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A sharp inequality for multilinear singular integral operators with non-smooth kernels

Abstract

In this paper, we establish a sharp inequality for some multilinear singular integral operators with non-smooth kernels. As an application, we obtain the weighted L p -norm inequality and LlogL-type inequality for the multilinear operators.

MSC:42B20, 42B25.

1 Definitions and results

As the development of singular integral operators and their commutators, multilinear singular integral operators have been well studied (see [16]). In this paper, we study some multilinear operator associated to the singular integral operators with non-smooth kernels as follows.

Definition 1 A family of operators D t , t>0, is said to be ‘approximations to the identity’ if, for every t>0, D t can be represented by the kernel a t (x,y) in the following sense:

D t (f)(x)= R n a t (x,y)f(y)dy

for every f L p ( R n ) with p1, and a t (x,y) satisfies

| a t ( x , y ) | h t (x,y)=C t n / 2 s ( | x y | 2 / t ) ,

where s is a positive, bounded and decreasing function satisfying

lim r r n + ϵ s ( r 2 ) =0

for some ϵ>0.

Definition 2 A linear operator T is called a singular integral operator with non-smooth kernel if T is bounded on L 2 ( R n ) and associated with a kernel K(x,y) such that

T(f)(x)= R n K(x,y)f(y)dy

for every continuous function f with compact support, and for almost all x not in the support of f.

  1. (1)

    There exists an ‘approximation to the identity’ { B t ,t>0} such that T B t has an associated kernel k t (x,y) and there exist c 1 , c 2 >0 so that

    | x y | > c 1 t 1 / 2 | K ( x , y ) k t ( x , y ) | dx c 2 for all y R n .
  2. (2)

    There exists an ‘approximation to the identity’ { A t ,t>0} such that A t T has an associated kernel K t (x,y) which satisfies

    | K t ( x , y ) | c 4 t n / 2 if |xy| c 3 t 1 / 2 ,

and

| K ( x , y ) K t ( x , y ) | c 4 t δ / 2 | x y | n δ if |xy| c 3 t 1 / 2

for some c 3 , c 4 >0, δ>0.

Let m j be positive integers (j=1,,l), m 1 ++ m l =m, and let b j be functions on R n (j=1,,l). Set, for 1jm,

R m j + 1 ( b j ;x,y)= b j (x) | α | m j 1 α ! D α b j (y) ( x y ) α .

The multilinear operator associated to T is defined by

T b (f)(x)= R n j = 1 l R m j + 1 ( b j ; x , y ) | x y | m K(x,y)f(y)dy.

Note that when m=0, T b is just the multilinear commutator of T and b j (see [7]). However, when m>0, T b is a non-trivial generalization of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [14]). Hu and Yang (see [8]) proved a variant sharp estimate for the multilinear singular integral operators. In [7], Pérez and Trujillo-Gonzalez proved a sharp estimate for the multilinear commutator when b j Osc exp L r j ( R n ) and noted that Osc exp L r j BMO. The main purpose of this paper is to prove a sharp function inequality for the multilinear singular integral operator with non-smooth kernel when D α b j BMO( R n ) for all α with |α|= m j . As an application, we obtain an L p (p>1) norm inequality and an LlogL-type inequality for the multilinear operators. In [912], the boundedness of a singular integral operator with non-smooth kernel is obtained. In [13], the boundedness of the commutator associated to the singular integral operator with non-smooth kernel is obtained. Our works are motivated by these papers.

First, let us introduce some notations. Throughout this paper, Q denotes a cube of R n with sides parallel to the axes. For any locally integrable function f, the sharp function of f is defined by

f # (x)= sup Q x 1 | Q | Q | f ( y ) f Q | dy,

where, and in what follows, f Q = | Q | 1 Q f(x)dx. It is well known that (see [14, 15])

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

We say that f belongs to BMO( R n ) if f # belongs to L ( R n ) and f BMO = f # L . Let M be a Hardy-Littlewood maximal operator defined by

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

For kN, we denote by M k the operator M iterated k times, i.e., M 1 (f)(x)=M(f)(x) and

M k (f)(x)=M ( M k 1 ( f ) ) (x)when k2.

