A note on commutators of strongly singular Calderón-Zygmund operators

Pu Zhang; Xiaomeng Zhu

doi:10.1515/math-2022-0498

Open Access Published by De Gruyter Open Access October 5, 2022

A note on commutators of strongly singular Calderón-Zygmund operators

Pu Zhang and Xiaomeng Zhu

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0498

Abstract

In this article, the authors consider the commutators of strongly singular Calderón-Zygmund operator with Lipschitz functions. A sufficient condition is given for the boundedness of the commutators from Lebesgue spaces L p ( R n ) to certain Campanato spaces C p , β ( R n ) .

Keywords: strongly singular Calderón-Zygmund operator; commutator; Lipschitz space; Campanato space

MSC 2010: 42B20; 42B35; 47B47

1 Introduction and result

Let T be the classical singular integral operator, and the commutator T b generated by T and a locally integrable function b is given by

T b f = b T ( f ) − T ( b f ) .

A well-known result by Coifman et al. [1] states that T b is bounded on L p ( R n ) for 1 < p < ∞ when b ∈ BMO ( R n ) . They also gave some characterization of BMO ( R n ) in virtue of the L p boundedness of the aforementioned commutator (see also [2,3]).

In 1978, Janson [2] studied the boundedness of the commutator T b when b ∈ Λ ˙ γ ( R n ) , the homogeneous Lipschitz space of order 0 < γ < 1 , which is the space of all functions b , such that

(1.1) ‖ b ‖ Λ ˙ γ ( R n ) = sup x , y ∈ R n x ≠ y ∣ b ( x ) − b ( y ) ∣ ∣ x − y ∣ γ < ∞ .

Janson proved that T b is bounded from L p ( R n ) to L q ( R n ) for 1 < p < q < ∞ if and only if b ∈ Λ ˙ γ ( R n ) with γ = n ( 1 / p − 1 / q ) . In 1995, Paluszyński [4] made a further study of the problem and proved that T b is bounded from L p ( R n ) to some Triebel-Lizorkin spaces F ˙ p γ , ∞ ( R n ) if and only if b ∈ Λ ˙ γ ( R n ) , for 1 < p < ∞ and 0 < γ < 1 .

In 2015, Zhang et al. [5] gave another kind of interesting results for T b when b belongs to Lipschitz spaces. They proved that T b is bounded from L p ( R n ) to C p , β ( R n ) if and only if b ∈ Λ ˙ γ ( R n ) , for 1 < p < ∞ , − n / p ≤ β < 0 and 0 < γ = β + n / p < 1 , where C p , β ( R n ) is Campanato space (see Definition 1.2).

On the other hand, motivated by the study of multiplier operator with symbol given by e i ∣ ξ ∣ α ∣ ξ ∣ β away from the origin ( 0 < α < 1 , β > 0 ) , Alvarez and Milman [6] introduced the following strongly singular Calderón-Zygmund operator.

Definition 1.1

[6] Let T : S → S ′ be a bounded linear operator. T is called a strongly singular Calderón-Zygmund operator if the following conditions are fulfilled.

T can be extended to a continuous operator from L 2 into itself.
There exists a continuous function K ( x , y ) on { ( x , y ) : x ≠ y } such that
∣ K ( x , y ) − K ( x , z ) ∣ + ∣ K ( y , x ) − K ( z , x ) ∣ ⩽ C ∣ y − z ∣ δ ∣ x − z ∣ n + δ / α ,
if 2 ∣ y − z ∣ α ≤ ∣ x − z ∣ for some 0 < δ ≤ 1 and 0 < α < 1 , and,
⟨ T f , g ⟩ = ∫ K ( x , y ) f ( y ) g ( x ) d y d x ,
for f , g ∈ S with disjoint supports.
For some ( 1 − α ) n / 2 ≤ η < n / 2 , both T and its conjugate operator T ∗ can be extended to continuous operators from L q ( R n ) into L 2 ( R n ) , where 1 / q = 1 / 2 + η / n .

