Open Access

Multilinear commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces

Journal of Inequalities and Applications20142014:258

https://doi.org/10.1186/1029-242X-2014-258

Received: 19 December 2013

Accepted: 30 June 2014

Published: 22 July 2014

Abstract

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogs of intrinsic square functions in the generalized weighted Morrey spaces M w p , φ ( l 2 ) . We study the boundedness of intrinsic square functions including the Lusin area integral, the Littlewood-Paley g-function and g λ -function, and their multilinear commutators on vector-valued generalized weighted Morrey spaces M w p , φ ( l 2 ) . In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on φ ( x , r ) without assuming any monotonicity property of φ ( x , r ) on r.

MSC:42B25, 42B35.

Keywords

intrinsic square functionsvector-valued generalized weighted Morrey spacesvector-valued inequalities A p weightsmultilinear commutatorsBMO

1 Introduction

It is well known that the commutator is an important integral operator and it plays a key role in harmonic analysis. In 1965, Calderon [1, 2] studied a kind of commutators, appearing in Cauchy integral problems of Lip-line. Let K be a Calderón-Zygmund singular integral operator and b B M O ( R n ) . A well-known result of Coifman et al. [3] states that the commutator operator [ b , K ] f = K ( b f ) b K f is bounded on L p ( R n ) for 1 < p < . The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [48]).

The classical Morrey spaces were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [710]. Recently, Komori and Shirai [11] first defined the weighted Morrey spaces L p , κ ( w ) and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces. Also, Guliyev [12, 13] introduced the generalized weighted Morrey spaces M w p , φ and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [1416]).

The intrinsic square functions were first introduced by Wilson in [17, 18]. They are defined as follows. For 0 < α 1 , let C α be the family of functions ϕ : R n R such that ϕ’s support is contained in { x : | x | 1 } , R n ϕ ( x ) d x = 0 , and for x , x R n ,
| ϕ ( x ) ϕ ( x ) | | x x | α .
For ( y , t ) R + n + 1 and f L 1 , loc ( R n ) , set
A α f ( t , y ) sup ϕ C α | f ϕ t ( y ) | ,
where ϕ t ( y ) = t n ϕ ( y t ) . Then we define the varying-aperture intrinsic square (intrinsic Lusin) function of f by the formula
G α , β ( f ) ( x ) = ( Γ β ( x ) ( A α f ( t , y ) ) 2 d y d t t n + 1 ) 1 2 ,

where Γ β ( x ) = { ( y , t ) R + n + 1 : | x y | < β t } . Denote G α , 1 ( f ) = G α ( f ) .

This function is independent of any particular kernel, such as Poisson kernel. It dominates pointwise the classical square function (Lusin area integral) and its real-variable generalizations. Although the function G α , β ( f ) depends on kernels with uniform compact support, there is pointwise relation between G α , β ( f ) with different β:
G α , β ( f ) ( x ) β 3 n 2 + α G α ( f ) ( x ) .

We can see details in [17].

The intrinsic Littlewood-Paley g-function and the intrinsic g λ function are defined, respectively, by
g α f ( x ) = ( 0 ( A α f ( y , t ) ) 2 d t t ) 1 2 , g λ , α f ( x ) = ( R + n + 1 ( t t + | x y | ) n λ ( A α f ( y , t ) ) 2 d y d t t n + 1 ) 1 2 .
When we say that f maps into l 2 , we mean that f ( x ) = ( f j ) j = 1 , where each f j is Lebesgue measurable and, for almost every x R n
f ( x ) l 2 = ( j = 1 | f j ( x ) | 2 ) 1 / 2 .

Let f = ( f 1 , f 2 , ) be a sequence of locally integrable functions on R n . For any x R n , Wilson [18] also defined the vector-valued intrinsic square functions of f by G α f ( x ) l 2 and proved the following result.

Theorem A Let 1 p < , 0 < α 1 , and w A p . Then the operators G α and g λ , α are bounded from L w p ( l 2 ) into itself for p > 1 and from L w 1 ( l 2 ) to W L w 1 ( l 2 ) .

Moreover, in [19], Lerner showed sharp L w p norm inequalities for the intrinsic square functions in terms of the A p characteristic constant of w for all 1 < p < . Also Huang and Liu [20] studied the boundedness of intrinsic square functions on weighted Hardy spaces. Moreover, they characterized the weighted Hardy spaces by intrinsic square functions. In [21] and [22], Wang and Liu obtained some weak type estimates on weighted Hardy spaces. In [23], Wang considered intrinsic functions and the commutators generated with BMO functions on weighted Morrey spaces. Let b = ( b 1 , , b m ) and b j , j = 1 , , m be locally integrable function on R n . Setting
A α , b f ( t , y ) sup ϕ C α | R n j = 1 m [ b j ( x ) b j ( z ) ] ϕ t ( y z ) f ( z ) d z | ,
the multilinear commutators are defined by
[ b , G α ] f ( x ) = ( Γ ( x ) ( A α , b f ( t , y ) ) 2 d y d t t n + 1 ) 1 2 ,
[ b , g α ] f ( x ) = ( 0 ( A α , b f ( t , y ) ) 2 d t t ) 1 2 ,
and
[ b , g λ , α ] f ( x ) = ( R + n + 1 ( t t + | x y | ) λ n ( A α , b f ( t , y ) ) 2 d y d t t n + 1 ) 1 2 .

In [23], Wang proved the following result.

Theorem B Let 1 < p < , 0 < α 1 , w A p , and b B M O ( R n ) . Then the commutator operators [ b , G α ] and [ b , g λ , α ] are bounded from L w p ( l 2 ) into itself.

Analogously the following result may be proved.

Theorem B′ Let 1 < p < , 0 < α 1 , w A p . Let also b = ( b 1 , , b m ) and b j B M O ( R n ) , j = 1 , , m . Then the multilinear commutator operators [ b , G α ] and [ b , g λ , α ] are bounded from L w p ( l 2 ) into itself.

