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Some estimates for the commutators of multilinear maximal function on Morrey-type space

  • Xiao Yu EMAIL logo , Pu Zhang and Hongliang Li
Published/Copyright: July 3, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

Keywords: Morrey-type space; multilinear maximal function; commutator; Orlicz function
MSC 2010: 42B20; 42B25

1 Introduction

Let b ∈ L loc 1 ( R n ) . We say that b belongs to the mean oscillation space BMO( R n ) if b satisfies

∥ b ∥ BMO ≔ sup Q 1 ∣ Q ∣ ∫ Q ∣ b ( y ) − b Q ∣ d y < ∞ ,

where Q denotes any cube of R n and b Q = 1 ∣ Q ∣ ∫ Q b ( x ) d x .

The commutator [ b , M ] ( f ) ( x ) , which is generated by the maximal function and the BMO function, can be stated as follows:

[ b , M ] ( f ) ( x ) = b M ( f ) ( x ) − M ( b f ) ( x ) .

Here, M ( f ) ( x ) is the classical Hardy-Littlewood maximal function defined as

M ( f ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ d y ,

where the supremum is taken over all cubes Q containing x .

For the study of [ b , M ] , one may see [1,2, 3,4] for more details. Here, we would like to point out that in [2], Bastero, Milman and Ruiz gave the sufficient and necessary conditions for the boundedness of [ b , M ] on L p ( R n ) . Precisely, they proved the following theorem.

Theorem A

[2] Let b be a real valued, locally integrable function in R n . Then, the following three assertions are equivalent.

  1. The commutator [ b , M ] is bounded on L p for 1 < p < ∞ .

  2. b is in BMO and b − belongs to L ∞ with b − = − min { b ( x ) , 0 } .

  3. For p ∈ ( 1 , ∞ ) , there is

    sup Q 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ p d x < ∞ ,

    where M Q ( b ) ( x ) = sup Q 0 ∋ x , Q 0 ⊂ Q 1 ∣ Q 0 ∣ ∫ Q 0 ∣ b ( t ) ∣ p d t 1 / p .

Later, Xie [4] extended Theorem A to the Morrey space. Here the Morrey space L p , λ ( R n ) is defined by

L p , λ ( R n ) = f ∈ L loc p ( R n ) : ∥ f ∥ L p , λ ( R n ) = sup x ∈ R n , t > 0 1 t λ ∫ B ( x , t ) ∣ f ( y ) ∣ p d y 1 / p < ∞ ,

where 0 ≤ λ < n and 1 ≤ p < ∞ .

Similarly, the weak Morrey space W L p , λ ( R n ) can be stated as follows:

W L p , λ ( R n ) = f ∈ L loc p ( R n ) : ∥ f ∥ W L p , λ ( R n ) = sup x ∈ R n , t > 0 , β > 0 β t λ / p ∣ { t ∈ Q ( x , t ) : ∣ f ( t ) ∣ > β } ∣ 1 / p < ∞ .

In 1938, Morrey [5] obtained the Hölder regularity of the solutions of elliptic equations by applying new technique. Here, the new technique was based on the estimates of the integrals over balls of the gradient of the solutions via the radius of the same ball. The classical Morrey spaces L p , λ ( R n ) , usually attributed to him, were introduced in the 1960s by Campanato, Peetre and Brudneii, independently. Readers may refer to [6] for more details.

Later, the boundedness of Hardy-Littlewood maximal function on Morrey space can be found in [7] and the maximal inequality in generalized Morrey-type spaces can be found in [8].

Obviously, if we choose λ = 0 , then L p , λ ( R n ) becomes the classical L p ( R n ) space.

On the other hand, the multilinear theory was also developed a lot in the past 20 years and readers may refer to [9,10]. for more details. Moreover, Pérez and Torres [11] introduced the commutators of multilinear Calderón-Zygmund operators defined as

T b → ( f → ) ( x ) = ∑ i = 1 m T b → i ( f → ) ( x ) ( b → = ( b 1 , … , b m ) ) ,

where

T b → i ( f → ) ( x ) = b i ( x ) T ( f → ) ( x ) − T ( f 1 , … , f i − 1 , b i f i , f i + 1 , … , f m ) ( x ) ,

b → = ( b 1 , … , b m ) and T is the multilinear Calderón-Zygmund operator which was first studied by Grafakos and Torres [9].

Later, Lerner et al. [10] introduced a new type of multiple weight classes A P → which is very adopted to the weighted norm estimates for the commutators of multilinear Calderón-Zygmund operators. Moreover, in order to give the characterization of A P → , Lerner et al. [10] introduced the following multilinear maximal function ℳ ( f → ) ( x ) defined as

ℳ ( f → ) ( x ) = sup Q ∋ x ∏ i = 1 m 1 ∣ Q ∣ ∫ Q ∣ f i ( y i ) ∣ d y i ,

where f → = ( f 1 , … , f m ) with m ∈ Z + ( m > 1 ) and the supremum is taken over all cubes Q containing x .

Obviously, it is easy to see ℳ ( f → ) ( x ) ≤ ∏ i = 1 m M ( f i ) ( x ) .

For any cube Q , we write Q m = Q × ⋯ × Q ︸ m . In 2015, Zhang [12] studied the commutator generated by the BMO function and the multilinear maximal functions [ b → , ℳ ] ( f → ) ( x ) and ℳ b → ( f → ) ( x ) as follows:

[ b → , ℳ ] ( f → ) ( x ) = ∑ j = 1 m [ b → , M ] j ( f → ) ( x ) , ℳ b → ( f → ) ( x ) = ∑ j = 1 m ℳ b j ( f → ) ( x ) .

Here

[ b → , M ] j ( f → ) ( x ) = b j ( x ) ℳ ( f → ) ( x ) − ℳ ( f 1 , … , f j − 1 , b j f j , f j + 1 , … , f m ) ( x )

and

ℳ b j ( f → ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ m ∫ Q m ∣ b j ( x ) − b j ( y j ) ∣ ∏ i = 1 m ∣ f i ( y i ) ∣ d y → ,

where y → = ( y 1 , … , y m ) .

Zhang [12] proved the weighted boundedness of [ b → , ℳ ] ( f → ) ( x ) and ℳ b → ( f → ) ( x ) on the product L p spaces.

Motivated by the above backgrounds, in this paper, we would like to extend Theorem A with [ b → , ℳ ] ( f → ) ( x ) on Morrey-type spaces. The first result of this paper can be regarded as follows.

Theorem 1.1

Let b i be a real valued, locally integrable function in R n with i = 1 , … , m . Then, the following three assertions are equivalent.

  1. The commutator [ b → , ℳ ] ( f → ) ( x ) is bounded from L p 1 , λ ( R n ) × ⋯ × L p m , λ ( R n ) to L p , λ ( R n ) for 1 < p i < ∞ and 1 / p = ∑ i = 1 m 1 / p i with p > 1 and 0 ≤ λ < n .

