhoupt weights. Definition 1.3. A non-negative measurable function w is said to belong to the class of Muckenhoupt weights, Aq (Rn ) with q ∈ [1, ∞), denoted by w ∈ Aq (Rn ) if, when q ∈ (1, ∞), [w]Aq (Rn )

1 := sup n B⊂R |B|

{

∫ w(x) dx B

1 |B|

∫

−q ′ /q

[w(y)]

}q/q′ < ∞,

dy

(1.3)

B

where 1/q + 1/q ′ = 1, or, when q = 1, [w]A1 (Rn )

1 := sup B⊂Rn |B|

{

∫ w(x) dx

−1

ess sup [w(y)]

} < ∞.

(1.4)

y∈B

B

Here the supremum is taken over all balls B ⊂ Rn . Let ⋃ A∞ (Rn ) :=

Aq (Rn ).

q∈[1,∞)

From the definition, it is easy to get If 1 < p < q < ∞, then A1 (Rn ) ⊂ Ap (Rn ) ⊂ Aq (Rn ).

(1.5)

Let w ∈ A∞ (Rn ) and q ∈ (0, ∞]. If q ∈ (0, ∞), then we let Lqw (Rn ) be the space of all measurable functions f such that {∫ }1/q q ∥f ∥Lqw (Rn ) := |f (x)| w(x) dx < ∞. (1.6) Rn

n ∞ n ∞ n When q = ∞, L∞ w (R ) is defined to be the same as L (R ) and, for any f ∈ Lw (R ), let n := ∥f ∥L∞ (Rn ) . ∥f ∥L∞ w (R )

We point out that if w ∈ A∞ (Rn ), then there exist p ∈ [1, ∞) and r ∈ (1, ∞) such that w ∈ Ap (Rn ) ∩ RHr (Rn ) (see, for example, [18, Chapter IV, Lemma 2.5]), where RHr (Rn ) denotes the reverse H¨ older class of weights w satisfying that there exists a positive constant C, depending on [w]Ap (Rn ) , such that, for any ball B ⊂ Rn , }1/r { ∫ ∫ 1 1 [w(x)]r dx ≤C w(x) dx; (1.7) |B| B |B| B moreover, there exist positive constants C1 ≤ C2 , depending on [w]Ap (Rn ) , such that, for any measurable sets E ⊂ B, ( )p ( )(r−1)/r w(E) |E| |E| C1 ≤ ≤ C2 (1.8) |B| w(B) |B|

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

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(see, for example, [18, Chapter IV, (1.6) and Theorem 2.9]). Then it is easy to get from (1.8) that w(2k B) ≲ 2kpn w(B).

(1.9)

The following multiple weight was introduced in [23], which is appropriate for multilinear operators. Definition 1.4 ([23]). Let p ∈ (0, ∞) and 1 ≤ p1 , . . . , pm < ∞ satisfying 1/p = 1/p1 + · · · + 1/pm . Given w ⃗ = (w1 , . . . , wm ) and P⃗ = (p1 , . . . , pm ), for x ∈ Rn , set vw⃗ (x) :=

m ∏

wi (x)p/pi .

i=1

We say that w ⃗ satisfies the AP⃗ condition if ( sup B 1 When pi = 1, ( |B|

1 |B|

∫ ∏ m

)1/p wi (x)

B

dx

B i=1 ′

∫

p/pi

)1/pi ′ ∫ m ( ∏ 1 1−pi ′ wi (x) dx < ∞. |B| B i=1

′

wi (x)1−pi dx)1/pi is understood as (inf B wi )−1 .

The main results of this paper are the following theorems. ∑m Theorem 1.5. Let 1/p = ⃗ ∈ AP⃗ . Let ⃗b ∈ (BMO(Rn ))m and µ be the i=1 1/pi with pi ∈ (1, ∞], w multilinear Marcinkiewicz integral. Then there exists a constant C such that ∥µ⃗b (f⃗)∥Lpν

w ⃗

(Rn )

≤ C∥⃗b∥(BMO(Rn ))m

m ∏

∥fi ∥Lpi (Rn ) ,

i=1

wi

where and in what follows, ∥⃗b∥(BMO(Rn ))m := supi=1,..., m ∥bi ∥BMO(Rn ) . For the endpoint case, we first have the following weighted weak L log L type estimate. Theorem 1.6. Let w ⃗ ∈ A(1,...,1) , ⃗b ∈ (BMO(Rn ))m . Let µ be the multilinear Marcinkiewicz integral. Then there exists a constant C, depending on ∥⃗b∥(BMO(Rn ))m , such that, for any t ∈ (0, ∞), νw⃗

({

x ∈ R : µ⃗b (f⃗)(x) > tm n

})

≤C

m (∫ ∏ i=1

Rn

[ ( )] )1/m |fi (x)| |fi (x)| + 1 + log wi (x) dx . t t

Remark 1.7. (i) Xue et al. [35] obtained the strong weighted boundedness of generalized commutator of multilinear square operators, including the iterated commutator introduced by P´erez et al. [31] and the commutator first studied by P´erez and Gonz´ alez [30]. Also, they obtained the weighted weak L log L type estimates of these commutators. Theorems 1.5 and 1.6 study the commutator introduced by Lerner et al. [23], which is different from the commutator studied in [35]. (ii) Since the kernel of multilinear Marcinkiewicz integral is more complex than that of multilinear Calder´ on–Zygmund operators, we cannot use the method of the proof of multilinear Calder´on–Zygmund operators as in [20,23,24,27]. To overcome this difficulty, we should decompose the integral region according to the geometric structure of the kernel in the proofs of Theorems 1.5 and 1.6, which is more subtle compared to that of the Calder´ on–Zygmund operators, see (2.5) and (2.6) for details. To state the endpoint boundedness on weighted Hardy space, we first recall some definitions.

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

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Let ϕ be a function in the Schwartz class, S(Rn ), satisfying ϕ(x) = 1 for all x ∈ B(⃗0, 1). The maximal function of a tempered distribution f ∈ S ′ (Rn ) is defined by setting, for any x ∈ Rn , Mϕ f (x) := sup |f ∗ ϕt (x)|,

(1.10)

t∈(0,∞)

where ϕt (·) := t1n ϕ(t−1 ·) for all t ∈ (0, ∞). Then the weighted Hardy space Hw1 (Rn ) is defined as the space of all tempered distributions f ∈ S ′ (Rn ) such that ∥f ∥Hw1 (Rn ) := ∥Mϕ f ∥L1w (Rn ) < ∞; see [17]. Notice that ∥ · ∥Hw1 (Rn ) defines a norm on Hw1 (Rn ), whose size depends on the choice of ϕ, but the space 1 Hw (Rn ) is independent of this choice. Definition 1.8. BMOw (Rn ) if

Let w ∈ A∞ (Rn ) and {

∥b∥BMOw (Rn ) := sup

B⊂Rn

∫

w(x) Rn 1+|x|n

1 w(B)

[∫ B∁

dx < ∞. A locally integrable function b is said to be in w(x) n dx |x − xB |

] [∫

]} |b(y) − bB | dy

< ∞,

(1.11)

B

where the supremum is taken over all balls B := B(xB , rB ) ⊂ Rn , with xB ∈ Rn and rB ∈ (0, ∞), and B ∁ := Rn \B. Here and hereafter, xB denotes the center of the ball B and rB its radius, ∫ ∫ 1 b(z) dz. w(B) := w(z) dz and bB := |B| B B It should be pointed out that the space BMOw (Rn ) has been considered first by Bloom [2] when studying the pointwise multipliers of weighted BMO spaces (see also [37]). In [25, Proposition 2.2], the authors proved BMOw (Rn ) ⊂ BMO(Rn ) and the inclusion is continuous. ∫ w(x) Also, they showed for any w ∈ A1+ δ (Rn ) with δ ∈ (0, 1] satisfying Rn 1+|x| n dx < ∞, the space n BMOw (Rn ) is not a trivial function space, moreover, any Lipschitz function b with compact support belongs to BMOw (Rn ) (see [25, Remark 1.2 (ii)]). Our second endpoint estimate for the commutator of Marcinkiewicz integral is as follows. Theorem 1.9. Let µ be the multilinear Marcinkiewicz integral and w ∈ A1+min{1/2,γ}/mn (Rn ) satisfying ∫ w(x) dx < ∞, where γ be as in (1.2). If ⃗b ∈ (BMOw (Rn ))m , then Rn 1+|x|n     µ⃗b (f⃗)

1/m

Lw

(Rn )

≤ C∥⃗b∥(BMOw (Rn ))m

m ∏

∥fi ∥Hw1 (Rn ) ,

(1.12)

i=1

where and in what follows, ∥⃗b∥(BMOw (Rn ))m := supi=1,...,m ∥bi ∥BMOw (Rn ) . Remark 1.10. (i) The results of Theorem 1.9 are new, even in the linear case. (ii) Since there are much differences when considering the linear and multilinear operators on Hardy space, we cannot use the method of [25]. Instead, we make use of the main idea of [19] in the proof. (iii) For weighted estimates on Hardy spaces, there is a possibility to have the weight only belonging to A∞ (Rn ), see [29] for example. However, the commutator is more singular than the associated Marcinkiewicz integral. Thus, we should use the atomic decomposition in the proof of Theorem 1.9 which leads to the condition w ∈ A1+min{1/2,γ}/mn (Rn ). Even in the linear case, we still cannot obtain the result for the weight belonging to A∞ (Rn ), see [25, Theorem 1.3]. Furthermore, it is still an open problem that whether can we replace the weight w in Theorem 1.9 by the multiple weights.