In 1986, Alvarez and Milman studied the boundedness of strongly singular Calderón-Zygmund operator on Lebesgue spaces and Hardy spaces in [6,7]. Later on, there are many authors discussed the mapping properties of strongly singular Calderón-Zygmund operators in various spaces. See, for instance, [8,9, 10,11]. We would like to note that, as stated in [6,7], the strongly singular Calderón-Zygmund operators include pseudo-differential operator with a symbol in the Hörmander class S α , δ − η , where 0 < δ ≤ α < 1 , ( 1 − α ) n / 2 ≤ η < n / 2 .

Now, we define the commutator generated by strongly singular Calderón-Zygmund operator T and a locally integrable function b as follows:

T b f ( x ) = b ( x ) T ( f ) ( x ) − T ( b f ) ( x ) .

The main result of Alvarez et al. in [8] yields the boundedness of T b on L p ( R n ) , 1 < p < ∞ , when b ∈ BMO ( R n ) . Afterward, the mapping properties of T b , when b belongs to BMO space or Lipschitz space, on Lebesgue spaces, Morrey spaces, Herz type spaces, and Hardy spaces have been studied by several authors. See [10,11,12, 13,14,15] for example.

In this article, we will continue the study of the commutator of strongly singular Calderón-Zygmund operator when the symbol b belongs to Lipschitz space. The aim is to extend some of the results in [5] to a strongly singular Calderón-Zygmund operator.

As usual, let B = B ( x 0 , r ) denote the ball centered at x 0 with radius r . For a > 0 , a B stands for the ball concentric with B having a times its radius, that is, a B = B ( x 0 , a r ) . Denote by ∣ B ∣ the Lebesgue measure of B and by χ B its characteristic function. For f ∈ L loc 1 ( R n ) , we write

f B = 1 ∣ B ∣ ∫ B f ( x ) d x .

Definition 1.2

Let 1 ≤ p < ∞ , − n / p ≤ β < 1 , the Campanato space C p , β ( R n ) is given by

C p , β ( R n ) = { f ∈ L loc p ( R n ) , ‖ f ‖ C p , β ( R n ) < ∞ } ,

where

‖ f ‖ C p , β ( R n ) ≔ sup B 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ f ( x ) − f B ∣ p d x 1 / p ,

and the supremum is taken over all balls B in R n .

Our result can be stated as follows.

Theorem 1.1

Let T be a strongly singular Calderón-Zygmund operator, and, α , η , and δ be as in Definition 1.1. Suppose that n ( 1 − α ) + 2 η 2 η < p < ∞ , − n / p ≤ β < 0 , and 0 < γ = β + n / p < 1 . If b ∈ Λ ˙ γ ( R n ) , then T b is bounded from L p ( R n ) to C p , β ( R n ) , that is, there exists a constant C > 0 such that for all f ∈ L p ( R n ) ,

‖ T b ( f ) ‖ C p , β ( R n ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) .

Remark 1.1

Theorem 1.1 gives a new kind of boundedness for commutator T b when b belongs to certain Lipschitz spaces, compared with the ( L p , L q ) -boundedness and the ( M p , β , M q , β + γ ) -boundedness of T b , when n ( 1 − α ) + 2 η 2 η < p < q < ∞ and 0 < γ = n 1 p − 1 q < 1 , obtained in [12, Corollary 1] and [10, Theorem 2.2], respectively.

2 Proof of Theorem 1.1

To prove Theorem 1.1, we need some known results. The first one is due to DeVore and Sharpley [16] and Janson et al. [17] (see also Paluszyński [4], Lemma 1.5).

Lemma 2.1

Let 0 < γ < 1 and b ∈ Λ ˙ γ ( R n ) , then for all 1 ≤ p < ∞ ,

‖ b ‖ Λ ˙ γ ( R n ) ≈ sup B 1 ∣ B ∣ γ / n 1 ∣ B ∣ ∫ B ∣ b ( x ) − b B ∣ p d x 1 / p ≈ sup B 1 ∣ B ∣ γ / n ‖ b − b B ‖ L ∞ ( B ) .

The next result is easy to check by using (1.1). See also DeVore and Sharpley [16], page 14.