  2. b i is in BMO and b i − belongs to L ∞ .

  3. For p ∈ ( 1 , ∞ ) , there is

    sup Q 1 ∣ Q ∣ ∫ Q ∣ b i ( x ) − M Q ( b i ) ( x ) ∣ p d x < ∞ ,

    where M Q ( b i ) ( x ) = sup Q 0 ∋ x , Q 0 ⊂ Q 1 ∣ Q 0 ∣ ∫ Q 0 ∣ b i ( t ) ∣ d t .

Remark 1.2

Obviously, our results improve the main results of [2, 4].

On the other hand, in 1995, Pérez [13] gave a counter example that the commutator [ b , T ] is not of weak type ( 1 , 1 ) and he proved that [ b , T ] satisfies the weak L log L -type estimates. For the study of the commutators on the endpoint case, one may see [10,12,14,15] for more details. Particularly in [12], Zhang proved the weak weighted L log L estimates for ℳ b → ( f → ) ( x ) . Recently, Wang [16] proved that the commutator generated by some integral operators is bounded from L L log L 1 , ω ( R n ) to W L 1 , ω ( R n ) in some sense and the definitions of W L 1 , ω ( R n ) and L L log L 1 , ω ( R n ) will be given in Sections 1 and 2, respectively.

Thus, it is natural to ask whether we can prove the endpoint estimates of the commutators generated by the multilinear maximal functions and BMO functions on Morrey-type space.

Before giving the second result of this paper, we introduce the generalized Morrey space L p , ω ( R n ) proposed by Nakai [8].

Definition A

[8] Let ω : R n × R → R + , 1 ≤ p < ∞ and Q ( a , r ) be the cube

{ x ∈ R n : ∣ x i − a i ∣ ≤ r / 2 , i = 1 , 2 , … , n }

whose edges have length r and are parallel to the coordinate axes. For Q = Q ( a , r ) , we denote k Q = Q ( a , k r ) and ω ( Q ) = ω ( a , r ) . Moreover, we suppose that ω satisfies

(1.1) ω ( a , 2 t ) ≤ C 0 ω ( a , t )

with 1 < C 0 < 2 n .

Then, the generalized Morrey space L p , ω ( R n ) ( 1 ≤ p < ∞ ) is defined as

L p , ω ( R n ) = f ∈ L loc p ( R n ) : ∥ f ∥ L p , ω ( R n ) = sup Q 1 ω ( Q ) ∫ Q ∣ f ( x ) ∣ p d x 1 / p < ∞ .

Similarly, the weak generalized Morrey space W L p , ω ( R n ) ( 1 ≤ p < ∞ ) is defined by

W L p , ω ( R n ) = f ∈ W L loc p ( R n ) : ∥ f ∥ L p , ω ( R n ) = sup Q sup β > 0 β ω ( Q ) 1 / p ∣ { x ∈ Q : ∣ f ( x ) ∣ > β } ∣ 1 / p < ∞ .

Obviously, if we choose ω ( Q ) = ∣ Q ∣ λ n with 0 ≤ λ < n , there is

(1.2) ω ( 2 Q ) = ∣ 2 Q ∣ λ n = 2 λ ∣ Q ∣ λ n < 2 n ω ( Q ) .

In this case, L p , ω ( R n ) and W L p , ω ( R n ) become L p , λ ( R n ) and W L p , λ ( R n ) with 0 ≤ λ < n , respectively.

The second result of this paper can be stated as follows.

Theorem 1.3

Let b i ∈ BMO ( R n ) and ω satisfy (1.1) with 1 < C 0 ≤ 2 γ for any 0 < γ < n . Then, for any β > 0 and cube B = B ( x , t ) with center x ∈ R n and radius t > 0 , there is

1 ω ( B ) m ∣ { z ∈ B ( x , t ) : ℳ b → ( f → ) ( z ) > β m } ∣ m ≤ C ∥ b → ∥ BMO ∏ i = 1 m Φ ∣ f i ∣ β L L log L 1 , ω ( R n ) ,

where Φ ( x ) = t ( 1 + log + t ) , ∥ b → ∥ BMO = sup 1 ≤ i ≤ m ∥ b i ∥ BMO and C is a positive constant depending on the dimension n .

Remark 1.4

By the definition of W L 1 m , ω ( R n ) , we may roughly say that ℳ b → ( f → ) ( x ) is bounded from L L log L 1 , ω ( R n ) × ⋯ × L L log L 1 , ω ( R n ) to W L 1 m , ω ( R n ) in some sense.

2 Orlicz space, generalized Hölder inequalities, weak L p spaces and another kind of multilinear maximal function

First, we introduce some facts and theory related to the Orlicz spaces. For more information about Orlicz spaces, one may refer to [17].

Suppose that Φ : [ 0 , ∞ ) → [ 0 , ∞ ) is a continuous, convex, increasing function with Φ ( 0 ) = 0 and Φ satisfies Φ ( t ) → ∞ if t → ∞ . Then, we say that Φ is a Young function.

For a function f defined on a cube Q , we define

∥ f ∥ Φ , Q = inf { λ > 0 : 1 ∣ Q ∣ ∫ Q Φ ∣ f ( y ) ∣ λ d y ≤ 1 } ,

which is called the mean Luxemburg norm of f on Q .

Obviously, if Φ ≤ Ψ , then

∥ f ∥ Φ , Q ≤ ∥ f ∥ Ψ , Q .

For a Young function Φ , we may define its complementary function Φ ¯ ( s ) as

Φ ¯ ( s ) = sup t > 0 { s t − Φ ( t ) } .

Obviously, Φ ( t ) = t ( 1 + log + t ) is a Young function and its complementary function Φ ¯ ( s ) ≈ e s (see [17]).

From [18], we know that if E , F , G are Young functions and satisfy

E − 1 ( t ) G − 1 ( t ) ≤ F − 1 ( t ) ( ∀ t > 0 ) .

Then, the following generalized Hölder inequality holds.

∥ f g ∥ F , Q ≤ 2 ∥ f ∥ E , Q ∥ g ∥ G , Q .

In this case, we have

1 ∣ Q ∣ ∫ Q ∣ f ( y ) g ( y ) ∣ d y ≤ ∥ f ∥ Φ , Q ∥ g ∥ Φ ¯ , Q .

When Φ ( t ) = t ( 1 + log + t ) , we write ∥ f ∥ L log L , Q = ∥ f ∥ Φ , Q and ∥ f ∥ exp L , Q = ∥ f ∥ Φ ¯ , Q . Thus, we get

(2.1) 1 ∣ Q ∣ ∫ Q ∣ f ( y ) g ( y ) ∣ d y ≤ ∥ f ∥ L log L , Q ∥ g ∥ exp L , Q .