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

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At last, we also obtain the endpoint boundedness of the commutator of multilinear Calder´on–Zygmund operators on weighted Hardy spaces. Definition 1.11. Let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values in the space of tempered distributions: T : φ(Rn ) × · · · × φ(Rn ) → φ′ (Rn ). We say that T is an m-linear Calder´ on–Zygmund operator if for some qj ∈ [1, ∞), it extends to a bounded ∑m multilinear operator from Lq1 (Rn ) × · · · × Lqm (Rn ) to Lq (Rn ), where 1/q = i=1 1/qi , and if there exists a function K, defined away from the diagonal x = y1 = · · · = ym in (Rn )m+1 , satisfying ∫ K(x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym ) dy1 · · · dym T (f1 , . . . , fm )(x) = (Rn )m

for all x ̸∈ ∩m i=1 supp fi ;

and

A |K(y0 , y1 , . . . , ym )| ≤ ∑m ( k,l=0 |yk − yl |)mn ε

A|yj − yj ′ | |K(y0 , . . . , yj , . . . , ym ) − K(y0 , . . . , yj ′ , . . . , ym )| ≤ ∑m ( k,l=0 |yk − yl |)mn+ε

(1.13)

for some positive constants A, ε and all 0 ≤ j ≤ m, whenever |yj − yj ′ | ≤ 1/2 max0≤k≤m |yj − yk |. Definition 1.12. Given a collection of locally integrable functions ⃗b = (b1 , . . . , bm ). If T is the m-linear Calder´ on–Zygmund operator, then the multilinear commutators of T are defined to be [⃗b, T ](f⃗)(x) =

m ∑

[bi , T ](f⃗)(x),

i=1

where each term is the commutator of T in the ith entry with bi , that is [bi , T ](f⃗)(x) = bi (x)T (f1 , . . . , fi , . . . , fm )(x) − T (f1 , . . . , bi fi , . . . , fm )(x). ε (Rn ) satisfying Theorem 1.13. Let T be an m-linear Calder´ on–Zygmund operator and w ∈ A1+ mn ∫ w(x) n m dx < ∞, where ε be as in (1.13). If ⃗b ∈ (BMOw (R )) , then Rn 1+|x|n

  ⃗  [b, T ](f⃗)

1/m

Lw

(Rn )

≤ C∥⃗b∥(BMOw (Rn ))m

m ∏

∥fi ∥Hw1 (Rn ) .

(1.14)

i=1

Finally we make some conventions on notations. Throughout the whole article, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. The symbol A ≲ B means that A ≤ CB. If A ≲ B and B ≲ A, then we write A ∼ B. For any measurable subset E of Rn , we denote by E ∁ the set Rn \ E and its characteristic function by χE . 2. Proof of Theorems 1.5 and 1.6 This section is devoted to proving Theorems 1.5 and 1.6. To this end, we first need some basic facts of the theory of Orlicz spaces that we will state without proof. For more information about these spaces the readers may consult [34] and [32].

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

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Let Φ : [0, ∞) → [0, ∞) be a Young function. That is, a continuous, convex, increasing function with Φ(0) = 0 and such that Φ(t) → ∞ as t → ∞. The Orlicz space LΦ (Rn ), is defined to be the set of measurable functions f such that for some λ > 0, ) ( ∫ |f (x)| dx < ∞ Φ λ Rn with the Luxemburg norm { ∫ ∥f ∥LΦ (Rn ) = inf λ > 0 :

( Φ

Rn

|f (x)| λ

)

} dx ≤ 1 .

The Φ-average of a function f over a cube Q is denoted by ∥f ∥Φ,Q , where { ( ) } ∫ 1 |f (x)| ∥f ∥Φ,Q := inf λ > 0 : Φ dx ≤ 1 . |Q| Q λ In this paper, we consider the Young functions Φ(t) := t(1 + log+ t), where log+ t = max{0, log t}. The corresponding averages will be denoted by ∥ · ∥Φ,Q = ∥ · ∥L(log L),Q . Since t ≤ t(1 + log+ t), it is easy to see that |f |Q ≤ ∥f ∥L(log L),Q .

(2.1)

Also, for any function b ∈ BMO(Rn ) and any nonnegative function f , we have ∫ 1 |b(y) − bQ |f (y) dy ≤ C∥b∥BMO(Rn ) ∥f ∥L(log L),Q , |Q| Q

(2.2)

see [23, (2.14)]. The following maximal functions are introduced by Lerner et al. [23], which also play the key role in our proof. Definition 2.1. Given f⃗ = (f1 , . . . , fm ), the maximal operator M is defined by ∫ m ∏ 1 M(f⃗)(x) = sup |fi (yi )| dyi , Q∋x i=1 |Q| Q where the supremum is taken over all cubes Q containing x. For δ > 0, let Mδ be the maximal function ( )1/δ ∫ 1 δ δ 1/δ |f (y)| dy . Mδ f (x) = M (|f | ) (x) = sup Q∋x |Q| Q Moreover, let M ♯ be the sharp maximal function of Fefferman–Stein [16], ∫ ∫ 1 1 M ♯ (f )(x) = sup inf |f (y) − c| dy ∼ sup |f (y) − fQ | dy, Q∋x |Q| Q Q∋x c |Q| Q and Mδ♯ (f )(x)

(

♯

δ

= M (|f | )(x)

)1/δ

( = sup Q∋x

1 |Q|

∫ ⏐ ⏐ )1/δ ⏐ ⏐ δ . ⏐|f (y)| − c⏐ dy Q

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

9

The maximal function ML(log L) (f⃗)(x) is defined by ML(log L) (f⃗)(x) = sup

m ∏

∥fi ∥L(log L),Q .

Q∋x i=1

Then by (2.1), M(f⃗)(x) ≤ ML(log L) (f⃗)(x). We will use the following form of result of Fefferman and Stein [16]. Let 0 < p, δ < ∞, w ∈ A∞ . Then there exists a constant C > 0 such that ∫ ∫ ( )p p (Mδ f (x)) w(x) dx ≤ C Mδ♯ f (x) w(x) dx, Rn

(2.3)

(2.4)

Rn

for all functions f for which the left-hand side is finite. To prove Theorems 1.5 and 1.6, we first need the following sharp maximal function estimate of multilinear commutator. Proposition 2.2. Let µ⃗b be a multilinear commutator with ⃗b ∈ (BMO(Rn ))m and let 0 < δ < 1/m, 0 < δ < ε. Then there exists a constant C > 0, depending on δ and ε, such that for any x ∈ Rn , ( ( ) ) ( ) Mδ♯ µ⃗b (f⃗) (x) ≤ C∥⃗b∥(BMO(Rn ))m ML(log L) (f⃗)(x) + Mε µ(f⃗) (x) for all m-tuples f⃗ = (f1 , . . . , fm ) of bounded measurable functions with compact support. To prove Proposition 2.2, we need the following Kolmogorov inequality. Lemma 2.3 (Kolmogorov Inequality [18, p. 485]). Let 0 < p < q < ∞, then there exists a constant C = Cp,q such that for any measurable functions f ∥f ∥Lp (Q, dx ) ≤ C∥f ∥Lq,∞ (Q, dx ) . |Q|

|Q|

Proof of Proposition 2.2. We take the point of view of vector-valued singular integral operators. Fixing x ∈ Rn , let H be the Hilbert space ⎧ ⎫ (∫ )1/2 2 ⎨ ⎬ ∞ |h(t)| H = h : ∥h∥ = dt < ∞ . ⎩ ⎭ t2m+1 0 Then µ(f⃗)(x) = ∥St (f⃗)(x)∥, and for ⃗b ∈ (BMO(Rn ))m , µ⃗b (f⃗)(x) =

m ∑

∥[bi , St ](f⃗)(x)∥,

i=1

where St (f⃗)(x) =

∫ (B(x,t))m

m ∏ Ω (x − y1 , . . . , x − ym ) fi (yi )dyi . (|x − y1 | + · · · + |x − ym |)m(n−1) i=1

By linearity it is enough to consider the operator with only one symbol. Fixing b ∈ BMO(Rn ) and consider the operator µb (f⃗)(x) = ∥b(x)St (f⃗)(x) − St (bf⃗)(x)∥.

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

10

Note that for any constant λ we also have µb (f⃗)(x) = ∥(b(x) − λ)St (f⃗)(x) − St ((b − λ)f⃗)(x)∥. Let Q be a cube centered at x. Taking Q∗ = 2Q, and λ = bQ∗ . We decompose fi = fi0 + fi∞ , where fi0 = fi χQ∗ , fi∞ = fi χ(Q∗ )∁ , i = 1, . . . , m. Then m ∏ i=1

∑

fi (yi ) =

αm f1α1 (y1 ) · · · fm (ym ) =

m ∏

fi0 (yi ) +

∑ ˜ α1 αm f1 (y1 ) · · · fm (ym )

i=1

{α1 ,...,αm }∈{0,∞}

∑ ˜ α1 αm =: f⃗0 + f1 (y1 ) · · · fm (ym ), ∑ where each term in ˜ contains at least one αi ̸= 0. ∑ αm )(x)∥. Then for z ∈ Q, Let c = ˜cα1 ,...,αm with cα1 ,...,αm = ∥St ((b − bQ∗ )f1α1 , . . . , fm |µb (f⃗)(z) − c| ⏐ ⏐ ∑ ⏐ ⏐ ˜ α1 αm ⃗ ⃗ ⏐ = ⏐∥(b(z) − bQ∗ )St (f )(z) − St ((b − bQ∗ )f )(z)∥ − ∥St ((b − bQ∗ )f1 , . . . , fm )(x)∥⏐⏐ ⏐ ∑ ⏐ ˜ αm )(z)∥ St ((b − bQ∗ )f1α1 , . . . , fm = ⏐⏐∥ − (b(z) − bQ∗ )St (f⃗)(z) + St ((b − bQ∗ )f⃗0 )(z) + ⏐ ∑ ⏐ ˜ αm − ∥St ((b − bQ∗ )f1α1 , . . . , fm )(x)∥⏐⏐ ∑ ˜ αm ≤∥ −(b(z) − bQ∗ )St (f⃗)(z) + St ((b − bQ∗ )f⃗0 )(z) + St ((b − bQ∗ )f1α1 , . . . , fm )(z) ∑ ˜ αm St ((b − bQ∗ )f1α1 , . . . , fm )(x) ∥ − ≤ ∥(b(z) − bQ∗ )St (f⃗)(z)∥ + ∥St ((b − bQ∗ )f⃗0 )(z)∥ ∑ ˜ αm αm + ∥St ((b − bQ∗ )f1α1 , . . . , fm )(z) − St ((b − bQ∗ )f1α1 , . . . , fm )(x)∥ = |b(z) − bQ∗ |µ(f⃗)(z) + µ((b − bQ∗ )f⃗0 )(z) ∑ ˜ αm αm + ∥St ((b − bQ∗ )f1α1 , . . . , fm )(z) − St ((b − bQ∗ )f1α1 , . . . , fm )(x)∥. Since 0 < δ < 1, taking the average over the cube Q, it follows )1/δ ( )1/δ ∫ ∫ 1 1 δ δ δ ⃗ ⃗ ||µb (f )(z)| − |c| | dz ≤ |µb (f )(z) − c| dz |Q| Q |Q| Q ( )1/δ ( )1/δ ∫ ∫ 1 1 ≲ |(b(z) − bQ∗ )µ(f⃗)(z)|δ dz + |µ((b − bQ∗ )f⃗0 )(z)|δ dz |Q| Q |Q| Q [ )δ ]1/δ ∫ (∑ 1 ˜ α1 α αm αm + ∥St ((b − bQ∗ )f1 , . . . , fm )(z) − St ((b − bQ∗ )f1 1 , . . . , fm )(x)∥ dz |Q| Q

(

=: I + II + III. We estimate each term separately. For any 0 < q < ε/δ, by H¨older’s inequality, ( I≲

1 |Q|

∫

δq ′

|b(z) − bQ∗ | Q

)1/δq′ ( dz

1 |Q|

∫

δq |µ(f⃗)(z)| dz

Q

≲ ∥b∥BMO(Rn ) Mδq (µ(f⃗))(x) ≲ ∥b∥BMO(Rn ) Mε (µ(f⃗))(x).

