Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces

Guanghui Lu; Miaomiao Wang; Shuangping Tao

doi:10.1515/agms-2023-0101

Article Open Access

Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces

Guanghui Lu , Miaomiao Wang and Shuangping Tao

Published/Copyright: November 23, 2023

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From the journal Analysis and Geometry in Metric Spaces Volume 11 Issue 1

Abstract

Let ( X , d , μ ) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space M p u ( μ ) , where 1 ≤ p < ∞ and u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions u 1 , u 2 , and u belong to W τ with τ ∈ ( 0 , 2 ) , we prove that the bilinear θ -type generalized fractional integral T ˜ θ , α is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) into spaces M q u ( μ ) , where u 1 u 2 = u , α ∈ ( 0 , 1 ) , and 1 q = 1 p 1 + 1 p 2 − 2 α with p 1 , p 2 ∈ ( 1 , 1 α ) , and also show that the T ˜ θ , α is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) into spaces M 1 u ( μ ) , where 1 = 1 p 1 + 1 p 2 − 2 α . Meanwhile, we prove that the commutator T ˜ θ , α , b 1 , b 2 formed by b 1 , b 2 ∈ RBMO ˜ ( μ ) and T ˜ θ , α is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) into spaces M q u ( μ ) , and it is also bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) into spaces M 1 u ( μ ) .

Keywords: non-homogeneous metric measure space; bilinear θ-type generalized fractional integral; commutator; space RBMO(μ); new generalized Morrey space

MSC 2010: 42B20; 42B25; 47B47; 30L99

1 Introduction

As we all know, to unify the spaces of homogeneous type in the sense of Coifman and Weiss [3,4] and the nondoubling measure spaces that satisfy the polynomial growth conditions [7,11,13,16,29], Hytönen [8] first introduced a new class of metric measure spaces satisfying the so-called upper doubling and geometrically doubling conditions. Also, the metric measure spaces are now called non-homogeneous metric measure spaces and simply denoted by ( X , d , μ ) . Since then, many studies focus on the various integral operators and function spaces on ( X , d , μ ) . For example, in 2020, Lin et al. [12] established the John-Nirenberg inequality for the regularized bounded lower oscillation RBLO ˜ ( μ ) . In 2021, Tao and Zhang [28] proved that the m th-order commutators ℳ m , b ρ formed by the log-Dini-type parametric Marcinkiewicz integral operators ℳ ρ and b ∈ RBMO ( μ ) are bounded on Lebesgue spaces L p ( μ ) and Morrey spaces M p q ( μ ) for 1 < p ≤ q < ∞ . In the same year, Zhao et al. [34] obtained some weak-type multiple weighted estimates for the iterated commutator T Π b → generated by b → = ( b 1 , ⋯ , b m ) ∈ [ RBMO ( μ ) ] m and a multilinear Calderón-Zygmund operator T . In 2022, Wang and Xie [31] showed that the multilinear strongly singular integral operator T is bounded from the product of Lebesgue spaces L p 1 ( μ ) × ⋯ × L p m ( μ ) into spaces L p ( μ ) , and it is bounded from the product of Morrey spaces M p 1 q 1 ( μ ) × ⋯ × M p m q m ( μ ) into spaces M p q ( μ ) , where 1 p = 1 p 1 + ⋯ + 1 p m and 1 q = 1 q 1 + ⋯ + 1 q m . Recently, Lu [17] obtained the boundedness of a bilinear Calderón-Zygmund operator T ˜ and its commutator T ˜ b 1 , b 2 generated by b 1 , b 2 ∈ RBMO ˜ ( μ ) and T ˜ on the product of generalized weighted Morrey spaces ℒ ω 1 p 1 , φ 1 ( μ ) × ℒ ω 2 p 2 , φ 2 ( μ ) . More studies on the various integral operators and function spaces on ( X , d , μ ) can be seen in [9,14,18,19,24,25,30].

In 1985, Yabuta [33] first introduced an θ -type Calderón-Zygmund operator T θ by a substantial weakening of the smoothness conditions on the kernel. Since then, many studies focus on the boundedness of various θ -type Calderón-Zygmund operators over non-homogeneous metric measure spaces [20–23,26,27]. However, in this article, we obtain the definition of the generalized Morrey space M p u ( μ ) , where 1 ≤ p < ∞ and u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) is a Lebesgue measurable function. Second, we prove that the bilinear θ -type generalized fractional integral T ˜ θ , α is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) into spaces M q u ( μ ) , where the measurable functions u 1 , u 2 , and u belonging to W τ satisfy u 1 u 2 = u , α ∈ ( 0 , 1 ) , 1 q = 1 p − 2 α , and 1 p = 1 p 1 + 1 p 2 for 1 < p 1 , p 2 < 1 α . Furthermore, we show that T ˜ θ , α is bounded from the product of spaces M p 1 u 1 ( μ ) × M p 2 u 2 ( μ ) into generalized weak Morrey space W M 1 u ( μ ) , where u 1 u 2 = u , α ∈ ( 0 , 1 ) , 1 = 1 p − 2 α , and 1 p = 1 p 1 + 1 p 2 . Finally, we prove that the commutator T ˜ θ , α , b 1 , b 2 generated by b 1 , b 2 ∈ RBMO ˜ ( μ ) and T ˜ θ , α is bounded on spaces M q u ( μ ) and W M 1 u ( μ ) .

Before stating the main results of this article, we need to recall some necessary notion and notations. The following definitions of the upper doubling and geometrically doubling conditions were introduced by Hytönen [8].

Definition 1.1

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 a positive constant C ( λ ) , 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 ) .

Here and in what follows, B ( x , r ) = { y ∈ X : d ( x , y ) < r } represents an open ball centered at x ∈ X with radius r > 0 .

Remark 1.2

Hytönen et al. [10] 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 ) .

Thus, in this article, we also assume that the λ defined as in (1.1) satisfies (1.2).

Definition 1.3

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 such that the cardinality of this covering is at most N 0 .

Remark 1.4

For a metric space ( X , d ) , Hytönen [8] proved that the geometrically doubling condition is equivalent to the following statement: for every ε ∈ ( 0 , 1 ) , any ball B ( x , r ) ⊂ X with x ∈ X and r ∈ ( 0 , ∞ ) contains at most N 0 ε − n 0 centers of disjoint balls { B ( x i , ε r ) } i , here and hereafter, N 0 is as in Definition 1.3 and n 0 ≔ log 2 N 0 .

A metric measure space ( X , d , μ ) is called a non-homogeneous metric measure space if ( X , d ) is geometrically doubling and ( X , d , μ ) is upper doubling.

The following notion of a discrete coefficient K ˜ B , S ( ρ ) , which is analogous to the one introduced by Tolsa [29], is from [1].

Definition 1.5

For any ρ ∈ ( 1 , ∞ ) and any two balls B and S with B ⊂ S ⊂ X , set

(1.3) K ˜ B , S ( ρ ) = 1 + ∑ k = − ⌊ log ρ 2 ⌋ N B , S ( ρ ) μ ( ρ k B ) λ ( c B , ρ k r B ) ,

where c B and r B represent the center and radius of the ball B , respectively; N B , S ( ρ ) is the smallest integer satisfying ρ N B , S ( ρ ) r B ≥ r S ; and for any x ∈ R , ⌊ x ⌋ represents the largest integer, which is smaller than or equals to x . In addition, other properties on the K ˜ B , S ( ρ ) can be seen in Remark 2.8 and Lemma 2.9 in [15].

We now recall the notion of an ( α , β ) -doubling ball introduced in [8].

Definition 1.6

A ball B ⊂ X is said to be an ( α , β ) -doubling if, for all α , β ∈ ( 1 , ∞ ) ,

μ ( α B ) ≤ β μ ( B ) .

The other properties on the ( α , β ) -doubling can be seen in Lemmas 3.2 and 3.3 of [8]. Furthermore, for any α ∈ ( 1 , ∞ ) and any ball B , we denote B ˜ α by the smallest ( α , β α ) -doubling ball of the form α j β with j ∈ Z + , where

(1.4) β α = α max { n 0 , ν } + [ max { 5 α , 30 } ] n 0 + [ max { 3 α , 30 } ] ν ,

ν = log 2 C ( λ ) and n 0 = log 2 N 0 . Particularly, we always assume that α = 6 in (1.4) and simply denote B ˜ 6 by B ˜ .

Fu et al. [5] obtained the following definition of regular BMO spaces associated with discrete coefficients K ˜ B , S ( ρ ) .

Definition 1.7

Let ρ ∈ ( 1 , ∞ ) and γ ∈ [ 1 , ∞ ) . A real-valued function f ∈ L loc 1 ( μ ) is said to be in the space RBMO ˜ ρ , γ ( μ ) if there exist a positive constant C and, for any ball B ⊂ X , a number f B such that

(1.5) 1 μ ( ρ B ) ∫ B ∣ f ( y ) − f B ∣ d μ ( y ) ≤ C

and, for any two balls B and S with B ⊂ S ,

(1.6) ∣ f B − f S ∣ ≤ C [ K ˜ B , S ( ρ ) ] γ ,

where f B represents the mean value of f over ball B , that is,

f B = 1 μ ( B ) ∫ B f ( y ) d μ ( y ) .

