Some estimates for commutators of bilinear pseudo-differential operators

Yanqi Yang; Shuangping Tao

doi:10.1515/math-2022-0517

Open Access Published by De Gruyter Open Access October 28, 2022

Some estimates for commutators of bilinear pseudo-differential operators

Yanqi Yang and Shuangping Tao

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0517

Abstract

We obtain a class of commutators of bilinear pseudo-differential operators on products of Hardy spaces by applying the accurate estimates of the Hörmander class. And we also prove another version of these types of commutators on Herz-type spaces.

Keywords: bilinear pseudo-differential operator; commutator; Lipschitz function; Hardy space; Herz-type space

MSC 2010: 42B20; 42B25; 47G30

1 Introduction and main results

Let T be a linear operator. Given a function b , the commutator [ T , b ] is defined by

[ T , b ] ( f ) ≔ T ( b f ) − b T ( f ) .

It is very interesting that T is a pseudo-differential operator because its theory plays an important role in many aspects of harmonic analysis and it has had quite a success in a linear setting. As one of the most meaningful branches, the study of bilinear pseudo-differential operators was motivated not only as generalizations of the theory of linear ones but also its natural appearance in Harmonic. This topic is continuous to attract many researchers.

Let b be a Lipschitz function and 1 < p < ∞ . The estimates of the form

(1.1) ‖ [ T , b ] ( f ) ‖ L p ≲ ‖ b ‖ Lip 1 ‖ f ‖ L p , for all f ∈ L p ( R n )

have been studied extensively. In particular, Calderón proved that (1.1) holds when T is a pseudo-differential operator whose kernel is homogeneous of degree of − n − 1 in [1]. Coifman and Meyer showed (1.1) when T = T σ and σ is a symbol in the Hörmander class S 1 , 0 1 in [2,3], and this result was later extended by Auscher and Taylor in [4] to σ ∈ ℬ S 1 , 1 1 , where the class ℬ S 1 , 1 1 , which contains S 1 , 0 1 modulo symbols associated with smoothing operators, consists of symbols whose Fourier transforms in the first n -dimensional variable are appropriately compactly supported. The method in the proofs of [2,3] mainly showed that, for each Lipschitz continuous function b on R n , [ T , b ] is a Calderón-Zygmund singular integral whose kernel constants are controlled by ‖ b ‖ Lip 1 . For another thing, Auscher and Taylor proved (1.1) in two different ways: one method is based on the paraproducts while another is based on the Calderón-Zygmund singular integral operator approach that relies on the T ( 1 ) theorem. For a more systematic study of these (and even more general) spaces, we refer the readers to [5,6].

Given a bilinear operator T and a function b , the following two kinds of commutators are, respectively, defined by

[ T , b ] 1 ( f , g ) = T ( b f , g ) − b T ( f , g )

and

[ T , b ] 2 ( f , g ) = T ( f , b g ) − b T ( f , g ) .

In 2014, Bényi and Oh proved that (1.1) is also valid for this bilinear setting in [7]. More precisely, given a bilinear pseudo-differential operator T σ with σ in the bilinear Hörmander class BS 1.0 1 and a Lipschitz function b on R n , it was proved in [7, Theorem 1] that [ T , b ] 1 and [ T , b ] 2 are bilinear Calderón-Zygmund operators. The main aim of this article is to study (1.1) of [ T σ , b ] j ( j = 1 , 2 ) on the products of Hardy spaces and Herz-type spaces with σ ∈ ℬ BS 1.1 1 . Before stating our main results, we need to recall some definitions and notations.

We say that a function b defined on R n is Lipschitz continuous if

‖ b ‖ Lip 1 ≔ sup x , y ∈ R n ∣ b ( x ) − b ( y ) ∣ ∣ x − y ∣ < ∞ .