)1/δq

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

11

For the second term II, using Kolmogorov inequality (Lemma 2.3) with δ < 1/m, the weak boundedness L1 × · · · × L1 → L1/m,∞ of µ [4, Theorem 1.1] and (2.2), we have 0 II ≲ ∥µ((b − bQ∗ )f10 , . . . , fm )∥L1/m,∞ (Q, dx ) |Q|

≲ ∥(b − bQ∗ )f10 ∥L1 (Q, dx ) |Q|

m ∏

∥fi0 ∥L1 (Q, dx )

i=2 m ∏

≲ ∥b∥BMO(Rn ) ∥f1 ∥L(log L),Q

|Q|

|fi |Q ≲ ∥b∥BMO(Rn ) ML(log L) (f⃗)(x).

i=2

Next we estimate III. At first, we denote III∞,...,∞ for the case that αi = ∞, i = 1, . . . , m. For 0 < δ < 1, by H¨ older’s inequality, we have ( )1/δ ∫ 1 ∞ ∞ ∞ ∞ δ ∗ ∗ ∥St ((b − bQ )f1 , . . . , fm )(z) − St ((b − bQ )f1 , . . . , fm )(x)∥ dz III∞,...,∞ := |Q| Q ∫ 1 ∞ ∞ ≤ ∥St ((b − bQ∗ )f1∞ , . . . , fm )(z) − St ((b − bQ∗ )f1∞ , . . . , fm )(x)∥ dz. |Q| Q To estimate the above term, we introduce some notations firstly. For fixed x, z ∈ Q and t > 0, denote Ξ (x, t) = {y ∈ Rn : |y − x| ≤ t, |y − z| ≤ t},

(2.5)

n

Ξ (z, t) = {y ∈ R : |y − z| ≤ t, |y − x| ≤ t}, Γ (x, t) = {y ∈ Rn : |y − x| ≤ t, |y − z| > t}, Γ (z, t) = {y ∈ Rn : |y − z| ≤ t, |y − x| > t}, ⃗ Θ(x, t) = Θ1 (x, t) × · · · × Θm (x, t), ⃗ Θ(z, t) = Θ1 (z, t) × · · · × Θm (z, t),

Θi (x, t) ∈ {Ξ (x, t), Γ (x, t)}, Θi (z, t) ∈ {Ξ (z, t), Γ (z, t)},

Note that Ξ (x, t) = Ξ (z, t) =: Ξ (t). For any fixed y ∈ Rn , denote Ξ (x, y) = {t > 0 : |y − x| ≤ t, |y − z| ≤ t}, Ξ (z, y) = {t > 0 : |y − z| ≤ t, |y − x| ≤ t}, Γ (x, y) = {t > 0 : |y − x| ≤ t, |y − z| > t}, Γ (z, y) = {t > 0 : |y − z| ≤ t, |y − x| > t}, ⃗ Θ(x, y) = Θ1 (x, y)× · · · × Θm (x, y), ⃗ Θ(z, y) = Θ1 (z, y)× · · · × Θm (z, y),

Θi (x, y) ∈ {Ξ (x, y), Γ (x, y)}, Θi (z, y) ∈ {Ξ (z, y), Γ (z, y)},

Note that Ξ (x, y) = Ξ (z, y) =: Ξ (y). With these notations at hand, we have ∞ ∞ ∥St ((b − bQ∗ )f1∞ , . . . , fm )(z) − St ((b − bQ∗ )f1∞ , . . . , fm )(x)∥ [∫ ⏐∫ m ∞⏐ ∏ Ω (z − y1 , . . . , z − ym ) ⏐ ∗ m = χ (z − y1 , . . . , z − ym )[b(y1 ) − bQ ] fi∞ (yi ) dyi ⏐ ∑m m(n−1) (B(0,t)) ⏐ n m 0 (R ) ( i=1 |z − yi |) i=1 ∫ Ω (x − y1 , . . . , x − ym ) m (x − y1 , . . . , x − ym ) − χ ∑m m(n−1) (B(0,t)) (Rn )m ( i=1 |x − yi |) ⎤1/2 ⏐2 m ⏐ ∏ dt ⏐ ×[b(y1 ) − bQ∗ ] fi∞ (yi ) dyi ⏐ 2m+1 ⎦ ⏐ t i=1

(2.6)

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

12

⎡ ∫ ⎢ ⎢ ≤⎣

⎞2

⎛ ∞

⎜ ⎜ ⎝

0

⎡ ∫ ⎢ +⎢ ⎣

∑

⃗ Θ(z,t)

⃗ Θ(z,t) ∃Θi (z,t)=Γ (z,t)

⎜ ⎜ ⎝

⎞2

⃗ Θ(x,t) ∃Θi (x,t)=Γ (x,t)

⎤1/2

m ∏

∫

∑

⎤1/2

m ∏ ⎟ |Ω (z − y1 , . . . , z − ym )| dt ⎥ ⎥ ∗| |b(y ) − b |fi∞ (yi )| dyi ⎟ 1 Q ∑m ⎠ t2m+1 ⎦ m(n−1) ( i=1 |z − yi |) i=1

⎛ ∞

0

⎡

∫

⃗ Θ(x,t)

⎟ |Ω (x − y1 , . . . , x − ym )| dt ⎥ ⎥ |b(y1 ) − bQ∗ | |fi∞ (yi )| dyi ⎟ ∑m ⎠ t2m+1 ⎦ m(n−1) ( i=1 |x − yi |) i=1

⏐ ⏐ ⏐ Ω (z − y , . . . , z − y ) Ω (x − y , . . . , x − y ) ⏐ ⏐ ⏐ 1 m 1 m +⎣ − ∑m ⏐ ∑m ⏐ m(n−1) m(n−1) ⏐ ⏐ 0 Ξ (t) ( ( i=1 |x − yi |) i=1 |z − yi |) ⎤1/2 )2 m ∏ dt ⎦ ×|b(y1 ) − bQ∗ | |fi∞ (yi )| dyi t2m+1 i=1 ∫

∞

(∫

=: III11 + III12 + III13 . ⃗ We discuss III11 at first. Notice that there are 2m − 1 terms in the summation over Θ(z, t). Moreover, ∑m 1 t > |z − yi |, i = 1, . . . , m, yields t > m i=1 |z − yi |. Also since t ∈ ∩Θi (z, yi ), x, z ∈ Q, ⏐ ⏐ | ∩ Θi (z, yi )| ≤ |Θi (z, yi )| = |Γ (z, yi )| ≤ ⏐|yi − x| − |yi − z|⏐ ≲ ℓ(Q). Then it follows from Minkowski’s inequality that ⎞1/2 ⎛ ⏐ ⏐2 ∫ ∫ m ⏐ Ω (z − y , . . . , z − y ) ⏐ ∏ dt ⏐ ⏐ 1 m ⎝ III11 ≲ fi∞ (yi )⏐ 2m+1 ⎠ dyi [b(y1 ) − bQ∗ ] ⏐ ∑m m(n−1) ⏐ t (Rn )m t∈∩Θi (z,yi ) ⏐ ( i=1 i=1 |z − yi |) ∫ m ∏ 1 |b(y1 ) − bQ∗ | |fi (yi )| ∑m dyi ≲ ℓ(Q)1/2 ∑ m(n−1) m+1/2 m ((Q∗ )∁ )m ( ( i=1 |z − yi |) i=1 i=1 |z − yi |) m ∞ ∫ ∏ ∑ |b(y1 ) − bQ∗ | |fi (yi )|dyi . ≲ ℓ(Q)1/2 ∑ mn+1/2 m k+1 Q)m \(2k Q)m ( i=1 k=1 (2 i=1 |z − yi |) Since z ∈ Q, (y1 , . . . , ym ) ∈ (2k+1 Q)m \ (2k Q)m , there exists i0 , 1 ≤ i0 ≤ m such that yi0 ̸∈ 2k Q, which ∑m yields |z − yi0 | > 2k ℓ(Q), so that i=1 |z − yi | > 2k ℓ(Q). Thus, ∞ ∫ m ∑ |b(y1 ) − bQ∗ | ∏ III11 ≲ |fi (yi )|dyi (2.7) k+1 Q)m \(2k Q)m 2kmn ℓ(Q)mn 2k/2 i=1 k=1 (2 )( ) ∫ ∫ ∞ m ( ∑ ∏ 1 1 −k/2 ≲ 2 |fi (yi )|dyi |b(y1 ) − bQ∗ | |f1 (y1 )|dy1 |2k+1 Q| 2k+1 Q |2k+1 Q| 2k+1 Q i=2 ≲

k=1 ∞ ∑

2−k/2

×

|fi |2k+1 Q

i=2

k=1

[

m ∏

1 |2k+1 Q|

∫

( k+1

) |b(y1 ) − b2k+1 Q ||f1 (y1 )| + |b2k+1 Q − bQ∗ | |f1 (y1 )| dy1

]

(2 ∞ Q ) m ∞ m ∑ ∏ ∑ ∏ −k/2 −k/2 ≲ ∥b∥BMO(Rn ) 2 ∥f1 ∥L(log L),2k+1 Q |fi |2k+1 Q + 2 k |fi |2k+1 Q k=1

i=2

k=1

i=1

≲ ∥b∥BMO(Rn ) ML(log L) (f⃗)(x), where in the fourth inequality, we make use of (2.2) and the last inequality follows from (2.1).