Lemma 2.2

[16] Let 0 < γ < 1 , b ∈ Λ ˙ γ ( R n ) , and B and B ′ be balls in R n . If B ′ ⊂ B , then

∣ b B ′ − b B ∣ ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ∣ B ∣ γ / n .

Now we recall the boundedness of strongly singular Calderón-Zygmund operator T on Lebesgue spaces. Let us observe that T is bounded from L ∞ to BMO ([6], Theorem 2.1), from L 1 to L 1 , ∞ ([7], Theorem 4.1), and from H 1 to L 1 ([12], Lemma 2), and note the assumption (3) in Definition 1.1, and by interpolation between these estimates, we achieve the following L p -boundedness of T . We refer to [12] (page 1052) and [11] (pages 42 and 43), for details.

Lemma 2.3

Let T be a strongly singular Calderón-Zygmund operator, and, α , η , and δ be the same as in Definition 1.1.

If 1 < p < ∞ , then T is bounded from L p ( R n ) to itself.
If n ( 1 − α ) + 2 η 2 η ≤ u < ∞ , then there is a positive number v satisfying 0 < u / v ≤ α , such that T is bounded from L u ( R n ) to L v ( R n ) .

Furthermore, the index v can be chosen as v = u q ′ 2 q ′ − u q ′ + 2 u − 2 when n ( 1 − α ) + 2 η 2 η ≤ u ≤ 2 and v = u q ′ 2 when 2 ≤ u < ∞ , where q is given in Definition 1.1 and q ′ is its conjugate index.

Now, let us prove Theorem 1.1.

Proof of Theorem 1.1

For any f ∈ L p ( R n ) , it suffices to prove

(2.1) 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − ( T b f ) B ∣ p d y 1 / p ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ,

for all balls B in R n .

For any ball B = B ( x 0 , r ) centered at x 0 with radius r , we divide the proof into two cases.

Case 1. The case when r > 1 . Denote by B ∗ = 8 B = B ( x 0 , 8 r ) the ball with the same center as B and 8 times the radius. Let f 1 = f χ B ∗ and f 2 = f − f 1 . For any real number c , by Minkowski’s inequality and Hölder’s inequality, we have

1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − ( T b f ) B ∣ p d y 1 / p ≤ 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − c ∣ p d y 1 / p + 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ ( T b f ) B − c ∣ p d y 1 / p ≤ 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − c ∣ p d y 1 / p + 1 ∣ B ∣ β / n ∣ ( T b f ) B − c ∣ ≤ 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − c ∣ p d y 1 / p + 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( z ) − c ∣ d z ≤ 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − c ∣ p d y 1 / p + 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( z ) − c ∣ p d z 1 / p ≤ 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − c ∣ p d y 1 / p .

Let c = − ( T ( ( b − b B ) f 2 ) ) B and notice that T b f = T b − b B f , one has

(2.2) 1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − ( T b f ) B ∣ p d y 1 / p ≤ 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b − b B f ( y ) + ( T ( ( b − b B ) f 2 ) ) B ∣ p d y 1 / p ≤ 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ ( b ( y ) − b B ) T f ( y ) ∣ p d y 1 / p + 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T ( ( b − b B ) f 1 ) ( y ) ∣ p d y 1 / p + 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T ( ( b − b B ) f 2 ) ( y ) − ( T ( ( b − b B ) f 2 ) ) B ∣ p d y 1 / p ≔ I 1 + I 2 + I 3 .

For I 1 , note that 1 < p < ∞ and 0 < γ = β + n / p < 1 , it follows from Lemmas 2.1 and 2.3 that

I 1 = 2 ∣ B ∣ γ / n ∫ B ∣ ( b ( y ) − b B ) T f ( y ) ∣ p d y 1 / p ≤ 2 ∣ B ∣ γ / n ‖ b − b B ‖ L ∞ ( B ) ∫ B ∣ T f ( y ) ∣ p d y 1 / p

≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ T f ‖ L p ( R n ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) .

