Open Access

Sharp boundedness for vector-valued multilinear integral operators

Journal of Inequalities and Applications20142014:315

https://doi.org/10.1186/1029-242X-2014-315

Received: 15 January 2014

Accepted: 11 July 2014

Published: 21 August 2014

Abstract

In this paper, the sharp inequalities for some vector-valued multilinear integral operators are obtained. As applications, we get the weighted L p ( p > 1 ) norm inequalities and an L log L -type estimate for the vector-valued multilinear operators.

MSC:42B20, 42B25.

Keywords

vector-valued multilinear operator Littlewood-Paley operator Marcinkiewicz operator Bochner-Riesz operator sharp estimate BMO A p -weight

1 Preliminaries and theorems

As the development of singular integral operators and their commutators, multilinear singular integral operators have been well studied (see [15]). In this paper, we study some vector-valued multilinear integral operators as follows.

Suppose that m j are positive integers ( j = 1 , , l ), m 1 + + m l = m and A j are functions on R n ( j = 1 , , l ). Let F t ( x , y ) be defined on R n × R n × [ 0 , + ) . Set
F t ( f ) ( x ) = R n F t ( x , y ) f ( y ) d y
and
F t A ( f ) ( x ) = R n j = 1 l R m j + 1 ( A j ; x , y ) | x y | m F t ( x , y ) f ( y ) d y
for every bounded and compactly supported function f, where
R m j + 1 ( A j ; x , y ) = A j ( x ) | α | m j 1 α ! D α A j ( y ) ( x y ) α .
Let H be the Banach space H = { h : h < } such that, for each fixed x R n , F t ( f ) ( x ) and F t A ( f ) ( x ) may be viewed as a mapping from [ 0 , + ) to H. For 1 < s < , the vector-valued multilinear operator related to F t is defined by
| T A ( f ) ( x ) | s = ( i = 1 ( T A ( f i ) ( x ) ) s ) 1 / s ,
where
T A ( f i ) ( x ) = F t A ( f i ) ( x ) ,
and F t satisfies: for fixed ε > 0 ,
F t ( x , y ) C | x y | n
and
F t ( y , x ) F t ( z , x ) C | y z | ε | x z | n ε
if 2 | y z | | x z | . Set
| T ( f ) ( x ) | s = ( i = 1 | T ( f i ) ( x ) | s ) 1 / s and | f | s = ( i = 1 | f i ( x ) | s ) 1 / s .

Suppose that | T | s is bounded on L p ( R n ) for 1 < p < and weak ( L 1 , L 1 ) -bounded.

Note that when m = 0 , T A is just a vector-valued multilinear commutator of T and A (see [6]). While when m > 0 , T A 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 studied by many authors (see [15]). In [7], Hu and Yang proved a variant sharp estimate for the multilinear singular integral operators. In [6], Pérez and Trujillo-Gonzalez proved a sharp estimate for some multilinear commutator. The main purpose of this paper is to prove a sharp inequality for the vector-valued multilinear integral operators. As applications, we obtain the weighted L p ( p > 1 ) norm inequalities and an L log L -type estimate for the vector-valued multilinear operators.

First, let us introduce some notations. Throughout this paper, Q will denote 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 x Q 1 | Q | Q | f ( y ) f Q | d y ,
where, and in what follows, f Q = | Q | 1 Q f ( x ) d x . It is well known that (see [8])
f # ( x ) sup x Q inf c C 1 | Q | Q | f ( y ) c | d y .
We say that f belongs to B M O ( R n ) if f # belongs to L ( R n ) and f B M O = f # L . For 0 < r < , we denote f r # by
f r # ( x ) = [ ( | f | r ) # ( x ) ] 1 / r .

Let M be the Hardy-Littlewood maximal operator, that is, M ( f ) ( x ) = sup x Q | Q | 1 Q | f ( y ) | d y . For k N , 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 ) for k 2 .