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

13

Since III12 follows the same lines as III11 , we have III12 ≲ ∥b∥BMO(Rn ) ML(log L) (f⃗)(x). Now we are in the position of calculating III13 . ⏐ ⏐ ⏐ Ω (z − y , . . . , z − y ) Ω (x − y , . . . , x − y ) ⏐ ⏐ 1 m 1 m ⏐ − ∑m ⏐ ⏐ ∑m m(n−1) m(n−1) ⏐ ⏐( ( i=1 |x − yi |) i=1 |z − yi |) ⏐ ⏐ ⏐ ⏐ 1 1 ⏐ ⏐ ≤ |Ω (z − y1 , . . . , z − ym )| ⏐ ∑m − ⏐ ∑ m(n−1) m(n−1) ⏐ m ⏐( ( i=1 |x − yi |) i=1 |z − yi |) 1 + ∑m |Ω (z − y1 , . . . , z − ym ) − Ω (x − y1 , . . . , x − ym )| =: A + B. m(n−1) ( i=1 |x − yi |) Since x, z ∈ Q and yi ∈ (Q∗ )∁ , by the mean value theorem, ⏐ ⏐ ⏐ ⏐ 1 1 |z − x| ⏐ ⏐ − ⏐ ∑m ⏐ ≤ ∑m ∑ mn−m+1 , m(n−1) m(n−1) ⏐ m ⏐( ( i=1 |x − yi |) ( i=1 |x − yi |) i=1 |z − yi |) which yields ℓ(Q) A ≲ ∑m mn−m+1 . ( i=1 |x − yi |) On the other hand, by making use of (1.2), |Ω (z − y1 , . . . , z − ym ) − Ω (x − y1 , . . . , x − ym )| ⏐ ⏐ ⏐ (z − y1 , . . . , z − ym ) (x − y1 , . . . , x − ym ) ⏐γ ⏐ ⏐ ∑m ∑m ≲⏐ − ⏐ |z − y | |x − y | i i i=1 i=1 γ

|z − x| ∼ ∑m γ. ( i=1 |x − yi |) Thus,

ℓ(Q)γ B ≲ ∑m . m(n−1)+γ ( i=1 |x − yi |)

Therefore, by Minkowski’s inequality and the similar arguments similar to (2.7), we obtain )1/2 ( (∫ ∫ ∞ ℓ(Q) dt III13 ≲ ∑m mn−m+1 2m+1 t ( i=1 |x − yi |) ((Q∗ )∁ )m max{|z−yi |,|x−yi |} ) ℓ(Q)γ + ∑m mn−m+γ ( i=1 |x − yi |) × |b(y1 ) − bQ∗ |

m ∏

|fi (yi )|dyi

i=1

∫ ∼ ((Q∗ )∁ )m

∫ m m ℓ(Q)|b(y1 ) − bQ∗ | ∏ ℓ(Q)γ |b(y1 ) − bQ∗ | ∏ |f (y )|dy + |fi (yi )|dyi ∑m ∑m i i i mn+γ mn+1 ( i=1 |x − yi |) ((Q∗ )∁ )m ( i=1 |x − yi |) i=1 i=1

≲ ∥b∥BMO(Rn ) ML(log L) (f⃗)(x). At last, we consider IIIα1 ,...,αm such that αj1 = · · · = αjl = 0 for some {j1 , . . . , jl } ⊂ {1, . . . , m}, where 1 ≤ l < m. Without loss of generality, we consider only the case α1 = · · · = αs = 0, 1 ≤ s < m, since the

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

14

other ones follow in analogous way. Then by the similar estimate as III∞,...,∞ , ( )1/δ ∫ 1 αm αm ∥St ((b − bQ∗ )f1α1 , . . . , fm )(z) − St ((b − bQ∗ )f1α1 , . . . , fm )(x)∥δ dz |Q| Q ∫ 1 αm αm ∥St ((b − bQ∗ )f1α1 , . . . , fm ≤ )(z) − St ((b − bQ∗ )f1α1 , . . . , fm )(x)∥ dz. |Q| Q Note that αm αm ∥St ((b − bQ∗ )f1α1 , . . . , fm )(z) − St ((b − bQ∗ )f1α1 , . . . , fm )(x)∥ [ ∫ ⏐∫ m ∞⏐ ∏ Ω (z − y1 , . . . , z − ym ) ⏐ α ∗ m (z − y , . . . , z − y )[b(y ) − b χ ] fi i (yi ) dyi = ⏐ 1 m 1 Q ⏐ (Rn )m (∑m |z − yi |)m(n−1) (B(0,t)) 0 i=1 i=1 ∫ Ω (x − y1 , . . . , x − ym ) m (x − y1 , . . . , x − ym )[b(y1 ) − bQ∗ ] − χ ∑m m(n−1) (B(0,t)) (Rn )m ( |x − y |) i i=1 ⎤1/2 ⏐2 m ⏐ ∏ α dt ⏐ × fi i (yi ) dyi ⏐ 2m+1 ⎦ ⏐ t i=1 ⎡ ⎛ ∫ ∞ ∫ ∑ ⎢ ⎜ |Ω (z − y1 , . . . , z − ym )| ⎜ |b(y1 ) − bQ∗ | ≲⎢ ∑m ⎣ ⎝ m(n−1) ⃗ Θ(z,t) ( 0 |z − y |) i ⃗ i=1 Θ(z,t) ∃Θi (z,t)=Γ (z,t)

×

s ∏

|fi0 (yi )|

i=1

⎡ ∫ ⎢ +⎢ ⎣

|fi∞ (yi )|

i=s+1

m ∏

⎤1/2 dt ⎦

)2 dyi

t2m+1

i=1

⎛ ∞

⎜ ⎜ ⎝

0

×

m ∏

s ∏

∫

∑ ⃗ Θ(x,t) ∃Θi (x,t)=Γ (x,t)

|fi0 (yi )|

m ∏

⃗ Θ(x,t)

|fi∞ (yi )|

m ∏

)2 dyi

i=1

i=s+1

i=1

|Ω (x − y1 , . . . , x − ym )| |b(y1 ) − bQ∗ | ∑m m(n−1) ( i=1 |x − yi |) ⎤1/2 dt ⎦ t2m+1

⏐ ⏐ ⏐ Ω (z − y , . . . , z − y ) Ω (x − y , . . . , x − y ) ⏐ ⏐ 1 m 1 m ⏐ + − ∑m ⏐ ∑m ⏐ m(n−1) m(n−1) ⏐ 0 Ξ (t) ⏐ ( ( i=1 |x − yi |) i=1 |z − yi |) ⎤1/2 )2 s m m ∏ ∏ ∏ dt ⎦ ×|b(y1 ) − bQ∗ | |fi0 (yi )| |fi∞ (yi )| dyi =: III21 + III22 + III23 . 2m+1 t i=1 i=s+1 i=1 [∫

∞

(∫

By Minkowski’s inequality and the arguments similar to (2.7), we obtain ∫ s ∏ III21 ≲ |b(y1 ) − bQ∗ | |fi (yi )| dyi (Q∗ )s

i=1

(∫

∫

dt

× ((Q∗ )∁ )m−s

≲ ℓ(Q)1/2

t∈∩Θi (z,yi )

t2m+1

∫ (Q∗ )s

∫ × ((Q∗ )∁ )m−s

|b(y1 ) − bQ∗ |

s ∏

)1/2

m |Ω (z − y1 , . . . , z − ym )| ∏ |fi (yi )| dyi (∑m )m(n−1) i=s+1 i=s+1 |z − yi |

|fi (yi )| dyi

i=1

∏m

i=s+1 |fi (yi )| dyi )m(n−1) ∑m m+1/2 ( i=1 |z − yi |) i=s+1 |z − yi |

(∑m

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

≲ ℓ(Q)1/2

∫ (Q∗ )s

|b(y1 ) − bQ∗ |

s ∏

15

∞ ∫ ∑

|fi (yi )| dyi

i=1

k=1

(2k+1 Q)m−s \(2k Q)m−s

∏m

|fi (yi )| )mn+1/2 dyi i=s+1 |z − yi | ∫ s ∞ ∏ ∑ |b(y1 ) − bQ∗ | |fi (yi )| dyi 2−k/2

× (∑ m ∫ ≲

i=s+1

(Q∗ )s ∞ ∑

i=1

∏m

(2k+1 Q)m−s

k=1

i=s+1 |fi (yi )| (2k ℓ(Q))mn

dyi

∫ ∫ m ∏ 1 1 ∗ |b(y ) − b |fi (yi )| dyi | |f (y )| dy 1 Q i i i |2k Q| 2k Q |2k+1 Q| 2k+1 Q i=1 i=s+1 k=1 (∞ ) m ∞ ∑ ∑ ∏ −k/2 −k/2 ≲ ∥b∥BMO(Rn ) 2 ∥f1 ∥L(log L),2k Q + 2 k|f1 |2k Q |fi |2k+1 Q ≲

2−k/2

s ∏

k=1

i=2

k=1

≲ ∥b∥BMO(Rn ) ML(log L) (f⃗)(x). Following similar lines of III21 , it is easy to get III22 ≲ ∥b∥BMO(Rn ) ML(log L) (f⃗)(x). Combining the methods of estimating III13 and III21 , it immediately yields III23 ≲ ∥b∥BMO(Rn ) ML(log L) (f⃗)(x). Therefore, we complete the proof. □ Similar to Proposition 2.2, we have the following results, the proof is omitted. Proposition 2.4. Let µ be the multilinear Marcinkiewicz integral, if δ ∈ (0, 1/m), then there exists a constant C > 0 such that Mδ♯ (µ(f⃗))(x) ≤ CM(f⃗)(x) for all m-tuples f⃗ = (f1 , . . . , fm ) of bounded measurable functions with compact support. By Propositions 2.2 and 2.4, we can prove the following proposition, which is the key step in the proof of Theorems 1.5 and 1.6. Proposition 2.5. Let p ∈ (0, ∞) and w ∈ A∞ (Rn ). Suppose ⃗b ∈ (BMO(Rn ))m , then there exists a constant C > 0 such that ∥µ⃗b (f⃗)∥Lpw (Rn ) ≤ C∥⃗b∥(BMO(Rn ))m ∥ML(log L) (f⃗)∥Lpw (Rn ) , (2.8) and ({ }) 1 n ⃗)(x) > tm w x ∈ R : µ ( f ⃗ 1 b t>0 Φ( t ) ({ }) 1 n ⃗)(x) > tm , ≤ C∥⃗b∥(BMO(Rn ))m sup w x ∈ R : M ( f L(log)L 1 t>0 Φ( t )

sup

(2.9)

for all f⃗ = (f1 , . . . , fm ) of bounded measurable functions with compact support. Before proving Proposition 2.5, we need the following lemma, which is [36, Corollary 1.5]. ∑m Lemma 2.6 ([36, Corollary 1.5]). Let 1/p = ⃗ ∈ AP⃗ and µ be the i=1 1/pi with 1 < pi ≤ ∞. Let w multilinear Marcinkiewicz integral. Then there exists a constant C > 0 such that ∥µ(f⃗)∥Lpν

w ⃗

(Rn )