The sharp maximal function M A (f) associated with the ‘approximations to the identity’ { A t ,t>0} is defined by

M A # (f)(x)= sup Q x 1 | Q | Q | f ( y ) A t Q ( f ) ( y ) | dy,

where t Q =l ( Q ) 2 and l(Q) denotes the side length of Q. For 0<r<, we denote M A # ( f ) r by

M A # ( f ) r = [ M A # ( | f | r ) ] 1 / r .

Let Φ be a Young function and Φ ˜ be the complementary associated to Φ. For a function f, we denote the Φ-average by

f Φ , Q =inf { λ > 0 : 1 | Q | Q Φ ( | f ( y ) | λ ) d y 1 }

and the maximal function associated to Φ by

M Φ (f)(x)= sup Q x f Φ , Q .

The Young functions used in this paper are Φ(t)=t ( 1 + log t ) r and Φ ˜ (t)=exp( t 1 / r ), the corresponding average and maximal functions are denoted by L ( log L ) r , Q , M L ( log L ) r and exp L 1 / r , Q , M exp L 1 / r . Following [11, 12, 16], we know the generalized Hölder inequality

1 | Q | Q | f ( y ) g ( y ) | dy f Φ , Q g Φ ˜ , Q

and the following inequality, for r, r j 1, j=1,,l with 1/r=1/ r 1 ++1/ r l , and any x R n , bBMO( R n ),

f L ( log L ) 1 / r , Q M L ( log L ) 1 / r ( f ) C M L ( log L ) l ( f ) C M l + 1 ( f ) , b b Q exp L r , Q C b BMO , | b 2 k + 1 Q b 2 Q | C k b BMO .

We denote the Muckenhoupt weights by A p for 1p< (see [14]).

We shall prove the following theorems.

Theorem 1 If T is a singular integral operator with non-smooth kernel as given in Definition  2, let D α b j BMO( R n ) for all α with |α|= m j and j=1,,l. Then there exists a constant C>0 such that for any f C 0 ( R n ), 0<r<1 and x ˜ R n ,

M A # ( T b ( f ) ) r ( x ˜ )C j = 1 l ( | α j | = m j D α j b j BMO ) M l + 1 (f)( x ˜ ).

Theorem 2 If T is a singular integral operator with non-smooth kernel as given in Definition  2, let D α b j BMO( R n ) for all α with |α|= m j and j=1,,l. Then T b is bounded on L p (w) for any 1<p< and w A p , that is,

T b ( f ) L p ( w ) C j = 1 l ( | α j | = m j D α j b j BMO ) f L p ( w ) .

Theorem 3 If T is a singular integral operator with non-smooth kernel as given in Definition  2, let w A 1 , D α b j BMO( R n ) for all α with |α|= m j and j=1,,l. Then there exists a constant C>0 such that for all λ>0,

w ( { x R n : | T b ( f ) ( x ) | > λ } ) C R n | f ( x ) | λ [ 1 + log + ( | f ( x ) | λ ) ] l w(x)dx.

2 Proof of the theorem

To prove the theorems, we need the following lemma.

Lemma 1 (see [1])

Let b be a function on R n and D α b L q ( R n ) for all α with |α|=m and some q>n. Then

| R m ( b ; x , y ) | C | x y | m | α | = m ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) | D α b ( z ) | q d z ) 1 / q ,

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

Lemma 2 ([[14], p.485])

Let 0<p<q< and for any function f0, we define that 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 E f χ E L p / χ E L r ,

where the sup is taken for all measurable sets E with 0<|E|<. Then

f W L q N p , q (f) ( q / ( q p ) ) 1 / p f W L q .

Lemma 3 (see [17])

Let r j 1 for j=1,,l, we denote that 1/r=1/ r 1 ++1/ r l . Then

1 | Q | Q | f 1 ( x ) f l ( x ) g ( x ) | dx f exp L r 1 , Q f exp L r l , Q g L ( log L ) 1 / r , Q .

Lemma 4 ([9, 10])

Let T be a singular integral operator with non-smooth kernel as given in Definition  2. Then T is bounded on L p ( R n ) for every 1<p< and bounded from L 1 ( R n ) to W L 1 ( R n ).