In this paper, we will consider the boundedness of the operators G α , g α , g λ , α and their multilinear commutators on vector-valued generalized weighted Morrey spaces. Let φ ( x , r ) be a positive measurable function on R n × R + and w be non-negative measurable function on R n . For any f L w p , loc ( l 2 ) , we denote by M w p , φ ( l 2 ) the vector-valued generalized weighted Morrey spaces, if
f M w p , φ ( l 2 ) = sup x R n , r > 0 φ ( x , r ) 1 w ( B ( x , r ) ) 1 p f ( ) l 2 L w p ( B ( x , r ) ) < .
When w 1 , then M w p , φ ( l 2 ) coincide the vector-valued generalized Morrey spaces M p , φ ( l 2 ) . There are many papers discussed the conditions on φ ( x , r ) to obtain the boundedness of operators on the generalized Morrey spaces. For example, in [24] (see, also [10]), by Guliyev the following condition was imposed on the pair ( φ 1 , φ 2 ) :
r φ 1 ( x , t ) d t t C φ 2 ( x , r ) ,
(1.1)
where C > 0 does not depend on x and r. Under the above condition, they obtained the boundedness of Calderón-Zygmund singular integral operators from M p , φ 1 ( R n ) to M p , φ 2 ( R n ) . Also, in [25] and [26], Guliyev et al. introduced a weaker condition: If 1 p < , there exists a constant C > 0 , such that, for any x R n and r > 0 ,
r ess inf t < s < φ 1 ( x , s ) s n p t n p + 1 d t C φ 2 ( x , r ) .
(1.2)

If the pair ( φ 1 , φ 2 ) satisfies condition (1.1), then ( φ 1 , φ 2 ) satisfied condition (1.2). But the opposite is not true. We can see Remark 4.7 in [26] for details.

Recently, in [12, 13] (see, also [1416]), Guliyev introduced a weighted condition: If  1 p < , there exists a constant C > 0 , such that, for any x R n and t > 0 ,
r ess inf t < s < φ 1 ( x , s ) w ( B ( x , s ) ) 1 p w ( B ( x , t ) ) 1 p d t t C φ 2 ( x , r ) .
(1.3)
In this paper, we will obtain the boundedness of the vector-valued intrinsic function, the intrinsic Littlewood-Paley g function, the intrinsic g λ function and their multilinear commutators on vector-valued generalized weighted Morrey spaces when w A p and the pair ( φ 1 , φ 2 ) satisfies condition (1.3) or the following inequalities:
r ln m ( e + t r ) ess inf t < s < φ 1 ( x , s ) w ( B ( x , s ) ) 1 p w ( B ( x , t ) ) 1 p d t t C φ 2 ( x , r ) ,
(1.4)

where C does not depend on x and r. Our main results in this paper are stated as follows.

Theorem 1.1 Let 1 p < , 0 < α 1 , w A p , and ( φ 1 , φ 2 ) satisfy condition (1.3). Then the operator G α is bounded from M w p , φ 1 ( l 2 ) to M w p , φ 2 ( l 2 ) for p > 1 and from M w 1 , φ 1 ( l 2 ) to W M w 1 , φ 2 ( l 2 ) .

Theorem 1.2 Let 1 p < , 0 < α 1 , w A p , λ > 3 + α n , and ( φ 1 , φ 2 ) satisfy condition (1.3). Then the operator g λ , α is bounded from M w p , φ 1 ( l 2 ) to M w p , φ 2 ( l 2 ) for p > 1 and from M w 1 , φ 1 ( l 2 ) to W M w 1 , φ 2 ( l 2 ) .

Theorem 1.3 Let 1 < p < , 0 < α 1 , w A p , and ( φ 1 , φ 2 ) satisfy condition (1.4). Let also b = ( b 1 , , b m ) and b j B M O ( R n ) , j = 1 , , m . Then [ b , G α ] is bounded from M w p , φ 1 ( l 2 ) to M w p , φ 2 ( l 2 ) .

Theorem 1.4 Let 1 < p < , 0 < α 1 , w A p , and ( φ 1 , φ 2 ) satisfy condition (1.4). Let also b = ( b 1 , , b m ) and b j B M O ( R n ) , j = 1 , , m . Then for λ > 3 + α n , [ b , g λ , α ] is bounded from M w p , φ 1 ( l 2 ) to M w p , φ 2 ( l 2 ) .

In [17], the author proved that the functions G α f and g α f are pointwise comparable. Thus, as a consequence of Theorem 1.1 and Theorem 1.3, we have the following results.

Corollary 1.5 Let 1 p < , 0 < α 1 , w A p , and ( φ 1 , φ 2 ) satisfy condition (1.3), then g α is bounded from M w p , φ 1 ( l 2 ) to M w p , φ 2 ( l 2 ) for p > 1 and from M w 1 , φ 1 ( l 2 ) to W M w 1 , φ 2 ( l 2 ) .

Corollary 1.6 Let 1 < p < , 0 < α 1 , w A p , and ( φ 1 , φ 2 ) satisfy condition (1.4). Let also b = ( b 1 , , b m ) and b j B M O ( R n ) , j = 1 , , m . Then [ b , g α ] is bounded from M w p , φ 1 ( l 2 ) to M w p , φ 2 ( l 2 ) .