Let Φ be a Young function and Φ ˜ be the complementary associated to Φ, we denote the Φ-average by, for a function f,
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 x Q f Φ , Q .
The Young functions to be used in this paper are Φ ( t ) = exp ( t r ) 1 and Ψ ( t ) = t log r ( t + e ) , the corresponding Φ-average and maximal functions are denoted by exp L r , Q , M exp L r and L ( log L ) r , Q , M L ( log L ) r . We have the following inequality, for any r > 0 and m N (see [6])
M ( f ) M L ( log L ) r ( f ) , M L ( log L ) m ( f ) M m + 1 ( f ) .
For r 1 , we denote that
b osc exp L r = sup Q b b Q exp L r , Q ,
the space Osc exp L r is defined by
Osc exp L r = { b L log 1 ( R n ) : b osc exp L r < } .
It has been known that (see [6])
b b 2 Q exp L r , 2 k Q C k b Osc exp L r .

It is obvious that Osc exp L r coincides with the B M O space if r = 1 , and Osc exp L r B M O if r > 1 . We denote the Muckenhoupt weights by A p for 1 p < (see [8]).

Now we state our main results as follows.

Theorem 1 Let 1 < s < , r j 1 and D α A j Osc exp L r j for all α with | α | = m j and j = 1 , , l . Define 1 / r = 1 / r 1 + + 1 / r l . Then, for any 0 < p < 1 , there exists a constant C > 0 such that for any f = { f i } C 0 ( R n ) and x R n ,
( | T A ( f ) | s ) p # ( x ) C j = 1 l ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ) .
Theorem 2 Let 1 < s < , r j 1 and D α A j Osc exp L r j for all α with | α | = m j and j = 1 , , l .
  1. (1)
    If 1 < p < and w A p , then
    | T A ( f ) | s L p ( w ) C j = 1 l ( | α j | = m j D α j A j Osc exp L r j ) | f | s L p ( w ) ;
     
  2. (2)
    If w A 1 . Define 1 / r = 1 / r 1 + + 1 / r l and Φ ( t ) = t log 1 / r ( t + e ) . Then there exists a constant C > 0 such that for all λ > 0 ,
    w ( { x R n : | T A ( f ) ( x ) | s > λ } ) C R n Φ [ λ 1 j = 1 l ( | α j | = m j D α j A j Osc exp L r j ) | f ( x ) | s ] w ( x ) d x .
     

Remark The conditions in Theorems 1 and 2 are satisfied by many operators.

Now we give some examples.

Example 1 Littlewood-Paley operators.

Fix ε > 0 and μ > ( 3 n + 2 ) / n . Let ψ be a fixed function which satisfies the following properties:
  1. (1)

    R n ψ ( x ) d x = 0 ,

     
  2. (2)

    | ψ ( x ) | C ( 1 + | x | ) ( n + 1 ) ,

     
  3. (3)

    | ψ ( x + y ) ψ ( x ) | C | y | ε ( 1 + | x | ) ( n + 1 + ε ) when 2 | y | < | x | .

