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Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator satisfying a variant of Hörmander’s condition

Journal of Inequalities and Applications20142014:57

https://doi.org/10.1186/1029-242X-2014-57

  • Received: 9 August 2013
  • Accepted: 13 January 2014
  • Published:

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for the multilinear operator associated with the singular integral operator satisfying a variant of Hörmander’s condition. As an application, we obtain the boundedness of the operator on weighted Lebesgue spaces.

MSC:42B20, 42B25.

Keywords

  • multilinear operator
  • singular integral operator
  • sharp maximal function
  • weighted BMO
  • weighted Lipschitz function

1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators and multilinear operators have been well studied. In [35], the authors proved that the commutators generated by the singular integral operators and BMO functions are bounded on L p ( R n ) for 1 < p < . Chanillo (see [6]) proved a similar result when singular integral operators are replaced by the fractional integral operators. In [7, 8], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and L p ( R n ) ( 1 < p < ) spaces are obtained. In [9, 10], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on L p ( R n ) ( 1 < p < ) spaces are obtained (also see [11]). In [12, 13], the authors studied some multilinear singular integral operators as follows (also see [14]):
T b ( f ) ( x ) = R m + 1 ( b ; x , y ) | x y | m K ( x , y ) f ( y ) d y ,

and they obtained some variant sharp function estimates and boundedness of the multilinear operators if D α b BMO ( R n ) for all α with | α | = m . In [15], some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators and their commutators is obtained (see [16, 17]). Motivated by these results, in this paper, we will study the multilinear operator generated by the singular integral operator satisfying a variant of Hörmander’s condition and the weighted Lipschitz and BMO functions, that is, D α b BMO ( w ) or D α b Lip β ( w ) for all α with | α | = m .

2 Preliminaries

First, let us introduce some notation. Throughout this paper, Q will denote a cube of R n with sides parallel to the axes. For a non-negative integrable function ω, let ω ( Q ) = Q ω ( x ) d x and ω Q = | Q | 1 Q ω ( x ) d x .

For any locally integrable function f, the sharp maximal function of f is defined by
M # ( f ) ( x ) = sup Q x 1 | Q | Q | f ( y ) f Q | d y .
It is well known that (see [1])
M # ( f ) ( x ) sup Q x inf c C 1 | Q | Q | f ( y ) c | d y .
Let
M ( f ) ( x ) = sup Q x 1 | Q | Q | f ( y ) | d y .

For η > 0 , let M η # ( f ) ( x ) = M # ( | f | η ) 1 / η ( x ) and M η ( f ) ( x ) = M ( | f | η ) 1 / η ( x ) .

For 0 < η < n , 1 p < and the non-negative weight function ω, set
M η , p , ω ( f ) ( x ) = sup Q x ( 1 ω ( Q ) 1 p η / n Q | f ( y ) | p ω ( y ) d y ) 1 / p
and
M ω ( f ) ( x ) = sup Q x 1 ω ( Q ) Q | f ( y ) | ω ( y ) d y .
The A p weight is defined by (see [1])
A p = { ω L loc 1 ( R n ) : sup Q ( 1 | Q | Q ω ( x ) d x ) ( 1 | Q | Q ω ( x ) 1 / ( p 1 ) d x ) p 1 < } , 1 < p <
and
A 1 = { ω L loc p ( R n ) : M ( ω ) ( x ) C w ( x ) , a.e. } .
Given a non-negative weight function ω. For 1 p < , the weighted Lebesgue space L p ( R n , ω ) is the space of functions f such that
f L p ( ω ) = ( R n | f ( x ) | p ω ( x ) d x ) 1 / p < .
Given the non-negative weight function ω, the weighted BMO space BMO ( ω ) is the space of functions b such that
b BMO ( ω ) = sup Q 1 ω ( Q ) Q | b ( y ) b Q | d y < .
For 0 < β < 1 , the weighted Lipschitz space Lip β ( ω ) is the space of functions b such that
b Lip β ( ω ) = sup Q 1 ω ( Q ) β / n ( 1 ω ( Q ) Q | b ( y ) b Q | p ω ( x ) 1 p d y ) 1 / p < .
Remark (1) It has been known that (see [18]), for b Lip β ( ω ) , ω A 1 and x Q ,
| b Q b 2 k Q | C k b Lip β ( ω ) ω ( x ) ω ( 2 k Q ) β / n .
  1. (2)

    Let b Lip β ( ω ) and ω A 1 . By [18], we know that spaces Lip β ( ω ) coincide and the norms b Lip β ( ω ) are equivalent with respect to different values 1 p < .

