Home Mean oscillation and boundedness of multilinear operator related to multiplier operator
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Mean oscillation and boundedness of multilinear operator related to multiplier operator

  • Qiaozhen Zhao and Dejian Huang EMAIL logo
Published/Copyright: August 5, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, the boundedness of certain multilinear operator related to the multiplier operator from Lebesgue spaces to Orlicz spaces is obtained.

Keywords: multilinear operator; multiplier operator; BMO space; Orlicz space
MSC 2010: 42B20; 42B25

1 Introduction and preliminaries

Let b be a locally integrable function on R n and T be an integral operator. For a suitable function f , the commutator generated by b and T is defined by [ b , T ] ( f ) = T ( b f ) − b T ( f ) . The investigation of the commutator begins with Coifman-Rochberg-Weiss pioneering study and classical result [1]. If T is the Calderón-Zygmund singular integral operator, a well known result in [1] states that the commutator [ b , T ] (where b ∈ B M O ) is bounded on L p ( R n ) ( 1 < p < ∞ ) . In fact, they even proved that this property characterizes B M O functions. There are two major reasons for considering the problem of commutators, one is that the boundedness of commutator can produce some characterizations of function space, and the other one is that the theory of commutator plays an important role in studying the regularity of solutions of the second order elliptic and parabolic partial differential equations. With the development of singular integral operators [2,3], their commutators have been well studied. In [1,4,5], the authors proved that the commutators generated by the singular integral operators and B M O functions are bounded on L p ( R n ) for 1 < p < ∞ . Chanillo [6] proved a similar result when singular integral operators are replaced by the fractional integral operators. In [7], Janson proved the boundedness of the commutators generated by the singular integral operators and B M O functions from Lebesgue spaces to Orlicz spaces; he pointed out that one can also characterize those functions for which the above commutator is bounded from L p to L q (for q ≠ p ). As it turned out, this is equivalent to the functions being in a certain Lipschitz space. In [8,9,10, 11,12], the authors studied the algebras of singular integral operators. Calderón [13] introduced the multilinear singular integral operator as follows:

T b ( f ) ( x ) = ∫ R n R m + 1 ( b ; x , y ) ∣ x − y ∣ m K ( x − y ) f ( y ) d y ,

where R m + 1 ( b ; x , y ) denotes the ( m + 1 )th remainder of Taylor series of the function b at x about y , which is a non-trivial and natural extension of the commutator if m = 0 .

In this paper, we will study a more complicated expression, which extends naturally the formula of the above commutator, and prove the analogue of Janson’s theorem for these “multilinear expressions,” following closely Janson’s method, that is, we will prove the boundedness of the multilinear operator associated with the multiplier operator from Lebesgue space to Orlicz space.

Throughout this paper, Q denotes a cube of R n with sides parallel to the axes. For any locally integrable function f , the sharp function of f is defined by

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

where and in what follows, f Q = ∣ Q ∣ − 1 ∫ Q f ( x ) d x . It is well known [2,3] that

f # ( x ) ≈ sup Q ∋ x inf c ∈ C 1 ∣ Q ∣ ∫ Q ∣ f ( y ) − c ∣ d y .

Let M be the Hardy-Littlewood maximal operator defined by

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

We denote M p f = ( M ( f p ) ) 1 / p for 0 < p < ∞ . For 1 ≤ r < ∞ and 0 < β < n , let

M β , r ( f ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ 1 − r β / n ∫ Q ∣ f ( y ) ∣ r d y 1 / r .

We say that f belongs to B M O ( R n ) if f # belongs to L ∞ ( R n ) and ∣ ∣ f ∣ ∣ B M O = ∣ ∣ f # ∣ ∣ L ∞ . More generally, let ρ be a non-decreasing positive function on [ 0 , + ∞ ) and define B M O ρ ( R n ) as the space of all functions f such that

1 ∣ Q ( x , r ) ∣ ∫ Q ( x , r ) ∣ f ( y ) − f Q ∣ d y ≤ C ρ ( r ) .

For β > 0 , the Lipschitz space Lip β ( R n ) is the space of functions f such that

∣ ∣ f ∣ ∣ Lip β = sup x ≠ y ∣ f ( x ) − f ( y ) ∣ / ∣ x − y ∣ β < ∞ .

Let m f denote the distribution function of f , that is, m f ( t ) = ∣ { x ∈ R n : ∣ f ( x ) ∣ > t } ∣ .

Let ρ be a non-decreasing convex function on [ 0 , + ∞ ) with ρ ( 0 ) = 0 . ρ − 1 denotes the inverse function of ρ . The Orlicz space L ρ ( R n ) is defined by the set of functions f such that ∫ R n ρ ( λ ∣ f ( x ) ∣ ) d x < ∞ for some λ > 0 , and the norm is given by

∣ ∣ f ∣ ∣ L ρ = inf λ > 0 λ − 1 1 + ∫ R n ρ ( λ ∣ f ( x ) ∣ ) d x .

2 Main results

In this paper, we will study the multilinear operator associated with the multiplier operator [14].

A bounded measurable function k defined on R n ⧹ { 0 } is called a multiplier. The multiplier operator T associated with k is defined by

T ( f ) ( x ) = k ( x ) f ˆ ( x ) , f ∈ S ( R n ) ,

where f ˆ denotes the Fourier transform of f and S ( R n ) is the Schwartz test function class. We recall the definition of the class M ( s , l ) . Denote by ∣ x ∣ ∼ t the fact that the value of x lies in the annulus { x ∈ R n : a t < ∣ x ∣ < b t } , where 0 < a ≤ 1 < b < ∞ are values specified in each instance.

