Some estimates for commutators of Littlewood-Paley g-functions

Guanghui Lu

doi:10.1515/math-2021-0051

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Some estimates for commutators of Littlewood-Paley g-functions

Guanghui Lu

Published/Copyright: August 27, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

The aim of this paper is to establish the boundedness of commutator [ b , g ˙ r ] generated by Littlewood-Paley g -functions g ˙ r and b ∈ RBMO ( μ ) on non-homogeneous metric measure space. Under assumption that λ satisfies ε -weak reverse doubling condition, the author proves that [ b , g ˙ r ] is bounded from Lebesgue spaces L p ( μ ) into Lebesgue spaces L p ( μ ) for p ∈ ( 1 , ∞ ) and also bounded from spaces L 1 ( μ ) into spaces L 1 , ∞ ( μ ) . Furthermore, the boundedness of [ b , g ˙ r ] on Morrey space M q p ( μ ) and on generalized Morrey L p , ϕ ( μ ) is obtained.

Keywords: non-homogeneous metric measure space; Littlewood-Paley g-function; commutator; space RBMO(μ); generalized Morrey space

MSC 2010: 42B25; 42B35; 30L99

1 Introduction

Although the metric measure spaces equipped with the polynomial growth conditions (see [1,2,3, 4,5]) and spaces of homogeneous type in the sense of Coifman and Weiss [6,7] are two important classes of function spaces in harmonic analysis, there exist no relations between the non-doubling measure spaces and spaces of homogeneous type. To solve this problem, Hytönen [8] first introduced the non-homogeneous metric measure spaces satisfying the so-called geometrically doubling and upper doubling conditions. From then on, many papers focus on the properties of function spaces and operators over non-homogeneous metric measure spaces. For example, Cao and Zhou [9] obtained the definition of Morrey space on non-homogeneous metric measure space and also proved that Hardy-Littlewood maximal operator, Calderón-Zygmund operator and fractional integral are bounded on Morrey space. Fu and Zhao [10] proved that generalized homogeneous Littlewood-Paley g -function is bounded from atomic Hardy space H atb 1 ( μ ) into space L 1 ( μ ) and also bounded from space RBMO ( μ ) into space RBLO ( μ ) . For more development of harmonic analysis on non-homogeneous metric measure space the readers can see [11,12,13, 14,15,16, 17,18,19] and references therein.

Let ( X , d , μ ) be a non-homogeneous metric measure space in the sense of Hytönen [8]. In this setting, the author proves that the commutator [ b , g ˙ r ] generated by b ∈ RBMO ( μ ) and generalized homogeneous Littlewood-Paley g -function g ˙ r is bounded on Lebesgue spaces L p ( μ ) for p ∈ ( 1 , ∞ ) and also bounded from spaces L 1 ( μ ) into spaces L 1 , ∞ ( μ ) . Furthermore, the boundedness of [ b , g ˙ r ] on Morrey space M q p ( μ ) and on generalized Morrey space L p , ϕ ( μ ) is also established in this paper. In 2016, Fu and Zhao obtained the boundedness of g ˙ r on atomic Hardy space H atb 1 ( μ ) and on space RBMO ( μ ) (see [10]). In 2021, Lu and Tao [11] proved that the g ˙ r is bounded on Lipschitz space Lip β ( μ ) and on generalized Morrey space L p , ϕ ( μ ) .

Before stating the main results of this paper, we first recall some necessary notions. The following definitions of geometrically doubling and upper doubling conditions are from [8].

Definition 1.1

A metric space ( X , d ) is said to be geometrically doubling if there exists some N 0 ∈ N such that, for any ball B ( x , r ) ⊂ X with x ∈ X and r ∈ ( 0 , ∞ ) , there exists a finite ball covering { B ( x i , r / 2 ) } i of B ( x , r ) such that the cardinality of this covering is at most N 0 .

Remark 1.2

Let ( X , d ) be a metric measure. Hytönen [8] showed that the geometrically doubling ( X , d ) is equivalent to the following statement: for any ε ∈ ( 0 , 1 ) and any ball B ( x , r ) ⊂ X with x ∈ X and r ∈ ( 0 , ∞ ) , there exists a finite ball covering { B ( x i , ε r ) } i of B ( x , r ) such that the cardinality of this covering is at most ε − n 0 , where n 0 ≔ log 2 N 0 .

Definition 1.3

A metric measure space ( X , d , μ ) is said to be upper doubling if μ is a Borel measure on X and there exist a dominating function λ : X × ( 0 , ∞ ) → ( 0 , ∞ ) and constant C ( λ ) > 0 , depending on λ , such that, for each x ∈ X , r → λ ( x , r ) is non-decreasing and for all x ∈ X and r ∈ ( 0 , ∞ ) ,

(1.1) μ ( B ( x , r ) ) ≤ λ ( x , r ) ≤ C ( λ ) λ ( x , r / 2 ) .