     
We denote that Γ ( x ) = { ( y , t ) R + n + 1 : | x y | < t } and the characteristic function of Γ ( x ) by χ Γ ( x ) . The Littlewood-Paley multilinear operators are defined by
g ψ A ( f ) ( x ) = ( 0 | F t A ( f ) ( x ) | 2 d t t ) 1 / 2 , S ψ A ( f ) ( x ) = [ Γ ( x ) | F t A ( f ) ( x , y ) | 2 d y d t t n + 1 ] 1 / 2
and
g μ A ( f ) ( x ) = [ R + n + 1 ( t t + | x y | ) n μ | F t A ( f ) ( x , y ) | 2 d y d t t n + 1 ] 1 / 2 ,
where
F t A ( f ) ( x ) = R n j = 1 l R m j + 1 ( A j ; x , y ) | x y | m ψ t ( x y ) f ( y ) d y , F t A ( f ) ( x , y ) = R n j = 1 l R m j + 1 ( A j ; x , z ) | x z | m f ( z ) ψ t ( y z ) d z
and ψ t ( x ) = t n ψ ( x / t ) for t > 0 . Set F t ( f ) ( y ) = f ψ t ( y ) . We also define that
g ψ ( f ) ( x ) = ( 0 | F t ( f ) ( x ) | 2 d t t ) 1 / 2 , S ψ ( f ) ( x ) = ( Γ ( x ) | F t ( f ) ( y ) | 2 d y d t t n + 1 ) 1 / 2
and
g μ ( f ) ( x ) = ( R + n + 1 ( t t + | x y | ) n μ | F t ( f ) ( y ) | 2 d y d t t n + 1 ) 1 / 2 ,
which are the Littlewood-Paley operators (see [9]). Let H be the space
H = { h : h = ( 0 | h ( t ) | 2 d t / t ) 1 / 2 < }
or
H = { h : h = ( R + n + 1 | h ( y , t ) | 2 d y d t / t n + 1 ) 1 / 2 < } .
Then, for each fixed x R n , F t A ( f ) ( x ) and F t A ( f ) ( x , y ) may be viewed as the mapping from [ 0 , + ) to H, and it is clear that
g ψ A ( f ) ( x ) = F t A ( f ) ( x ) , g ψ ( f ) ( x ) = F t ( f ) ( x ) , S ψ A ( f ) ( x ) = χ Γ ( x ) F t A ( f ) ( x , y ) , S ψ ( f ) ( x ) = χ Γ ( x ) F t ( f ) ( y )
and
g μ A ( f ) ( x ) = ( t t + | x y | ) n μ / 2 F t A ( f ) ( x , y ) , g μ ( f ) ( x ) = ( t t + | x y | ) n μ / 2 F t ( f ) ( y ) .

It is easy to see that g ψ , S ψ and g μ satisfy the conditions of Theorems 1 and 2 (see [1012]), thus Theorems 1 and 2 hold for g ψ A , S ψ A and g μ A .

Example 2 Marcinkiewicz operators.

Fix λ > max ( 1 , 2 n / ( n + 2 ) ) and 0 < γ 1 . Let Ω be homogeneous of degree zero on R n with S n 1 Ω ( x ) d σ ( x ) = 0 . Assume that Ω Lip γ ( S n 1 ) . The Marcinkiewicz multilinear operators are defined by
μ Ω A ( f ) ( x ) = ( 0 | F t A ( f ) ( x ) | 2 d t t 3 ) 1 / 2 , μ S A ( f ) ( x ) = [ Γ ( x ) | F t A ( f ) ( x , y ) | 2 d y d t t n + 3 ] 1 / 2
and
μ λ A ( f ) ( x ) = [ R + n + 1 ( t t + | x y | ) n λ | F t A ( f ) ( x , y ) | 2 d y d t t n + 3 ] 1 / 2 ,
where
F t A ( f ) ( x ) = | x y | t j = 1 l R m j + 1 ( A j ; x , y ) | x y | m Ω ( x y ) | x y | n 1 f ( y ) d y
and
F t A ( f ) ( x , y ) = | y z | t j = 1 l R m j + 1 ( A j ; y , z ) | y z | m Ω ( y z ) | y z | n 1 f ( z ) d z .
Set
F t ( f ) ( x ) = | x y | t Ω ( x y ) | x y | n 1 f ( y ) d y .
We also define that
μ Ω ( f ) ( x ) = ( 0 | F t ( f ) ( x ) | 2 d t t 3 ) 1 / 2 , μ S ( f ) ( x ) = ( Γ ( x ) | F t ( f ) ( y ) | 2 d y d t t n + 3 ) 1 / 2
and
μ λ ( f ) ( x ) = ( R + n + 1 ( t t + | x y | ) n λ | F t ( f ) ( y ) | 2 d y d t t n + 3 ) 1 / 2 ,
which are the Marcinkiewicz operators (see [13]). Let H be the space
H = { h : h = ( 0 | h ( t ) | 2 d t / t 3 ) 1 / 2 < }
or
H = { h : h = ( R + n + 1 | h ( y , t ) | 2 d y d t / t n + 3 ) 1 / 2 < } .
Then it is clear that
μ Ω A ( f ) ( x ) = F t A ( f ) ( x ) , μ Ω ( f ) ( x ) = F t ( f ) ( x ) , μ S A ( f ) ( x ) = χ Γ ( x ) F t A ( f ) ( x , y ) , μ S ( f ) ( x ) = χ Γ ( x ) F t ( f ) ( y )
and
μ λ A ( f ) ( x ) = ( t t + | x y | ) n λ / 2 F t A ( f ) ( x , y ) , μ λ ( f ) ( x ) = ( t t + | x y | ) n λ / 2 F t ( f ) ( y ) .