Definition 1

Let l ≥ 0 be a real number and 1 ≤ s ≤ 2 , then the multiplier k satisfies the condition M ( s , l ) , if

∫ ∣ ξ ∣ ∼ R ∣ D α k ( ξ ) ∣ s d ξ 1 s < C R n / s − ∣ α ∣

for all R > 0 and multi-indices α with ∣ α ∣ ≤ l , when l is a positive integer. In addition, if

∫ ∣ ξ ∣ ∼ R ∣ D α k ( ξ ) − D α k ( ξ − z ) ∣ s d ξ 1 s ≤ C ∣ z ∣ R γ R n s − ∣ α ∣

for all ∣ z ∣ < R / 2 and all multi-indices α with ∣ α ∣ = [ l ] , the integer part of l when l is not an integer.

Denote D ( R n ) = { ϕ ∈ S ( R n ) : supp ( ϕ ) is compact } and D ˆ 0 ( R n ) = { ϕ ∈ S ( R n ) : ϕ ˆ ∈ D ( R n ) with ϕ ˆ vanishes in a neightborhood of the origin } . The boundedness of T on L p ( R n ) is proved by [14].

Definition 2

For a real number l ˜ ≥ 0 and 1 ≤ s ˜ < ∞ , then K verifies the condition M ˜ ( s ˜ , l ˜ ) , and K ∈ M ˜ ( s ˜ , l ˜ ) if

∫ ∣ x ∣ ∼ R ∣ D α ˜ K ( x ) ∣ s ˜ d x 1 s ˜ ≤ C R n / s ˜ − n − ∣ α ˜ ∣ , R > 0 ,

for all multi-indices ∣ α ∣ ˜ ≤ l ˜ . In addition, if

∫ ∣ x ∣ ∼ R ∣ D α ˜ K ( x ) − D α ˜ K ( x − z ) ∣ s ˜ d x 1 s ˜ ≤ C ∣ z ∣ R v R n s ˜ − n − u , if 0 < v < 1 ,

∫ ∣ x ∣ ∼ R ∣ D α ˜ K ( x ) − D α ˜ K ( x − z ) ∣ s ˜ d x 1 s ˜ ≤ C ∣ z ∣ R log R ∣ z ∣ R n s ˜ − n − u , if v = 1 ,

for all ∣ z ∣ < R 2 , R > 0 , and all multi-indices α ˜ with ∣ α ˜ ∣ = u , where u denotes the largest integer strictly less than l ˜ with l ˜ = u + v .

Lemma 1

[14] Let k ∈ M ( s , l ) , 1 ≤ s ≤ 2 , and l > n s . Then the associated mapping T , defined a priori for f ∈ D ˆ 0 ( R n ) , T ( f ) ( x ) = ( f ∗ K ) ( x ) , extends to a bounded mapping from L p ( R n ) into itself for 1 < p < ∞ and K ( x ) = k ˇ ( x ) .

Lemma 2

[14] Suppose k ∈ M ( s , l ) , 1 ≤ s ≤ 2 . Given 1 ≤ s ˜ < ∞ , let r ≥ 1 be such that 1 r = max 1 s , 1 − 1 s ˜ . Then K ∈ M ˜ ( s ˜ , l ˜ ) , where l ˜ = l − n r .

Lemma 3

[14] Let 1 ≤ s < ∞ , suppose that l is a positive real number with l > n / r , 1 / r = max { 1 / s , 1 − 1 / s ˜ } , and k ∈ M ( s , l ) . Then there is a positive constant a , such that

∫ Q k ∣ K ( x − z ) − K ( x Q − z ) ∣ s ˜ d z 1 / s ˜ ≤ C 2 − k a ( 2 k h ) − n / s ˜ ′ .

Now we define the multilinear operator associated with the multiplier operator as follows.

Definition 3

Let l and m j be the positive integers ( j = 1 , … , l ), m 1 + ⋯ + m l = m and b j be the functions on R n ( j = 1 , … , l ). For 1 ≤ j ≤ l , set

R m j + 1 ( b j ; x , y ) = b j ( x ) − ∑ ∣ α ∣ ≤ m j 1 α ! D α b j ( y ) ( x − y ) α .

Let T be the multiplier operator. By Lemma 1, T ( f ) ( x ) = ( K ∗ f ) ( x ) for K ( x ) = k ˇ ( x ) . The multilinear operator associated with T is defined by

T b ( f ) ( x ) = ∫ R n ∏ j = 1 l R m j + 1 ( b j ; x , y ) ∣ x − y ∣ m K ( x − y ) f ( y ) d y .

Note that T b is just the multilinear commutators of T and b when m = 0 [4,5]. It is well known that multilinear operator, as a non-trivial extension of commutator, is of great importance in harmonic analysis and has been widely studied by many authors [1,3,4, 5,6,7,9,10,11, 12,15,16, 17,18,19, 20,21,22, 23,24]. The main purpose of this paper is to prove the boundedness of the multilinear operator T b from Lebesgue space to Orlicz space.

We shall prove the following main theorem in Section 3.

Theorem

Let 0 < β ≤ 1 , 1 < p < n / l β , and φ , ψ be two non-decreasing positive functions on [ 0 , + ∞ ) with ( ψ l ) − 1 ( t ) = t 1 / p φ l ( t − 1 / n ) . Suppose that ψ is convex, ψ ( 0 ) = 0 , ψ ( 2 t ) ≤ C ψ ( t ) . Let T be the multiplier operator as Definition 3. Then T b is bounded from L p ( R n ) to L ψ l ( R n ) if D α b j ∈ B M O ( R n ) for all α with ∣ α ∣ = m j and j = 1 , … , l .

