Jun 26, 2016 - By using the properties of the variable ex- ponents ... Fractional Integral, Variable Kernel, Commutator, Variable Exponent, Lipschitz Space,.
Applied Mathematics, 2016, 7, 1165-1182 Published Online June 2016 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2016.710104
Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz Spaces Afif Abdalmonem1,2, Omer Abdalrhman1,3, Shuangping Tao1* 1
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China Faculty of Science, University of Dalanj, Dalanj, Sudan 3 College of Education, Shendi University, Shendi, Sudan 2
Received 25 April 2016; accepted 26 June 2016; published 29 June 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents p ( ⋅ ) , q ( ⋅ ) . By using the properties of the variable exponents Lebesgue spaces, the boundedness of the fractional integral operator and their commutator generated by Lipschitz function is obtained on those Herz spaces.
Keywords Fractional Integral, Variable Kernel, Commutator, Variable Exponent, Lipschitz Space, Herz Spaces
1. Introduction
( )
(
)
Let 0 < µ < n , Ω ∈ L∞ n × Lr S n−1 ( r ≥ 1) is homogenous of degree zero on n , S n−1 denotes the unit sphere in n . If (i) For any x, z ∈ n , one has Ω ( x, λ z ) = Ω ( x, z ) ; (ii) Ω
L∞
=: sup n × Lr S n −1
( ) (
)
x∈ n
(∫
Ω ( x , z ′ ) dσ ( z ′ ) r
s n −1
)
1 r
0 . We first define the Lebesgue spaces with variable exponent. Definition 2.1. see [1] Let p ( ⋅) : E → [1, ∞ ) be a measurable function. The Lebesgue space with variable
exponent Lp(⋅) ( E ) is defined by
f ( x) p⋅ = L ( ) ( E ) f is measurable : ∫E η
p( x )
p( ⋅) The space Lloc ( E ) is defined by
{
( K ) for all compact K ⊂ E}
Lloc( ) ( E ) = f is measurable : f ∈ L p⋅
dx < ∞ for some constant η > 0
p( ⋅)
The Lebesgue spaces Lp(⋅) ( E ) is a Banach spaces with the norm defined by
f
L
p ( ⋅)
(E)
f ( x) = inf η > 0 : ∫E η
p( x )
dx ≤ 1
We denote = p− essinf { p ( x )= : x ∈ E} , p+ esssup { p ( x ) : x ∈ E} .
Then ( E ) consists of all p ( ⋅) satisfying p− > 1 and p+ < ∞ . Let M be the Hardy-Littlewood maximal operator. We denote B ( E ) to be the set of all function p ( ⋅) ∈ ( E ) satisfying the M is bounded on Lp(⋅) ( E ) . Definition 2.2. see [17] Let p ( ⋅) , q ( ⋅) ∈ ( E ) . The mixed Lebesgue sequence space with variable exponent l
q( ⋅)
(L ( ) ) p⋅
is the collection of all sequences
{ f j}
∞ j =0
{f }
∞
j
j =0
of the measurable functions on n such that
f j ∞ = inf η > 0 : Q q(⋅) p(⋅) ≤ 1 < ∞ q ⋅ p ⋅ l (L ) µ l ( )( L ( ) ) j =0
∞ f j ( x) ∞ inf µ j ; ∫R n = Q q ( ⋅) p ( ⋅) { f j } ∑ 1 j =0 l (L ) j =0 q( x ) µj Noticing q+ < ∞ , we see that
)
(
Q q ( ⋅) l
{
(
f = ) ({ } ) ∑ ∞
∞
p ⋅ L ()
j
j =0
}
fj
j =0
q( ⋅)
p( x )
dx ≤ 1
p ( ⋅)
q ⋅ L ()
Let Bk = x ∈ n : x ≤ 2 k , Ck = Bk \ Bk −1 , χ k = χ Ck , k ∈
( )
Definition 2.3. see [16] Let α ∈ , p ( ⋅) , q ( ⋅) ∈ n . The homogeneous Herz space with variable exponent K αp(,q⋅)(⋅) n is defined by
( )
( )
α ,q ⋅ K p(⋅)( ) n =
{
(
)
() f ∈ LLoc n \ {0} : f p⋅
where
1167
()
}
( ) 0 : ∑ η k =−∞
( ) satisfying ( q ) ≤ ( q ) K ( ) ( ) ⊂ K ( ) ( ) ( )
Remark 2.1. see [16] (1) If q1 ( ⋅) , q2 ( ⋅) ∈ n α ,q1(⋅)
1 +
α ,q2 ⋅
n
p⋅
( )
(2) If q1 ( ⋅) , q2 ( ⋅) ∈ n
and
( q1 )+ ≤ ( q2 )+ ,
∑
k =−∞
q2 ( ⋅) p ( ⋅)
≤
∞
∑
k =−∞
q ⋅ L 2( )
p ( ⋅)
q⋅ L ()
≤ 1
, then
n
q2 ( ⋅) ∈ n q1 ( ⋅)
( )
then
( )
2kα f χ k η
q( ⋅)
p⋅
and Remark 2.2, for any f ∈ K αp(,q⋅)(⋅) n , we have ∞
2 +
2kα f χ k η
q1( ⋅)
ph p ( ⋅)
L 1( ) q ⋅
∞ ≤∑ k =−∞
and
q2 ( ⋅) ≥ 1 . Thus, by Lemma 3.7 q1 ( ⋅)
2kα f χ k η
p*
≤1 p ( ⋅) q1 ( ⋅) L
q1 ( ⋅) ph
where
q2 ( ⋅) 2kα f χ k ≤1 , η q1 ( ⋅) − ph = 2kα f χ k q2 ( ⋅) >1 , q ( ⋅) η 1 + ph , min h∈ p* = max p , h∈ h α ,q1(⋅)
This implies that K p(⋅)
∞
∑ah ≤ 1 h =0 ∞
∑ah > 1 h =0
( ) ⊂ K ( ) ( ) . n
α ,q2 (⋅)
n
p⋅
Remark 2.2. Let h ∈ , ah ≥ 0,1 ≤ ph < ∞ . then
∞ ∑ah ≤ ∑ah = h 0= h 0 ∞
p*
where
ph , min h∈ p* = max p , h∈ h
∞
∑ah ≤ 1 h =0 ∞
∑ah > 1 h =0
( )
Definition 2.4. see [18] For 0 < β ≤ 1 , the Lipschitz space Lipβ n
n = Lip f : f Lip = β β
( )
sup x , y∈ n , x ≠ y
is defined by
f ( x) − f ( y) x− y
β
3. Properties of Variable Exponent
< ∞
(1.1)
( )
In this section we state some properties of variable exponent belonging to the class B n Ω ∈ L∞ n × Lr S n−1 . Proposition 3.1. see [1] If p ( ⋅) ∈ n satisfies
( )
(
)
( )
p ( x) − p ( y) ≤
−C , x − y ≤1 2 log ( x − y )