Moreover, Hytönen et al. [12] showed that there exists another dominating function λ ˜ such that λ ˜ ≤ λ , C ( λ ˜ ) ≤ C ( λ ) and, for all x , y ∈ X with d ( x , y ) ≤ r ,

(1.2) λ ˜ ( x , r ) ≤ C ( λ ) λ ˜ ( y , r ) .

Here and in what follows, we always assume that λ satisfies (1.2).

Although the doubling measure condition is not assumed uniformly for all balls on ( X , d , μ ) , Hytönen [8] showed that there exist many balls satisfying the ( α , β ) -doubling condition, i.e., let α , β ∈ ( 1 , ∞ ) , a ball B ⊂ X is said to be ( α , β ) -doubling if μ ( α B ) ≤ β μ ( B ) . Throughout this paper, for any α ∈ ( 1 , ∞ ) and ball B , the smallest ( α , β α ) -doubling ball of the form α j B with j ∈ N is denoted by B ˜ α , where

β α ≔ α 3 ( max { n , ν } ) + 3 0 n + 3 0 ν .

We always denote by B ˜ the smallest ( 6 , β 6 ) -doubling ball of the form B ˜ 6 in this paper.

We now recall the definition of coefficient K B , S introduced in [8], which is very close to the quantity K Q , R introduced by Tolsa [3], that is, for any two balls B ⊂ S , define

(1.3) K B , S ≔ 1 + ∫ ( 2 S ) ⧹ B 1 λ ( c ( B ) , d ( x , c ( B ) ) ) d μ ( x ) ,

where c ( B ) represents the center of ball B . For more properties of the K B , S , we can see [12, Lemma 2.1].

The following regularized bounded mean oscillation (RBMO) space was from [8].

Definition 1.4

Let ρ > 1 . A function f ∈ L loc 1 ( μ ) is said to be in the space RBMO ( μ ) if there exist a positive constant C and, for any ball B , a number f B such that

(1.4) 1 μ ( ρ B ) ∫ B ∣ f ( y ) − f B ∣ d μ ( y ) ≤ C

and, for any two balls B and S such that B ⊂ S ,

(1.5) ∣ f B − f S ∣ ≤ C K B , S .

The infimum of the positive constants C satisfying both (1.4) and (1.5) is defined to be the RBMO ( μ ) norm of f and denoted by ‖ f ‖ RBMO ( μ ) . Moreover, Hytönen [8] also showed that the space RBMO ( μ ) is independent of the choice of the constant ρ ∈ ( 1 , ∞ ) .

In 2015, Tan and Li [13] gave an approximation of the identity S ≔ { S k } k ∈ Z associated with ( 2 , 2 ( C λ + 1 ) ) -doubling balls { Q x , k } x ∈ supp ( μ ) , k ∈ Z on ( X , d , μ ) , which are integral operators associated with kernels S k ( x , y ) on X × X satisfying the following conditions:

S k ( x , y ) = S k ( y , x ) for all x , y ∈ supp ( μ ) .
For any k ∈ Z and x ∈ supp ( μ ) , S k 1 ( x ) = 1 = S k ∗ 1 ( x ) , where S k ∗ is the adjoint operator of S k .
For each k ∈ Z and x ∈ supp ( μ ) , supp ( S k ( x , ⋅ ) ) ⊂ Q x , k − 1 .
For all x , y ∈ X and k ∈ Z , if x ≠ y or Q x , k ≠ { x } , then there exists a non-negative constant C such that
0 ≤ S k ( x , y ) ≤ C λ ( x , r ( Q x , k ) + r ( Q y , k ) + d ( x , y ) ) .
For all x , x ˜ , y ∈ X and k ∈ Z , if x , x ˜ ∈ Q x 0 , k for some x 0 ∈ supp ( μ ) and x ≠ x ˜ , then there exists a positive constant C such that
∣ S k ( x , y ) − S k ( x ˜ , y ) ∣ ≤ d ( x , x ˜ ) r ( Q x 0 , k ) ε C λ ( x , r ( Q x , k ) + r ( Q y , k ) + d ( x , y ) ) ,
where ε ∈ ( 0 , ∞ ) and Q x , k represents a fixed doubling ball center as x of generation k .

Moreover, Tan and Li [13] showed that the aforementioned results through (A-1) to (A-5) are still correct if the ( 2 , 2 ( C λ + 1 ) ) -doubling balls are replaced by ( 6 , β 6 ) -doubling balls.

We now recall the definition of generalized homogeneous Littlewood-Paley g -function introduced in [10].

