Multilinear Square Functions with Kernels of Dini's Type

Hindawi Publishing Corporation Journal of Function Spaces Volume 2016, Article ID 4876146, 11 pages http://dx.doi.org/10.1155/2016/4876146

Research Article Multilinear Square Functions with Kernels of Dini’s Type Zengyan Si1 and Qingying Xue2 1

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

2

Correspondence should be addressed to Zengyan Si; [email protected] Received 9 December 2015; Accepted 31 January 2016 Academic Editor: Rodolfo H. Torres Copyright © 2016 Z. Si and Q. Xue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let 𝑇 be a multilinear square function with a kernel satisfying Dini(1) condition and let 𝑇∗ be the corresponding multilinear maximal square function. In this paper, first, we showed that 𝑇 is bounded from 𝐿1 × ⋅ ⋅ ⋅ × 𝐿1 to 𝐿1/𝑚,∞ . Secondly, we obtained that if each 𝑝𝑖 > 1, then 𝑇 and 𝑇∗ are bounded from 𝐿𝑝1 (𝜔1 ) × ⋅ ⋅ ⋅ × 𝐿𝑝𝑚 (𝜔𝑚 ) to 𝐿𝑝 (]𝜔⃗ ) and if there is 𝑝𝑖 = 1, then 𝑇 and 𝑇∗ are bounded 𝑝/𝑝 𝑚 from 𝐿𝑝1 (𝜔1 ) × ⋅ ⋅ ⋅ × 𝐿𝑝𝑚 (𝜔𝑚 ) to 𝐿𝑝,∞ (]𝜔⃗ ), where ]𝜔⃗ = ∏𝑖=1 𝜔𝑖 𝑖 . Furthermore, we established the weighted strong and weak type ∗ boundedness for 𝑇 and 𝑇 on weighted Morrey type spaces, respectively.

1. Introduction and Main Results Let 𝑚(𝑡) ∈ 𝐿∞ and 𝑎 ∈ BMO. In 1978, Coifman and Meyer [1] introduced a class of multilinear operators as a multilinearization of Littlewood-Paley 𝑔-function as follows: ∞

B (𝑎, 𝑓) = ∫ (𝑓 ∗ 𝜙𝑡 ) (𝑎 ∗ Φ𝑡 ) 0

𝑚 (𝑡) 𝑑𝑡, 𝑡

(1)

̂ and Φ ̂ have compact support with 0 ∉ supp Φ. ̂ They where 𝜙 studied the 𝐿2 estimate of B by using the notion of Carleson measures. In 1982, letting 𝑚 ≥ 2 and 𝑝 ≥ 1, Yabuta [2] obtained the 𝐿𝑝 boundedness and BMO type estimates of B by weakening the assumptions in [1]. In 2002, Sato and Yabuta [3] studied the (𝐿𝑝1 × ⋅ ⋅ ⋅ × 𝐿𝑝𝑚 , 𝐿𝑝 ) boundedness of the following multilinear Littlewood-Paley 𝑔-function: ∞ 𝑚

⃗ (𝑥) = ∫ ∏ ((𝜙 ) ∗ 𝑓) (𝑥) 𝑇𝑔 (𝑓) 𝑖 𝑡 0

𝑖=1

𝑑𝑡 . 𝑡

(2)

The kernels in (1) and (2) are restricted to separable variable kernels. Thus, efforts have been made to study the above operators with kernels of nonseparated type. In 2015, Xue et al. [4] introduced and studied the weighted estimates

for the following multilinear Littlewood-Paley 𝑔-function with convolution type kernel: ∞

⃗ (𝑥) = (∫ 𝑔 (𝑓)

0

󵄨2 𝑑𝑡 1/2 󵄨󵄨 󵄨󵄨𝜓𝑡 ∗ 𝑓⃗ (𝑥)󵄨󵄨󵄨 󵄨 𝑡) , 󵄨

(3)

where 𝜓𝑡 ∗ 𝑓⃗ (𝑥) = ∫

𝑚

𝜓𝑡 (𝑦1 , . . . , 𝑦𝑚 ) ∏𝑓𝑗 (𝑥 − 𝑦𝑗 ) 𝑑𝑦𝑗 , (4) 𝑚

(R𝑛 )

𝑗=1

with 𝜓𝑡 (𝑦1 , . . . , 𝑦𝑚 ) = (1/𝑡𝑚𝑛 )𝜓(𝑦1 /𝑡, . . . , 𝑦𝑚 /𝑡). The importance of multilinear Littlewood-Paley 𝑔function and related multilinear Littlewood-Paley type estimates were shown in PDE and other fields (see [5–10]). For other recent works about multilinear Littlewood-Paley type operators, see [11, 12] and the references therein. In this paper, our aim is to study the boundedness of multilinear square functions with more rough kernels. To begin with, we give some notations and introduce some definitions. The following multiple weights classes 𝐴 𝑝⃗ were introduced and studied by Lerner et al. [13].

2

Journal of Function Spaces

Definition 1 (𝐴 𝑝⃗ weights class [13]). Let 1 ≤ 𝑝1 , . . . , 𝑝𝑚 < ∞, 1/𝑝 = 1/𝑝1 + ⋅ ⋅ ⋅ + 1/𝑝𝑚 . Given 𝜔⃗ = (𝜔1 , . . . , 𝜔𝑚 ), set ]𝜔⃗ = 𝑝/𝑝𝑖 . One says that 𝜔⃗ satisfies the 𝐴 𝑝⃗ condition if ∏𝑚 𝑖=1 𝜔𝑖 sup ( 𝑄

𝑚

1/𝑝 𝑚

1 𝑝/𝑝 ∫ ∏𝜔 𝑖 ) |𝑄| 𝑄 𝑖=1 𝑖

∏( 𝑖=1

⃗ (𝑥) = sup 𝑇 (𝑓) ⃗ (𝑥) , 𝑇∗ (𝑓) 𝛿

(11)

󵄨󵄨 󵄨󵄨 󵄨󵄨∫ 𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) 󵄨󵄨 2 2 0 󵄨󵄨 ∑𝑚 𝑖=1 |𝑥−𝑦𝑖 | >𝛿 󵄨 1/2 󵄨󵄨2 𝑚 󵄨󵄨 𝑑𝑡 󵄨 ⋅ ∏𝑓𝑗 (𝑦𝑗 ) 𝑑𝑦1 , . . . , 𝑑𝑦𝑚 󵄨󵄨󵄨 ) . 󵄨󵄨 𝑡 𝑗=1 󵄨

(12)

𝛿>0

1/𝑝𝑖󸀠

1 1−𝑝󸀠 ∫ 𝜔 𝑖) |𝑄| 𝑄 𝑖

(5)

where ∞

⃗ (𝑥) = (∫ 𝑇𝛿 (𝑓)

< ∞, 1−𝑝𝑖󸀠 1/𝑝𝑖󸀠

when 𝑝𝑖 = 1 and ((1/|𝑄|) ∫𝑄 𝜔𝑖

)

is understood as

−1

(inf 𝑄 𝜔𝑖 ) .

Definition 2 (Dini(𝑎) condition). Suppose that 𝜔(𝑡) : [0, ∞) 󳨃→ [0, ∞) is a nondecreasing function with 0 < 𝜔(1) < ∞. For 𝑎 > 0, one says that 𝜔 ∈ Dini(𝑎), if 𝜔𝑎 (𝑡) (6) 𝑑𝑡 < ∞. 𝑡 0 Definition 3 (kernels of type 𝜔(𝑡)). For any 𝑡 ∈ (0, ∞), a locally integrable function 𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) defined away from the diagonal 𝑥 = 𝑦1 = ⋅ ⋅ ⋅ = 𝑦𝑚 in (R𝑛 )𝑚+1 is called a kernel of type 𝜔(𝑡), if there is a positive constant 𝐴 > 0, such that |𝜔|Dini(𝑎) = ∫

1

1/2 ∞ 󵄨 󵄨2 𝑑𝑡 ) (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 )󵄨󵄨󵄨 𝑡 0

≤

The corresponding multilinear maximal square function 𝑇∗ is defined by

(7)

𝐴

1/2 ∞ 󵄨 󵄨2 𝑑𝑡 (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑧, 𝑦1 , . . . , 𝑦𝑚 ) − 𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 )󵄨󵄨󵄨 ) 𝑡 0

|𝑧 − 𝑥| ≤ 󵄨) , 󵄨󵄨 𝑚𝑛 𝜔 ( 𝑚 󵄨󵄨󵄨 𝑚 󵄨󵄨 󵄨 󵄨 ∑𝑗=1 󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨󵄨󵄨 (∑𝑗=1 󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨) whenever |𝑧 − 𝑥| ≤ (1/2)max𝑚 𝑗=1 |𝑥 − 𝑦𝑗 |; and

The aim of this paper is to study the bounded properties of multilinear square function 𝑇 and multilinear maximal square function 𝑇∗ with nonconvolution type kernels. It should be pointed out that the methods used in [4, 11, 12] do not work for Littlewood-Paley operators with more general nonconvolution type kernels, for the reason that the estimates there rely heavily on the convolution type kernels and the well-known Marcinkiewicz function studied in [14]. We formulate the main results of this paper as follows. Theorem 4. Let 𝑇 be a multilinear square function of type 𝜔(𝑡) and 𝜔 ∈ 𝐷𝑖𝑛𝑖(1). Then, 𝑇 can be extended to be a bounded operator from 𝐿1 × ⋅ ⋅ ⋅ × 𝐿1 to 𝐿1/𝑚,∞ .

󵄨󵄨 󵄨󵄨 𝑚𝑛 ; 󵄨 󵄨 (∑𝑚 𝑗=1 󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨)

𝐴

We always assume that 𝑇 and 𝑇∗ can be extended to be a bounded operator from 𝐿𝑞1 × ⋅ ⋅ ⋅ × 𝐿𝑞𝑚 to 𝐿𝑞 for some 1 < 𝑞, 𝑞1 , . . . , 𝑞𝑚 < ∞ with 1/𝑞1 + ⋅ ⋅ ⋅ + 1/𝑞𝑚 = 1/𝑞.