≤C

m ∏ i=1

∥fi ∥Lpi (Rn ) . wi

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

16

Proof of Proposition 2.5. It is enough to prove the result for only one symbol, that is, we consider (∫ ∞ )1/2 2 dt ⃗ ⃗ µb (f ) = |[b, Gt ](f )(x)| . t 0 At first, we prove (2.8). We assume the right hand side of (2.8) is finite, since otherwise there is nothing needs to be proved. Taking exponent 0 < δ < ε < 1/m, then by Propositions 2.2, 2.4 and (2.3), we have ∥µb (f⃗)∥Lpw (Rn ) ≤ ∥Mδ (µb (f⃗))∥Lpw (Rn ) ≲ ∥Mδ♯ (µb (f⃗))∥Lpw (Rn ) ( ) ≲ ∥b∥BMO(Rn ) ∥ML(log L) (f⃗)∥Lpw (Rn ) + ∥Mε (µ(f⃗))∥Lpw (Rn ) ( ) ≲ ∥b∥BMO(Rn ) ∥ML(log L) (f⃗)∥Lpw (Rn ) + ∥Mε♯ (µ(f⃗))∥Lpw (Rn ) ) ( ≲ ∥b∥BMO(Rn ) ∥ML(log L) (f⃗)∥Lpw (Rn ) + ∥M(f⃗)∥Lpw (Rn )

(2.10)

≲ ∥b∥BMO(Rn ) ∥ML(log L) (f⃗)∥Lpw (Rn ) , where in the second and the fourth inequality we use (2.4), so we need ∥Mδ (µb (f⃗))∥Lpw (Rn ) and ∥Mε (µ(f⃗))∥Lpw (Rn ) be finite. We will show them as follows. Note that it is enough to prove the right hand of (2.8) is finite, that is, ∥ML(log L) (f⃗)∥Lpw (Rn ) < ∞, since otherwise there is nothing needs to be proved. Note that since w ∈ A∞ (Rn ), there exists p0 ∈ (max{1, pm}, ∞) such that w ∈ Ap0 (Rn ) and ε < p/p0 < 1/m, by the boundedness of Hardy–Littlewood maximal function, we have p /p ∥Mε (µ(f⃗))∥Lpw (Rn ) ≤ ∥Mp/p0 (µ(f⃗))∥Lpw (Rn ) = ∥M (µ(f⃗)p/p0 )∥L0p0 (Rn )

(2.11)

w

p /p ≤ C∥µ(f⃗)p/p0 ∥L0p0 (Rn ) = ∥µ(f⃗)∥Lpw (Rn ) . w

Then it is suffice to prove ∥µ(f⃗)∥Lpw (Rn ) is finite for all bounded measurable functions f⃗ with compact support and ∥ML(log L) (f⃗)∥Lpw (Rn ) is finite. The proof is as follows. Taking a ball B = B(0, R) centered at the origin with radius R. Without loss of generality, we assume supp fi ⊂ B(0, R) for i = 1, . . . , m. By the reverse H¨older’s inequality (1.7), w ∈ Lqloc (Rn ) for q sufficient close to 1 such that pq ′ ∈ (1/m, ∞). Thus, it follows from H¨older’s inequality and Lemma 2.6 that ∥µ(f⃗)∥Lpw (2B) ≤

(∫

)1/pq′ (∫ pq ′ ⃗ |µ(f )(x)| dx

2B

≲ ∥µ(f⃗)∥Lpq′ (Rn ) ≲

q

)1/pq

w(x) dx

(2.12)

2B m ∏

∥fi ∥Lqi (Rn ) < ∞,

i=1

∑m where 1/pq ′ = i=1 qi . On the other hand, for |x| > 2R, yi ∈ B(0, R), we have maxi=1,...,m {|x − yi |} ∼ ∑m i=1 |x − yi |, |x − yi | ∼ |x|, i = 1, . . . , m. Then by Minkowski’s inequality and (2.3), ⎞1/2 )2 m ∏ dt |Ω (x − y , . . . , x − y )| 1 m ⎠ |fi (yi )| dyi |µ(f⃗)(x)| ≤ ⎝ ∑m 2m+1 m(n−1) t 0 (B(x,t)∩B(0,R))m ( |x − y |) i i=1 i=1 (∫ )1/2 ∫ m ∞ ∏ dt 1 ≤ |fi (yi )| dyi ∑m 2m+1 m(n−1) (B(0,R))m maxi=1,...,m {|x−yi |} t ( i=1 |x − yi |) i=1 ∫ m ∏ 1 ∑m ∼ |fi (yi )| dyi mn (B(0,R))m ( i=1 |x − yi |) i=1 ⎛

∫

∞

(∫

(2.13)

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

∫ ∼ (B(0,R))m

≲

m ∏

1 n |x| i=1

17

m 1 ∏ |fi (yi )| dyi mn |x| i=1

∫

|fi (yi )| dyi ≲ M(f⃗)(x) ≲ ML(log L) (f⃗)(x).

B(0,|x|)

From the assumption that ∥ML(log L) (f⃗)∥Lp (w)(Rn ) is finite, we have ∥µ(f⃗)∥Lp ((2B)∁ ) ≲ ∥ML(log L) (f⃗)∥Lp ((2B)∁ ) < ∞. w

w

Thus, we have proved ∥Mε (µ(f⃗))∥Lpw (Rn ) is finite. Next, we show ∥Mδ (µb (f⃗))∥Lpw (Rn ) is finite. Similar to the argument of (2.11), it suffices to prove ∥µb (f⃗)∥Lpw (Rn ) is finite. At first, we assume b is bounded. Then, by Minkowski’s inequality, ⎛ ⎞1/2 ⏐2 ∫ ∞ ⏐⏐∫ m ⏐ ∏ dt Ω (x − y , . . . , x − y ) ⏐ ⏐ 1 m µb (f⃗)(x) = ⎝ fi (yi ) dyi ⏐ 2m+1 ⎠ (b(x) − b(y1 )) ⏐ ⏐ (B(x,t))m (∑m |x − yi |)m(n−1) ⏐ t 0 i=1

i=1

≤ |b(x)|µ(f⃗)(x) + µ(bf1 , f2 , . . . , fm )(x) ≲ µ(f⃗)(x) + µ(bf1 , f2 , . . . , fm )(x). Thus, following the similar arguments as (2.12), by Lemma 2.6 and b is bounded, we have ∥µb (f⃗)∥Lpw (2B) ≲ ∥µ(f⃗)∥Lpw (2B) + ∥µ(bf1 , f2 , . . . , fm )∥Lpw (2B) (∫ )1/pq′ (∫ )1/pq pq ′ ⃗ ≲ |µ(f )(x)| dx w(x) dx 2B

2B

(∫

pq ′

|µ(bf1 , . . . , fm )(x)|

+

)1/pq′ (∫ dx

2B

≲

m ∏

q

)1/pq

w(x) dx

2B

∥fi ∥Lqi (Rn ) + ∥b∥L∞ (Rn )

m ∏

∥fi ∥Lqi (Rn ) < ∞. w

w

i=1

i=1

On the other hand, for |x| > 2R, note that b is bounded, then similar to the argument of (2.13), we have µb (f⃗)(x) ≲ ∥b∥L∞ (Rn ) ML(log L) (f⃗)(x). From the assumption ∥ML(log L) (f⃗)∥Lpw (Rn ) is finite, we thus have ∥µb (f⃗)∥Lp ((2B)∁ ) ≲ ∥ML(log L) (f⃗)∥Lp ((2B)∁ ) < ∞. w

w

Thus, we proved ∥µb (f⃗)∥Lpw (Rn ) is finite when b is bounded. For general b, we use the limiting argument as [23, p1254]. Let {bj } be a sequence of functions such that ⎧ b(x) > j, ⎨j, bj (x) = b(x), |b(x)| ≤ j, ⎩ −j, b(x) < −j. Then bj is a sequence of bounded functions, bj converges pointwisely to b almost everywhere and ∥bj ∥BMO(Rn ) ≤ C∥b∥BMO(Rn ) . Thus, by making use of the above conclusions that (2.10) holds for bounded function b, we have ∥µbj (f⃗)∥Lpw (Rn ) ≲ ∥bj ∥BMO(Rn ) ∥ML(log L) (f⃗)∥Lpw (Rn ) ≲ ∥b∥BMO(Rn ) ∥ML(log L) (f⃗)∥Lpw (Rn ) . Then for bounded and compact support functions fi , i = 1, . . . , m, it follows from the boundedness of µ and the dominated convergence theorem that µbj (f⃗) converges to µb (f⃗) in Lp for p ∈ (1, ∞). Thus, there exists a subsequence {bj ′ } ⊂ {bj } such that µbj ′ (f⃗)(x) converges to µb (f⃗)(x) almost everywhere. Then by Fatou’s lemma, we get the required estimate. The proof of (2.9) is the similar to that of (2.8). The details being omitted. See also [23, (3.20)]. □

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

18

By Proposition 2.5, we can prove Theorems 1.5 and 1.6. Proof of Theorems 1.5 and 1.6. By the same argument as [23, Theorems 3.18 and 3.16] with (3.19) and (3.20) there replaced by (2.8) and (2.9), we finish the proof of Theorems 1.5 and 1.6. □ 3. The endpoint estimates of commutator on weighted Hardy spaces In this section, we prove Theorem 1.9. We first need the following endpoint boundedness of Marcinkiewicz integral operator. Proposition 3.1. Let Ω be as in Definition 1.1 and 0 < γ < 1 be as in (1.2). Suppose w ∈ A1+min{1/2,γ}/mn (Rn ). Then there exists a constant C > 0 such that ∥µ(f⃗)∥L1/m (Rn ) ≤ C w

m ∏

∥fi ∥Hw1 (Rn ) .