By the definition of ∥ f ∥ Φ , Q , it is easy to see that

(2.2) 1 ∣ Q ∣ ∫ Q ∣ f ( x ) ∣ d x = ∥ f ∥ L 1 , Q ≤ ∥ f ∥ L log L , Q .

Moreover, for b ∈ BMO ( R n ) , using the John-Nirenberg inequality, we have ∥ b − b Q ∥ exp L , Q ≤ C ∥ b ∥ BMO . Thus, (2.1) can be written as follows:

(2.3) 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ( b ( y ) − b Q ) ∣ d y ≤ C ∥ f ∥ L log L , Q ∥ b ∥ BMO .

Next, following [16], we introduce the generalized Morrey space L L log L 1 , ω ( R n ) related to L log L type associated with ω , that is,

L L 1 log L 1 , ω ( R n ) = f ∈ L loc ( R n ) : ∥ f ∥ L L log L 1 , ω ( R n ) = sup Q ∣ Q ∣ ω ( Q ) ∥ f ∥ L log L , Q < ∞ .

Obviously, it is easy to see that L L log L 1 , ω ( R n ) ⊂ L 1 , ω ( R n ) from (2.1) (see [16]).

Now, we introduce the definition of the weak L p space (see [19]).

Suppose that X is a measure space and has the normal Lebesgue measure. For 0 < p < ∞ , the weak L p ( X ) is defined as the set of all Lebesgue measurable functions f , such that

∥ f ∥ L p , ∞ ( X ) = sup { λ d f ( λ ) 1 / p : λ > 0 } < ∞ ,

where the definition of d f ( γ ) is

d f ( λ ) = ∣ { x ∈ X : ∣ f ( x ) ∣ > λ } ∣ .

Finally, we introduce another kind of multilinear maximal function ℳ c ( f → ) ( x ) as follows:

ℳ c ( f → ) ( x ) = sup r > 0 1 ∣ Q ( x , r ) ∣ m ∫ Q ( x , r ) m ∏ j = 1 m ∣ f j ( y j ) ∣ d y → .

Moreover, we define the commutator of ℳ c b j ( f → ) ( x ) as

ℳ c b j ( f → ) ( x ) = sup r > 0 1 ∣ Q ( x , r ) ∣ m ∫ Q ( x , r ) m ∣ b j ( x ) − b j ( y j ) ∣ ∏ i = 1 m ∣ f i ( y i ) ∣ d y → .

Then, it is easy to see [12]

(2.4) ℳ c ( f → ) ( x ) ∼ ℳ ( f → ) ( x ) and ℳ c b j ( f → ) ( x ) ∼ ℳ b j ( f → ) ( x ) .

3 Some useful lemmas

In this section, we would like to give some lemmas, which is very useful throughout this paper. For some techniques to deal with commutators of operators on Morrey-type spaces, one may see [4,12,16,20,21,22] for more details.

Lemma 3.1

Suppose that f ∈ BMO ( R n ) and r 1 , r 2 ∈ R + , then for any 1 < p < ∞ and j ∈ Z + , we have

  1. 1 ∣ Q ( x 0 , r 1 ) ∣ ∫ Q ( x 0 , r 1 ) ∣ f ( x ) − f Q ( x 0 , r 2 ) ∣ d x ≤ C 1 + ∣ log r 1 r 2 ∣ ∥ f ∥ BMO .

  2. ∥ f ∥ BMO ∼ sup Q 1 ∣ Q ∣ ∫ Q ∣ f ( x ) − f Q ∣ p d x 1 / p .

  3. ∣ f Q − f 2 j + 1 Q ∣ ≤ C ( j + 1 ) ∥ f ∥ BMO .

The proof of Lemma 3.1 is very standard and can be found in many papers.

Lemma 3.2

Suppose that 1 / p = ∑ i = 1 m 1 / p i with p i > 1 . If b i ∈ BMO with i = 1 , … , m , we obtain that ℳ b → ( f → ) ( x ) is bounded from L p 1 , λ ( R n ) × ⋯ × L p m , λ ( R n ) to L p , λ ( R n ) with 0 ≤ λ < n . That is,

∥ ℳ b → ( f → ) ∥ L p , λ ( R n ) ≤ C ∥ b → ∥ BMO ∏ i = 1 m ∥ f i ∥ L p i , λ ( R n )

with ∥ b → ∥ BMO = sup i ∥ b i ∥ BMO .

Proof

By the definition of ℳ b → ( f → ) ( x ) and the fact ℳ c b i ( f → ) ( x ) ∼ ℳ b i ( f → ) ( x ) , it suffices to consider ℳ c b 1 ( f → ) ( x ) . For any cube B = B ( x , t ) , we split each f i = f i 0 + f i ∞ where f i 0 = f i χ 2 B and 2 B = B ( x , 2 t ) . Then, we get

∥ ℳ c b 1 ( f → ) ∥ L p ( B ) ≤ ∥ ℳ c b 1 ( f 0 → ) χ B ∥ L p + ∑ ′ ∥ ℳ c b 1 ( f 1 α 1 , … , f m α m ) χ B ∥ L p ≔ I + I I .

where α 1 , … , α m ∈ { 0 , ∞ } and each term in the sum ∑ ′ contains at least one α i = ∞ and one α j = 0 .

For I , by the boundedness of ℳ c b 1 ( f → ) ( x ) on the product L p spaces (see [12]), it is easy to see

(3.1) I ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m ∥ f i ∥ L p i , λ t λ / p .

To estimate I I , first, we consider the case α 1 = ⋯ = α m = ∞ . For any cube Q = Q ( z , r ) with z ∈ B ( x , t ) and using the fact y 1 ∈ B ( x , 2 t ) c and z ∈ B ( x , t ) , we have

sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ f 1 ∞ ( y 1 ) ∣ d y 1 = sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ⋂ B ( x , 2 t ) c ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ f 1 ( y 1 ) ∣ d y 1 ≤ C sup r > 2 t ∫ Q ( z , r ) ⋂ B ( x , 2 t ) c ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ f 1 ( y 1 ) ∣ ∣ y 1 − z ∣ n d y 1

≤ C sup r > 2 t ∫ Q ( z , r ) ⋂ B ( x , 2 t ) c ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ f 1 ( y 1 ) ∣ ∣ y 1 − x ∣ n d y 1 ≤ C sup r > 2 t ∫ B ( x , 2 t ) c ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ f 1 ( y 1 ) ∣ ∣ y 1 − x ∣ n d y 1 ≤ ∫ B ( x , 2 t ) c ∣ f 1 ( y 1 ) ∣ ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ y 1 − x ∣ n d y 1 ≤ ∫ B ( x , 2 t ) c ∣ f 1 ( y 1 ) ∣ ∣ b 1 ( z ) − ( b 1 ) B ∣ ∣ y 1 − x ∣ n d y 1 + ∫ B ( x , 2 t ) c ∣ f 1 ( y 1 ) ∣ ∣ b 1 ( y 1 ) − ( b 1 ) B ∣ ∣ y 1 − x ∣ n d y 1 ≕ A 1 + A 2 .