Let δ ≥ 0 , ρ > 0 and m ∈ R . An infinitely differentiable function σ : R n × R n × R n → C belongs to the bilinear Hörmander class BS ρ , δ m if for all multi-indices α , β , γ ∈ N 0 n there exists a positive constant C α , β , γ such that

∣ ∂ x α ∂ ξ β ∂ η γ σ ( x , ξ , η ) ∣ ≤ C ( 1 + ∣ ξ ∣ + ∣ η ∣ ) m + δ ∣ α ∣ − ρ ( ∣ β ∣ + ∣ γ ∣ ) .

Given a σ ( x , ξ , η ) ∈ BS ρ , δ m , the bilinear pseudo-differential operator associated with σ is defined by

T σ ( f , g ) ( x ) = ∫ R n ∫ R n σ ( x , ξ , η ) f ˆ ( ξ ) g ˆ ( η ) e 2 π i x ⋅ ( ξ + η ) d ξ d η , for all x ∈ R n , f , g ∈ S ( R n ) .

In 1980, Meyer [8] first introduced the linear BS 1 , 1 m , and corresponding boundedness of [ T σ , a ] j ( j = 1 , 2 ) is obtained by Bényi and Oh in [7], that is given m ∈ R and r > 0 , an infinitely differentiable function σ : R n × R n × R n → C belongs to ℬ r BS 1.1 m if

σ ∈ B S 1 , 1 m , supp ( σ ˆ 1 ) ⊂ { ( τ , ξ , η ) ∈ R 3 n : ∣ τ ∣ ≤ r ( ∣ ξ ∣ + ∣ η ∣ ) } ,

where σ ˆ 1 denotes the Fourier transform of σ with respect to its first variable in R n , that is, σ ˆ 1 ( τ , ξ , η ) = σ ( ⋅ , ξ , η ) ^ ( τ ) . for all τ , ξ , η ∈ R n . The class ℬ BS 1.1 m is defined as

ℬ BS 1 , 1 m = ⋃ r ∈ ( 0 , 1 7 ) ℬ r BS 1 , 1 m .

Recently, many authors are interested in bilinear operators, which is a natural generalization of linear case. With further research, Bényi and Naibo proved the boundedness for the commutators of bilinear pseudo-differential operators and Lipschitz functions with σ ∈ ℬ BS 1.1 1 on the Lebesgue spaces in [9]. In 2018, Tao and Li proved that the boundedness of the commutators of bilinear pseudo-differential operators was also true on the classical and generalized Morrey spaces in [10]. Motivated by the results mentioned above, a natural and interesting problem is to consider whether or not (1.1) is true on the products of Hardy spaces and Herz-type spaces with σ ∈ ℬ BS 1.1 1 . The purpose of this article is to give a surely an answer. Our proofs are based on the pointwise estimates of the sharp maximal function proved in the next section.

Suppose that σ ∈ ℬ BS 1 , 1 1 . Let K and K j denote the kernel of T σ and [ T σ , b ] j ( j = 1 , 2 ) , respectively. We have

K ( x , y , z ) = ∫ ∫ e i ξ ⋅ ( x − y ) e i η ⋅ ( x − z ) σ ( x , ξ , η ) d ξ d η , K 1 ( x , y , z ) = ( b ( y ) − b ( x ) ) K ( x , y , z ) , K 2 ( x , y , z ) = ( b ( z ) − b ( x ) ) K ( x , y , z ) .

Then the following consequences are true.

Theorem A

[7, Lemma 3] If x ≠ y or x ≠ z , then we have

∣ ∂ x α ∂ y β ∂ z γ K ( x , y , z ) ∣ ≤ C α , β , γ ( ∣ x − y ∣ + ∣ x − z ∣ ) − 2 n − 1 − ∣ α ∣ − ∣ β ∣ − ∣ γ ∣ ,
∣ K j ( x , y , z ) ∣ ≲ ‖ b ‖ Lip 1 ( ∣ x − y ∣ + ∣ x − z ∣ + ∣ y − z ∣ ) − 2 n .