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and
A. Abdalmonem et al.
C , y ≥ x log ( e + x )
p ( x) − p ( y) ≤
( )
then, we have p ( ⋅) ∈ B n .
( )
( )
(
)
Proposition 3.2. see [15] Suppose that p1 ( ⋅) ∈ B n , Ω ∈ L∞ n × Lr S n−1 . Let 0 < µ ≤ define the variable exponent p2 ( ⋅) by:
n
( p1 )+
, and
1 1 µ = . Then we have that for all f ∈ Lp1(⋅) n , − p1 ( x ) p2 ( x ) n
( )
TΩ ,µ f
( )
p ⋅ L 2 ( ) n
≤c f
( )
( )
p ⋅ L 1( ) n
( )
( )
(
)
Proposition 3.3. Suppose that p1 ( ⋅) ∈ B n , b ∈ Lipβ n , 0 < β ≤ 1 , Ω ∈ L∞ n × Lr S n−1 . Let µ + mβ 1 1 n , and define the variable exponent p2 ( ⋅) by: − = . Then 0 < µ + mβ ≤ p1 ( x ) p2 ( x ) n ( p1 )+
b m , TΩ,µ f
( )
p ⋅ L 2 ( ) n
≤C b
m
f
Lipβ
( )
p ⋅ L 1( ) n
Proof m = b , TΩ,µ f ( x )
∫
Ω ( x, x − y ) n
x− y
n− µ
(b ( x ) − b ( y ))
Ω ( x, x − y )
≤∫
n
x− y
≤C b ≤C b
n− µ
(b ( x ) − b ( y ))
Ω ( x, x − y )
Lipβ
∫
Lipβ
T Ω , µ + mβ
(f)
T Ω , µ + mβ
(f)L
n
m
x− y
f ( y ) dy f ( y ) dy
m
f ( y ) dy
n − µ − mβ
By Proposition 3.2, we get
b m , TΩ,µ f
( )
p ⋅ L 2 ( ) n
≤C b
Lipβ
p2 ( ⋅)
≤C b
m Lipβ
f
( ).
p ⋅ L 1( ) n
Now, we need recall some lemmas Lemma 3.1. see [13] Given p ( ⋅) : n → [1, ∞ ) have that for all function f and g,
∫
n
f ( x ) g ( x ) dx ≤ C f
( ) g
p⋅ L ( ) n
( )
p′ ⋅ L ( ) n
( )
(
Lemma 3.2. see [19] Suppose that 0 < µ < n , r > 1 , Ω ∈ L∞ n × Lr S n−1 dition. If there exists an 0 < α 0 < 1 2 such that y < α 0 R then
)
satisfies the Lr -Dini con-
1
r n r −n+ µ y Ω ( x, x − y ) Ω ( x, x ) r + dx ≤ CR − ∫ n n µ µ − − R R< x 0 such that for all balls B in and all measurable subset S ⊂ R n
χB
p⋅ L ( ) n
χS
p⋅ L ( ) n
χS
p ⋅ L 1( ) n
χB
p ⋅ L 1( ) n
( )
( ) ( ) ( )
≤C
B S
,
χS
p′ ⋅ L 1( ) n
χB
p′ ⋅ L 1( ) n
( ) ( )
δ1
S ≤ C , B
δ2
S ≤ C B
( )
Lemma 3.6. see [13] If p ( ⋅) ∈ B n , there exist a constant C > 0 such that for any balls B in n . we have
1 χB B
( ) χB
p⋅ L ( ) n
( ) ≤C
p′ ⋅ L ( ) n
( )
Lemma 3.7. see [16] Let p ( ⋅) , q ( ⋅) ∈ n . If f ∈ Lp(⋅)q(⋅) , then
min
(f
q+
p⋅q⋅ L () ()
, f
q−
p⋅q⋅ L () ()
)≤
q ( ⋅)
f
Lp ( ⋅ )
4. Main Theorems and Their Proof
≤ max
(f
q+
p⋅q⋅ L () ()
( ) ( )
, f
(
q−
p⋅q⋅ L () ()
)
)(
)
Theorem 1. Suppose that 0 < µ < n , µ − nδ 2 < α < nδ1 , Ω ∈ L∞ n × Lr S n−1 r > p2+ , n and define the variq1 ( ⋅) , q2 ( ⋅) ∈ n with ( q2 )− ≥ ( q1 )+ . And let p1 ( ⋅) ∈ B n satisfy 0 < µ ≤ p ( 1 )+
( )
1
able exponent p2 ( x ) by
µ . Then the operators TΩ,µ is bounded from K αp1,q(⋅1)(⋅) ( n ) to = p2 ( x ) n 1
−
p1 ( x ) α ,q2 ( ⋅) n K p2 (⋅) . Theorem 2. Let b ∈ Lipβ n , m ∈ . Suppose that 0 < µ < n, ( µ + mβ ) − nδ 2 < α < nδ1 ,
( )
∞
Ω∈L
( ) × L (S n
0 < µ + mβ ≤
r
n −1
( ) )( r > p ) , q (⋅) , q (⋅) ∈ ( ) + 2
n
1
2
with
( q2 )− ≥ ( q1 )+ . If
( )
p1 ( ⋅) ∈ B n
satisfy
n 1 1 µ + mβ and define the variable exponent p2 ( x ) by − = . Then the comp1 ( x ) p2 ( x ) n ( p1 )+
( )
mutators b m , TΩ ,µ is bounded from K αp1,q(⋅1)(⋅) n Proof of Theorem1: Let f ∈ K αp1,q(⋅1)(⋅) n . We write
( )
to K αp2,q(⋅2)(⋅) n .