Definition 1.5

Let k ∈ Z , r ∈ [ 2 , ∞ ) , D k ( x , y ) ≔ S k ( x , y ) − S k − 1 ( x , y ) for all x , y ∈ X , and D k be the corresponding integral operator associated with the kernel D k ( x , y ) . Then, the generalized homogeneous Littlewood-Paley g -function g ˙ r is defined by

(1.6) g ˙ r ( f ) ( x ) ≔ ∑ k ∈ Z ∣ D k ( x , ⋅ ) f ( x ) ∣ r 1 r , for any x ∈ X .

Given a function b ∈ RBMO ( μ ) , the commutator [ b , g ˙ r ] , which is generated by b and g ˙ r as in (1.6), is defined by

(1.7) [ b , g ˙ r ] ( x ) ≔ ∑ k ∈ Z ∣ D k ( x , ⋅ ) ( b ( x ) − b ( ⋅ ) ) f ( x ) ∣ r 1 r ,

where x ∈ X and f ∈ L b ∞ ( μ ) being the space of all L ∞ ( μ ) functions with bounded support.

The following definition of generalized Morrey space is from [14].

Definition 1.6

Let k > 1 and 1 < p < ∞ . Suppose that ϕ is an increasing function on ( 0 , ∞ ) . Then, the generalized Morrey space L p , ϕ ( μ ) is defined by

L p , ϕ ( μ ) ≔ { f ∈ L loc p ( μ ) : ‖ f ‖ L p , ϕ ( μ ) < ∞ } ,

where

(1.8) ‖ f ‖ L p , ϕ ( μ ) = sup B [ ϕ ( μ ( k B ) ) ] − 1 p ∫ B ∣ f ( x ) ∣ p d μ ( x ) 1 p .

Remark 1.7

From [14], Lu and Tao showed that the norm ‖ f ‖ M q p ( μ ) is independent of the choice of parameter k for k > 1 .
If we take ϕ ( t ) = t 1 − q p with t > 0 and 1 < q ≤ p < ∞ , then L p , ϕ ( μ ) defined as in (1.8) is just the Morrey space M q p ( μ ) , which was introduced by Cao and Zhou [9], that is, let k > 1 and 1 < q ≤ p < ∞ , then Morrey space M q p ( μ ) is defined by
M q p ( μ ) = { f ∈ L loc q ( μ ) : ‖ f ‖ M q p ( μ ) < ∞ } ,
where
(1.9) ‖ f ‖ M q p ( μ ) ≔ sup B [ μ ( k B ) ] 1 p − 1 q ∫ B ∣ f ( y ) ∣ q d μ ( y ) 1 q .

We now recall the following ε -weak reverse doubling condition introduced in [16].

Definition 1.8

Let ε ∈ ( 0 , ∞ ) . A dominating function λ is said to satisfy the ε -weak reverse doubling condition if, for all r ∈ ( 0 , 2 diam ( X ) ) and a ∈ ( 1 , 2 diam ( X ) / r ) , there exists a number C ( a ) ∈ [ 1 , ∞ ) , depending only on a and X , such that, for all x ∈ X ,

λ ( x , a r ) ≥ C ( a ) λ ( x , r )

and, moreover,

(1.10) ∑ k = 1 ∞ 1 [ C ( a k ) ] ε < ∞ .

The main theorems of this paper are stated as follows:

Theorem 1.9

Let b ∈ RBMO ( μ ) and r ∈ [ 2 , ∞ ) . Suppose that λ satisfies the ε -weak reverse doubling condition defined as in Definition 1.8. Then, there exists a constant C > 0 such that, for all f ∈ L p ( μ ) with p ∈ ( 1 , ∞ ) ,

‖ [ b , g ˙ r ] ( f ) ‖ L p ( μ ) ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ L p ( μ ) .

Theorem 1.10

‖ [ b , g ˙ r ] ( f ) ‖ L 1 , ∞ ( μ ) ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ L 1 ( μ ) .

Theorem 1.11

Let b ∈ RBMO ( μ ) , r ∈ [ 2 , ∞ ) and 1 < q ≤ p < ∞ . Suppose that λ satisfies the weak reverse doubling condition defined as in Definition 1.8. Then, there exists a constant C > 0 such that, for all f ∈ M q p ( μ ) ,

‖ [ b , g ˙ r ] ( f ) ‖ M q p ( μ ) ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ M q p ( μ ) .

Theorem 1.12

Let b ∈ RBMO ( μ ) , r ∈ [ 2 , ∞ ) and p ∈ ( 1 , ∞ ) . Suppose that ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) is an increasing function and the mapping s ↦ ϕ ( s ) s is almost decreasing, namely, there is a constant C > 0 such that, the following inequality

(1.11) ϕ ( w ) w ≤ C ϕ ( s ) s

holds for s ≥ w . Then, there exists a positive constant C such that, for all f ∈ L p , ϕ ( μ ) ,

‖ [ b , g ˙ r ] ( f ) ‖ L p , ϕ ( μ ) ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ L p , ϕ ( μ ) .