Let m ≥ 1 be a positive integer and K ( x , y 1 , … , y m ) be a locally integrable function defined away from the diagonal x = y 1 = ⋯ = y m in ( R n ) m + 1 and C be a positive constant. We say that K is a multilinear Calderón-Zygmund kernel if it satisfies the size condition that for all ( x , y 1 , … , y m ) ∈ ( R m ) m + 1 with x ≠ y s for some 1 ≤ s ≤ m ,

∣ K ( x , y 1 , … , y m ) ∣ ≤ C ( ∣ x − y 1 ∣ + ⋯ + ∣ x − y m ∣ ) − m n

and satisfies the regularity condition that

∣ K ( x , y 1 , … , y m ) − K ( x ′ , y 1 , … , y m ) ∣ ≤ C ∣ x − x ′ ∣ ( ∣ x − y 1 ∣ + ⋯ + ∣ x − y m ∣ ) − m n − 1

whenever max 1 ≤ k ≤ m ∣ x − y k ∣ ≥ 2 ∣ x − x ′ ∣ , and also that for each fixed k with 1 ≤ k ≤ m ,

(1.2) ∣ K ( x , y 1 , … , y k − 1 , y k , y k + 1 , … , y m ) − K ( x , y 1 , … , y k − 1 , y k ′ , y k + 1 , … , y m ) ∣ ≤ C ∣ x − x ′ ∣ ( ∣ x − y 1 ∣ + ⋯ + ∣ x − y m ∣ ) − m n − 1

whenever max 1 ≤ s ≤ m ∣ x − y s ∣ ≥ 2 ∣ y k − y k ′ ∣ .

The statements of our main theorems are presented as follows.

Theorem 1.1

Let σ ∈ ℬ BS 1 , 1 1 and b be a Lipschitz function on R n . Suppose that for all bounded functions a i supported on some cubes in R n with ∫ R n a i ( x ) d x = 0 for i = 1 , 2 ,

∫ R m [ T σ , b ] j ( a 1 , a 2 ) ( x ) x α d x = 0

for every multi-index α with ∣ α ∣ ≤ n , where α = ( α 1 , … , α n ) ∈ ( N ∪ { 0 } ) n and ∣ α ∣ = ∑ i = 1 n α i . Then [ T σ , b ] j ( j = 1 , 2 ) extends boundedly from H p 1 ( R n ) × H p 2 ( R n ) into H p ( R n ) for p 1 , p 2 ∈ n / ( n + 1 ) , 1 and p with 1 / p = 1 / p 1 + 1 / p 2 and ⌊ n ( 1 / p − 1 ) = m ⌋ , where ⌊ s ⌋ for any s ∈ R denotes the integer not greater than s .

Theorem 1.2

Let σ ∈ ℬ BS 1 , 1 1 and b be a Lipschitz function on R n . Suppose 0 ≤ p 1 , p 2 ≤ 1 , 1 < q 1 , q 2 < ∞ , 1 / p = 1 / p 1 + 1 / p 2 , 1 / q = 1 / q 1 + 1 / q 2 . If [ T σ , b ] j ( j = 1 , 2 ) is bounded from L q 1 × L q 2 into L q , ∞ with controlled by ‖ a ‖ Lip 1 , then [ T σ , b ] j ( j = 1 , 2 ) is bounded from K ˙ q 1 n ( 1 − 1 / q 1 ) , p 1 ( R n ) × K ˙ q 2 n ( 1 − 1 / q 2 ) , p 2 ( R n ) into W K ˙ q 2 n ( 2 − 1 / q ) , p ( R n ) .

Throughout this article, for 1 ≤ p ≤ ∞ , p ′ is the conjugate index of p , that is, 1 / p + 1 / p ′ = 1 . B ( x , R ) denotes the ball centered at x with radius R > 0 and f B = 1 ∣ B ( x , R ) ∣ ∫ B ( x , R ) f ( y ) d y . The boundedness of commutators on product of Hardy spaces is presented in Section 2. The boundedness of commutators on product of Herz-type spaces is given in Section 3.