(8)

Theorem 5. Let 𝑇 be a multilinear square function of type 𝜔(𝑡) and 𝜔 ∈ 𝐷𝑖𝑛𝑖(1). Let 𝜔⃗ ∈ 𝐴 𝑝⃗ and 1/𝑝 = 1/𝑝1 + ⋅ ⋅ ⋅ + 1/𝑝𝑚 . Then, one has the following weighted estimates: (i) If 1 < 𝑝1 , . . . , 𝑝𝑚 < ∞, then 𝑚 󵄩󵄩 ⃗ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑇𝑓󵄩󵄩 𝑝 ≤ 𝐶 ∏ 󵄩󵄩𝑓𝑖 󵄩󵄩𝐿𝑝𝑖 (𝜔𝑖 ) . 󵄩 󵄩𝐿 (]𝜔⃗ )

∞

(13)

𝑖=1

󵄨 (∫ 󵄨󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑗 , . . . , 𝑦𝑚 ) 0

(ii) If 1 ≤ 𝑝1 , . . . , 𝑝𝑚 < ∞, then 1/2

󵄨2 𝑑𝑡 − 𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑗󸀠 , . . . , 𝑦𝑚 )󵄨󵄨󵄨󵄨 ) 𝑡 󵄨󵄨 󵄨 󵄨󵄨𝑦𝑗 − 𝑦𝑗󸀠 󵄨󵄨󵄨 𝐴 󵄨 󵄨 ), ≤ 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑚𝑛 𝜔 ( 𝑚 󵄨󵄨󵄨 󵄨 󵄨 𝑥 − 𝑦 ∑ 𝑥 − 𝑦 ) (∑𝑚 𝑗 󵄨󵄨󵄨 𝑗=1 󵄨󵄨 𝑗 󵄨󵄨 𝑗=1 󵄨󵄨

(9)

whenever |𝑦𝑗 − 𝑦𝑗󸀠 | ≤ (1/2)max𝑚 𝑗=1 |𝑥 − 𝑦𝑗 |. We say that 𝑇 is a multilinear square function with a kernel of type 𝜔(𝑡), if 󵄨 ∞ 󵄨󵄨 ⃗ (𝑥) = (∫ 󵄨󵄨󵄨󵄨∫ 𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) 𝑇 (𝑓) 󵄨 0 󵄨󵄨 (R𝑛 )𝑚 󵄨 (10) 1/2 󵄨󵄨2 𝑚 󵄨󵄨 𝑑𝑡 ⋅ ∏𝑓𝑗 (𝑦𝑗 ) 𝑑𝑦1 , . . . , 𝑑𝑦𝑚 󵄨󵄨󵄨󵄨 ) , 󵄨󵄨 𝑡 𝑗=1 󵄨 ∞ 𝑛 whenever 𝑥 ∉ ⋂𝑚 𝑗=1 supp 𝑓𝑗 and each 𝑓𝑗 ∈ 𝐶𝑐 (R ).

𝑚 󵄩󵄩 ⃗ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑇𝑓󵄩󵄩 𝑝,∞ ≤ 𝐶 ∏ 󵄩󵄩𝑓𝑖 󵄩󵄩𝐿𝑝𝑖 (𝜔𝑖 ) . 󵄩 󵄩𝐿 (]𝜔⃗ )

(14)

𝑖=1

Theorem 6. Let 𝑇∗ be a multilinear maximal square function of type 𝜔(𝑡) and 𝜔 ∈ 𝐷𝑖𝑛𝑖(1). Let 1/𝑝 = 1/𝑝1 + ⋅ ⋅ ⋅ + 1/𝑝𝑚 and 𝜔⃗ ∈ 𝐴 𝑝⃗ . Then, one has the following: (i) If 1 < 𝑝1 , . . . , 𝑝𝑚 < ∞, then 𝑚 󵄩󵄩 ∗ ⃗ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑇 𝑓󵄩󵄩 𝑝 ≤ 𝐶 ∏ 󵄩󵄩𝑓𝑖 󵄩󵄩𝐿𝑝𝑖 (𝜔𝑖 ) . 󵄩 󵄩𝐿 (]𝜔⃗ )

(15)

𝑖=1

(ii) If 1 ≤ 𝑝1 , . . . , 𝑝𝑚 < ∞, then 𝑚 󵄩󵄩 ∗ ⃗ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑇 𝑓󵄩󵄩 𝑝,∞ 󵄩 󵄩𝐿 (]𝜔⃗ ) ≤ 𝐶∏ 󵄩󵄩𝑓𝑖 󵄩󵄩𝐿𝑝𝑖 (𝜔𝑖 ) .

(16)

𝑖=1

Remark 7. When 𝜔(𝑥) = 𝑥𝛾 for some 𝛾 > 0, Theorems 4 and 5 were proved in [15].

Journal of Function Spaces

3

Remark 8. The theory on multilinear Calder´on-Zygmund operators has attracted much attention recently. In 2009, Maldonado and Naibo [16] established the weighted norm inequalities, with the Muckenhoupt weights, for bilinear Calder´on-Zygmund operators of type 𝜔(𝑡). In 2014, Lu and Zhang [17] obtained some multiple-weighted norm inequalities for multilinear Calder´on-Zygmund operators of type 𝜔(𝑡) and related operators. The paper is organized as follows. Section 2 contains some preliminary information and proofs of Theorems 4–6. In Section 3, we establish the weighted strong and weak type boundedness of 𝑇 and 𝑇∗ on Morrey type spaces.

𝜆 󵄨 󵄨 𝐸3 = {𝑥 ∈ R𝑛 \ Ω∗ : 󵄨󵄨󵄨𝑇 (𝑔1 , 𝑏2 , . . . , 𝑔𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝑚 } ; 2 .. . 𝜆 󵄨 󵄨 𝐸2𝑚 = {𝑥 ∈ R𝑛 \ Ω∗ : 󵄨󵄨󵄨𝑇 (𝑏1 , 𝑏2 , . . . , 𝑏𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝑚 } . 2 (18) It follows from property (P4) that 𝑚 𝑚 󵄨󵄨 󵄨󵄨 󵄨 ∗󵄨 −1/𝑚 󵄨󵄨 ∗ 󵄨󵄨 . 󵄨󵄨Ω 󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨󵄨Ω𝑗 󵄨󵄨󵄨󵄨 ≤ 𝐶∑ ∑ 󵄨󵄨󵄨𝑄𝑗,𝑘𝑗 󵄨󵄨󵄨 ≤ 𝐶 (𝛾𝜆) 󵄨 󵄨 𝑗=1

2. Proofs of the Main Theorems First, we give the proof of Theorem 4. Proof. The basic idea of the following arguments is essentially taken from [15, 17]. Set 𝐵 = ‖𝑇‖𝐿𝑞1 ×⋅⋅⋅×𝐿𝑞𝑚 →𝐿𝑞 . Without loss of generality, we can assume that ‖𝑓𝑗 ‖𝐿1 = 1, where 𝑗 = 1, . . . , 𝑚. For any fixed number 𝜆 > 0, we need to show that there is constant 𝐶 > 0 such that 󵄨 󵄨 󵄨󵄨 𝑛 󵄨 󵄨󵄨{𝑥 ∈ R : 󵄨󵄨󵄨𝑇 (𝑓1 , . . . , 𝑓𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝜆}󵄨󵄨󵄨 (17) ≤ 𝐶 (𝐴 + 𝐵)1/𝑚 𝜆−1/𝑚 . We perform Calder´on-Zygmund decomposition to each function 𝑓𝑗 at level (𝛾𝜆)1/𝑚 , where 𝛾 is a positive number to be determined later. Then, we obtain a sequence of pairwise disjoint cubes {𝑄𝑗,𝑘𝑗 }∞ 𝑘𝑗 =1 and decomposition 𝑓𝑗 = 𝑔𝑗 + 𝑏𝑗 = 𝑔𝑗 + ∑𝑘𝑗 𝑏𝑗,𝑘𝑗 . Moreover, we have (P1) supp(𝑏𝑗,𝑘𝑗 ) ⊂ 𝑄𝑗,𝑘𝑗 , (P2) ∫R𝑛 𝑏𝑗,𝑘𝑗 (𝑥)𝑑𝑥 = 0, (P3) ∫R𝑛 |𝑏𝑗,𝑘𝑗 (𝑥)|𝑑𝑥 ≤ 𝐶(𝛾𝜆)1/𝑚 |𝑄𝑗,𝑘𝑗 |, (P4) ∑𝑘𝑗 |𝑄𝑗,𝑘𝑗 | ≤ 𝐶(𝛾𝜆)1/𝑚 ,

𝑗=1 𝑘𝑗

(19)

By the 𝐿𝑞1 × ⋅ ⋅ ⋅×𝐿𝑞𝑚 → 𝐿𝑞,∞ boundedness of 𝑇 and property (P6), we get 󵄩 󵄩𝑞 󵄨󵄨 󵄨󵄨 𝑚 𝑞 −𝑞 󵄩 󵄩𝑞 󵄨󵄨𝐸1 󵄨󵄨 ≤ (2 𝐵) 𝜆 󵄩󵄩󵄩𝑔1 󵄩󵄩󵄩𝐿𝑞1 (R𝑛 ) ⋅ ⋅ ⋅ 󵄩󵄩󵄩𝑔𝑚 󵄩󵄩󵄩𝐿𝑞𝑚 (R𝑛 ) ≤ 𝐶𝐵𝑞 𝛾𝑞−1 𝜆−1/𝑚 .