The infimum of the aforementioned constant C satisfying (1.5) and (1.6) is defined to be RBMO ˜ ρ , γ ( μ ) norm of f and denoted by ‖ f ‖ RBMO ˜ ρ , γ ( μ ) . Moreover, Fu et al. [5] showed that the space RBMO ˜ ρ , γ ( μ ) is independent of the choice of ρ > 1 and γ ≥ 1 . Hence, in what follows, the space RBMO ˜ ρ , γ ( μ ) is denoted by the RBMO ˜ ( μ ) .

The following notion of a bilinear θ -type generalized fractional integrals is from [22].

Definition 1.8

Let θ be a nonnegative and non-decreasing function defined on ( 0 , ∞ ) satisfying

(1.7) ∫ 0 1 θ ( t ) t log 1 t d t < ∞ .

Let α ∈ ( 0 , 1 ) . A locally integrable function K θ , α is called a bilinear θ -type generalized fractional integral kernel if there exists some positive constant C such that:

for all x , y 1 , y 2 ∈ X with x ≠ y j and j = 1 , 2 ,
(1.8) ∣ K θ , α ( x , y 1 , y 2 ) ∣ ≤ C ∑ j = 1 2 [ λ ( x , d ( x , y j ) ) ] 1 − α − 2 ;
there exists a constant c ∈ ( 0 , ∞ ) such that, for all x , x ′ , y 1 , y 2 ∈ X with c d ( x , x ′ ) ≤ max { d ( x , y 1 ) , d ( x , y 2 ) } ,
(1.9) ∣ K θ , α ( x , y 1 , y 2 ) − K θ , α ( x ′ , y 1 , y 2 ) ∣ ≤ C θ d ( x , x ′ ) ∑ j = 1 2 d ( x , y j ) ∑ j = 1 2 [ λ ( x , d ( x , y j ) ) ] 1 − α − 2 ;
there exists a constant c ∈ ( 0 , ∞ ) such that, for all x , y 1 ′ , y 1 , y 2 ∈ X with c d ( y 1 , y 1 ′ ) ≤ max { d ( x , y 1 ) , d ( x , y 2 ) } ,
(1.10) ∣ K θ , α ( x , y 1 , y 2 ) − K θ , α ( x , y 1 ′ , y 2 ) ∣ ≤ C θ d ( y 1 , y 1 ′ ) ∑ j = 1 2 d ( x , y j ) ∑ j = 1 2 [ λ ( x , d ( x , y j ) ) ] 1 − α − 2 .

Remark 1.9

Without loss of generality, for the simplicity, we may assume that c = 2 in (1.9) and (1.10).
If take α ≡ 0 in Definition 1.8, then the bilinear θ -type generalized fractional integral kernel K θ , α is just the bilinear θ -type Calderón-Zygmund kernel K θ introduced in [32].
If take α ≡ 0 and θ ( t ) = t δ with t > 0 and δ > 0 in Definition 1.8, then the bilinear θ -type generalized fractional kernel K θ , α is just the bilinear Calderón-Zygmund kernel K introduced by Zhao et al. [34].

Let L b ∞ ( μ ) be the space of all L ∞ ( μ ) functions with bounded support. A bilinear operator T ˜ θ , α is called a bilinear θ -type generalized fractional integral with kernel K θ , α satisfying (1.8), (1.9), and (1.10) if, for all f 1 , f 2 ∈ L b ∞ ( μ ) and x ∈ X \ ( supp ( f 1 ) ⋂ supp ( f 2 ) ) ,

(1.11) T ˜ θ , α ( f 1 , f 2 ) ( x ) = ∫ X 2 K θ , α ( x , y 1 , y 2 ) f 1 ( y 1 ) f 2 ( y 2 ) d μ ( y 1 ) d μ ( y 2 ) .

Given b 1 , b 2 ∈ RBMO ˜ ( μ ) , the commutator T ˜ θ , α , b 1 , b 2 formed by b 1 , b 2 , and T ˜ θ , α is defined by:

(1.12) T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ( x ) = b 1 ( x ) b 2 ( x ) T ˜ θ , α ( f 1 , f 2 ) ( x ) − b 1 ( x ) T ˜ θ , α ( f 1 , b 2 ( ⋅ ) f 2 ) ( x ) − b 2 ( x ) T ˜ θ , α ( b 1 ( ⋅ ) f 1 , f 2 ) ( x ) + T ˜ θ , α ( b 1 ( ⋅ ) f 1 , b 2 ( ⋅ ) f 2 ) ( x ) .

Equivalently, T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ( x ) can be formally written as:

∫ X 2 K θ , α ( x , y 1 , y 2 ) ( b 1 ( x ) − b 1 ( y 1 ) ) ( b 2 ( x ) − b 2 ( y 2 ) ) f 1 ( y 1 ) f 2 ( y 2 ) d μ ( y 1 ) d μ ( y 2 ) .

Also, the commutators T ˜ θ , α , b 1 and T ˜ θ , α , b 2 , respectively, are defined by:

(1.13) T ˜ θ , α , b 1 ( f 1 , f 2 ) ( x ) = b 1 ( x ) T ˜ θ , α ( f 1 , f 2 ) ( x ) − T ˜ θ , α ( b 1 ( ⋅ ) f 1 , f 2 ) ( x ) ,

(1.14) T ˜ θ , α , b 2 ( f 1 , f 2 ) ( x ) = b 2 ( x ) T ˜ θ , α ( f 1 , f 2 ) ( x ) − T ˜ θ , α ( f 1 , b 2 ( ⋅ ) f 2 ) ( x ) .

We now state the definition of a generalized Morrey space as follows.

Definition 1.10

Let 1 ≤ p < ∞ and u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) be a Lebesgue measurable function. Then, the generalized Morrey space M p u ( μ ) is defined by:

M p u ( μ ) = { f ∈ L loc p ( μ ) : ‖ f ‖ M p u ( μ ) < ∞ } ,

where

(1.15) ‖ f ‖ M p u ( μ ) = sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ f χ B ( x , r ) ‖ L p ( μ ) .

Also, we denote by W M p u ( μ ) the generalized weak Morrey space of all locally integrable functions satisfying

(1.16) ‖ f ‖ W M p u ( μ ) = sup x ∈ X , r > 0 sup t > 0 [ u ( x , r ) ] − 1 t μ ( { y ∈ B ( x , r ) : ∣ f ( y ) ∣ > t } ) 1 p < ∞ .

Remark 1.11

If we take u ( x , r ) ≔ [ μ ( B ( x , η r ) ) ] 1 p − 1 q with 1 < q ≤ p < ∞ and η > 1 in (1.15), then the generalized Morrey space M p u ( μ ) is just the Morrey space M p q ( μ ) introduced by Cao and Zhou [2].
If we take u ( x , r ) ≔ [ ϕ ( μ ( η B ( x , r ) ) ) ] 1 p with ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) being an increasing function in (1.15), then the generalized Morrey space M p u ( μ ) is just the generalized Morrey space L p , ϕ ( μ ) introduced in the study by Lu and Tao [24]. In other words, spaces L p , ϕ ( μ ) ⊂ M p u ( μ ) .

We now recall the following definition of a weak reverse doubling condition [6].

Definition 1.12

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 some number C ( a ) ∈ [ 1 , ∞ ) , only depending on a and X , such that, for all x ∈ X ,

(1.17) λ ( x , a r ) ≥ C ( a ) λ ( x , r )

and, moreover,

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

Also, we need to establish the following notion.

Definition 1.13

Let τ ∈ ( 0 , 2 ) , ε ∈ ( 0 , ∞ ) , and the dominating function λ satisfy ε -weak reverse doubling condition. We say that a Lebesgue measurable function u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) belongs to W τ if, for all x ∈ X and r > 0 , there exists some positive constant C such that u ( x , r ) satisfies

(1.19) ∑ k = 0 ∞ λ ( x , r ) λ ( x , 6 k + 1 r ) τ u ( x , 6 k + 1 r ) ≤ C u ( x , r )

and

(1.20) C ≤ u ( x , r ) , for any x ∈ X and r ≥ 1 .

Remark 1.14

We note that Definition 1.13 is meaningful. Indeed, let 1 < q < p < ∞ , τ ≔ 1 q , and u ( x , r ) ≔ [ λ ( x , r ) ] 1 q − 1 p for x ∈ X and r ∈ ( 0 , ∞ ) . Then, by Definition 1.12 and (1.19), we deduce that

∑ k = 0 ∞ λ ( x , r ) λ ( x , 6 k + 1 r ) τ u ( x , 6 k + 1 r ) = ∑ k = 0 ∞ λ ( x , r ) λ ( x , 6 k + 1 r ) 1 q [ λ ( x , 6 k + 1 r ) ] 1 q − 1 p ≤ ∑ k = 0 ∞ [ λ ( x , r ) ] 1 q [ C ( 6 k + 1 ) ] 1 p [ λ ( x , r ) ] 1 p ≤ C [ λ ( x , r ) ] 1 q − 1 p ≤ C u ( x , r ) .

