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

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.

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)dy,

and they obtained some variant sharp function estimates and boundedness of the multilinear operators if D α bBMO( 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 bBMO(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)dx and ω Q = | Q | 1 Q ω(x)dx.

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 | dy.

It is well known that (see [1])

M # (f)(x) sup Q x inf c C 1 | Q | Q |f(y)c|dy.

Let

M(f)(x)= sup Q x 1 | Q | Q |f(y)|dy.

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, 1p< 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)dy.

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 1p<, 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 |dy<.

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 xQ,

| b Q b 2 k Q |Ck 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 1p<.

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)|dy,

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 fR 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)dy.

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(xy) 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(xy)f(y)dy.

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(xy)f(y)dy.

Note that the commutator [b,T](f)=bT(f)T(bf) 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 f0. We define, for 1/r=1/p1/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 bBMO(ω). Then

| b Q b 2 j Q |Cj b BMO ( ω ) ω Q j ,

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

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 bBMO(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 dxC b BMO ( ω ) r Q ν ( x ) r / p dx.

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 (dμ) for any p <r< p (1+δ), where dμ= ω r / p dx.

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 0η<n, 1s<p<n/η, 1/q=1/pη/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)dxC R n M Φ , η # (f) ( x ) p ω(x)dx.

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 |xy|.

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 bBMO(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 bBMO(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/pβ/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)=bT(f)T(bf) 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)dy. 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)dx

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(r1)/(rs), 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 |xy|| x 0 y| for xQ 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 k0,

| 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)dy. 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)dx A p / r (ν ( x ) r / p dx) 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. □

References

  1. Garcia-Cuerva J, Rubio de Francia JL North-Holland Math. 16. In Weighted Norm Inequalities and Related Topics. Elsevier, Amsterdam; 1985.

    Google Scholar 

  2. Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.

    Google Scholar 

  3. Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611–635. 10.2307/1970954

    MathSciNet  Article  Google Scholar 

  4. Pérez C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal. 1995, 128: 163–185. 10.1006/jfan.1995.1027

    MathSciNet  Article  Google Scholar 

  5. Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174

    Article  Google Scholar 

  6. Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002

    MathSciNet  Article  Google Scholar 

  7. Janson S: Mean oscillation and commutators of singular integral operators. Ark. Mat. 1978, 16: 263–270. 10.1007/BF02386000

    MathSciNet  Article  Google Scholar 

  8. Paluszynski M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J. 1995, 44: 1–17.

    MathSciNet  Article  Google Scholar 

  9. Bloom S: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 1985, 292: 103–122. 10.1090/S0002-9947-1985-0805955-5

    Article  Google Scholar 

  10. Hu B, Gu J: Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz spaces. J. Math. Anal. Appl. 2008, 340: 598–605. 10.1016/j.jmaa.2007.08.034

    MathSciNet  Article  Google Scholar 

  11. He YX, Wang YS: Commutators of Marcinkiewicz integrals and weighted BMO. Acta Math. Sin. Chin. Ser. 2011, 54: 513–520.

    Google Scholar 

  12. Cohen J, Gosselin J:On multilinear singular integral operators on R n . Stud. Math. 1982, 72: 199–223.

    MathSciNet  Google Scholar 

  13. Cohen J, Gosselin J: A BMO estimate for multilinear singular integral operators. Ill. J. Math. 1986, 30: 445–465.

    MathSciNet  Google Scholar 

  14. Ding Y, Lu SZ: Weighted boundedness for a class of rough multilinear operators. Acta Math. Sin. 2001, 17: 517–526.

    MathSciNet  Article  Google Scholar 

  15. Grubb DJ, Moore CN: A variant of Hörmander’s condition for singular integrals. Colloq. Math. 1997, 73: 165–172.

    MathSciNet  Google Scholar 

  16. Lu DQ, Liu LZ: M 2 -Type sharp estimates and weighted boundedness for commutators related to singular integral operators satisfying a variant of Hörmander’s condition. Bol. Soc. Parana. Mat. 2013, 31: 99–114.

    MathSciNet  Google Scholar 

  17. Trujillo-Gonzalez R: Weighted norm inequalities for singular integral operators satisfying a variant of Hörmander’s condition. Comment. Math. Univ. Carol. 2003, 44: 137–152.

    MathSciNet  Google Scholar 

  18. Garcia-Cuerva J Dissert. Math. 162. Weighted Hp Spaces 1979.

    Google Scholar 

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Correspondence to Changhong Wu.

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CW carried out all of the paper, MZ participated in the proof of Theorem 2. All authors read and approved the final manuscript.

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Wu, C., Zhang, M. Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator satisfying a variant of Hörmander’s condition. J Inequal Appl 2014, 57 (2014). https://doi.org/10.1186/1029-242X-2014-57

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Keywords

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