2. Definition of Function Spaces with Variable Exponent

Volume 4, Issue 3–A (2016), 29–40.

ISSN: 2347-1557

Available Online: http://ijmaa.in/

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International Journal of Mathematics And its Applications

Commutators of Fractional Integral with Variable Kernel on Variable Exponent Herz and Lebesgue Spaces

Research Article

Afif Abdalmonem^1,2∗, Omer Abdalrhman^1,3 and Shuangping Tao¹

1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, P.R. China.

2 Faculty of Science, University of Dalanj, Dalanj, South kordofan, Sudan.

3 College of Education, Shendi University, Shendi, River Nile State, Sudan.

Abstract: In this paper, we study the boundedness for commutators of the fractional integral with variable kernel on Lebesgue spaecs with variable exponent. Also, we study the boundedness of the fractional integral operator and their commutator generated by BMO function is obtained on those Herz spaces with two variable exponent p(·), q(·).

MSC: 42B25, 42B30.

Keywords: Fractional integral, variable kernel, commutator, variable exponent, BMO function, Lebesgue spaecs, Herz spaces.

c

JS Publication.

1. Introduction

Let Sⁿ⁻¹(n ≥ 2) be the unit sphere in Rⁿwith normalized Lebesgue measure dσ(x⁰). We say a function Ω(x, z) defined on Rⁿ× Rⁿ belongs to L^∞(Rⁿ) × L^r(Sⁿ⁻¹)(r ≥ 1), if

(i) For any x, z ∈ Rⁿ, one has Ω(x, λz) = Ω(x, z);

(ii) kΩk_L∞(Rⁿ)×L^r(Sⁿ⁻¹):= sup

x∈Rⁿ

(R

sⁿ⁻¹|Ω(x, z⁰)|^rdσ(z⁰))^1r< ∞

For 0 ≤ µ < n the fractional integral operator with variable kernel TΩ,µis defined by

TΩ,µf (x) = Z

Rⁿ

Ω(x, x − y)

|x − y|^n−µf (y)dy, (1)

And the commutators of the fractional integral is defined by

[b^m, TΩ,µ]f (x) = Z

Rⁿ

Ω(x, x − y)

|x − y|^n−µ(b(x) − b(y))^mf (y)dy, (2)

Now we need the further assumption for Ω(x, z). It satisfies

Z

sⁿ⁻¹

Ω(x, z⁰)dσ(z⁰) = 0, ∀x ∈ Rⁿ

∗ E-mail: [email protected]

For r ≥ 1, we say Kernel function Ω(x, z) satisfies the L^r- Dini condition, if Ω meets the conditions (i),(ii) and Z 1

0

ωr(δ) δ dδ < ∞

where ωr(δ) denotes the integral modulus of continuity of order r of Ω defined by

ωr(δ) = sup

x∈Rⁿ,|ρ|<δ

( Z

sⁿ⁻¹

|Ω(x, ρz⁰) − Ω(x, z⁰)|^rdσ(z⁰))^1r

where ρ is the a rotation in Rⁿ

|ρ| = sup

z⁰∈Sⁿ⁻¹

|ρz⁰− z⁰|

The corresponding fractional maximal operator with variable kernel is defined by

MΩ,µf (x) = sup

Q3x

1

|Q|^1−µn Z

Q

|Ω(x, x − y)||f (y)|dy (3)

And the commutators of the fractional maximal operator with variable kernel is defined by

[b^m, MΩ,µ]f (x) = sup

Q3x

1

|Q|^1−µn Z

Q

|Ω(y)||f (x, x − y)[b(x) − b(y)]^m|dy (4)

