Boundedness of Littlewood-Paley operators with variable ... · Abstract. Let Ω ∈L ... 1}. In...

Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)Volume 15 (2020), 295 – 313

BOUNDEDNESS OF LITTLEWOOD-PALEYOPERATORS WITH VARIABLE KERNEL ON THE

WEIGHTED HERZ-MORREY SPACES WITHVARIABLE EXPONENT

Afif Abdalmonem, Omer Abdalrhman and Shuangping Tao

Abstract. Let Ω ∈ L∞(Rn) × L2(Sn−1) be a homogeneous function of degree zero. In this

article, we obtain some boundedness of the parameterized Littlewood−Paley operators with variable

kernels on weighted Herz-Morrey spaces with variable exponent. As a supplement, the boundedness

of fractional integral operators with variable kernel is also obtained on these spaces.

1 Introduction

It is well known that the boundedness of Littlewood−Paley operators on functionspaces are one of the very important tools not only in harmonic analysis but alsoin potential theory and in partial differential equations (see [1],[2],[2],[23],[27],[33])for details). In 2004, Ding, Lin and Shao [3] investigated the L2-boundedness for aclass of Marcinkiewicz integral operators with variable kernels µΩ and µΩ,s relatedto the Littlewood−Paley function µ∗

Ω,λ and the area integral g∗λ. In 2006, the au-thors [4] have proved the Lp-boundedness of the Littlewood−Paley operators withvariable kernels. In 2009, Xue and Ding [5] established the weighted estimate forLittlewood−Paley operators and their commutators.

In 1960, Hormander [6] introduced the parameterized Littlewood−Paley opera-tors for the first time. Now, let us recall the definitions of the parameterized Lusinarea integral and the Littlewood-Paley g∗λ function.

Let Sn−1(n ≥ 2) be the unit sphere in Rn with normalized Lebesgue measuredσ(x′). Denote Ω(x, z) ∈ L∞(Rn)×Lr(Sn−1)(r ≥ 1) to be a homogeneous functionof degree zero and ∫

sn−1

Ω(x, z′)dσ(z′) = 0, for all x ∈ Rn, (1)

2020 Mathematics Subject Classification: 43A15; 43A85Keywords: Parameterized Littlewood−Paley operators; variable kernel; weighted Herz space;

Muckenhoupt weight; variable exponent.

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296 A. Abdalmonem, O. Abdalrhman and S. Tao

where Ω satisfies the following conditions:

(i) For any x, z ∈ Rn and any λ > 0, one has Ω(x, λz) = Ω(x, z);

(ii) ∥Ω∥L∞(Rn)×Lr(Sn−1) := supr≥0,y∈Rn

(∫sn−1 |Ω(rz′ + y, z′)|rdσ(z′)

) 1r < ∞.

The parameterized Littlewood−Paley operators µρΩ,s and µ∗,ρ

Ω,λ with variable ker-nels, which are related to the Lusin area integral and the Littlewood−Paley g∗λfunction are defined by

µρΩ,s(f)(x) =

⎛⎝∫ ∫Γ(x)

1tρ∫|y−z|≤t

Ω(y, y − z)

|y − z|n−ρf(z)dz

2dydt

tn+1

⎞⎠ 12

and

µ∗,ρΩ,λ(f)(x) =

⎛⎝∫ ∫Rn+1+

(t

t+ |x− y|

)λn 1tρ∫|y−z|≤t

Ω(y, y − z)

|y − z|n−ρf(z)dz

2dydt

tn+1

⎞⎠ 12

,

where Γ(x) = (y, t) ∈ Rn+1+ : |x− y| < t and λ > 1.

In 2013, Wei and Tao [7] investigated the boundedness of parameterized Little-wood −Paley operators on weighted weak Hardy spaces. Lin and Xuan [8] estab-lished the boundedness for commutators of parameterized Littlewood−Paley oper-ators and area integrals on weighted Lebesgue spaces Lp(w).

On the other hand, the theory of the variable exponent function spaces has beenrapidly developed after the work [9] where Kovacik and Rakosnık have clarifiedfundamental properties of Lebesgue spaces with variable exponent. After that, manyresearchers have been interested in the theory of the variable exponent spaces (see[10]-[14]).