(20)

Thus, we have 󵄨󵄨 󵄨󵄨{𝑥 ∈ R𝑛 󵄨

𝑚

2 󵄨 ⃗ (𝑥)󵄨󵄨󵄨󵄨 > 𝜆}󵄨󵄨󵄨󵄨 ≤ 𝐶∑ 󵄨󵄨󵄨𝐸 󵄨󵄨󵄨 + 𝐶 󵄨󵄨󵄨Ω∗ 󵄨󵄨󵄨 : 󵄨󵄨󵄨󵄨𝑇 (𝑓) 󵄨 󵄨 󵄨 𝑠󵄨 󵄨 󵄨 𝑠=1

𝑚

2

(21)

−1/𝑚 󵄨 󵄨 ≤ 𝐶∑ 󵄨󵄨󵄨𝐸𝑠 󵄨󵄨󵄨 + 𝐶𝐵𝑞 𝛾𝑞−1/𝑚 𝜆−1/𝑚 + 𝐶 (𝛾𝜆) . 𝑠=2

To complete the proof, we need to estimate |𝐸𝑠 | for 2 ≤ 𝑠 ≤ 2𝑚 . Suppose that for some 1 ≤ 𝑙 ≤ 𝑚 there are 𝑙 bad functions and 𝑚 − 𝑙 good functions appearing in 𝑇(ℎ1 , . . . , ℎ𝑚 ), where ℎ𝑗 ∈ {𝑔𝑗 , 𝑏𝑗 }. For simplicity, we assume that the bad functions appear at the entries 1, . . . , 𝑙, and denote the corresponding term by |𝐸𝑠(𝑙) | to distinguish it from the other terms. That is, we will consider 󵄨󵄨 (𝑙) 󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨𝐸𝑠 󵄨󵄨 = 󵄨󵄨{𝑥 󵄨 󵄨 󵄨󵄨

(P5) ‖𝑏𝑗 ‖𝐿1 (R𝑛 ) ≤ 𝐶, 󸀠

(P6) ‖𝑔𝑗 ‖𝐿𝑠 (R𝑛 ) ≤ 𝐶(𝛾𝜆)1/(𝑚𝑠 ) for 1 ≤ 𝑠 ≤ ∞. Let 𝑐𝑗,𝑘𝑗 be the center of cube 𝑄𝑗,𝑘𝑗 and let 𝑙(𝑄𝑗,𝑘𝑗 ) be its ∗ ∗ side length. For 𝑗 = 1, . . . , 𝑚, set Ω∗ = ⋃𝑚 𝑗=1 Ω𝑗 , where Ω𝑗 = ∗ ∗ ⋃𝑘𝑗 𝑄𝑗,𝑘𝑗 and 𝑄𝑗,𝑘𝑗 = 8√𝑛𝑄𝑗,𝑘𝑗 . And let 𝜆 󵄨 󵄨 𝐸1 = {𝑥 ∈ R : 󵄨󵄨󵄨𝑇 (𝑔1 , 𝑔2 , . . . , 𝑔𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝑚 } ; 2 𝜆 󵄨󵄨 𝑛 ∗ 󵄨󵄨 𝐸2 = {𝑥 ∈ R \ Ω : 󵄨󵄨𝑇 (𝑏1 , 𝑔2 , . . . , 𝑔𝑚 ) (𝑥)󵄨󵄨 > 𝑚 } ; 2 𝑛

󵄨 󵄨 ∈ R𝑛 \ Ω∗ : 󵄨󵄨󵄨𝑇 (𝑏1 , 𝑏2 , . . . , 𝑏𝑙 , 𝑔𝑙+1 , . . . , 𝑔𝑚 ) (𝑥)󵄨󵄨󵄨 >

𝜆 󵄨󵄨󵄨󵄨 }󵄨 , 2𝑚 󵄨󵄨󵄨

and the other terms can be estimated similarly. We will show that |𝐸𝑠(𝑙) | ≤ 𝐶𝐴(𝛾𝜆)−1/𝑚 . Obviously, 𝑇(𝑏1 , 𝑏2 , . . . , 𝑏𝑙 , 𝑔𝑙+1 , . . . , 𝑔𝑚 )(𝑥) can be controlled by

1/2 󵄨󵄨 󵄨󵄨2 𝑙 𝑚 󵄨󵄨 󵄨󵄨 𝑑𝑡 ∑ (∫ 󵄨󵄨󵄨∫ 𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) ∏𝑏𝑖,𝑘𝑖 (𝑦𝑖 ) ∏ 𝑔𝑖 (𝑦𝑖 ) 𝑑𝑦⃗ 󵄨󵄨󵄨 ) . 󵄨󵄨 𝑡 0 󵄨󵄨󵄨 R𝑛𝑚 𝑖=1 𝑘1 ,...,𝑘𝑙 𝑖=𝑙+1 󵄨 ∞

(22)

(23)

4

Journal of Function Spaces

By properties (P4) and (P6) and Minkowski’s inequality, we can further control the above term by 1/2 󵄨󵄨 󵄨󵄨2 𝑙 𝑚 󵄨󵄨 󵄨󵄨 𝑑𝑡 󵄨 󵄨 ) ∑ (∫ 󵄨󵄨∫ (𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) − 𝐾𝑡 (𝑥, 𝑐1,𝑘1 , . . . , 𝑦𝑚 )) ∏𝑏𝑖,𝑘𝑖 (𝑦𝑖 ) ∏ 𝑔𝑖 (𝑦𝑖 ) 𝑑𝑦⃗ 󵄨󵄨 󵄨󵄨 𝑡 0 󵄨󵄨󵄨 R𝑛𝑚 𝑖=1 𝑘1 ,...,𝑘𝑙 𝑖=𝑙+1 󵄨 ∞

𝑚 ∞ 󵄨 󵄨2 𝑑𝑡 1/2 𝑙 (∫ 󵄨󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) − 𝐾𝑡 (𝑥, 𝑐1,𝑘1 , . . . , 𝑦𝑚 )󵄨󵄨󵄨󵄨 ) ∏𝑏𝑖,𝑘𝑖 (𝑦𝑖 ) ∏ 𝑔𝑖 (𝑦𝑖 ) 𝑑𝑦⃗ 𝑡 0 R𝑛𝑚 𝑖=1 𝑖=𝑙+1

≤ ∑ ∫ 𝑘1 ,...,𝑘𝑙

(𝑚−𝑙)/𝑚

≤ 𝐶 (𝛾𝜆)

(24)

∞ 󵄨 󵄨2 𝑑𝑡 1/2 𝑙 (∫ 󵄨󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) − 𝐾𝑡 (𝑥, 𝑐1,𝑘1 , . . . , 𝑦𝑚 )󵄨󵄨󵄨󵄨 ) ∏𝑏𝑖,𝑘𝑖 (𝑦𝑖 ) 𝑑𝑦.⃗ 𝑡 0 R𝑛𝑚 𝑖=1

∑ ∫ 𝑘1 ,...,𝑘𝑙

𝑖𝑟 Let Q𝑟,𝑘 = (2𝑖𝑟 +2 √𝑛𝑄𝑟,𝑘𝑟 ) \ (2𝑖𝑟 +1 √𝑛𝑄𝑟,𝑘𝑟 ) for 𝑟 = 𝑟 1, . . . , 𝑙 and 𝑖𝑟 = 1, 2, . . .. It was proved that R𝑛 \ 𝑄∗ ⊂

𝑖

𝑙 𝑟 ⋃∞ 𝑖1 ,...,𝑖𝑙 =1 ⋂𝑟=1 Q𝑟,𝑘 ; see [17, p. 105]. Assume that 𝑄1,𝑘1 has the 𝑟

smallest length in {𝑄𝑖,𝑘𝑖 }𝑙𝑖=1 ; then, by (9) one has

󵄨 󵄨󵄨 󵄨󵄨𝑦1 − 𝑐1,𝑘1 󵄨󵄨󵄨 1 󵄨󵄨 󵄨󵄨2 𝑑𝑡 1/2 󵄨 ) 𝑑𝑥 󵄨 (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) − 𝐾𝑡 (𝑥, 𝑐1,𝑘1 , . . . , 𝑦𝑚 )󵄨󵄨󵄨 ) 𝑑𝑥 ≤ 𝐶 ∫ ∫ 󵄨󵄨 󵄨󵄨 𝑚𝑛 𝜔 ( 𝑚 󵄨󵄨󵄨 𝑚 󵄨󵄨 𝑡 0 R𝑛 \𝑄∗ R𝑛 \𝑄∗ (∑ 󵄨 󵄨 𝑥 − 𝑦 ∑ 𝑗 󵄨󵄨󵄨 𝑗=1 󵄨󵄨 𝑗=1 󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨) ∞

∞

≤𝐶 ∑ ∫

𝑖

𝑙 𝑟 𝑖1 ,...,𝑖𝑙 =1 ⋂𝑟=1 Q𝑟,𝑘𝑟

󵄨󵄨 󵄨 ∞ 󵄨󵄨𝑦1 − 𝑐1,𝑘1 󵄨󵄨󵄨 1 1 󵄨 󵄨 ) 𝑑𝑥 ≤ 𝐶 ∑ 𝜔 (2−𝑖𝑙 ) ∫ 𝜔 ( 󵄨 󵄨 𝑚𝑛 󵄨 󵄨 󵄨󵄨 𝑚𝑛 𝑑𝑥. 𝑚 𝑖𝑟 𝑚 󵄨 𝑚 󵄨󵄨 𝑙 󵄨 󵄨 󵄨 󵄨 󵄨 ⋂𝑟=1 Q𝑟,𝑘 (∑ 󵄨 󵄨 ∑𝑗=1 󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨 (∑𝑗=1 󵄨󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨󵄨) 𝑖1 ,...,𝑖𝑙 =1 𝑟 𝑗=1 󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨)

(25)

This together with Chebyshev’s inequality gives 󵄨󵄨 (𝑙) 󵄨󵄨 2𝑚 󵄨 󵄨󵄨 󵄨󵄨𝐸𝑠 󵄨󵄨 ≤ ∫ 󵄨𝑇 (𝑏1 , 𝑏2 , . . . , 𝑏𝑙 , 𝑔𝑙+1 , . . . , 𝑔𝑚 ) (𝑥)󵄨󵄨󵄨 𝑑𝑥 󵄨 󵄨 𝜆 R𝑛 \𝑄∗ 󵄨 ≤

∞ 2𝑚 󵄨 󵄨2 𝑑𝑡 1/2 𝑙 (𝑚−𝑙)/𝑚 (𝛾𝜆) ) ∏𝑏𝑖,𝑘𝑖 (𝑦𝑖 ) 𝑑𝑦⃗ 𝑑𝑥 ∫ ∑ ∫ (∫ 󵄨󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) − 𝐾𝑡 (𝑥, 𝑐1,𝑘1 , . . . , 𝑦𝑚 )󵄨󵄨󵄨󵄨 𝜆 𝑡 0 R𝑛 \𝑄∗ 𝑘 ,...,𝑘 R𝑛𝑚 𝑖=1 1

≤

2𝑚 (𝑚−𝑙)/𝑚 (𝛾𝜆) ∑ 𝜆 𝑘 ,...,𝑘 1

∞

∑ 𝜔 (2−𝑖𝑙 ) ∫

𝑙 𝑖1 ,...,𝑖𝑙 =1

(R𝑛 )𝑚−𝑙

∫

∏𝑙𝑟=1 𝑄𝑟,𝑘𝑟

where ∏𝑙𝑟=1 𝑄𝑟,𝑘𝑟 = 𝑄1,𝑘1 × ⋅ ⋅ ⋅ × 𝑄𝑙,𝑘𝑙 . Now, by repeating the same arguments as in [17, p. 106– 108], we can easily obtain |𝐸𝑠(𝑙) | ≤ 𝐶𝐴𝛾(𝛾𝜆)−1/𝑚 . Thus, setting 𝛾 = (𝐴 + 𝐵)−1 , we get 󵄨󵄨󵄨{𝑥 ∈ R𝑛 : 󵄨󵄨󵄨𝑇 (𝑓) ⃗ (𝑥)󵄨󵄨󵄨󵄨 > 𝜆}󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 −1/𝑚

≤ 𝐶𝐴𝛾 (𝛾𝜆)

(26)

𝑙

−1/𝑚

+ 𝐶𝐵𝑞 𝛾𝑞−1/𝑚 𝜆−1/𝑚 + 𝐶 (𝛾𝜆)

(27)

= 𝐶 (𝐴 + 𝐵)1/𝑚 𝜆−1/𝑚 . This completes the proof of Theorem 4. In order to prove Theorem 5, the following lemmas are needed.