Finally, we make some conventions on notation. Throughout the paper, C represents a positive constant, which is independent of the main parameters. For any subset E of X , we use χ E to denote its characteristic function. Given any q ∈ ( 1 , ∞ ) , let q ′ ≔ q / ( q − 1 ) denote its conjugate index. For any ball B , c ( B ) and r ( B ) represent the center and radius of ball B , respectively. Furthermore, m B ( f ) denotes the mean value of function f over ball B , that is, m B ( f ) = 1 μ ( B ) ∫ B f ( y ) d μ ( y ) .

2 Preliminaries

To prove the main theorems of this paper, we should recall some necessary results in this section (see [10,14,16,17]).

Lemma 2.1

Let ( X , d , μ ) be a non-homogeneous metric measure space satisfying the weak reverse doubling condition, r ∈ [ 2 , ∞ ) and p ∈ ( 1 , ∞ ) . Then, there exists a constant C > 0 such that, for all f ∈ L p ( μ ) ,

(2.1) ‖ g ˙ r ( f ) ‖ L p ( μ ) ≤ C ‖ f ‖ L p ( μ ) .

Corollary 2.2

If f ∈ RBMO ( μ ) , then there exists a constant C > 0 such that, for any ball B , τ ∈ ( 1 , ∞ ) and s ∈ [ 1 , ∞ ) ,

(2.2) 1 μ ( τ B ) ∫ B ∣ f ( y ) − f B ∣ s d μ ( y ) 1 s ≤ C ‖ f ‖ RBMO ( μ ) .

Lemma 2.3

Let p ∈ ( 1 , ∞ ) , t ∈ ( 1 , p ) and ρ ∈ [ 5 , ∞ ) . The following maximal operators, respectively, defined for all f ∈ L loc 1 ( μ ) and x ∈ X ,
(2.3) M t , ρ f ( x ) ≔ sup B ∋ x 1 μ ( ρ B ) ∫ B ∣ f ( y ) ∣ t d μ ( y ) 1 t
and
N f ( x ) ≔ sup B ∋ x B doubling 1 μ ( B ) ∫ B ∣ f ( y ) ∣ d μ ( y )
are bounded on L p ( μ ) and also bounded from L 1 ( μ ) into L 1 , ∞ ( μ ) .
For all f ∈ L loc 1 ( μ ) , it holds true that ∣ f ( x ) ∣ ≤ N f ( x ) for μ -a.e x ∈ X .

Lemma 2.4

Let f ∈ L loc 1 ( μ ) satisfy ∫ X f ( x ) d μ ( x ) = 0 when ‖ μ ‖ ≔ μ ( x ) < ∞ . Assume that, for some p , q satisfying 1 < q ≤ p < ∞ , inf { 1 , N f } ∈ M q p ( μ ) . Then, there exists a constant C > 0 ,

(2.4) ‖ N f ‖ M q p ( μ ) ≤ C ‖ M ♯ f ‖ M q p ( μ ) ,

where the sharp maximal operator M ♯ is defined by, for all f ∈ L loc 1 ( μ ) and x ∈ X ,

(2.5) M ♯ f ( x ) ≔ sup B ∋ x 1 μ ( 6 B ) ∫ B ∣ f ( y ) − m B ˜ ( f ) ∣ d μ ( y ) + sup x ∈ B ⊂ S B , S ( 6 , β 6 ) - doubling ∣ m B ( f ) − m S ( f ) ∣ K B , S .

Lemma 2.5

Let ρ > 1 , 1 < t < p < ∞ and ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) be an increasing function. Suppose that M t , ρ is defined as in (2.3) and the mapping s ↦ ϕ ( s ) s satisfies the condition (1.11). Then, there exists a positive constant C > 0 such that, for all f ∈ L p , ϕ ( μ ) ,

‖ M t , ρ f ‖ L p , ϕ ( μ ) ≤ C ‖ f ‖ L p , ϕ ( μ ) .

3 Proof of Theorems 1.9–1.12

Proof of Theorem 1.9

For any r ∈ [ 2 , ∞ ) , p ∈ ( 1 , ∞ ) , f ∈ L p ( μ ) and x ∈ X , we first claim that

(3.1) M ♯ ( [ b , g ˙ r ] ( f ) ) ( x ) ≤ C ‖ b ‖ RBMO ( μ ) { M r 2 , 6 ( f ) ( x ) + M r , 6 ( f ) ( x ) + M r , 6 ( g ˙ r ( f ) ) ( x ) } .