That is, u ( x , r ) ∈ W τ .

The main theorems of this article are stated as follows.

Theorem 1.15

Let α ∈ ( 0 , 1 ) , θ be a nonnegative and non-decreasing function on ( 0 , ∞ ) and T ˜ θ , α be a bilinear generalized fractional integral defined as in (1.11). Suppose that the measurable functions u 1 , u 2 , and u satisfy (1.19) and u 1 u 2 = u . Then, there exists some positive constant C such that, for all f i ∈ M p i u i ( μ ) with i = 1 , 2 ,

‖ T ˜ θ , α ( f 1 , f 2 ) ‖ M q u ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ,

where 1 q = 1 p 1 + 1 p 2 − 1 2 α with 1 < p 1 , p 2 < 1 α .

Theorem 1.16

Let α ∈ ( 0 , 1 ) , θ be a nonnegative and non-decreasing function on ( 0 , ∞ ) and T ˜ θ , α be a bilinear generalized fractional integral defined as in (1.11). Suppose that the measurable functions u 1 , u 2 , and u satisfy (1.19) and u 1 u 2 = u . Then there exists some positive constant C such that, for all f i ∈ M p i u i ( μ ) with i = 1 , 2 ,

‖ T ˜ θ , α ( f 1 , f 2 ) ‖ W M 1 u ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ,

where 1 = 1 p 1 + 1 p 2 − 1 2 α with 1 < p 1 , p 2 < 1 α .

Theorem 1.17

Let b 1 , b 2 ∈ RBMO ˜ ( μ ) , α ∈ ( 0 , 1 ) , θ be a nonnegative and non-decreasing function defined on ( 0 , ∞ ) and T ˜ θ , α be a bilinear generalized fractional integral defined as in (1.11). Suppose that the measurable functions u i ( i = 1 , 2 ) satisfy

(1.21) ∑ k = 0 ∞ ( k + 1 ) λ ( x , r ) λ ( x , 6 k + 1 r ) τ u i ( x , 6 k + 1 r ) ≤ C u i ( x , r ) , f o r 0 < τ < 2 ,

and u 1 u 2 = u . Then, there exists some positive constant C such that, for all f i ∈ M p i u i ( μ ) with i = 1 , 2 ,

‖ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ‖ M q u ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ,

where 1 q = 1 p 1 + 1 p 2 − 1 2 α with 1 < p 1 , p 2 < 1 α .

Theorem 1.18

Let b 1 , b 2 ∈ RBMO ˜ ( μ ) , α ∈ ( 0 , 1 ) , θ be a nonnegative and non-decreasing function defined on ( 0 , ∞ ) , and T ˜ θ , α be a bilinear generalized fractional integral defined as in (1.11). Suppose that measurable functions u i ( i = 1 , 2 ) satisfy (1.21) and u 1 u 2 = u . Then, there exists some positive constant C such that, for all f i ∈ M p i u i ( μ ) with i = 1 , 2 ,

‖ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ‖ W M 1 u ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ,

where 1 = 1 p 1 + 1 p 2 − 1 2 α with 1 < p 1 , p 2 < 1 α .

Finally, we make some conventions on notation. Throughout this article, let C or c be a positive constant being independent of the main parameters, but it may vary from line to line. Moreover, we use C ( ρ , ν ) or C ˜ ( ρ , ν ) to denote a positive constant only depending on the main parameters ρ and ν . Given any p ∈ [ 1 , ∞ ) , let p ′ ≔ p ∕ ( p − 1 ) denote its conjugate index. For any measurable subset E ⊂ X , we use χ E to denote its characteristic function, and

m E ( f ) = 1 μ ( E ) ∫ E f ( x ) d μ ( x )

indicates the mean value of functions f over E .

2 Preliminaries

To prove the main theorems of this article, in this section, we mainly recall some necessary results. The following useful properties of the coefficient K ˜ B , S ( ρ ) is from [5].

Lemma 2.1

Let ( X , d , μ ) be a non-homogeneous metric measure space.

For any ρ ∈ ( 1 , ∞ ) , there exist positive constants C ( ρ ) , C ˜ ( ρ , ν ) , and C ( ρ , ν ) such that, for any balls B , R , and S with B ⊂ R ⊂ S ,
K ˜ B , R ( ρ ) ≤ C ( ρ ) K ˜ B , S ( ρ ) , K ˜ R , S ( ρ ) ≤ C ˜ ( ρ , ν ) K ˜ B , S ( ρ ) , and K ˜ B , S ( ρ ) ≤ K ˜ B , R ( ρ ) + C ( ρ , ν ) K ˜ R , S ( ρ ) .
For any γ ∈ [ 1 , ∞ ) and ρ ∈ ( 1 , ∞ ) , there exists a positive constant C ( γ , ρ ) , depending on γ and ρ , such that, for any two balls B and S with B ⊂ S and r S ≤ γ r B ,
K ˜ B , S ( ρ ) ≤ C ( γ , ρ ) .
For any ρ ∈ ( 1 , ∞ ) , there exists a positive constant C ( ρ , ν ) , depending on ρ and ν , such that, for any ball B , K ˜ B , B ˜ ρ ( ρ ) ≤ C ( ρ , ν ) .

We now recall the following characterizations of spaces RBMO ˜ ( μ ) introduced in the study by Lin et al. [15].

Lemma 2.2

Let τ , ρ ∈ ( 1 , ∞ ) . For a μ -locally integrable function f defined on X , the following statements are mutually equivalent:

f ∈ RBMO ˜ ( μ ) ;
there exists some positive constant C such that, for any ball B,
1 μ ( τ B ) ∫ B ∣ f ( y ) − m B ˜ ρ ( f ) ∣ d μ ( y ) ≤ C ,
and, for any two ( ρ , β ρ ) -doubling balls B and S with B ⊂ S ,
∣ m B ( f ) − m S ( f ) ∣ ≤ C K ˜ B , S ( ρ ) .
Moreover, the infimum of the aforementioned constant C is equivalent to ‖ f ‖ RBMO ˜ ( μ ) .

Corollary 2.3

Let ( X , d , μ ) be a non-homogeneous metric measure space. Then, for every ρ ∈ ( 1 , ∞ ) and p ∈ [ 1 , ∞ ) , there exists some positive constant C such that, for all f ∈ RBMO ˜ ( μ ) and ball B,

(2.1) 1 μ ( ρ B ) ∫ B ∣ f ( x ) − m B ˜ ( f ) ∣ p d μ ( x ) 1 p ≤ C ‖ f ‖ RBMO ˜ ( μ ) .

Also, we need the following two lemmas on the T ˜ θ , α and T ˜ θ , α , b 1 , b 2 introduced in [22].

Lemma 2.4

Let α ∈ ( 0 , 1 ) , θ be a nonnegative and non-decreasing function on ( 0 , ∞ ) and satisfy (1.7) and T ˜ θ , α be defined as in (1.11). Suppose that the λ satisfies weak reverse doubling. Then, there exists a positive constant C such that, for all f i ∈ L p i ( μ ) with i = 1 , 2 ,

‖ T ˜ θ , α ( f 1 , f 2 ) ‖ L q ( μ ) ≤ C ‖ f 1 ‖ L p 1 ( μ ) ‖ f 2 ‖ L p 2 ( μ ) ,

where 1 q = 1 p 1 + 1 p 2 − 2 α with 1 < p 1 , p 2 < 1 α .

Lemma 2.5

Let b 1 , b 2 ∈ RBMO ˜ ( μ ) , α ∈ ( 0 , 1 ) , and θ be a non-negative and non-decreasing function on ( 0 , ∞ ) with satisfying (1.7). Suppose that T ˜ θ , α is as in (1.11), and λ satisfies ε -weak reverse doubling condition. Then, there exists some positive constant C such that, for all f i ∈ L p i ( μ ) with i = 1 , 2 ,

‖ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ‖ L q ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ L p 1 ( μ ) ‖ f 2 ‖ L p 2 ( μ ) ,

where 1 q = 1 p − 2 α and 1 p = 1 p 1 + 1 p 2 with 1 < p 1 , p 2 < 1 α .

Finally, we establish the following lemma on W τ .

Lemma 2.6

Let τ ∈ ( 0 , 2 ) and u ( ⋅ , ⋅ ) ∈ W τ . Then,

there exists some positive constant C ˜ such that, for any x ∈ X and r > 0 ,
u ( x , 6 r ) ≤ C ˜ u ( x , r ) ;
for any ball B ⊂ X and 1 ≤ p < ∞ ,
‖ χ B ‖ M p u ( μ ) < ∞ .

Proof

(a) From (1.1) and (1.19), it then follows that

λ ( x , r ) λ ( x , 6 r ) τ u ( x , 6 r ) ≤ C u ( x , r ) ⇔ u ( x , 6 r ) ≤ C λ ( x , 6 r ) λ ( x , r ) τ u ( x , r ) ⇔ u ( x , 6 r ) ≤ C C ( λ ) λ ( x , r ) λ ( x , r ) τ u ( x , r ) ⇔ u ( x , 6 r ) ≤ C ˜ u ( x , r ) ,

where C ˜ ≔ C C ( λ ) τ .