     
Definition 1 Let Φ = { ϕ 1 , , ϕ l } be a finite family of bounded functions in R n . For any locally integrable function f, the Φ sharp maximal function of f is defined by
M Φ # ( f ) ( x ) = sup Q x inf { c 1 , , c l } 1 | Q | Q | f ( y ) i = 1 l c i ϕ i ( x Q y ) | d y ,
where the infimum is taken over all m-tuples { c 1 , , c l } of complex numbers and x Q is the center of Q. For η > 0 , let
M Φ , η # ( f ) ( x ) = sup Q x inf { c 1 , , c l } ( 1 | Q | Q | f ( y ) i = 1 l c j ϕ i ( x Q y ) | η d y ) 1 / η .

Remark We note that M Φ # f # if l = 1 and ϕ 1 = 1 .

Definition 2 Given a positive and locally integrable function f in R n , we say that f satisfies the reverse Hölder’s condition (write this as f R H ( R n ) ), if for any cube Q centered at the origin we have
0 < sup x Q f ( x ) C 1 | Q | Q f ( y ) d y .

In this paper, we will study some singular integral operators as follows (see [15]).

Definition 3 Let K L 2 ( R n ) and satisfy
K L C , | K ( x ) | C | x | n ,
there exist functions B 1 , , B l L loc 1 ( R n { 0 } ) and Φ = { ϕ 1 , , ϕ l } L ( R n ) such that | det [ ϕ j ( y i ) ] | 2 R H ( R n l ) , and for a fixed δ > 0 and any | x | > 2 | y | > 0 ,
| K ( x y ) i = 1 l B i ( x ) ϕ i ( y ) | C | y | δ | x y | n + δ .
For f C 0 , we define the singular integral operator related to the kernel K by
T ( f ) ( x ) = R n K ( x y ) f ( y ) d y .
Moreover, let m be the positive integer and b be the function on R n . Set
R m + 1 ( b ; x , y ) = b ( x ) | α | m 1 α ! D α b ( y ) ( x y ) α .
The multilinear operator related to the operator T is defined by
T b ( f ) ( x ) = R n R m + 1 ( b ; x , y ) | x y | m K ( x y ) f ( y ) d y .

Note that the commutator [ b , T ] ( f ) = b T ( f ) T ( b f ) is a particular operator of the multilinear operator T b if m = 0 . The multilinear operator T b are the non-trivial generalizations of the commutator. It is well known that commutators and multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [1214]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator T b . As the application, we obtain the weighted L p -boundedness for the multilinear operator T b .

We give some preliminary lemmas.

Lemma 1 (see [[1], p.485])

Let 0 < p < q < and for any function f 0 . We define, for 1 / r = 1 / p 1 / q ,
f W L q = sup λ > 0 λ | { x R n : f ( x ) > λ } | 1 / q , N p , q ( f ) = sup Q f χ Q L p / χ Q L r ,
where the sup is taken for all measurable sets Q with 0 < | Q | < . Then
f W L q N p , q ( f ) ( q / ( q p ) ) 1 / p f W L q .

Lemma 2 (see [15])

Let T be the singular integral operator as Definition  2. Then T is bounded on L p ( R n , ω ) for ω A p with 1 < p < , and weak ( L 1 , L 1 ) bounded.

Lemma 3 (see [9])

Let b BMO ( ω ) . Then
| b Q b 2 j Q | C j b BMO ( ω ) ω Q j ,

where ω Q j = max 1 i j | 2 i Q | 1 2 i Q ω ( x ) d x .

Lemma 4 (see [9])

Let ω A p , 1 < p < . Then there exists ε > 0 such that ω r / p A r for any p r p + ε .

Lemma 5 (see [9])

Let b BMO ( ω ) , ω = ( μ ν 1 ) 1 / p , μ , ν A p and p > 1 . Then there exists ε > 0 such that for p r p + ε ,
Q | b ( x ) b Q | r μ ( x ) r / p d x C b BMO ( ω ) r Q ν ( x ) r / p d x .

Lemma 6 (see [9])

Let ω A p , 1 < p < . Then there exists 0 < δ < 1 such that ω 1 r / p A p / r ( d μ ) for any p < r < p ( 1 + δ ) , where d μ = ω r / p d x .

Lemma 7 (see [9])

Let μ , ν A p , ω = ( μ ν 1 ) 1 / p , 1 < p < . Then there exists 1 < q < p such that
ω Q ( ν Q ) 1 / q ( 1 | Q | Q ω ( x ) q ν ( x ) q / q d x ) 1 / q C .

Lemma 8 (see [1, 6])

Let 0 η < n , 1 s < p < n / η , 1 / q = 1 / p η / n and ω A 1 . Then
M η , s , ω ( f ) L q ( ω ) C f L p ( ω ) .

