Definition 1.1. Let T be a bounded linear operator from S ( ℝ n ) to S ′ ( ℝ n ) (see [1] , [2] ). T is called a standard operator if T satisfies the following conditions:

1) T extends to a bounded linear operator on L 2 ( ℝ n ) .

2) There exists a function K ( x , y ) defined by { ( x , y ) ∈ ( ℝ n ) × ( ℝ n ) ; x ≠ y } satisfies

| K ( x , y ) | ≤ C / | x − y | n , (1.1)

where C > 0 .

3) 〈 T f , g 〉 = ∫ ( ℝ n ) ∫ ( ℝ n ) K ( x , y ) f ( y ) g ( x ) d x d y , for f , g ∈ S ( ℝ n ) with supp ( f ) ∩ supp ( g ) = ∅

A standard operator T is called a γ -Calderón - Zygmund operator if K is a standard kernel satisfies:

| K ( x , y ) − K ( z , y ) | ≤ C | x − z | γ / | x − y | n + γ ; (1.2)

| K ( y , x ) − K ( y , z ) | ≤ C | x − z | γ / | x − y | n + γ , (1.3)

if | x − z | < 1 2 | x − y | for some 0 < γ ≤ 1 .

The bounded mean oscillation BMO space and BMO norm are defined, respectively, by

B M O ( ℝ n ) = { b ∈ L l o c 1 ( ℝ n ) : ‖ b ‖ B M O ( ℝ n ) < ∞ } , (1.4)

‖ b ‖ B M O ( ℝ n ) = sup B : ball 1 / | B | ∫ B | b ( x ) − b B | d x . (1.5)

The commutator of the Calderón-Zygmund operator is defined by

[ b , T ] f ( x ) = b ( x ) T f ( x ) − T ( b f ) ( x ) . (1.6)

In 1983, J.-L. Jouné proved γ -Calderón - Zygmund operator is bounded on L p ( ℝ n ) in [3] . Coifman, Rochberg and Weiss proved that commutator [b,T] is bounded on L p ( ℝ n ) ( 1 < p < 1 ) (see [4] ).

Kovácik and Rákosník introduced Lebesgue spaces and Sobolev spaces with variable exponents (see [5] ). The function spaces with variable exponent has been recently obtained an increasing interest by a number of authors since many applications are found in many different fields, for example, in fluid dynamics (see [6] ), image restoration (see [7] [8] [9] ) and differential equations.

Herz spaces play an important role in harmonic analysis. After they were introduced in [10] , the boundedness of some operators and some characteriza- tions of Herz spaces with variable exponents were studied extensively (see [11] - [16] ). In 2015, Wang and Tao introduced the Herz spaces with two variable exponents p ( . ) , q ( . ) , and studied the parameterized Littlewood-Paley operators and their commutators on Herz spaces with variable exponents in [17] .

In this paper, we will discuss the boundedness of the Calderón-Zygmund operator T and their commutator [ b , T ] are bounded on Herz spaces with two variable exponents p ( . ) , q ( . ) .

In this section we recall some definitions. Let Ω be a measurable set in ℝ n with | Ω | > 0 . We firstly recall the definition of the Lebesgue spaces with variable exponent.

Definition 2.1. [5] Let p ( ⋅ ) : Ω → [ 1 , ∞ ) be a measurable function. The Lebesgue space with variable exponent L p ( ⋅ ) ( Ω ) is defined by

L p ( ⋅ ) ( Ω ) = { f ismeasurable : ∫ Ω ( | f ( x ) | η ) p ( x ) d x < ∞ forsomeconstant η > 0 } . (2.1)

For all compact K ⊂ Ω , the space L l o c p ( ⋅ ) ( Ω ) is defined by

L l o c p ( ⋅ ) ( Ω ) = {   f ismeasurable : f ∈ L p ( ⋅ ) ( K ) } . (2.2)