It is easy to see that μ Ω , μ S and μ λ satisfy the conditions of Theorems 1 and 2 (see [13, 14]), thus Theorems 1 and 2 hold for μ Ω A , μ S A and μ λ A .

Example 3 Bochner-Riesz operators.

Let δ > ( n 1 ) / 2 , B t δ ( f ˆ ) ( ξ ) = ( 1 t 2 | ξ | 2 ) + δ f ˆ ( ξ ) and B t δ ( z ) = t n B δ ( z / t ) for t > 0 . Set
F δ , t A ( f ) ( x ) = R n j = 1 l R m j + 1 ( A j ; x , y ) | x y | m B t δ ( x y ) f ( y ) d y .
The maximal Bochner-Riesz multilinear operators are defined by
B δ , A ( f ) ( x ) = sup t > 0 | B δ , t A ( f ) ( x ) | .
We also define that
B δ , ( f ) ( x ) = sup t > 0 | B t δ ( f ) ( x ) | ,
which is the maximal Bochner-Riesz operator (see [15]). Let H be the space H = { h : h = sup t > 0 | h ( t ) | < } , then
B δ , A ( f ) ( x ) = B δ , t A ( f ) ( x ) , B δ ( f ) ( x ) = B t δ ( f ) ( x ) .

It is easy to see that B δ , A satisfies the conditions of Theorems 1 and 2 (see [16]), thus Theorems 1 and 2 hold for B δ , A .

2 Some lemmas

We give some preliminary lemmas.

Lemma 1 ([3])

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

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

Lemma 2 ([[8], p.485])

Let 0 < p < q < and for any function f 0 , we define that, 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 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 ([6])

Let r j 1 for j = 1 , , m , we denote that 1 / r = 1 / r 1 + + 1 / r m . Then
1 | Q | Q | f 1 ( x ) f m ( x ) g ( x ) | d x f exp L r 1 , Q f exp L r m , Q g L ( log L ) 1 / r , Q .

3 Proof of the theorem

It is only to prove Theorem 1.