(b) It suffices to show that B ≔ B ( 0 , R ) ∈ M p u ( μ ) , where R > 0 . For any x ∈ X and r > 0 , when r ≥ 1 , Definition 1.10 and (1.20) give

1 u ( x , r ) ‖ χ B χ B ( x , r ) ‖ L p ( μ ) ≤ 1 u ( x , r ) ‖ χ B ‖ L p ( μ ) ≤ C ‖ χ B ( 0 , R ) ‖ L p ( μ )

for some C > 0 .

If 0 < r < 1 , χ B ⋂ χ B ( y , r ) ≠ ∅ when ∣ y ∣ < R + r < 1 + R ; moreover, if χ B ⋂ χ B ( x , r ) ≠ ∅ , then B ( x , r ) ⊂ B ( 0 , R + 2 ) . Definition 1.10 and (1.19) yield

1 u ( x , r ) ‖ χ B χ B ( x , r ) ‖ L p ( μ ) ≤ 1 u ( x , r ) ‖ χ B ( x , r ) ‖ L p ( μ ) ≤ u ( x , 1 ) u ( x , r ) 1 u ( x , 1 ) ‖ χ B ( x , 1 ) ‖ L p ( μ ) ≤ C 1 u ( x , 1 ) ‖ χ B ( x , 1 ) ‖ L p ( μ ) ≤ C ‖ χ B ( 0 , 2 + R ) ‖ L p ( μ ) .

Combining the aforementioned two inequalities, we deduce that χ B ∈ M p u ( μ ) .□

3 Proofs of Theorems 1.15 and 1.16

Here we prove Theorems 1.15 and 1.16.

Proof of Theorem 1.15

Let B = B ( x , r ) be the open ball centered at x ∈ X with radius r > 0 . Represent functions f i as

(3.1) f i = f i 1 + f i ∞ = f i χ 6 B + f i χ X \ ( 6 B ) , i = 1 , 2 .

Then, write

sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 , f 2 ) χ B ( x , r ) ‖ L p ( μ ) ≤ sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 1 , f 2 1 ) χ B ( x , r ) ‖ L q ( μ ) + sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 1 , f 2 ∞ ) χ B ( x , r ) ‖ L q ( μ ) + sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 ∞ , f 2 1 ) χ B ( x , r ) ‖ L q ( μ ) + sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 ∞ , f 2 ∞ ) χ B ( x , r ) ‖ L q ( μ ) = D 1 + D 2 + D 3 + D 4 .

From (1.15), (1.19), Lemma 2.4, Lemma 2.6 (a), and u 1 u 2 = u , it then follows that

sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 1 , f 2 1 ) χ B ( x , r ) ‖ L q ( μ ) ≤ C sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ f 1 χ B ( x , 6 r ) ‖ L p 1 ( μ ) ‖ f 2 χ B ( x , 6 r ) ‖ L p 1 ( μ ) ≤ C sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) u 2 ( x , 6 r ) [ u 1 ( x , 6 r ) ] − 1 ‖ f 1 χ B ( x , 6 r ) ‖ L p 1 ( μ ) [ u 2 ( x , 6 r ) ] − 1 ‖ f 2 χ B ( x , 6 r ) ‖ L p 1 ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , r ) u 2 ( x , r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

To estimate D 2 , we first consider ∣ T ˜ θ , α ( f 1 1 , f 2 ∞ ) ( y ) ∣ with y ∈ B ( x , r ) . By applying (1.2), (1.8), Hölder’s inequality, and (1.15), we have

∣ T ˜ θ , α ( f 1 1 , f 2 ∞ ) ( y ) ∣ ≤ ∫ X 2 ∣ K θ , α ( y , z 1 , z 2 ) ∣ ∣ f 1 1 ( z 1 ) ∣ ∣ f 2 ∞ ( z 2 ) ∣ d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 ∣ f 1 1 ( z 1 ) ∣ ∣ f 2 ∞ ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ 6 B ∣ f 1 ( z 1 ) ∣ d μ ( z 1 ) ∫ X \ ( 6 B ) ∣ f 2 ( z 2 ) ∣ [ λ ( y , d ( y , z 2 ) ) ] 2 − 2 α d μ ( z 2 ) ≤ C ∫ 6 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 [ μ ( 6 B ) ] 1 − 1 p 1 × ∑ k = 1 ∞ ∫ 6 k + 1 B \ ( 6 k B ) ∣ f 2 ( z 2 ) ∣ [ λ ( x , d ( x , z 2 ) ) ] 2 − 2 α d μ ( z 2 ) ≤ C [ u 1 ( x , 6 r ) ] − 1 ∫ 6 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 × ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ d μ ( z 2 ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 × ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 6 k + 1 B ) ] 1 − 1 p 2 ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ∑ k = 1 ∞ [ μ ( 6 k + 1 B ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ∑ k = 1 ∞ [ λ ( x , 6 k + 1 r ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) u 1 ( x , 6 r ) [ λ ( x , 6 r ) ] 1 − 1 p 1 ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ∑ k = 1 ∞ [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) u 1 ( x , 6 r ) .

Furthermore, from (1.15) and (1.19), it then follows that

D 2 = sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 1 , f 2 ∞ ) χ B ( x , r ) ‖ L q ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) [ μ ( B ) ] 1 q × ∑ k = 1 ∞ [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ [ λ ( x , r ) ] 1 + 1 q − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ [ λ ( x , r ) ] 1 + 1 p − 2 α − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ [ λ ( x , r ) ] 1 + 1 p 2 − 2 α [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , r ) u 2 ( x , r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ,

where 0 < α < 1 , 1 q = 1 p − 2 α , and 1 p = 1 p 1 + 1 p 2 with 1 < p 1 , p 2 < 1 α .

With an argument similar to that used in the estimate of D 2 , it is easy to obtain that

D 3 ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Now, let us turn D 4 . For any y ∈ B ( x , r ) , by applying (1.8), Hölder’s inequality, and (1.15), we have

∣ T ˜ θ , α ( f 1 ∞ , f 2 ∞ ) ( y ) ∣ ≤ C ∫ ( X \ ( 6 B ) ) 2 ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∑ k = 1 ∞ ∫ ( 6 k + 1 B \ ( 6 k B ) ) 2 ∏ j = 1 2 ∣ f j ( z j ) ∣ [ λ ( y , d ( y , z j ) ) ] 1 − α d μ ( z 1 ) d μ ( z 2 ) ≤ C ∑ k = 1 ∞ ∏ j = 1 2 ∫ 6 k + 1 B \ ( 6 k B ) ∣ f j ( z j ) ∣ [ λ ( x , d ( x , z j ) ) ] 1 − α d μ ( z j ) ≤ C ∑ k = 1 ∞ ∏ j = 1 2 1 [ λ ( x , 6 k r ) ] 1 − α ∫ 6 k + 1 B ∣ f j ( z j ) ∣ d μ ( z j ) ≤ C ∑ k = 1 ∞ ∏ j = 1 2 1 [ λ ( x , 6 k r ) ] 1 − α ∫ 6 k + 1 B ∣ f j ( z j ) ∣ p j d μ ( z j ) 1 p j [ μ ( 6 k + 1 B ) ] 1 − 1 p j ≤ C ∑ k = 1 ∞ ∏ j = 1 2 [ μ ( 6 k + 1 B ) ] 1 − 1 p j [ λ ( x , 6 k r ) ] 1 − α u j ( x , 6 k + 1 r ) × [ u j ( x , 6 k + 1 r ) ] − 1 ∫ 6 k + 1 B ∣ f j ( z j ) ∣ p j d μ ( z j ) 1 p j ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ∑ k = 1 ∞ ∏ j = 1 2 [ λ ( x , 6 k + 1 r ) ] 1 − 1 p j [ λ ( x , 6 k r ) ] 1 − α u j ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ∑ k = 1 ∞ ∏ j = 1 2 1 [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) .

Furthermore, it follows, from (1.1), (1.15), (1.19), and 1 q = 1 p 1 + 1 p 2 − 2 α , that

D 4 = sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( f 1 ∞ , f 2 ∞ ) χ B ( x , r ) ‖ L q ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ χ B ( x , r ) ‖ L q ( μ ) × ∑ k = 1 ∞ ∏ j = 1 2 1 [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ ∏ j = 1 2 [ μ ( B ( x , r ) ) ] 1 q [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ ∏ j = 1 2 [ λ ( x , r ) ] 1 p j − α [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ [ λ ( x , r ) ] 1 p 1 − α [ λ ( x , 6 k r ) ] 1 p 1 − α u 1 ( x , 6 k + 1 r ) × ∑ k = 1 ∞ [ λ ( x , r ) ] 1 p 2 − α [ λ ( x , 6 k r ) ] 1 p 2 − α u 2 ( x , 6 k + 1 r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , r ) u 2 ( x , r ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ,

where we use the following inequality:

(3.2) ∑ j = 1 ∞ a j b j ≤ ∑ j = 1 ∞ a j r 1 r ∑ j = 1 ∞ b j r ′ 1 r ′ ≤ ∑ j = 1 ∞ a j ∑ j = 1 ∞ b j , for a j , b j > 0 and r > 1 .