The Lebesgue spaces L p ( ⋅ ) ( Ω ) is a Banach spaces with the norm defined by

‖ f ‖ L p ( ⋅ ) ( Ω ) = inf { η > 0 : ∫ Ω ( | f ( x ) | η ) p ( x ) d x ≤ 1 } . (2.3)

We denote p − = e s s inf { p ( x ) : x ∈ Ω } , p + = e s s sup { p ( x ) : x ∈ Ω } . Then P ( Ω ) consists of all p ( ⋅ ) satisfying p − > 1 and p + < ∞ . Let M be the Hardy-Littlewood maximal operator. We denote B ( Ω ) to be the set of all function p ( ⋅ ) ∈ P ( Ω ) satisfying the M is bounded on L p ( ⋅ ) ( Ω ) .

Definition 2.2. [18] Let p ( ⋅ ) , q ( ⋅ ) ∈ P ( Ω ) . The mixed Lebesgue sequence space with variable exponent l q ( ⋅ ) ( L p ( ⋅ ) ) is the collection of all sequences { f j } j = 0 ∞ of the measurable functions on ℝ n such that

‖ { f j } j = 0 ∞ ‖ l q ( ⋅ ) ( L p ( ⋅ ) ) = inf { η > 0 : Q l q ( ⋅ ) ( L p ( ⋅ ) ) ( { f j ζ } j = 0 ∞ ) ≤ 1 } < ∞ , Q l q ( ⋅ ) ( L p ( ⋅ ) ) ( { f j } j = 0 ∞ ) = ∑ j = 0 ∞ inf { ζ j > 0 ; ∫ R n ( | f j ( x ) | ζ j 1 q ( x ) ) p ( x ) d x ≤ 1 } . (2.4)

Let B k = { x ∈ ℝ n : | x | ≤ 2 k } , C k = B k \ B k − 1 , χ k = χ C k , k ∈ ℤ . , for q + < ∞ , we have that

Q l q ( ⋅ ) ( L p ( ⋅ ) ) ( { f j } j = 0 ∞ ) = ∑ j = 0 ∞ ‖ | f j | q ( ⋅ ) ‖ L p ( ⋅ ) q ( ⋅ ) . (2.5)

Let B k = { x ∈ ℝ n : | x | ≤ 2 k } , C k = B k \ B k − 1 , χ k = χ C k , k ∈ ℤ .

Definition 2.3. [17] Let α ∈ ℝ n , q ( ⋅ ) , p ( ⋅ ) ∈ P ( ℝ n ) . The homogeneous Herz space with variable exponent K ˙ p ( ⋅ ) α , q ( ⋅ ) ( ℝ n ) is defined by

K ˙ p ( ⋅ ) α , q ( ⋅ ) ( ℝ n ) = { f ∈ L l o c p ( ⋅ ) ( ℝ n \ { 0 } ) : ‖ f ‖ K ˙ p ( ⋅ ) α , q ( ⋅ ) ( ℝ n ) < ∞ } .

Equipped the norm

‖ f ‖ K ˙ p ( ⋅ ) α , q ( ⋅ ) ( ℝ n ) = ‖ { 2 k α | f χ k | } k = 0 ∞ ‖ l q ( ⋅ ) ( L p ( ⋅ ) ) = inf { η > 0 : ∑ k = − ∞ ∞ ‖ ( 2 k α | f χ k | η ) q ( ⋅ ) ‖ L p ( ⋅ ) q ( ⋅ ) ≤ 1 } .

Remark 2.1. [17] Let q 1 ( ⋅ ) , q 2 ( ⋅ ) ∈ P ( ℝ n ) satisfying ( q 1 ) + ≤ ( q 2 ) + and satisfy the following results:

1) K ˙ p ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) ⊂ K ˙ p ( ⋅ ) α , q 2 ( ⋅ ) ( ℝ n ) .