Remark

  1. If l = 1 and ψ − 1 ( t ) = t 1 / p φ ( t − 1 / n ) , then T b and S b are all bounded from L p ( R n ) to L ψ ( R n ) under the conditions of Lemmas 1 and 2.

  2. If l = 1 , φ ( t ) ≡ 1 and ψ ( t ) = t p for 1 < p < ∞ , then T b is bounded on L p ( R n ) if D α b ∈ B M O φ ( R n ) for all α with ∣ α ∣ = m .

  3. If l = 1 , ψ ( t ) = t q and φ ( t ) = t n ( 1 / p − 1 / q ) for 1 < p < q < ∞ , then, by B M O t β ( R n ) = Lip β ( R n ) [11, Lemma 4], T b is bounded from L p ( R n ) to L q ( R n ) if D α b ∈ Lip n ( 1 / p − 1 / q ) ( R n ) for all α with ∣ α ∣ = m .

3 Proof of Theorem

We begin with the following preliminary lemmas.

Lemma 4

[14] Let T be the multiplier operator as Definition 3. Then T is bounded on L p ( R n ) for 1 < p < ∞ .

Lemma 5

[7] Let ρ be a non-decreasing positive function on [ 0 , + ∞ ) and η be an infinitely differentiable function on R n with compact support such that ∫ R n η ( x ) d x = 1 . Denote that b t ( x ) = ∫ R n b ( x − t y ) η ( y ) d y . Then ∣ ∣ b − b t ∣ ∣ B M O ≤ C ρ ( t ) ∣ ∣ b ∣ ∣ B M O ρ .

Lemma 6

[7] Let 0 < β < 1 or β = 1 and ρ be a non-decreasing positive function on [ 0 , + ∞ ) . Then ∣ ∣ b t ∣ ∣ Lip β ≤ C t − β ρ ( t ) ∣ ∣ b ∣ ∣ B M O ρ .

Lemma 7

[7] Suppose 1 ≤ p 2 < p < p 1 < ∞ , ρ is a non-increasing function on [ 0 , + ∞ ) , B is a linear operator such that m B ( f ) ( t 1 / p 1 ρ ( t ) ) ≤ C t − 1 if ∣ ∣ f ∣ ∣ L p 1 ≤ 1 and m B ( f ) ( t 1 / p 2 ρ ( t ) ) ≤ C t − 1 if ∣ ∣ f ∣ ∣ L p 2 ≤ 1 . Then ∫ 0 ∞ m B ( f ) ( t 1 / p ρ ( t ) ) d t ≤ C if ∣ ∣ f ∣ ∣ L p ≤ ( p / p 1 ) 1 / p .

Lemma 8

[6] Suppose that 0 < β < n , 1 ≤ r < p < n / β , and 1 / q = 1 / p − β / n . Then ∣ ∣ M β , r ( f ) ∣ ∣ L q ≤ C ∣ ∣ f ∣ ∣ L p .

Lemma 9

[11] Let b be a function on R n and D α A ∈ L q ( R n ) for all α with ∣ α ∣ = m and some q > n . Then

∣ R m ( b ; x , y ) ∣ ≤ C ∣ x − y ∣ m ∑ ∣ α ∣ = m 1 ∣ Q ˜ ( x , y ) ∣ ∫ Q ˜ ( x , y ) ∣ D α b ( z ) ∣ q d z 1 / q ,

where Q ˜ is the cube centered at x and having side length 5 n ∣ x − y ∣ .

In order to prove the theorem of the paper, we need the following Key Lemma.

Key Lemma. Let T be the multiplier operator as Definition 3. Suppose that Q = Q ( x 0 , d ) is a cube with supp f ⊂ ( 2 Q ) c and x , x ˜ ∈ Q .

  1. If D α b j ∈ B M O ( R n ) for all α with ∣ α ∣ = m j , j = 1 , … , l , then

    ∣ T b ( f ) ( x ) − T b ( f ) ( x 0 ) ∣ ≤ C ∏ j = 1 l ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) for any r > 1 ;

  2. If 0 < β ≤ 1 and D α b j ∈ Lip β ( R n ) for all α with ∣ α ∣ = m j and j = 1 , … , l , then

    ∣ T b ( f ) ( x ) − T b ( f ) ( x 0 ) ∣ ≤ C ∏ j = 1 l ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M l β , r ( f ) ( x ˜ ) for any r > 1 .

Proof

Without loss of generality, we may assume l = 2 . Let Q ˜ = 5 n Q and b ˜ ( x ) = b ( x ) − ∑ ∣ α ∣ = m 1 α ! ( D α b ) Q ˜ x α , then R m ( b ; x , y ) = R m ( b ˜ ; x , y ) and D α b ˜ = D α b − ( D α b ) Q ˜ for ∣ α ∣ = m . For supp f ⊂ ( 2 Q ) c and x , x ˜ ∈ Q , we write