Once (3.1) is proved, by applying Lemma 2.3 and Theorem 4.2 in [15], we have

‖ [ b , g ˙ r ] ( f ) ‖ L p ( μ ) ≤ ‖ N ( [ b , g ˙ r ] ( f ) ) ‖ L p ( μ ) ≤ C ‖ M ♯ ( [ b , g ˙ r ] ( f ) ) ‖ L p ( μ ) ≤ C ‖ b ‖ RBMO ( μ ) { ‖ M r , 6 ( g ˙ r ( f ) ) ‖ L p ( μ ) + ‖ M r 2 , 6 ( f ) ‖ L p ( μ ) + ‖ M r , 6 ( f ) ‖ L p ( μ ) } ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ L p ( μ ) ,

which is just the desired conclusion.

To show (3.1), by the definition of sharp maximal function M ♯ , there exists a family of numbers { b B } B such that, for all x and B with B ∋ x ,

(3.2) 1 μ ( 6 B ) ∫ B ∣ g ˙ r , b ( f ) ( x ) − h B ∣ d μ ( x ) ≤ C ‖ b ‖ RBMO ( μ ) { M r 2 , 6 ( f ) ( x ) + M r , 6 ( f ) ( x ) + M r , 6 ( g ˙ r ( f ) ) ( x ) }

and, for all balls B , S satisfying B ⊂ S and B ∋ x ,

(3.3) ∣ h B − h S ∣ ≤ C K B , S ‖ b ‖ RBMO ( μ ) { M r , 6 ( g ˙ r ( f ) ) ( x ) + M r 2 , 6 ( f ) ( x ) + M r , 6 ( f ) ( x ) } ,

where

h B ≔ m B ( g ˙ r ( ( b − b B ) f χ X ⧹ 6 B ) ) and h S ≔ m S ( g ˙ r ( ( b − b S ) f χ X ⧹ 6 S ) ) .

To prove (3.2), for a fixed ball B and x with x ∈ B , write

[ b , g ˙ r ] ( f ) ( x ) ≤ C ∣ b ( x ) − b B ∣ ∑ k ∈ Z ∣ D k f ( x ) ∣ r 1 r + C ∑ k ∈ Z ∣ D k ( b ( ⋅ ) − b B ) f 1 ( x ) ∣ r 1 r + C ∑ k ∈ Z ∣ D k ( b ( ⋅ ) − b B ) f 2 ( x ) ∣ r 1 r ≤ C { ∣ b ( x ) − b B ∣ g ˙ r ( f ) ( x ) + g ˙ r ( ( b ( ⋅ ) − b B ) f 1 ) ( x ) + g ˙ r ( ( b ( ⋅ ) − b B ) f 2 ) ( x ) } ,

where f 1 ≔ f χ 6 B and f 2 ≔ f − f 1 .

By applying Hölder inequality, (2.1) and (2.2), we obtain that

(3.4) 1 μ ( 6 B ) ∫ B ∣ b ( x ) − b B ∣ ∣ g ˙ r ( f ) ( x ) ∣ d μ ( x ) ≤ 1 μ ( 6 B ) ∫ B ∣ g ˙ r ( f ) ( x ) ∣ r d μ ( x ) 1 r 1 μ ( 6 B ) ∫ B ∣ b ( x ) − b B ∣ r ′ d μ ( x ) 1 r ′ ≤ C ‖ b ‖ RBMO ( μ ) M r , 6 ( g ˙ r ( f ) ) ( x ) .

To estimate g ˙ r ( ( b ( ⋅ ) − b B ) f 1 ) , take s ≔ r . From Hölder inequality, (2.1) and (2.2), it follows that

(3.5) 1 μ ( 6 B ) ∫ B ∣ g ˙ r ( ( b ( ⋅ ) − b B ) f 1 ) ( x ) ∣ d μ ( x ) ≤ 1 [ μ ( 6 B ) ] 1 s ∫ B ∣ g ˙ r ( ( b ( ⋅ ) − b B ) f 1 ) ( x ) ∣ s d μ ( x ) 1 s ≤ C [ μ ( 6 B ) ] 1 s ‖ ( b ( ⋅ ) − b B ) f 1 ‖ L s ( μ ) ≤ C 1 μ ( 6 B ) ∫ 6 B ∣ b ( y ) − b B ∣ s s ′ d μ ( y ) 1 s s ′ 1 μ ( 6 B ) ∫ 6 B ∣ f ( y ) ∣ s s d μ ( y ) 1 s s ≤ C ‖ b ‖ RBMO ( μ ) M r , 6 ( f ) ( x ) .