Combining the estimates of D 1 , D 2 , and D 3 , the proof of Theorem 1.15 is completed.□

Proof of Theorem 1.16

For any y ∈ B ( x , r ) , by (1.8) and ( 1.12 ′ ) , we have

(3.3) ∣ T ˜ θ , α ( f 1 , f 2 ) ( y ) ∣ ≤ ∫ X 2 ∣ K θ , α ( y , z 1 , z 2 ) ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( y , d ( y , z 1 ) ) ] 1 − α [ λ ( y , d ( y , z 2 ) ) ] 1 − α d μ ( z 1 ) d μ ( z 2 ) ≤ C I α ( ∣ f 1 ∣ ) ( y ) I α ( ∣ f 2 ∣ ) ( y ) ,

where I α represents the fractional integral operator [6], i.e., for any y ∈ X , define

I α ( f ) ( y ) = ∫ X f ( z ) [ λ ( y , d ( y , z ) ) ] 1 − α d μ ( z ) , 0 < α < 1 .

Taking parameters 1 < q 1 and q 2 < ∞ such that

1 q 1 = 1 p 1 − α , 1 q 2 = 1 p 1 − α .

Since 1 = 1 p − 2 α and 1 p = 1 p 1 + 1 p 2 , we note that

1 q 1 + 1 q 2 = 1 p 1 + 1 p 2 − 2 α = 1 .

Using this, (1.15), (1.16), (3.3), Hölder’s inequality, and the ( L p ( μ ) , L q ( μ ) ) -boundedness of I α in [6], we deduce

‖ T ˜ θ , α ( f 1 , f 2 ) ‖ W M 1 u ( μ ) = sup x ∈ X , r > 0 sup t > 0 [ u ( x , r ) ] − 1 t μ ( { y ∈ B ( x , r ) : ∣ T ˜ θ , α ( f 1 , f 2 ) ( y ) ∣ > t } ) ≤ sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∫ B ( x , r ) ∣ T ˜ θ , α ( f 1 , f 2 ) ( y ) ∣ d μ ( y ) ≤ C sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∫ B ( x , r ) ∣ I α ( ∣ f 1 ∣ ) ( y ) I α ( ∣ f 2 ∣ ) ( y ) ∣ d μ ( y ) ≤ C sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ I α ( ∣ f 1 ∣ ) χ B ( x , r ) ‖ L q 1 ( μ ) ‖ I α ( ∣ f 2 ∣ ) χ B ( x , r ) ‖ L q 2 ( μ ) ≤ C sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , r ) u 2 ( x , r ) × [ u 1 ( x , r ) ] − 1 ‖ f 1 χ B ( x , r ) ‖ L p 1 ( μ ) [ u 2 ( x , r ) ] − 1 ‖ f 2 χ B ( x , r ) ‖ L p 2 ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Hence, the proof of Theorem 1.16 is completed.□

4 Proofs of Theorems 1.17 and 1.18

We prove Theorems 1.17 and 1.18.

Proof of Theorem 1.17

Let B = B ( x , r ) be the open ball centered at x with radius r . Decompose f i using the same form in (3.1) as f i = f i 1 + f i ∞ , where f i 1 = f i χ 6 B with i = 1 , 2 . Then, by the linearity of T ˜ θ , α , b 1 , b 2 and the Minkowski inequality, write

‖ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ‖ M q u ( μ ) ≤ ‖ T ˜ θ , α , b 1 , b 2 ( f 1 1 , f 2 1 ) ‖ M q u ( μ ) + ‖ T ˜ θ , α , b 1 , b 2 ( f 1 1 , f 2 ∞ ) ‖ M q u ( μ ) + ‖ T ˜ θ , α , b 1 , b 2 ( f 1 ∞ , f 2 1 ) ‖ M q u ( μ ) + ‖ T ˜ θ , α , b 1 , b 2 ( f 1 ∞ , f 2 ∞ ) ‖ M q u ( μ ) = E 1 + E 2 + E 3 + E 4 .

From (1.15), Lemmas 2.5 and 2.6 (a) and u 1 u 2 = u , it then follows that

‖ T ˜ θ , α , b 1 , b 2 ( f 1 1 , f 2 1 ) ‖ M q u ( μ ) = sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α , b 1 , b 2 ( f 1 1 , f 2 1 ) χ B ( x , r ) ‖ L p ( μ ) ≤ sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α , b 1 , b 2 ( f 1 1 , f 2 1 ) ‖ L p ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ f 1 1 ‖ L p 1 ( μ ) ‖ f 2 1 ‖ L p 2 ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) u 2 ( x , 6 r ) × [ u 1 ( x , 6 r ) ] − 1 ‖ f 1 χ 6 B ‖ L p 1 ( μ ) [ u 2 ( x , 6 r ) ] − 1 ‖ f 2 χ 6 B ‖ L p 2 ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 u 1 ( x , r ) u 2 ( x , r ) u ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

For any y ∈ B ( x , r ) , since

T ˜ θ , α , b 1 , b 2 ( f 1 1 , f 2 ∞ ) ( y ) = ( b 1 ( y ) − ( b 1 ) 6 B ) ( b 2 ( y ) − ( b 2 ) 6 B ) T ˜ θ , α ( f 1 1 , f 2 ∞ ) ( y ) + ( b 1 ( y ) − ( b 1 ) 6 B ) T ˜ θ , α ( f 1 1 , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) + ( b 2 ( y ) − ( b 2 ) 6 B ) T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 1 , f 2 ∞ ) ( y ) + T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 1 , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ,

then, by the Minkowski inequality, write

E 2 = ‖ T ˜ θ , α , b 1 , b 2 ( f 1 1 , f 2 ∞ ) ‖ M q u ( μ ) ≤ ‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) T ˜ θ , α ( f 1 1 , f 2 ∞ ) ‖ M q u ( μ ) + ‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) T ˜ θ , α ( f 1 1 , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) + ‖ ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 1 , f 2 ∞ ) ‖ M q u ( μ ) + ‖ T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 1 , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) = E 2 1 + E 2 2 + E 2 3 + E 2 4 .

To estimate E 2 1 , take 1 q 1 ≔ 1 p 1 − α and 1 q 2 ≔ 1 p 2 − α . Moreover, since 1 q = 1 p − 2 α and 1 p = 1 p 1 + 1 p 2 , then we have 1 q = 1 q 1 + 1 q 2 . By this, the estimate of ∣ T ˜ θ , α ( f 1 1 , f 2 ∞ ) ( y ) ∣ in D 2 , Hölder’s inequality, (1.19), (2.1), Lemma 2.6 (a), and u 1 u 2 = u , we have

‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) T ˜ θ , α ( f 1 1 , f 2 ∞ ) ‖ M q u ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) u 1 ( x , 6 r ) × ∫ B ( x , r ) ∣ ( b 1 ( y ) − ( b 1 ) 6 B ) ( b 2 ( y ) − ( b 2 ) 6 B ) ∣ q d μ ( y ) 1 q ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) u 1 ( x , 6 r ) × ∫ B ( x , r ) ∣ b 1 ( y ) − ( b 1 ) 6 B ∣ q 1 d μ ( y ) 1 q 1 ∫ B ( x , r ) ∣ b 2 ( y ) − ( b 2 ) 6 B ∣ q 2 d μ ( y ) 1 q 2 ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) × sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) u 1 ( x , 6 r ) [ μ ( 6 B ( x , r ) ) ] 1 q ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) × sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ [ λ ( x , 6 r ) ] 1 + 1 p 2 − 2 α [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) u 1 ( x , 6 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) × sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , r ) u 2 ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

To estimate E 2 2 , we first consider ∣ T ˜ θ , α ( f 1 1 , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ∣ for any y ∈ B ( x , r ) . From (1.1), (1.8), Hölder’s inequality, (1.15), and (2.1), it follows that

∣ T ˜ θ , α ( f 1 1 , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ∣ ≤ ∫ X 2 ∣ K θ , α ( y , z 1 , z 2 ) ∣ ∣ b 2 ( z ) − ( b 2 ) 6 B ∣ ∣ f 1 1 ( z 1 ) ∣ ∣ f 2 ∞ ( z 2 ) ∣ d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 ∣ b 2 ( z ) − ( b 2 ) 6 B ∣ ∣ f 1 1 ( z 1 ) ∣ ∣ f 2 ∞ ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ 6 B ∣ f 1 ( z 1 ) ∣ d μ ( z 1 ) ∫ X \ ( 6 B ) ∣ b 2 ( z ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( y , d ( y , z 2 ) ) ] 2 − 2 α d μ ( z 2 ) ≤ C ∫ 6 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 [ μ ( 6 B ) ] 1 − 1 p 1 ∑ k = 1 ∞ ∫ 6 k + 1 B \ ( 6 k B ) ∣ b 2 ( z ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( x , d ( x , z 2 ) ) ] 2 − 2 α d μ ( z 2 ) ≤ C [ u 1 ( x , 6 r ) ] − 1 ∫ 6 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 × ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α ∫ 6 k + 1 B ∣ b 2 ( z ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 2 )

≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α ( ∣ ( b 2 ) 6 B − ( b 2 ) 6 k + 1 B ∣ × ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ d μ ( z 2 ) + ∫ 6 k + 1 B ∣ b 2 ( z ) − ( b 2 ) 6 k + 1 B ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 2 ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α × k ‖ b 2 ‖ RBMO ˜ ( μ ) ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 6 k + 1 B ) ] 1 − 1 p 2 + ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 ∫ 6 k + 1 B ∣ b 2 ( z ) − ( b 2 ) 6 k + 1 B ∣ p 2 ′ d μ ( z 2 ) 1 p 2 ′

≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α × k ‖ b 2 ‖ RBMO ˜ ( μ ) ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 6 k + 1 B ) ] 1 − 1 p 2 + C ‖ b 2 ‖ RBMO ˜ ( μ ) ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 6 k + 2 B ) ] 1 − 1 p 2 ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 ∑ k = 1 ∞ ( k + 1 ) 1 [ λ ( x , 6 k r ) ] 2 − 2 α × [ u 2 ( x , 6 k + 1 r ) ] − 1 ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 u 2 ( x , 6 k + 1 r ) [ μ ( 6 k + 2 B ) ] 1 − 1 p 2

≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) u 1 ( x , 6 r ) [ μ ( 6 B ) ] 1 − 1 p 1 × ∑ k = 1 ∞ ( k + 1 ) [ μ ( 6 k + 2 B ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) u 1 ( x , 6 r ) [ λ ( x , 6 r ) ] 1 − 1 p 1 × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 k + 2 r ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ,

Furthermore, by applying (1.15), (1.21), and (2.1), we obtain

E 2 2 = ‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) T ˜ θ , α ( f 1 1 , ( b 2 − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ∫ B ( x , r ) ∣ ( b 1 ( y ) − ( b 1 ) 6 B ) ∣ q d μ ( y ) 1 q ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) [ μ ( 6 B ) ] 1 q ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) × sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 + 1 q − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) × sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 + 1 p 2 − 2 α [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , r ) u 2 ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ,

where we use the following fact introduced in [6]:

(4.1) ∣ b B − b 6 k + 1 B ∣ ≤ C k ‖ b ‖ RBMO ˜ ( μ ) .

With an argument similar to that used in the estimate of E 2 2 , it is easy to obtain that

E 2 3 ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Now, let us consider E 2 4 . For any y ∈ B ( x , r ) , by applying (1.1), (1.8), Hölder’s inequality, (1.15), (2.1), and (4.1), we have

∣ T ˜ θ , α ( ( b 1 − ( b 1 ) 6 B ) f 1 1 , ( b 2 − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ∣ ≤ ∫ X 2 ∣ K θ , α ( y , z 1 , z 2 ) ∣ ∣ b 1 ( z 1 ) − ( b 1 ) 6 B ∣ ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 1 1 ( z 1 ) ∣ ∣ f 2 ∞ ( z 2 ) ∣ d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 ∣ b 1 ( z 1 ) − ( b 1 ) 6 B ∣ ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 1 1 ( z 1 ) ∣ ∣ f 2 ∞ ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ 6 B ∣ b 1 ( z 1 ) − ( b 1 ) 6 B ∣ ∣ f 1 ( z 1 ) ∣ d μ ( z 1 ) ∫ X \ ( 6 B ) ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( y , d ( y , z 2 ) ) ] 2 − 2 α d μ ( z 2 ) ≤ C ∫ 6 B ∣ b 1 ( z 1 ) − ( b 1 ) 6 B ∣ ∣ f 1 ( z 1 ) ∣ d μ ( z 1 ) ∑ k = 1 ∞ ∫ 6 k + 1 B \ ( 6 k B ) ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( x , d ( x , z 2 ) ) ] 2 − 2 α d μ ( z 2 ) ≤ C ∫ 6 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 ∫ 6 B ∣ b 1 ( z 1 ) − ( b 1 ) 6 B ∣ p 1 ′ d μ ( z 1 ) 1 p 1 ′ × ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α ∫ 6 k + 1 B ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 2 ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) [ u 1 ( x , 6 r ) ] − 1 ∫ 6 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 [ μ ( 12 B ) ] 1 − 1 p 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α ∣ ( b 2 ) 6 B − ( b 2 ) 6 k + 1 B ∣ ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ d μ ( z 2 ) + ∫ 6 k + 1 B ∣ b 2 ( z 2 ) − ( b 2 ) 6 k + 1 B ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 2 )

≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) [ μ ( 12 B ) ] 1 − 1 p 1 u 1 ( x , 6 r ) ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α × k ‖ b 2 ‖ RBMO ˜ ( μ ) ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 6 k + 1 B ) ] 1 − 1 p 2 + ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 ∫ 6 k + 1 B ∣ b 2 ( z 2 ) − ( b 2 ) 6 k + 1 B ∣ p 2 ′ d μ ( z 2 ) 1 p 2 ′ ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) [ μ ( 12 B ) ] 1 − 1 p 1 u 1 ( x , 6 r ) ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 2 − 2 α × k [ u 2 ( x , 6 k + 1 r ) ] − 1 ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 u 2 ( x , 6 k + 1 r ) [ μ ( 6 k + 1 B ) ] 1 − 1 p 2 + [ u 2 ( x , 6 k + 1 r ) ] − 1 ∫ 6 k + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 u 2 ( x , 6 k + 1 r ) [ μ ( 2 × 6 k + 1 B ) ] 1 − 1 p 2 x≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) [ μ ( 12 B ) ] 1 − 1 p 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ μ ( 2 × 6 k + 1 B ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) [ μ ( 12 B ) ] 1 − 1 p 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 2 × 6 k + 1 r ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) [ λ ( x , 2 × 6 r ) ] 1 − 1 p 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 2 × 6 k + 1 r ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) [ λ ( x , 6 r ) ] 1 − 1 p 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 k r ) ] 1 − 1 p 2 [ λ ( x , 6 k r ) ] 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) .

Furthermore, it follows, from (1.15), (1.21), 1 q = 1 p 1 + 1 p 2 − 2 α , and u 1 u 2 = u , that

E 2 4 = ‖ T ˜ θ , α ( ( b 1 − ( b 1 ) 6 B ) f 1 1 , ( b 2 − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) = sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ T ˜ θ , α ( ( b 1 − ( b 1 ) 6 B ) f 1 1 , ( b 2 − ( b 2 ) 6 B ) f 2 ∞ ) χ B ( x , r ) ‖ L q ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) [ μ ( B ( x , r ) ) ] 1 q ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) [ λ ( x , 6 r ) ] 1 q ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , 6 r ) × ∑ k = 1 ∞ ( k + 1 ) [ λ ( x , 6 r ) ] 1 + 1 p 1 − 2 α [ λ ( x , 6 k r ) ] 1 + 1 p 2 − 2 α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 u 1 ( x , r ) u 2 ( x , r ) u ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Combining the estimates of E 2 1 , E 2 2 , E 2 3 , and E 2 4 , we deduce

E 2 ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Similarly, we also have E 3 ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Finally, let us turn E 4 . For any y ∈ B ( x , r ) , since

T ˜ θ , α , b 1 , b 2 ( f 1 ∞ , f 2 ∞ ) ( y ) = ( b 1 ( y ) − ( b 1 ) 6 B ) ( b 2 ( y ) − ( b 2 ) 6 B ) T ˜ θ , α ( f 1 ∞ , f 2 ∞ ) ( y ) + ( b 1 ( y ) − ( b 1 ) 6 B ) T ˜ θ , α ( f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) + ( b 2 ( y ) − ( b 2 ) 6 B ) T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 ∞ , f 2 ∞ ) ( y ) + T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ,

then, by the Minkowski inequality, write

E 4 = ‖ T ˜ θ , α , b 1 , b 2 ( f 1 ∞ , f 2 ∞ ) ‖ M q u ( μ ) ≤ ‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) T ˜ θ , α ( f 1 ∞ , f 2 ∞ ) ‖ M q u ( μ ) + ‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) T ˜ θ , α ( f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) + ‖ ( b 2 ( y ) − ( b 2 ) 6 B ) T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 ∞ , f 2 ∞ ) ‖ M q u ( μ ) + ‖ T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) = E 4 1 + E 4 2 + E 4 3 + E 4 4 .

To estimate E 4 1 , take 1 q 1 ≔ 1 p 1 − α and 1 q 2 ≔ 1 p 2 − α . Because 1 q = 1 p − 2 α and 1 p = 1 p 1 + 1 p 2 , then, we deduce

1 q 1 + 1 q 2 = 1 p 1 + 1 p 2 − 2 α = 1 q .