T b ( f ) ( x ) − T b ( f ) ( x 0 ) = ∫ R n K ( x − y ) ∣ x − y ∣ m − K ( x 0 − y ) ∣ x 0 − y ∣ m ∏ j = 1 2 R m j ( b ˜ j ; x , y ) f ( y ) d y + ∫ R n ( R m 1 ( b ˜ 1 ; x , y ) − R m 1 ( b ˜ 1 ; x 0 , y ) ) R m 2 ( b ˜ 2 ; x , y ) ∣ x 0 − y ∣ m K ( x 0 − y ) f ( y ) d y + ∫ R n ( R m 2 ( b ˜ 2 ; x , y ) − R m 2 ( b ˜ 2 ; x 0 , y ) ) R m 1 ( b ˜ 1 ; x 0 , y ) ∣ x 0 − y ∣ m K ( x 0 − y ) f ( y ) d y − ∑ ∣ α 1 ∣ = m 1 1 α 1 ! ∫ R n R m 2 ( b ˜ 2 ; x , y ) ( x − y ) α 1 ∣ x − y ∣ m K ( x − y ) − R m 2 ( b ˜ 2 ; x 0 , y ) ( x 0 − y ) α 1 ∣ x 0 − y ∣ m K ( x 0 − y ) D α 1 b ˜ 1 ( y ) f ( y ) d y − ∑ ∣ α 2 ∣ = m 2 1 α 2 ! ∫ R n R m 1 ( b ˜ 1 ; x , y ) ( x − y ) α 2 ∣ x − y ∣ m K ( x − y ) − R m 1 ( b ˜ 1 ; x 0 , y ) ( x 0 − y ) α 2 ∣ x 0 − y ∣ m K ( x 0 − y ) D α 2 b ˜ 2 ( y ) f ( y ) d y + ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 α 1 ! α 2 ! ∫ R n ( x − y ) α 1 + α 2 ∣ x − y ∣ m K ( x − y ) − ( x 0 − y ) α 1 + α 2 ∣ x 0 − y ∣ m K ( x 0 − y ) D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) f ( y ) d y = I 1 + I 2 + I 3 + I 4 + I 5 + I 6 .

  1. By Lemma 9 and the following inequality [3], for b ∈ B M O ( R n ) ,

    ∣ b Q 1 − b Q 2 ∣ ≤ C log ( ∣ Q 2 ∣ / ∣ Q 1 ∣ ) ∣ ∣ b ∣ ∣ B M O for Q 1 ⊂ Q 2 .

Note that ∣ x − y ∣ ∼ ∣ x 0 − y ∣ for x ∈ Q and y ∈ R n ⧹ Q ˜ , then, for x ∈ Q and y ∈ 2 k + 1 Q ⧹ 2 k Q with k ≥ 1 ,

∣ R m ( b ˜ ; x , y ) ∣ ≤ C ∣ x − y ∣ m ∑ ∣ α ∣ = m ( ∣ ∣ D α b ∣ ∣ B M O + ∣ ( D α b ) Q ˜ ( x , y ) − ( D α b ) Q ˜ ∣ ) ≤ C k ∣ x − y ∣ m ∑ ∣ α ∣ = m ∣ ∣ D α b ∣ ∣ B M O .

For I 1 , by the conditions on K , Lemmas 2 and 3, we obtain

∣ I 1 ∣ ≤ ∑ k = 1 ∞ ∫ 2 k + 1 Q ⧹ 2 k Q 1 ∣ x − y ∣ m − 1 ∣ x 0 − y ∣ m ∏ j = 1 2 ∣ R m j ( b ˜ j ; x , y ) ∣ ∣ f ( y ) ∣ d y + ∑ k = 1 ∞ ∫ 2 k + 1 Q ⧹ 2 k Q ∣ K ( x − y ) − K ( x 0 − y ) ∣ ∣ x 0 − y ∣ − m ∏ j = 1 2 ∣ R m j ( b ˜ j ; x , y ) ∣ ∣ f ( y ) ∣ d y ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∑ k = 1 ∞ k 2 ∫ 2 k + 1 Q ⧹ 2 k Q ∣ x − x 0 ∣ ∣ x 0 − y ∣ ∣ K ( x − y ) ∣ ∣ f ( y ) ∣ d y + C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∑ k = 1 ∞ k 2 ∫ 2 k + 1 Q ˜ ∣ f ( y ) ∣ r d y 1 / r ∫ 2 k + 1 Q ⧹ 2 k Q ∣ K ( x − y ) − K ( x 0 − y ) ∣ r ′ d y 1 / r ′ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∑ k = 1 ∞ k 2 ( 2 − k + 2 − k a ) 1 ∣ 2 k + 1 Q ∣ ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

For I 2 , by the formula [11]

R m ( b ˜ ; x , y ) − R m ( b ˜ ; x 0 , y ) = ∑ ∣ γ ∣ < m 1 γ ! R m − ∣ γ ∣ ( D γ b ˜ ; x , x 0 ) ( x − y ) γ

and Lemma 9, we have

∣ R m ( b ˜ ; x , y ) − R m ( b ˜ ; x 0 , y ) ∣ ≤ C ∑ ∣ γ ∣ < m ∑ ∣ α ∣ = m ∣ x − x 0 ∣ m − ∣ γ ∣ ∣ x − y ∣ ∣ γ ∣ ∣ ∣ D α b ∣ ∣ B M O ,

thus,

∣ I 2 ∣ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∑ k = 1 ∞ ∫ 2 k + 1 Q ⧹ 2 k Q k ∣ x − x 0 ∣ ∣ x 0 − y ∣ ∣ K ( x 0 − y ) ∣ ∣ f ( y ) ∣ d y ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∑ k = 1 ∞ k 2 − k 1 ∣ 2 k + 1 Q ∣ ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