∫

𝑑𝑥

𝑖

𝑟 ⋂𝑙𝑟=1 Q𝑟,𝑘

𝑟

𝑙

󵄨󵄨 󵄨󵄨 𝑚𝑛 ∏𝑏𝑖,𝑘𝑖 (𝑦𝑖 ) 𝑑𝑦,⃗ 󵄨 󵄨 (∑𝑚 𝑖=1 𝑗=1 󵄨󵄨𝑥 − 𝑦𝑗 󵄨󵄨)

Lemma 9 (see [15]). Suppose that supp 𝑓𝑖 ⊂ 𝐵(0, 𝑅); then, there is a constant 𝐶 < ∞ such that, for all |𝑥| > 2𝑅, the following inequality holds: ⃗ (𝑥) ≤ 𝐶M (𝑓) ⃗ (𝑥) . 𝑇 (𝑓)

(28)

Lemma 10. Let 𝑇 be a multilinear square function of type 𝜔(𝑡) and 𝜔 ∈ 𝐷𝑖𝑛𝑖(1). For any 0 < 𝛿 < 1/𝑚, there is a constant 𝐶 < ∞ such that for any bounded and compact supported 𝑓𝑗 , 𝑗 = 1, . . . , 𝑚, the following inequality holds: ♯ ⃗ (𝑥) ≤ 𝐶M (𝑓) ⃗ (𝑥) . 𝑀𝛿 𝑇 (𝑓)

(29)

Journal of Function Spaces

5

Proof. The proof of Lemma 10 involves a routine application of the method used in Lemma 4.1 in [15]. For the sake of similarity, we sketch the proof. Given 0 < 𝛿 < 1/𝑚, for a fixed point 𝑥 ∈ R𝑛 and a cube 𝑄 ∋ 𝑥, it is sufficient to show that there exists a constant 𝑐𝑄 such that 1/𝛿 1 󵄨 ⃗ (𝑧) − 𝑐 󵄨󵄨󵄨󵄨𝛿 𝑑𝑧) ≤ 𝐶M𝑓⃗ (𝑥) . ( ∫ 󵄨󵄨󵄨󵄨𝑇 (𝑓) 𝑄󵄨 |𝑄| 𝑄

𝑚

𝑐𝑄,𝑡 = ∑ ∫

R𝑛𝑚

𝛼,⃗ 𝛼⃗ =0̸ ⃗

𝐾𝑡 (𝑥, 𝑦1 , . . . , 𝑦𝑚 ) ∏𝑓𝑗 (𝑦𝑗 ) 𝑑𝑦,⃗ 𝑗=1

1/2

∞

󵄨 󵄨2 𝑑𝑡 𝑐𝑄 = (∫ 󵄨󵄨󵄨𝑐𝑄,𝑡 󵄨󵄨󵄨 ) 𝑡 0

.

Therefore, (30)

For each 𝑖 = 1, . . . , 𝑚, let 𝛼⃗ = (𝛼1 , . . . , 𝛼𝑚 ) satisfying 𝛼𝑖 = 0 or ∞. We next introduce two notations:

(

1/𝛿 1 󵄨 ⃗ (𝑧) − 𝑐 󵄨󵄨󵄨󵄨𝛿 𝑑𝑧) ≤ 𝐶𝐼 ⃗ + 𝐶∑ 𝐼 , ∫ 󵄨󵄨󵄨󵄨𝑇 (𝑓) 𝑄󵄨 𝛼⃗ 0 |𝑄| 𝑄 𝛼⃗ =0̸ ⃗

𝛿/2 󵄨󵄨 󵄨󵄨2 𝑚 󵄨󵄨 󵄨󵄨 𝑑𝑡 1 0 󵄨 󵄨 𝐼0⃗ = ( ) 𝑑𝑧) ∫ (∫ 󵄨󵄨∫ 𝐾 (𝑧, 𝑦)⃗ ∏𝑓𝑗 (𝑦𝑗 ) 𝑑𝑦⃗ 󵄨󵄨󵄨 |𝑄| 𝑄 0 󵄨󵄨󵄨 R𝑛𝑚 𝑡 󵄨󵄨 𝑡 𝑗=1 󵄨 󵄨

1/𝛿

,

𝛿/2 󵄨 󵄨󵄨2 𝑚 ∞ 󵄨󵄨 󵄨󵄨 𝑑𝑡 1 󵄨󵄨 𝛼𝑗 ⃗ ∏𝑓𝑗 (𝑦𝑗 ) 𝑑𝑦⃗ 󵄨󵄨󵄨󵄨 𝐼𝛼⃗ = ∑ ( ) 𝑑𝑧) ∫ (∫ 󵄨󵄨󵄨∫ (𝐾𝑡 (𝑧, 𝑦)⃗ − 𝐾𝑡 (𝑥, 𝑦)) |𝑄| 󵄨󵄨 𝑡 0 󵄨󵄨 R𝑛𝑚 𝑄 𝑗=1 ⃗ 𝛼,⃗ 𝛼⃗ =0̸ 󵄨 󵄨

By using the boundedness of 𝑇, we get immediately that ⃗ 𝐼0⃗ ≤ 𝐶M(𝑓)(𝑥). Using the smooth condition (8), we obtain 1/2 ∞ 󵄨 󵄨2 𝑑𝑡 (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑧, 𝑦)⃗ − 𝐾𝑡 (𝑥, 𝑦)⃗ 󵄨󵄨󵄨 ) 𝑡 0 R𝑛𝑚

𝑗=1 ∞

⋅∑∫

𝑚−𝑙 𝑘=1 (Q𝑘 )

𝑄(𝑥,𝛿)𝑙

󵄨 󵄨 ∏ 󵄨󵄨󵄨󵄨𝑓𝑖0 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨󵄨

𝑖∈I

𝐴 |𝑧 − 𝑥| 󵄨󵄨 󵄨󵄨 𝑚𝑛 𝜔 ( ∑ 𝑐 󵄨󵄨󵄨𝑧 − 𝑦 󵄨󵄨󵄨 ) (∑𝑖∈I𝑐 󵄨󵄨𝑧 − 𝑦𝑖 󵄨󵄨) 𝑖󵄨 𝑖∈I 󵄨

𝑖∈I𝑐

𝑖∈I

𝑘=1 (Q𝑘

)𝑚−𝑙

𝜔 (2 ) 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 𝑘 󵄨󵄨𝑚 ∏ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨 𝑑𝑦𝑖 󵄨󵄨2 𝑄󵄨󵄨 𝑖∈I𝑐

∞ 1 󵄨󵄨 0 󵄨 󵄨𝑓 (𝑦 )󵄨󵄨 𝑑𝑦 ≤ 𝐶 ∑ 𝜔 (2−𝑘 ) 󵄨 𝑘 󵄨𝑚 ∏ ∫ 󵄨󵄨2 𝑄󵄨󵄨 𝑖∈I 𝑄(𝑥,𝛿) 󵄨󵄨 𝑖 𝑖 󵄨󵄨 𝑖 𝑘=1 󵄨 󵄨 󵄨󵄨 󵄨󵄨 ∞ ⃗ (𝑥) . ⋅ ∏ ∫ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨 𝑑𝑦𝑖 ≤ 𝐶 |𝜔|Dini(1) M (𝑓) 𝑘 𝑖∈I𝑐 2 𝑄

(34) Then, by Minkowski’s inequality, it yields that

𝛿

Proof of Theorem 5. Theorem 5 follows from using Lemmas 9 and 10 and repeating the same steps as in [15], here, we omit the proof. To prove Theorem 6, we need some preliminary lemmas. Lemma 11 (see [18]). If 𝜔 ∈ 𝐴 𝑝 and 𝑝 ≥ 1, then 𝑀 maps from 𝐿𝑝,∞ (𝜔) to 𝐿𝑝,∞ (𝜔).

.

−𝑘

∞

1/2 𝑚 ∞ 1 󵄨 𝛼 󵄨 󵄨 󵄨2 𝑑𝑡 ⃗ 𝑑𝑧) 𝐼𝛼⃗ ≤ 𝐶 ( ) ∏ 󵄨󵄨󵄨󵄨𝑓𝑗 𝑗 (𝑦𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑦) ∫ (∫ (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑧, 𝑦)⃗ − 𝐾𝑡 (𝑥, 𝑦)⃗ 󵄨󵄨󵄨 𝑡 |𝑄| 𝑄 R𝑛𝑚 0 𝑗=1

Thus, we have finished the proof of Lemma 10.