For A 1 , we decompose it as

A 1 ≤ ∑ j = 1 ∞ ∫ 2 j + 1 B ⧹ 2 j B ∣ f 1 ( y 1 ) ∣ ∣ b 1 ( z ) − ( b 1 ) B ∣ ∣ y 1 − x ∣ n d y 1

with 2 j + 1 B = B ( x , 2 j + 1 t ) .

As it is easy to see 2 j t 2 ≤ ∣ x − y 1 ∣ ≤ 2 j + 1 t n 2 , which implies

C 1 ∣ 2 j + 1 B ∣ ≤ 1 ∣ x − y 1 ∣ n ≤ C 2 ∣ 2 j B ∣ ,

where C 1 and C 2 are positive constants depending on the dimension n and 2 j + 1 B = B ( x , 2 j + 1 t ) . Thus, we conclude that there exists a constant C depending on the dimension n , such that

(3.2) 1 ∣ x − y 1 ∣ n ∼ C ∣ 2 j + 1 B ∣ .

Then, we have

A 1 ≤ ∣ b 1 ( z ) − ( b 1 ) B ∣ ∫ B ( x , 2 t ) c ∣ f 1 ( y 1 ) ∣ ∣ y 1 − x ∣ n d y 1 ≤ ∣ b 1 ( z ) − ( b 1 ) B ∣ ∑ j = 1 ∞ ∫ 2 j + 1 B ⧹ 2 j B ∣ f 1 ( y 1 ) ∣ ∣ y 1 − x ∣ n d y 1 ≤ ∣ b 1 ( z ) − ( b 1 ) B ∣ ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ f 1 ( y 1 ) ∣ d y 1 ≤ ∣ b 1 ( z ) − ( b 1 ) B ∣ ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ f 1 ( y 1 ) ∣ p 1 d y 1 1 / p 1 ∣ 2 j + 1 B ∣ 1 − 1 p 1 ≤ C ∣ b 1 ( z ) − ( b 1 ) B ∣ ∥ f 1 ∥ L p 1 , λ ∑ j = 1 ∞ ∣ 2 j + 1 B ∣ 1 p 1 λ n − 1 ≤ C ∣ b 1 ( z ) − ( b 1 ) B ∣ ∥ f 1 ∥ L p 1 , λ ∣ B ∣ 1 p 1 λ n − 1

with the constant C depending on the dimension n .

For the estimates of A 2 , we decompose it as

A 2 ≤ ∑ j = 1 ∞ ∫ 2 j + 1 B ⧹ 2 j B ∣ f 1 ( y 1 ) ∣ ∣ b 1 ( y 1 ) − ( b 1 ) B ∣ ∣ y 1 − x ∣ − n d y 1 .

Then, using Lemma 3.1, the Hölder inequality and (3.2) again, there is

A 2 ≤ ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ f 1 ( y 1 ) ∣ ∣ b 1 ( y 1 ) − ( b 1 ) B ∣ d y 1 ≤ ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ f 1 ( y 1 ) ∣ p 1 d y 1 1 / p 1 ∫ 2 j + 1 B ∣ b 1 ( y 1 ) − ( b 1 ) B ∣ p 1 p 1 − 1 d y 1 1 − 1 p 1 ≤ C ∥ f 1 ∥ L p 1 , λ ∑ j = 1 ∞ ∣ 2 j + 1 B ∣ 1 p 1 λ n − 1 ≤ C ∥ f 1 ∥ L p 1 , λ ∥ b 1 ∥ BMO ∣ B ∣ 1 p 1 λ n − 1 ,

where the positive constant C is depending on the dimension n .

Thus, we obtain

sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ f 1 ∞ ( y 1 ) ∣ d y 1 ≤ C ( ∥ b 1 ∥ BMO + ∣ b 1 ( z ) − ( b 1 ) B ∣ ) ∥ f 1 ∥ L p 1 , λ ∣ B ∣ 1 p 1 λ n − 1 .

Moreover, for ∫ B ( x , 2 t ) c ∣ f i ( y i ) ∣ ∣ y i − x ∣ n d y i with i = 2 , 3 , … , m , there is

∫ B ( x , 2 t ) c ∣ f i ( y i ) ∣ ∣ y i − x ∣ n d y i ≤ ∑ j = 1 ∞ ∫ 2 j + 1 B ⧹ 2 j B ∣ f i ( y i ) ∣ ∣ y i − x ∣ n d y i ≤ ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ f i ( y i ) ∣ d y i ≤ ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ f i ( y i ) ∣ p i d y i 1 / p i ∣ 2 j + 1 B ∣ 1 − 1 p i ≤ C ∥ f i ∥ L p i , λ ∑ j = 1 ∞ ∣ 2 j + 1 B ∣ 1 p i λ n − 1 ≤ C ∥ f i ∥ L p i , λ ∣ B ∣ 1 p i λ n − 1 ,

which implies

(3.3) M c ( f i ∞ ) ( z ) ≔ sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ d y i ≤ C ∥ f i ∥ L p i , λ ∣ B ∣ 1 p i λ n − 1 .

Combining the aforementioned estimates, we may obtain

∫ R n ∣ ℳ c b 1 ( f ∞ → ) ( z ) ∣ p χ B ( z ) d z 1 / p ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m ∥ f i ∥ L p i , λ ∣ B ∣ 1 p i λ n − 1 ∫ B d z 1 / p + ∏ i = 1 m ∥ f i ∥ L p i , λ ∣ B ∣ λ n p i ∣ B ∣ − 1 ∫ B ∣ b 1 ( z ) − ( b 1 ) B ∣ d z 1 / p ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m ∥ f i ∥ L p i , λ t λ / p .

Finally, for the case that α j 1 = ⋯ = α j l = 0 for some { j 1 , … , j l } ⊂ { 1 , … , m } where 1 ≤ l < m , we only consider the case α 1 = ∞ , α 2 = ∞ and α 3 = ⋯ = α m = 0 since the other cases follow in a similar way. Using (2.4) with m = 1 , the Hölder inequality and the aforementioned estimates, there is

∥ ℳ b 1 ( f 1 ∞ , f 2 ∞ , f 3 0 , … , f m 0 ) χ B ∥ L p ≤ ∥ M b 1 ( f 1 ∞ ) χ B ∥ L p 1 ∥ M ( f 2 ∞ ) χ B ∥ L p 2 ∥ M ( f 3 0 ) χ B ∥ L p 3 ⋯ ∥ M ( f m 0 ) χ B ∥ L p m ≤ C ∥ f 1 ∥ L p 1 , ω ∣ B ∣ λ / n p 1 ∣ B ∣ − 1 p 1 ∫ B ( ∥ b 1 ∥ BMO + ∣ b 1 ( z ) − ( b 1 ) B ∣ ) d z 1 / p 1 × ∣ B ∣ λ / n p 2 ∥ f 2 ∥ L p 2 , ω ∣ B ∣ − 1 / p 2 ∣ B ∣ 1 / p 2 ∏ i = 3 m ∥ f i ∥ L p i , λ ∣ B ∣ 1 p i λ n ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m ∥ f i ∥ L p i , λ t λ / p ,

where M b ( f ) ( x ) is the commutator generated by the M ( f ) ( x ) and the symbol b , that is,

M b ( f ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( y ) ∣ ∣ f ( y ) ∣ d y .

