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

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.

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 bBMO( R n ). A well-known result of Coifman et al. [3] states that the commutator operator [b,K]f=K(bf)bKf 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<α1, 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)dx=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 :|xy|<βt}. 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 1p<, 0<α1, 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 (yz)f(z)dz|,

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<α1, w A p , and bBMO( 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<α1, w A p . Let also b =( b 1 ,, b m ) and b j BMO( 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 w1, 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 1p<, 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 dtC 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 1p<, 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 1p<, 0<α1, 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 1p<, 0<α1, 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<α1, w A p , and ( φ 1 , φ 2 ) satisfy condition (1.4). Let also b =( b 1 ,, b m ) and b j BMO( 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<α1, w A p , and ( φ 1 , φ 2 ) satisfy condition (1.4). Let also b =( b 1 ,, b m ) and b j BMO( 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 1p<, 0<α1, 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<α1, w A p , and ( φ 1 , φ 2 ) satisfy condition (1.4). Let also b =( b 1 ,, b m ) and b j BMO( 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, w1 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 AB to express that there is a positive constant C independent of all essential variables such that ACB. 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 1p< and 0λn.

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 1p<, 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 fW 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 w1, 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 w1 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)dx, 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(2B)Dw(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 1p<

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 Mw(x)Cw(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 1p<, then w Δ 2 . Moreover, for all λ>1

    w(λB) λ n p [ w ] A p w(B).
  2. (2)

    If w A , then w Δ 2 . Moreover, for all λ>1

    w(λB) 2 λ n [ w ] A w(B).
  3. (3)

    If w A p for some 1p, then there exist C>0 and δ>0 such that for any ball B and a measurable set SB,

    w ( S ) w ( B ) C ( | S | | B | ) δ .

We are going to use the following result on the boundedness of the Hardy operator:

(Hg)(t):= 1 t 0 t g(r)dμ(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)Hg(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 cA.

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)dμ(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 BMO( 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 ) | dy<,

where b L 1 loc ( R n ) and

b B ( x , r ) = 1 | B ( x , r ) | B ( x , r ) b(y)dy.

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)dy,

where

b B ( x , r ) , w = 1 w ( B ( x , r ) ) B ( x , r ) b(y)w(y)dy.

Remark 3.6 (1) The John-Nirenberg inequality: there are constants C 1 , C 2 >0, such that for all bBMO( 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 1p< 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 1p< 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 w1.

The following lemma was proved in [13].

Lemma 3.7 (i) Let w A and bBMO( R n ). Let also 1p<, 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 bBMO( 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<α1, 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<α1, 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 BMO( R n ), j=1,,m.

Now we are in a position to prove the theorems.

Lemma 4.3 Let 1p<, 0<α1, 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), 2BB( x 0 ,2r). 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+II.

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

Iw ( 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 dz.

Since xB( x 0 ,r), (y,t)Γ(x), we have |zx||zy|+|yx|2t, and

r|z x 0 || x 0 x||xz||xy|+|yz|2t.

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 |zx||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 1p<, 0<α1, 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:

III ( 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 |xz||yz|+|xy| 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 |zx|| 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<α1, w A p , b =( b 1 ,, b m ), and b i BMO( 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 xB 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 )