Remark 1.7 Note that, in the scalar valued case and for m = 1 , w 1 Theorems 1.1-1.4 and Corollaries 1.5-1.6 was proved in [27]. Also, in the scalar valued case and m = 1 , w A p , and φ 1 ( x , r ) = φ 2 ( x , r ) w ( B ( x , r ) ) κ 1 p , 0 < κ < 1 Theorems 1.1-1.4 and Corollaries 1.5-1.6 were proved by Wang in [23, 28]. If φ ( x , r ) w ( B ( x , r ) ) κ 1 p , then the vector-valued generalized weighed Morrey space M w p , φ ( l 2 ) coincides with the vector-valued weighed Morrey space L w p , κ ( l 2 ) and the pair ( w ( B ( x , r ) ) κ 1 p , w ( B ( x , r ) ) κ 1 p ) satisfies the two conditions (1.3) and (1.4). Indeed, by Lemma 3.1 there exist C > 0 and δ > 0 such that for all x R n and t > r :
w ( B ( x , t ) ) C ( t r ) n δ w ( B ( x , r ) ) .
Then
r ess inf t < s < w ( B ( x , s ) ) κ p w ( B ( x , t ) ) 1 / p d t t r ln m ( e + t r ) ess inf t < s < w ( B ( x , s ) ) κ p w ( B ( x , t ) ) 1 / p d t t = r ln m ( e + t r ) w ( B ( x , t ) ) κ 1 p d t t r ln m ( e + t r ) ( ( t r ) n δ w ( B ( x , r ) ) ) κ 1 p d t t = w ( B ( x , r ) ) κ 1 p r ln m ( e + t r ) ( t r ) n δ κ 1 p d t t = w ( B ( x , r ) ) κ 1 p 1 ln m ( e + τ ) τ n δ κ 1 p d τ τ w ( B ( x , r ) ) κ 1 p .

Throughout this paper, we use the notation A B to express that there is a positive constant C independent of all essential variables such that A C B . Moreover, C may be different from place to place.

2 Vector-valued generalized weighted Morrey spaces

The classical Morrey spaces M p , λ were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [29, 30].

We denote by M p , λ ( l 2 ) M p , λ ( R n , l 2 ) the vector-valued Morrey space, the space of all vector-valued functions f L p , loc ( l 2 ) with finite quasinorm
f M p , λ ( l 2 ) = sup x R n , r > 0 r λ p f L p ( B ( x , r ) , l 2 ) ,

where 1 p < and 0 λ n .

Note that M p , 0 ( l 2 ) = L p ( l 2 ) and M p , n ( l 2 ) = L ( l 2 ) . If λ < 0 or λ > n , then M p , λ ( l 2 ) = Θ , where Θ is the set of all vector-valued functions equivalent to 0 on R n .

We define the vector-valued generalized weighed Morrey spaces as follows.

Definition 2.1 Let 1 p < , φ be a positive measurable vector-valued function on R n × ( 0 , ) and w be non-negative measurable function on R n . We denote by M w p , φ ( l 2 ) the vector-valued generalized weighted Morrey space, the space of all vector-valued functions f L w p , loc ( l 2 ) with finite norm
f M w p , φ ( l 2 ) = sup x R n , r > 0 φ ( x , r ) 1 w ( B ( x , r ) ) 1 p f L w p ( B ( x , r ) , l 2 ) ,
where L w p ( B ( x , r ) , l 2 ) denotes the vector-valued weighted L p -space of measurable functions f for which
f L w p ( B ( x , r ) ) f χ B ( x , r ) L w p ( R n ) = ( B ( x , r ) f ( y ) l 2 p w ( y ) d y ) 1 p .
Furthermore, by W M w p , φ ( l 2 ) we denote the vector-valued weak generalized weighted Morrey space of all functions f W L w p , loc ( l 2 ) for which
f W M w p , φ ( l 2 ) = sup x R n , r > 0 φ ( x , r ) 1 w ( B ( x , r ) ) 1 p f W L w p ( B ( x , r ) , l 2 ) < ,
where W L w p ( B ( x , r ) , l 2 ) denotes the weak L w p -space of measurable functions f for which
f W L w p ( B ( x , r ) , l 2 ) f χ B ( x , r ) W L w p ( l 2 ) = sup t > 0 t ( { y B ( x , r ) : f ( y ) l 2 > t } w ( y ) d y ) 1 p .
Remark 2.2
  1. (1)

    If w 1 , then M 1 p , φ ( l 2 ) = M p , φ ( l 2 ) is the vector-valued generalized Morrey space.

     
  2. (2)

    If φ ( x , r ) w ( B ( x , r ) ) κ 1 p , then M w p , φ ( l 2 ) = L w p , κ ( l 2 ) is the vector-valued weighted Morrey space.

     
  3. (3)

    If φ ( x , r ) v ( B ( x , r ) ) κ p w ( B ( x , r ) ) 1 p , then M w p , φ ( l 2 ) = L v , w p , κ ( l 2 ) is the vector-valued two weighted Morrey space.

     
  4. (4)

    If w 1 and φ ( x , r ) = r λ n p with 0 < λ < n , then M w p , φ ( l 2 ) = L p , λ ( l 2 ) is the vector-valued Morrey space and W M w p , φ ( l 2 ) = W L p , λ ( l 2 ) is the vector-valued weak Morrey space.

     
  5. (5)

    If φ ( x , r ) w ( B ( x , r ) ) 1 p , then M w p , φ ( l 2 ) = L w p ( l 2 ) is the vector-valued weighted Lebesgue space.

     

3 Preliminaries and some lemmas

By a weight function, briefly weight, we mean a locally integrable function on R n which takes values in ( 0 , ) almost everywhere. For a weight w and a measurable set E, we define w ( E ) = E w ( x ) d x , and denote the Lebesgue measure of E by | E | and the characteristic function of E by χ E . Given a weight w, we say that w satisfies the doubling condition if there exists a constant D > 0 such that for any ball B, we have w ( 2 B ) D w ( B ) . When w satisfies this condition, we write for brevity w Δ 2 .

If w is a weight function, we denote by L w p ( l 2 ) L w p ( R n , l 2 ) the vector-valued weighted Lebesgue space defined by finiteness of the norm
f L w p ( l 2 ) = ( R n f ( x ) l 2 p w ( x ) d x ) 1 p < , if  1 p <

and by f L w ( l 2 ) = ess sup x R n f ( x ) l 2 w ( x ) if p = .