Combining the estimates of I and II, we finish the proof of Lemma 3.2 from the definition of L p , λ ( R n ) .□

Using the similar ideas and estimates in the proof of Lemma 3.2, we can easily get the boundedness of ℳ ( f → ) ( x ) on L p , λ ( R n ) and we still give a sketch and simple proof of the following lemma for the sake of completeness.

Lemma 3.3

Let 1 / p = ∑ i = 1 m 1 / p i with p i > 1 . Then, ℳ ( f → ) ( x ) is bounded from L p 1 , λ ( R n ) × ⋯ × L p m , λ ( R n ) to L p , λ ( R n ) with 0 ≤ λ < n .

Proof

We only give some main steps of this proof. It suffices to study ℳ c ( f → ) ( x ) . Following the proof of Lemma 3.2, we split each f i = f i 0 + f i ∞ with f i 0 = f i χ 2 B . Then, we decompose ℳ c ( f → ) ( z ) as

ℳ c ( f → ) ( z ) = ℳ c ( f 0 → ) ( z ) + ∑ ′ ℳ c ( f 1 α 1 , … , f m α m ) ( z ) ≔ I 1 + I 2 ,

where α 1 , … , α m ∈ { 0 , ∞ } and each term in the sum ∑ ′ contains at least one α i = ∞ .

Thus, it suffices to show the following three inequalities:

(3.4) ∥ ℳ c ( f 1 0 , … , f m 0 ) ∥ L p ( B ) ≤ C ∏ i = 1 m ∥ f i ∥ L p i , λ t λ / p ,

(3.5) ∥ ℳ c ( f 1 ∞ , … , f m ∞ ) ∥ L p ( B ) ≤ C ∏ i = 1 m ∥ f i ∥ L p i , λ t λ / p

and

(3.6) ∥ ℳ c ( f 1 α 1 , f 2 α 2 , … , f m α m ) ∥ L p ( B ) ≤ C ∏ i = 1 m ∥ f i ∥ L p i , λ t λ / p ,

where each term in (3.6) contains at least one α i = ∞ and one α j = 0 .

By checking the proof of Lemma 3.2, we know that (3.4) follows from the boundedness of ℳ ( f → ) ( x ) on product L p spaces (see [10]) and (3.5) follows from (3.3) and the multilinear Hölder inequality on L p spaces.

For (3.6), without loss of generality, using (3.3) and the similar estimates of ∥ ℳ b 1 ( f 1 ∞ , f 2 ∞ , f 3 0 , … , f m 0 ) χ B ∥ L p in the proof of Lemma 3.2, we know that (3.6) is also true.□

Lemma 3.4

[12] Let b → = ( b 1 , … , b m ) and f → = ( f 1 , … , f m ) be two collections of locally integrable functions, then

∣ [ b → , ℳ ] ( f → ) ( x ) ∣ ≤ ℳ b → ( f → ) ( x ) + 2 ∑ i = 1 m b i − ( x ) ℳ ( f → ) ( x ) .

Lemma 3.5

[22] Let 1 ≤ p < ∞ and 0 ≤ λ ≤ n . Then for any B ( x 0 , t 0 ) , we have

∥ χ B ( x 0 , t 0 ) ∥ L p , λ ( R n ) ≤ C t n − λ p ,

where C is a positive constant only depending on the dimension n .

4 Proof of Theorem 1.1.

In this section, we will give the proof of Theorem 1.1. First, using Lemmas 3.2–3.4, we know that (II) implies (I).

Next, we show ( I ) ⇒ ( III ) and we only prove the case for b 1 . Choose f i ( x ) = χ Q ( x ) ∈ L p i , λ with any cube Q . Then, using Lemma 3.5, we have

∥ [ b → , ℳ ] ( f → ) ∥ L p , λ ≤ C ∏ i = 1 m ∥ f i ∥ L p i , λ ≤ C ∏ i = 1 m t n − λ p i = C t n − λ p = C ∣ Q ∣ n − λ n p ,

where C is a positive constant depending on the dimension n .

From [12, p. 991], we have

ℳ ( b 1 χ Q , χ Q , … , χ Q ) ( x ) = M Q ( b 1 ) ( x )

for x ∈ Q and

M Q ( b 1 ) ( x ) = sup Q 0 ∋ x , Q 0 ⊂ Q 1 ∣ Q 0 ∣ ∫ Q 0 ∣ b 1 ( x ) ∣ d x .

From [2], we know that M Q ( b i ) ≥ b i with i = 1 , 2 , … , m . Then, for f → = ( χ Q ( x ) , … , χ Q ( x ) ) , we have

1 ∣ Q ∣ λ n ∫ Q ∣ M Q ( b 1 ) ( x ) − b 1 ( x ) ∣ p d x 1 / p ≤ 1 ∣ Q ∣ λ n ∫ Q ∑ i = 1 m ∣ M Q ( b i ) ( x ) − b i ( x ) ∣ p d x 1 / p ≤ C ∥ [ b → , ℳ ] ( f → ) ∥ L p , λ ≤ C ∣ Q ∣ n − λ n p .

Thus, we obtain

1 ∣ Q ∣ ∫ Q ∣ M Q ( b 1 ) ( x ) − b 1 ( x ) ∣ p d x 1 / p ≤ C ,

which implies ( I ) ⇒ ( III ) .

Finally, ( III ) → ( II ) can be found in [2].

5 Proof of Theorem 1.3.

Proof of Theorem 1.3. Obviously, it suffices to consider ℳ c b 1 ( f → ) ( x ) . As in the above section, for any cube B = B ( x , t ) , we split each f i as f i = f i 0 + f i ∞ with f i 0 = f i χ 2 B . Thus, for any β > 0 , there is

1 ω ( B ) m ∣ { z ∈ B ( x , t ) : ℳ c b 1 ( f → ) ( z ) > β m } ∣ m ≤ 1 ω ( B ) m z ∈ B ( x , t ) : ℳ c b 1 ( f 0 → ) ( z ) > β m 4 m + 1 ω ( B ) m z ∈ B ( x , t ) : ℳ c b 1 ( f ∞ → ) ( z ) > β m 4 m + ∑ ′ 1 ω ( B ) m z ∈ B ( x , t ) : ℳ c b 1 ( f 1 α 1 , … , f m α m ) ( z ) > β m 2 m ≔ I + II + III ,

where α 1 , … , α m ∈ { 0 , ∞ } and each term in the sum ∑ ′ contains at least one α i = ∞ and α j = 0 .