Similarly,

∣ I 3 ∣ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

For I 4 , similar to the proof of I 1 and I 2 , we get

∣ I 4 ∣ ≤ C ∑ ∣ α 1 ∣ = m 1 ∫ R n ⧹ 2 Q ( x − y ) α 1 ∣ x − y ∣ m − ( x 0 − y ) α 1 ∣ x 0 − y ∣ m ∣ K ( x − y ) ∣ ∣ R m 2 ( b ˜ 2 ; x , y ) ∣ ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ f ( y ) ∣ d y + C ∑ ∣ α 1 ∣ = m 1 ∫ R n ⧹ 2 Q ∣ R m 2 ( b ˜ 2 ; x , y ) − R m 2 ( b ˜ 2 ; x 0 , y ) ∣ ∣ ( x 0 − y ) α 1 K ( x − y ) ∣ ∣ x 0 − y ∣ m ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ f ( y ) ∣ d y

+ C ∑ ∣ α 1 ∣ = m 1 ∫ R n ⧹ 2 Q ∣ K ( x − y ) − K ( x 0 − y ) ∣ ( x 0 − y ) α 1 ∣ x 0 − y ∣ m ∣ R m 2 ( b ˜ 2 ; x 0 , y ) ∣ ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ f ( y ) ∣ d y ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∑ k = 1 ∞ k 2 ( 2 − k + 2 − k a ) 1 ∣ 2 k + 1 Q ∣ ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

Similarly,

∣ I 5 ∣ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

For I 6 , taking 1 < s 1 , s 2 < ∞ such that 1 / r + 1 / s 1 + 1 / s 2 = 1 , then

∣ I 6 ∣ ≤ C ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 ∫ R n ⧹ 2 Q ( x − y ) α 1 + α 2 K ( x − y ) ∣ x − y ∣ m − ( x 0 − y ) α 1 + α 2 K ( x 0 − y ) ∣ x 0 − y ∣ m × ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ D α 2 b ˜ 2 ( y ) ∣ ∣ f ( y ) ∣ d y ≤ C ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 ∑ k = 1 ∞ ( 2 − k + 2 − k a ) 1 ∣ 2 k + 1 Q ∣ ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r × 1 ∣ 2 k + 1 Q ∣ ∫ 2 k + 1 Q ∣ D α 1 b ˜ 1 ( y ) ∣ s 1 d y 1 / s 1 1 ∣ 2 k + 1 Q ∣ ∫ 2 k + 1 Q ∣ D α 2 b ˜ 2 ( y ) ∣ s 2 d y 1 / s 2 ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∑ k = 1 ∞ k 2 ( 2 − k + 2 − k a ) M r ( f ) ( x ˜ ) ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

Thus,

∣ T b ( f ) ( x ) − T b ( f ) ( x 0 ) ∣ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

  1. For b ∈ Lip β ( R n ) , by Lemma 9 and the following inequality:

∣ b ( x ) − b Q ∣ ≤ 1 ∣ Q ∣ ∫ Q ∣ ∣ b ∣ ∣ Lip β ∣ x − y ∣ β d y ≤ ∣ ∣ b ∣ ∣ Lip β ( ∣ x − x 0 ∣ + d ) β ,

we get

∣ R m ( b ˜ ; x , y ) ∣ ≤ C ∑ ∣ α ∣ = m ∣ ∣ D α b ∣ ∣ Lip β ( ∣ x − y ∣ + d ) m + β

and

∣ R m ( b ˜ ; x , y ) − R m ( b ˜ ; x 0 , y ) ∣ ≤ C ∑ ∣ α ∣ = m ∣ ∣ D α b ∣ ∣ Lip β ( ∣ x − y ∣ + d ) m + β ,

then, similar to the proof of (I),

∣ I 1 ∣ ≤ ∑ k = 1 ∞ ∫ 2 k + 1 Q ⧹ 2 k Q 1 ∣ x − y ∣ m − 1 ∣ x 0 − y ∣ m ∣ K ( x − y ) ∣ ∏ j = 1 2 ∣ R m j ( b ˜ j ; x , y ) ∣ ∣ f ( y ) ∣ d y + ∑ k = 1 ∞ ∫ 2 k + 1 Q ⧹ 2 k Q ∣ K ( x − y ) − K ( x 0 − y ) ∣ ∣ x 0 − y ∣ − m ∏ j = 1 2 ∣ R m j ( b ˜ j ; x , y ) ∣ ∣ f ( y ) ∣ d y ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β ∑ k = 1 ∞ ( 2 − k + 2 − k a ) 1 ∣ 2 k + 1 Q ∣ 1 − 2 β r / n ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M 2 β , r ( f ) ( x ˜ ) , ∣ I 2 + I 3 ∣ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β ∑ k = 1 ∞ ( 2 − k + 2 − k a ) 1 ∣ 2 k + 1 Q ∣ 1 − 2 β r / n ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M 2 β , r ( f ) ( x ˜ ) , ∣ I 4 ∣ ≤ C ∑ ∣ α 1 ∣ = m 1 ∫ R n ⧹ 2 Q ( x − y ) α 1 ∣ x − y ∣ m − ( x 0 − y ) α 1 ∣ x 0 − y ∣ m ∣ K ( x , x − y ) ∣ ∣ R m 2 ( b ˜ 2 ; x , y ) ∣ ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ f ( y ) ∣ d y + C ∑ ∣ α 1 ∣ = m 1 ∫ R n ⧹ 2 Q ∣ R m 2 ( b ˜ 2 ; x , y ) − R m 2 ( b ˜ 2 ; x 0 , y ) ∣ ∣ ( x 0 − y ) α 1 K ( x , x − y ) ∣ ∣ x 0 − y ∣ m ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ f ( y ) ∣ d y + C ∑ ∣ α 1 ∣ = m 1 ∫ R n ⧹ 2 Q ∣ K ( x , x − y ) − K ( x 0 , x 0 − y ) ∣ ( x 0 − y ) α 1 ∣ x 0 − y ∣ m ∣ R m 2 ( b ˜ 2 ; x 0 , y ) ∣ ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ f ( y ) ∣ d y ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β ∑ k = 1 ∞ ( 2 − k + 2 − k a ) 1 ∣ 2 k + 1 Q ∣ 1 − 2 β r / n ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M 2 β , r ( f ) ( x ˜ ) , ∣ I 5 ∣ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M 2 β , r ( f ) ( x ˜ ) ,