2 Boundedness on product of Hardy spaces

Definition 2.1

[11] Let p ∈ ( 0 , 1 ] . The Hardy space H p ( R n ) is defined by

H p ( R n ) ≡ { f ∈ S ( R n ) : φ + ( f ) ≡ sup t > 0 ∣ φ t ∗ f ∣ ∈ L p ( R n ) } ,

where φ ∈ S ( R n ) with ∫ R n φ ( x ) d x = 1 , and for any y ∈ R n and t ∈ ( 0 , ∞ ) , φ t ( y ) = t − n φ ( y / t ) . Moreover, define

‖ f ‖ H p ( R n ) ≡ ‖ φ + ( f ) ‖ L p ( R n ) .

It is known that the definition of Hardy space H p ( R n ) does not depend on the choice of φ (see [11]).

Definition 2.2

[12] Let p ∈ ( 0 , 1 ] and q ∈ [ 1 , ∞ ] with p ≠ q . For s ∈ Z satisfying s ≥ ⌊ n ( 1 / p − 1 ) ⌋ , a real-valued function a ( x ) is called a ( p , q , s ) -atom centered at x 0 if

a ∈ L q ( R n ) and is supported in a cube Q centered at x 0 ;
‖ a ‖ L q ( R n ) ≤ ∣ Q ∣ 1 / q − 1 / p ;
∫ R n a ( x ) x α d x = 0 for every multi-index α with ∣ α ∣ ≤ s .

Let H fin p , q , s ( R n ) be the set of all finite linear combinations of ( p , q , s ) -atoms. For any f ∈ H fin p , q , s ( R n ) , define

‖ f ‖ H fin p , q , s ( R n ) ≡ ∑ i = 1 k ∣ λ i ∣ p 1 / p : f = ∑ i = 1 k λ i a i , k ∈ N , { a i } i = 1 k are ( p , q , s ) -atoms .

Denote by C ( R n ) the set of all continuous. Meda et al. proved the following result in [13], which ensures that a bounded linear operator on H fin p , q , s ( R n ) with q < ∞ or H fin p , q , s ( R n ) ∩ C ( R n ) can be extended to be a bounded operator on H p ( R n ) .

Lemma 2.1

[13] Let p ∈ ( 0 , 1 ] , q ∈ [ 1 , ∞ ] with p ≠ q and s ∈ Z satisfying s ≥ ⌊ n ( 1 / p − 1 ) ⌋ . The quasi-norms ‖ ⋅ ‖ H fin p , q , s ( R n ) and ‖ ⋅ ‖ H p ( R n ) are equivalent on H fin p , q , s ( R n ) where q < ∞ and on H fin p , q , s ( R n ) ∩ C ( R n ) when q = ∞ .

To prove Theorem 1.2, we need the boundedness of [ T σ , a ] j ( j = 1 , 2 ) on products of Lebesgue spaces.

Lemma 2.2

[9] If σ ∈ ℬ BS 1 , 1 m and b is a Lipschitz function on R n , then the commutators [ T σ , b ] j ( j = 1 , 2 ) are bilinear Calderón-Zygmund operators. In particular, [ T σ , b ] j , j = 1 , 2 are bounded from L p 1 × L p 2 into L p for 1 p = 1 p 1 + 1 p 2 and 1 < p 1 , p 2 < ∞ and verify appropriate end-point boundedness properties. Moreover, the corresponding norms of the operators are controlled by ‖ b ‖ Lip 1 .