For I , using [12, Theorem 1.9] and (2.2), we have

I ≤ C ∥ b 1 ∥ BMO 1 ω ( B ) m ∏ i = 1 m ∫ R n Φ ∣ f i 0 ( y i ) ∣ β d y i = C ∥ b 1 ∥ BMO ω ( 2 B ) m ω ( B ) m ∏ i = 1 m ∣ 2 B ∣ ω ( 2 B ) ⋅ 1 ∣ 2 B ∣ ∫ 2 B Φ ∣ f i ( y i ) ∣ β d y i ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m Φ ∣ f i ∣ β L L log L 1 , ω ( R n ) .

To give the estimates of II, for any Q = Q ( z , r ) with z ∈ B ( x , t ) , we show the following decomposition:

sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ b 1 ( z ) − b 1 ( y 1 ) ∣ ∣ f 1 ∞ ( y 1 ) ∣ d y 1 ∏ i = 2 m 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ d y i ≤ sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ b 1 ( z ) − ( b 1 ) B ∣ ∣ f 1 ∞ ( y 1 ) ∣ d y 1 ∏ i = 2 m 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ d y i + sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ ( b 1 ) B − b 1 ( y 1 ) ∣ ∣ f 1 ∞ ( y 1 ) ∣ d y 1 ∏ i = 2 m 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ d y i ≕ F + H .

First, we have

1 ω ( B ) m z ∈ B : F > β m 8 m ≤ 1 ω ( B ) ∫ B ∏ i = 1 m sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ d y i β 1 / m ∣ b 1 ( z ) − ( b 1 ) B ∣ 1 / m d z m .

As y i ∈ ( 2 B ) c ⋂ Q ( z , r ) and z ∈ B ( x , t ) . Following the discussion in Section 3 and using (3.2), we get

sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ β d y i ≤ C β sup r > 2 t ∫ ( 2 B ) c ⋂ Q ( z , r ) ∣ f i ( y i ) ∣ ∣ y i − z ∣ n d y i ≤ C β ∫ ( 2 B ) c ∣ f i ( y i ) ∣ ∣ y i − x ∣ n d y i ≤ C β ∑ j = 1 ∞ ∫ 2 j + 1 B ⧹ 2 j B ∣ f i ( y i ) ∣ ∣ y i − x ∣ n d y i ≤ C β ∑ j = 1 ∞ 1 ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ f i ( y i ) ∣ d y i ≤ C ∑ j = 1 ∞ 1 ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B Φ ∣ f i ( y i ) ∣ β d y i .

Moreover, as ω satisfies (1.1), there is

(5.1) ω ( 2 j + 1 B ) ω ( B ) = ω ( 2 j + 1 B ) ω ( 2 j B ) ω ( 2 j B ) ω ( 2 j − 1 B ) ⋯ ω ( 2 B ) ω ( 2 B ) ≤ C 0 j + 1 .

Recall the condition of Theorem 1.3 and the fact 1 < C 0 ≤ 2 γ with 0 < γ < n . Thus, using the Hölder inequality and (2.2), we obtain

1 ω ( B ) m z ∈ B : F > β m 8 m ≤ 1 ω ( B ) ∫ B ∣ b 1 ( z ) − ( b 1 ) B ∣ 1 / m ∏ i = 1 m ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B Φ ∣ f i ( y i ) ∣ β d y i 1 / m d z m ≤ C 1 ∣ B ∣ ∫ B ∣ b 1 ( z ) − ( b 1 ) B ∣ d z ∏ i = 1 m ∑ j = 1 ∞ ∣ B ∣ ∣ 2 j + 1 B ∣ ω ( B ) ∫ 2 j + 1 B Φ ∣ f i ( y i ) ∣ β d y i 1 / m m ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m ∑ j = 1 ∞ ∣ B ∣ ω ( 2 j + 1 B ) ∣ 2 j + 1 B ∣ ω ( B ) ⋅ ∣ 2 j + 1 B ∣ ω ( 2 j + 1 B ) ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B Φ ∣ f i ( y i ) ∣ β d y i 1 / m m ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m ∑ j = 1 ∞ ∣ B ∣ ω ( 2 j + 1 B ) ∣ 2 j + 1 B ∣ ω ( B ) Φ ∣ f i ∣ β L log L , 2 j + 1 B 1 / m m ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m Φ ∣ f i ∣ β L L log L 1 , ω ( R n ) ,

where C is a positive constant depending on γ and the dimension n .

For H , we have

1 ω ( B ) m z ∈ B : H > β m 4 m ≤ C 1 ω ( B ) ∫ B sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ ( b 1 ) B − b 1 ( y i ) ∣ ∣ f 1 ∞ ( y 1 ) ∣ β d y 1 1 / m × ∏ i = 2 m sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ β d y i 1 / m d z m .

Using (3.2) again, we know that there exists a positive constant C depending on the dimension n , such that

sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ ( b 1 ) B − b 1 ( y i ) ∣ ∣ f 1 ∞ ( y 1 ) ∣ β d y 1 1 / m ≤ ∫ ( 2 B ) c ∣ ( b 1 ) B − b 1 ( y 1 ) ∣ ∣ f 1 ( y 1 ) ∣ β ∣ x − y 1 ∣ n d y 1 1 / m ≤ ∑ j = 1 m ∫ 2 j + 1 B ⧹ 2 j B ∣ b 1 ( y 1 ) − ( b 1 ) B ∣ ∣ f 1 ( y 1 ) ∣ β ∣ x − y 1 ∣ n d y 1 1 / m ≤ ∑ j = 1 m C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ⧹ 2 j B ∣ b 1 ( y 1 ) − ( b 1 ) B ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m ≤ ∑ j = 1 m C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ⧹ 2 j B ∣ ( b 1 ) 2 j + 1 B − ( b 1 ) B ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m + ∑ j = 1 m C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ⧹ 2 j B ∣ ( b 1 ) 2 j + 1 B − b 1 ( y 1 ) ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m .

Thus, we obtain

1 ω ( B ) m z ∈ B : H > β m 4 m ≤ H 1 + H 2 ,

where

H 1 = 1 ω ( B ) ∫ B ∑ j = 1 m C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ⧹ 2 j B ∣ ( b 1 ) 2 j + 1 B − ( b 1 ) B ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m ∏ i = 2 m sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ β d y i 1 / m d z m

and

H 2 = 1 ω ( B ) ∫ B ∑ j = 1 m C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ⧹ 2 j B ∣ ( b 1 ) 2 j + 1 B − b 1 ( y 1 ) ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m ∏ i = 2 m sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ β d y i 1 / m d z m .