2) If q 2 ( ⋅ ) q 1 ( ⋅ ) ∈ P ( ℝ n ) and q 2 ( ⋅ ) q 1 ( ⋅ ) ≥ 1 . For any f ∈ K ˙ p ( ⋅ ) α , q ( ⋅ ) ( ℝ n ) , by using Lemma 3.7 and Remark 2.2, we have

∑ k = − ∞ ∞ ‖ ( 2 k α | f χ k | η ) q 2 ( ⋅ ) ‖ L p ( ⋅ ) q 2 ( ⋅ ) ≤ ∑ k = − ∞ ∞ ‖ ( 2 k α | f χ k | η ) q 1 ( ⋅ ) ‖ L p ( ⋅ ) q 1 ( ⋅ ) p v ≤ { ∑ k = − ∞ ∞ ‖ ( 2 k α | f χ k | η ) q 1 ( ⋅ ) ‖ L p ( ⋅ ) q 1 ( ⋅ ) p h } p * ≤ 1.

where

p v = { ( q 2 ( ⋅ ) q 1 ( ⋅ ) ) − , 2 k α | f χ k | η ≤ 1 , ( q 2 ( ⋅ ) q 1 ( ⋅ ) ) + , 2 k α | f χ k | η > 1.

p * = { min v ∈ ℕ p v , ∑ v = 0 ∞ a v ≤ 1 , max v ∈ ℕ p v , ∑ v = 0 ∞ a v > 1.

This implies that K ˙ p ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) ⊂ K ˙ p ( ⋅ ) α , q 2 ( ⋅ ) ( ℝ n ) .

Remark 2.2. Let v ∈ ℕ , a v ≥ 0 , 1 ≤ p v < ∞ . Then we have

∑ v = 0 ∞ a v ≤ ( ∑ v = 0 ∞ a h ) p * ,

where

p * = { min v ∈ ℕ p v , ∑ v = 0 ∞ a v ≤ 1 , max v ∈ ℕ p v , ∑ v = 0 ∞ a v > 1.

In this section, we recall some properties and some lemmas of variable exponent belonging to the class B ( ℝ n ) .

Proposition 3.1. [19] If p ( ⋅ ) ∈ P ( ℝ n ) satisfies

| p ( x ) − p ( y ) | ≤ − C Log ( | x − y | ) , | x − y | ≤ 1 / 2 ; (3.1)

| p ( x ) − p ( y ) | ≤ C Log ( e + | x | ) , | y | ≥ | x | . (3.2)

Hence we have p ( ⋅ ) ∈ B ( ℝ n ) .

Lemma 3.1. [5] Given p ( ⋅ ) : ℝ n → [ 1 , ∞ ) have that for all functions f and g ,

∫ ℝ n | f ( x ) g ( x ) | d x ≤ C ‖ f ‖ L p ( ⋅ ) ( ℝ n ) ‖ g ‖ L p ′ ( ⋅ ) ( ℝ n ) . (3.3)

where C p = 1 + 1 p − − 1 p + .

Lemma 3.2. [5] Suppose that p ( ⋅ ) , p 1 ( ⋅ ) , p 2 ( ⋅ ) ∈ P ( ℝ n ) , for any f ∈ L p 1 ( ⋅ ) ( ℝ n ) , g ∈ L p 2 ( ⋅ ) ( ℝ n ) , when 1 p ( ⋅ ) = 1 p 2 ( ⋅ ) + 1 p 1 ( ⋅ ) , we get

‖ f ( x ) g ( x ) ‖ L p ( ⋅ ) ( ℝ n ) ≤ C ‖ g ( x ) ‖ L p 2 ( ℝ n ) ‖ f ( x ) ‖ L p 1 ( ⋅ ) ( ℝ n ) , (3.4)

where C p 1 , p 2 = [ 1 + 1 p 1 − − 1 p 1 + ] 1 p − .