Many results about the fractional integral operator with variable kernel have been achieved [1–5]. In 1971, Muckenhoupt and Wheeden [6] had proved the operator TΩ,µwas bounded from L^pto L^q. In 1991, Kov´aˇcik and R´akosn´ik [7] introduced variable exponents Lebesgue and Sobolev spaces as a new method for dealing with nonlinear Dirichet boundary value problem. In 2004, Zhang Pu and Zhao Kai [8] introduced the Commutators of fractional integral operators with variable kernel on Hardy spaces. In the last twenty years, more and more researchers have been interested in the theory of the variable exponent function space and its applications [9–15]. In 2012, Wu Huiling and Lan Jiacheng [16] studied the bonudedness property of TΩ,µ with a rough kernel on variable exponents Lebesgue spaces. In 2010, M .Izuki [3] discussed the Commutators of fractional integrals on Lebesgue and Herz spaces. Recently, L.Wang and S.Tao [17] introduced the class of Herz spaces with two variable exponents. Throughout this paper |E| denotes the Lebesgue measure, χE means he characteristic function of a measurable set S ⊂ Rⁿ. C always means a positive constant independent of the main parameters and may change from one occurrence to another.

2. Definition of Function Spaces with Variable Exponent

In this section we define the Lebesgue spaces with variable exponent and Herz spaces with two variable exponent, and also define the mixed Lebesgue sequence spaces. Let E be a measurable set in Rⁿ with |E| > 0. We first define the Lebesgue spaces with variable exponent.

Definition 2.1 ([1]). Let p(·) : E → [1, ∞) be a measurable function. The Lebesgue space with variable exponent L^p(·)(E) is defined by L^p(·)(E) = {f is measurable :R

E(^{|f (x)|}_η )^p(x)dx < ∞ for some constant η > 0}. The space L^p(·)_loc(E) is defined by L^p(·)_loc(E) = {f is measurable : f ∈ L^p(·)(K) for all compact K ⊂ E}. The Lebesgue spaces L^p(·)(E) is a Banach spaces with the norm defined by

kf k_Lp(·)(E)= inf{η > 0 : Z

E

(|f (x)|

η )^p(x)dx ≤ 1}

We denote p−= essinf {p(x) : x ∈ E}, p+ = esssup{p(x) : x ∈ E}. Then P(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(·) ∈ P(E) satisfying the M is bounded on L^p(·)(E).

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

{fj}^∞j=0

l^q(·)(L^p(·))= inf



η > 0 : Q_lq(·)(L^p(·))

n_f

j µ

o∞ j=0



≤ 1



< ∞

Q_lq(·)(L^p(·))({fj}^∞j=0) =

∞

X

j=0

inf





 µj;

Z

Rⁿ





|fj(x)|

µ

1 q(x) j





p(x)

dx ≤ 1







Noticing q+< ∞, we see that

Qlq(·)(Lp(·) )({fj}^∞j=0) =

∞

X

j=0

|fj|^q(·)

L p(·) q(·)

Let Bk= {x ∈ Rⁿ: |x| ≤ 2^k}, Ck= Bk\Bk−1, χk= χC_k, k ∈ Z.

Definition 2.3 ([17]). Let α ∈ R, q(·), p(·) ∈ P(Rⁿ). The homogeneous Herz space with two variable exponent ˙K_p(·)^α,q(·)(Rⁿ) is defined by

K˙_p(·)^α,q(·)(Rⁿ) = {f ∈ L^p(·)_Loc(Rⁿ\{0}) : kf k_K˙α,q(·)

p(·) (Rⁿ)< ∞}

Where

kf k_˙

K_p(·)^α,q(·)(Rⁿ)=

{2^kα|f χk|}^∞k=0

l^q(·)(L^p(·))= inf (

η > 0 :

∞

X

k=−∞

 2^kα|f χk| η

q(·) L

p(·) q(·)

≤ 1 )

.

Remark 2.4 ([17]).

(1) If q1(·), q2(·) ∈ P(Rⁿ) satisfying (q1)+≤ (q2)+, then ˙K_p(·)^α,q1(·)(Rⁿ) ⊂ ˙K_p(·)^α,q2(·)(Rⁿ).

