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Sharp boundedness for vector-valued multilinear integral operators

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 LlogL-type estimate for the vector-valued multilinear operators.

MSC:42B20, 42B25.

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)dy

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)dy

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|yz||xz|. 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 LlogL-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 | dy,

where, and in what follows, f Q = | Q | 1 Q f(x)dx. It is well known that (see [8])

f # (x) sup x Q 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 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)|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) for k2.

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 mN (see [6])

M(f) M L ( log L ) r (f), M L ( log L ) m (f) M m + 1 (f).

For r1, 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 Ck b Osc exp L r .

It is obvious that Osc exp L r coincides with the BMO space if r=1, and Osc exp L r BMO if r>1. We denote the Muckenhoupt weights by A p for 1p< (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>(3n+2)/n. Let ψ be a fixed function which satisfies the following properties:

  1. (1)

    R n ψ(x)dx=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 :|xy|<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(1,2n/(n+2)) and 0<γ1. Let Ω be homogeneous of degree zero on R n with S n 1 Ω( x )dσ( 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)dy

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)dz.

Set

F t (f)(x)= | x y | t Ω ( x y ) | x y | n 1 f(y)dy.

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>(n1)/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 (xy)f(y)dy.

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 |xy|.

Lemma 2 ([[8], 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 ([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 ) | dx 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 xQ 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 xQ 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 |xy|| x 0 y| for xQ 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.

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Pan, J., Tong, L. Sharp boundedness for vector-valued multilinear integral operators. J Inequal Appl 2014, 315 (2014). https://doi.org/10.1186/1029-242X-2014-315

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Keywords

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