i=1

Before giving the proof of Proposition 3.1, we make some remarks. Remark 3.2. This result is better than Theorem 1.1 in [21], since we do not impose any restrictions on m and n. To prove Proposition 3.1, we need the following lemma. Lemma 3.3 ([10, Lemma 3.4]). Let w ∈ A∞ (Rn ) and p ∈ (0, 1). Then there is a constant C = C(p, n, w) n 1 n such that for all finite collection {Qk }m k=1 of cubes in R and all nonnegative functions gk ∈ Lw (R ) with supp gk ⊂ Qk we have m ( m   ) ∫ ∑ ∑   1     gk  ≤C gk (x)w(x) dx χQk  .     p  p w(Qk ) Qk k=1

k=1

Lw (Rn )

Lw (Rn )

Proof of Proposition 3.1. We use the atomic decomposition of Hw1 (Rn ) spaces. Since finite sums of atoms are dense in Hw1 (Rn ), we will work with such sums and obtain estimates independent of the number of terms ∑ in each sum. We assume each fi , i = 1, . . . , m, is a finite sum of Hw1 (Rn )-atoms, fi = k λi,k ai,k , where ai,k are (Hw1 , ∞)-atoms, which means supp ai,k ⊂ Qi,k , (3.1) ∫ ai,k (x) dx = 0, (3.2) Rn −1 n ≤ w(Qi,k ) ∥ai,k ∥L∞ , w (R )

(3.3)

then we get (∫ ∥ai,k ∥Lqw (Rn ) =

)1/q q

|a(y)| w(y)dy

≤ ∥ai,k ∥L∞ (Rn ) w(Qi,k )1/q ≤ w(Qi,k )1/q−1 ,

(3.4)

Qi,k

for any q ∈ [1, ∞). ˜ be the cube with the same center as Q and 8√n its side length. Let ℓ(Q) denote the For a cube Q, let Q side length of Q. Using multilinearity, we write ∑ µ(f1 , . . . , fm )(x) = λ1,k1 · · · λm,km µ(a1,k1 , . . . , am,km )(x). k1 ,...,km

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

19

Set |µ(f1 , . . . , fm )(x)| ≤ I1 (x) + I2 (x),

(3.5)

where ∑

I1 (x) =

|λ1,k1 | · · · |λm,km | |µ(a1,k1 , . . . , am,km )(x)|χQ˜ 1,k

k1 ,...,km

∑

I2 (x) =

|λ1,k1 | · · · |λm,km | |µ(a1,k1 , . . . , am,km )(x)|χQ˜ ∁

1

1,k1

k1 ,...,km

(x), ˜ ∩···∩Q m,km ˜∁ ∪···∪Q m,k

(x). m

Now, let us begin to discuss I1 . For fixed k1 , . . . , km , assume that ˜ 1,k ∩ · · · ∩ Q ˜ m,k ̸= ∅, Q m 1 since otherwise there is nothing needs to be proved. Suppose that Q1,k1 has the smallest size among all these cubes. We take a cube Gk1 ,...,km such that ˜ ˜ ˜ 1,k ∩ · · · ∩ Q ˜ m,k ⊂ Gk ,...,k ⊂ G ˜ k ,...,k ⊂ Q Q 1,k1 ∩ · · · ∩ Qm,km m m m 1 1 1 and w(Gk1 ,...,km ) ≥ Cw(Q1,k1 ). By H¨ older’s inequality, Lemma 2.6, and (3.4), we have ∫ 1 |µ(a1,k1 , . . . , am,km )(x)|w(x) dx w(Gk1 ,...,km ) Gk ,...,k m

1

−1/2

≲ w(Gk1 ,...,km )

∥µ(a1,k1 , . . . , am,km )∥L2w (Rn )

≲ w(Gk1 ,...,km )−1/2 ∥a1,k1 ∥L2w (Rn )

m ∏

n ∥ai,ki ∥L∞ w (R )

i=2 m ∏ 1/2−1

≲ w(Gk1 ,...,km )−1/2 w(Q1,k1 )

w(Qi,ki )−1

i=2

≲

(m ∏

) w(Qi,ki )−1

.

i=1

By Lemma 3.3, we have ∥I1 ∥L1/m (Rn ) w

    m ∏  ∑  −1   ≲ |λ1,k1 | · · · |λm,km | w(Qi,ki ) χQ˜˜ · · · χQ˜˜ 1,k1 m,km  k1 ,...,km  1/m n i=1 Lw (R )  ⎛ ⎞ m  ∏ ∑  ⎝ ≲ |λi,ki |w(Qi,ki )−1 χQ˜˜ ⎠   i,ki i=1 ki  1/m n Lw (R )  ⎛ ⎞ ∏  m ∑   −1  ⎝ ⎠ ∼ |λ |w(Q ) w(x)χ ˜ i,ki i,ki ˜   Q i,ki  1/m n i=1 ki L (R ) ⎛ ⎞  m  ∑ ∏   −1 ⎝  ⎠ ≲ |λ |w(Q ) w(x)χ ˜ i,ki i,ki ˜   Q i,ki  1 n i=1  ki L (R )

(3.6)

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

20

≲

m ∏ i=1

⎞ ⎛ ∑ ˜ )⎠ ⎝ |λi,ki |w(Qi,ki )−1 w(Q i,ki ki

⎞ ⎛ m ∏ ∑ ⎝ ≲ |λi,ki |⎠ . i=1

ki

Let A be a nonempty subset of {1, . . . , m}, and we denote the cardinality of A by |A|. Then 1 ≤ |A| ≤ m. Set A∁ = {1, . . . , m} \ A. For A = {1, . . . , m}, we define ⎞ ( ) ⎛ ⋂ ⋂ ⋂ ˜ i,k ⎠ = ˜ ∁i,k . ˜ ∁i,k ∩ ⎝ Q Q Q i

i

i∈A

Then we have

i

i∈A

i∈A∁

⎛( ˜ ∁1,k ∪ · · · ∪ Q ˜ ∁m,k = Q m 1

) ⋂

⋃ ⎝ A⊂{1,...,m}

i∈A

˜ ∁i,k Q i

⎛ ∩⎝

⎞⎞ ⋂

˜ i,k ⎠⎠ . Q i

i∈A∁

For fixed 1 ≤ r ≤ m, without loss of generality, we consider the particular case, that is, by permuting the indices, we assume x ∈ Er , where ˜ ∁1,k ∩ · · · ∩ Q ˜ ∁r,k ) ∩ (Q ˜ r+1,k ˜ Er = (Q r+1 ∩ · · · ∩ Qm,km ). r 1

(3.7)

Denoting the center of Qi,ki by ci,ki . Using the vanish condition of atom a1,k1 (3.2) and Minkowski’s ˜∁ ∪ · · · ∪ Q ˜∁ inequality, we have for x ∈ Q 1,k1 m,km , µ(a[1,k1 , . .⏐. , am,km()(x) ∫ ∞ ⏐∫ Ω (x − y1 , . . . , x − ym )χ(B(0,1))m ((x − y1 )/t, . . . , (x − ym )/t) ⏐ = ⏐ ∑m m(n−1) ⏐ (Rn )m 0 ( i=1 |x − yi |) ⎤1/2 ⏐2 ) m ⏐ ∏ Ω (x − c1,k1 , . . . , x − ym )χ(B(0,1))m ((x − c1,k1 )/t, . . . , (x − ym )/t) dt ⏐ ai,ki (yi )dyi ⏐ 2m+1 ⎦ − ∑m m(n−1) ⏐ t (|x − c1,k1 | + i=2 |x − yi |) i=1 ⎛ ∫ ∞ ⏐⏐ ∫ ⏐ Ω (x − y1 , . . . , x − ym )χ(B(0,1))m ((x − y1 )/t, . . . , (x − ym )/t) ⎝ ≤ ⏐ ∑m m(n−1) ⏐ n m (R ) 0 ( i=1 |x − yi |) ⎞1/2 ⏐2 m Ω (x − c1,k1 , . . . , x − ym )χ(B(0,1))m ((x − c1,k1 )/t, . . . , (x − ym )/t) ⏐⏐ dt ⎠ ∏ |ai,ki (yi )|dyi . − ⏐ 2m+1 ∑m m(n−1) ⏐ t (|x − c1,k | + |x − yi |) i=1 i=2

1

Note that the above square function is equal to ⎛ ⏐ ⏐2 ∫ ⏐ Ω (x − y , . . . , x − y ) ⏐ Ω (x − c , . . . , x − y ) ⏐ ⏐ 1,k m 1 m 1 ⎝ − ⏐ ∑m ⏐ ∑ m(n−1) m(n−1) ⏐ m ⏐ max {|x−yi |,|x−c1,k |}≤t ( (|x − c1,k1 | + i=2 |x − yi |) 1 i=1 |x − yi |) 1≤i≤m ⎞1/2 dt ⎠ × 2m+1 t ⎛ ⎞1/2 ⏐ ⏐ ∫ |x−c1,k | ⏐ Ω (x − y , . . . , x − y ) ⏐2 dt 1 ⏐ ⏐ 1 m ⎠ +⎝ ⏐ ⏐ ∑m m(n−1) ⏐ t2m+1 max1≤i≤m {|x−yi |} ⏐ ( |x − y |) i i=1 ⎛ ⎞1/2 ⏐ ⏐2 ∫ |x−y1 | ⏐ Ω (x − c ⏐ , . . . , x − y ) dt ⏐ ⏐ 1,k m 1 ⎠ =: I21 + I22 + I23 . +⎝ ⏐ ∑m ⏐ m(n−1) ⏐ t2m+1 max2≤i≤m {|x−yi |,|x−c1,k |} ⏐ ( |x − yi |) 1

i=1

(3.8)

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

21

From (1.2), we see that |Ω (x − y1 , . . . , x − ym ) − Ω (x − c1,k1 , . . . , x − ym )| ⏐γ ⏐ ⏐ (x − y1 , . . . , x − ym ) ℓ(Q1,k1 )γ (x − c1,k1 , . . . , x − ym ) ⏐⏐ ⏐ ∑m ∑m ∑ ≲ − ≲⏐ γ. m |x − c1,k1 | + i=2 |x − yi | ⏐ ( i=1 |x − yi |) i=1 |x − yi | ∑m For fixed Er , x ∈ Er , yi ∈ Qi,ki , we have max1≤i≤m {|x − yi |, |x − c1,k1 |} ∼ i=1 |x − yi |. Thus, I21 is controlled by (∫