(2) If q1(·), q2(·) ∈ P(Rⁿ) and (q1)+ ≤ (q2)+, then ^q_q^2(·)

1(·) ∈ P(Rⁿ) and ^q_q^2(·)

1(·) ≥ 1. Thus, by Lemma 3.7 and Remark 2.2, for any f ∈ ˙K_p(·)^α,q(·)(Rⁿ), we have

∞

X

k=−∞

 2^kα|f χk| η

q₂(·) L

p(·) q2(·)

≤

∞

X

k=−∞

 2^kα|f χk| η

q₁(·)

p_h

L p(·) q1(·)

≤

( _∞

X

k=−∞

 2^kα|f χk| η

q₁(·)

p_h

L p(·) q1(·)

)p∗

≤ 1

Where

ph=





 (^q_q^2(·)

1(·))−, ^{2kα|f χ}_η ^k| ≤ 1 (^q_q^2(·)

1(·))+, ^{2kα|f χ}_η ^k| > 1

p∗=









 minh∈Nph,

∞

P

h=0

ah≤ 1 max

h∈N ph,

∞

P

h=0

ah> 1

This implies that ˙K_p(·)^α,q1(·)(Rⁿ) ⊂ ˙K_p(·)^α,q2(·)(Rⁿ).

Remark 2.5. Let h ∈ N, ah≥ 0, 1 ≤ ph< ∞. Then

∞

X

h=0

ah≤

∞

X

h=0

ah

!p∗

Where

p∗=









 minh∈Nph,

∞

P

h=0

ah≤ 1 max

h∈Nph,

∞

P

h=0

ah> 1

3. Properties of Variable Exponent

In this section we state some properties of variable exponent belonging to the class B(Rⁿ) and Ω ∈ L^∞(Rⁿ) × L^r(Sⁿ⁻¹).

Proposition 3.1 ([1]). If p(·) ∈ P(Rⁿ) satisfies

|p(x) − p(y)| ≤ −C

log(|x − y|), |x − y| ≤ 1/2

|p(x) − p(y)| ≤ C

log(e + |x|), |y| ≥ |x|

then, we have p(·) ∈ B(Rⁿ).

Proposition 3.2 ([3]). Suppose that p1(·) ∈ B(Rⁿ), 0 < µ ≤ _(pⁿ

1)₊. And define the variable exponent p2(·) by : _p¹

1(x)−

1

p₂(x)= ^µ_n. Then we have that

k[b, Iµ]f k_Lp2(·)(Rn)≤ CkbkBM Okf k_Lp1(·)(Rn)

For all f ∈ L^p1(·)(Rⁿ) and b ∈ BMO.

Now, we need recall some lemmas.

Lemma 3.3 ([1]).

(1). Given p(·) : Rⁿ→ [1, ∞) have that for all function f and g,

Z

Rⁿ

|f (x)g(x)|dx ≤ Ckf k_Lp(·)(Rⁿ)kgk_Lp0 (·)(Rⁿ)

(2). If p(·), q(·), r(·) ∈ Rⁿ, define p(·) by : _p(·)¹ =_q(·)¹ +_r(·)¹ . Then there exists a constant C such that for all f ∈ L^q(·)(Rⁿ), g ∈ L^r(·)(Rⁿ), we have

kf gk_Lp(·) ≤ Ckf k_Lq(·)(Rⁿ)kgk_Lr(·)(Rⁿ)

Lemma 3.4. Suppose that 0 < µ < n , 1 < s⁰< ⁿ_µ, Ω ∈ L^∞(Rⁿ) × L^r(Sⁿ⁻¹) is homogeneous of degree zero on Rⁿ. Then we have

|MΩ,µ,bf (x)| ≤ CkΩk_L∞(Rⁿ)×L^r(Sⁿ⁻¹)



M_µs0,b(|f |^s0)(x)¹

s0 .