∣ I 6 ∣ ≤ C ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 ∫ R n ⧹ 2 Q ( x − y ) α 1 + α 2 K ( x , y ) ∣ x − y ∣ m − ( x 0 − y ) α 1 + α 2 K ( x 0 , y ) ∣ x 0 − y ∣ m ∣ D α 1 b ˜ 1 ( y ) ∣ ∣ D α 2 b ˜ 2 ( y ) ∣ ∣ f ( y ) ∣ d y ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β ∑ k = 1 ∞ ( 2 − k + 2 − k a ) 1 ∣ 2 k + 1 Q ∣ 1 − 2 β r / n ∫ 2 k + 1 Q ∣ f ( y ) ∣ r d y 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M 2 β , r ( f ) ( x ˜ ) .

Thus,

∣ T b ( f ) ( x ) − T b ( f ) ( x 0 ) ∣ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M β , r ( f ) ( x ˜ ) . □

Now we are in a position to prove the main theorem.

Proof of Theorem

Without loss of generality, we assume l = 2 . We prove the theorem in several steps. First, if D α b j ∈ B M O ( R n ) for all α with ∣ α ∣ = m j and j = 1 , … , l , we prove

(1) ( T b ( f ) ) # ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f )

for any 1 < r < ∞ . Fix a cube Q = Q ( x 0 , d ) and x ˜ ∈ Q . Let Q ˜ = 5 n Q and b ˜ j ( x ) = b j ( x ) − ∑ ∣ α ∣ = m 1 α ! ( D α b j ) Q ˜ x α , then R m ( b j ; x , y ) = R m ( b ˜ j ; x , y ) and D α b ˜ j = D α b j − ( D α b j ) Q ˜ for ∣ α ∣ = m j . For f 1 = f χ Q ˜ and f 2 = f χ R n ⧹ Q ˜ , we write

T b ( f ) ( x ) = ∫ R n ∏ j = 1 2 R m j ( b ˜ j ; x , y ) ∣ x − y ∣ m K ( x − y ) f 1 ( y ) d y − ∑ ∣ α 1 ∣ = m 1 1 α 1 ! ∫ R n R m 2 ( b ˜ 2 ; x , y ) ( x − y ) α 1 D α 1 b ˜ 1 ( y ) ∣ x − y ∣ m K ( x − y ) f 1 ( y ) d y − ∑ ∣ α 2 ∣ = m 2 1 α 2 ! ∫ R n R m 1 ( b ˜ 1 ; x , y ) ( x − y ) α 2 D α 2 b ˜ 2 ( y ) ∣ x − y ∣ m K ( x − y ) f 1 ( y ) d y + ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 α 1 ! α 2 ! ∫ R n ( x − y ) α 1 + α 2 D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) ∣ x − y ∣ m K ( x − y ) f 1 ( y ) d y + ∫ R n ∏ j = 1 2 R m j + 1 ( b j ; x , y ) ∣ x − y ∣ m K ( x − y ) f 2 ( y ) d y = T ∏ j = 1 2 R m j ( b ˜ j ; x , ⋅ ) ∣ x − ⋅ ∣ m f 1 − T ∑ ∣ α 1 ∣ = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ⋅ ) ( x − ⋅ ) α 1 D α 1 b ˜ 1 ∣ x − ⋅ ∣ m f 1 − T ∑ ∣ α 2 ∣ = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ⋅ ) ( x − ⋅ ) α 2 D α 2 b ˜ 2 ∣ x − ⋅ ∣ m f 1 + T ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 α 1 ! α 2 ! ( x − ⋅ ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 ∣ x − ⋅ ∣ m f 1 + T b ( f 2 ) ( x ) ,

then

∣ T b ( f ) ( x ) − T b ( f 2 ) ( x 0 ) ∣ ≤ T ∏ j = 1 2 R m j ( b ˜ j ; x , ⋅ ) ∣ x − ⋅ ∣ m f 1 + T ∑ ∣ α 1 ∣ = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ⋅ ) ( x − ⋅ ) α 1 D α 1 b ˜ 1 ∣ x − ⋅ ∣ m f 1 + T ∑ ∣ α 2 ∣ = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ⋅ ) ( x − ⋅ ) α 2 D α 2 b ˜ 2 ∣ x − ⋅ ∣ m f 1 + T ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 α 1 ! α 2 ! ( x − ⋅ ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 ∣ x − ⋅ ∣ m f 1 + ∣ T b ( f 2 ) ( x ) − T b ( f 2 ) ( x 0 ) ∣ = L 1 ( x ) + L 2 ( x ) + L 3 ( x ) + L 4 ( x ) + L 5 ( x )

and

1 ∣ Q ∣ ∫ Q ∣ T b ( f ) ( x ) − T b ( f 2 ) ( x 0 ) ∣ d x ≤ 1 ∣ Q ∣ ∫ Q L 1 ( x ) d x + 1 ∣ Q ∣ ∫ Q L 2 ( x ) d x + 1 ∣ Q ∣ ∫ Q L 3 ( x ) d x + 1 ∣ Q ∣ ∫ Q L 4 ( x ) d x + 1 ∣ Q ∣ ∫ Q L 5 ( x ) d x = L 1 + L 2 + L 3 + L 4 + L 5 .