From this, the estimate of ∣ T ˜ θ , α ( f 1 ∞ , f 2 ∞ ) ( y ) ∣ in D 4 , Hölder’s inequality, (1.21), (2.1), and (3.2), it follows that

E 4 1 = ‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) T ˜ θ , α ( f 1 ∞ , f 2 ∞ ) ‖ M q u ( μ ) ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ ∏ j = 1 2 1 [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) × ∫ B ( x , r ) ∣ ( b 1 ( y ) − ( b 1 ) 6 B ) ( b 2 ( y ) − ( b 2 ) 6 B ) ∣ q d μ ( y ) 1 q ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ ∏ j = 1 2 1 [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) × ∫ B ( x , r ) ∣ b 1 ( y ) − ( b 1 ) 6 B ∣ q 1 d μ ( y ) 1 q 1 ∫ B ( x , r ) ∣ b 2 ( y ) − ( b 2 ) 6 B ∣ q 1 d μ ( y ) 1 q 2 ≤ C ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ ∏ j = 1 2 1 [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) × ∫ B ( x , r ) ∣ b 1 ( y ) − ( b 1 ) B ∣ q 1 d μ ( y ) 1 q 1 + ∣ ( b 1 ) B − ( b 1 ) 6 B ∣ [ μ ( B ( x , r ) ) ] 1 q 1 × ∫ B ( x , r ) ∣ b 2 ( y ) − ( b 2 ) B ∣ q 2 d μ ( y ) 1 q 2 + ∣ ( b 2 ) B − ( b 2 ) 6 B ∣ [ μ ( B ( x , r ) ) ] 1 q 2 ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 × ∑ k = 1 ∞ ∏ j = 1 2 [ μ ( 6 B ) ] 1 q j [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 × ∑ k = 1 ∞ ∏ j = 1 2 [ λ ( x , 6 r ) ] 1 p j − α [ λ ( x , 6 k r ) ] 1 p j − α u j ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 × ∑ k = 1 ∞ λ ( x , 6 r ) λ ( x , 6 k r ) 1 p 1 − α u 1 ( x , 6 k + 1 r ) ∑ k = 1 ∞ λ ( x , 6 r ) λ ( x , 6 k r ) 1 p 2 − α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 u 1 ( x , r ) u 2 ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

To estimate E 4 2 , we first consider ∣ T ˜ θ , α ( f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ∣ with y ∈ B ( x , r ) . By applying (1.1), (1.8), (1.15), Hölder’s inequality, (2.1), and (4.1), we have

∣ T ˜ θ , α ( f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ∣ ≤ ∫ X 2 ∣ K θ , α ( y , z 1 , z 2 ) ∣ ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 1 ∞ ( z 1 ) ∣ ∣ f 2 ∞ ( z 2 ) ∣ d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 \ ( 6 B ) 2 ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 \ ( 6 B ) 2 ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( y , d ( y , z 1 ) ) ] 1 − α [ λ ( y , d ( y , z 2 ) ) ] 1 − α d μ ( z 1 ) d μ ( z 2 ) ≤ C ∑ k = 1 ∞ ∫ 6 k + 1 B \ ( 6 k B ) ∣ f 1 ( z 1 ) ∣ [ λ ( x , d ( x , z 1 ) ) ] 1 − α d μ ( z 1 ) × ∑ j = 1 ∞ ∫ 6 j + 1 B \ ( 6 j B ) ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( x , d ( x , z 2 ) ) ] 1 − α d μ ( z 2 ) ≤ C ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 − α ∫ 6 k + 1 B ∣ f 1 ( z 1 ) ∣ d μ ( z 1 ) × ∑ j = 1 ∞ 1 [ λ ( x , 6 j r ) ] 1 − α ∫ 6 j + 1 B ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 2 )

≤ C ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 − α ∫ 6 k + 1 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 [ μ ( 6 k + 1 B ) ] 1 − 1 p 1 × ∑ j = 1 ∞ 1 [ λ ( x , 6 j r ) ] 1 − α ∫ 6 j + 1 B ∣ b 2 ( z 2 ) − ( b 2 ) 6 j + 1 B ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 2 ) + ∣ ( b 2 ) 6 B − ( b 2 ) 6 j + 1 B ∣ ∫ 6 j + 1 B ∣ f 2 ( z 2 ) ∣ d μ ( z 2 ) ≤ C ∑ k = 1 ∞ [ μ ( 6 k + 1 B ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 − α [ u ( x , 6 k + 1 r ) ] − 1 ∫ 6 k + 1 B ∣ f 1 ( z 1 ) ∣ p 1 d μ ( z 1 ) 1 p 1 u ( x , 6 k + 1 r ) × ∑ j = 1 ∞ 1 [ λ ( x , 6 j r ) ] 1 − α ∫ 6 j + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 × ∫ 6 j + 1 B ∣ b 2 ( z 2 ) − ( b 2 ) 6 j + 1 B ∣ p 2 ′ d μ ( z 2 ) 1 p 2 ′ + C j ‖ b 2 ‖ RBMO ˜ ( μ ) ∫ 6 j + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 6 j + 1 B ) ] 1 − 1 p 2 ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ∑ k = 1 ∞ [ λ ( x , 6 k + 1 r ) ] 1 − 1 p 1 [ λ ( x , 6 k r ) ] 1 − α u ( x , 6 k + 1 r ) × ∑ j = 1 ∞ 1 [ λ ( x , 6 j r ) ] 1 − α ∫ 6 j + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 2 × 6 j + 1 B ) ] 1 − 1 p 2 + j ∫ 6 j + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 [ μ ( 6 j + 1 B ) ] 1 − 1 p 2

≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 p 1 − α u ( x , 6 k + 1 r ) × ∑ j = 1 ∞ ( j + 1 ) [ μ ( 2 × 6 j + 1 B ) ] 1 − 1 p 2 [ λ ( x , 6 j r ) ] 1 − α ∫ 6 j + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 p 1 − α u ( x , 6 k + 1 r ) × ∑ j = 1 ∞ ( j + 1 ) [ μ ( 2 × 6 j + 1 B ) ] 1 − 1 p 2 [ λ ( x , 6 j r ) ] 1 − α u ( x , 6 j + 1 r ) × [ u ( x , 6 j + 1 r ) ] − 1 ∫ 6 j + 1 B ∣ f 2 ( z 2 ) ∣ p 2 d μ ( z 2 ) 1 p 2 ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 p 1 − α u ( x , 6 k + 1 r ) × ∑ j = 1 ∞ ( j + 1 ) [ λ ( x , 2 × 6 j + 1 r ) ] 1 − 1 p 2 [ λ ( x , 6 j r ) ] 1 − α u ( x , 6 j + 1 r ) ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 p 1 − α u ( x , 6 k + 1 r ) × ∑ j = 1 ∞ ( j + 1 ) 1 [ λ ( x , 6 j r ) ] 1 p 2 − α u ( x , 6 j + 1 r ) .

Furthermore, it follows, from (1.15), (1.21), 1 q = 1 p 1 + 1 p 2 − 2 α , and u 1 u 2 = u , that

E 4 2 = ‖ ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) T ˜ θ , α ( f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) = sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∫ B ( x , r ) ∣ ( b 1 ( y ) − ( b 1 ) 6 B ) T ˜ θ , α ( f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ∣ q d μ ( y ) 1 q ≤ C ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 p 1 − α u ( x , 6 k + 1 r ) × ∑ j = 1 ∞ ( j + 1 ) 1 [ λ ( x , 6 j r ) ] 1 p 2 − α u ( x , 6 j + 1 r ) ∫ B ( x , r ) ∣ b 1 ( y ) − ( b 1 ) 6 B ∣ q d μ ( y ) 1 q ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 [ μ ( 6 B ) ] 1 q × ∑ k = 1 ∞ 1 [ λ ( x , 6 k r ) ] 1 p 1 − α u ( x , 6 k + 1 r ) ∑ j = 1 ∞ ( j + 1 ) 1 [ λ ( x , 6 j r ) ] 1 p 2 − α u ( x , 6 j + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 × ∑ k = 1 ∞ λ ( x , 6 r ) λ ( x , 6 k r ) 1 p 1 − α u ( x , 6 k + 1 r ) ∑ j = 1 ∞ ( j + 1 ) λ ( x , 6 r ) λ ( x , 6 j r ) 1 p 2 − α u ( x , 6 j + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 u 1 ( x , r ) u 2 ( x , r ) u ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

With an argument similar to that used in the estimate of E 4 2 , it is easy to obtain that

E 4 3 ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Finally, let us estimate E 4 4 . For any y ∈ B ( x , r ) , by (1.1), (1.8), (1.15), Hölder’s inequality, (2.1), and (4.1), we have