Lemma 2.3

Let σ ∈ ℬ BS 1 , 1 1 and b be a Lipschitz function on R n . Suppose that a 1 is a ( p 1 , ∞ , 0 ) -atom supported on Q 1 and a 2 be a ( p 2 , ∞ , 0 ) -atom supported on Q 2 , with p 1 , p 2 ∈ ( n / n + 1 , 1 ] . Then

for any y ∈ ( 2 Q 1 ) c ,
∣ [ T σ , b ] j ( a 1 , a 2 ) ( y ) ∣ ≲ ‖ b ‖ Lip 1 ∣ Q 2 ∣ − 1 p 2 ∣ Q 1 ∣ 1 + 1 n − 1 p 1 ∣ y − x Q 1 ∣ n + 1 ,
while for any y ∈ ( 2 Q 2 ) c ,
∣ [ T σ , b ] j ( a 1 , a 2 ) ( y ) ∣ ≲ ‖ b ‖ Lip 1 ∣ Q 1 ∣ − 1 p 2 ∣ Q 2 ∣ 1 + 1 n − 1 p 1 ∣ y − x Q 2 ∣ n + 1 ;
for any y ∈ ( 2 Q 1 ) c ∩ y ∈ ( 2 Q 2 ) c ,
∣ [ T σ , b ] j ( a 1 , a 2 ) ( y ) ∣ ≲ ‖ b ‖ Lip 1 min ∣ Q 1 ∣ 1 + 1 n − 1 p 1 ∣ Q 2 ∣ 1 − 1 p 2 ( ∣ y − x Q 1 ∣ + ∣ y − x Q 2 ∣ ) 2 n + 1 , ∣ Q 2 ∣ 1 + 1 n − 1 p 2 ∣ Q 1 ∣ 1 − 1 p 1 ( ∣ y − x Q 1 ∣ + ∣ y − x Q 2 ∣ ) 2 n + 1 ;
for any cube R ∈ R n
∫ R φ + [ T σ , b ] j ( a 1 , a 2 ) ( x ) d x ≲ ‖ b ‖ Lip 1 ∣ R ∣ 1 2 min ∣ Q 1 ∣ 1 2 − 1 p 1 ∣ Q 2 ∣ − 1 p 2 , ∣ Q 1 ∣ − 1 p 2 ∣ Q 2 ∣ 1 2 − 1 p 1 .

Proof of Lemma 2.3

To obtain the conclusion of Lemma 2.3, we only prove (ii) and (iii). Together with (1.2), Theorem A and the vanishing moment of a 1 , it follows that for any y ∈ ( 2 Q 1 ) c ∩ y ∈ ( 2 Q 2 ) c ,

∣ [ T σ , b ] j ( a 1 , a 2 ) ( y ) ∣ ≲ ∫ ( R n ) 2 ∣ K j ( y , z 1 , z 2 ) − K j ( y , x Q 1 , z 2 ) ∣ ∣ a 1 ( z 1 ) ∣ ∣ a 2 ( z 2 ) ∣ d z 1 d z 2 ≲ ‖ b ‖ Lip 1 ∫ ( R n ) 2 ∣ z 1 − x Q 1 ∣ ( ∣ y − z 1 ∣ + ∣ y − z 2 ∣ ) 2 n + 1 ∣ a 1 ( z 1 ) ∣ ∣ a 2 ( z 2 ) ∣ d z 1 d z 2 ≲ ‖ b ‖ Lip 1 l ( Q 1 ) ( ∣ y − x Q 1 ∣ + ∣ y − x Q 2 ∣ ) 2 n + 1 ∏ k = 1 2 ‖ a k ‖ L 1 ( R n ) ≲ ‖ b ‖ Lip 1 ∣ Q 1 ∣ 1 + 1 n − 1 p 1 ∣ Q 2 ∣ 1 − 1 p 2 ( ∣ y − x Q 1 ∣ + ∣ y − x Q 2 ∣ ) 2 n + 1 .