Now, for L 1 , if x ∈ Q and y ∈ 2 Q , by Lemma 9, we get

R m ( b ˜ ; x , y ) ≤ C ∣ x − y ∣ m ∑ ∣ α ∣ = m ∣ ∣ D α b ∣ ∣ B M O ,

thus, by the L r boundedness of T (see Lemma 1) and Hölder’s inequality, we obtain

L 1 ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O 1 ∣ Q ∣ ∫ Q ∣ T ( f 1 ) ( x ) ∣ d x ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O 1 ∣ Q ∣ ∫ R n ∣ T ( f 1 ) ( x ) ∣ r d x 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O 1 ∣ Q ∣ ∫ R n ∣ f 1 ( x ) ∣ r d x 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O 1 ∣ Q ˜ ∣ ∫ Q ˜ ∣ f ( x ) ∣ r d x 1 / r ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

For L 2 , denoting r = u v for 1 < u , v < ∞ and 1 / v + 1 / v ′ = 1 , we have

L 2 ≤ C ∑ ∣ α 2 ∣ = m 2 ∣ ∣ D α 2 b 2 ∣ ∣ B M O ∑ ∣ α 1 ∣ = m 1 ∑ ∣ α ∣ = m 1 ∣ Q ∣ ∫ Q ∣ T ( D α 1 b ˜ 1 f 1 ) ( x ) ∣ d x ≤ C ∑ ∣ α 2 ∣ = m 2 ∣ ∣ D α 2 b 2 ∣ ∣ B M O ∑ ∣ α 1 ∣ = m 1 1 ∣ Q ∣ ∫ R n ∣ T ( D α 1 b ˜ 1 f 1 ) ( x ) ∣ u d x 1 / u ≤ C ∑ ∣ α 2 ∣ = m 2 ∣ ∣ D α 2 b 2 ∣ ∣ B M O ∑ ∣ α 1 ∣ = m 1 1 ∣ Q ∣ ∫ R n ∣ D α 1 b ˜ 1 ( x ) ∣ ∣ f 1 ( x ) ∣ u d x 1 / u ≤ C ∑ ∣ α 2 ∣ = m 2 ∣ ∣ D α 2 b 2 ∣ ∣ B M O 1 ∣ Q ˜ ∣ ∫ Q ˜ ∣ f ( x ) ∣ u v d x 1 / u v ∑ ∣ α 1 ∣ = m 1 1 ∣ Q ˜ ∣ ∫ Q ˜ ∣ D α 1 b ˜ 1 ( x ) − ( D α 1 b ˜ 1 ) Q ˜ ∣ u v ′ d x 1 / u v ′ ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

For L 3 , similar to the proof of L 2 , we get

L 3 ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

Similarly, for L 4 , denoting r = u w for 1 < u , v 1 , v 2 , w < ∞ and 1 / v 1 + 1 / v 2 + 1 / w = 1 , by Holder’s inequality, we obtain

L 4 ≤ C ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 ∣ Q ∣ ∫ Q ∣ T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) ( x ) ∣ d x ≤ C ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 ∣ Q ∣ ∫ R n ∣ T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) ( x ) ∣ u d x 1 / u ≤ C ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 ∣ Q ∣ − 1 / u ∫ R n ∣ D α 1 b ˜ 1 ( x ) D α 2 b ˜ 2 ( x ) f 1 ( x ) ∣ u d x 1 / u ≤ C ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 ∣ Q ˜ ∣ ∫ Q ˜ ∣ D α 1 b ˜ 1 ( x ) ∣ u v 1 d x 1 / u v 1 1 ∣ Q ˜ ∣ ∫ Q ˜ ∣ D α 2 b ˜ 2 ( x ) ∣ u v 2 d x 1 / u v 2 1 ∣ Q ˜ ∣ ∫ Q ˜ ∣ f ( x ) ∣ u w d x 1 / u w ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

For L 5 , by Key Lemma, we have

L 3 ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

Put these estimates together and take the supremum over all Q such that x ˜ ∈ Q , we have

( T b ( f ) ) # ( x ˜ ) ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O M r ( f ) ( x ˜ ) .

Thus, taking r such that 1 < r < p , we obtain

(2) ∣ ∣ T b ( f ) ∣ ∣ L p ≤ C ∣ ∣ ( T b ( f ) ) # ∣ ∣ L p ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∣ ∣ M r ( f ) ∣ ∣ L p ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O ∣ ∣ f ∣ ∣ L p .

Second, if D α b j ∈ Lip β ( R n ) for all α with ∣ α ∣ = m j and j = 1 , … , l , we prove

(3) ( T b ( f ) ) # ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M 2 β , r ( f )

for any r with 1 < r < n / 2 β . In fact, for x ∈ Q and y ∈ 2 Q , by Lemma 9, we have

∣ R m ( b ˜ ; x , y ) ∣ ≤ C ∣ x − y ∣ m ∑ ∣ α ∣ = m sup z ∈ 2 Q ∣ D α b ( z ) − ( D α b ) Q ∣ ≤ C ∣ x − y ∣ m ∣ Q ∣ β / n ∑ ∣ α ∣ = m ∣ ∣ D α b ∣ ∣ Lip β .