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 A ( f ) ( x ) | s C 0 | p d x ) 1 / p C j = 1 l ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( 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 A ˜ j ( x ) = A j ( x ) | α | = m 1 α ! ( D α A j ) Q ˜ x α , then R m ( A j ; x , y ) = R m ( A ˜ j ; x , y ) and D α A ˜ j = D α A j ( D α A j ) Q ˜ for | α | = m j . We split f = g + h = { g i } + { h i } for g i = f i χ Q ˜ and h i = f i χ R n Q ˜ . Write
F t A ( f i ) ( x ) = R n j = 1 2 R m j + 1 ( A ˜ j ; x , y ) | x y | m F t ( x , y ) f i ( y ) d y = R n j = 1 2 R m j + 1 ( A ˜ j ; x , y ) | x y | m F t ( x , y ) h i ( y ) d y + R n j = 1 2 R m j ( A ˜ j ; x , y ) | x y | m F t ( x , y ) g i ( y ) d y | α 1 | = m 1 1 α 1 ! R n R m 2 ( A ˜ 2 ; x , y ) ( x y ) α 1 | x y | m D α 1 A ˜ 1 ( y ) F t ( x , y ) g i ( y ) d y | α 2 | = m 2 1 α 2 ! R n R m 1 ( A ˜ 1 ; x , y ) ( x y ) α 2 | x y | m D α 2 A ˜ 2 ( y ) F t ( x , y ) g i ( 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 A ˜ 1 ( y ) D α 2 A ˜ 2 ( y ) | x y | m F t ( x , y ) g i ( y ) d y .
Then, by Minkowski’s inequality, we have
[ 1 | Q | Q | | T A ( f ) ( x ) | s | T A ˜ ( h ) ( x 0 ) | s | p d x ] 1 / p [ 1 | Q | Q | F t A ( f ) ( x ) s F t A ˜ ( h ) ( x 0 ) s | p d x ] 1 / p [ 1 | Q | Q ( i = 1 F t A ( f i ) ( x ) F t A ˜ ( h i ) ( x 0 ) s ) p / s d x ] 1 / p [ C | Q | Q ( i = 1 R n j = 1 2 R m j ( A ˜ j ; x , y ) | x y | m F t ( x , y ) g i ( y ) d y s ) p / s d x ] 1 / p + [ C | Q | Q ( i = 1 | α 1 | = m 1 1 α 1 ! × R n R m 2 ( A ˜ 2 ; x , y ) ( x y ) α 1 | x y | m D α 1 A ˜ 1 ( y ) F t ( x , y ) g i ( y ) d y s ) p / s d x ] 1 / p + [ C | Q | Q ( i = 1 | α 2 | = m 2 1 α 2 ! × R n R m 1 ( A ˜ 1 ; x , y ) ( x y ) α 2 | x y | m D α 2 A ˜ 2 ( y ) F t ( x , y ) g i ( y ) d y s ) p / s d x ] 1 / p + [ C | Q | Q ( i = 1 | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! × R n ( x y ) α 1 + α 2 D α 1 A ˜ 1 ( y ) D α 2 A ˜ 2 ( y ) | x y | m F t ( x , y ) g i ( y ) d y s ) p / s d x ] 1 / p + [ C | Q | Q ( i = 1 F t A ˜ ( h i ) ( x ) F t A ˜ ( h i ) ( x 0 ) s ) p / s d x ] 1 / p : = I 1 + I 2 + I 3 + I 4 + I 5 .
Now, let us estimate I 1 , I 2 , I 3 , I 4 and I 5 , respectively. First, for x Q and y Q ˜ , by Lemma 1, we get
R m j ( A ˜ j ; x , y ) C | x y | m j | α j | = m j D α j A j Osc exp L r j .
Thus, by Lemma 2 and the weak type ( 1 , 1 ) of | T | s , we obtain
I 1 C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) ( 1 | Q | Q | T ( g ) ( x ) | s p d x ) 1 / p = C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) | Q | 1 | T ( g ) | s χ Q L p | Q | 1 / p 1 C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) | Q | 1 | T ( g ) | s W L 1 C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) | Q | 1 | g | s L 1 C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M ( | f | s ) ( x ˜ ) C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