For H 1 , using Lemma 3.1, we have

H 1 ≤ 1 ω ( B ) ∣ B ∣ ∏ i = 2 m sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ∞ ( y i ) ∣ β d y i 1 / m × ∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ ( b 1 ) B − ( b 1 ) 2 j + 1 B ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m m ≤ C ω ( B ) ∣ B ∣ ∏ i = 2 m ∑ j = 1 ∞ 1 ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B Φ ∣ f i ( y i ) ∣ β d y i 1 / m × ∑ j = 1 ∞ 1 ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ ( b 1 ) B − ( b 1 ) 2 j + 1 B ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m m ≤ C ∏ i = 2 m ∑ j = 1 ∞ ∣ B ∣ ω ( 2 j + 1 B ) ∣ 2 j + 1 B ∣ ω ( B ) ⋅ ∣ 2 j + 1 B ∣ ω ( 2 j + 1 B ) ⋅ 1 ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B Φ ∣ f i ( y i ) ∣ β d y i 1 / m × ∑ j = 1 ∞ ω ( 2 j + 1 B ) ( j + 1 ) ∣ B ∣ ω ( 2 j + 1 B ) ω ( B ) ∣ 2 j + 1 B ∣ ∥ b 1 ∥ BMO ∫ 2 j + 1 B ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m m ≤ C ∥ b 1 ∥ BMO ∏ i = 2 m ∑ j = 1 ∞ ∣ B ∣ ω ( 2 j + 1 B ) ∣ 2 j + 1 B ∣ ω ( B ) ⋅ ∣ 2 j + 1 B ∣ ω ( 2 j + 1 B ) Φ ∣ f i ∣ β L log L , 2 j + 1 B 1 / m × ∑ j = 1 ∞ ω ( 2 j + 1 B ) ( j + 1 ) ∣ B ∣ ω ( B ) ∣ 2 j + 1 B ∣ ∣ 2 j + 1 B ∣ ω ( 2 j + 1 B ) 1 ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B Φ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m m ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m Φ ∣ f i ∣ β L L log L 1 , ω ( R n ) ,

where C is a positive constant depending on γ and the dimension n .

For H 2 , there is

H 2 ≤ ∣ B ∣ ω ( B ) ∏ i = 2 m sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i 0 ( y i ) ∣ β d y i 1 / m ∑ j = 1 m C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ b 1 ( y 1 ) − ( b 1 ) 2 j + 1 B ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 1 / m m .

Then, using (2.3), we have

∑ j = 1 ∞ C ∣ 2 j + 1 B ∣ ∫ 2 j + 1 B ∣ b 1 ( y 1 ) − ( b 1 ) 2 j + 1 B ∣ ∣ f 1 ( y 1 ) ∣ β d y 1 ≤ C ∑ j = 1 ∞ ∥ b 1 ( ⋅ ) − ( b 1 ) 2 j + 1 B ∥ exp L , 2 j + 1 B Φ ∣ f 1 ∣ β L log L , 2 j + 1 B ≤ C ∥ b 1 ∥ BMO ∑ j = 1 ∞ Φ ∣ f 1 ∣ β L log L , 2 j + 1 B .

Thus, we obtain

H 2 ≤ C ∥ b 1 ∥ BMO ∣ B ∣ ω ( B ) ∏ i = 2 m sup r > 0 1 ∣ Q ( z , r ) ∣ ∫ Q ( z , r ) ∣ f i ( y i ) ∣ β d y i 1 / m ∑ j = 1 ∞ Φ ∣ f 1 ∣ β L log L , 2 j + 1 B 1 / m m ≤ C ∥ b 1 ∥ BMO ∏ i = 2 m ∣ B ∣ ω ( B ) 1 ∣ Q ∣ ∫ Q ∣ f i ( y i ) ∣ β d y i 1 / m ∑ j = 1 ∞ ω ( 2 j + 1 B ) ∣ B ∣ ∣ 2 j + 1 B ∣ ∣ ω ( B ) ∣ ∣ 2 j + 1 B ∣ ω ( 2 j + 1 B ) Φ ∣ f 1 ∣ β L log L , 2 j + 1 B 1 / m m .

Similar to the estimates of H 1 , it is easy to see

H 2 ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m Φ ∣ f i ∣ β L L log L 1 , ω ( R n ) ,

where C is a positive constant depending on γ and the dimension n .

Combining the estimates of H 1 and H 2 , we get

1 ω ( B ) m z ∈ B : H > β m 4 m ≤ C ∥ b 1 ∥ BMO ∏ i = 1 m Φ ∣ f i ∣ β L L log L 1 , ω ( R n ) .

For III, without loss of generality, we only consider α 1 = α 2 = 0 and α 3 = α 4 = α 5 = ⋯ = α m = ∞ .

Recall the definition of M b ( f ) ( x ) and M ( f ) ( x ) . Then, using the Hölder inequality for weak L p space (see [19, p. 16]) and (2.4) with m = 1 , we obtain

1 ω ( B ) m z ∈ B : sup r > 0 1 ∣ Q ∣ ∫ Q ∣ b 1 ( y 1 ) − b 1 ( z ) ∣ ∣ f 1 0 ( y 1 ) ∣ d y 1 1 ∣ Q ∣ ∫ Q ∣ f 2 0 ( y 2 ) ∣ d y 2 × 1 ∣ Q ∣ ∫ Q ∣ f 3 ∞ ( y 3 ) ∣ d y 3 1 ∣ Q ∣ ∫ Q ∣ f 4 ∞ ( y 4 ) ∣ d y 4 ⋯ 1 ∣ Q ∣ ∫ Q ∣ f m ∞ ( y m ) ∣ d y m > β m m ≤ 1 ω ( B ) m ∣ { z ∈ B : M b 1 ( f 1 0 ) M ( f 2 0 ) M ( f 3 ∞ ) ⋯ M ( f m ∞ ) > β m } ∣ m ≤ 1 ω ( B ) m 1 β m ∥ M b 1 ( f 1 0 ) M ( f 2 0 ) M ( f 3 ∞ ) ⋯ M ( f m ∞ ) ∥ L 1 m , ∞ ( B ) ≤ C 1 ω ( B ) m 1 β m ∥ M b 1 ( f 1 0 ) ∥ L 1 , ∞ ( B ) ∥ M ( f 2 0 ) ∥ L 1 , ∞ ( B ) ∥ M ( f 3 ∞ ) ⋯ M ( f m ∞ ) ∥ L 1 m − 2 , ∞ ( B ) .

