Some estimates for commutators of bilinear pseudo-differential operators

Theorem A

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

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

2. ∣ K j ( x , y , z ) ∣ ≲ ‖ b ‖ Lip 1 ( ∣ x − y ∣ + ∣ x − z ∣ + ∣ y − z ∣ ) − 2 n .

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 ,

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 ) .

Definition 2.1

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

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

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

1. a ∈ L q ( R n ) and is supported in a cube Q centered at x 0 ;

2. ‖ a ‖ L q ( R n ) ≤ ∣ Q ∣ 1 / q − 1 / p ;

3. ∫ R n a ( x ) x α d x = 0 for every multi-index α with ∣ α ∣ ≤ s .

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 = ∞ .

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

1. 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 ;

2. 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 ;

3. 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 ,

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

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 ,

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

and similarly,

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 ,

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

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

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

For this goal, we write