( )
= f ( x)
∞
∞
j =−∞
j =−∞
= f ( x) χ j ∑ f j ( x) ∑
( )
From definition of K αp(,q⋅)(⋅) n
TΩ ,µ , f
( )
α , q ( ⋅) K p ⋅2 n 2(
)
∞ 2kα TΩ ,µ ( f ) χ k inf η > 0 : ∑ = η k =−∞
Since
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q2 ( ⋅) p2 ( ⋅)
q ⋅ L 2( )
≤ 1
2kα TΩ,µ ( f ) χ k η
A. Abdalmonem et al.
q2 ( ⋅) p2 ( ⋅)
q ⋅ L 2( )
kα ∞ 2 ∑ TΩ,µ ( f j ) χ k j =−∞ ≤ η11 + η12 + η13 kα 2 +
q2 ( ⋅)
kα 2 ≤
p2 ( ⋅)
q ⋅ L 2( )
∑ TΩ,µ ( f j ) χ k j =−∞ η11 k −2
q2 ( ⋅)
p2 ( ⋅)
q ⋅ L 2( )
q2 ( ⋅)
∑ TΩ,µ ( f j ) χ k j= k +2 η13 p2 ( ⋅) q ⋅ L 2( )
kα 2 ∑ TΩ,µ ( f j ) χ k j= k −1 + η12 p2 ( ⋅) q2 ( ⋅) L k +1
q2 ( ⋅)
∞
where
q ⋅ p ⋅ l 2( ) L 2( )
(
)
q ⋅ p ⋅ l 2( ) L 2( )
(
)
(
)
∞
∑ TΩ,µ ( f j ) χ k j = k −1 k =−∞ k +1
η12 = 2kα
η13 = 2kα
∞
∑ TΩ,µ ( f j ) χ k j =−∞ k =−∞ k −2
η11 = 2kα
∞
∑ TΩ,µ ( f j ) χ k j= k + 2 k =−∞ ∞
q ⋅ p ⋅ l 2( ) L 2( )
And η = η11 + η12 + η13 , thus ∞
∑
k =−∞
2kα TΩ,µ ( f j ) χ k η
q2 ( ⋅)
≤C L
p2 ( ⋅) q2 ( ⋅)
That is TΩ ,µ ( f )
( )
α , q ( ⋅) K p ⋅2 n 2(
)
This implies only to prove η11 ,η12 ,η13 ≤ C f
≤ Cη= C [η11 + η12 + η13 ] .
( ) . Denote η1 = f
α , q ( ⋅) K p ⋅1 n 1( )
( )
α , q ( ⋅) K p ⋅1 n 1( )
Now we consider η12 . Applying Lemma 3.7
∞
∑
k =−∞
kα 2
q2 ( ⋅)
2kα ∑ TΩ,µ ( f j ) χ k ∞ j= k −1 ≤C ∑ η1 k =−∞ p2 ( ⋅) q ⋅ L 2( ) k +1
k +1
∑T ( f )χ
j k Ω ,µ j= k −1
η1
k +1 2kα T ( f ) χ j k Ω ,µ ≤C ∑ ∑ η1 k =−∞ j = k −1 ∞
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( q12 )k
p ⋅ L 2( )
p ⋅ L 2( )