The generalization of the Muckenhoupt weights with variable exponent Ap(·) hasbeen considered in ([15]-[18]). We note that the equivalence between the Mucken-houpt condition and the boundedness of the Hardy-Littlewood maximal operator onweighted Lebesgue spaces with variable exponent were discussed in ([15], [16]). Afterthat, Cruz-Uribe and Wang [19] proved the boundedness of some classical operatorson weighted Lebesgue spaces with variable exponent Lp(·)(w).

Very recently, Mitsuo and Takahiro [20] introduced the weighted Herz spaceswith variable exponent, and also studied the boundedness of fractional integrals onthose spaces.

In this paper, by applying the theory of Banach function spaces and the variableexponent Muckenhoupt weight theory, we establish the boundedness of parameter-ized Littlewood− Paley operators with variable kernels on weighted Herz-Morreyspaces with variable exponent MKα,q

p(·)(w). The similar results for the boundednessof the fractional integral operators with variable kernel are also discussed. Let E bea Lebesgue measurable set in Rn with measure |E| > 0, χE means its characteristicfunction. We shall recall some definitions.

******************************************************************************Surveys in Mathematics and its Applications 15 (2020), 295 – 313




Boundedness of Littlewood-Paley operators 297

Definition 1 ([2]). Let p(·) : E → [1,∞) be a measurable function, the variableexponent Lebesgue space is defined by

Lp(·)(E) = f is measurable :

∫E

(|f(x)|η

)p(x)

dx < ∞ for some constant η > 0.

The space Lp(·)loc (E) is defined by

Lp(·)loc (E) = f is measurable : f ∈ Lp(·)(K) for all compact K ⊂ E.

The Lebesgue spaces Lp(·)(E) is a Banach spaces with the norm defined by

∥f∥Lp(·)(E) = inf

η > 0 :

∫E

(|f(x)|η

)p(x)

dx ≤ 1

.

We denote p− = ess infp(x) : x ∈ E, p+ = ess supp(x) : x ∈ E, then P(E)consists of all p(·) satisfying p− > 1 and p+ < ∞.

Definition 2 ([32]). Let p(·) : Rn → [1,∞). A measurable function p(·) is said tobe globally log-Hoder continuous if it satisfies

(1) |p(x)− p(y)| ≤ 1

−log(|x− y|), x, y ∈ Rn, |x− y| ≤ 1/2;

(2) |p(x)− p∞| ≤ 1

log(e+ |x|), x ∈ Rn,

for some p∞ ≥ 1. The set of p(·) satisfying (1) and (2) is denoted by LH(Rn). Weknow that, if p(·) ∈ LH(Rn)∩P(Rn), the Hardy−Littlewood maximal operator M isbounded on Lp(·)(Rn) (see[34]).

Definition 3 ([24]). Suppose that p(·) ∈ P(Rn) and w is a weight function. Theweighted Lebesgue spaces with variable exponent Lp(·)(w) is the set of all complex-valued measurable function f such that fw1/p(·) ∈ Lp(·)(Rn). The space Lp(·)(w) isa Banach space equipped with the norm

∥f∥Lp(·)(w) = ∥fw1/p(·)∥Lp(·) .

p′(·) is the conjugate of p(·) such that 1p′(·)+

1p(·) = 1. Next, we introduce the classical

Muckenhoupt Ap weight.

Definition 4 ([21]). Let 1 < p < ∞, then w ∈ Ap for every cube Q,(1

|Q|

∫Qw(x)dx

)(1

|Q|

∫Qw(x)1−p′dx

)p−1

≤ C < ∞.

We say that w ∈ A1 if it satisfies Mw(x) ≤ w(x) for all x ∈ Rn. The set of A1

consists of all Muckenhoupt A1 weights.






Definition 5 ([15],[19]). Given p(·) ∈ P(Rn) and a weight w, then w ∈ Ap(·) if

supB:ball

|B|−1∥w1/p(·)χB∥Lp(·)(w)∥w−1/p(·)χB∥Lp′(·)(w) < ∞.

Definition 6 ([19]). Given p1(·), p2(·) ∈ P(Rn) and 1/p1(x)− 1/p2(x) = µ/n suchthat 0 < µ < n. Then w ∈ Ap1(·),p2(·) if

∥wχB∥Lp2(·)∥w−1χB∥Lp′1(·)≤ |B|

n−µn

holds for all balls B ∈ Rn.