We recall that a weight function w is in the Muckenhoupt class A p [31], 1 < p < , if
[ w ] A p : = sup B [ w ] A p ( B ) = sup B ( 1 | B | B w ( x ) d x ) ( 1 | B | B w ( x ) 1 p d x ) p 1 < ,
where the sup is taken with respect to all the balls B and 1 p + 1 p = 1 . Note that, for all balls B, by Hölder’s inequality
[ w ] A p ( B ) 1 / p = | B | 1 w L 1 ( B ) 1 / p w 1 / p L p ( B ) 1 .

For p = 1 , the class A 1 is defined by the condition M w ( x ) C w ( x ) with [ w ] A 1 = sup x R n M w ( x ) w ( x ) , and for p = , A = 1 p < A p and [ w ] A = inf 1 p < [ w ] A p .

Lemma 3.1 ([32])

  1. (1)
    If w A p for some 1 p < , then w Δ 2 . Moreover, for all λ > 1
    w ( λ B ) λ n p [ w ] A p w ( B ) .
     
  2. (2)
    If w A , then w Δ 2 . Moreover, for all λ > 1
    w ( λ B ) 2 λ n [ w ] A w ( B ) .
     
  3. (3)
    If w A p for some 1 p , then there exist C > 0 and δ > 0 such that for any ball B and a measurable set S B ,
    w ( S ) w ( B ) C ( | S | | B | ) δ .
     
We are going to use the following result on the boundedness of the Hardy operator:
( H g ) ( t ) : = 1 t 0 t g ( r ) d μ ( r ) , 0 < t < ,

where μ is a non-negative Borel measure on ( 0 , ) .

Theorem 3.2 ([33])

The inequality
ess sup t > 0 ω ( t ) H g ( t ) c ess sup t > 0 v ( t ) g ( t )
holds for all functions g non-negative and non-increasing on ( 0 , ) if and only if
A : = sup t > 0 ω ( t ) t 0 t d μ ( r ) ess sup 0 < s < r v ( s ) < ,

and c A .

We also need the following statement on the boundedness of the Hardy type operator:
( H 1 g ) ( t ) : = 1 t 0 t ln m ( e + t r ) g ( r ) d μ ( r ) , 0 < t < ,

where μ is a non-negative Borel measure on ( 0 , ) .

Theorem 3.3 The inequality
ess sup t > 0 ω ( t ) H 1 g ( t ) c ess sup t > 0 v ( t ) g ( t )
holds for all functions g non-negative and non-increasing on ( 0 , ) if and only if
A 1 : = sup t > 0 ω ( t ) t 0 t ln m ( e + t r ) d μ ( r ) ess sup 0 < s < r v ( s ) < ,

and c A 1 .

Note that Theorem 3.3 can be proved analogously to Theorem 4.3 in [34].

Definition 3.4 B M O ( R n ) is the Banach space modulo constants with the norm defined by
b = sup x R n , r > 0 1 | B ( x , r ) | B ( x , r ) | b ( y ) b B ( x , r ) | d y < ,
where b L 1 loc ( R n ) and
b B ( x , r ) = 1 | B ( x , r ) | B ( x , r ) b ( y ) d y .

Lemma 3.5 ([35], Theorem 5, p.236)

Let w A . Then the norm is equivalent to the norm
b , w = sup x R n , r > 0 1 w ( B ( x , r ) ) B ( x , r ) | b ( y ) b B ( x , r ) , w | w ( y ) d y ,
where
b B ( x , r ) , w = 1 w ( B ( x , r ) ) B ( x , r ) b ( y ) w ( y ) d y .
Remark 3.6 (1) The John-Nirenberg inequality: there are constants C 1 , C 2 > 0 , such that for all b B M O ( R n ) and β > 0
| { x B : | b ( x ) b B | > β } | C 1 | B | e C 2 β / b , B R n .
(2) For 1 p < the John-Nirenberg inequality implies that
b sup B ( 1 | B | B | b ( y ) b B | p d y ) 1 p
(3.1)
and for 1 p < and w A
b sup B ( 1 w ( B ) B | b ( y ) b B | p w ( y ) d y ) 1 p .
(3.2)
Note that by the John-Nirenberg inequality and Lemma 3.1 (part 3) it follows that
w ( { x B : | b ( x ) b B | > β } ) C 1 δ w ( B ) e C 2 β δ / b
for some δ > 0 . Hence
B | b ( y ) b B | p w ( y ) d y = p 0 β p 1 w ( { x B : | b ( x ) b B | > β } ) d β p C 1 δ w ( B ) 0 β p 1 e C 2 β δ / b d β = C 3 w ( B ) b p ,

where C 3 > 0 depends only on C 1 δ , C 2 , p, and δ, which implies (3.2).

Also (3.1) is a particular case of (3.2) with w 1 .

The following lemma was proved in [13].

Lemma 3.7 (i) Let w A and b B M O ( R n ) . Let also 1 p < , x R n , k > 0 , and r 1 , r 2 > 0 . Then
( 1 w ( B ( x , r 1 ) ) B ( x , r 1 ) | b ( y ) b B ( x , r 2 ) , w | k p w ( y ) d y ) 1 p C ( 1 + | ln r 1 r 2 | ) k b k ,

where C > 0 is independent of f, w, x, r 1 , and r 2 .

(ii) Let w A p and b B M O ( R n ) . Let also 1 < p < , x R n , k > 0 , and r 1 , r 2 > 0 . Then
( 1 w 1 p ( B ( x , r 1 ) ) B ( x , r 1 ) | b ( y ) b B ( x , r 2 ) , w | k p w ( y ) 1 p d y ) 1 p C ( 1 + | ln r 1 r 2 | ) k b k ,

where C > 0 is independent of f, w, x, r 1 , and r 2 .