(33)

1/𝛿

󵄨 󵄨 󵄨 󵄨 ⋅ ∏ 󵄨󵄨󵄨𝑓𝑖∞ (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 ≤ 𝐶 ∫ ∏ 󵄨󵄨󵄨󵄨𝑓𝑖0 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨󵄨 𝑄(𝑥,𝛿)𝑙 ⋅∑∫

∫

(32)

where

∞

𝑚 󵄨 𝛼 󵄨 ⋅ ∏ 󵄨󵄨󵄨󵄨𝑓𝑗 𝑗 (𝑦𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑦⃗ ≤ 𝐶 ∫

(31)

1/𝛿

⃗ (𝑥) . ≤ 𝐶 |𝜔|Dini(1) M (𝑓)

(35)

Lemma 12 (see [13]). Let 𝑝⃗ = (𝑝1 , . . . , 𝑝𝑚 ) with 1/𝑝 = 1/𝑝1 + ⋅ ⋅ ⋅ + 1/𝑝𝑚 and 1 ≤ 𝑝1 , . . . , 𝑝𝑚 . (1) If 1 < 𝑝𝑗 < ∞ for all 𝑗 = 1, . . . , 𝑚, then M is bounded from 𝐿𝑝1 (𝜔1 ) × ⋅ ⋅ ⋅ × 𝐿𝑝𝑚 (𝜔𝑚 ) to 𝐿𝑝 (]𝜔⃗ ) if and only if 𝜔⃗ = (𝜔1 , . . . , 𝜔𝑚 ) ∈ 𝐴 𝑝⃗ . (2) If 1 ≤ 𝑝𝑗 < ∞ for all 𝑗 = 1, . . . , 𝑚, then M is bounded from 𝐿𝑝1 (𝜔1 ) × ⋅ ⋅ ⋅ × 𝐿𝑝𝑚 (𝜔𝑚 ) to 𝐿𝑝,∞ (]𝜔⃗ ) if 𝜔⃗ = (𝜔1 , . . . , 𝜔𝑚 ) ∈ 𝐴 𝑝⃗ .

6

Journal of Function Spaces

Lemma 13 (see [19]). If 𝜔 ∈ 𝐴 𝑝 and 𝑝 > 1, then 𝑀 maps from 𝐿𝑝 (𝜔) to 𝐿𝑝 (𝜔). Lemma 14. Let 𝑇 be a multilinear square function of type 𝜔(𝑡) and 𝜔 ∈ 𝐷𝑖𝑛𝑖(1). For any 𝜂 > 0, there is a constant 𝐶 < ∞ depending on 𝜂 such that for all 𝑓⃗ in any product of 𝐿𝑞𝑗 (R𝑛 ) spaces, with 1 ≤ 𝑞𝑗 < ∞, the following inequality holds for all 𝑥 ∈ R𝑛 : ⃗ (𝑥) ≤ 𝐶 (𝑀 (𝑇 (𝑓)) ⃗ (𝑥) + M (𝑓) ⃗ (𝑥)) . 𝑇∗ (𝑓) 𝜂

(36)

Proof. The basic idea is due to [18, 20]. Set 𝑈𝛿 = {𝑦⃗ ∈ 2 2 (𝑄(𝑥, 𝛿))𝑚 : ∑𝑚 𝑖=1 |𝑥 − 𝑦𝑖 | > 𝛿 }. For a fixed point 𝑥 and a cube 𝑄(𝑥, 𝛿) centered at 𝑥 with radius 𝛿, it is clear that

if and only if |𝑥 − 𝑦𝑖 | ≤ 𝛿. Using the smooth condition (8), we obtain that 󵄨󵄨 ̃ ⃗ (𝑥) − 𝑇 ⃗ (𝑧)󵄨󵄨󵄨󵄨 ̃ 𝛿 (𝑓) 󵄨󵄨𝑇𝛿 (𝑓) 󵄨 󵄨 ∞

≤ 𝐶 (∫ (∫

((𝑄(𝑥,𝛿))𝑚 )𝑐

0

2

𝑚

󵄨󵄨 󵄨 󵄨󵄨𝐾𝑡 (𝑧, 𝑦)⃗ − 𝐾𝑡 (𝑥, 𝑦)⃗ 󵄨󵄨󵄨

𝑑𝑡 󵄨 󵄨 ⋅ ∏ 󵄨󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 ) ) 𝑡 𝑖=1

1/2

1/2 ∞ 󵄨 󵄨2 𝑑𝑡 (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑧, 𝑦)⃗ − 𝐾𝑡 (𝑥, 𝑦)⃗ 󵄨󵄨󵄨 ) 𝑡 0 ((𝑄(𝑥,𝛿))𝑚 )𝑐

≤ 𝐶∫ 𝑚

󵄨 󵄨 ⋅ ∏ 󵄨󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 ≤ 𝐶 ∑ ∫

󵄨󵄨 ∗ ⃗ 󵄨 󵄨󵄨𝑇 (𝑓) (𝑥)󵄨󵄨󵄨 󵄨 󵄨

𝑖=1

1/2 󵄨󵄨2 𝑚 ∞ 󵄨󵄨󵄨 󵄨󵄨 𝑑𝑡 󵄨 ≤ sup (∫ 󵄨󵄨󵄨∫ 𝐾𝑡 (𝑥, 𝑦)⃗ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨 ) 󵄨󵄨 𝑡 0 󵄨󵄨󵄨 𝑈𝛿 𝛿>0 𝑖=1 󵄨

󵄨󵄨2

𝑚 ∞ 󵄨󵄨󵄨 󵄨󵄨 𝑑𝑡 󵄨 + sup (∫ 󵄨󵄨󵄨∫ 𝐾𝑡 (𝑥, 𝑦)⃗ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨 ) 𝑚 )𝑐 󵄨 󵄨󵄨 𝑡 0 ((𝑄(𝑥,𝛿)) 󵄨 𝛿>0 𝑖=1 󵄨 󵄨

⃗ I 𝑦∈𝑅

∞

(37) 1/2

𝑚−𝑙 𝑘=1 (Q𝑘 )

󵄨 󵄨 ∏ 󵄨󵄨󵄨󵄨𝑓𝑖0 (𝑦𝑖 )󵄨󵄨󵄨󵄨 𝑑𝑦𝑖

𝑖∈I

𝐴 |𝑧 − 𝑥| 󵄨󵄨 󵄨󵄨 𝑚𝑛 𝜔 ( ∑ 𝑐 󵄨󵄨󵄨𝑧 − 𝑦 󵄨󵄨󵄨 ) (∑𝑖∈I𝑐 󵄨󵄨𝑧 − 𝑦𝑖 󵄨󵄨) 𝑖󵄨 𝑖∈I 󵄨

(40)

󵄨 󵄨 󵄨 󵄨 ⋅ ∏ 󵄨󵄨󵄨𝑓𝑖∞ (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 ≤ 𝐶 ∑ ∫ ∏ 󵄨󵄨󵄨󵄨𝑓𝑖0 (𝑦𝑖 )󵄨󵄨󵄨󵄨 𝑑𝑦𝑖 𝑙 𝑄(𝑥,𝛿)

.

By using the size condition (7) and Minkowski’s inequality, we get 1/2 󵄨󵄨 󵄨󵄨2 𝑚 󵄨󵄨 󵄨󵄨 𝑑𝑡 (∫ 󵄨󵄨󵄨∫ 𝐾𝑡 (𝑥, 𝑦)⃗ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨 ) 󵄨󵄨 𝑡 0 󵄨󵄨󵄨 𝑈𝛿 𝑖=1 󵄨 󵄨󵄨 󵄨󵄨 2 𝑚 ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨2 𝑑𝑡 ≤ 𝐶 󵄨󵄨󵄨∫ (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑧, 𝑦)⃗ 󵄨󵄨󵄨 ) ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑈𝛿 0 𝑡 𝑖=1 󵄨 󵄨 󵄨󵄨 󵄨 𝑚 󵄨󵄨 𝐴 󵄨󵄨 󵄨 ≤ 𝐶 󵄨󵄨󵄨∫ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨 𝑚𝑛 󵄨 󵄨 𝑚 󵄨󵄨 𝑈𝛿 (∑𝑖=1 󵄨󵄨𝑦𝑖 − 𝑥󵄨󵄨) 𝑖=1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨

⋅∑∫

𝑄(𝑥,𝛿)𝑙

𝑖∈I𝑐

⃗ I 𝑦∈𝑅

∞

𝜔 (2−𝑘 ) 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 𝑘 󵄨󵄨𝑚 ∏ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨 𝑑𝑦𝑖 󵄨󵄨2 𝑄󵄨󵄨 𝑖∈I𝑐

⋅∑∫

𝑚−𝑙 𝑘=1 (Q𝑘 )

𝑖∈I

∞ 1 󵄨󵄨 0 󵄨 󵄨𝑓 (𝑦 )󵄨󵄨 𝑑𝑦 ≤ 𝐶 ∑ ∑ 𝜔 (2−𝑘 ) 󵄨 𝑘 󵄨𝑚 ∏ ∫ 󵄨󵄨2 𝑄󵄨󵄨 𝑖∈I 𝑄(𝑥,𝛿) 󵄨󵄨 𝑖 𝑖 󵄨󵄨 𝑖 ⃗ I 𝑘=1 𝑦∈𝑅 󵄨 󵄨

∞

⋅∏∫

(38)

𝑖∈I𝑐

2𝑘 𝑄

󵄨󵄨 ∞ 󵄨 ⃗ (𝑥) , 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 ≤ 𝐶 |𝜔|Dini(1) M (𝑓)

where Q𝑘 = (2𝑘 𝑄) \ (2𝑘−1 𝑄), for 𝑘 = 1, 2, . . . , ∞. Thus, we obtain

𝑚

1 ⃗ (𝑥) . 𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 ≤ 𝐶M (𝑓) ∫ 𝑛 𝛿 𝑄(𝑥,𝛿) 𝑖=1

≤ 𝐶∏

0 We are ready to estimate the second term. Set 𝑓⃗ = (𝑓1 𝜒𝑄, . . . , 𝑓𝑚 𝜒𝑄). For any 𝑧 ∈ 𝑄(𝑥, 𝛿/2), we introduce an ̃𝛿 : operator 𝑇

⃗ (𝑧) ̃ 𝛿 (𝑓) 𝑇 1/2 󵄨󵄨 󵄨󵄨2 𝑚 󵄨󵄨 󵄨󵄨 𝑑𝑡 󵄨 󵄨 (39) 𝐾𝑡 (𝑧, 𝑦)⃗ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨 = (∫ 󵄨󵄨∫ ) 󵄨󵄨 𝑡 0 󵄨󵄨󵄨 ((𝑄(𝑥,𝛿))𝑚 )𝑐 𝑖=1 󵄨 ∞

⃗ (𝑧) + 𝑇 (𝑓⃗ 0 ) (𝑧) . ≤ 𝑇 (𝑓) Let 𝑅I be the sets in (R𝑛 )𝑚 , where I fl {𝑖1 , . . . , 𝑖𝑙 } ⊆ {1, . . . , 𝑚}, such that for 𝑦⃗ = (𝑦1 , . . . , 𝑦𝑚 ) ∈ 𝑅I we have 𝑖 ∈ I