Next we estimate I 2 . Again we note that 1 < p < ∞ and 0 < γ = β + n / p < 1 . By Lemmas 2.1, 2.2, and 2.3, we deduce

I 2 ≤ 2 ∣ B ∣ γ / n ‖ T ( ( b − b B ) f 1 ) ‖ L p ( R n ) ≤ C ∣ B ∣ γ / n ‖ ( b − b B ) f 1 ‖ L p ( R n ) ≤ C ∣ B ∣ γ / n { ‖ ( b − b B ∗ ) f ‖ L p ( B ∗ ) + ‖ ( b B ∗ − b B ) f ‖ L p ( B ∗ ) } ≤ C ∣ B ∣ γ / n { ‖ b − b B ∗ ‖ L ∞ ( B ∗ ) + ∣ b B ∗ − b B ∣ } ‖ f ‖ L p ( R n ) ≤ C ∣ B ∣ γ / n { C ∣ B ∗ ∣ γ / n ‖ b ‖ Λ ˙ γ ( R n ) + C ‖ b ‖ Λ ˙ γ ( R n ) ∣ B ∗ ∣ γ / n } ‖ f ‖ L p ( R n ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) .

Now, let us consider I 3 . Since for any w , y ∈ B = B ( x 0 , r ) and any z ∈ ( B ∗ ) c one has 2 ∣ y − w ∣ α < ∣ z − w ∣ , it follows from Definition 1.1 that

(2.3) ∣ T ( ( b − b B ) f 2 ) ( y ) − T ( ( b − b B ) f 2 ) ( w ) ∣ ≤ ∫ R n ∣ K ( y , z ) − K ( w , z ) ∣ ∣ ( b ( z ) − b B ) f 2 ( z ) ∣ d z = ∫ ( B ∗ ) c ∣ K ( y , z ) − K ( w , z ) ∣ ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z ≤ C ∫ ( B ∗ ) c ∣ y − w ∣ δ ∣ z − w ∣ n + δ / α ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z ≤ C ∑ k = 1 ∞ ∫ 2 k B ∗ \ 2 k − 1 B ∗ ∣ y − w ∣ δ ∣ z − w ∣ n + δ / α ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z ≤ C r δ − δ / α ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∫ 2 k B ∗ ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z .

Observe that the last term of (2.3) is always independent of w and y , for any w , y ∈ B . Then we can write

(2.4) I 3 = 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T ( ( b − b B ) f 2 ) ( y ) − ( T ( ( b − b B ) f 2 ) ) B ∣ p d y 1 / p ≤ 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B 1 ∣ B ∣ ∫ B ∣ T ( ( b − b B ) f 2 ) ( y ) − T ( ( b − b B ) f 2 ) ( w ) ∣ d w p d y 1 / p ≤ 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B 1 ∣ B ∣ ∫ B C r δ − δ / α ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∫ 2 k B ∗ ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z d w p d y 1 / p ≤ C r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∫ 2 k B ∗ ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z ≤ C r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∫ 2 k B ∗ ∣ b ( z ) − b 2 k B ∗ ∣ ∣ f ( z ) ∣ d z + C r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∫ 2 k B ∗ ∣ b 2 k B ∗ − b B ∣ ∣ f ( z ) ∣ d z ≔ I 3 , 1 + I 3 , 2 .

Applying Hölder’s inequality, Lemma 2.1, and noting that 0 < γ = β + n / p < 1 , we obtain

I 3 , 1 ≤ C r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∫ 2 k B ∗ ∣ b ( z ) − b 2 k B ∗ ∣ p ′ d z 1 / p ′ ∫ 2 k B ∗ ∣ f ( z ) ∣ p d z 1 / p ≤ C r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∣ 2 k B ∗ ∣ γ / n + 1 / p ′ ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ β / n ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) r δ ( 1 − 1 / α ) ∑ k = 1 ∞ 2 k ( β − δ / α ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ,

where in the last step we made use of the fact that r δ ( 1 − 1 / α ) ≤ 1 since r > 1 and δ ( 1 − 1 / α ) < 0 and the fact that the series ∑ k = 1 ∞ 2 k ( β − δ / α ) is convergent since β − δ / α < 0 .