Proposition 3.2. [20] Let p ( ⋅ ) ∈ B ( ℝ n ) and T be a Calderón - Zygmund operator. Then we have

‖ T f ‖ L p ( ⋅ ) ( ℝ n ) ≤ C ‖ f ‖ L p ( ⋅ ) ( ℝ n ) . (3.5)

Lemma 3.3. [20] Let p ( ⋅ ) ∈ B ( ℝ n ) , b ∈ BMO function and T be a Calderón - Zygmund operator.Then

‖ [ b , T ] f ‖ L p ( ⋅ ) ( ℝ n ) ≤ C ‖ b ‖ BMO ( ℝ n ) ‖ f ‖ L p ( ⋅ ) ( ℝ n ) (3.6)

Lemma 3.4. [11] Let b ∈ BMO ( ℝ n ) . If i , j ∈ ℤ with i < j , then we have

1. C − 1 ‖ b ‖ BMO ( ℝ n ) ≤ sup B 1 ‖ χ B ‖ L p ( ⋅ ) ( ℝ n ) ‖ ( b − b B ) χ B ‖ L p ( ⋅ ) ( ℝ n ) ≤ C ‖ b ‖ BMO ( ℝ n ) .

2. ‖ ( b − b B i ) χ B j ‖ L q ( ⋅ ) ( ℝ n ) ≤ C ( j − i ) ‖ b ‖ BMO ( ℝ n ) ‖ χ B j ‖ L q ( ⋅ ) ( ℝ n ) .

Lemma 3.5. [21] Let p u ( ⋅ ) ∈ B ( ℝ n ) ( u = 1 , 2 ) , then there exist constants 0 < ι u 1 , ι u 2 < 1 , and C > 0 such that for all balls B ⊂ ℝ n and all measurable subset R ⊂ B ,

‖ χ R ‖ L p u ( ⋅ ) ( ℝ n ) ‖ χ B ‖ L p u ( ⋅ ) ( ℝ n ) ≤ C ( | R | | B | ) ι u 1 , ‖ χ R ‖ L p ′ u ( ⋅ ) ( ℝ n ) ‖ χ B ‖ L p ′ u ( ⋅ ) ( ℝ n ) ≤ C ( | R | | B | ) ι u 2 . (3.7)

Lemma 3.6. [11] If p ( ⋅ ) ∈ B ( ℝ n ) , there exist a constant C > 0 such that for any balls B in ℝ n , we have

1 | B | ‖ χ B ‖ L p ( ⋅ ) ( ℝ n ) ‖ χ B ‖ L p ′ ( ⋅ ) ( ℝ n ) ≤ C . (3.8)

Lemma 3.7. [17] Suppose that p ( ⋅ ) , q ( ⋅ ) ∈ P ( B n ) . If f ∈ L p ( ⋅ ) q ( ⋅ ) , then

min ( ‖ f ‖ L p ( ⋅ ) q ( ⋅ ) q + , ‖ f ‖ L p ( ⋅ ) q ( ⋅ ) q − ) ≤ ‖ | f | q ( ⋅ ) ‖ L p ( ⋅ ) ≤ max ( ‖ f ‖ L p ( ⋅ ) q ( ⋅ ) q + , ‖ f ‖ L p ( ⋅ ) q ( ⋅ ) q − ) . (3.9)

Theorem 4.1. Suppose that p 1 ( ⋅ ) ∈ B ( ℝ n ) , q 1 ( ⋅ ) , q 2 ( ⋅ ) ∈ P ( ℝ n ) with ( q 2 ) − ≥ ( q 1 ) + . If − n ι 12 < α < n ι 11 with ι 11 , ι 12 as defined in Lemma 3.5, then the operator T is bounded from K ˙ p 1 ( ⋅ ) α , q 2 ( ⋅ ) ( ℝ n ) to K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) .

Proof Let h ( x ) ∈ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) . We write

h ( x ) = ∑ j = − ∞ ∞ h ( x ) χ j = ∑ j = − ∞ ∞ h j ( x ) .