Proof. Applying H¨older’s inequality, we get that

|MΩ,µ,bf (x)| = sup

Q3x

1

|Q|^1−nµ Z

Q

|Ω(y)||f (x, x − y)b(x) − b(y)|dy

≤ sup

Q3x

1

|Q|^1−nµ

Z

Q

|Ω(y)|^sdy

¹_s



|f (x, x − y)b(x) − b(y)|^s0dy¹

s0

≤ CkΩkL^∞(Rⁿ)×L^r(Sⁿ⁻¹)sup

Q3x

1

|Q|^1−µs0n

×

|f (x, x − y)b(x) − b(y)|^s0dy_s0¹

≤ CkΩkL^∞(Rⁿ)×L^r(Sⁿ⁻¹)



M_µs0,b(|f |^s0)(x)¹

s0

The proof of Lemma 3.2 is finished.

Lemma 3.5 ([19]). Suppose that x ∈ Rⁿ, the variable function ˜q(x) is defined by _p(x)¹ = ¹_q+_q(x)˜¹ , then for all measurable function f and g, we have

kf (x)g(x)k_Lp(·)(Rⁿ)≤ Ckg(x)kL^q(Rⁿ)kf (x)k_Lq(·)˜ (Rⁿ).

Lemma 3.6 ([20]). Suppose that p(·) ∈ B(Rⁿ) and 0 < p⁻≤ p⁺< ∞.

(1). For any cube and |Q| ≤ 2ⁿ, all the χ ∈ Q, then : kχQk_Lp(·) ≈ |Q|^1/p(x)

(2). For any cube and |Q| ≥ 1, then kχQk_Lp(·)≈ |Q|^1/p∞ where p∞= limx→∞p(x).

Lemma 3.7 ([21]). If p(·) ∈ B(Rⁿ), then there exist constants δ1, δ2, C > 0 such that for all balls B in Rⁿand all measurable subset S ⊂ R

kχBk_Lp(·)(Rⁿ)

kχSk_Lp(·)(Rⁿ)

≤ C|B|

|S|, kχSk

L^p01(·)(Rⁿ)

kχBk

L^p01(·) (Rⁿ)

≤ C



|S|

|B|

δ₁

, kχSk_Lp1(·)(Rn)

kχBk_Lp1(·)(Rn)

≤ C



|S|

|B|

δ₂

.

Lemma 3.8 ([14]). If p(·) ∈ B(Rⁿ), there exist a constant C > 0 such that for any balls B in Rⁿ. We have

1

|B|kχBk_Lp(·)(Rⁿ)kχBk_Lp0 (·)(Rⁿ)≤ C.

Lemma 3.9 ([17]). Let p(·), q(·) ∈ P(Rⁿ).If f ∈ L^p(·)q(·), then

min(kf k^q+

L^p(·)q(·) , kf k^q−

L^p(·)q(·)) ≤ k|f |^q(·)k_Lp(·)≤ max(kf k^q+

L^p(·)q(·) , kf k^q−

L^p(·)q(·)).

Lemma 3.10 ([9]). Let ∈ BMO, n is a positive integer, and there constants C > 0, such that for any l, j ∈ Z with l > j

(1). C⁻¹kbkⁿ∗ ≤ sup

B 1 kχ_Bk

Lp(·) (Rn)

k(b − bB)ⁿχBk_Lp(·)(Rⁿ)≤ Ckbkⁿ∗

(2). k(b − bBj)ⁿχB_kk_Lp(·)(Rⁿ)≤ C(l − j)ⁿkbkⁿ∗kχB_kk_Lp(·)(Rⁿ).

Lemma 3.11 ([10]). For any ε > 0 with 0 < µ − ε < µ + ε, we have

|TΩ,µ,bf (x)| ≤ C[MΩ,µ+ε,bf (x)]¹²[MΩ,µ−ε,bf (x)]¹².