Similarly, applying the vanishing moment of a 2 , we obtain

∣ [ T σ , b ] j ( a 1 , a 2 ) ( y ) ∣ ≲ ∣ b ‖ Lip 1 ∣ Q 2 ∣ 1 + 1 n − 1 p 2 ∣ Q 1 ∣ 1 − 1 p 1 ( ∣ y − x Q 1 ∣ + ∣ y − x Q 2 ∣ ) 2 n + 1 ,

and conclusion (ii) follows directly.

To prove conclusion (iii), we first observe that for any g ∈ L loc 1 ( R n ) and x ∈ R n ,

φ + ( g ) ( x ) ≲ M ( g ) ( x ) ,

where M is the Hardy-Littlewood maximal operator. By Hölder’s inequality, the fact φ + is bounded on L 2 ( R n ) , Lemma 2.1 and the size condition of a 1 and a 2 , we have

∫ R φ + [ T σ , b ] j ( a 1 , a 2 ) ( x ) d x ≲ ∣ R ∣ 1 2 ‖ φ + [ T σ , b ] j ( a 1 , a 2 ) ‖ L 2 ( R n ) ≲ ‖ b ‖ Lip 1 ∣ R ∣ 1 2 ‖ a 1 ‖ L 2 ( R n ) ‖ a 1 ‖ L ∞ ( R n ) ≲ ‖ b ‖ Lip 1 ∣ R ∣ 1 2 ∣ Q 1 ∣ 1 2 − 1 p 1 ∣ Q 2 ∣ − 1 p 2 ,

and similarly,

∫ R φ + [ T σ , b ] j ( a 1 , a 2 ) ( x ) d ≲ ‖ b ‖ Lip 1 ∣ R ∣ 1 2 ∣ Q 1 ∣ − 1 p 1 ∣ Q 2 ∣ 1 2 − 1 p 2 .

This leads conclusion (iii) and completes the proof of Lemma 2.3.□

Lemma 2.4

[14] Let p ∈ ( 0 , 1 ] . Then there exists a positive constant C p such that for all finite collections of cubes { Q k } k = 1 K in R n and all nonnegative integrable functions g k with supp ( g k ) ∈ Q k ,

∑ k = 1 K g k L p ( R n ) ≤ C p ∑ k = 1 K 1 ∣ Q k ∣ ∫ Q k g k ( x ) d x χ Q k ∗ L p ( R n ) ,

where Q k ∗ denotes the cube with the same center as Q k and 2 n its side-length.

Proof of Theorem 1.1

By Lemma 2.1 and a density argument, it suffices to prove that [ T σ , b ] j ( j = 1 , 2 ) is bounded from ( H fin p 1 , ∞ , 0 ( R n ) ∩ C ( R n ) ) × ( H fin p 2 , ∞ , 0 ( R n ) ∩ C ( R n ) ) into H p ( R n ) , i = 1 , 2 , we decompose f i as

f i ( x ) = ∑ k i λ i , k i a i , k i ( x ) ,

where a i , k i are ( p i , ∞ , 0 ) -atoms in Definition 2.2. This means that a i , k i are functions supported in cubes Q i , k i and satisfy the properties ‖ a i , k i ‖ L ∞ ( R n ) ≤ ∣ Q i , k i ∣ − 1 / p i and ∫ Q i , k i a i , k i d x = 0 . Without loss of generality, we may assume that for fixed Q 1 , k 1 and Q 2 , k 2 , l ( Q 1 , k 1 ) ≤ l ( Q 2 , k 2 ) . It is easy to see that there exists a cube R k 1 , k 2 such that

( Q 1 , k 1 ∗ ∩ Q 2 , k 2 ∗ ) ⊂ R k 1 , k 2 ⊂ R k 1 , k 2 ∗ ⊂ ( Q 1 , k 1 ∗ ∗ ∩ Q 2 , k 2 ∗ ∗ )

and ∣ R k 1 , k 2 ∣ ≥ C 1 ∣ Q 1 , k 1 ∣ , where C 1 ∈ ( 0 , 1 ) is a constant independent of R k 1 , k 2 and Q i , k i with i = 1 , 2 . Our purpose is to show

(2.1) ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] L p ( R n ) ≲ ‖ b ‖ Lip 1 ∏ i = 1 2 ∑ k i ∣ λ i , k i ∣ p i 1 / p i .