Lemma 9 (see [15, 17])

Let 1 < p < , 0 < η < , ω A and Φ = { ϕ 1 , , ϕ l } L ( R n ) such that | det [ ϕ j ( y i ) ] | 2 R H ( R n l ) . Then, for any smooth function f for which the left-hand side is finite,
R n M η ( f ) ( x ) p ω ( x ) d x C R n M Φ , η # ( f ) ( x ) p ω ( x ) d x .

Lemma 10 (see [13])

Let b be a function on R n and D α A L s ( R n ) for all α with | α | = m and any s > n . Then
| R m ( b ; x , y ) | C | x y | m | α | = m ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) | D α b ( z ) | s d z ) 1 / s ,

where Q ˜ is the cube centered at x and having side length 5 n | x y | .

3 Theorems and proofs

We shall prove the following theorems.

Theorem 1 Let T be the singular integral operator as Definition  3, 1 < p < , μ , ν A p , ω = ( μ ν 1 ) 1 / p , 0 < η < 1 and D α b BMO ( ω ) for all α with | α | = m . Then there exist a constant C > 0 , ε > 0 , 0 < δ < 1 , 1 < q < p and p < r < min ( p + ε , p ( 1 + δ ) ) such that, for any f C 0 ( R n ) and x ˜ R n ,
M Φ , η # ( T b ( f ) ) ( x ˜ ) C | α | = m D α b BMO ( ω ) ( [ M ν ( | ω T ( f ) | q ) ( x ˜ ) ] 1 / q + [ M ν r / p ( | ω f | r ) ( x ˜ ) ] 1 / r + [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q ) .
Theorem 2 Let T be the singular integral operator as Definition  3, ω A 1 , 0 < η < 1 , 1 < r < , 0 < β < 1 and D α b Lip β ( ω ) for all α with | α | = m . Then there exists a constant C > 0 such that, for any f C 0 ( R n ) and x ˜ R n ,
M Φ , η # ( T b ( f ) ) ( x ˜ ) C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) M β , r , ω ( f ) ( x ˜ ) .

Theorem 3 Let T be the singular integral operator as Definition  3, 1 < p < , μ , ν A p , ω = ( μ ν 1 ) 1 / p and D α b BMO ( ω ) for all α with | α | = m . Then T b is bounded from L p ( R n , μ ) to L p ( R n , ν ) .

Theorem 4 Let T be the singular integral operator as Definition  3, ω A 1 , 0 < β < 1 , 1 < p < n / β , 1 / q = 1 / p β / n and D α b Lip β ( ω ) for all α with | α | = m . Then T b is bounded from L p ( R n , ω ) to L q ( R n , ω 1 q ) .

Corollary Let [ b , T ] ( f ) = b T ( f ) T ( b f ) be the commutator generated by the singular integral operator T as Definition  2 and b. Then Theorems 1-4 hold for [ b , T ] .