Similar to the proof of (1) and by Key Lemma, we obtain

1 ∣ Q ∣ ∫ Q ∣ T b ( f ) ( x ) − T b ( f ) ( x 0 ) ∣ d x ≤ 1 ∣ Q ∣ ∫ Q T ∏ j = 1 2 R m j ( b ˜ j ; x , ⋅ ) ∣ x − ⋅ ∣ m f 1 d x + C 1 ∣ Q ∣ ∫ Q T ∑ ∣ α 1 ∣ = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ⋅ ) ( x − ⋅ ) α 1 D α 1 b ˜ 1 ∣ x − ⋅ ∣ m f 1 d x

+ C 1 ∣ Q ∣ ∫ Q T ∑ ∣ α 2 ∣ = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ⋅ ) ( x − ⋅ ) α 2 D α 2 b ˜ 2 ∣ x − ⋅ ∣ m f 1 d x + C 1 ∣ Q ∣ ∫ Q T ∑ ∣ α 1 ∣ = m 1 , ∣ α 2 ∣ = m 2 1 α 1 ! α 2 ! ( x − ⋅ ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 ∣ x − ⋅ ∣ m f 1 d x + 1 ∣ Q ∣ ∫ Q ∣ T b ( f 2 ) ( x ) − T b ( f 2 ) ( x 0 ) ∣ d x ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β 1 ∣ Q ∣ 1 / r − 2 β / n ∫ Q ˜ ∣ f ( x ) ∣ r d x 1 / r + 1 ∣ Q ∣ ∫ Q ∣ T b ( f 2 ) ( x ) − T b ( f 2 ) ( x 0 ) ∣ d x ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β M 2 β , r ( f ) ( x ˜ ) .

Thus, (3) holds. Taking 1 < r < p < n / 2 β , 1 / q = 1 / p − 2 β / n , by Lemma 8 we obtain

(4) ∣ ∣ T b ( f ) ∣ ∣ L q ≤ C ∣ ∣ ( T b ( f ) ) # ∣ ∣ L q ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β ∣ ∣ M 2 β , r ( f ) ∣ ∣ L q ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β ∣ ∣ f ∣ ∣ L p .

Now we verify that T b satisfies the conditions of Lemma 6. In fact, for any 1 < p i < n / 2 β , 1 / q i = 1 / p i − 2 β / n ( i = 1 , 2 ) , and ∣ ∣ f ∣ ∣ L p i ≤ 1 , note that T b ( f ) ( x ) = T b − b s ( f ) ( x ) + T b s ( f ) ( x ) and D α ( b s ) = ( D α b ) s with D α ( b j − b j s ) ∈ B M O ( R n ) and D α b j s ∈ Lip β ( R n ) , by (2) and Lemma 5, we obtain

∣ ∣ T b − b s ( f ) ∣ ∣ L p i ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j ( b j − b j s ) ∣ ∣ B M O ∣ ∣ f ∣ ∣ L p i ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j − ( D α j b j ) s ∣ ∣ B M O ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O φ φ 2 ( s ) ,

and by (4) and Lemma 6, we obtain

∣ ∣ T b s ( f ) ∣ ∣ L q i ≤ C ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ Lip β ∣ ∣ f ∣ ∣ L p i ≤ C s − 2 β φ 2 ( s ) ∏ j = 1 2 ∑ ∣ α j ∣ = m j ∣ ∣ D α j b j ∣ ∣ B M O φ .

Thus, for s = t − 1 / n ,

m T b ( f ) ( ( ψ 2 ) − 1 ( t ) ) ≤ m T b ( f ) ( t 1 / p i φ 2 ( t − 1 / n ) ) ≤ m T b − b s ( f ) ( t 1 / p i φ 2 ( t − 1 / n ) / 2 ) + m T b s ( f ) ( t 1 / p i φ 2 ( t − 1 / n ) / 2 ) ≤ C φ 2 ( s ) t 1 / p i φ 2 ( s ) p i + s − 2 β φ 2 ( s ) t 1 / p i φ 2 ( s ) q i = C t − 1 .

For ∣ ∣ f ∣ ∣ L p ≤ ( p / p 1 ) 1 / p , taking 1 < p 2 < p < p 1 < n / 2 β and by Lemma 7, we obtain

∫ R n ψ 2 ( ∣ T b ( f ) ( x ) ∣ ) d x = ∫ 0 ∞ m T b ( f ) ( ( ψ 2 ) − 1 ( t ) ) d t ≤ C ,

then, ∣ ∣ T b ( f ) ∣ ∣ L ψ 2 ≤ C .□

Acknowledgments

The authors express their gratitude to the anonymous referees for their valuable comments that largely polish this paper.

  1. Funding information: This work was supported by the School-level Scientific Research Projects of Jiaxing Nanyang Vocational and Technical College (Grant: p30018ky013).

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2020-06-28
Accepted: 2021-06-03
Published Online: 2021-08-05

© 2021 Qiaozhen Zhao and Dejian Huang, published by De Gruyter

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

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  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-2021-0056
Keywords for this article
multilinear operator; multiplier operator; BMO space; Orlicz space
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  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
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  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
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  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
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  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
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  38. Inviscid, zero Froude number limit of the viscous shallow water system
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  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
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  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
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  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
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