Together with the aforementioned estimates, to get (3.2), we still need to estimate the difference ∣ g ˙ r ( ( b ( ⋅ ) − b B ) f 2 ) − h B ∣ . For all y 1 , y 2 ∈ B , write

∣ g ˙ r ( ( b ( ⋅ ) − b B ) f 2 ) ( y 1 ) − h B ∣ ≤ 1 μ ( B ) ∫ B ∑ k ∈ Z ∫ X D k ( y 1 , z ) ( b ( z ) − b B ) f 2 ( z ) d μ ( z ) r 1 r − ∑ k ∈ Z ∫ X D k ( y 2 , z ) ( b ( z ) − b B ) f 2 ( z ) d μ ( z ) r 1 r d μ ( y 2 ) ≤ 1 μ ( B ) ∫ B ∑ k ∈ Z ∫ X ( D k ( y 1 , z ) − D k ( y 2 , z ) ) ( b ( z ) − b B ) f 2 ( z ) d μ ( z ) r 1 r d μ ( y 2 ) ,

therefore, we only consider

∑ k ∈ Z ∫ X ( D k ( y 1 , z ) − D k ( y 2 , z ) ) ( b ( z ) − b B ) f 2 ( z ) d μ ( z ) r 1 r .

From Fubini-Tonelli theorem, Hölder inequality, (1.1), (1.6) and (2.2), it follows that

(3.6) ∑ k ∈ Z ∫ X ( D k ( y 1 , z ) − D k ( y 2 , z ) ) ( b ( z ) − b B ) f 2 ( z ) d μ ( z ) r 1 r ≤ ∫ X ⧹ 6 B ∑ k ∈ Z ∣ D k ( y 1 , z ) − D k ( y 2 , z ) ∣ r ∣ b ( z ) − b B ∣ r ∣ f ( z ) ∣ r d μ ( z ) 1 r ≤ C ∑ i = − ∞ H B y 1 − 3 ∫ Q y 1 , i ⧹ Q y 1 , i + 1 d ( y 1 , y 2 ) r ( Q y 1 , i ) r ε ∣ b ( z ) − b B ∣ r ∣ f ( z ) ∣ r [ λ ( y 1 , d ( y 1 , z ) ) ] r d μ ( z ) 1 r + C ∫ Q y 1 , H B y 1 − 2 ⧹ 6 B ∣ b ( z ) − b B ∣ r ∣ f ( z ) ∣ r [ λ ( y 1 , d ( y 1 , z ) ) ] r d μ ( z ) 1 r ≤ C ∑ i = 3 ∞ ∫ Q y 1 , H B y 1 − i ⧹ Q y 1 , H B y 1 − i + 1 r ( B ) r ( Q y 1 , H B y 1 − i + 3 ) r ε ∣ b ( z ) − b B ∣ r ∣ f ( z ) ∣ r [ λ ( y 1 , d ( y 1 , z ) ) ] r d μ ( z ) 1 r + C ∫ Q y 1 , H B y 1 − 2 ⧹ 6 B ∣ b ( z ) − b B ∣ r ∣ f ( z ) ∣ r [ λ ( y 1 , d ( y 1 , z ) ) ] r d μ ( z ) 1 r ≤ C ∑ i = 3 ∞ r ( B ) r ( Q y 1 , H B y 1 − i + 3 ) r ε 1 [ λ ( y 1 , r ( Q y 1 , H B y 1 − i + 1 ) ) ] r ∫ Q y 1 , H B y 1 − i ∣ b ( z ) − b B ∣ r ∣ f ( z ) ∣ r d μ ( z ) 1 r + C λ ( y 1 , r ( 6 B ) ) ∫ Q y 1 , H B y 1 − 2 ∣ b ( z ) − b B ∣ r ∣ f ( z ) ∣ r d μ ( z ) 1 r

≤ C ∑ i = 3 ∞ r ( B ) r ( Q y 1 , H B y 1 − i + 3 ) r ε 1 [ λ ( y 1 , r ( Q y 1 , H B y 1 − i + 1 ) ) ] r × ∫ Q y 1 , H B y 1 − i ∣ b ( z ) − b Q y 1 , H B y 1 − i ∣ r ∣ f ( z ) ∣ r d μ ( z ) + ∣ b B − b Q y 1 , H B y 1 − i ∣ r ∫ Q y 1 , H B y 1 − i ∣ f ( z ) ∣ r d μ ( z ) 1 r + C λ ( y 1 , r ( 6 B ) ) ∣ b Q y 1 , H B y 1 − 2 − b B ∣ r ∫ Q y 1 , H B y 1 − 2 ∣ f ( z ) ∣ r d μ ( z ) + ∫ Q y 1 , H B y 1 − 2 ∣ b ( z ) − b Q y 1 , H B y 1 − 2 ∣ r ∣ f ( z ) ∣ r d μ ( z ) 1 r ≤ C ∑ i = 3 ∞ 1 1 0 ( r ( i − 3 ) ε ) 1 [ λ ( y 1 , r ( Q y 1 , H B y 1 − i + 1 ) ) ] r ‖ b ‖ RBMO ( μ ) r [ μ ( 6 Q y 1 , H B y 1 − i ) ] M r 2 , 6 r ( f ) ( x ) + i r ‖ b ‖ RBMO ( μ ) r [ μ ( 6 Q y 1 , H B y 1 − i ) ] M r , 6 r ( f ) ( x ) 1 r + C λ ( y 1 , r ( 6 B ) ) { ‖ b ‖ RBMO ( μ ) r [ μ ( 6 Q y 1 , H B y 1 − 2 ) ] M r , 6 r ( f ) ( x ) + ‖ b ‖ RBMO ( μ ) r [ μ ( 6 Q y 1 , H B y 1 − i ) ] M r 2 , 6 r ( f ) ( x ) } 1 r ≤ C ‖ b ‖ RBMO ( μ ) { M r 2 , 6 ( f ) ( x ) + M r , 6 ( f ) ( x ) } ,