ℓ(Q1,k1 )γ ∑m m(n−1)+γ ( i=1 |x − yi |)

∞

)1/2

dt

max1≤i≤m {|x−yi |,|x−c1,k |} 1

For I22 , note that |x − c1,k1 | ∼ max1≤i≤m |x − yi | ∼ (∫

1

I22 ≲ ∑m m(n−1) ( i=1 |x − yi |) ≲ ∑m m(n−1) ( i=1 |x − yi |)

i=1

|x − yi |, we thus have

|x−c1,k |

|x−c1,k |

1

1

t2m+1

|x−y1 |

)1/2

1

1

max1≤i≤m {|x−yi |}

(∫

1

∑m

t2m+1

ℓ(Q1,k1 )γ ∼ ∑m mn+γ . ( i=1 |x − yi |)

t2m+1 )1/2

dt

dt ℓ(Q1,k1 )1/2 ≲ ∑m . mn+1/2 ( i=1 |x − yi |)

The estimate of I23 is similar as that of I22 , thus, ℓ(Q1,k1 )1/2 . I23 ≲ ∑m mn+1/2 ( i=1 |x − yi |) )β

ℓ(Q

Therefore, (3.8) is controlled by a multiple of ∑m 1,k1 mn+β , where β = min{1/2, γ}. ( i=1 |x−yi |) Thus, by the fact that Q1,k1 is the smallest cube among all the cubes, we have µ(a1,k1 , . . . , am,km )(x) ∫ m ∏ ℓ(Q1,k1 )β |ai,ki (yi )|dyi ≲ ∑m mn+β (Rn )m ( |x − y |) i i=1 i=1 ∫ m ∏ ℓ(Q1,k1 )β ∼ |ai,ki (yi )|dyi (∑r )mn+β ∑m (Rn )m i=1 i=1 |x − ci,ki | + i=r+1 |x − yi | r ∫ m ∏ ∏ β ≤ ℓ(Q1,k1 ) |ai,ki (yi )|dyi ∥ai,ki ∥L∞ i=1

Qi,k

∫ ×

(∑r

i

i=r+1 m ∏

1 ∑m

)mn+β

|x − ci,ki | + i=r+1 |x − yi | i=r+1 r m ∏ ∏ 1 ≲ ℓ(Q1,k1 )β |Qi,ki | w(Qi,ki )−1 (∑r )rn+β i=1 i=1 i=1 |x − ci,ki | r m ∏ ∏ ℓ(Qi,ki )n+β/r ≲ w(Qi,ki )−1 , ( )n+β/r i=1 |x − ci,ki | + ℓ(Qi,ki ) i=1 (Rn )m−r

where we use

∫ Rn

for

(Hw1 , ∞)-atom

i=1

|a(y)| dy ≤ bn−M w(Q)−1 (b + |y − c|)M

∫ Rn

dyi

1 dy (1 + |y|)M

a if M > n, b is any positive number and c ∈ Rn in the third inequality.

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

22

Summing over all possible 1 ≤ r ≤ m and all possible combinations of subset of {1, . . . , m} of size r, we obtain the pointwise estimate µ(a1,k1 , . . . , am,km )(x) ≲

m ∏

ℓ(Qi,ki )n+β/m w(Qi,ki )−1 ( )n+β/m i=1 |x − ci,ki | + ℓ(Qi,ki )

(3.9)

˜∁ ∪ · · · ∪ Q ˜∁ for all x ∈ Q 1,k1 m,km . Hence,     m n+β/m −1  ∏  ∑ ℓ(Qi,ki ) w(Qi,ki ) |λ | · · · |λ | ∥I2 ∥L1/m (Rn ) ≲  1,k1 m,km ( )n+β/m    w k1 ,...,km  1/m n i=1 |x − ci,ki | + ℓ(Qi,ki ) Lw (R )     m ∑ ∏  ℓ(Qi,ki )n+β/m w(Qi,ki )−1   |λ | ≲ i,k ( )n+β/m  i     1 n |x − c | + ℓ(Q ) i,ki i,ki i=1 ki Lw (R ) m ∫ ∏ ∑ ℓ(Qi,ki )n+β/m w(Qi,ki )−1 |λi,ki | ( ≲ )n+β/m w(x) dx n |x − ci,ki | + ℓ(Qi,ki ) i=1 R ki ⎡ ∫ m ∑ ∏ w(x) ≲ |λi,ki |ℓ(Qi,ki )n+β/m w(Qi,ki )−1 ⎣ ( )n+β/m dx 2Qi,k |x − c | + ℓ(Q ) i,k i,k i=1 ki i i i ⎤ ∫ ∞ ∑ w(x) + ( )n+β/m dx⎦ . j+1 Q j i,ki \2 Qi,ki |x − ci,ki | + ℓ(Qi,ki ) j=1 2 Since w ∈ A1+β/mn (Rn ), there exists q ∈ (1, 1 + β/mn) such that w ∈ Aq (Rn ). Thus, it is easy to get ⎤ ⎡ m ∑ ∞ j ˜ ∏ ∑ Q ) w(2 i,ki ⎦ |λi,ki |ℓ(Qi,ki )n+β/m w(Qi,ki )−1 ⎣ ∥I2 ∥L1/m (Rn ) ≲ 1+β/mn j(n+β/m) |Q w 2 i,ki | i=1 ki j=0 [ ] m ∑ ∏ w(Qi,ki ) n+β/m −1 ≲ |λi,ki |ℓ(Qi,ki ) w(Qi,ki ) 1+β/mn |Qi,ki | i=1 ki ≲

m ∑ ∏

|λi,ki | ≲

m ∏

∥fi ∥Hw1 (Rn ) .

i=1

i=1 ki

Therefore, we complete the proof. □ In order to prove Theorem 1.9, we also need the following lemma. Lemma 3.4 ([25, Lemma 2.3]). Let w ∈ A∞ (Rn ) and q ∈ [1, ∞). Then there exists a positive constant C such that, for any f ∈ BMO(Rn ) and any ball B := B(xB , rB ) ⊂ Rn with some xB ∈ Rn and rB ∈ (0, ∞), [

1 w(B)

∫

]1/q q |f (x) − fB | w(x) dx ≤ C∥f ∥BMO(Rn ) .

B

Now we are preparing to prove Theorem 1.9. Proof of Theorem 1.9. We decompose the multilinear commutator µb (f⃗) as (3.5) into two parts, for x ∈ Rn , we have |µb (f1 , . . . , fm )(x)| ≤ I1 (x) + I2 (x),

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

23

where I1 (x) =

∑

|λ1,k1 | · · · |λm,km | |µb (a1,k1 , . . . , am,km )(x)|χQ˜ 1,k

k1 ,...,km

I2 (x) =

∑

|λ1,k1 | · · · |λm,km | |µb (a1,k1 , . . . , am,km )(x)|χQ˜ ∁

1

1,k1

k1 ,...,km

(x), ˜ ∩···∩Q m,km ˜∁ ∪···∪Q m,k

(x). m

Similar to the proof of (3.6), with Lemma 2.6 there replaced by Theorem 1.5, we have ∥I1 ∥L1/m (Rn ) ≲ ∥b∥BMO(Rn ) w

m ∏

∥fi ∥Hw1 (Rn ) .

i=1

1/m

Then, we estimate I2 (x). The Lw (Rn ) norm of I2 (x) is controlled by      ∑    |λ | · · · |λ |(b(x) − b )|µ(a , . . . , a , . . . , a )|χ 1,k1 m,km Ql,k 1,k1 l,kl m,km ∁ ∁ ˜ ˜   Q ∪···∪ Q l 1,k1 m,km  k1 ,...,km 1/m Lw (Rn )    ∑     + |λ | · · · |λ | |µ(a , . . . , (b − b )a , . . . , a )|χ 1,k1 m,km 1,k1 Ql,k l,kl m,km ∁ ∁ ˜ ˜   Q ∪···∪ Q l 1,k1 m,km  k1 ,...,km 1/m Lw

(Rn )

:= ∥I21 ∥L1/m (Rn ) + ∥I22 ∥L1/m (Rn ) . w

w

Since we have the pointwise estimate (3.9), then     m n+β/m −1  ∏  ∑ ℓ(Qi,ki ) w(Qi,ki ) ∥I21 ∥L1/m (Rn ) ≲  |λ | · · · |λ |(b(x) − b ) 1,k1 m,km Ql,k ( )n+β/m    l w  1/m n k1 ,...,km i=1 |x − ci,ki | + ℓ(Qi,ki ) Lw (R ) ⏐⏐ ⎛ ⎞ ⏐⏐ m ⏐⏐ ∏ ∑ |λ |ℓ(Q )n+β/m w(Q )−1 ⏐⏐ i,ki i,ki i,ki ⎝ ≲ ⏐⏐ ( )n+β/m ⎠ ⏐⏐ i=1 |x − ci,ki | + ℓ(Qi,ki ) ki ⏐⏐ i̸=l ⎛ ⎞⏐⏐ ⏐⏐ ∑ |λl,kl |(b(x) − bQl,k )ℓ(Ql,kl )n+β/m w(Ql,kl )−1 ⏐⏐ l ⏐⏐ ⎠ ×⎝ ( )n+β/m ⏐⏐ ⏐⏐ 1/m n |x − cl,kl | + ℓ(Ql,kl ) kl Lw (R )     ∏ ∑  ℓ(Qi,ki )n+β/m w(x)  ≲ |λ | i,ki ( )n+β/m     1 n w(Qi,ki ) |x − ci,ki | + ℓ(Qi,ki ) i̸=l  ki L (R )     ℓ(Ql,kl )n+β/m w(x)(b(x) − bQl,k )  ∑ l × |λ | l,kl ( )n+β/m     1 n  kl w(Ql,kl ) |x − cl,kl | + ℓ(Ql,kl ) L (R ) ⎛ ⎞    n+β/m  ∏∑ ℓ(Q ) w(x)   i,k i ⎠ ≲⎝ |λi,ki | (   |x − c | + ℓ(Q ))n+β/m  w(Qi,ki ) i̸=l ki

⎛ ∑ ℓ(Ql,kl )n+β/m ×⎝ |λl,kl | w(Ql,kl ) kl

:= J1 · J2 . Next, we estimate J1 , J2 respectively.

i,ki

i,ki

L1 (Rn )

    w(x)(b(x) − bQl,k )   l (   |x − c | + ℓ(Q ))n+β/m  l,kl

l,kl

L1 (Rn )