Definition 7 ([19]). Let p(·) ∈ P(Rn) and w be a weight. We say that (p(·), w) isan M -pair if the maximal operator M is bounded on Lp(·)(w) and Lp′(·)(w−1).

Now, we need give the definition of weighted Herz space with variable exponent.For all k ∈ Z, we denote Bk = x ∈ Rn : |x| ≤ 2k, Ck = Bk\Bk−1, χk = χCk

.

Definition 8 ([20]). Suppose that p(·) ∈ P(Rn), 0 < q < ∞, α ∈ R. The ho-mogeneous weighted Herz space with variable exponent Kα,q

p(·)(w) is the collection of

f ∈ Lp(·)loc (R

n\0, w) such that

∥f∥Kα,qp(·)(w) :=

( ∞∑k=−∞

2αqk∥fχk∥qLp(·)(w)

)1/q

< ∞.

Definition 9. Suppose that p(·) ∈ P(Rn), α ∈ R, 0 ≤ λ < ∞, 0 < q < ∞. The

homogeneous weighted Herz-Morrey space with variable exponent MKα,λq,p(·)(w) is the

collection of f ∈ Lp(·)loc (R

n\0, w) such that

∥f∥MKα,λ

q,p(·)(w):= sup

k0∈Z2−k0λ

(k0∑

k=−∞2kαq∥fχk∥qLp(·)(w)

)1/q

< ∞.

Obviously, when λ = 0, then MKα,0q,p(·)(w) = Kα,q

p(·)(w). Obviously, when λ = 0, then

MKα,0q,p(·)(w) = Kα,q

p(·)(w).

Definition 10. We say a kernel function Ω(x, z) satisfies the Lr-Dini condition(r ≥ 1), if ∫ 1

0

ωr(δ)

δ(1 + | log δ|σ)dδ < ∞, (2)

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

ωr(δ) = supx∈Rn,|ρ|<δ

⎛⎝ ∫sn−1

|Ω(x, ρz′)− Ω(x, z′)|rdσ(z′)

⎞⎠ 1r

,

where ρ is the rotation in Rn, ∥ρ∥ = supz′∈Sn−1

∥ρz′ − z′∥.






2 Preliminaries and notations

In order to prove our main theorems, we need the following Lemmas.

Lemma 11 ([23]). Suppose that X ⊂ M is a Banach function space(1) (The generalized Holder’s inequality) For all f ∈ X and g ∈ X ′, we have∫

Rn

|f(x)g(x)|dx ≤ ∥f∥X∥g∥X .

(2) For all f ∈ X, we have

sup

∫Rn

f(x)g(x)dx

: ∥g∥X′ ≤ 1

= ∥f∥X .

In particular, space (X ′)′ = X.

As an application of the generalized Holder’s inequality above, we have the fol-lowing Lemma.

Lemma 12. Let X be a Banach function space, we have

1 ≤ 1

|B|∥χB∥X∥χB∥X′ ,

holds for all balls B.

Lemma 13 ([18]). Let X be a Banach function space. If the Hardy-Littlewoodmaximal operator M is weakly bounded on X, that is

∥χMf>λ∥X ≤ λ−1∥f∥X ,

it holds for all f ∈ X and λ > 0. Then we get

supB:Ball

1

|B|∥χB∥X∥χB∥X′ < ∞.

Remark 14. ([24]) The weighted Banach function space X(Rn,W ) is a Banachfunction space equipped the norm ∥f∥X(Rn,W ) := ∥fW∥X . The associated space ofX(Rn,W ) is a Banach function space and equals X ′(Rn,W−1).

Remark 15. If p1(·) ∈ P(Rn), by comparing the definition of the weighted Banachfunction space with weighted variable Lebesgue space, we have

(1) If X = Lp1(·)(Rn) and W = w, then we obtain Lp1(·)(Rn, w) = Lp1(·)(wp1(·)).(2) If X = Lp′1(·)(Rn) and W = w−1, by Remark 14, then we obtain

Lp′1(·)(Rn, w−1) = Lp′1(·)(w−p′1(·)) = (Lp1(·)(wp1(·)))′.






Lemma 16 ([25]). Suppose that X is a Banach space. Let M be bounded on theassociated space X ′. Then there exists a constant 0 < δ < 1 such that

∥χE∥X∥χB∥X

≤(|E||B|

)δ

holds for all balls B and all measurable sets E ⊂ B.