1/2 󵄨󵄨 󵄨󵄨2 𝑚 󵄨󵄨 󵄨󵄨 𝑑𝑡 󵄨 󵄨 (∫ 󵄨󵄨∫ 𝐾 (𝑥, 𝑦)⃗ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨 ) 󵄨󵄨 𝑡 0 󵄨󵄨󵄨 ((𝑄(𝑥,𝛿))𝑚 )𝑐 𝑖=1 󵄨 ∞

(41)

󵄨 󵄨 ⃗ (𝑧)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑇 (𝑓⃗ 0 ) (𝑧)󵄨󵄨󵄨󵄨 . ⃗ (𝑥) + 󵄨󵄨󵄨󵄨𝑇 (𝑓) ≤ 𝐶M (𝑓) 󵄨 󵄨󵄨 󵄨 󵄨󵄨 Raising the above inequality to the power 𝜂, integrating over 𝑧 ∈ 𝑄(𝑥, 𝛿/2), and dividing by |𝑄|, we conclude that 𝜂/2 󵄨󵄨2 𝑚 ∞ 󵄨󵄨󵄨 󵄨󵄨 𝑑𝑡 󵄨 (∫ 󵄨󵄨󵄨∫ 𝐾 (𝑥, 𝑦)⃗ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 󵄨󵄨󵄨 ) 󵄨󵄨 𝑡 0 󵄨󵄨󵄨 ((𝑄(𝑥,𝛿))𝑚 )𝑐 𝑖=1 󵄨 ⃗ 󵄨󵄨󵄨󵄨𝜂 ) (𝑥) ⃗ (𝑥))𝜂 + 𝑀 (󵄨󵄨󵄨󵄨𝑇 (𝑓) ≤ 𝐶 (M (𝑓) 󵄨 󵄨 𝜂 󵄨 󵄨 0 1 󵄨 󵄨 + ∫ 󵄨󵄨󵄨𝑇 (𝑓⃗ ) (𝑧)󵄨󵄨󵄨󵄨 𝑑𝑧. |𝑄| 𝑄 󵄨󵄨 󵄨

(42)

Journal of Function Spaces

7

Next we estimate the last term in (42). By Theorem 4, we know that 𝑇 is bounded from 𝐿1 × ⋅ ⋅ ⋅ × 𝐿1 to 𝐿1/𝑚,∞ ; then we can deduce that 󵄨󵄨 ∞ 󵄨󵄨𝜂 󵄨󵄨 0 󵄨 ∫ 󵄨󵄨󵄨󵄨𝑇 (𝑓⃗ ) (𝑧)󵄨󵄨󵄨󵄨 𝑑𝑧 = 𝑚𝜂 ∫ 𝜆𝑚𝜂−1 󵄨󵄨󵄨{𝑧 󵄨󵄨 󵄨 󵄨 0 𝑄 󵄨󵄨 󵄨󵄨 0 󵄨󵄨1/𝑚 󵄨 ∈ 𝑄 : 󵄨󵄨󵄨󵄨𝑇 (𝑓⃗ ) (𝑧)󵄨󵄨󵄨󵄨 > 𝜆}󵄨󵄨󵄨 𝑑𝜆 󵄨󵄨 󵄨 󵄨 (43)

∞

≤ 𝑚𝜂 ∫ 𝜆𝑚𝜂−1 0

{ 𝑐 𝑚 󵄩 󵄩 ⋅ min {|𝑄| , (∏ 󵄩󵄩󵄩󵄩𝑓𝑗 𝜒𝑄(𝑥,𝛿) 󵄩󵄩󵄩󵄩𝐿1 ) 𝜆 𝑗=1 {

1/𝑚

} } 𝑑𝜆. }

1/𝑚 , we obtain that Letting 𝑅 = 𝑐(∏𝑚 𝑗=1 ‖𝑓𝑗 𝜒𝑄(𝑥,𝛿) ‖𝐿1 )

󵄨󵄨𝜂 󵄨󵄨 0 1 ∫ 󵄨󵄨󵄨󵄨𝑇 (𝑓⃗ ) (𝑧)󵄨󵄨󵄨󵄨 𝑑𝑧 |𝑄| 𝑄 󵄨 󵄨 𝑅/|𝑄| ∞ 𝐶 𝜆𝑚𝜂−1 |𝑄| 𝑑𝜆 + ∫ 𝜆𝑚𝜂−1 𝑅 𝑑𝜆) ≤ (∫ |𝑄| 0 𝑅/|𝑄|

𝑚𝜂

≤ 𝐶𝑅

1−𝑚𝜂

|𝑄|

𝑚

1 󵄨 󵄨 ≤ 𝐶 (∏ ∫ 󵄨󵄨󵄨󵄨𝑓𝑗 (𝑦𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑦𝑗 ) 𝑗=1 |𝑄| 𝑄

𝜂

3. Weighted Boundedness on Morrey Type Spaces The classical Morrey space was first introduced by Morrey in [21] to study the local behavior of solutions to second order elliptic partial differential equations. Later, Komori and Shirai [22] introduced the weighted Morrey space L𝑝,𝑘 (𝜔) for 1 ≤ 𝑝 < ∞ and investigated the boundedness of classical operators, including Hardy-Littlewood maximal operator, Calder´on-Zygmund operator, and fractional integral operator. In order to deal with the multilinear case 𝑚 ≥ 2, Wang and Yi [23] extended the range 1 ≤ 𝑝 < ∞ to 0 < 𝑝 < ∞. Motivated by the works on multilinear Calder´onZygmund operators and multilinear square functions, as demonstrated in [4, 12, 15, 24–26], we are going to study the boundedness of multilinear square function of type 𝜔(𝑡) on weighted Morrey type spaces. Let 0 < 𝑝 < ∞ and 0 < 𝑘 < 1 and let 𝜔 be a weighted function on R𝑛 . Then, the weighted Morrey space L𝑝,𝑘 is defined by 𝑝 󵄩 󵄩 L𝑝,𝑘 = {𝑓 ∈ 𝐿 loc (𝜇) : 󵄩󵄩󵄩𝑓󵄩󵄩󵄩L𝑝,𝑘 < ∞} ,

(44)

(48)

where 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩L𝑝,𝑘 = sup ( 𝑄

1 𝜔 (𝑄)𝑘

󵄨𝑝 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 𝜔 (𝑥) 𝑑𝑥) 𝑄

1/𝑝

.

(49)

Furthermore, the weighted weak Morrey space 𝑊L𝑝,𝑘 is defined by

𝜂

⃗ (𝑥)) , ≤ 𝐶 (M (𝑓) where we have used the fact 𝑚𝜂 < 1 (it suffices to prove the lemma for 𝜂 arbitrarily small). Finally, if we insert estimate (44) into (42) and raise to the power 1/𝜂, we obtain the desired estimate. This finishes the proof of Lemma 14. Proof of Theorem 6. Theorem 6 follows by using Lemmas 13 and 14. Using the pointwise estimate for 𝑇∗ in Lemma 14, we obtain that 󵄩󵄩 ∗ ⃗ 󵄩󵄩 󵄩󵄩𝑇 (𝑓)󵄩󵄩 𝑝 󵄩𝐿 (]𝜔⃗ ) 󵄩

󵄩 󵄩 𝑊L𝑝,𝑘 = {𝑓 measurable : 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑊L𝑝,𝑘 < ∞} ,

(50)

where 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝑊L𝑝,𝑘 = sup sup 𝑄

𝜆>0

1 𝜔 (𝑄)

1/𝑝 󵄨 󵄨 𝜆 ⋅ ({𝑥 ∈ 𝑄 : 󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 > 𝜆}) . 𝑘/𝑝

(51)

The main results in this section are the following Theorem.

󵄩 󵄩󵄩 ⃗ 󵄩󵄩󵄩 ⃗ 󵄩󵄩󵄩󵄩 󵄩 ≤ 𝐶 (󵄩󵄩󵄩󵄩𝑀𝜂 (𝑇 (𝑓)) 󵄩𝐿𝑝 (]𝜔⃗ ) + 󵄩󵄩M (𝑓)󵄩󵄩𝐿𝑝 (]𝜔⃗ ) ) .

(45)

𝑝/𝑝

Notice that ]𝜔⃗ ∈ 𝐴 𝑚𝑝 for all 𝜔⃗ ∈ 𝐴 𝑝⃗ (see [13, Theorem 3.6]). By Lemma 13, we have 󵄩󵄩 󵄩󵄩 ⃗ 󵄩󵄩󵄩 ⃗ 󵄩󵄩󵄩󵄩 󵄩 󵄩󵄩𝑀𝜂 (𝑇 (𝑓)) 󵄩𝐿𝑝 (]𝜔⃗ ) ≤ 𝐶 󵄩󵄩𝑇 (𝑓)󵄩󵄩𝐿𝑝 (]𝜔⃗ ) . 󵄩

(46)

Thus, we obtain the desired estimates by applying Theorem 5 and Lemma 12: 𝑚

󵄩󵄩 ∗ ⃗ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑇 (𝑓)󵄩󵄩 𝑝 󵄩 󵄩 󵄩 󵄩𝐿 (]𝜔⃗ ) ≤ 𝐶∏ 󵄩󵄩𝑓𝑗 󵄩󵄩𝐿𝑝𝑗 (𝜔𝑗 ) . 𝑗=1

Theorem 15. Let 𝑇 be a multilinear square function of type 𝜔(𝑡) and 𝜔 ∈ 𝐷𝑖𝑛𝑖(1). Suppose that 𝜔⃗ ∈ 𝐴 𝑝⃗ and ]𝜔⃗ =

(47)

If we use Lemma 11, by using the same arguments, we can get the weak type estimates; we omit its proof here.