Lemma 5 (see [9, 12])

For any γ>0, there exists a constant C>0 independent of γ such that

| { x R n : M ( f ) ( x ) > D λ , M A # ( f ) ( x ) γ λ } | Cγ | { x R n : M ( f ) ( x ) > λ } |

for λ>0, where D is a fixed constant which only depends on n. Thus

M ( f ) L p C M A # ( f ) L p

for every f L p ( R n ), 1<p<.

Lemma 6 Let { A t ,t>0} be an ‘approximation to the identity’ and bBMO( R n ). Then, for every f L p ( R n ), p>1 and x ˜ R n ,

sup Q x ˜ 1 | Q | Q | A t Q ( ( b b Q ) f ) ( x ) | dxC b BMO M 2 (f)( x ˜ ),

where t Q =l ( Q ) 2 and l(Q) denotes the side length of Q.

Proof We write, for any cube Q with x ˜ Q,

1 | Q | Q | A t Q ( ( b b Q ) f ) ( x ) | d x 1 | Q | Q R n h t Q ( x , y ) | ( b ( y ) b Q ) f ( y ) | d y d x 1 | Q | Q 2 Q h t Q ( x , y ) | ( b ( y ) b Q ) f ( y ) | d y d x + k = 1 1 | Q | Q 2 k + 1 Q 2 k Q h t Q ( x , y ) | ( b ( y ) b Q ) f ( y ) | d y d x = I 1 + I 2 .

We have, by the generalized Hölder inequality,

I 1 C 1 | Q | | 2 Q | Q 2 Q | ( b ( y ) b Q ) f ( y ) | d y d x C b b Q exp L , 2 Q f L ( log L ) , 2 Q C b BMO M 2 ( f ) ( x ˜ ) .

For I 2 , notice for xQ and y 2 k + 1 Q 2 k Q, then |xy| 2 k 1 t Q and h t Q (x,y)C s ( 2 2 ( k 1 ) ) | Q | , then

I 2 C k = 1 s ( 2 2 ( k 1 ) ) 1 | Q | 2 Q 2 k + 1 Q | ( b ( y ) b Q ) f ( y ) | d y d x C k = 1 2 k n s ( 2 2 ( k 1 ) ) 1 | 2 k + 1 Q | 2 k + 1 Q | ( b ( y ) b Q ) f ( y ) | d y C k = 1 2 k n s ( 2 2 ( k 1 ) ) b b Q exp L , 2 k + 1 Q f L ( log L ) , 2 k + 1 Q C k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) b BMO M 2 ( f ) ( x ˜ ) C b BMO M 2 ( f ) ( x ˜ ) ,

where the last inequality follows from

k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) C k = 1 2 ( k 1 ) ε <

for some ϵ>0. This completes the proof. □

Proof of Theorem 1 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 ) | r | A t Q T b ( f ) ( x ) | r | d x ) 1 / r C j = 1 l ( | α j | = m j D α j b j BMO ) M l + 1 (f)(x).

Without loss of generality, we may assume l=2. Fix a cube Q=Q( x 0 ,d) and x ˜ Q. Let Q ˜ =5 n Q and b ˜ j (x)= b j (x) | α | = m j 1 α ! ( D α b j ) Q ˜ x α , then R m j ( b j ;x,y)= R m j ( b ˜ j ;x,y) and D α b ˜ j = D α b j ( D α b j ) Q ˜ for |α|= m j . We write, for f=f χ Q ˜ +f χ R n Q ˜ = f 1 + f 2 ,