For this goal, we write

∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] L p ( R n ) ≲ ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] χ Q 1 , k 1 ∗ ∩ Q 2 , k 2 ∗ L p ( R n ) + ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] χ ( Q 1 , k 1 ∗ ) c ∩ ( Q 2 , k 2 ∗ ) c L p ( R n ) + ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] χ ( Q 1 , k 1 ∗ ) c ∩ Q 2 , k 2 ∗ L p ( R n ) + ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] χ Q 1 , k 1 ∗ ∩ ( Q 2 , k 2 ∗ ) c L p ( R n ) ≔ ∑ i = 1 4 E i .

It follows from Lemma 2.4, (iii) of Lemma 2.3 and Hölder’s inequality that

E 1 ≲ ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 ∣ 1 ∣ R k 1 , k 2 ∣ ∫ R k 1 , k 2 φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] ( x ) d x χ R k 1 , k 2 ∗ L p ( R n ) ≲ ‖ b ‖ Lip 1 ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 ∣ 1 ∣ R k 1 , k 2 ∣ 1 / 2 ∣ Q 1 , k 1 ∣ 1 2 − 1 p 1 ∣ Q 2 , k 2 ∣ − 1 p 2 χ R k 1 , k 2 ∗ L p ( R n ) ≤ ‖ b ‖ Lip 1 ∏ i = 1 2 ∑ k i ∣ λ i , k i ∣ ∣ Q i , k i ∣ − i / p i χ Q i , k i ∗ ∗ ≤ ‖ b ‖ Lip 1 ∏ i = 1 2 ∑ k i ∣ λ i , k i ∣ p i 1 / p i .

We now turn to estimate E 2 . Since [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) has vanishing moment up to order n , we can subtract the Taylor polynomial P n of the function φ ( x − y ) at the point ( x − x Q 1 , k 1 ) with x ∈ ( Q 1 , k 1 ∗ ) c ∩ ( Q 2 , k 2 ∗ ) c and y ∈ R n . Set

Δ ( t , x , y ) ≔ φ t ( x , y ) − P t n ( x − x Q 1 , k 1 ) ,

and for all x ∈ ( Q 1 , k 1 ∗ ) c ∩ ( Q 2 , k 2 ∗ ) c , write

φ + [ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ] ( x ) ≲ sup t > 0 ∫ 2 Q 1 , k 1 ∩ 2 Q 2 , k 2 Δ ( t , x , y ) [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ( y ) d y

+ sup t > 0 ∫ ( 2 Q 1 , k 1 ) c ∩ ( 2 Q 2 , k 2 ) c Δ ( t , x , y ) [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ( y ) d y + sup t > 0 ∫ ( 2 Q 1 , k 1 ) c ∩ 2 Q 2 , k 2 Δ ( t , x , y ) [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ( y ) d y + sup t > 0 ∫ 2 Q 1 , k 1 ∩ ( 2 Q 2 , k 2 ) c Δ ( t , x , y ) [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ( y ) d y ≔ ∑ i = 1 4 Φ i ( a 1 , k 1 , a 2 , k 2 ) ( x ) .

Thus,

E 2 ≲ ∑ i = 1 4 ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 ∣ Φ i ( a 1 , k 1 , a 2 , k 2 ) χ ( Q 1 , k 1 ∗ ) c ∩ ( Q 2 , k 2 ∗ ) c L p ( R n ) ≔ E 2 i .