⎞ ⎠

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

24

For J1 , since w ∈ A1+

β mn

(Rn ), there exists q ∈ (1, 1 +

β mn )

such that w ∈ Aq (Rn ). Thus,

    w(x)   (   |x − c | + ℓ(Q ))n+β/m  i,ki i,ki L1 ∫ ∞ ∫ ∑ w(x) w(x) ≤ ( )n+β/m dx + ( )n+β/m dx j j−1 ˜ ˜ ˜ Qi,k Qi,k j=1 2 Qi,ki \2 i |x − ci,ki | + ℓ(Qi,ki ) i |x − ci,ki | + ℓ(Qi,ki ) ∞ ∑ ˜ i,k ) w(Qi,ki ) w(2j Q i ≲ , ≲ 1+β/mn 1+β/mn j(n+β/m) |Q |Qi,ki | i,ki | j=0 2 where the last inequality follows from (1.9). Hence, J1 ≲

∏

⎛ ⎞ ∑ ⎝ |λi,ki |⎠ .

i̸=l

ki

For J2 , by Lemma 3.4 and (1.9),     w(x)(b(x) − bQl,k )   l (  ) n+β/m  |x − c | + ℓ(Q )  l,kl l,kl L1 ∫ ∞ ∫ ∑ w(x)|b(x) − bQl,k | w(x)|b(x) − bQl,k | l l ≤ ( )n+β/m dx + ( )n+β/m dx jQ j−1 Q ˜ ˜ ˜ 2 \2 Q |x − c | + ℓ(Q ) |x − c | + ℓ(Q ) l,kl l,kl l,kl l,kl l,kl l,kl l,kl j=1 ∞ ∫ ∑ w(x)|b(x) − bQl,k | w(Q ) l,kl l . ≲ ( )n+β/m dx ≲ ∥b∥BMO(Rn ) 1+β/m j 1 ˜ |Q | l,k j=0 2 Ql,kl n l ˜ l,k | 2 j |Q l Thus, J2 ≲

∑

|λl,kl |∥b∥BMO(Rn ) .

kl

Therefore, ∥I21 ∥L1/m (Rn ) ≲ ∥b∥BMO(Rn ) w

m ∏

⎛

⎞ ∑

⎝ i=1

|λi,ki |⎠ .

(3.10)

ki

Now, we are in the position of considering I22 . For any (Hw1 (Rn ), ∞)-atom a, by Proposition 3.1, it suffices to prove ∥(b − bQ )a∥Hw1 (Rn ) ≲ ∥b∥BMOw (Rn ) . This was proved by Liang et al. [25, Theorem 1.3]. Hence, ∥I22 ∥L1/m (Rn ) ≲ ∥b∥BMOw (Rn ) w

m ∏

⎛

⎞ ∑

⎝ i=1

|λi,ki |⎠ .

ki

Together with (3.10) and (3.11), we immediately get ∥I2 ∥L1/m (Rn ) ≲ ∥b∥BMOw (Rn ) w

m ∏ i=1

⎛ ⎞ ∑ ⎝ |λi,ki |⎠ . ki

(3.11)

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

25

In conclusion, summing the estimates for ∥I1 ∥L1/m (Rn ) and for ∥I2 ∥L1/m (Rn ) , and taking the limit, we w w complete the proof of Theorem 1.9. □ 4. The endpoint boundedness of commutator of Calderón–Zygmund operator We prove Theorem 1.13 in the last part of this paper. In order to prove Theorem 1.13, we need the following lemmas. ∑m Lemma 4.1 ([23, Theorem 3.18]). Let 1/p = ⃗ ∈ AP⃗ and i=1 1/pi with pi ∈ (1, ∞], i = 1, . . . , m, w ⃗b ∈ (BMO(Rn ))m . Then there exists a constant C such that ∥[⃗b, T ](f⃗)∥Lpν

w ⃗

(Rn )

≤ C∥⃗b∥(BMO(Rn ))m

m ∏

∥fi ∥Lpi (Rn ) .

i=1

wi

Lemma 4.2 ([3, Theorem 7.2]). Let w ∈ A1+δ/n (Rn ) and X be a Banach space. Assume that T is a linear operator defined on the space of finite linear combinations of continuous (Hw1 (Rn ), ∞)-atoms with the property that } { sup ∥T (a)∥X : a is a continuous (Hw1 (Rn ), ∞)-atom < ∞. Then T admits a unique continuous extension to a bounded linear operator from Hw1 (Rn ) into X . Now we are ready to give the proof of Theorem 1.13. Proof of Theorem 1.13. The proof of this theorem is similar as that of Theorem 1.9. Thus, we only show the differences. We decompose the multilinear commutator [b, T ](f⃗) as (3.5) into two parts, for x ∈ Rn , |[b, T ](f1 , . . . , fm )(x)| ≤ I1 (x) + I2 (x), where I1 (x) =

∑

|λ1,k1 | · · · |λm,km | |[b, T ](a1,k1 , . . . , am,km )(x)|χQ˜ 1,k

(x), ˜ ∩···∩Q m,km

|λ1,k1 | · · · |λm,km | |[b, T ](a1,k1 , . . . , am,km )(x)|χQ˜ ∁

˜∁ ∪···∪Q m,k

1

k1 ,...,km

I2 (x) =

∑

1,k1

k1 ,...,km

(x).

m

Similar to the proof of (3.6), with Lemma 2.6 there replaced by Lemma 4.1, we have ∥I1 ∥L1/m (Rn ) ≲ ∥b∥BMO(Rn ) w

m ∏

∥fi ∥Hw1 (Rn ) .

i=1

1/m

Next, we estimate I2 (x). The Lw (Rn ) norm of I2 (x) is controlled by      ∑    |λ | · · · |λ |(b(x) − b )|T (a , . . . , a , . . . , a )|χ 1,k m,k Q 1,k l,k m,k ∁ ∁ ˜ ˜ m m 1 1 l,kl l   Q ∪···∪ Q 1,k1 m,km  k1 ,...,km 1/m Lw (Rn )    ∑     + |λ | · · · |λ | |T (a , . . . , (b − b )a , . . . , a )|χ 1,k1 m,km 1,k1 Ql,k l,kl m,km ∁ ∁ ˜ ˜   Q ∪···∪ Q l 1,k1 m,km  k1 ,...,km 1/m Lw

:= ∥I21 ∥L1/m (Rn ) + ∥I22 ∥L1/m (Rn ) . w

w

(Rn )

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

26

Let us estimate I21 . Using (3.2), (1.13), we have |T (a1,k1 , . . . , am,km )(x)| ⏐ ⏐∫ ⏐ ⏐ ⏐ ⏐ (K(x, y1 , . . . , ym ) − K(x, c1,k1 , . . . , ym )) a1,k1 (y1 ) · · · am,km (ym )dy1 · · · dym ⏐ =⏐ ⏐ ⏐ (Rn )m ∫ ε |y1 − c1,k1 | ≲ ∑m mn+ε |a1,k1 (y1 )| · · · |am,km (ym )|dy1 · · · dym . (Rn )m ( i=1 |x − yi |) ˜∁ ∩ · · · ∩ Q ˜ ∁ ) ∩ (Q ˜ r+1,k ˜ Let Er be as in (3.7). For x ∈ Er = (Q 1,k1 r,kr r+1 ∩ · · · ∩ Qm,km ), yi ∈ Qi,ki , 1 ≤ i ≤ r, we have 1 |x − yi | ≥ |x − ci,ki | − |yi − ci,ki | ≥ |x − ci,ki |. 2 Thus, by (3.1) and (3.3), |T (a1,k1 , . . . , am,km )(x)| ∫ ∫ ε ≲ |y1 − c1,k1 | |a1,k1 (y1 )|dy1 Rn

r ∏

(Rn )r−1 i=2

|ai,ki (yi )|dyi

∏m

|aj,kj (yj )|dyj )mn+ε ∑m (Rn )m−r i=1 |x − ci,ki | + j=r+1 |x − yj | 2 ( r ) ∏m ∫ −1 ∏ j=r+1 w(Qj,kj ) ε −1 −1 ≲ w(Q1,k1 ) |y1 − c1,k1 | dy1 w(Qi,ki ) |Qi,ki | (∑r )nr+ε Q1,k i=2 i=1 |x − ci,ki | 1 ε m r ∏ |Qi,k | ∏ |Q1,k1 | n 1 i ≲ (∑r )nr+ε , w(Qi,ki ) j=r+1 w(Qj,kj ) i=1 i=1 |x − ci,ki | ∫

×

j=r+1

( ∑ r 1

˜∁ ˜∁ Since Q1,k1 is the smallest cube among {Qi,ki }, 1 ≤ i ≤ r, and x ∈ Q 1,k1 ∩ · · · ∩ Qr,kr , x ∈ ˜ r+1,k ˜ Q r+1 ∩ · · · ∩ Qm,km , we get |T (a1,k1 , . . . , am,km )(x)| m r 1+ ε ∏ ∏ |Qi,ki | nr 1 ≲ ( )n+ rε w(Q j,kj ) j=r+1 i=1 w(Qi,ki ) |x − ci,ki | + ℓ(Qi,ki ) ε m 1+ ∏ |Qi,ki | nr ≲ ( )n+ rε . i=1 w(Qi,ki ) |x − ci,ki | + ℓ(Qi,ki ) Summing over all possible 1 ≤ r ≤ m and all possible combinations of subset of {1, . . . , m} of size r we obtain the pointwise estimate |T (a1,k1 , . . . , am,km )(x)| ≲

ε 1+ nm

m ∏ i=1

|Qi,ki |

ε ( )n+ m w(Qi,ki ) |x − ci,ki | + ℓ(Qi,ki )

˜∁ ∪ · · · ∪ Q ˜∁ for all x ∈ Q 1,k1 m,km . Then, similar to the proof of (3.10), we have ⎛ ⎞ m ∏ ∑ ⎝ ∥I21 ∥L1/m (Rn ) ≲ ∥b∥BMO(Rn ) |λi,ki |⎠ . w

i=1

ki

The estimate of I22 is the same as (3.11). Hence, we have ∥I2 ∥L1/m (Rn ) ≲ ∥b∥BMOw (Rn ) w

m ∏ i=1

⎛ ⎞ ∑ ⎝ |λi,ki |⎠ . ki

S. He and Y. Liang / Nonlinear Analysis 195 (2020) 111727

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