T b ( f ) ( x ) = R n j = 1 2 R m j + 1 ( b ˜ j ; x , y ) | x y | m K ( x , y ) f ( y ) d y = R n j = 1 2 R m j ( b ˜ j ; x , y ) | x y | m K ( x , y ) f 1 ( y ) d y | α 1 | = m 1 1 α 1 ! R n R m 2 ( b ˜ 2 ; x , y ) ( x y ) α 1 D α 1 b ˜ 1 ( y ) | x y | m K ( x , y ) f 1 ( y ) d y | α 2 | = m 2 1 α 2 ! R n R m 1 ( b ˜ 1 ; x , y ) ( x y ) α 2 D α 2 b ˜ 2 ( y ) | x y | m K ( x , y ) f 1 ( y ) d y + | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! R n ( x y ) α 1 + α 2 D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) | x y | m K ( x , y ) f 1 ( y ) d y + R n j = 1 2 R m j + 1 ( b ˜ j ; x , y ) | x y | m K ( x , y ) f 2 ( y ) d y = T ( j = 1 2 R m j ( b ˜ j ; x , ) | x | m f 1 ) T ( | α 1 | = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ) ( x ) α 1 D α 1 b ˜ 1 | x | m f 1 ) T ( | α 2 | = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ) ( x ) α 2 D α 2 b ˜ 2 | x | m f 1 ) + T ( | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! ( x ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 | x | m f 1 ) + T ( j = 1 2 R m j + 1 ( b ˜ j ; x , ) | x | m f 2 )

and

A t Q T b ( f ) ( x ) = R n j = 1 2 R m j ( b ˜ j ; x , y ) | x y | m K t ( x , y ) f 1 ( y ) d y | α 1 | = m 1 1 α 1 ! R n R m 2 ( b ˜ 2 ; x , y ) ( x y ) α 1 D α 1 b ˜ 1 ( y ) | x y | m K t ( x , y ) f 1 ( y ) d y | α 2 | = m 2 1 α 2 ! R n R m 1 ( b ˜ 1 ; x , y ) ( x y ) α 2 D α 2 b ˜ 2 ( y ) | x y | m K t ( x , y ) f 1 ( y ) d y + | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! R n ( x y ) α 1 + α 2 D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) | x y | m K t ( x , y ) f 1 ( y ) d y + R n j = 1 2 R m j + 1 ( b ˜ j ; x , y ) | x y | m K t ( x , y ) f 2 ( y ) d y = A t Q T ( j = 1 2 R m j ( b ˜ j ; x , ) | x | m f 1 ) A t Q T ( | α 1 | = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ) ( x ) α 1 D α 1 b ˜ 1 | x | m f 1 ) A t Q T ( | α 2 | = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ) ( x ) α 2 D α 2 b ˜ 2 | x | m f 1 ) + A t Q T ( | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! ( x ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 | x | m f 1 ) + A t Q T ( j = 1 2 R m j + 1 ( b ˜ j ; x , ) | x | m f 2 ) ,

then

[ 1 | Q | Q | | T b ( f ) ( x ) | r | A t Q T b ( f ) ( x ) | r | d x ] 1 / r [ 1 | Q | Q | T b ( f ) ( x ) A t Q T b ( f ) ( x ) | r d x ] 1 / r [ C | Q | Q | T ( j = 1 2 R m j ( b ˜ j ; x , ) | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | T ( | α 1 | = m 1 R m 2 ( b ˜ 2 ; x , ) ( x ) α 1 D α 1 b ˜ 1 | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | T ( | α 2 | = m 2 R m 1 ( b ˜ 1 ; x , ) ( x ) α 2 D α 2 b ˜ 2 | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | T ( | α 1 | = m 1 , | α 2 | = m 2 Q ( x ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | A t Q T ( j = 1 2 R m j ( b ˜ j ; x , ) | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | A t Q T ( | α 1 | = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ) ( x ) α 1 D α 1 b ˜ 1 | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | A t Q T ( | α 2 | = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ) ( x ) α 2 D α 2 b ˜ 2 | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | A t Q T ( | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! ( x ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 | x | m f 1 ) | r d x ] 1 / r + [ C | Q | Q | ( T A t Q T ) ( j = 1 2 R m j + 1 ( b ˜ j ; x , ) | x | m f 2 ) | r d x ] 1 / r : = I 1 + I 2 + I 3 + I 4 + I 5 + I 6 + I 7 + I 8 + I 9 .