Lemma 17 ([20]). Suppose that p1(·) ∈ LH(Rn) ∩ P(Rn) and wp1(·) ∈ A1. Let Mbe a bounded on Lp1(·)(wp1(·)) and Lp′1(·)(w−p′1(·)), then there exist constants δ1, δ2 ∈(0, 1) such that

∥χS∥Lp1(·)(wp1(·))

∥χB∥Lp1(·)(wp1(·))

≤ C

(|S||B|

)δ1

,∥χS∥(Lp1(·)(wp1(·)))′

∥χB∥(Lp1(·)(wp1(·)))′≤ C

(|S||B|

)δ2

holds for all balls B and all measurable sets S ⊂ B.

Lemma 18 ([26]). Suppose that Ω ∈ L∞(Rn) × L2(Sn−1) satisfies (1) and (2),λ > 2, 2ρ − n > 0, 1 < p < ∞. Then for all f ∈ Lq(w) there exists C > 0independent of f such that

∥µρΩ,sf∥Lq(w) ≤ C∥f∥Lq(w)

and

∥µ∗,ρΩ,λf∥Lq(w) ≤ C∥f∥Lq(w).

Lemma 19 ([19]). Assume that for p0, 1 < p0 < ∞ and every w0 ∈ Ap0,∫Rn

f(x)p0w0(x)dx ≤∫Rn

g(x)p0w0(x)dx, (f, g) ∈ F .

Then for any M -pair (p(·), w),

∥f∥Lp(·)(w) ≤ C∥g∥Lp(·)(w), (f, g) ∈ F ,

Lemma 19 holds for p0 = 1 and the maximal operator is bounded on Lp′(·)(w−1). Weknow the Hardy−Littlewood maximal operator is bounded on Lp(·)(w) and Lp′(·)(w−p′(·))(see [25]).

Combining Lemma 18 with Lemma 19, we obtain the following conclusion.

Corollary 20. Let p(·) ∈ LH(Rn) ∩ P(Rn), Ω ∈ L∞(Rn) × Lr(Sn−1)(r ≥ 1) andw ∈ Ap(·). Then the parameterized Littlewood−Paley type operators µρ

Ω,s and µ∗,ρΩ,λ

with variable kernel are bounded on Lp(·)(w).






3 Main Theorems and their proofs

In this section, we will prove the boundedness of the parameterized Littlewood−Paleyoperators with variable kernels on variable weighted Herz-Morrey spaces.

Theorem 21. Let p(·) ∈ LH(Rn)∩P(Rn), 0 < q1 ≤ q2 < ∞, λ > 2, 2ρ−n > 0 andΩ ∈ L∞(Rn)× L2(Sn−1) satisfies (1) and (2). If wp1(·) ∈ A1 and −nδ1 < α < nδ2,where δ1, δ2 are the constants in Lemma 17, then the operator µΩ,s is bounded from

MKα,λq2,p1(·)(w

p1(·)) to MKα,λq1,p1(·)(w

p1(·)).

Theorem 22. Let p(·) ∈ LH(Rn)∩P(Rn), 0 < q1 ≤ q2 < ∞, λ > 2, 2ρ−n > 0 andΩ ∈ L∞(Rn)× L2(Sn−1) satisfies (1) and (2). If wp1(·) ∈ A1 and −nδ1 < α < nδ2,where δ1, δ2 are the constants in Lemma 17, then the operator µ∗

Ω,λ is bounded from

MKα,q2p1(·)(w

p1(·)) to MKα,q1p1(·)(w

p1(·)).

Remark 23. As is well known that, µρΩ,sf(x) ≤ 2nλµ∗,ρ

Ω,λf(x) (see [27], p.89). There-fore, we give only the proof of Theorem 22.

Proof. Let f ∈ MKα,λq2,p1(·)(w

p1(·)). By the Jensen’s inequality, we have

∥µ∗,ρΩ,λf∥

q1

MKα,λq2,p1(·)

(wp1(·))≤ sup

k0∈Z2−k0λq1

(k0∑

k=−∞2αq2k∥(µ∗,ρ

Ω,λf)χk∥q2Lp1(·)(wp1(·))

)q1/q2

≤ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k∥(µ∗,ρΩ,λf)χk∥q1Lp1(·)(wp1(·))

.