4. Main Theorems and Their Proof

Theorem 4.1. Suppose that b ∈ BMO, p1(·) ∈ B(Rⁿ), Ω ∈ L^∞(Rⁿ) × L^r(Sⁿ⁻¹). Let 0 < β ≤ _(pⁿ

1)₊ and define the variable exponent p2(·) by : _p¹

1(x)−_p¹

2(x)=_n^µ, Then we have that

k[b, TΩ,µ]f k_Lp2(·)(Rn)≤ Ckbk∗kf k_Lp1(·)(Rn)

For all f ∈ L^p1(·)(Rⁿ).

Proof. Let 0 < ε < min(µ, n − µ), and r(·) : Rⁿ−→ [1, +∞). And let

1

p1(·)− 1

r(·)p₂(·) 2

=µ − ε 2

1

p1(·)− 1

r⁰(·)p2(·) 2

=µ + ε 2 By Lemma 3.11, we have

|TΩ,µ,bf (x)| ≤ C[MΩ,µ+ε,bf (x)]¹²[MΩ,µ−ε,bf (x)]¹²

Applying the generalized H¨older’s Inequality, we get that

kTΩ,µ,bf (x)k_Lp2(·)(Rn)≤ Ck(MΩ,µ+ε,bf )¹²k_Lp2(·)r0 (·)k(MΩ,µ−ε,bf )¹²k_Lp2(·)r(·)

By Lemma 3.3 and Proposition 3.2, we obtain that

k(MΩ,µ−ε,bf )¹²k_Lp2(·)r(·)≤ CkM(µ−ε)s⁰,b(|f |^s0)k

1 2

L p2(·)r(·)

2s0

≤ Ckbk∗¹²k|f |^s0k¹²

L p1(x)

2s0

≤ Ckbk∗¹²kf k¹²

Lp1(·)(Rⁿ) (5)

As the same way, we can concluded that

k(MΩ,µ+ε,bf )¹²k_Lp2(·)r0 (·)≤ Ckbk∗¹²kf k¹²

L^p1(·)(Rⁿ) (6)

Therefore by (5) and (6), we have that

k[b, TΩ,µ]f k_Lp2(·)(Rn)≤ Ckbk∗kf k_Lp1(·)(Rn).

Theorem 4.2. Let b ∈ BM O(Rⁿ), m ∈ N. Suppose that 0 < µ < n, µ − nδ2< α < nδ1+ⁿ_r, Ω ∈ L^∞(Rⁿ) × L^r(Sⁿ⁻¹)(r >

p⁺₂), q1(·), q2(·) ∈ P(Rⁿ) with (q2)−≥ (q1)+. If p1(·) ∈ B(Rⁿ) satisfy 0 < µ ≤ _(pⁿ

1)₊ and define the variable exponent p2(x) by _p¹

1(x)−_p¹

2(x)= ^µ_n. Then the commutators [b^m, TΩ,µ] is bounded from ˙K_p^α,q1(·)

1(·) (Rⁿ) to ˙K_p^α,q2(·)

2(·) (Rⁿ).

Proof. Let b ∈ BMO, f ∈ ˙K_p^α,q1(·)

1(·) (Rⁿ). We write

f (x) =

∞

X

j=−∞

f (x)χk=

∞

X

j=−∞

fj(x)

From definition of ˙K_p(·)^α,q(·)(Rⁿ)

k[b^m, TΩ,µ](f )χkk_˙

K^α,q2(·)

p2(·) (Rⁿ)= inf (

η > 0 :

∞

X

k=−∞

 2^kα|[b^m, TΩ,µ](f )χk| η

^q2(·) L

p2(·) q2(·)

≤ 1 )

Since

 2^kα|[b^m, TΩ,µ](f )χk| η

^q2(·) L

p2(·) q2(·)

≤







 2^kα|

∞

P

j=−∞

[b^m, TΩ,µ](f )χk| η21+ η22+ η23









q₂(·) L

p2(·) q2(·)

≤









 2^kα|

k−2

P

j=−∞

[b^m, TΩ,µ](f )χk| η31











q2(·) L

p2(·) q2(·)