According to the endpoint estimates of M b (see [1]), (2.2) and the definition of L L log L 1 , ω ( R n ) , it is easy to get

1 ω ( B ) β ∥ M b 1 ( f 1 0 ) ∥ L 1 , ∞ ( B ) ≤ C ∥ b 1 ∥ BMO 1 ω ( B ) β ∫ 2 B Φ ( ∣ f 1 ∣ ) d y 1 ≤ C ∥ b 1 ∥ BMO Φ ∣ f 1 ∣ β L L log L 1 , ω ( R n ) .

For ∥ M ( f 2 0 ) ∥ L 1 , ∞ ( B ) , noting the fact M ( f ) ( x ) is bounded from L 1 ( R n ) to L 1 , ∞ ( R n ) , it is easy to see

1 ω ( B ) β ∥ M ( f 2 0 ) ∥ L 1 , ∞ ( B ) ≤ C Φ ∣ f 2 ∣ β L L log L 1 , ω ( R n ) .

Finally, using the fact L 1 ( B ) ↪ L 1 , ∞ ( B ) and adopting some similar estimates in the proof of II, we can easily obtain

1 ω ( B ) m − 2 β m − 2 ∥ M ( f 3 ∞ ) ⋯ M ( f m ∞ ) ∥ L 1 m − 2 , ∞ ( B ) ≤ C ∏ i = 3 m Φ ∣ f i ∣ β L L log L 1 , ω ( R n ) .

Combining the estimates of I, II and III, we finish the proof of Theorem 1.3.

Acknowledgment

The authors would like to express their gratitude to the anonymous referee for his/her valuable suggestions which improve the quality of this paper a lot.

  1. Funding information: Xiao Yu was supported by the National Natural Science Foundation of China (No. 11961056), the Natural Science Foundation of Jiangxi Province (No. 20192BAB201004) and the Science and Technology Project of Jiangxi Provincial Department of Education (No. GJJ190890). Pu Zhang was supported by the National Natural Science Foundation of China (No. 11571160).

  2. Author contributions: Yu wrote the first version and finished the revised version of this paper. Zhang proposed the problem of Theorem 1.1 and gave some useful ideas. Li pointed out several mistakes in the first version of this paper and gave some useful suggestions. All authors read and approved the final manuscript.

  3. Conflict of interest: Authors state no conflict of interest.

References

[1] M. Agcayazi , A. Gogatishvili , K. Koca , and R. Mustafayev , A note on maximal commutators and commutators of maximal functions, J. Math. Soc. Japan 67 (2015), no. 2, 581–593, https://doi.org/10.2969/jmsj/06720581. Search in Google Scholar

[2] J. Bastero , M. Milman , and F. J. Ruiz , Commutators for the maximal and sharp functions, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3329–3334. 10.1090/S0002-9939-00-05763-4Search in Google Scholar

[3] G. Hu , H. Lin , and D. Yang , Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal. 2008 (2008), 237937, https://doi.org/10.1155/2008/237937. Search in Google Scholar

[4] C. Xie , Some estimates of commutators, Real. Anal. Exch. 36 (2010/2011), no. 2, 405–416, https://doi.org/10.14321/realanalexch.36.2.0405 . 10.14321/realanalexch.36.2.0405Search in Google Scholar

[5] C. B. Morrey , On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166, https://doi.org/10.1090/S0002-9947-1938-1501936-8. Search in Google Scholar

[6] H. Triebel , Hybrid Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics, Switzerland, 2015. 10.4171/150Search in Google Scholar

[7] F. Chiarenza and F. Frasca , Morrey spaces and Hardy-Littlewood maximal function, Rend. Math. Appl. 7 (1987), no. 3–4, 273–279. Search in Google Scholar

[8] E. Nakai , Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95–103, https://doi.org/10.1002/mana.19941660108. Search in Google Scholar

[9] L. Grafakos and R. Torres , Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164, https://doi.org/10.1006/aima.2001.2028. Search in Google Scholar

[10] A. K. Lerner , S. Ombrosi , C. Pérez , R. H. Torres , and R. Trujillo-González , New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222–1264, https://doi.org/10.1016/j.aim.2008.10.014 . 10.1016/j.aim.2008.10.014Search in Google Scholar

[11] C. Pérez and R. H. Torres , Sharp maximal function estimates for multilinear singular integrals, Contemp. Math. 320 (2003), 323–331, https://doi.org/10.1090/conm/320/05615. Search in Google Scholar

[12] P. Zhang , Multiple weighted estimates for commutators of multilinear maximal function, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 6, 973–994, https://doi.org/10.1007/s10114-015-4293-6. Search in Google Scholar

[13] C. Pérez , Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), no. 1, 163–185, https://doi.org/10.1006/jfan.1995.1027. Search in Google Scholar

[14] S. Chen and H. Wu , Multiple weighted estimates for commutators of multilinear fractional integral operators, Sci. China Math. 56 (2013), no. 9, 1879–1894, https://doi.org/10.1007/s11425-013-4607-z. Search in Google Scholar

[15] Y. Ding , S. Lu , and P. Zhang , Weak estimates for commutators of fractional integral operators, Sci. China Ser. A-Math. 44 (2001), no. 7, 877–888, https://doi.org/10.1007/BF02880137. Search in Google Scholar

[16] H. Wang , Weighted inequalities for fractional integral operators and linear commutators in the Morrey-type spaces, J. Inequal. Appl. 2017 (2017), 6, https://doi.org/10.1186/s13660-016-1279-6. Search in Google Scholar PubMed PubMed Central

[17] M. M. Rao and Z. D. Ren , Theory of Orlicz Spaces, Marcel Dekker, New York, 1991. Search in Google Scholar

[18] R. O’Neil , Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc. 115 (1965), 300–328, https://doi.org/10.1090/S0002-9947-1965-0194881-0 . 10.1090/S0002-9947-1965-0194881-0Search in Google Scholar

[19] L. Grafakos , Classical and Fourier Analysis, Springer, Prentice Hall, New York, 2009. 10.1007/978-0-387-09432-8Search in Google Scholar

[20] L. Softova , Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 757–766, https://doi.org/10.1007/s10114-005-0628-z. Search in Google Scholar

[21] L. Softova , Morrey-type regularity of solutions to parabolic problems with discontinuous data, Manuscripta Math. 136 (2011), no. 3–4, 365–382, https://doi.org/10.1007/s00229-011-0447-8. Search in Google Scholar

[22] Y. Komori and T. Mizuhara , Notes on commutators and Morrey spaces, Hokkaido Math. J. 32 (2003), no. 2, 345–353, https://doi.org/10.14492/hokmj/1350657528. Search in Google Scholar
Received: 2020-10-25
Revised: 2021-01-22
Accepted: 2021-01-27
Published Online: 2021-07-03

© 2021 Xiao Yu et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0031
Keywords for this article
Morrey-type space; multilinear maximal function; commutator; Orlicz function
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  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
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  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
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  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
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  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
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  38. Inviscid, zero Froude number limit of the viscous shallow water system
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  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
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  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
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  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
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  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
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  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
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  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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