Now, let us estimate I 1 , I 2 , I 3 , I 4 , I 5 , I 6 , I 7 , I 8 and I 9 , respectively. First, for xQ and y Q ˜ , by Lemma 1, we get

R m ( b ˜ j ;x,y)C | x y | m | α j | = m D α j b j BMO ,

by Lemma 2 and the weak type (1,1) of T (Lemma 4), we obtain

I 1 C j = 1 2 ( | α j | = m j D α j b j BMO ) ( 1 | Q | R n | T ( f 1 ) ( x ) | r d x ) 1 / r C j = 1 2 ( | α j | = m j D α j b j BMO ) | Q | 1 T ( f 1 ) χ Q L r | Q | 1 / r 1 C j = 1 2 ( | α j | = m j D α j b j BMO ) | Q | 1 T ( f 1 ) W L 1 C j = 1 2 ( | α j | = m j D α j b j BMO ) | Q ˜ | 1 f 1 L 1 C j = 1 2 ( | α j | = m j D α j b j BMO ) M ( f ) ( x ˜ ) .

For I 2 , we get, by Lemma 2 and the generalized Hölder inequality,

I 2 C | α 2 | = m 2 D α 2 b 2 BMO | α 1 | = m 1 ( 1 | Q | R n | T ( D α 1 b ˜ 1 f 1 ) ( x ) | r d x ) 1 / r C | α 2 | = m 2 D α 2 b 2 BMO | α 1 | = m 1 | Q | 1 T ( D α 1 b ˜ 1 f 1 ) χ Q L r | Q | 1 / r 1 C | α 2 | = m 2 D α 2 b 2 BMO | α 1 | = m 1 | Q | 1 T ( D α 1 b ˜ 1 f 1 ) W L 1 C | α 2 | = m 2 D α 2 b 2 BMO | α 1 | = m 1 | Q ˜ | 1 D α 1 b ˜ 1 f 1 L 1 C | α 2 | = m 2 D α 2 b 2 BMO | α 1 | = m 1 D α 1 b 1 ( D α b 1 ) Q ˜ exp L , Q ˜ f L ( log L ) , Q ˜ C j = 1 2 ( | α j | = m j D α j b j BMO ) M 2 ( f ) ( x ˜ ) .

For I 3 , similar to the proof of I 2 , we get

I 3 C j = 1 2 ( | α | = m j D α b j BMO ) M 2 (f)( x ˜ ).

Similarly, for I 4 , taking r, r 1 , r 2 1 such that 1/r=1/ r 1 +1/ r 2 , we obtain, by Lemma 3 and the generalized Hölder inequality,

I 4 C | α 1 | = m 1 , | α 2 | = m 2 ( 1 | Q | R n | T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) ( x ) | r d x ) 1 / r C | α 1 | = m 1 , | α 2 | = m 2 | Q | 1 T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) χ Q L r | Q | 1 / r 1 C | α 1 | = m 1 , | α 2 | = m 2 | Q | 1 T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) W L 1 C | α 1 | = m 1 , | α 2 | = m 2 | Q | 1 D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 L 1 C | α 1 | = m 1 , | α 2 | = m 2 j = 1 2 D α j b j ( D α j b j ) Q ˜ exp L r j , Q ˜ f L ( log L ) 1 / r , Q ˜ C j = 1 2 ( | α | = m j D α b j BMO ) M 3 ( f ) ( x ˜ ) .

For I 5 , I 6 , I 7 and I 8 , by Lemma 6, we get

I 5 + I 6 + I 7 + I 8 C j = 1 2 ( | α j | = m j D α j b j BMO ) 1 | Q | Q | A t Q T ( f 1 ) ( x ) | d x + C | α 2 | = m 2 D α 2 b 2 BMO | α 1 | = m 1 1 | Q | Q | A t Q T ( D α 1 b ˜ 1 f 1 ) ( x ) | d x + C | α 1 | = m 1 D α 1 b 1 BMO | α 2 | = m 2 1 | Q | Q | A t Q T ( D α 2 b ˜ 2 f 1 ) ( x ) | d x + C | α 1 | = m 1 , | α 2 | = m 2 1 | Q | Q | A t Q T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) ( x ) | d x C j = 1 2 ( | α | = m j D α b j BMO ) M 3 ( f ) ( x ˜ ) .