Proof of Theorem 1 It suffices to prove for f C 0 ( R n ) and some constant C 0 , the following inequality holds:
( 1 | Q | Q | T b ( f ) ( x ) C 0 | η d x ) 1 / η C | α | = m D α b BMO ( ω ) ( [ M ν ( | ω T ( f ) | q ) ( x ˜ ) ] 1 / q + [ M ν r / p ( | ω f | r ) ( x ˜ ) ] 1 / r + [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q ) ,
where Q is any a cube centered at x 0 , C 0 = j = 1 l c j ϕ j ( x 0 x ) and c j = R n K ( x 0 , y ) | x 0 y | m B j ( x 0 y ) f 2 ( y ) d y . Fix a cube Q = Q ( x 0 , d ) and x ˜ Q . Let Q ˜ = 5 n Q and b ˜ ( x ) = b ( x ) | α | = m 1 α ! ( D α b ) Q ˜ x α , then R m ( b ; x , y ) = R m ( b ˜ ; x , y ) and D α b ˜ = D α b ( D α b ) Q ˜ for | α | = m . We write, for f 1 = f χ Q ˜ and f 2 = f χ R n Q ˜ ,
T b ( f ) ( x ) = R n R m ( b ˜ ; x , y ) | x y | m K ( x , x y ) f 1 ( y ) d y | α | = m 1 α ! R n ( x y ) α D α b ˜ ( y ) | x y | m K ( x , x y ) f 1 ( y ) d y + R n R m + 1 ( b ˜ ; x , y ) | x y | m K ( x , x y ) f 2 ( y ) d y = T ( R m ( b ˜ ; x , ) | x | m f 1 ) T ( | α | = m 1 α ! ( x ) α D α b ˜ | x | m f 1 ) + T b ˜ ( f 2 ) ( x ) ,
then
( 1 | Q | Q | T b ( f ) ( x ) C 0 | η d x ) 1 / η C ( 1 | Q | Q | T ( R m ( b ˜ ; x , ) | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T ( | α | = m ( x ) α D α b ˜ | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T b ˜ ( f 2 ) ( x ) C 0 | η d x ) 1 / η = I 1 + I 2 + I 3 .
For I 1 , noting that ω A 1 , w satisfies the reverse of Hölder’s inequality:
( 1 | Q | Q ω ( x ) p 0 d x ) 1 / p 0 C | Q | Q ω ( x ) d x
for all cubes Q and some 1 < p 0 < (see [1]). We take s = r p 0 / ( r + p 0 1 ) in Lemma 10 and have 1 < s < r and p 0 = s ( r 1 ) / ( r s ) , then by Lemma 10 and Hölder’s inequality, we obtain
| R m ( b ; x , y ) | C | x y | m | α | = m ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) | D α b ( z ) | s d z ) 1 / s C | x y | m | α | = m | Q ˜ | 1 / s ( Q ˜ ( x , y ) | D α b ( z ) | s ω ( z ) s ( 1 r ) / r ω ( z ) s ( r 1 ) / r d z ) 1 / s C | x y | m | α | = m | Q ˜ | 1 / s ( Q ˜ ( x , y ) | D α b ( z ) | r ω ( z ) 1 r d z ) 1 / r × ( Q ˜ ( x , y ) ω ( z ) s ( r 1 ) / ( r s ) d z ) ( r s ) / r s C | x y | m | α | = m | Q ˜ | 1 / s D α b BMO ( ω ) ω ( Q ˜ ) 1 / r | Q ˜ | ( r s ) / r s × ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) ω ( z ) p 0 d z ) ( r s ) / r s C | x y | m | α | = m D α b BMO ( ω ) | Q ˜ | 1 / q ω ( Q ˜ ) 1 / r | Q ˜ | 1 / s 1 / r × ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) ω ( z ) d z ) ( r 1 ) / r C | x y | m | α | = m D α b BMO ( ω ) | Q ˜ | 1 / q ω ( Q ˜ ) 1 / r | Q ˜ | 1 / s 1 / r ω ( Q ˜ ) 1 1 / r | Q ˜ | 1 / r 1 C | x y | m | α | = m D α b BMO ( ω ) ω ( Q ˜ ) | Q ˜ | ,
thus, by Lemma 7, we obtain
I 1 C | Q | Q | T ( R m ( b ˜ ; x , ) | x | m f 1 ) | d x C | α | = m D α b BMO ( ω ) ω ( Q ˜ ) | Q ˜ | 1 | Q | Q | T ( f ) ( y ) | ω ( y ) ν ( y ) 1 / q ω ( y ) 1 ν ( y ) 1 / q d y C | α | = m D α b BMO ( ω ) ω Q ˜ ( 1 | Q | Q | ω ( y ) T ( f ) ( y ) | q ν ( y ) d y ) 1 / q × ( 1 | Q | Q ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) ω Q ( ν Q ) 1 / q ( 1 ν ( Q ) Q | ω ( y ) T ( f ) ( y ) | q ν ( y ) d y ) 1 / q × ( 1 | Q | Q ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) [ M ν ( | ω T ( f ) | q ) ( x ˜ ) ] 1 / q × ω Q ( ν Q ) 1 / q ( 1 | Q | Q ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) [ M ν ( | ω T ( f ) | q ) ( x ˜ ) ] 1 / q .