For I 2 , note that χ Q exp L r 2 , Q C , similar to the proof of I 1 and by using Lemma 3, we get
I 2 C | α 2 | = m 2 D α 2 A 2 Osc exp L r 2 | α 1 | = m 1 ( 1 | Q | R n | T ( D α 1 A ˜ 1 g ) ( x ) | s p d x ) 1 / p C | α 2 | = m 2 D α 2 A 2 Osc exp L r 2 | α 1 | = m 1 | Q | 1 | T ( D α 1 A ˜ 1 g ) ( x ) | s χ Q W L 1 C | α 2 | = m 2 D α 2 A 2 Osc exp L r 2 | α 1 | = m 1 1 | Q | R n | D α 1 A ˜ 1 ( x ) | | g ( x ) | s d x C | α 2 | = m 2 D α 2 A 2 Osc exp L r 2 χ Q exp L r 2 , Q × | α 1 | = m 1 D α 1 A 1 ( D α 1 A 1 ) Q ˜ exp L r 1 , Q ˜ | f | s L ( log L ) 1 / r , Q ˜ C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
For I 3 , similar to the proof of I 2 , we get
I 3 C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
Similarly, for I 4 , by using Lemma 3, we get
I 4 C | α 1 | = m 1 , | α 2 | = m 2 ( 1 | Q | R n | T ( D α 1 A ˜ 1 D α 2 A ˜ 2 f 1 ) ( x ) | s p d x ) 1 / p C | α 1 | = m 1 , | α 2 | = m 2 | Q | 1 | T ( D α 1 A ˜ 1 D α 2 A ˜ 2 g ) | s χ Q W L 1 C | α 1 | = m 1 , | α 2 | = m 2 1 | Q | R n | D α 1 A ˜ 1 ( x ) D α 2 A ˜ 2 ( x ) | | g ( x ) | s d x C | α 1 | = m 1 D α 1 A 1 ( D α 1 A 1 ) Q ˜ exp L r 1 , Q ˜ × | α 2 | = m 2 D α 2 A 2 ( D α 2 A 2 ) Q ˜ exp L r 2 , Q ˜ | f | s L ( log L ) 1 / r , Q ˜ C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
For I 5 , we write
F t A ˜ ( h i ) ( x ) F t A ˜ ( h i ) ( x 0 ) = R n ( F t ( x , y ) | x y | m F t ( x 0 , y ) | x 0 y | m ) j = 1 2 R m j ( A ˜ j ; x , y ) h i ( y ) d y + R n ( R m 1 ( A ˜ 1 ; x , y ) R m 1 ( A ˜ 1 ; x 0 , y ) ) R m 2 ( A ˜ 2 ; x , y ) | x 0 y | m F t ( x 0 , y ) h i ( y ) d y + R n ( R m 2 ( A ˜ 2 ; x , y ) R m 2 ( A ˜ 2 ; x 0 , y ) ) R m 1 ( A ˜ 1 ; x 0 , y ) | x 0 y | m F t ( x 0 , y ) h i ( y ) d y | α 1 | = m 1 1 α 1 ! R n [ R m 2 ( A ˜ 2 ; x , y ) ( x y ) α 1 | x y | m F t ( x , y ) R m 2 ( A ˜ 2 ; x 0 , y ) ( x 0 y ) α 1 | x 0 y | m F t ( x 0 , y ) ] × D α 1 A ˜ 1 ( y ) h i ( y ) d y | α 2 | = m 2 1 α 2 ! R n [ R m 1 ( A ˜ 1 ; x , y ) ( x y ) α 2 | x y | m F t ( x , y ) R m 1 ( A ˜ 1 ; x 0 , y ) ( x 0 y ) α 2 | x 0 y | m F t ( x 0 , y ) ] × D α 2 A ˜ 2 ( y ) h i ( y ) d y + | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! R n [ ( x y ) α 1 + α 2 | x y | m F t ( x , y ) ( x 0 y ) α 1 + α 2 | x 0 y | m F t ( x 0 , y ) ] × D α 1 A ˜ 1 ( y ) D α 2 A ˜ 2 ( y ) h i ( y ) d y = I 5 ( 1 ) + I 5 ( 2 ) + I 5 ( 3 ) + I 5 ( 4 ) + I 5 ( 5 ) + I 5 ( 6 ) .
By Lemma 1, we know that, for x Q and y 2 k + 1 Q ˜ 2 k Q ˜ ,
| R m j ( A ˜ j ; x , y ) | C | x y | m j | α j | = m j ( D α j A Osc exp L r j + | ( D α j A ) Q ˜ ( x , y ) ( D α j A ) Q ˜ | ) C k | x y | m j | α j | = m j D α j A Osc exp L r j .
Note that | x y | | x 0 y | for x Q and y R n Q ˜ , we obtain, by the condition of F t ,
I 5 ( 1 ) C R n ( | x x 0 | | x 0 y | m + n + 1 + | x x 0 | ε | x 0 y | m + n + ε ) j = 1 2 R m j ( A ˜ j ; x , y ) | h i ( y ) | d y C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) × k = 0 2 k + 1 Q ˜ 2 k Q ˜ k 2 ( | x x 0 | | x 0 y | n + 1 + | x x 0 | ε | x 0 y | n + ε ) | f i ( y ) | d y C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) k = 1 k 2 ( 2 k + 2 ε k ) 1 | 2 k Q ˜ | 2 k Q ˜ | f i ( y ) | d y .