For I 9 , we write

( T A t Q T ) ( j = 1 2 R m j + 1 ( b ˜ j ; x , ) | x | m f 2 ) = R n j = 1 2 R m j + 1 ( b ˜ j ; x , y ) | x y | m ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y = R n j = 1 2 R m j ( b ˜ j ; x , y ) | x y | m ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y | α 1 | = m 1 1 α 1 ! R n D α 1 b ˜ 1 ( y ) ( x y ) α 1 R m 2 ( b ˜ 2 ; x , y ) | x y | m ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y | α 2 | = m 2 1 α 2 ! R n D α 2 b ˜ 2 ( y ) ( x y ) α 2 R m 1 ( b ˜ 1 ; x , y ) | x y | m ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y + | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! R n D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) ( x y ) α 1 + α 2 | x y | m ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y = I 9 ( 1 ) + I 9 ( 2 ) + I 9 ( 3 ) + I 9 ( 4 ) .

By Lemma 1 and the following inequality (see [15])

| b Q 1 b Q 2 |Clog ( | Q 2 | / | Q 1 | ) b BMO for  Q 1 Q 2 ,

we know that for xQ and y 2 k + 1 Q ˜ 2 k Q ˜ ,

| R m ( b ˜ ; x , y ) | C | x y | m | α | = m ( D α b BMO + | ( D α b ) Q ˜ ( x , y ) ( D α b ) Q ˜ | ) C k | x y | m | α | = m D α b BMO .

Note that |xy|d= t 1 / 2 and |xy|| x 0 y| for xQ and y R n Q ˜ . By the conditions on K and K t , we obtain

| I 9 ( 1 ) | = k = 0 2 k + 1 Q ˜ 2 k Q ˜ j = 1 2 | R m j ( b ˜ j ; x , y ) | | x y | m | K ( x , y ) K t ( x , y ) | | f ( y ) | d y C j = 1 2 ( | α | = m j D α b j BMO ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ k 2 d δ | x 0 y | n + δ | f ( y ) | d y C j = 1 2 ( | α | = m j D α b j BMO ) k = 1 k 2 2 δ k 1 | 2 k Q ˜ | 2 k Q ˜ | f ( y ) | d y C j = 1 2 ( | α | = m j D α b j BMO ) M ( f ) ( x ˜ ) .

For I 9 ( 2 ) , we get, by the generalized Hölder inequality,

| I 9 ( 2 ) | C ( | α 2 | = m 2 D α 2 b 2 BMO ) | α 1 | = m 1 k = 0 2 k + 1 Q ˜ 2 k Q ˜ k d δ | x 0 y | n + δ | D α 1 b ˜ 1 ( y ) | | f ( y ) | d y C ( | α 2 | = m 2 D α 2 b 2 BMO ) × | α 1 | = m 1 k = 1 k 2 δ k D α 1 b 1 ( D α 1 b 1 ) Q ˜ exp L , 2 k Q ˜ f L ( log L ) , 2 k Q ˜ C j = 1 2 ( | α | = m j D α b j BMO ) M 2 ( f ) ( x ˜ ) .

Similarly,

| I 9 ( 3 ) | C j = 1 2 ( | α | = m j D α b j BMO ) M 2 (f)( x ˜ ).

For I 9 ( 4 ) , taking r, r 1 , r 2 1 such that 1/r=1/ r 1 +1/ r 2 , by Lemma 3 and the generalized Hölder inequality, we get

| I 9 ( 4 ) | C | α 1 | = m 1 , | α 2 | = m 2 k = 0 2 k + 1 Q ˜ 2 k Q ˜ d δ | x 0 y | n + δ | D α 1 b ˜ 1 ( y ) | | D α 2 b ˜ 2 ( y ) | | f ( y ) | d y C | α 1 | = m 1 , | α 2 | = m 2 k = 1 j = 1 2 D α j b j ( D α j b j ) Q ˜ exp L r j , 2 k Q ˜ f L ( log L ) 1 / r , 2 k Q ˜ C j = 1 2 ( | α | = m j D α b j BMO ) M 3 ( f ) ( x ˜ ) .

Thus

| I 5 |C j = 1 2 ( | α | = m j D α b j BMO ) M 3 (f)( x ˜ ).

This completes the proof of Theorem 1. □

By Theorem 1 and the L p (w)-boundedness of M l + 1 , we may obtain the conclusions of Theorem 2. By Theorem 1 and [16, 17], we may obtain the conclusions of Theorem 3.