+









 2^kα|

k+1

P

j=k−1

[b^m, TΩ,µ](f )χk| η32











q₂(·) L

p2(·) q2(·)

+







 2^kα|

∞

P

j=k+2

[b^m, TΩ,µ](f )χk| η33









q₂(·) L

p2(·) q2(·)

Where

η21= (

2^kα|

k−2

X

j=−∞

[b^m, TΩ,µ](f )χk| )^∞

k=−∞

lq2(·)(Lp2(·))

η22=





 2^kα|

k+1

X

j=k−1

[b^m, TΩ,µ](f )χk|







∞

k=−∞

lq2(·)(Lp2(·))

η23=





 2^kα|

∞

X

j=k+2

[b^m, TΩ,µ](f )χk|







∞

k=−∞

lq2(·)(Lp2(·))

And η = η21+ η22+ η23, thus

∞

X

k=−∞

 2^kα|[b^m, TΩ,µ](fj)χk| η

^q2(·) L

p2(·) q2(·)

≤ C

That is

k[b^m, TΩ,µ](f )χkk_˙

K^α,q2(·)

p2(·) (Rⁿ)≤ Cη ≤ C[η21+ η22+ η23] Hence η21, η22, η23≤ Ckbk^m∗kf k_˙

K^α,q1(·)

p1(·) (Rⁿ). Denote η1 = kf k_˙

K^α,q1(·)

p1(·) (Rⁿ). Now we consider η22. Applying Lemma 3.9, we get

∞

X

k=−∞









 2^kα|

k+1

P

j=k−1

TΩ,µ(fj)χk| η1kbk^m∗











q₂(·) L

p2(·) q2(·)

≤ C

∞

X

k=−∞

2^kα|

k+1

P

j=k−1

[b^m, TΩ,µ](fj)χk| η1kbk^m∗

(q¹₂)k

Lp2(·)

≤ C

∞

X

k=−∞





k+1

X

j=k−1

2^kα|[b^m, TΩ,µ](fj)χk| η1kbk^m∗

Lp2(·)





(q¹₂)k

Where

(q₂¹)k =























(q2)− ,







2^kα| ^k+1P j=k−1

[b^m,T_Ω,µ](f_j)χ_k|

η₁kbk^m_∗







q2(·) L

p2(·) q2(·)

≤ 1

(q2)+ ,







2^kα| k+1

P j=k−1

[b^m,T_Ω,µ](f_j)χ_k|

η₁kbk^m_∗







q₂(·) L

p2(·) q2(·)

> 1

By the Theorem 4.1, we get

∞

X

k=−∞









 2^kα|

k+1

P

j=k−1

[b^m, TΩ,µ](fj)χk| η1kbk^m∗











q₂(·) L

p2(·) q2(·)

≤ C

∞

X

k=−∞

k+1

X

j=k−1

2^kα|fj| η1

(q¹₂)k

Lp1(·)

Since f ∈ ˙K_p^α,q1(·)

1(·) (Rⁿ), then we have

2^jα|f χ_j| η1

L^p(·)

≤ 1, and

∞

X

j=−∞

 2^jα|f χj| η1

^q1(·) L

p1(·) q1(·)

≤ 1

Again by Lemma 3.9 and Remark 2.5, we have

∞

X

k=−∞









 2^kα|

k+1

P

j=k−1

[b^m, TΩ,µ](fj)χk| η1kbk^m∗











q₂(·) L

p2(·) q2(·)

≤ C

∞

X

k=−∞

 2^kα|f χk| η1

^q1(·)

(q12 )k (q1)+

Lp1 (·) Lq1 (·)

≤ C







∞

X

k=−∞

 2^kα|f χk| η1

q1(·) _{Lp1 (·)}

Lq1 (·)







q∗

≤ C

Hence (p1)+, (p2)−≤ (q¹2)k, and q∗= min

k∈N (q¹₂)k

(q1)₊ , this implies that

η22≤ Cη1kbk^m∗ ≤ Ckbk^m∗kf k_˙

K^α,q1(·)

p1(·) (Rⁿ) (7)