By Definition 2.3, we have

#Math_135# (4.1)

Since

‖ ( 2 k α | T ( h ) χ k | η ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ ‖ ( 2 k α | ∑ j = − ∞ ∞ T ( h j ) χ k | ∑ i = 1 3 η 1 i ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ ‖ ( 2 k α | ∑ j = − ∞ k − 2 T ( h j ) χ k | η 11 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) + ‖ ( 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | η 12 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) + ‖ ( 2 k α | ∑ j = k + 2 ∞ T ( h j ) χ k | η 13 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) , (4.2)

where

η 11 = ‖ { 2 k α | ∑ j = − ∞ k − 2 T ( h j ) χ k | } k = − ∞ ∞ ‖ l q 2 ( ⋅ ) ( L p 1 ( ⋅ ) ) , (4.3)

η 12 = ‖ { 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | } k = − ∞ ∞ ‖ l q 2 ( ⋅ ) ( L p 1 ( ⋅ ) ) , (4.4)

η 13 = ‖ { 2 k α | ∑ j = k + 2 ∞ T ( h j ) χ k | } k = − ∞ ∞ ‖ l q 2 ( ⋅ ) ( L p 1 ( ⋅ ) ) ,

and

η = ∑ i = 1 3 η 1 i .

Thus,

∑ k = − ∞ ∞ ‖ ( 2 k α | T ( h ) χ k | η ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C .

We easily see that

‖ T ( h ) ‖ K ˙ p 1 ( ⋅ ) α , q 2 ( ⋅ ) ( ℝ n ) ≤ C η = C ∑ i = 1 3 η 1 i . (4.6)

This implies that we only need to prove η 11 , η 12 , η 13 ≤ C ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) . Denote η 10 = ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) .

First, we consider η 12 . By virtue of Lemma 3.7, we get

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ ∑ k = − ∞ ∞ ‖ 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | η 10 ‖ L p 1 ( ⋅ ) ( q 2 1 ) k ≤ ∑ k = − ∞ ∞ ( ‖ 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | η 10 ‖ L p 1 ( ⋅ ) ) ( q 2 1 ) k , (4.7)

where,

( q 2 1 ) k = { ( q 2 ) − , ‖ ( 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ 1 , ( q 2 ) + , ‖ ( 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) > 1.

In the above, we use the Proposition 3.2 and Remark 2.2. Since h ( x ) ∈ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) , we have ‖ 2 k α | h χ k | η 10 ‖ L p 1 ( ⋅ ) ≤ 1 and ∑ k = − ∞ ∞ ‖ ( 2 k α | h χ k | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ≤ 1 , we get

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = k − 2 k + 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C ∑ k = − ∞ ∞ ( ∑ j = k − 2 k + 2 ‖ 2 k α | h j | η 10 ‖ L p 1 ( ⋅ ) ) ( q 2 1 ) k ≤ C ∑ k = − ∞ ∞ ‖ 2 k α | h χ k | η 10 ‖ L p 1 ( ⋅ ) ( q 2 1 ) k ≤ C ∑ k = − ∞ ∞ ‖ ( 2 k α | h χ k | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ( q 2 1 ) k ( q 1 ) + ≤ C { ∑ k = − ∞ ∞ ‖ ( 2 k α | h χ k | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) } q * ≤ C .

Here ( p 1 ) + ≤ ( p 2 ) − ≤ ( q 2 1 ) k and q * = min k ∈ N ( q 2 1 ) k ( q 1 ) + . That is

η 12 ≤ C η 10 ≤ C ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) . (4.8)

Let us now turn to estimate η 11 . Noting that x ∈ A j and j ≤ k − 2 , by the generalized Hölder's inequality and the Minkowski’s inequality, we get

| T h j ( x ) | ≤ ∫ A j | K ( x , y ) h j ( y ) | d y ≤ C ∫ A j | h j ( y ) | / | x − y | n d y ≤ C 2 − k n ∫ A j | h j ( y ) | d y ≤ C 2 − k n ‖ h j ‖ L 1 ( ℝ n ) . (4.9)

By Lemmas 3.5-3.7 and the fact that ‖ 2 j α | h χ j | η 10 ‖ L p 1 ( ⋅ ) q 1 ≤ 1 , we easily see that

where

( q 2 2 ) k = { ( q 2 ) − , ‖ ( 2 k α | ∑ j = − ∞ k − 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ 1 , ( q 2 ) + , ‖ ( 2 k α | ∑ j = − ∞ k − 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) > 1.