( q12 )k
A. Abdalmonem et al.
where
( q2 )− , 1 q2 k = ( q2 )+ ,
( )
kα 2
∑ TΩ,µ ( f j ) χ k j = k −1 η1
kα 2
∑ TΩ,µ ( f j ) χ k j = k −1 η1
k +1
k +1
q2 ( ⋅)
≤1 p2 ( ⋅)
q ⋅ L 2( )
q2 ( ⋅)
>1 p2 ( ⋅)
q ⋅ L 2( )
By the Proposition 3.2, we get
∞
∑
k =−∞
kα 2
∑ TΩ,µ ( f j ) χ k j = k −1 η1 k +1
( )
Since f ∈ K αp1,q(⋅1)(⋅) n , then we have
q2 ( ⋅)
η1
k =−∞ j = k −1
L
( q12 )k p ⋅ L 1( )
p2 ( ⋅) q2 ( ⋅)
L
p ( ⋅)
2 jα f χ j η1
∑
2kα f j
≤ 1 , and
η1
j =−∞
k +1
∑∑
≤C
2 jα f χ j
∞
∞
q1( ⋅)
L
p1 ( ⋅) q1 ( ⋅)
≤1
By Lemma 3.7 and Remark 2.2, we get
∞
∑
k = −∞
kα 2
∑ TΩ, µ ( f j ) χ k j= k −1 η1 k +1
q2 ( ⋅)
≤C
∞
∑
k = −∞
L
2 kα f χ k η1
( p1 )+ , ( p2 )− ≤ ( q12 ) k , and
q* = min k∈
q1 ( ⋅)
( q1 )+ p ⋅ L 1( )
L 1( )
p2 ( ⋅) q2 ( ⋅)
q ⋅
∞ ≤C ∑ k = −∞ Hence
( q12 ) k
q*
kα
2 f χk η1
q1 ( ⋅)
≤C p ⋅ L 1( ) q ⋅ L 1( )
( q ) k , this implies that 1 2
( q1 )+
η12 ≤ Cη1 ≤ C f
( )
α , q ( ⋅) K p ⋅1 n 1( )
Now, we estimate of η11 using size condition of f j and Minkowski inequality, when j ≤ k − 1 we get,
TΩ,µ ( f j ) χ k
p ⋅ L 2( )
( ) n
≤ ∫B f j ( y ) j
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Ω ( x, x − y ) x− y
n− µ
−
Ω ( x, x ) x
n− µ
χk
dy p ⋅ L 2( )
( ) n
A. Abdalmonem et al.
1
Since r > p2+ we define the variable exponent
Ω ( x, x − y ) x− y
n− µ
−
Ω ( x, x ) x
p2 ( x )
χk
n− µ
x− y
( )
p ⋅ L 2 ( ) n
1 = p 2 ( x )
According Lemma 3.4 and the formula
( )
p x L 2 ( ) n
≈ χ Bk
1 1 , by Lemma 3.3 we get + r p 2 ( x )
Ω ( x, x − y )
≤
n− µ
x− y
−
Ω ( x, x ) n− µ
x
Ω ( x, x − y )
≤
χ Bk
=
n− µ
−
r
( )
L
Ω ( x, x ) n− µ
x
χk
( )
p x L 2 ( ) n
n
( )
χ Bk
( )
p x L 2 ( ) n
Lr n
1 1 − , then we have p2 ( x ) r −1 r
Bk
( )
p x L 2 ( ) n
≈ χ Bk
−1 µ − r n
Bk
( )
p x L 1( ) n
(1.2)
By Lemma 3.2, we get
Ω ( x, x − y ) x− y
n− µ
−
Ω ( x, x ) x
n −n+ µ
≤ CR r
n− µ
( )
Lr n
≤ CR
n −n+ µ r
≤C 2
( k −1)
n
r
y k −1 + 2 2(
j −k )β
y 2k −1
∫
ωr ( δ ) dδ δ
y 2k
1 ωr ( δ ) dδ 1 + ∫ 0 δ
−n+ µ
It follows that
TΩ,µ ( f j ) χ k
( )
p ⋅ L 2 ( ) n
≤ C 2− kn ∫B f j ( y ) dy χ Bk
(1.3)
( )
p x L 1( ) n
j
By the Equation (1.3) and using Lemmas 3.1, 3.5, 3.6, 3.7, we can obtain
∞
∑
k =−∞
kα 2
∑ TΩ,µ ( f j ) χ k j =−∞ η1 k −2
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
k −2
∑ 2− kn
j =−∞
k −2
∑
j =−∞
≤C
η1
1
L
( )
L
n
∞ k −2 j − k nδ −α ≤ C ∑ ∑ 2( )( 1 ) k =−∞ j =−∞
( ) χ Bk n
∑ TΩ,µ ( f j ) χ k
∑
j =−∞
η1
p2 ( ⋅) q2 ( ⋅)
p ⋅ L 2( )
χ k Lp1(⋅)
χBj p ⋅ L 1( )
2
∞
( q22 )k
k −2
kα
k =−∞
fj
fj
η1
q2 ( ⋅)
p′ ⋅ L 1( )
( q22 )k
( q22 )k
p′ ⋅ L 1( )
2α j f j χ k η1
q1( ⋅)
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
p ⋅q ⋅ L 1( ) 1( ) ( n ) 1
( q1 )+
where
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( q22 )k
k −2
∑
j =−∞
fj
η1
p ⋅ L 1( )
k −2
∑ 2( j −k )nδ
1
j =−∞
( ) n
χBj
p′ ⋅ L 1( )
η1 Lp1(⋅) n ( ) fj
2− kn χ Bk
( q22 )k
Lp1 ( ⋅ )
