The aim of this paper is to study the boundedness of Calderón-Zygmund operator and their commutator on Herz Spaces with two variable exponents p(.), q(.). By applying the properties of the Lebesgue spaces with variable exponent, the boundedness of the Calderón-Zygmund operator and the commutator generated by BMO function and Calderón-Zygmund operator is obtained on Herz space.
Definition 1.1. Let T be a bounded linear operator from S ( ℝ n ) to S ′ ( ℝ n ) (see [
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 [
Kovácik and Rákosník introduced Lebesgue spaces and Sobolev spaces with variable exponents (see [
Herz spaces play an important role in harmonic analysis. After they were introduced in [
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. [
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. [
‖ { 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. [
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. [
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. [
| 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. [
∫ 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. [
‖ 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. [
‖ T f ‖ L p ( ⋅ ) ( ℝ n ) ≤ C ‖ f ‖ L p ( ⋅ ) ( ℝ n ) . (3.5)
Lemma 3.3. [
‖ [ b , T ] f ‖ L p ( ⋅ ) ( ℝ n ) ≤ C ‖ b ‖ BMO ( ℝ n ) ‖ f ‖ L p ( ⋅ ) ( ℝ n ) (3.6)
Lemma 3.4. [
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. [
‖ 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. [
1 | B | ‖ χ B ‖ L p ( ⋅ ) ( ℝ n ) ‖ χ B ‖ L p ′ ( ⋅ ) ( ℝ n ) ≤ C . (3.8)
Lemma 3.7. [
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