For I 2 , we know ν r / p A r by Lemma 4, thus
( 1 | Q | Q ν ( x ) r / p d x ) 1 / r C ( 1 | Q | Q ν ( x ) r / p d x ) 1 / r ,
then, by the weak ( L 1 , L 1 ) boundedness of T (see Lemma 2) and Kolmogorov’s inequality (see Lemma 1), we obtain, by Lemma 5,
I 2 C | α | = m ( 1 | Q | Q | T ( D α b ˜ f 1 ) ( x ) | η d x ) 1 / η C | α | = m | Q | 1 / η 1 | Q | 1 / η T ( D α b ˜ f 1 ) χ Q L η χ Q L η / ( 1 η ) C | α | = m 1 | Q | T ( D α b ˜ f 1 ) W L 1 C | α | = m 1 | Q | R n | D α b ˜ ( x ) f 1 ( x ) | d x = C | α | = m 1 | Q | Q ˜ | D α b ( x ) ( D α b ) Q ˜ | μ ( x ) 1 / p | f ( x ) | ω ( x ) ν ( x ) 1 / p d x C | α | = m ( 1 | Q ˜ | Q ˜ | ( D α b ( x ) ( D α b ) Q ˜ ) | r μ ( x ) r / p d x ) 1 / r × ( 1 | Q ˜ | Q ˜ | f ( x ) | r ω ( x ) r ν ( x ) r / p d x ) 1 / r C | α | = m D α b BMO ( ω ) ( 1 | Q ˜ | Q ˜ ν ( x ) r / p d x ) 1 / r ( 1 | Q ˜ | Q ˜ | f ( x ) ω ( x ) | r ν ( x ) r / p d x ) 1 / r C | α | = m D α b BMO ( ω ) ( 1 | Q ˜ | Q ˜ ν ( x ) r / p d x ) 1 / r × ( 1 | Q ˜ | Q ˜ | f ( x ) ω ( x ) | r ν ( x ) r / p d x ) 1 / r C | α | = m D α b BMO ( ω ) ( 1 ν ( Q ˜ ) r / p Q ˜ | f ( x ) ω ( x ) | r ν ( x ) r / p d x ) 1 / r C | α | = m D α b BMO ( ω ) [ M ν r / p ( | ω f | r ) ( x ˜ ) ] 1 / r .
For I 3 , note that | x y | | x 0 y | for x Q and y R n Q , we write
| T b ˜ ( f 2 ) ( x ) C 0 | R n | R m ( b ˜ ; x , y ) | x y | m R m ( b ˜ ; x 0 , y ) | x 0 y | m | | K ( x y ) | | f 2 ( y ) | d y + R n | R m + 1 ( b ˜ ; x 0 , y ) | | x 0 y | m | K ( x y ) j = 1 l B j ( x 0 y ) ϕ j ( x 0 x ) | | f 2 ( y ) | d y + C | α | = m R n | ( x y ) α | x y | m ( x 0 y ) α | x 0 y | m | | K ( x y ) | | D α b ˜ ( y ) | | f 2 ( y ) | d y = I 3 ( 1 ) ( x ) + I 3 ( 2 ) ( x ) + I 3 ( 3 ) ( x ) .
For I 3 ( 1 ) , by the formula (see [13]):
R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) = | γ | < m 1 γ ! R m | γ | ( D γ b ˜ ; x , x 0 ) ( x y ) γ
and Lemma 10, we have, similar to the proof of I 1 and for k 0 ,
| R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | C | γ | < m | α | = m | x x 0 | m | γ | | x y | | γ | D α b BMO ( ω ) ω ( 2 k Q ˜ ) | 2 k Q ˜ |
and
| R m ( b ˜ ; x , y ) | C | x y | m | α | = m D α b BMO ( ω ) ω ( 2 k Q ˜ ) | 2 k Q ˜ | ,
thus
I 3 ( 1 ) ( x ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ | R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | | K ( x y ) | | x y | m | f ( y ) | d y + k = 0 2 k + 1 Q ˜ 2 k Q ˜ | 1 | x y | m 1 | x 0 y | m | | R m ( b ˜ ; x 0 , y ) | | f ( y ) | d y C | α | = m D α b BMO ( ω ) k = 0 ω ( 2 k + 1 Q ˜ ) | 2 k + 1 Q ˜ | 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 | f ( y ) | d y C | α | = m D α b BMO ( ω ) k = 1 ω 2 k Q ˜ d ( 2 k d ) n + 1 2 k Q ˜ | f ( y ) | ω ( y ) ν ( y ) 1 / q ω ( y ) 1 ν ( y ) 1 / q d y C | α | = m D α b BMO ( ω ) k = 1 2 k ω 2 k Q ˜ ( 1 | 2 k Q ˜ | 2 k Q ˜ | ω ( y ) f ( y ) | q ν ( y ) d y ) 1 / q × ( 1 | 2 k Q | 2 k Q ˜ ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) k = 1 2 k ω 2 k Q ˜ ( ν 2 k Q ˜ ) 1 / q × ( 1 ν ( 2 k Q ˜ ) 2 k Q ˜ | ω ( y ) f ( y ) | q ν ( y ) d y ) 1 / q × ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q k = 1 k 2 k × ω 2 k Q ( ν 2 k Q ) 1 / q ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q k = 1 k 2 k C | α | = m D α b BMO ( ω ) [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q .