3 Applications

In this section we shall apply Theorems 1, 2 and 3 of the paper to the holomorphic functional calculus of linear elliptic operators. First, we review some definitions regarding the holomorphic functional calculus (see [9]). Given 0θ<π, define

S θ = { z C : | arg ( z ) | θ } {0}

and its interior by S θ 0 . Set S ˜ θ = S θ {0}. A closed operator L on some Banach space E is said to be of type θ if its spectrum σ(L) S θ and if for every ν(θ,π], there exists a constant C ν such that

|η| ( η I L ) 1 C ν ,η S ˜ θ .

For ν(0,π], let

H ( S μ 0 ) = { f : S θ 0 C : f  is holomorphic and  f L < } ,

where f L =sup{|f(z)|:z S μ 0 }. Set

Ψ ( S μ 0 ) = { g H ( S μ 0 ) : s > 0 , c > 0  such that  | g ( z ) | c | z | s 1 + | z | 2 s } .

If L is of type θ and g H ( S μ 0 ), we define g(L)L(E) by

g(L)= ( 2 π i ) 1 Γ ( η I L ) 1 g(η)dη,

where Γ is the contour {ξ=r e ± i ϕ :r0} parameterized clockwise around S θ with θ<ϕ<μ. If, in addition, L is one-to-one and has a dense range, then, for f H ( S μ 0 ),

f(L)= [ h ( L ) ] 1 (fh)(L),

where h(z)=z ( 1 + z ) 2 . L is said to have a bounded holomorphic functional calculus on the sector S μ if

g ( L ) N g L

for some N>0 and for all g H ( S μ 0 ).

Now, let L be a linear operator on L 2 ( R n ) with θ<π/2 so that (L) generates a holomorphic semigroup e z L , 0|arg(z)|<π/2θ. Applying Theorem 6 of [9], we get the following.

Theorem 4 Assume the following conditions are satisfied:

  1. (i)

    The holomorphic semigroup e z L , 0|arg(z)|<π/2θ is represented by the kernels a z (x,y) which satisfy, for all ν>θ, an upper bound

    | a z ( x , y ) | c ν h | z | (x,y)

for x,y R n , and 0|arg(z)|<π/2θ, where h t (x,y)=C t n / 2 s( | x y | 2 /t) and s is a positive, bounded and decreasing function satisfying

lim r r n + ϵ s ( r 2 ) =0.
  1. (ii)

    The operator L has a bounded holomorphic functional calculus in L 2 ( R n ), that is, for all ν>θ and g H ( S μ 0 ), the operator g(L) satisfies

    g ( L ) ( f ) L 2 c ν g L f L 2 .

Then, for D α b j BMO( R n ) for all α with |α|= m j and j=1,,l, the multilinear operator g ( L ) b associated to g(L) and b j satisfies:

  1. (a)

    For 0<r<1 and x ˜ R n ,

    M A # ( g ( L ) b ( f ) ) r ( x ˜ )C j = 1 l ( | α j | = m j D α j b j BMO ) M l + 1 (f)( x ˜ );
  2. (b)

    g ( L ) b is bounded on L p (w) for any 1<p< and w A p , that is,

    g ( L ) b ( f ) L p ( w ) C j = 1 l ( | α j | = m j D α j b j BMO ) f L p ( w ) ;
  3. (c)

    There exists a constant C>0 such that for all λ>0 and w A 1 ,

    w ( { x R n : | g ( L ) b ( f ) ( x ) | > λ } ) C R n | f ( x ) | λ [ 1 + log + ( | f ( x ) | λ ) ] l w(x)dx.

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Acknowledgements

Project supported by Hunan Provincial Natural Science Foundation of China (12JJ6003) and Scientific Research Fund of Hunan Provincial Education Departments (12K017).

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Gu, G., Cai, M. A sharp inequality for multilinear singular integral operators with non-smooth kernels. J Inequal Appl 2013, 439 (2013). https://doi.org/10.1186/1029-242X-2013-439

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Keywords

  • multi-linear operator
  • singular integral operator with non-smooth kernel
  • sharp inequality
  • BMO
  • A p -weight