Now, we estimate of η21. Let x ∈ Ck j ≤ k − 2, then |x − y| ∼ |x|, we show that

|[b^m, TΩ,µ]fj(x)| ≤ C Z

C_j

   

Ω(x, x − y)

|x − y|^n−µ    

|(b(x) − b(y))|^m|fj(y)|dy,

≤ 2^−k(n−µ) Z

C_j

|Ω(x, x − y)||(b(x) − b(y))|^m|fj(y)|dy

≤ C2^−k(n−µ)

"

|b(x) − bj|^m Z

C_j

|Ω(x, x − y)||fj(y)|dy + Z

C_j

|Ω(x, x − y)||(b(x) − bj)|^m|fj(y)|dy

#

Applying the generalized H¨older’s Inequality, we know that

|[b^m, TΩ,µ]fj(x)| ≤ C2^−k(n−µ)kfjk_Lp1(·)h

|(b(x) − bj)|^mkΩ(x, x − y)χjk

L^p01(·)+ kΩ(x, x − y)(b(y) − bj)^mχjk

L^p01(·)

i (8)

Define the variable exponent _p´¹

1(·) =¹_r+ ¹

pf⁰₁(·) by Lemma 3.5, then we have

kΩ(x, x − y)χjk

L^p01 ≤ kΩ(x, x − y)kL^rkχjk

L^p0f1(·)≤







2^k

Z

2^k−1

rⁿ⁻¹dr(

Z

sⁿ⁻¹

|Ω(x, y⁰)|^rdσ(y⁰)







1 r

kχjk

L^p0f1(·)

According to Lemma 3.6 and the formula ¹

pe⁰₁(x) =_p0¹

1(x)−¹_r, then we get

kΩ(x, x − y)χjk

L^p01 ≤ 2^knr kΩk_L∞(Rⁿ)×L^r(Sⁿ⁻¹)kχjk

L^p0f1(·)≤ C2^knr 2^−jnr kχB_jk

L^p01(·)≤ C2^(k−j)nrkχB_jk

L^p01(·) (9)

Similarly, by Lemma 3.10, we have

kΩ(x, x − y)(b(y) − bj)^mχjk

L^p01(·)≤ kΩ(x, x − y)kL^rk(b(y) − bj)^mχjk

L^p0f1(·)

≤ C2^knr (k − j)^mkbk^m∗kχjk

L^p0f1(·)≤ C2^(k−j)nr(k − j)^mkbk^m∗kχB_jk

L^p01(·) (10)

By (9), (10), and Lemma 3.8, we get

k[b^m, TΩ,µ]fj, χkk_Lp2(·)≤ C2^−k(n−µ)2^(k−j)nrkfjk_Lp1(·)h

k(b(x) − bj)^mχkk_Lp2(·)kχB_jk

L^p01(·)

+(k − j)^mkbk^m∗kχB_jk

L^p01(·)kχB_kk_Lp2(·)i

≤ C2^−k(n−µ)2^(k−j)nr(k − j)^mkbk^m∗kfjk_Lp1(·)kχB_jk

L^p01(·)kχB_kk_Lp2(·) (11)

By (11), and using Lemmas 3.7-3.9, and

 2^jα|f χj| η1

^q1(·) L

p1(·) q1(·)

≤ 1.