For I 3 ( 2 ) , we get
I 3 ( 2 ) ( x ) C k = 0 2 k + 1 Q ˜ 2 k Q ˜ | R m ( b ˜ ; x 0 , y ) | | x 0 y | m | K ( x y ) j = 1 l B j ( x 0 y ) ϕ j ( x 0 x ) | | f ( y ) | d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | D α b ˜ ( y ) | | ( x 0 y ) α | | x 0 y | m × | K ( x y ) j = 1 l B j ( x 0 y ) ϕ j ( x 0 x ) | | f ( y ) | d y C | α | = m D α b BMO ( ω ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ ω ( 2 k + 1 Q ˜ ) | 2 k + 1 Q ˜ | | x x 0 | δ | x 0 y | n + δ | f ( y ) | d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | D α b ( y ) ( D α b ) 2 k + 1 Q ˜ | | x x 0 | δ | x 0 y | n + δ | f ( y ) | d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | ( D α b ) 2 k + 1 Q ˜ ( D α b ) Q ˜ | | x x 0 | δ | x 0 y | n + δ | f ( y ) | d y C | α | = m D α b BMO ( ω ) [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q k = 1 k 2 k δ × ω 2 k Q ˜ ( ν 2 k Q ˜ ) 1 / q ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) q ν ( y ) q / q d y ) 1 / q + C k = 1 2 k δ ( 1 | 2 k Q ˜ | 2 k Q ˜ | D α b ( y ) ( D α b ) 2 k Q ˜ | r μ ( y ) r / p d y ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ | f ( y ) | r ω ( y ) r ν ( y ) r / p d y ) 1 / r + C | α | = m D α b BMO ( ω ) k = 1 k 2 k δ ω 2 k Q ˜ ( 1 | 2 k Q ˜ | 2 k Q ˜ | ω ( y ) f ( y ) | q ν ( y ) d y ) 1 / q × ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) k = 1 2 k δ ( 1 | 2 k Q ˜ | 2 k Q ˜ ν ( y ) r / p d y ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ | f ( y ) ω ( y ) | r ν ( y ) r / p d y ) 1 / r + C | α | = m D α b BMO ( ω ) [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q k = 1 k 2 k δ × ω 2 k Q ˜ ( ν 2 k Q ˜ ) 1 / q ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) ( [ M ν r / p ( | ω f | r ) ( x ˜ ) ] 1 / r + [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q ) .
Similarly, we have
I 3 ( 3 ) ( x ) C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | D α b ( y ) ( D α b ) 2 k + 1 Q ˜ | | x x 0 | | x 0 y | n + 1 | f ( y ) | d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | ( D α b ) 2 k + 1 Q ˜ ( D α b ) Q ˜ | | x x 0 | | x 0 y | n + 1 | f ( y ) | d y C | α | = m D α b BMO ( ω ) k = 1 2 k ( 1 | 2 k Q ˜ | 2 k Q ˜ ν ( y ) r / p d y ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ | f ( y ) ω ( y ) | r ν ( y ) r / p d y ) 1 / r + C | α | = m D α b BMO ( ω ) [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q k = 1 k 2 k × ω 2 k Q ˜ ( ν 2 k Q ˜ ) 1 / q ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) q ν ( y ) q / q d y ) 1 / q C | α | = m D α b BMO ( ω ) ( [ M ν r / p ( | ω f | r ) ( x ˜ ) ] 1 / r + [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q ) .
Thus
I 3 C | α | = m D α b BMO ( ω ) ( [ M ν r / p ( | ω f | r ) ( x ˜ ) ] 1 / r + [ M ν ( | ω f | q ) ( x ˜ ) ] 1 / q ) .