( q22 )k
A. Abdalmonem et al.
( q2 )− , q22 k = ( q2 )+ ,
( )
( )
Since f ∈ K αp1,q(⋅1)(⋅) n , then we have
∑ TΩ,µ ( f j ) χ k j =−∞ η1
kα 2
∑ TΩ,µ ( f j ) χ k j =−∞ η1
k −2
k −2
2 jα f χ j
∑
j =−∞
≤1 p2 ( ⋅)
q ⋅ L 2( )
q2 ( ⋅)
>1 p2 ( ⋅)
q ⋅ L 2( )
L
p ( ⋅)
2 jα f χ j η1
q1( ⋅) p1 ( ⋅) q ⋅ L 1( )
≤1
( q1 )+ < 1 , then we have ∞
∑
k =−∞
kα 2
∑ TΩ,µ ( f j ) χ k j =−∞ η1 k −2
q2 ( ⋅)
k −2 j −k nδ −α ≤ C ∑ ∑ 2( )( 1 ) k =−∞ j =−∞ ∞
p2 ( ⋅)
q ⋅ L 2( )
∞ ≤C∑ j =−∞
where q* = min k∈
If
q2 ( ⋅)
≤ 1 , and
η1 ∞
Now if
kα 2
2α j f j χ k η1
( q22 )k
2α j f j χ k η1
q1( ⋅)
q1( ⋅)
( q1 )+ p1 ( ⋅) q1 ( ⋅) n L ( ) q*
j − k nδ −α ∑ 2( )( 1 ) ≤ C k = j +2 p ⋅q ⋅ L 1( ) 1( ) ( n ) ∞
(q ) k 2 2
( q1 )+
( q1 )+ ≥ 1 , then we have ∞
∑
k =−∞
kα 2
∑ TΩ,µ ( f j ) χ k j =−∞ η1 k −2
∞ k − 2 ( j − k )( nδ −α ) ( q2 )+ 1 2 ≤C ∑ ∑2 k =−∞ j =−∞ ∞ ≤C∑ j =−∞
2 f j χk η1 αj
q2 ( ⋅)
p2 ( ⋅)
q ⋅ L 2( )
( q22 )k
2α j f j χ k η1
q1( ⋅)
q1( ⋅)
∞
∑2
p ⋅q ⋅ L 1( ) 1( )
( )
( j −k )( nδ1 −α )
k = j +2
n
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( q22 )+
( q1 )+ k −2 ( ( q1 )+ )′ ( ( q1 )+ )′ ( j −k )( nδ1 −α ) 2 × ∑2 =−∞ j p ⋅q ⋅ L 1( ) 1( ) ( n )
( q1 )+ 2
q*
≤C
where q* = min k∈
A. Abdalmonem et al.
( q ) k , this implies that 2 2
( q1 )+
η11 ≤ Cη1 ≤ C f
( )
α , q ( ⋅) K p ⋅1 n 1( )
Finally, we estimate η13 by Lemma 3.7, we get
∞
∑
k =−∞
kα 2
≤ ∑ 2kα k =−∞ ∞
q2 ( ⋅)
2kα ∑ TΩ,µ ( f j ) χ k ∞ j= k +2 ≤C ∑ η1 k =−∞ p2 ( ⋅) q ⋅ L 2( ) ∞
∞
∑
1
j = k +2
η1
Ω ( x, x − y )
∫
x− y
Cj
n− µ
( q23 )k
∞
∑ T ( f )χ
Ω ,µ j k j= k +2
η1 p ⋅ L 2( )
f j ( y ) dy χ k L p ( ⋅) 2
(1.4)
( )
q23 k
Note that, when x ∈ Ck , j ≥ k + 2 , then x − y ~ x . Therefore, applying the generalized Hölder’s Inequality, we have
Ω ( x, x − y )
∫
x− y
Cj
Define the variable exponent
∫ Cj
∫
x− y
n− µ
Ω ( x, x − y ) x− y
n−µ
≤ C2
− j(n−µ )
≤ C2
− j(n−µ )
≤ C2
− j(n−µ )
fj fj fj
f j ( y ) dy ≤ f j
Cj
Ω ( x, x − y )
p ⋅ L 1( )
x− y
n− µ
χj p′ ⋅ L 1( )
p ⋅ L 1( )
p ⋅ L 1( )
p ⋅ L 1( )
Ω ( x, x − y ) x− y
p ⋅ L 1( )
Ω ( x, x − y )
χj Lr
x− y
n− µ
p′ ⋅ L 1( )
f j ( y ) dy
According Lemma 3.4 and the formula
∫
f j ( y ) dy ≤ f j
1 1 1 by Lemma 3.3, then we have = + p1′ ( ⋅) r p1′ ( ⋅)
Ω ( x, x − y )
Cj
n− µ
n− µ
χj χj χj
p′ ⋅ L 1( )
2 j n −1 ∫ r dr 2 j −2
(∫
s n−1
Ω ( x, y ′ )
r
1
r dσ ( y ′ )
)
jn p1′ ( ⋅)
L
p′ ⋅ L 1( )
1 = p1′ ( ⋅)
2r Ω 2
( ) (
L∞ n × Lr S n−1
)
jn r
1 1 − , we have p1′ ( ⋅) r
f j ( y ) dy ≤ 2
− j( n− µ )
fj
p ⋅ L 1( )
2 jα f χ j From Equations (1.4), (1.5) and using Lemma 3.7, and η1
1175
χBj χBj
p′ ⋅ L 1( )
≈ χBj
p′ ⋅ L 1( )
Bj
−1 r
(1.5)
p′ ⋅ L 1( )
q1( ⋅)
L
p1 ( ⋅) q1 ( ⋅)
Then we get
≤ 1 we can obtain
A. Abdalmonem et al. ∞
∑
k =−∞
kα 2
∑ TΩ,µ ( f j ) χ k j= k + 2 η1 ∞
≤ C ∑ 2kα k =−∞ ∞
Note that −kµ χ Bk p2 ( ⋅) ≤ C 2
χ Bk
L
p ⋅ L 1( )
∞
∑2
q2 ( ⋅)
p2 ( ⋅)
q ⋅ L 2( )
fj
− j( n− µ )
η1
j = k +2
χBj
( )
p ⋅ L 1( ) n
p′ ⋅ L 1( )
χ Bk
p2 ( ⋅) L
( q23 )k
see [9].