Note that kχB_kk_Lp2(·) ≤ C2^−kµkχB_kk_Lp1(·), then (see [22])

∞

X

k=−∞









 2^kα|

k−2

P

j=−∞

[b^m, TΩ,µ](fj)χk| η1kbk^m∗











q₂(·) L

p2(·) q2(·)

≤ C

∞

X

k=−∞









 2^kα|

k−2

P

j=−∞

[b^m, TΩ,µ](fj)χk| η1kbk^m∗











q₂(·) Lp2(·)

≤ C

∞

X

k=−∞

"

2^kα

k−2

X

j=−∞

2^−k(n−µ)2^(k−j)nr(k − j)^m

fj

η1

Lp1(·)(Rⁿ)

kχB_jk

L^p01(·)kχB_kk_Lp2(·)

#(q²₂)k

≤ C

∞

X

k=−∞

"

2^kα

k−2

X

j=−∞

(k − j)^m2^(k−j)nr

fj

η1

Lp1(·)(Rⁿ)

kχB_jk

L^p01(·)

kχB_kk

L^p01(·)

#(q²₂)k

≤ C

∞

X

k=−∞

"

2^kα

k−2

X

j=−∞

(k − j)^m2^{(j−k)(nδ1−nr})

fj

η1

L^p1(·)(Rⁿ)

#^(q22)k

≤ C

∞

X

k=−∞





k−2

X

j=−∞

(k − j)^m2^{(j−k)(nδ1−nr−α})

 |2^αjfjχk| η1

^q1(·)

1 (q1)+

Lp1(·)q1(·)(Rⁿ)





(q²₂)k

Where

(q²₂)k =





















 (q2)−,







2^kα| ^k−2P j=−∞

[b^m,T_Ω,µ](f_j)χ_k|

η₁kbk^m_∗







q₂(·) L

p2(·) q2(·)

≤ 1

(q2)+,







2^kα| ^k−2P j=−∞

[b^m,T_Ω,µ](f_j)χ_k|

η₁kbk^m_∗







q2(·) L

p2(·) q2(·)

> 1

Since f ∈ ˙K_p^α,q1(·)

1(·) (Rⁿ), then we have

2^jα|f χ_j| η₁

L^p(·)

≤ 1, and

∞

X

j=−∞

 2^jα|f χj| η1

q₁(·) L

p1(·) q1(·)

≤ 1

Now if (q1)+< 1, and α < nδ1+ⁿ_r, then we have

∞

P

k=−∞







2^kα| ^k−2P j=−∞

[b^m,T_Ω,µ](f_j)χ_k|

η₁







q2(·) L

p2(·) q2(·)

≤ C

∞

X

k=−∞





k−2

X

j=−∞

(k − j)^m2^{(j−k)(nδ1−nr−α)}

 |2^αjfjχk| η1

q₁(·)

Lp1(·)q1(·)(Rⁿ)





(q22 )k (q1)+

≤ C





∞

X

j=−∞

 |2^αjfjχk| η1

^q1(·)

L^p1(·)q1(·)(Rⁿ)

∞

X

k=j+2

(k − j)^m2^{(j−k)(nδ1−nr−α)}





q∗

≤ C

Where q∗= min

k∈N (q₂²)k

(q₁)₊. If (q1)+≥ 1, and α < nδ1+ⁿ_r, then we have

∞

P

k=−∞







2^kα| k−2

P j=−∞

[b^m,T_Ω,µ](f_j)χ_k|

η₁







q₂(·) L

p2(·) q2(·)

≤ C

∞

X

k=−∞





k−2

X

j=−∞

(k − j)^m2^{(j−k)(nδ1−nr−α)(q2)+2}

 |2^αjfjχk| η1

^q1(·)

Lp1(·)q1(·)(Rⁿ)





(q12 )k (q1)+

×

" _k−2 X

j=−∞

(k − j)^m2^{(j−k)(nδ1−nr−α)((q1)+)}

0 2

#

(q22 )+

((q1)+)0

≤ C





∞

X

j=−∞

 |2^αjfjχk| η1

^q1(·)

L^p1(·)q1(·)(Rⁿ)

∞

X

k=j+2

(k − j)^m2^{(j−k)(nδ1−nr−α)}

(q1)+

2





q∗

≤ C

Where q∗= min

k∈N (q₂²)k

(q1)₊, this implies that

η21≤ Cη1kbk^m∗ ≤ Ckbk^m∗kf k_˙

K^α,q1(·)

p1(·) (Rⁿ) (12)