Our assumption p 1 , p 2 ∈ ( n / ( n + 1 ) , 1 ] and 1 / p = 1 / p 1 + 1 / p 2 with ⌊ n ( 1 / p − 1 ) ⌋ = n guarantee that we can choose θ ∈ ( 0 , 1 ) such that p 1 > n ( n + θ ) and p 2 > n ( n + 1 − θ ) . On the other hand, we can verify that for all x ∈ ( Q 1 , k 1 ∗ ) c ∩ ( Q 2 , k 2 ∗ ) c and y ∈ 2 Q 1 , k 1 ∩ 2 Q 1 , k 1 , ∣ x − x Q 1 , k 1 ∣ ∼ ∣ x − y ∣ ∼ ∣ x − x Q 2 , k 2 ∣ , and then for all t ∈ ( 0 , ∞ ) ,

∣ Δ ( t , x , y ) ∣ ≲ ∣ y − x Q 1 , k 1 ∣ n + 1 ∣ x − x Q 1 , k 1 ∣ 2 n + 1 ≲ ∣ Q 1 , k 1 ∣ 1 2 + θ n ∣ x − x Q 1 , k 1 ∣ n + θ ∣ Q 1 , k 1 ∣ 1 2 + 1 n − θ n ∣ x − x Q 1 , k 1 ∣ n + 1 − θ ,

since we assume that l ( Q 1 , k 1 ) ≤ l ( Q 2 , k 2 ) . This, together with Lemma 2.2 and Hölder’s inequality, tells us that

E 21 ≲ ∫ R n ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 ∣ ∣ Q 1 , k 1 ∣ 1 2 + θ n ∣ x − x Q 1 , k 1 ∣ n + θ ∣ Q 2 , k 2 ∣ 1 2 + 1 n − θ n ∣ x − x Q 2 , k 2 ∣ n + 1 − θ × ∫ 2 Q 1 , k 1 ∩ 2 Q 2 , k 2 ∣ [ T σ , b ] j ( a 1 , k 1 , a 2 , k 2 ) ( y ) ∣ d y χ ( Q 1 , k 1 ∗ ) c ∩ ( Q 2 , k 2 ∗ ) c ( x ) p d x 1 p ≲ ∫ R n ∑ k 1 ∑ k 2 ∣ λ 1 , k 1 ∣ ∣ λ 2 , k 2 ∣ ∣ Q 1 , k 1 ∣ 1 2 + θ n ∣ x − x Q 1 , k 1 ∣ n + θ ∣ Q 2 , k 2 ∣ 1 2 + 1 n − θ n ∣ x − x Q 2 , k 2 ∣ n + 1 − θ × ‖ b ‖ Lip 1 ‖ a 1 , k 1 ‖ L 2 ( R n ) ‖ a 2 , k 2 ‖ L 2 ( R n ) χ ( Q 1 , k 1 ∗ ) c ∩ ( Q 2 , k 2 ∗ ) c ( x ) ] p d x } 1 p ≲ ‖ a 1 , k 1 ‖ L 2 ( R n ) ∑ k 1 ∣ λ 1 , k 1 ∣ p 1 ∫ ( Q 1 , k 1 ∗ ) c ∣ Q 1 , k 1 ∣ ( 1 + θ n − 1 p 1 ) p 1 ∣ x − x Q 1 , k 1 ∣ ( n + θ ) p 1 d x 1 / p 1 × ‖ b ‖ Lip 1 ‖ a 1 , k 1 ‖ L 2 ( R n ) ∑ k 2 ∣ λ 2 , k 2 ∣ p 2 ∫ ( Q 2 , k 2 ∗ ) c ∣ Q 2 , k 2 ∣ ( 1 + 1 n − θ n − 1 p 2 ) p 1 ∣ x − x Q 2 , k 2 ∣ ( n + 1 − θ ) p 2 d x 1 / p 2 ≲ ‖ b ‖ Lip 1 ∏ i = 1 2 ∑ k i ∣ λ i , k i ∣ p i 1 / p i .

The estimate for E 22 is cumbersome but straightforward. For i , s = 0 , 1 , … , let

I i s