∣ T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ( y ) ∣ ≤ C ∫ X 2 \ ( 6 B ) 2 ∣ b 1 ( z 1 ) − ( b 1 ) 6 B ∣ ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∑ k = 1 ∞ ∫ ( 6 k + 1 B ) 2 \ ( 6 k B ) 2 ∣ b 1 ( z 1 ) − ( b 1 ) 6 B ∣ ∣ b 2 ( z 2 ) − ( b 2 ) 6 B ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( y , d ( y , z 1 ) ) ] 1 − α [ λ ( y , d ( y , z 2 ) ) ] 1 − α d μ ( z 1 ) d μ ( z 2 ) ≤ C ∑ k = 1 ∞ ∏ i = 1 2 ∫ 6 k + 1 B \ ( 6 k B ) ∣ b i ( z i ) − ( b i ) 6 B ∣ ∣ f i ( z i ) ∣ [ λ ( x , d ( x , z i ) ) ] 1 − α d μ ( z i ) ≤ C ∑ k = 1 ∞ ∏ i = 1 2 1 [ λ ( x , 6 k r ) ] 1 − α ∫ 6 k + 1 B ∣ b i ( z i ) − ( b i ) 6 B ∣ ∣ f i ( z i ) ∣ d μ ( z i ) ≤ C ∑ k = 1 ∞ ∏ i = 1 2 1 [ λ ( x , 6 k r ) ] 1 − α ∣ ( b i ) 6 B − ( b i ) 6 k + 1 B ∣ ∫ 6 k + 1 B ∣ f i ( z i ) ∣ d μ ( z i ) + ∫ 6 k + 1 B ∣ b i ( z i ) − ( b i ) 6 k + 1 B ∣ ∣ f i ( z i ) ∣ d μ ( z i ) ≤ C ∑ k = 1 ∞ ∏ i = 1 2 1 [ λ ( x , 6 k r ) ] 1 − α C k ‖ b i ‖ RBMO ˜ ( μ ) ∫ 6 k + 1 B ∣ f i ( z i ) ∣ p i d μ ( z i ) 1 p i [ μ ( 6 k + 1 B ) ] 1 − 1 p i + ∫ 6 k + 1 B ∣ f i ( z i ) ∣ p i d μ ( z i ) 1 p i ∫ 6 k + 1 B ∣ b i ( z i ) − ( b i ) 6 k + 1 B ∣ p i ′ d μ ( z i ) 1 p i ′ ≤ C ∑ k = 1 ∞ ∏ i = 1 2 1 [ λ ( x , 6 k r ) ] 1 − α C k ‖ b i ‖ RBMO ˜ ( μ ) ∫ 6 k + 1 B ∣ f i ( z i ) ∣ p i d μ ( z i ) 1 p i [ μ ( 6 k + 1 B ) ] 1 − 1 p i + C ‖ b i ‖ RBMO ˜ ( μ ) ∫ 6 k + 1 B ∣ f i ( z i ) ∣ p i d μ ( z i ) 1 p i [ μ ( 2 × 6 k + 1 B ) ] 1 − 1 p i ≤ C ∑ k = 1 ∞ ∏ i = 1 2 ( k + 1 ) ‖ b i ‖ RBMO ˜ ( μ ) [ μ ( 2 × 6 k + 1 B ) ] 1 − 1 p i [ λ ( x , 6 k r ) ] 1 − α u i ( x , 6 k + 1 r ) × [ u i ( x , 6 k + 1 r ) ] − 1 ∫ 6 k + 1 B ∣ f i ( z i ) ∣ p i d μ ( z i ) 1 p i ≤ C ∑ k = 1 ∞ ∏ i = 1 2 ( k + 1 ) ‖ b i ‖ RBMO ˜ ( μ ) ‖ f i ‖ M p i u i ( μ ) [ λ ( x , 6 k + 1 r ) ] 1 − 1 p i [ λ ( x , 6 k r ) ] 1 − α u i ( x , 6 k + 1 r ) ≤ C ∑ k = 1 ∞ ∏ i = 1 2 ( k + 1 ) ‖ b i ‖ RBMO ˜ ( μ ) ‖ f i ‖ M p i u i ( μ ) 1 [ λ ( x , 6 k r ) ] 1 p i − α u i ( x , 6 k + 1 r ) .

From this, (1.15), (1.21), 1 q = 1 p 1 + 1 p 2 − 2 α , (3.2), and u 1 u 2 = u , it follows that

E 4 4 = ‖ T ˜ θ , α ( ( b 1 ( ⋅ ) − ( b 1 ) 6 B ) f 1 ∞ , ( b 2 ( ⋅ ) − ( b 2 ) 6 B ) f 2 ∞ ) ‖ M q u ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) × sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ ∏ i = 1 2 ( k + 1 ) [ μ ( B ) ] 1 p i − α [ λ ( x , 6 k r ) ] 1 p i − α u i ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) × sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∑ k = 1 ∞ ∏ i = 1 2 ( k + 1 ) [ λ ( x , r ) ] 1 p i − α [ λ ( x , 6 k r ) ] 1 p i − α u i ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 × ∑ k = 1 ∞ ( k + 1 ) λ ( x , r ) λ ( x , 6 k r ) 1 p 1 − α u 1 ( x , 6 k + 1 r ) × ∑ k = 1 ∞ ( k + 1 ) λ ( x , r ) λ ( x , 6 k r ) 1 p 2 − α u 2 ( x , 6 k + 1 r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 u 1 ( x , r ) u 2 ( x , r ) u ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Combining the estimates for E 4 1 , E 4 2 , E 4 3 , E 1 , E 2 and E 3 , the proof of Theorem 1.17 is completed.□

Proof of Theorem 1.18

For any y ∈ B ( x , r ) , by (1.8) and ( 1.12 ′ ) , we have

(4.2) ∣ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ( y ) ∣ ≤ ∫ X 2 ∣ K θ , α ( y , z 1 , z 2 ) ∣ ∣ b 1 ( y ) − b 1 ( z 1 ) ∣ ∣ b 2 ( y ) − b 2 ( z 2 ) ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 ∣ b 1 ( y ) − b 1 ( z 1 ) ∣ ∣ b 2 ( y ) − b 2 ( z 2 ) ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ ∑ j = 1 2 [ λ ( y , d ( y , z j ) ) ] 1 − α 2 d μ ( z 1 ) d μ ( z 2 ) ≤ C ∫ X 2 ∣ b 1 ( y ) − b 1 ( z 1 ) ∣ ∣ b 2 ( y ) − b 2 ( z 2 ) ∣ ∣ f 1 ( z 1 ) ∣ ∣ f 2 ( z 2 ) ∣ [ λ ( y , d ( y , z 1 ) ) ] 1 − α [ λ ( y , d ( y , z 2 ) ) ] 1 − α d μ ( z 1 ) d μ ( z 2 ) ≤ C ∏ i = 1 2 ∫ X ∣ b i ( y ) − b i ( z i ) ∣ ∣ f i ( z i ) ∣ [ λ ( y , d ( y , z i ) ) ] 1 − α d μ ( z i ) ≤ C ∏ i = 1 2 [ b , I α ] ( ∣ f i ∣ ) ( y ) ,

where the commutator [ b , I α ] is defined by:

[ b , I α ] f ( y ) ≔ ∫ X ∣ b ( y ) − b ( z ) ∣ [ λ ( y , d ( y , z ) ) ] 1 − α f ( z ) d μ ( z ) , for any y ∈ X .

From (1.15), (1.16), (4.2), Hölder’s inequality, the ( L p ( μ ) , L q ( μ ) ) -boundedness of the commutator [ b , I α ] introduced in [23], u 1 u 2 = u , 1 q = 1 q 1 + 1 q 2 , 1 q 1 = 1 p 1 − α and 1 q 2 = 1 p 2 − α . It then follows that

‖ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ‖ W M 1 u ( μ ) = sup x ∈ X , r > 0 sup t > 0 [ u ( x , r ) ] − 1 t μ ( { y ∈ B ( x , r ) : ∣ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ( y ) ∣ > t } ) ≤ sup x ∈ X , r > 0 sup t > 0 [ u ( x , r ) ] − 1 t ∫ B ( x , r ) ∣ T ˜ θ , α , b 1 , b 2 ( f 1 , f 2 ) ( y ) ∣ t d μ ( y ) ≤ C sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ∫ B ( x , r ) ∣ [ b , I α ] ( ∣ f 1 ∣ ) ( y ) [ b , I α ] ( ∣ f 2 ∣ ) ( y ) ∣ d μ ( y ) ≤ C sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ [ b , I α ] ( ∣ f 1 ∣ ) χ B ( x , r ) ‖ L q 1 ( μ ) ‖ [ b , I α ] ( ∣ f 2 ∣ ) χ B ( x , r ) ‖ L q 2 ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) sup x ∈ X , r > 0 [ u ( x , r ) ] − 1 ‖ f 1 χ B ( x , r ) ‖ L p 1 ( μ ) ‖ f 2 χ B ( x , r ) ‖ L p 2 ( μ ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) sup x ∈ X , r > 0 u 1 ( x , r ) u 2 ( x , r ) u ( x , r ) ≤ C ‖ b 1 ‖ RBMO ˜ ( μ ) ‖ b 2 ‖ RBMO ˜ ( μ ) ‖ f 1 ‖ M p 1 u 1 ( μ ) ‖ f 2 ‖ M p 2 u 2 ( μ ) .

Hence, the proof of Theorem 1.18 is completed.□

Acknowledgments

The authors would also like to thank the anonymous referee for his/her careful reading that helped improving the presentation of this article.

Funding information: This work was supported by the National Natural Science Foundation of China (Grant No. 12201500), the Youth Science and Technology Fund of Gansu Province (Grant No. 22JR5RA173), and the Young Teachers’ Scientific Research Ability Promotion Project of Northwest Normal University (Grant No. NWNU-LKQN2020-07).
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-01-30

Revised: 2023-07-28

Accepted: 2023-08-24

Published Online: 2023-11-23

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

Articles in the same Issue

https://doi.org/10.1515/agms-2023-0101

Keywords for this article

non-homogeneous metric measure space; bilinear θ-type generalized fractional integral; commutator; space RBMO(μ); new generalized Morrey space

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