Then we have
∞
∑
k =−∞
kα 2
∑ TΩ,µ ( f j ) χ k j= k + 2 η1
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
∞
∞
∑ 2 jµ
j = k +2
∞
j= k +2
∞
∑ 2( k
p2 ( ⋅)
q ⋅ L 2( )
fj
η1
∑ 2( j k ) µ −
q2 ( ⋅)
p ⋅ L 1( )
( )
2− jn χ B j
n
η1
( ) χBj
p ⋅ L 1( ) n
− j )( nδ 2 − µ )
j= k +2
∞ ∞ k − j nδ − µ +α ) ≤ C ∑ ∑ 2( )( 2 k =−∞ j = k + 2
p ⋅ L 1( ) p ⋅ L 1( )
χ Bk
fj
p′ ⋅ L 1( )
η1 Lp1(⋅) n ( )
2− k µ χ Bk
( q23 )k
p ⋅ L 1( )
( q23 )k
fj
2α j f j χ k η1
q1( ⋅)
p ⋅q ⋅ L 1( ) 1( ) ( n ) 1
( q1 )+
( q23 )k
where
( q2 )− , q23 k = ( q2 )+ ,
( )
Since
( q2 )− ≥ ( q1 )+
kα 2
∑ TΩ,µ ( f j ) χ k j= k + 2 η1
kα 2
∑ TΩ,µ ( f j ) χ k j= k + 2 η1
∞
∞
and α > µ − nδ 2 , as the same η11 we have
1176
q2 ( ⋅)
≤1 p2 ( ⋅)
q ⋅ L 2( )
q2 ( ⋅)
>1 p2 ( ⋅)
q ⋅ L 2( )
( q23 )k
η13 ≤ Cη1 ≤ C f
A. Abdalmonem et al.
( )
α , q ( ⋅) K p ⋅1 n 1( )
This completes the proof Theorem 1. Proof of Theorem 2 Let b ∈ Lipβ n , f ∈ K αp1,q(⋅1)(⋅) n . We write
( )
( )
∞
∞
j =−∞
j =−∞
= f ( x ) χk ∑ f j ( x ) ∑
= f ( x)
( )
From definition of K αp(,q⋅)(⋅) n
b , TΩ,µ ( f ) χ k m
( )
α , q ( ⋅) K p ⋅2 n 2(
)
2kα b m , TΩ,µ ( f ) χ k ∞ = inf η > 0 : ∑ η k =−∞
q2 ( ⋅)
p2 ( ⋅)
q ⋅ L 2( )
≤ 1
Since
2kα b m , TΩ ,µ ( f ) χ k η kα 2 ≤
q2 ( ⋅)
p2 ( ⋅)
q ⋅ L 2( )
∑ b , TΩ,µ ( f ) χ k j =−∞ η21 + η22 + η23
kα 2 +
∞
q2 ( ⋅)
m
p2 ( ⋅)
q ⋅ L 2( )
∑ b , TΩ,µ ( f ) χ k j = k −1 η22 k +1
kα 2 ≤
q2 ( ⋅)
m
p2 ( ⋅)
q ⋅ L 2( )
∑ b , TΩ,µ ( f ) χ k j =−∞ η21 k −2
kα 2 +
q2 ( ⋅)
m
∑ b , TΩ,µ ( f ) χ k j= k + 2 η23 ∞
p2 ( ⋅)
q ⋅ L 2( )
q2 ( ⋅)
m
p2 ( ⋅)
q ⋅ L 2( )
where ∞
η21 = 2kα
∑ bm , TΩ,µ ( f ) χ k j =−∞ k =−∞
η22 = 2kα
∑ b , TΩ,µ ( f ) χ k j = k −1 k =−∞
η23 = 2kα
k −2
k +1
q ⋅ p ⋅ l 2( ) L 2( )
(
)
q ⋅ p ⋅ l 2( ) L 2( )
(
)
(
)
∞
m
∞
∑ bm , TΩ,µ ( f ) χ k j= k + 2 k =−∞ ∞
q ⋅ p ⋅ l 2( ) L 2( )
And η = η21 + η22 + η23 . The similar to prove of Theorem 1
b m , TΩ ,µ ( f ) χ k Hence η21 ,η22 ,η23 ≤ C b
m
( ) f
Lipβ n
( )
α , q ( ⋅) K p ⋅2 n 2(
)
( ) . Denote η1 = f
α , q ( ⋅) K p ⋅1 n 1(
)
≤ Cη ≤ C [η21 + η22 + η23 ]
1( )
p( ⋅)
First we estimate η22 . Note that b m , TΩ ,µ is bonuded on L
1177
( )