4 Proofs of main theorems

Before proving the main theorems, we need the following lemmas.

Lemma 4.1 [23]

For j Z + , denote
G α , 2 j ( f ) ( x ) = ( 0 | x y | 2 j t ( A α f ( y , t ) ) 2 d y d t t n + 1 ) 1 2 .
Let 0 < α 1 , 1 < p < , and w A p . Then any j Z + , we have
G α , 2 j ( f ) L w p 2 j ( 3 n 2 + α ) G α ( f ) L w p .
This lemma is easy by the following inequality, which is proved in [17]:
G α , β ( f ) ( x ) β 3 n 2 + α G α ( f ) ( x ) .

By a similar argument to [2], we can get the following lemma.

Lemma 4.2 Let 1 < p < , 0 < α 1 , and w A p , then the multilinear commutator [ b , G α ] is bounded from L w p ( l 2 ) to itself whenever b = ( b 1 , , b m ) and b j B M O ( R n ) , j = 1 , , m .

Now we are in a position to prove the theorems.

Lemma 4.3 Let 1 p < , 0 < α 1 , and w A p .

Then, for p > 1 , the inequality
G α f L w p ( B , l 2 ) ( w ( B ) ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t

holds for any ball B = B ( x 0 , r ) and for all f L w p , loc ( l 2 ) .

Moreover, for p = 1 the inequality
G α f W L w 1 ( B , l 2 ) w ( B ) 2 r f L w 1 ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 d t t

holds for any ball B = B ( x 0 , r ) and for all f L w 1 , loc ( l 2 ) .

Proof The main ideas of these proofs come from [13]. For arbitrary x R n , set B = B ( x 0 , r ) , 2 B B ( x 0 , 2 r ) . We decompose f = f 0 + f , where f 0 ( y ) = f ( y ) χ 2 B ( y ) , f ( y ) = f ( y ) f 0 ( y ) . Then
G α f L w p ( B ( x 0 , r ) , l 2 ) G α f 0 L w p ( B ( x 0 , r ) , l 2 ) + G α f L p ( B ( x 0 , r ) , l 2 ) : = I + I I .
First, let us estimate I. By Theorem A, we obtain
I G α f 0 L w p ( l 2 ) f 0 L w p ( l 2 ) = f L w p ( 2 B , l 2 ) .
(4.1)
On the other hand,
f L w p ( 2 B , l 2 ) | B | f L w p ( 2 B , l 2 ) 2 r d t t n + 1 | B | 2 r f L w p ( B ( x 0 , t ) , l 2 ) d t t n + 1 w ( B ) 1 p w 1 / p L p ( B ) 2 r f L w p ( B ( x 0 , t ) , l 2 ) d t t n + 1 w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) w 1 / p L p ( B ( x 0 , t ) ) d t t n + 1 [ ω ] A p 1 / p w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
(4.2)
Therefore from (4.1) and (4.2) we get
I w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
(4.3)
Then let us estimate II:
f ϕ t ( y ) l 2 = t n | y z | t ϕ ( y z t ) f ( z ) d z l 2 t n | y z | t f ( z ) l 2 d z .
Since x B ( x 0 , r ) , ( y , t ) Γ ( x ) , we have | z x | | z y | + | y x | 2 t , and
r | z x 0 | | x 0 x | | x z | | x y | + | y z | 2 t .
So, we obtain
G α f ( x ) l 2 ( Γ ( x ) ( t n | y z | t f ( z ) l 2 d z ) 2 d y d t t n + 1 ) 1 2 ( t > r / 2 | x y | < t ( | x z | 2 t f ( z ) l 2 d z ) 2 d y d t t 3 n + 1 ) 1 2 ( t > r / 2 ( | z x | 2 t f ( z ) l 2 d z ) 2 d t t 2 n + 1 ) 1 2 .
By Minkowski’s and Hölder’s inequalities and | z x | | z x 0 | | x 0 x | 1 2 | z x 0 | , we have
G α f ( x ) l 2 R n ( t > | z x | 2 d t t 2 n + 1 ) 1 2 f ( z ) l 2 d z | z x 0 | > 2 r f ( z ) l 2 | z x | n d z | z x 0 | > 2 r f ( z ) l 2 | z x 0 | n d z = | z x 0 | > 2 r f ( z ) l 2 | z x 0 | + d t t n + 1 d z = 2 r + 2 r < | z x 0 | < t f ( z ) l 2 d z d t t n + 1 2 r f ( z ) l 2 L w p ( B ( x 0 , t ) ) w 1 L p ( B ( x 0 , t ) ) d t t n + 1 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
Thus,
G α f L w p ( B , l 2 ) w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
(4.4)
By combining (4.3) and (4.4), we have
G α f L w p ( B , l 2 ) w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .

 □

Proof of Theorem 1.1 By Lemma 4.3 and Theorem 3.2 we have for p > 1
G α f M w p , φ 2 ( l 2 ) sup x 0 R n , r > 0 φ 2 ( x 0 , r ) 1 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t = sup x 0 R n , r > 0 φ 2 ( x 0 , r ) 1 0 r 1 f L w p ( B ( x 0 , t 1 ) , l 2 ) ( w ( B ( x 0 , t 1 ) ) ) 1 p d t t = sup x 0 R n , r > 0 φ 2 ( x 0 , r 1 ) 1 r 1 r 0 r f L w p ( B ( x 0 , t 1 ) , l 2 ) ( w ( B ( x 0 , t 1 ) ) ) 1 p d t t sup x 0 R n , r > 0 φ 1 ( x 0 , r 1 ) 1 ( w ( B ( x 0 , r 1 ) ) ) 1 p f L w p ( B ( x 0 , r 1 ) , l 2 ) = sup x 0 R n , r > 0 φ 1 ( x 0 , r ) 1 ( w ( B ( x 0 , r ) ) ) 1 p f L w p ( B ( x 0 , r ) , l 2 ) = f M w p , φ 1 ( l 2 )
and for p = 1
G α f W M w 1 , φ 2 ( l 2 ) sup x 0 R n , r > 0 φ 2 ( x 0 , r ) 1 r f L w 1 ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 d t t = sup x 0 R n , r > 0 φ 2 ( x 0 , r ) 1 0 r 1 f L w 1 ( B ( x 0 , t 1 ) , l 2 ) ( w ( B ( x 0 , t 1 ) ) ) 1 d t t = sup x 0 R n , r > 0 φ 2 ( x 0 , r 1 ) 1 r 1 r 0 r f L w 1 ( B ( x 0 , t 1 ) , l 2 ) ( w ( B ( x 0 , t 1 ) ) ) 1 d t t sup x 0 R n , r > 0 φ 1 ( x 0 , r 1 ) 1 ( w ( B ( x 0 , r 1 ) ) ) 1 f L w 1 ( B ( x 0 , r 1 ) , l 2 ) = sup x 0 R n , r > 0 φ 1 ( x 0 , r ) 1 ( w ( B ( x 0 , r ) ) ) 1 f L w 1 ( B ( x 0 , r ) , l 2 ) = f M w 1 , φ 1 ( l 2 ) .

 □

Lemma 4.4 Let 1 p < , 0 < α 1 , λ > 3 + α n , and w A p . Then, for p > 1 , the inequality
g λ , α ( f ) L w p ( B , l 2 ) ( w ( B ) ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t

holds for any ball B = B ( x 0 , r ) and for all f L w p , loc ( l 2 ) .

Moreover, for p = 1 the inequality
g λ , α ( f ) W L w 1 ( B , l 2 ) w ( B ) 2 r f L w 1 ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 d t t

holds for any ball B = B ( x 0 , r ) and for all f L w 1 , loc ( l 2 ) .

Proof From the definition of g λ , α ( f ) , we readily see that
g λ , α ( f ) ( x ) l 2 = ( 0 R n ( t t + | x y | ) n λ ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 ( 0 | x y | < t ( t t + | x y | ) n λ ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 + ( 0 | x y | t ( t t + | x y | ) n λ ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 : = I I I + I V .
First, let us estimate III:
I I I ( 0 | x y | < t ( t t + | x y | ) n λ ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 G α f ( x ) l 2 .
Now, let us estimate IV:
I V ( j = 1 0 2 j 1 t | x y | 2 j t ( t t + | x y | ) n λ ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 ( j = 1 0 2 j 1 t | x y | 2 j t 2 j n λ ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 j = 1 2 j n λ ( 0 | x y | 2 j t ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 : = j = 1 2 j n λ G α , 2 j ( f ) ( x ) l 2 .
Thus,
g λ , α ( f ) L w p ( B , l 2 ) G α f L w p ( B , l 2 ) + j = 1 2 j n λ 2 G α , 2 j ( f ) L w p ( B , l 2 ) .
(4.5)
By Lemma 4.3, we have
G α f L w p ( B , l 2 ) ( w ( B ) ) 1 p 2 r f L w p ( B ( x 0 , t ) ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
(4.6)
In the following, we will estimate G α , 2 j ( f ) L w p ( B , l 2 ) . We divide G α , 2 j ( f ) L w p ( B , l 2 ) into two parts,
G α , 2 j ( f ) L w p ( B , l 2 ) G α , 2 j ( f 0 ) L w p ( B , l 2 ) + G α , 2 j ( f ) L w p ( B , l 2 ) ,
(4.7)
where f 0 ( y ) = f ( y ) χ 2 B ( y ) , f ( y ) = f ( y ) f ( y ) . For the first part, by Lemma 4.1,
G α , 2 j ( f 0 ) L w p ( B , l 2 ) 2 j ( 3 n 2 + α ) G α ( f 0 ) L w p ( l 2 ) 2 j ( 3 n 2 + α ) f L w p ( B , l 2 ) 2 j ( 3 n 2 + α ) w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
(4.8)
For the second part,
G α , 2 j ( f ) ( x ) l 2 = ( 0 | x y | 2 j t ( A α f ( y , t ) ) 2 d y d t t n + 1 ) l / 2 l 2 = ( 0 | x y | 2 j t ( sup ϕ C α | f ϕ t ( y ) | ) 2 d y d t t n + 1 ) 1 2 l 2 ( 0 | x y | 2 j t ( | z y | t f ( z ) l 2 d z ) 2 d y d t t 3 n + 1 ) 1 2 .
Since | x z | | y z | + | x y | 2 j + 1 t , we get
G α , 2 j ( f ) ( x ) l 2 ( 0 | x y | 2 j t ( | x z | 2 j + 1 t f ( z ) l 2 d z ) 2 d y d t t 3 n + 1 ) 1 2 ( 0 ( | z x | 2 j + 1 t f ( z ) l 2 d z ) 2 2 j n d t t 2 n + 1 ) 1 2 2 j n 2 R n ( t | x z | 2 j + 1 f ( z ) l 2 2 d t t 2 n + 1 ) 1 2 d z 2 3 j n 2 | x 0 z | > 2 r f ( z ) l 2 | x z | n d z .
For | z x | | x 0 z | | x x 0 | | x 0 z | 1 2 | x 0 z | = 1 2 | x 0 z | , so by Fubini’s theorem and Hölder’s inequality, we obtain
G α , 2 j ( f ) ( x ) l 2 2 3 j n 2 | x 0 z | > 2 r f ( z ) l 2 | x 0 z | n d z = 2 3 j n 2 | x 0 z | > 2 r f ( z ) l 2 | x 0 z | d t t n + 1 d z 2 3 j n 2 2 r | x 0 z | < t f ( z ) l 2 d z d t t n + 1 2 3 j n 2 2 r f ( ) l 2 L 1 ( B ( x 0 , t ) ) d t t n + 1 2 3 j n 2 2 r f ( ) l 2 L w p ( B ( x 0 , t ) ) w 1 L p ( B ( x 0 , t ) ) d t t n + 1 2 3 j n 2 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
So,
G α , 2 j ( f ) L w p ( B , l 2 ) 2 3 j n 2 w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
(4.9)
Combining (4.7), (4.8), and (4.9), we have
G α , 2 j ( f ) L w p ( B , l 2 ) 2 j ( 3 n 2 + α ) w ( B ) 1 p 2 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t .
(4.10)
Thus,
g λ , α ( f ) L w p ( B , l 2 ) G α f L w p ( B , l 2 ) + j = 1 2 j n λ 2 G α , 2 j ( f ) L w p ( B , l 2 ) .
(4.11)