𝑖 with 1/𝑝 = 1/𝑝1 + ⋅ ⋅ ⋅ + 1/𝑝𝑚 . For 1/𝑚 < 𝑝 < ∞ ∏𝑚 𝑖=1 𝜔𝑖 and 0 < 𝑘 < 1, the following two weighted inequalities hold:

(i) If 1 < 𝑝1 , . . . , 𝑝𝑚 < ∞, then 𝑚 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∗ ⃗ 󵄩󵄩 󵄩󵄩𝑓𝑗 󵄩󵄩 𝑝𝑗 ,𝑘 󵄩󵄩𝑇 (𝑓)󵄩󵄩 𝑝,𝑘 ≤ 𝐶 ∏ 󵄩 󵄩L (𝑤𝑗 ) . 󵄩L (]𝜔⃗ ) 󵄩 𝑗=1

(52)

(ii) If 1 ≤ 𝑝1 , . . . , 𝑝𝑚 < ∞ and min{𝑝1 , . . . , 𝑝𝑚 } = 1, then 𝑚 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∗ ⃗ 󵄩󵄩 󵄩󵄩𝑓𝑗 󵄩󵄩 𝑝𝑗 ,𝑘 󵄩󵄩𝑇 (𝑓)󵄩󵄩 𝑝,𝑘 ≤ 𝐶 ∏ 󵄩 󵄩L (𝑤𝑗 ) . 󵄩𝑊L (]𝜔⃗ ) 󵄩 𝑗=1

(53)

Remark 16. Theorem 15 also holds with 𝑇∗ replaced by 𝑇.

8

Journal of Function Spaces

In order to prove Theorem 15, we will use the following lemmas.

Then, we have

Lemma 17 (see [23]). Let 𝑚 ≥ 2, 𝑝1 , . . . , 𝑝𝑚 ∈ [1, ∞), and 𝑝 ∈ (0, ∞) with 1/𝑝 = 1/𝑝1 + ⋅ ⋅ ⋅ + 1/𝑝𝑚 . Assume that 𝑝/𝑝𝑖 . Then, for any ball 𝜔1 , . . . , 𝜔𝑚 ∈ 𝐴 ∞ and set ]𝜔⃗ = ∏𝑚 𝑖=1 𝜔𝑖 𝐵, there exists a constant 𝐶 > 0 such that

(

𝑚

∏ (∫ 𝜔𝑖 (𝑥) 𝑑𝑥) 𝑖=1

𝑝/𝑝𝑖

𝐵

≤ 𝐶 ∫ ]𝜔⃗ (𝑥) 𝑑𝑥.

(54)

𝐵

Lemma 18 (see [27]). Let 𝜔 ∈ 𝐴 𝑝 with 1 ≤ 𝑝 < ∞. Then, for any ball 𝐵, there exists an absolute constant 𝐶 > 0 such that 𝜔 (2𝐵) ≤ 𝐶𝜔 (𝐵) .

(55)

Lemma 19 (see [27]). Let 𝜔 ∈ 𝐴 ∞ . Then, for any ball 𝐵 and all measurable subsets 𝐸 of 𝐵, there exists 𝛿 > 0 such that 𝛿 𝜔 (𝐸) |𝐸| ≤ 𝐶( ) . 𝜔 (𝐵) |𝐵|

(56)

𝑚

𝑚

𝑖=1

𝑖=1

𝛼

∏𝑓𝑖 (𝑦𝑖 ) = ∏𝑓𝑖0 (𝑦𝑖 ) + ∑ 𝑓1 1 (𝑦1 ) ⋅ ⋅ ⋅ 𝑓𝑚𝛼𝑚 (𝑦𝑚 ) , 𝛼,⃗ 𝛼⃗ =0̸ ⃗

where 𝛼⃗ = (𝛼1 , . . . , 𝛼𝑚 ), 𝛼𝑖 = 0 or ∞ for 𝑖 = 1, . . . , 𝑚.

(57)

󵄨𝑝 󵄨 ∫ 󵄨󵄨󵄨𝑇∗ (𝑓1 , . . . , 𝑓𝑚 ) (𝑥)󵄨󵄨󵄨 ]𝜔⃗ (𝑥) 𝑑𝑥) 𝐵

]𝜔⃗ (𝐵)𝑘/𝑝 ≤𝐶

1 ]𝜔⃗ (𝐵)

𝑘/𝑝

⋅ ]𝜔⃗ (𝑥) 𝑑𝑥)

(∫

𝐵

] 𝛼,⃗ 𝛼⃗ =0̸ ⃗ 𝜔⃗

(58)

(𝐵)

⋅ ]𝜔⃗ (𝑥) 𝑑𝑥)

󵄨𝑝 󵄨󵄨 ∗ 0 󵄨󵄨𝑇 (𝑓1 , . . . , 𝑓𝑚0 ) (𝑥)󵄨󵄨󵄨 󵄨 󵄨

1/𝑝

1

+ ∑

𝑘/𝑝

1/𝑝

󵄨𝑝 󵄨 𝛼 (∫ 󵄨󵄨󵄨𝑇∗ (𝑓1 1 , . . . , 𝑓𝑚𝛼𝑚 ) (𝑥)󵄨󵄨󵄨 𝐵

= 𝐶 (𝐼0⃗ + ∑ 𝐼𝛼⃗ ) . 𝛼,⃗ 𝛼⃗ =0̸ ⃗

It was shown in [13, Theorem 3.6] that ]𝜔⃗ ∈ 𝐴 𝑚𝑝 . This fact together with Theorem 6(i) and Lemmas 18 and 17 yields that 𝐼0⃗
𝛿2 𝛿>0 𝑖=1 󵄨 1/2 𝑚 ∞ 󵄨 󵄨2 𝑑𝑡 (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦)⃗ 󵄨󵄨󵄨 ) ∏𝑓𝑖∞ (𝑦𝑖 ) 𝑑𝑦⃗ 𝑡 0 R𝑛𝑚 𝑖=1

≤ 𝐶∫

∞

≤ 𝐶∑ ∫

𝑗+1 𝑗 𝑚 𝑗=1 (2 𝐵\2 𝐵)

∞ 𝑚

(∑𝑚 𝑖=1

∞ 𝑚 𝑚 1 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑚𝑛 ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦⃗ ≤ 𝐶∑ ∏ 󵄨󵄨󵄨2𝑗+1 𝐵󵄨󵄨󵄨 ∫ 𝑗+1 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨 𝑑𝑦𝑖 2 𝐵 󵄨󵄨𝑥 − 𝑦𝑖 󵄨󵄨) 𝑖=1 󵄨 𝑗=1 𝑖=1 󵄨

1

󵄨󵄨 󵄨𝑝 ≤ 𝐶∑ ∏ 󵄨󵄨 𝑗+1 󵄨󵄨 (∫ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑖 𝜔𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 ) 𝑗+1 𝐵󵄨󵄨 2 𝐵 𝑗=1 𝑖=1 󵄨󵄨2

1/𝑝𝑖

(∫

2𝑗+1 𝐵

1−𝑝𝑖󸀠

𝜔𝑖 (𝑦𝑖 )

𝑑𝑦𝑖 )

1/𝑝𝑖󸀠

󵄨󵄨 𝑗+1 󵄨󵄨1/𝑝+∑𝑚𝑖=1 (1−1/𝑝𝑖 ) 𝑚 󵄨󵄨2 𝐵󵄨󵄨 𝑘/𝑝𝑖 1 󵄩 󵄩 󵄨 ≤ 𝐶∑ 󵄨 𝑗+1 󵄨𝑚 󵄨 ∏ (󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) 𝜔𝑖 (2𝑗+1 𝐵) ) 1/𝑝 󵄨 󵄨 𝑗+1 2 𝐵 ]𝜔⃗ (2 𝐵) 𝑗=1 󵄨󵄨 𝑖=1 󵄨󵄨 ∞

𝑘/𝑝𝑖

𝑗+1 ∞ 𝐵) ∏𝑚 𝑖=1 𝜔𝑖 (2 󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ∑ ( 1/𝑝 ]𝜔⃗ (2𝑗+1 𝐵) 𝑖=1 𝑗=1 𝑚

𝑚

∞

𝑖=1

𝑗=1

(𝑘−1)/𝑝 󵄩 󵄩 ) ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ∑ ]𝜔⃗ (2𝑗+1 𝐵) .

(60)

Journal of Function Spaces

9

Then, Lemma 19 implies that

Combining the above estimates and then taking the supremum over all balls 𝐵 ∈ R𝑛 , we complete the proof of Theorem 15(i). We are now in a position to demonstrate (ii). For any 𝜆 > 0, we have 1/𝑝 󵄨 󵄨 ]𝜔⃗ ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨𝑇∗ (𝑓1 , . . . , 𝑓𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝜆}) ≤ 𝐶]𝜔⃗ ({𝑥

󵄨 󵄨 𝐼𝛼⃗ ≤ 𝐶]𝜔⃗ (𝐵)(1−𝑘)/𝑝 󵄨󵄨󵄨𝑇∗ (𝑓1∞ , . . . , 𝑓𝑚∞ ) (𝑥)󵄨󵄨󵄨 𝑚 ∞ ] (𝐵)(1−𝑘)/𝑝 󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ∑ 𝜔⃗ 𝑗+1 𝐵)(1−𝑘)/𝑝 𝑖=1 𝑗=1 ]𝜔⃗ (2 𝛿(1−𝑘)/𝑝

∞

𝑚

|𝐵| 󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ∑ ( 󵄨󵄨 𝑗+1 󵄨󵄨 ) 2 𝐵󵄨󵄨 󵄨 𝑖=1 𝑗=1 󵄨

(61)

1/𝑝 󵄨 󵄨 ∈ 𝐵 : 󵄨󵄨󵄨󵄨𝑇∗ (𝑓10 , . . . , 𝑓𝑚0 ) (𝑥)󵄨󵄨󵄨󵄨 > 𝜆})

󵄨 󵄨 𝛼 + 𝐶 ∑ ]𝜔⃗ ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨𝑇∗ (𝑓1 1 , . . . , 𝑓𝑚𝛼𝑚 ) (𝑥)󵄨󵄨󵄨

𝑚

󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) . 𝑖=1

1/𝑝

It remains for us to consider 𝐼𝛼⃗ with 𝛼⃗ = (𝛼1 , . . . , 𝛼𝑚 ), 𝛼𝑖 = 0 or ∞ for 𝑖 = 1, . . . , 𝑚. We may assume that 𝛼1 = ⋅ ⋅ ⋅ = 𝛼𝑙 = ∞ and 𝛼𝑙+1 = ⋅ ⋅ ⋅ = 𝛼𝑚 = 0. Minkowski’s inequality and the size condition (7) imply that 󵄨 󵄨󵄨 ∗ ∞ 0 󵄨󵄨𝑇 (𝑓1 , . . . , 𝑓𝑙∞ , 𝑓𝑙+1 , . . . , 𝑓𝑚0 ) (𝑥)󵄨󵄨󵄨󵄨 󵄨

𝑚

𝑖=1

𝑖=𝑙+1

(R𝑛 \2𝐵)𝑙

∫

(2𝐵)𝑚−𝑙

𝑚

(62)