For I 3 , 2 , noting that 0 < γ = β + n / p < 1 and applying Lemma 2.2 and Hölder’s inequality, we have

I 3 , 2 = C r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∫ 2 k B ∗ ∣ b 2 k B ∗ − b B ∣ ∣ f ( z ) ∣ d z ≤ C r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ‖ b ‖ Λ ˙ γ ( R n ) ∣ 2 k B ∗ ∣ γ / n ∫ 2 k B ∗ ∣ f ( z ) ∣ d z ≤ C ‖ b ‖ Λ ˙ γ ( R n ) r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ ∣ 2 k B ∗ ∣ γ / n ∫ 2 k B ∗ ∣ f ( z ) ∣ p d z 1 / p ∣ 2 k B ∗ ∣ 1 − 1 / p ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) r δ ( 1 − 1 / α ) ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ∗ ∣ β / n ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) r δ ( 1 − 1 / α ) ∑ k = 1 ∞ 2 k ( β − δ / α ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ,

where in the last step we also made use of the fact that r δ ( 1 − 1 / α ) ≤ 1 and ∑ k = 1 ∞ 2 k ( β − δ / α ) is convergent. This, together with the estimates for I 3 , 1 , yields

I 3 ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) .

Combining the estimates for I 1 , I 2 , and I 3 leads to (2.1) for the case r > 1 .

Case 2. The case 0 < r ≤ 1 . Set B ˜ = B ( x 0 , r α ) and denote B ˜ ∗ = 8 B ˜ = B ( x 0 , 8 r α ) . Let f 3 = f χ B ˜ ∗ and f 4 = f − f 3 . For the same reason as that in (2.2), we deduce that

1 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T b f ( y ) − ( T b f ) B ∣ p d y 1 / p ≤ 2 ∣ B ∣ β / n + 1 / p ∫ B ∣ ( b ( y ) − b B ) T f ( y ) ∣ p d y 1 / p + 2 ∣ B ∣ β / n + 1 / p ∫ B ∣ T ( ( b − b B ) f 3 ) ( y ) ∣ p d y 1 / p + 2 ∣ B ∣ β / n + 1 / p ∫ B ∣ T ( ( b − b B ) f 4 ) ( y ) − ( T ( ( b − b B ) f 4 ) ) B ∣ p d y 1 / p ≔ J 1 + J 2 + J 3 .

Similar to I 1 , we have

J 1 = 2 ∣ B ∣ γ / n ∫ B ∣ ( b ( y ) − b B ) T f ( y ) ∣ p d y 1 / p ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) .

To estimate J 2 , we first observe that, by Lemma 2.3, there is an s satisfying 0 < p / s ≤ α such that T is bounded from L p to L s since n ( 1 − α ) + 2 η 2 η < p < ∞ . This, together with Hölder’s inequality, Lemmas 2.1 and 2.2, gives

J 2 = 2 ∣ B ∣ β / n + 1 / p ∫ B ∣ T ( ( b − b B ) f 3 ) ( y ) ∣ p d y 1 / p ≤ 2 ∣ B ∣ β / n + 1 / s ‖ T ( ( b − b B ) f 3 ) ‖ L s ( R n ) ≤ C ∣ B ∣ β / n + 1 / s ‖ ( b − b B ) f 3 ‖ L p ( R n ) ≤ C ∣ B ∣ β / n + 1 / s { ‖ ( b − b B ∗ ˜ ) f ‖ L p ( B ˜ ∗ ) + ‖ ( b B ∗ ˜ − b B ) f ‖ L p ( B ˜ ∗ ) } ≤ C ∣ B ∣ β / n + 1 / s { ‖ b − b B ∗ ˜ ‖ L ∞ ( B ˜ ∗ ) + ∣ b B ∗ ˜ − b B ∣ } ‖ f ‖ L p ( R n ) ≤ C ∣ B ∣ β / n + 1 / s ∣ B ˜ ∗ ∣ γ / n ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ≤ C r ( α − 1 ) β + ( α / p − 1 / s ) n ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ,

where the last two steps follow from the condition 0 < γ = β + n / p < 1 and the fact that r ( α − 1 ) β + ( α / p − 1 / s ) n ≤ 1 since 0 < r ≤ 1 and ( α − 1 ) β + ( α / p − 1 / s ) n > 0 .