Since λ > 3 + α n , by (4.6), (4.10), and (4.11), we have the desired lemma. □

Proof of Theorem 1.2 From inequality (4.5) we have
g λ , α ( f ) M w p , φ 2 ( l 2 ) G α f M w p , φ 2 ( l 2 ) + j = 1 2 j n λ 2 G α , 2 j ( f ) M w p , φ 2 ( l 2 ) .
(4.12)
By Theorem 1.1, we have
G α f M w p , φ 2 ( l 2 ) f M w p , φ 1 ( l 2 ) .
(4.13)
In the following, we will estimate G α , 2 j ( f ) M w p , φ 2 ( l 2 ) . Thus, by substitution of variables and Theorem 3.2, we get
G α , 2 j ( f ) M w p , φ 2 ( l 2 ) 2 j ( 3 n 2 + α ) sup x 0 R n , r > 0 φ 2 ( x 0 , r ) 1 r f L w p ( B ( x 0 , t ) , l 2 ) ( w ( B ( x 0 , t ) ) ) 1 p d t t = 2 j ( 3 n 2 + α ) sup x 0 R n , r > 0 φ 2 ( x 0 , r 1 ) 1 r 1 r 0 r f L w p ( B ( x 0 , t 1 ) , l 2 ) ( w ( B ( x 0 , t 1 ) ) ) 1 p d t t 2 j ( 3 n 2 + α ) sup x 0 R n , r > 0 φ 1 ( x 0 , r 1 ) 1 ( w ( B ( x 0 , r 1 ) ) ) 1 p f L w p ( B ( x 0 , r 1 ) , l 2 ) = 2 j ( 3 n 2 + α ) f M w p , φ 1 ( l 2 ) .
(4.14)

Since λ > 3 + α n , by (4.12), (4.13), and (4.14), we have the desired theorem. □

Lemma 4.5 Let 1 < p < , 0 < α 1 , w A p , b = ( b 1 , , b m ) , and b i B M O ( R n ) , i = 1 , , m . Then the inequality
[ b , G α ] f L w p ( B , l 2 ) ( w ( B ) ) 1 p 2 r ln m ( e + t r ) f L w p ( B ( x 0 , τ ) , l 2 ) ( w ( B ( x 0 , τ ) ) ) 1 p d τ τ

holds for any ball B = B ( x 0 , r ) and for all f L w p , loc ( l 2 ) .