Thus, by Minkowski’s inequality, we get
( i = 1 I 5 ( 1 ) s ) 1 / s C j = 1 2 ( | α | = m j D α A j Osc exp L r j ) × k = 1 k 2 ( 2 k + 2 ε k ) 1 | 2 k Q ˜ | 2 k Q ˜ | f ( y ) | s d y C j = 1 2 ( | α | = m j D α A j Osc exp L r j ) M ( | f | s ) ( x ˜ ) C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
For I 5 ( 2 ) , by the formula (see [3])
R m j ( A ˜ ; x , y ) R m j ( A ˜ ; x 0 , y ) = | β | < m j 1 β ! R m j | β | ( D β A ˜ ; x , x 0 ) ( x y ) β
and Lemma 1, we have
| R m j ( A ˜ ; x , y ) R m j ( A ˜ ; x 0 , y ) | C | β | < m j | α | = m j | x x 0 | m j | β | | x y | | β | D α A Osc exp L r j ,
thus
( i = 1 I 5 ( 2 ) s ) 1 / s C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ k | x x 0 | | x 0 y | n + 1 | f ( y ) | s d y C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
Similarly,
( i = 1 I 5 ( 3 ) s ) 1 / s C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
For I 5 ( 4 ) , similar to the proof of I 5 ( 1 ) , I 5 ( 2 ) and I 2 , we get
( i = 1 I 5 ( 4 ) s ) 1 / s C | α 1 | = m 1 R n Q ˜ ( x y ) α 1 F t ( x , y ) | x y | m ( x 0 y ) α 1 F t ( x 0 , y ) | x 0 y | m × | R m 2 ( A ˜ 2 ; x , y ) | | D α 1 A ˜ 1 ( y ) | | f ( y ) | s d y + C | α 1 | = m 1 R n Q ˜ | R m 2 ( A ˜ 2 ; x , y ) R m 2 ( A ˜ 2 ; x 0 , y ) | × ( x 0 y ) α 1 F t ( x 0 , y ) | x 0 y | m | D α 1 A ˜ 1 ( y ) | | f ( y ) | s d y C | α 2 | = m 2 D α 2 A 2 Osc exp L r 2 × | α 1 | = m 1 k = 1 k ( 2 k + 2 ε k ) 1 | 2 k Q ˜ | 2 k Q ˜ | D α 1 A ˜ 1 ( y ) | | f ( y ) | s d y C | α 2 | = m 2 D α 2 A 2 Osc exp L r 2 | α 1 | = m 1 k = 1 k ( 2 k + 2 ε k ) × D α 1 A 1 ( D α 1 A 1 ) Q ˜ exp L r 1 , 2 k Q ˜ | f | s L ( log L ) 1 / r , 2 k Q ˜ C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
Similarly,
( i = 1 I 5 ( 5 ) s ) 1 / s C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
For I 5 ( 6 ) , by using Lemma 3, we obtain
( i = 1 I 5 ( 6 ) s ) 1 / s C | α 1 | = m 1 , | α 2 | = m 2 R n Q ˜ ( x y ) α 1 + α 2 F t ( x , y ) | x y | m ( x 0 y ) α 1 + α 2 F t ( x 0 , y ) | x 0 y | m × | D α 1 A ˜ 1 ( y ) | | D α 2 A ˜ 2 ( y ) | | f ( y ) | s d y C | α 1 | = m 1 , | α 2 | = m 2 k = 1 ( 2 k + 2 ε k ) 1 | 2 k Q ˜ | 2 k Q ˜ | D α 1 A ˜ 1 ( y ) | | D α 2 A ˜ 2 ( y ) | | f ( y ) | s d y C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .
Thus
I 5 C j = 1 2 ( | α j | = m j D α j A j Osc exp L r j ) M L ( log L ) 1 / r ( | f | s ) ( x ˜ ) .

This completes the proof of Theorem 1. □

By Theorem 1 and the L p -boundedness of M L ( log L ) 1 / r , we may obtain the conclusions (1), (2) of Theorem 2.

Declarations

Authors’ Affiliations

(1)
Hunan Mechanical and Electrical Polytechnic

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© Pan and Tong; licensee Springer. 2014

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