These results complete the proof of Theorem 1. □

Proof of Theorem 2 It suffices to prove for f C 0 ( R n ) and some constant C 0 that the following inequality holds:
( 1 | Q | Q | T b ( f ) ( x ) C 0 | η d x ) 1 / η C | α | = m D α b Lip β ( ω ) w ( x ˜ ) M β , r , ω ( f ) ( x ˜ ) ,
where Q is any cube centered at x 0 , C 0 = j = 1 m c j ϕ j ( x 0 x ) and c j = R n K ( x 0 , y ) | x 0 y | m B j ( x 0 y ) f 2 ( y ) d y . Fix a cube Q = Q ( x 0 , d ) and x ˜ Q . Similar to the proof of Theorem 1, we have, for f 1 = f χ Q ˜ and f 2 = f χ R n Q ˜ ,
( 1 | Q | Q | T b ( f ) ( x ) C 0 | η d x ) 1 / η C ( 1 | Q | Q | T ( R m ( b ˜ ; x , ) | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T ( | α | = m ( x ) α D α b ˜ | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T b ˜ ( f 2 ) ( x ) C 0 | η d x ) 1 / η = J 1 + J 2 + J 3 .
For J 1 and J 2 , by using the same argument as in the proof of Theorem 1, we get
| R m ( b ˜ ; x , y ) | C | x y | m | α | = m | Q ˜ | 1 / q ( Q ˜ ( x , y ) | D α b ˜ ( z ) | q ω ( z ) q ( 1 r ) / r ω ( z ) q ( r 1 ) / r d z ) 1 / q C | x y | m | α | = m | Q ˜ | 1 / q ( Q ˜ ( x , y ) | D α b ˜ ( z ) | r ω ( z ) 1 r d z ) 1 / r × ( Q ˜ ( x , y ) ω ( z ) q ( r 1 ) / ( r q ) d z ) ( r q ) / r q C | x y | m | α | = m | Q ˜ | 1 / q D α b Lip β ( ω ) ω ( Q ˜ ) β / n + 1 / r | Q ˜ | ( r q ) / r q × ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) ω ( z ) p 0 d z ) ( r q ) / r q C | x y | m | α | = m D α b Lip β ( ω ) | Q ˜ | 1 / q ω ( Q ˜ ) β / n + 1 / r | Q ˜ | 1 / q 1 / r × ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) ω ( z ) d z ) ( r 1 ) / r C | x y | m | α | = m D α b Lip β ( ω ) | Q ˜ | 1 / q ω ( Q ˜ ) β / n + 1 / r | Q ˜ | 1 / q 1 / r ω ( Q ˜ ) 1 1 / r | Q ˜ | 1 / r 1 C | x y | m | α | = m D α b Lip β ( ω ) ω ( Q ˜ ) β / n ω ( x ˜ ) ,
thus
J 1 C | α | = m D α b Lip β ( ω ) ω ( Q ˜ ) β / n ω ( x ˜ ) | Q | 1 / s ( R n | f 1 ( x ) | s d x ) 1 / s J 1 C | α | = m D α b Lip β ( ω ) ω ( Q ˜ ) β / n ω ( x ˜ ) | Q | 1 / s ( Q ˜ | f ( x ) | r ω ( x ) d x ) 1 / r J 1 × ( Q ˜ ω ( x ) s / ( r s ) d x ) ( r s ) / r s J 1 C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) | Q ˜ | 1 / s ω ( Q ˜ ) 1 / r ( 1 ω ( Q ˜ ) 1 r β / n Q ˜ | f ( x ) | r ω ( x ) d x ) 1 / r J 1 × ( 1 | Q ˜ | Q ˜ ω ( x ) s / ( r s ) d x ) ( r s ) / r s ( 1 | Q ˜ | Q ˜ ω ( x ) d x ) 1 / r | Q ˜ | 1 / s ω ( Q ˜ ) 1 / r J 1 C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) M β , r , ω ( f ) ( x ˜ ) , J 2 C | α | = m 1 | Q | Q ˜ | D α b ( x ) ( D α b ) Q ˜ | ω ( x ) 1 / r | f ( x ) | ω ( x ) 1 / r d x J 2 C | α | = m 1 | Q | ( Q ˜ | D α b ( x ) ( D α b ) Q ˜ | r ω ( x ) 1 r d x ) 1 / r ( Q ˜ | f ( x ) | r ω ( x ) d x ) 1 / r J 2 C | α | = m 1 | Q | D α b Lip β ( ω ) ω ( Q ˜ ) β / n + 1 / r ω ( Q ˜ ) 1 / r β / n J 2 × ( 1 ω ( Q ˜ ) 1 r β / n Q ˜ | f ( x ) | r ω ( x ) d x ) 1 / r J 2 C | α | = m D α b Lip β ( ω ) ω ( Q ˜ ) | Q ˜ | M β , r , ω ( f ) ( x ˜ ) J 2 C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) M β , r , ω ( f ) ( x ˜ ) .
For J 3 , we have
| R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | C | γ | < m | α | = m | x x 0 | m | γ | | x y | | γ | × D α b Lip β ( ω ) ω ( x ˜ ) ω ( 2 k Q ˜ ) β / n ,
thus
| T b ˜ ( f 2 ) ( x ) C 0 | k = 0 2 k + 1 Q ˜ 2 k Q ˜ | R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | | K ( x y ) | | x y | m | f ( y ) | d y + k = 0 2 k + 1 Q ˜ 2 k Q ˜ | 1 | x y | m 1 | x 0 y | m | | R m ( b ˜ ; x 0 , y ) | | K ( x y ) | | f ( y ) | d y + C k = 0 2 k + 1 Q ˜ 2 k Q ˜ | R m ( b ˜ ; x 0 , y ) | | x 0 y | m | K ( x y ) j = 1 l B j ( x 0 y ) ϕ j ( x 0 x ) | | f ( y ) | d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | D α b ˜ ( y ) | | ( x 0 y ) α | | x 0 y | m × | K ( x y ) j = 1 l B j ( x 0 y ) ϕ j ( x 0 x ) | | f ( y ) | d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | ( x y ) α | x y | m ( x 0 y ) α | x 0 y | m | | K ( x y ) | | D α b ˜ ( y ) | | f ( y ) | d y C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) k = 0 ω ( 2 k + 1 Q ˜ ) β / n × 2 k + 1 Q ˜ 2 k Q ˜ ( | x x 0 | | x 0 y | n + 1 + | x x 0 | δ | x 0 y | n + δ ) | f ( y ) | d y + C | α | = m k = 1 ( | x x 0 | | x 0 y | n + 1 + | x x 0 | δ | x 0 y | n + δ ) × 2 k Q ˜ | D α b ( y ) ( D α b ) 2 k Q ˜ | w ( y ) 1 / r | f ( y ) | ω ( y ) 1 / r d y + C | α | = m k = 1 ( | x x 0 | | x 0 y | n + 1 + | x x 0 | δ | x 0 y | n + δ ) × 2 k Q ˜ | ( D α b ) 2 k Q ˜ ( D α b ) Q ˜ | | f ( y ) | ω ( y ) 1 / r ω ( y ) 1 / r d y C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) k = 1 ( d ( 2 k d ) n + 1 + d δ ( 2 k d ) n + δ ) × ω ( 2 k Q ˜ ) β / n ( 2 k Q ˜ | f ( y ) | r ω ( y ) d x ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) 1 / ( r 1 ) d y ) ( r 1 ) / r ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) d y ) 1 / r | 2 k Q ˜ | ω ( 2 k Q ˜ ) 1 / r + C | α | = m k = 1 ( d ( 2 k d ) n + 1 + d δ ( 2 k d ) n + δ ) × ( 2 k Q ˜ | ( D α b ( y ) ( D α b ) 2 k Q ˜ ) | r ω ( y ) 1 r d y ) 1 / r ( 2 k Q ˜ | f ( y ) | r ω ( y ) d y ) 1 / r + C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) k = 1 k ω ( 2 k Q ˜ ) β / n ( d ( 2 k d ) n + 1 + d δ ( 2 k d ) n + δ ) × ( 2 k Q ˜ | f ( y ) | r ω ( y ) d x ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) 1 / ( r 1 ) d y ) ( r 1 ) / r ( 1 | 2 k Q ˜ | 2 k Q ˜ ω ( y ) d y ) 1 / r | 2 k Q ˜ | ω ( 2 k Q ˜ ) 1 / r C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) k = 1 k ( 2 k + 2 k δ ) × ( 1 ω ( 2 k Q ˜ ) 1 r β / n 2 k Q ˜ | f ( y ) | r ω ( y ) d x ) 1 / r + C | α | = m D α b Lip β ( ω ) k = 1 ( 2 k + 2 k δ ) × ω ( 2 k Q ˜ ) | 2 k Q ˜ | ( 1 ω ( 2 k Q ˜ ) 1 r β / n 2 k Q ˜ | f ( y ) | r ω ( y ) d x ) 1 / r C | α | = m D α b Lip β ( ω ) ω ( x ˜ ) M β , r , ω ( f ) ( x ˜ ) .