where H B y 1 represents the largest integer k satisfying B ⊂ B y 1 , k for any ball B and y 1 ∈ B ∩ supp ( μ ) . Moreover, we also need some known facts proved in [10].

X ⧹ 6 B = ⋃ i = 3 ∞ ( Q y 2 , H B y 1 − i ⧹ Q y 2 , H B y 1 − i + 1 ) ∪ ( Q y 2 , H B y 1 − i + 1 ⧹ 6 B )

and, for all y 1 , y 2 ∈ B and z ∈ X ⧹ Q y 2 , H B y 1 − 2 ,

∑ k ∈ Z ∣ D k ( y 1 , z ) − D k ( y 2 , z ) ∣ r ≤ C d ( y 1 , y 2 ) r ( Q y 1 , H B y 1 − i + 3 ) r ε 1 [ λ ( y 1 , d ( y 1 , z ) ) ] r .

Combining the estimates for (3.4), (3.5) and (3.6), we get (3.2).

Now, we show the condition (3.3). Consider two balls B , S ⊂ X satisfying B ⊂ S and x ∈ B , and let N ≔ N B , S + 1 . Write

∣ h B − h S ∣ ≤ ∣ m B ( g ˙ r ( ( b − b S ) f χ 6 N B ⧹ 6 B ) ) ∣ + ∣ m S ( g ˙ r ( ( b − b S ) f χ 6 N B ⧹ 6 S ) ) ∣ + ∣ m B ( g ˙ r ( ( b − b S ) f χ X ⧹ 6 N B ) ) − m S ( g ˙ r ( ( b − b S ) f χ X ⧹ 6 N B ) ) ∣ + ∣ m B ( g ˙ r ( ( b S − b B ) f χ X ⧹ 6 B ) ) ∣ ≕ E 1 + E 2 + E 3 + E 4 .

With an argument similar to that used in the estimate of (3.6), it is easy to see that

E 3 ≤ C ‖ b ‖ RBMO ( μ ) { M r 2 , 6 ( f ) ( x ) + M r , 6 ( f ) ( x ) } .

For y ∈ B , by applying Fubini-Tonelli Theorem, Minkowski inequality, Hölder inequality and (2.2), we can deduce that

E 1 ≤ 1 μ ( B ) ∫ B ∣ g ˙ r ( ( b − b S ) f χ 6 N B ⧹ 6 B ) ( y ) ∣ d μ ( y ) ≤ 1 μ ( B ) ∫ B ∫ 6 N B ⧹ 6 B ∑ k ∈ Z ∣ D ( y , z ) ∣ r ∣ b ( z ) − b S ∣ r ∣ f ( z ) ∣ r d μ ( z ) 1 r d μ ( y )

≤ 1 μ ( B ) ∫ 6 N B ⧹ 6 B ∫ B ∑ k ∈ Z ∣ D ( y , z ) ∣ r 1 r d μ ( y ) ∣ b ( z ) − b S ∣ ∣ f ( z ) ∣ d μ ( z ) ≤ C ∑ j = 1 N − 1 ∫ 6 j + 1 B ⧹ 6 j B ∣ b ( z ) − b S ∣ λ ( c ( B ) , d ( c ( B ) , z ) ) ∣ f ( z ) ∣ d μ ( z ) ≤ C ∑ j = 1 N − 1 1 λ ( c ( B ) , 6 j r B ) ∫ 6 j + 1 B ∣ b ( z ) − b 6 j + 1 B ∣ ∣ f ( z ) ∣ d μ ( z ) + ∣ b 6 j + 1 B − b S ∣ ∫ 6 j + 1 B ∣ f ( z ) ∣ d μ ( z ) ≤ C K B , S ‖ b ‖ RBMO ( μ ) M r , 6 ( f ) ( x ) ∑ j = 1 N − 1 μ ( 6 j + 1 B ) λ ( c ( B ) , 6 j r ( B ) ) ≤ C K B , S ‖ b ‖ RBMO ( μ ) M r , 6 ( f ) ( x ) .