󵄨 󵄨 ≤ 𝐶 ∏ ∫ 󵄨󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 2𝐵 𝑖=𝑙+1

∞ 𝑙 1 ⋅ ∑ 󵄨 𝑗+1 󵄨𝑚 ∫ ∏𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦1 ⋅ ⋅ ⋅ 𝑑𝑦𝑙 󵄨 𝐵󵄨󵄨󵄨 (2𝑗+1 𝐵\2𝑗 𝐵)𝑙 𝑖=1 𝑗=1 󵄨󵄨2 ∞ 𝑚

󵄨󵄨 󵄨 ≤ 𝐶∑ ∏ 󵄨󵄨 𝑗+1 󵄨󵄨 ∫ 󵄨𝑓 (𝑦 )󵄨󵄨 𝑑𝑦 𝐵󵄨󵄨 2𝑗+1 𝐵 󵄨 𝑖 𝑖 󵄨 𝑖 𝑗=1 𝑖=1 󵄨󵄨2 ∞

𝑖=1

𝑗=1

𝐼𝛼⃗

𝑚

∞

|𝐵| 󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ∑ ( 󵄨󵄨 𝑗+1 󵄨󵄨 ) 2 𝐵󵄨󵄨 𝑖=1 𝑗=1 󵄨󵄨 𝑚

󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) . 𝑖=1

𝜆

(63)

(65)

󵄩󵄩 󵄩󵄩 𝐶]𝜔⃗ (2𝐵)𝑘/𝑝 ∏𝑚 𝑖=1 󵄩 󵄩𝑓𝑖 󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) 𝜆 󵄩󵄩 󵄩󵄩 𝐶]𝜔⃗ (𝐵)𝑘/𝑝 ∏𝑚 𝑖=1 󵄩 󵄩𝑓𝑖 󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 )

. 𝜆 We assume that 𝑝1 = ⋅ ⋅ ⋅ = 𝑝𝑙 = 1 and 𝑝𝑙+1 , . . . , 𝑝𝑚 > 1. If 𝛼⃗ ≠ 0,⃗ recall that in (60) and (62) we have proved the following fact: 󵄨󵄨 ∗ 𝛼1 󵄨 𝛼 󵄨󵄨𝑇 (𝑓1 , . . . , 𝑓𝑚𝑚 ) (𝑥)󵄨󵄨󵄨

𝑚 1 󵄨󵄨 󵄨 ⋅ ∏ 󵄨󵄨 𝑗+1 󵄨󵄨 ∫ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 𝑗+1 𝐵 2 𝐵 󵄨 󵄨 2 󵄨 󵄨 𝑖=𝑙+1

󵄨󵄨 ∗ ∞ 󵄨 0 󵄨󵄨𝑇 (𝑓1 , . . . , 𝑓𝑙∞ , 𝑓𝑙+1 , . . . , 𝑓𝑚0 ) (𝑥)󵄨󵄨󵄨󵄨 󵄨

𝑚 ∞ ] (𝐵)(1−𝑘)/𝑝 󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ∑ 𝜔⃗ 𝑗+1 𝐵)(1−𝑘)/𝑝 𝑖=1 𝑗=1 ]𝜔⃗ (2

(2𝐵)

𝜆 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓𝑖 󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 )

𝑘/𝑝𝑖

∞ 𝑙 1 󵄨󵄨 󵄨 ≤ 𝐶∑ ∏ 󵄨󵄨 𝑗+1 󵄨󵄨 ∫ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 𝑗+1 𝐵 2 𝐵 󵄨 󵄨 2 󵄨 𝑗=1 𝑖=1 󵄨

Together with Lemma 19, we obtain

≤ 𝐶]𝜔⃗ (𝐵)

𝐶∏𝑚 𝑖=1 𝜔𝑖

Hence, we may obtain 󵄨 󵄨󵄨 ∗ 𝛼1 𝛼 󵄨󵄨𝑇 (𝑓1 , . . . , 𝑓𝑚𝑚 ) (𝑥)󵄨󵄨󵄨

(𝑘−1)/𝑝 󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ∑]𝜔⃗ (2𝑗+1 𝐵) .

(1−𝑘)/𝑝

≤

1/𝑝𝑖 󵄨󵄨𝑝𝑖 󵄨󵄨 𝐶∏𝑚 𝑖=1 (∫2𝐵 󵄨󵄨𝑓𝑖 (𝑥)󵄨󵄨 𝜔𝑖 (𝑥) 𝑑𝑥)

∞ 𝑚 1 󵄨󵄨 󵄨 ≤ 𝐶∑ ∏ 󵄨󵄨 𝑗+1 󵄨󵄨 ∫ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑑𝑦𝑖 . 𝑗+1 𝐵 2 𝐵 󵄨 󵄨 2 󵄨 𝑗=1 𝑖=1 󵄨

1

𝑚

Using Theorem 6(ii) and Lemmas 18 and 17, we obtain 1/𝑝 󵄨 󵄨 𝑃0⃗ = ]𝜔⃗ ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨󵄨𝑇∗ (𝑓10 , . . . , 𝑓𝑚0 ) (𝑥)󵄨󵄨󵄨󵄨 > 𝜆})

≤

0 ∏𝑙𝑖=1 𝑓𝑖∞ (𝑦𝑖 ) ∏𝑚 𝑖=𝑙+1 𝑓𝑖 (𝑦𝑖 ) 𝑑𝑦⃗ 󵄨󵄨 󵄨󵄨 𝑚𝑛 (∑𝑚 𝑖=1 󵄨󵄨𝑥 − 𝑦𝑖 󵄨󵄨)

= 𝐶 (𝑃0⃗ + ∑ 𝑃𝛼⃗ ) . 𝛼,⃗ 𝛼⃗ =0̸ ⃗

≤

⋅ ∏𝑓𝑖∞ (𝑦𝑖 ) ∏ 𝑓𝑖0 (𝑦𝑖 ) 𝑑𝑦⃗ ≤ 𝐶∫

> 𝜆})

≤

1/2 ∞ 󵄨 󵄨2 𝑑𝑡 (∫ 󵄨󵄨󵄨𝐾𝑡 (𝑥, 𝑦)⃗ 󵄨󵄨󵄨 ) 𝑡 0 R𝑛𝑚

≤ 𝐶∫ 𝑙

(64)

𝛼,⃗ 𝛼⃗ =0̸ ⃗

∞ 𝑙 1 󵄨󵄨 󵄨 ≤ 𝐶∑ ∏ 󵄨󵄨 𝑗+1 󵄨󵄨 ∫ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝜔𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 𝑗+1 𝐵 2 𝐵 󵄨 󵄨 2 󵄨 𝑗=1 𝑖=1 󵄨

𝛿(1−𝑘)/𝑝

−1

⋅ ( inf𝑗+1 𝜔𝑖 (𝑦𝑖 )) 𝑦𝑖 ∈2

𝐵

𝑚 1/𝑝𝑖 1 󵄨󵄨 󵄨𝑝 ⋅ ∏ 󵄨󵄨 𝑗+1 󵄨󵄨 (∫ 󵄨󵄨𝑓𝑖 (𝑦𝑖 )󵄨󵄨󵄨 𝑖 𝜔𝑖 (𝑦𝑖 ) 𝑑𝑦𝑖 ) 2 𝐵󵄨󵄨 2𝑗+1 𝐵 𝑖=𝑙+1 󵄨󵄨

(66)

10

Journal of Function Spaces ⋅ (∫

2𝑗+1 𝐵

𝜔𝑖 (𝑦𝑖 )

∞

1−𝑝𝑖󸀠

𝑑𝑦𝑖 )

⋅ ∑ ]𝜔⃗ (2𝑗+1 𝐵)

(𝑘−1)/𝑝

1

∞

]𝜔⃗ (𝐵)(1−𝑘)/𝑝

𝑚

⋅∑

]𝜔⃗ (𝐵)(1−𝑘)/𝑝

𝑗=1 ]𝜔⃗

(2𝑗+1 𝐵)

𝑚

𝑖=1

]𝜔⃗ (𝐵)(1−𝑘)/𝑝

(𝑘−1)/𝑝

1 ]𝜔⃗ (𝐵)(1−𝑘)/𝑝

𝛿(1−𝑘)/𝑝

∞ |𝐵| ⋅ ∑ ( 󵄨󵄨 𝑗+1 󵄨󵄨 ) 𝐵󵄨󵄨 𝑗=1 󵄨󵄨2

1

𝑖=1

𝑖=1

󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ⋅

⋅

𝑚

󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 )

󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 )

𝑗=1

⋅

1/𝑝𝑖󸀠

𝑚

󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) 𝑖=1

. (67)

To prove Theorem 15(ii), we may assume that 󵄨 󵄨 𝛼 ]𝜔⃗ ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨𝑇∗ (𝑓1 1 , . . . , 𝑓𝑚𝛼𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝜆}) > 0. Otherwise, there is nothing needing to be proved. Then, by the above estimates, we obtain that 󵄨 󵄨 𝛼 ]𝜔⃗ ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨𝑇∗ (𝑓1 1 , . . . , 𝑓𝑚𝛼𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝜆}) 󵄩󵄩 󵄩󵄩𝑝 𝐶]𝜔⃗ (𝐵) ∏𝑚 𝑖=1 󵄩 󵄩𝑓𝑖 󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) ≤ . 𝜆𝑝 ⋅ ]𝜔⃗ (𝐵)1−𝑘

(68)

(69)

Therefore, we get 1 ]𝜔⃗ (𝐵)𝑘/𝑝

⋅𝜆

1/𝑝 󵄨 󵄨 𝛼 ⋅ ]𝜔⃗ ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨𝑇∗ (𝑓1 1 , . . . , 𝑓𝑚𝛼𝑚 ) (𝑥)󵄨󵄨󵄨 > 𝜆})

(70)

𝑚

󵄩 󵄩 ≤ 𝐶∏ 󵄩󵄩󵄩𝑓𝑖 󵄩󵄩󵄩L𝑝𝑖 ,𝑘 (𝑤𝑖 ) . 𝑖=1

This completes the proof of Theorem 15.

Competing Interests The authors declare that they have no competing interests.

Acknowledgments The first author was supported by the National Natural Science Foundation of China (no. 11401175 and no. 11501169). The second author was supported partly by NSFC (no. 11471041), the Fundamental Research Funds for the Central Universities (no. 2014kJJCA10), and NCET-13-0065.

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