Therefore, if ( q 1 ) + < 1 and ( p 1 ) + ≤ ( p 2 ) − ≤ ( q 2 2 ) k , we can get

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = − ∞ k − 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C { ∑ j = − ∞ ∞ ‖ ( | 2 j α h χ j | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ∑ k = j + 2 ∞ 2 ( k − j ) ( α − n ι 11 ) } q * ≤ C ,

where q * = min k ∈ ℕ ( q 2 1 ) k ( q 1 ) + .

If ( q 1 ) + ≥ 1 and ( q 2 2 ) k ≥ ( q 2 ) − ≥ ( q 2 ) + ≥ 1 . By Remark 2.2 and applying the generalized Hölder’s inequality, we obtain

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = − ∞ k − 2 T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C ∑ k = − ∞ ∞ { ∑ j = − ∞ k − 2 ( k − j ) 2 ( k − j ) ( α − n ι 11 ) ( q 1 ) + / 2 ‖ ( | 2 j α h χ j | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) } ( q 2 2 ) k ( q 1 ) + × ( ∑ j = − ∞ k − 2 2 ( k − j ) ( α − n ι 11 ) ( ( q 1 ) + ) ′ / 2 ) ( q 2 2 ) k ( ( q 1 ) + ) ′ ≤ C { ∑ j = − ∞ ∞ ‖ ( | 2 j α h χ j | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ∑ k = j + 2 ∞ 2 ( k − j ) ( α − n ι 11 ) ( q 1 ) + / 2 } q * ≤ C ,

where q * = min k ∈ ℕ ( q 2 2 ) k ( q 1 ) + .

Hence, we see that

η 11 ≤ C η 10 ≤ C ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) . (4.11)

Finally, we estimate η 13 . Noting that for each x ∈ A j and j ≥ k + 2 , we have

| T h j ( x ) | ≤ ∫ A j | K ( x , y ) h j ( y ) | d y ≤ C ∫ A j | h j ( y ) | / | x − y | n d y ≤ C 2 − j n ‖ h j ‖ L 1 ( ℝ n ) . (4.12)

By Lemma 3.7 and ‖ 2 j α | h χ j | η 10 ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ≤ 1 , we get

where

( q 2 3 ) k = { ( q 2 ) − , ‖ ( 2 k α | ∑ j = k + 2 ∞ T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ 1 , ( q 2 ) + , ‖ ( 2 k α | ∑ j = k + 2 ∞ T ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) > 1.

Then we have η 13 ≤ C η 10 ≤ C ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) , by using the same argument in η 11 . Thus, we prove Theorem 4.1. �

Theorem 4.2. Let b ∈ BMO ( ℝ n ) . Suppose that p 1 ( ⋅ ) ∈ B ( ℝ n ) , q 1 ( ⋅ ) , q 2 ( ⋅ ) ∈ P ( ℝ n ) with ( q 2 ) − ≥ ( q 1 ) + . If − n ι 12 < α < n ι 11 with ι 11 , ι 12 as defined in lemma 3.5, then the commutator [ b , T ] is bounded from K ˙ p 1 ( ⋅ ) α , q 2 ( ⋅ ) ( ℝ n ) to K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) .