α , q ( ⋅) K p ⋅1 n
( ) n
(Proposition 3.3), similarly to esti-
A. Abdalmonem et al.
mate for η12 in the proof of the Theorem 1, we get that ∞
∑
k =−∞
kα 2
∑ b , TΩ,µ ( f j ) χ k j = k −1 η1 k +1
q2 ( ⋅)
m
≤C p2 ( ⋅)
q ⋅ L 2( )
That is
η22 ≤ Cη1 b
m ( ) ≤ C b Lipβ ( n ) f
m
( )
α , q ( ⋅) K p ⋅1 n
Lipβ n
1(
)
Now, we estimate of η21 . Using size condition of f j and Minkowski inequality, when j ≤ k − 1 we get,
∫
b m , TΩ,µ = ≤
∫
Ω ( x, x − y ) n
n− µ
x− y
Ω ( x, x − y ) n
x− y
≤ b
m
≤ b
m
∫
Lipβ
∫
Lipβ
n− µ
(b ( x ) − b ( y )) mβ
x− y
Ω ( x, x − y ) n
x− y x− y
Lipβ
f ( y ) dy
n − µ n − µ − mβ
Ω ( x, x − y ) n
f ( y ) dy
m
b
f ( y ) dy
m
n − µ n −( µ + m β )
f ( y ) dy
We have that
b m , TΩ,µ χ k
p ⋅ L 2( )
≤C b
Ω ( x, x − y )
∫B f j ( y )
m Lipβ
x− y
j
−
n −( µ + m β )
Ω ( x, x ) x
χk
n −( µ + m β )
(1.6)
dy p ⋅ L 2( )
( ) n
The similar way to estimate of TΩ,µ in the proof of Theorem 1, we get that
b m , TΩ,µ χ k
p ⋅ L 2( )
≤C b
2− kn ∫ f j ( y ) dy χ Bk
m Lipβ
(1.7)
( )
p x L 1( ) n
Bj
By (1.7) and lemma 3.7, we obtain that
∞
∑
k =−∞
kα 2
∑ b , TΩ,µ ( f j ) χ k j =−∞ m η1 b Lip β k −2
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
≤C
∑ 2kα k =−∞ ∞
k −2
∑ 2− kn
j =−∞
k −2
∑
j =−∞
k −2
fj
η1
1
( )
L
fj
η1
∑ 2(
j =−∞
q2 ( ⋅)
m
( )
p ⋅ L 1( ) n
j − k )nδ1
fj
n
≤C
k =−∞
L
p2 ( ⋅) q2 ( ⋅)
χ k Lp1(⋅)
χBj
∞
∑
p′ ⋅ L 1( )
( )
χ Bk
( q22 )k
η1 Lp1(⋅) n ( )
q22 k
kα 2
≤ C ∑ 2kα k =−∞ ∞
p′ ⋅ L 1( )
( q22 )k
−1
∑ b , TΩ,µ ( f j ) χ k j =−∞ m η1 b Lip β k −2
k −2
∑
j =−∞
fj
η1
≤ C ∑ 2kα k =−∞ ∞
∞ k −2 j − k nδ −α ≤ C ∑ ∑ 2( )( 1 ) k =−∞ j =−∞
1178
q2 ( ⋅)
m
p ⋅ L 1( )
( )
k −2
∑
j =−∞
χBj
n
p′ ⋅ L 1( )
2− kn χ Bk
χBj
fj
η1
p ⋅ L 2( )
( ) χ Bk
p ⋅ L 1( ) n
2α j f j χ k η1
q1( ⋅)
p′ ⋅ L 1( )
p ⋅ L 1( )
( q22 )k
p′ ⋅ L 1( )
p ⋅q ⋅ L 1( ) 1( ) ( n ) 1
( q1 )+
( q22 )k
( q22 )k
A. Abdalmonem et al.
where
( q2 )− , 2 q2 k = ( q2 )+ ,
( )
q2 ( ⋅)
kα 2
∑ bm , TΩ,µ ( f j ) χ k j =∞ m η1 b Lip β
kα 2
∑ bm , TΩ,µ ( f j ) χ k j = k −1 m η1 b Lip β
k −2
≤1 p2 ( ⋅)
q ⋅ L 2( )
∞
q2 ( ⋅)
>1 p2 ( ⋅)
q ⋅ L 2( )
Since ( q2 )− ≥ ( q1 )+ and α > −nδ1 , the similar way to estimate η11 in the proof of Theorem1, we can obtain that
∞
∑
k =−∞
kα 2
∑ b , TΩ,µ ( f j ) χ k j =−∞ η1 k −2
∞ k −2 j − k nδ −α ≤ C ∑ ∑ 2( )( 1 ) k =−∞ j =−∞ where q* = min k∈
q2 ( ⋅)
m
2 f j χk η1 αj
p2 ( ⋅)
q ⋅ L 2( )
q1( ⋅)
q*
≤C p ⋅q ⋅ L 1( ) 1( ) ( n )
( q ) k , this implies that 2 2
( q1 )+
η21 ≤ Cη1 b
m
m ( ) ≤ C b Lipβ ( n ) f
( )
α , q ( ⋅) K p ⋅1 n
Lipβ n
1(
)
Finally, we estimate η23 . Note that, when x ∈ Ck , j ≥ k + 2 , then x − y ~ y , we can obtain that
b m , TΩ,µ = ≤
∫ ∫
≤ b
Ω ( x, x − y ) n
x− y
n− µ
Ω ( x, x − y ) n
m Lipβ
x− y
∫
n− µ
(b ( x ) − b ( y )) x− y
mβ
Ω ( x, x − y ) n
x− y
n −( µ + m β )
m
b
f ( y ) dy
m
Lipβ
f ( y ) dy
f ( y ) dy
Then we have
b m , TΩ,µ ≤ b
m Lipβ
∫
Ω ( x, x − y ) n
x− y
n −( µ + m β )
f ( y ) dy
(1.8)
Applying the generalized Hölder’s Inequality, we get
∫ Cj
Ω ( x, x − y ) x− y
n −( µ + m β )
f j ( y ) dy ≤ f j
1179
p ⋅ L 1( )
Ω ( x, x − y ) x− y
n −( µ + m β )
χj p′ ⋅ L 1( )