This completes the proof of Theorem 2. □

Proof of Theorem 3 Notice that ν r / p A r + 1 r / p A p and ν ( x ) d x A p / r ( ν ( x ) r / p d x ) by Lemma 6, thus, by Theorem 1, Lemmas 2 and 9,
R n | T b ( f ) ( x ) | p ν ( x ) d x R n | M η ( T b ( f ) ) ( x ) | p ν ( x ) d x C R n | M Φ , η # ( T b ( f ) ) ( x ) | p ν ( x ) d x C | α | = m D α b BMO ( ω ) R n ( [ M ν ( | ω T ( f ) | q ) ( x ) ] p / q + [ M ν r / p ( | ω f | r ) ( x ) ] p / r + [ M ν ( | ω f | q ) ( x ) ] p / q ) ν ( x ) d x C | α | = m D α b BMO ( ω ) ( R n | ω ( x ) f ( x ) | p ν ( x ) d x + R n | ω ( x ) T ( f ) ( x ) | p ν ( x ) d x ) = C | α | = m D α b BMO ( ω ) ( R n | f ( x ) | p μ ( x ) d x + R n | T ( f ) ( x ) | p μ ( x ) d x ) C | α | = m D α b BMO ( ω ) R n | f ( x ) | p μ ( x ) d x .

This completes the proof of Theorem 3. □

Proof of Theorem 4 Choose 1 < r < p in Theorem 2 and notice ω 1 q A 1 , then we have, by Lemmas 8 and 9,
T b ( f ) L q ( ω 1 q ) M η ( T b ( f ) ) L q ( ω 1 q ) C M Φ , η # ( T b ( f ) ) L q ( ω 1 q ) C | α | = m D α b Lip β ( ω ) ω M β , r , ω ( f ) L q ( ω 1 q ) = C | α | = m D α b Lip β ( ω ) M β , r , ω ( f ) L q ( ω ) C | α | = m D α b Lip β ( ω ) f L p ( ω ) .

This completes the proof of Theorem 4. □

Declarations

Authors’ Affiliations

(1)
Orient Science and Technology College, Hunan Agriculture University, Changsha, 410128, P.R. China

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