Finally, let us consider J 3 . Since 0 < r ≤ 1 and 2 ∣ y − w ∣ α < ∣ z − w ∣ for any w , y ∈ B = B ( x 0 , r ) and z ∈ ( B ˜ ∗ ) c , similar to (2.3), we have

∣ T ( ( b − b B ) f 4 ) ( y ) − T ( ( b − b B ) f 4 ) ( w ) ∣ ≤ ∫ R n ∣ K ( y , z ) − K ( w , z ) ∣ ∣ ( b ( z ) − b B ) f 4 ( z ) ∣ d z ≤ C ∫ ( B ˜ ∗ ) c ∣ y − w ∣ δ ∣ z − w ∣ n + δ / α ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z ≤ C ∑ k = 1 ∞ ∫ 2 k B ˜ ∗ \ 2 k − 1 B ˜ ∗ ∣ y − w ∣ δ ∣ z − w ∣ n + δ / α ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z ≤ C ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∫ 2 k B ˜ ∗ ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z .

Reasoning as in (2.4), we can also write

J 3 = 2 ∣ B ∣ β / n 1 ∣ B ∣ ∫ B ∣ T ( ( b − b B ) f 4 ) ( y ) − ( T ( ( b − b B ) f 4 ) ) B ∣ p d y 1 / p ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∫ 2 k B ˜ ∗ ∣ b ( z ) − b B ∣ ∣ f ( z ) ∣ d z ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∫ 2 k B ˜ ∗ ∣ b ( z ) − b 2 k B ˜ ∗ ∣ ∣ f ( z ) ∣ d z + C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∫ 2 k B ˜ ∗ ∣ b 2 k B ˜ ∗ − b B ∣ ∣ f ( z ) ∣ d z ≔ J 3 , 1 + J 3 , 2 .

To estimate J 3 , 1 , we first observe the fact that r β ( α − 1 ) ≤ 1 since 0 < r ≤ 1 and β ( α − 1 ) > 0 and the series ∑ k = 1 ∞ 2 k ( β − δ / α ) is convergent since β − δ / α < 0 . Then by Hölder’s inequality and Lemma 2.1 and noticing that 0 < γ = β + n / p < 1 , we obtain

J 3 , 1 = C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∫ 2 k B ˜ ∗ ∣ b ( z ) − b 2 k B ˜ ∗ ∣ ∣ f ( z ) ∣ d z ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∫ 2 k B ˜ ∗ ∣ b ( z ) − b 2 k B ˜ ∗ ∣ p ′ d z 1 / p ′ ‖ f ‖ L p ( R n ) ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∣ 2 k B ˜ ∗ ∣ γ / n + 1 / p ′ ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ β / n ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) r β ( α − 1 ) ∑ k = 1 ∞ 2 k ( β − δ / α ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) .

For J 3 , 2 , applying Lemma 2.2 and Hölder’s inequality, and observing again the fact that r β ( α − 1 ) ≤ 1 , the series ∑ k = 1 ∞ 2 k ( β − δ / α ) is convergent and 0 < γ = β + n / p < 1 , and we have

J 3 , 2 = C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∫ 2 k B ˜ ∗ ∣ b 2 k B ˜ ∗ − b B ∣ ∣ f ( z ) ∣ d z ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∣ 2 k B ˜ ∗ ∣ γ / n ‖ b ‖ Λ ˙ γ ( R n ) ∫ 2 k B ˜ ∗ ∣ f ( z ) ∣ d z ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ ∣ 2 k B ˜ ∗ ∣ γ / n ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( 2 k B ˜ ∗ ) ∣ 2 k B ˜ ∗ ∣ 1 / p ′ ≤ C ∣ B ∣ β / n ∑ k = 1 ∞ 2 − k δ / α ∣ 2 k B ˜ ∗ ∣ β / n ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) r β ( α − 1 ) ∑ k = 1 ∞ 2 k ( β − δ / α ) ≤ C ‖ b ‖ Λ ˙ γ ( R n ) ‖ f ‖ L p ( R n ) .

The estimate for J 3 , 2