A. Abdalmonem et al.
1 1 1 by Lemma 3.3, then we have = + p1′ ( ⋅) r p1′ ( ⋅)
Define the variable exponent
Ω ( x, x − y )
∫
n − ( µ + mβ )
x− y
Cj
≤ fj
p ⋅ L 1( )
f j ( y ) dy
Ω ( x, x − y )
− j n − ( µ + mβ ) ) fj ≤ C2 (
p ⋅ L 1( )
− j n − ( µ + mβ ) ) fj ≤ C2 (
p ⋅ L 1( )
− j n − ( µ + mβ ) ) fj ≤ C2 (
p ⋅ L 1( )
According Lemma 3.4 and the formula
∫ Cj
Ω ( x, x − y ) x− y
n −( µ + m β )
χj Lr
x− y
χj χj χj
1 = p1′ ( ⋅)
n − ( µ + mβ )
p′ ⋅ L 1( )
2 j n −1 ∫ r dr 2 j −2
p′ ⋅ L 1( )
(∫
s n−1
Ω ( x, y ′ )
r
1
r dσ ( y ′ )
)
jn p′ ⋅ L 1( )
p′ ⋅ L 1( )
2r Ω 2
( ) (
L∞ n × Lr S n−1
)
jn r
1 1 − , we have p1′ ( ⋅) r
− j n −( µ + m β ) ) f j ( y ) dy ≤ 2 ( fj − j n −( µ + m β ) ) ≤2 ( fj
χBj
p ⋅ L 1( )
χBj
p ⋅ L 1( )
χBj
p′ ⋅ L 1( )
≈ χBj jn
p′ ⋅ L 1( )
2r 2
p′ ⋅ L 1( )
Bj
−1 r
. Then we get
− jn r
p′ ⋅ L 1( )
By (1.8), we can obtain that
b m , TΩ,µ ≤ C 2− j( n−( µ +mβ ) ) b
m Lipβ
fj
p ⋅ L 1( )
χBj
(1.9)
p′ ⋅ L 1( )
Then by (1.9) and Lemma 3.7, we have
∞
∑
k =−∞
kα 2
q2 ( ⋅)
kα 2 ∑ b , TΩ,µ ( f j ) χ k ∞ j= k +2 ≤ C ∑ m η1 b Lip k =−∞ β p2 ( ⋅) q ⋅ L 2( ) ∞
m
q2 ( ⋅)
∑ b , TΩ,µ ( f j ) χ k j= k +2 m η1 b Lip β p ⋅ L 2( ) ∞
m
where
( q2 )− , 3 (q2 )k = ( q2 )+ ,
kα 2
∑ b , TΩ,µ ( f j ) χ k j= k + 2 m η1 b Lip β
kα 2
∑ b , TΩ,µ ( f j ) χ k j= k + 2 m η1 b Lip β
∞
∞
Furthermore, when α > ( µ + mβ ) − nδ 2 , note that to estimate η11 , we get
1180
q2 ( ⋅)
m
≤1 p2 ( ⋅)
q ⋅ L 2( )
q2 ( ⋅)
m
χ Bk
p ⋅ L 2( )
≤ C2
− k ( µ + mβ )
>1 p2 ( ⋅)
q ⋅ L 2( )
χ Bk
p ⋅ L 1( )
see [9], the similar way
∞
∑
k =−∞
kα 2
∑ bm , TΩ,µ ( f j ) χ k j= k + 2 m η1 b Lip β ∞
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
≤ C ∑ 2kα k =−∞ ∞
∞
∑2 ∞
∑2
η1
∞
η1
∑ 2( − )(
j k µ + mβ )
∞
∑ 2(
j − k )( µ + mβ )
∞
∑
∞
k =−∞
k∈
p ⋅ L 1( ) p ⋅ L 1( )
( ) χBj
η1 Lp1(⋅) n ( )
2α j f j χ k η1
∑ bm , TΩ,µ ( f j ) χ k j= k + 2 m η1 b Lip β ∞
∞ ∞ ( k − j ) nδ −( µ + mβ )+α ) ≤C ∑ ∑ 2 ( 2 k =−∞ j = k + 2 where q* = min
χ Bk
p ⋅ L 1( )
p ⋅ L 1( ) n
− k ( µ + mβ )
( q23 )k
p ⋅ L 1( )
( q23 )k
χ Bk
( q23 )k
p ⋅ L 1( )
( q23 )k
( k − j ) nδ −( µ + mβ ) ) f j 2 ( 2
∞ ∞ ( k − j ) nδ −( µ + mβ )+α ) ≤C ∑ ∑ 2 ( 2 k =−∞ j = k + 2 We can conclude that
∑
−1
χ Bk
fj
2
p′ ⋅ L 1( )
χBj
( )
j = k +2
kα 2
( )
p2 ( ⋅) L
χ Bk
p′ ⋅ L 1( )
χBj
p ⋅ L 1( ) n
p ⋅ L 1( ) n
η1
j = k +2
χBj
( )
p ⋅ L 1( ) n
fj
η1
j = k +2
p2 ( ⋅)
q ⋅ L 2( )
fj
− j ( n −( µ + m β ) )
j = k +2
q2 ( ⋅)
fj
− j ( n −( µ + m β ) )
j = k +2
A. Abdalmonem et al.
q1( ⋅)
p ⋅q ⋅ L 1( ) 1( ) ( n ) 1
( q1 )+
( q23 )k
q2 ( ⋅)
p2 ( ⋅)
q ⋅ L 2( )
2 f j χk η1 αj
q1( ⋅)
q*
≤C p ⋅q ⋅ L 1( ) 1( ) ( n )
( q ) k , this implies that 3 2
( q1 )+
η23 ≤ Cη1 b
m
m ( ) ≤ C b Lipβ ( n ) f
Lipβ n
This completes the proof Theorem 2.
Competing Interests The authors declare that they have no competing interests.
1181
( )
α , q ( ⋅) K p ⋅1 n 1(
)
( q23 )k
A. Abdalmonem et al.
Acknowledgements This paper is supported by National Natural Foundation of China (Grant No. 11561062).
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