Proof Let h ( x ) ∈ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) , b ∈ BMO ( ℝ n ) .We write

h ( x ) = ∑ j = − ∞ ∞ h ( x ) χ j = ∑ j = − ∞ ∞ h j ( x )

By virtue of the definition of K ˙ p ( ⋅ ) α , q ( ⋅ ) ( ℝ n ) , we have

‖ [ b , T ] ( h ) ‖ K ˙ p 1 ( ⋅ ) α , q 2 ( ⋅ ) ( ℝ n ) = inf { η > 0 : ∑ k = − ∞ ∞ ‖ ( 2 k α | [ b , T ] ( h ) χ k | η ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ 1 } . (4.14)

Since

‖ ( 2 k α | [ b , T ] ( h ) χ k | η ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ ‖ ( 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | ∑ i = 1 3 η 2 i ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ ‖ ( 2 k α | ∑ j = − ∞ ∞ [ b , T ] ( h j ) χ k | η 21 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) + ‖ ( 2 k α | ∑ j = k − 2 k + 2 [ b , T ] ( h j ) χ k | η 22 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) + ‖ ( 2 k α | ∑ j = k + 2 ∞ [ b , T ] ( h j ) χ k | η 23 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) . (4.15)

Let

η 21 = ‖ { 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | } k = − ∞ ∞ ‖ l q 2 ( ⋅ ) ( L p 1 ( ⋅ ) ) , (4.16)

η 22 = ‖ { 2 k α | ∑ j = k − 2 k + 2 [ b , T ] ( h j ) χ k | } k = − ∞ ∞ ‖ l q 2 ( ⋅ ) ( L p 1 ( ⋅ ) ) , (4.17)

η 23 = ‖ { 2 k α | ∑ j = k + 2 ∞ [ b , T ] ( h j ) χ k | } k = − ∞ ∞ ‖ l q 2 ( ⋅ ) ( L p 1 ( ⋅ ) ) , (4.18)

and

η = ∑ i = 1 3 η 2 i .

Therefore, we can obtain

∑ k = − ∞ ∞ ‖ ( 2 k α | [ b , T ] ( h ) χ k | η ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C .

Thus it follows that,

‖ [ b , T ] ( h ) ‖ K ˙ p 1 ( ⋅ ) α , q 2 ( ⋅ ) ( ℝ n ) ≤ C η = C ∑ i = 1 3 η 1 i . (4.20)

Hence η 21 , η 22 , η 23 ≤ C ‖ b ‖ BMO ( ℝ n ) ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) . Denoting η 10 = C ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) , firstly we estimate η 22 as in Theorem 4.1. Applying Lemma 3.3, we imme- diately arrive at

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = k − 2 k + 2 [ b , T ] ( h j ) χ k | η 10 ‖ b ‖ BMO ( ℝ n ) ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C .

So we can get that

η 21 ≤ C η 10 ‖ b ‖ BMO ( ℝ n ) ≤ C ‖ b ‖ BMO ( ℝ n ) ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) . (4.21)

Next we estimate η 21 , Let x ∈ A j , j ≤ k − 2 .

| [ b , T ] h j | ≤ ∫ A j | K ( x , y ) ( b ( x ) − b ( y ) ) h j ( y ) | d y ≤ C ∫ A j | ( b ( x ) − b ( y ) ) h j ( y ) | / | x − y | n d y ≤ C 2 − n k | b ( x ) − b B j | ∫ A j | h j ( y ) | d y + ∫ A j | b B j − b ( y ) | | h j ( y ) | d y ≤ C 2 − n k | b ( x ) − b B j | ‖ h j ‖ L 1 ( ℝ n ) + ‖ b ( ⋅ ) − ( b B j ) h j ‖ L 1 ( ℝ n ) . (4.22)

Thus, from Lemmas 3.4-3.7, We obtain that

Therefore, we get

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = − ∞ ∞ [ b , T ] ( h j ) χ k | η 10 ‖ b ‖ BMO ( ℝ n ) ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C ∑ k = − ∞ ∞ { ∑ j = − ∞ k − 2 ( k − j ) 2 ( k − j ) ( α − n ι 11 ) ‖ ( | 2 j α h χ j | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ( ℝ n ) 1 ( q 1 ) + } ( q 2 2 ) k , (4.23)

where

( q 2 2 ) k = { ( q 2 ) − , ‖ ( 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ 1 , ( q 2 ) + , ‖ ( 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) > 1.