Similarly, it follows that E 2 ≤ C K B , S ‖ b ‖ RBMO ( μ ) M r , 6 ( f ) ( x ) .

Now, let us estimate E 4 . For any y ∈ B , from (1.5), Fubini-Tonelli Theorem, Minkowski inequality and Hölder inequality, we have

1 μ ( B ) ∫ B ∣ g ˙ r ( ( b B − b S ) f χ X ⧹ 6 B ) ( y ) ∣ d μ ( y ) ≤ C K B , S ‖ b ‖ RBMO ( μ ) 1 μ ( B ) ∫ B ∣ g ˙ r ( f χ X ⧹ 6 B ) ( y ) ∣ d μ ( y ) + ∫ B ∣ g ˙ r ( f χ 6 B ) ( y ) ∣ d μ ( y ) ≤ C K B , S ‖ b ‖ RBMO ( μ ) { M r , 6 ( g ˙ ( f ) ) ( x ) + M r , 6 ( f ) ( x ) } . □

Proof of Theorem 1.10

By applying Lemma 2.3 and (3.1), it is obvious to see that

‖ [ b , g ˙ r ] ( f ) ‖ L 1 , ∞ ( μ ) ≤ ‖ N ( [ b , g ˙ r ] ( f ) ) ‖ L 1 , ∞ ( μ ) ≤ C ‖ M ♯ ( [ b , g ˙ r ] ( f ) ) ‖ L 1 , ∞ ( μ ) ≤ C ‖ b ‖ RBMO ( μ ) { ‖ M r , 6 ( g ˙ r ( f ) ) ‖ L 1 , ∞ ( μ ) + ‖ M r 2 , 6 ( f ) ‖ L 1 , ∞ ( μ ) + ‖ M r , 6 ( f ) ‖ L 1 , ∞ ( μ ) } ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ L 1 ( μ ) ,

which is the desired result.□

Proof of Theorem 1.11

By applying (2.4), (3.1) and Theorem 14 (see [9]), we have

‖ [ b , g ˙ r ] ( f ) ‖ M q p ( μ ) ≤ ‖ N ( [ b , g ˙ r ] ( f ) ) ‖ M q p ( μ ) ≤ C ‖ M ♯ ( [ b , g ˙ r ] ( f ) ) ‖ M q p ( μ ) ≤ C ‖ b ‖ RBMO ( μ ) { ‖ M r , 6 ( g ˙ r ( f ) ) ‖ M q p ( μ ) + ‖ M r 2 , 6 ( f ) ‖ M q p ( μ ) + ‖ M r , 6 ( f ) ‖ M q p ( μ ) } ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ M q p ( μ ) . □

Proof of Theorem 1.12

From Remark 1.7 (1), we assume k = 6 in (1.8). By applying Definition 1.6, Lemma 2.3 (2), Theorem 4.2 in [15] and (3.1), we can deduce that

‖ [ b , g ˙ r ] ( f ) ‖ L p , ϕ ( μ ) ≤ sup B C [ ϕ ( μ ( 6 B ) ) ] 1 p ‖ N ( [ b , g ˙ r ] ( f ) ) ‖ L p ( μ , B ) ≤ sup B C [ ϕ ( μ ( 6 B ) ) ] 1 p ‖ M ♯ ( [ b , g ˙ r ] ( f ) ) ‖ L p ( μ , B ) ≤ C ‖ b ‖ RBMO ( μ ) { ‖ M r , 6 ( g ˙ r ( f ) ) ‖ L p , ϕ ( μ ) + ‖ M r 2 , 6 ( f ) ‖ L p , ϕ ( μ ) + ‖ M r , 6 ( f ) ‖ L p , ϕ ( μ ) } ≤ C ‖ b ‖ RBMO ( μ ) ‖ f ‖ L p , ϕ ( μ ) . □

Funding information: This research was supported by the Scientific Startup Foundation for Doctors of Northwest Normal University (0002020203), Young Teachers’ Scientific Research Ability Promotion Project of Northwest Normal University (NWNU-LKQN2020-07) and Innovation Fund Project for Higher Education of Gansu Province (2020A-010).
Conflict of interest: Author states no conflict of interest.

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Received: 2020-03-07

Revised: 2021-03-01

Accepted: 2021-05-30

Published Online: 2021-08-27

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

Articles in the same Issue

https://doi.org/10.1515/math-2021-0051

Keywords for this article

non-homogeneous metric measure space; Littlewood-Paley g-function; commutator; space RBMO(μ); generalized Morrey space

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