Proof We decompose f = f 0 + f , where f 0 = f χ 2 B and f = f f 0 . Then
[ b , G α ] f L w p ( B , l 2 ) [ b , G α ] f 0 L w p ( B , l 2 ) + [ b , G α ] f L w p ( B , l 2 ) .
Denote by b = i = 1 m b j . By Lemma 4.2, we have
[ b , G α ] f 0 L w p ( B , l 2 ) b f 0 L w p ( l 2 ) = b f L w p ( 2 B , l 2 ) b w ( B ) 1 p 2 r f L w p ( B ( x 0 , τ ) , l 2 ) ( w ( B ( x 0 , τ ) ) ) 1 p d τ τ .
(4.15)
For the term [ b , G α ] f L w p ( B , l 2 ) , without loss of generality, we can assume m = 2 . Thus, the operator [ b , G α ] f can be divided into four parts:
| [ b , G α ] f ( x ) | ( Γ ( x ) sup ϕ C α | R n ϕ t ( y z ) ( b 1 ( z ) ( b 1 ) B , ω ) ( b 2 ( z ) ( b 2 ) B , ω ) f ( z ) d z | 2 d y d t t n + 1 ) 1 / 2 + | b 1 ( x ) ( b 1 ) B , ω | ( Γ ( x ) sup ϕ C α | R n ϕ t ( y z ) ( b 2 ( z ) ( b 2 ) B , ω ) f ( z ) d z | 2 d y d t t n + 1 ) 1 / 2 + | b 2 ( x ) ( b 2 ) B , ω | ( Γ ( x ) sup ϕ C α | R n ϕ t ( y z ) ( b 1 ( z ) ( b 1 ) B , ω ) f ( z ) d z | 2 d y d t t n + 1 ) 1 / 2 | ( b 1 ( x ) ( b 1 ) B , ω ) ( b 2 ( x ) ( b 2 ) B , ω ) | ( Γ ( x ) sup ϕ C α | R n ϕ t ( y z ) f ( z ) d z | 2 d y d t t n + 1 ) 1 / 2 I 1 ( x ) + I 2 ( x ) + I 3 ( x ) + I 4 ( x ) .
For x B we have
[ b , G α ] f ( x ) l 2 I 1 ( x ) l 2 + I 2 ( x ) l 2 + I 3 ( x ) l 2 + I 4 ( x ) l 2 ( C 2 B ) | b 1 ( z ) ( b 1 ) B , ω | | b 2 ( z ) ( b 2 ) B , ω | f ( z ) l 2 | x 0 z | n d z + | b 1 ( x ) ( b 1 ) B , ω | ( C 2 B ) | b 2 ( z ) ( b 2 ) B , ω | f ( z ) l 2 | x 0 z | n d z + | b 2 ( x ) ( b 2 ) B , ω | ( C 2 B ) | b 1 ( z ) ( b 1 ) B , ω | f ( z ) l 2 | x 0 z | n d z + | ( b 1 ( x ) ( b 1 ) B , ω ) | | ( b 2 ( x ) ( b 2 ) B , ω ) | ( C 2 B ) f ( z ) l 2 | x 0 z | n d z .
Then
[ b , G α ] f L w p ( B , l 2 ) ( B ( ( C 2 B ) i = 1 2 | b i ( z ) ( b i ) B , ω | | x 0 z | n f ( z ) l 2 d z ) p ω ( x ) d x ) 1 / p + ( B | b 1 ( x ) ( b 1 ) B , ω | ( ( C 2 B ) | b 2 ( z ) ( b 2 ) B , ω | | x 0 z | n f ( z ) l 2 d z ) p ω ( x ) d x ) 1 / p + ( B | b 2 ( x ) ( b 2 ) B , ω | ( ( C 2 B ) | b 1 ( z ) ( b 1 ) B , ω | | x 0 z | n f ( z ) l 2 d z ) p ω ( x ) d x ) 1 / p + ( B ( ( C 2 B ) i = 1 2 | b i ( x ) ( b i ) B , ω | | x 0 z | n f ( z ) l 2 d z ) p ω ( x ) d x ) 1 / p I 1 + I 2 + I 3 + I 4 .
Let us estimate I 1 :
I 1 = ω ( B ) 1 / p ( C 2 B ) i = 1 2 | b i ( z ) ( b i ) B , ω | | x 0 z | n f ( z ) l 2 d z ω ( B ) 1 / p ( C 2 B ) i = 1 2 | b i ( z ) ( b i ) B , ω | f ( z ) l 2 | x 0 z | d τ τ n + 1 d z ω ( B ) 1 / p 2 r 2 r | x 0 z | < τ i = 1 2 | b i ( z ) ( b i ) B , ω | f ( z ) l 2 d z d τ τ n + 1 ω ( B ) 1 / p 2 r B ( x 0 , τ ) i = 1 2 | b i ( z ) ( b i ) B , ω | f ( z ) l 2 d z d τ τ n + 1 .
Applying Hölder’s inequality and by Lemma 3.7, we get
I 1 ω ( B ) 1 p 2 r i = 1 2 ( B ( x 0 , τ ) | b i ( z ) ( b i ) B , ω | 2 p ω ( z ) 1 2 p d z ) 1 2 p f L w p ( B ( x 0 , τ ) , l 2 ) d τ τ n + 1 i = 1 2 b j ω ( B ) 1 / p 2 r ( 1 + ln τ r ) 2 ω 1 / p L w p ( B ( x 0 , τ ) , l 2 ) f L w p ( B ( x 0 , τ ) , l 2 ) d τ τ n + 1 b ω ( B ) 1 / p 2 r ln 2 ( e + τ r ) f L w p ( B ( x 0 , τ ) , l 2 ) ω ( B ( x 0 , τ ) ) 1 / p d τ τ .
Let us estimate I 2 :
I 2 = ( B | b 1 ( x ) ( b 1 ) B , ω | p ω ( x ) d x ) 1 / p ( C 2 B ) | b 2 ( z ) ( b 2 ) B , ω | | x 0 z | n f ( z ) l 2 d z b 1 ω ( B ) 1 / p [ ( C 2 B ) | b 2 ( z ) ( b 2 ) B , ω | f ( z ) l 2 | x 0 z | d τ τ n + 1 d z ] b 1 ω ( B ) 1 / p 2 r 2 r | x 0 z | τ | b 2 ( z ) ( b 2 ) B , ω | f ( z ) l 2 d z d τ τ n + 1 b 1 ω ( B ) 1 / p 2 r B ( x 0 , τ ) | b 2 ( z ) ( b 2 ) B , ω | f ( z ) l 2 d z d τ τ n + 1 .
Applying Hölder’s inequality and by Lemma 3.7, we get
I 2 b 1 ω ( B ) 1 p 2 r ( B ( x 0 , τ ) | b 2 ( z ) ( b 2 ) B , ω | p ω ( z ) 1 p d z ) 1 p f L w p ( B ( x 0 , τ ) , l 2 ) d τ τ n + 1 i = 1 2 b j ω ( B ) 1 / p 2 r ( 1 + ln τ r ) ω 1 / p L w p ( B ( x 0 , τ ) , l 2 ) f L w p ( B ( x 0 , τ ) , l 2 ) d τ τ n + 1 b ω ( B ) 1 / p 2 r ln 2 ( e + τ r ) f L w p ( B ( x 0 , τ ) , l 2 ) ω ( B ( x 0 , τ ) ) 1 / p d τ τ .
In the same way, we shall get the result of I 3 :
I 3 b ω ( B ) 1 / p 2 r ln 2 ( e + τ r ) f L w p ( B ( x 0 , τ ) , l 2 ) ω ( B ( x 0 , τ ) ) 1 / p d τ τ .
In order to estimate I 4 note that
I 4 = ( B i = 1 2 | b i ( x ) ( b i ) B , ω | p ω ( x ) d x ) 1 / p ( C 2 B ) f ( z ) l 2 | x 0 z | n d z i = 1 2 ( B | b i ( x ) ( b i ) B , ω | 2 p ω ( x ) d x ) 1 / 2 p ( C 2 B ) f ( z ) l 2 | x 0 z | n d z .
By Lemma 3.7, we get
I 4 b ω ( B ) 1 / p ( C 2 B ) f ( z ) l 2 | x 0 z | n d z .
Applying Hölder’s inequality, we get
( C 2 B ) f ( z ) l 2 | x 0 z | n d z 2 r f L w p ( B ( x 0 , τ ) , l 2 ) ω 1 / p L w p ( B ( x 0 , τ ) , l 2 ) d τ τ n + 1