This, for ( q 1 ) + < 1 , ( p 1 ) + ≤ ( p 2 ) − ≤ ( q 2 2 ) k , along with Remark 2.2, tells us that

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | η 10 ‖ b ‖ B M O ( ℝ n ) ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C { ∑ j = − ∞ ∞ ‖ ( | 2 j α h χ j | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ∑ k = j + 2 ∞ ( k − j ) 2 ( k − j ) ( α − n ι 11 ) } q * ≤ C ,

where q * = min k ∈ N ( q 2 2 ) k ( q 1 ) + .

If ( q 1 ) + ≤ 1 , it is follows from Remark 2.2 and Hölder’s inequality that

∑ k = − ∞ ∞ ‖ ( 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | η 10 ‖ b ‖ BMO ( ℝ n ) ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ C ∑ k = − ∞ ∞ { ∑ j = − ∞ k − 2 ( k − j ) 2 ( k − j ) ( α − n ι 11 ) ( q 1 ) + / 2 ‖ ( | 2 j α h χ j | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) } ( q 2 2 ) k ( q 1 ) + × ( ∑ j = − ∞ k − 2 ( k − j ) 2 ( k − j ) ( α − n ι 11 ) ( ( q 1 ) + ) ′ / 2 ) ( q 2 2 ) k ( ( q 1 ) + ) ′ ≤ C { ∑ j = − ∞ ∞ ‖ ( | 2 j α h χ j | η 10 ) q 1 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 1 ( ⋅ ) ∑ k = j + 2 ∞ ( k − j ) 2 ( k − j ) ( α − n ι 11 ) ( q 1 ) + / 2 } q * ≤ C ,

where q * = min k ∈ N ( q 2 2 ) k ( q 1 ) + .

This implies that

η 21 ≤ C η 10 ‖ b ‖ BMO ( ℝ n ) ≤ C ‖ b ‖ BMO ( ℝ n ) ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) . (4.24)

Finally we estimate η 23 , for any x ∈ A j , j ≥ k + 2 , by the same way to argument in η 21 , we obtain that

| [ b , T ] h j | ≤ ∫ A j | K ( x , y ) ( b ( x ) − b ( y ) ) h j ( y ) | d y ≤ C ∫ A j | ( b ( x ) − b ( y ) ) h j ( y ) | / | x − y | n d y ≤ C 2 − n j | b ( x ) − b B k | ∫ A j | h j ( y ) | d y + ∫ A j | b B k − b ( y ) | | h j ( y ) | d y ≤ C 2 − n j | b ( x ) − b B j | ‖ h j ‖ L 1 ( ℝ n ) + ‖ b ( ⋅ ) − ( b B j ) h j ‖ L 1 ( ℝ n ) , (4.25)

and

where

( q 2 3 ) k = { ( q 2 ) − , ‖ ( 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) ≤ 1 , ( q 2 ) + , ‖ ( 2 k α | ∑ j = − ∞ k − 2 [ b , T ] ( h j ) χ k | η 10 ) q 2 ( ⋅ ) ‖ L p 1 ( ⋅ ) q 2 ( ⋅ ) > 1.

Hence, we arrive at that η 23 ≤ C η 10 ‖ b ‖ BMO ( ℝ n ) ≤ C ‖ b ‖ BMO ( ℝ n ) ‖ h ‖ K ˙ p 1 ( ⋅ ) α , q 1 ( ⋅ ) ( ℝ n ) by the similar argument in the proof Theorem 4.1.

This completes the proof of Theorem 4.2. �

This paper is supported by National Natural Foundation of China (Grant No. 11561062).

Abdalrhman, O., Abdalmonem, A. and Tao, S.P. (2017) Boundedness of Calderón-Zygmund Operator and Their Commutator on Herz Spaces with Variable Exponent. Applied Mathematics, 8, 428-443. https://doi.org/10.4236/am.2017.84035