Denote fj := fχj for each j ∈ Z, then f :=∑∞

j=−∞ fj , so we have

∥µ∗,ρΩ,λf∥

q1

MKα,λq2,p1(·)

(wp1(·))≤ sup

k0∈Z2−k0λq1

k0∑k=−∞

2αq1k∥(µ∗,ρΩ,λf)χk∥q1Lp1(·)(wp1(·))

≤ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞

∥(µ∗,ρΩ,λfj)χk∥Lp1(·)(wp1(·))

⎞⎠q1

+ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k+2∑j=k−2


⎞⎠q1

+ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ ∞∑j=k+2


⎞⎠q1

=: F1 + F2 + F3.






First, we consider F2. Using Corollary 20 and −2 ≤ k − j ≤ 2, it is easy to get

F2 = supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k+2∑j=k−2


⎞⎠q1

≤ C supk0∈Z

2−k0λq1

k0∑k=−∞

⎛⎝ k+2∑j=k−2

2α(k−j)2αj∥fj∥Lp1(·)(wp1(·))

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·)).

Now we need to consider µ∗,ρΩ,λfj . Applying the Minkowski inequality, we conclude

that

|µ∗,ρΩ,λ(fj)(x)| =

⎛⎝∫ ∞

0

∫Rn

(t

t+ |x− y|

)λn 1tρ∫|y−z|≤t

Ω(y, y − z)

|y − z|n−ρfj(z)dz

2dydt

tn+1

⎞⎠ 12

≤∫Rn

fj(z)

(∫ ∞

0

∫|y−z|≤t

(t

t+ |x− y|

)λn |Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ+n+1

) 12

dz

≤∫Rn

fj(z)

(∫ |x−z|

0

∫|y−z|≤t

(t

t+ |x− y|

)λn |Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ+n+1

) 12

dz

+

∫Rn

fj(z)

(∫ ∞

|x−z|

∫|y−z|≤t

(t

t+ |x− y|

)λn |Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ+n+1

) 12

dz.

For Ω ∈ L∞(Rn)× L2(Sn−1) and 2ρ− n > 0, the following inequality holds

∫|y−z|≤t

|Ω(y, y − z)|2

|y − z|2n−2ρdy ≤

∫sn−1

∫ t

0

|Ω(sy′ + z, y′)|2

s2n−2ρsn−1dsdσ(y′)

≤ ∥Ω∥2L∞(Rn)×L2(Sn−1)t2ρ−n.

Since |x− z| ≤ |x− y|+ |y− z| ≤ |x− y|+ t. For λ > 2, taking 0 < δ < (λ− 2)n, so






we have

∫ |x−z|

0

∫|y−z|≤t

(t

t+ |x− y|

)λn |Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ+n+1

≤∫ |x−z|

0

∫|y−z|≤t

(t

t+ |x− y|

)λn−2n−δ 1

|x− z|2n+δ

|Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ−n−δ+1

≤ 1

|x− z|2n+δ

∫ |x−z|

0

∫|y−z|≤t

|Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ−n−δ+1

≤∥Ω∥2L∞(Rn)×L2(Sn−1)

|x− z|2n+δ

∫ |x−z|

0tδ−1dt

≤ C|x− z|−2n.

If we take 1 < λ1 < 2, then λ1n− n > 0 and λ1n− 2n < 0, we have

∫ ∞

|x−z|

∫|y−z|≤t

(t

t+ |x− y|

)λn |Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ+n+1

≤∫ ∞

|x−z|

∫|y−z|≤t

|x− z|−λ1n |Ω(y, y − z)|2

|y − z|2n−2ρ

dydt

t2ρ−λ1n+n+1

≤∫ ∞

|x−z||x− z|−λ1n

∫|y−z|≤t

|Ω(y, y − z)|2

|y − z|2n−λ1n

dydt

tn+1

≤∫ ∞

|x−z||x− z|−λ1n

∫sn−1

∫ t

0

|Ω(y′, (y − z)′)|2

s2n−λ1nsn−1dsdσ(y′)

dt

tn+1

≤ C∥Ω∥2L∞(Rn)×L2(Sn−1)|x− z|−λ1n

∫ ∞

|x−z|tλ1n−2n−1dt

≤ C|x− z|−2n.

Combining the above two estimates, we obtain

µ∗Ω,λ(f)(x) ≤

∫Rn

|f(z)||x− z|n

dz. (3)

Next, we consider F1. Noting that for x ∈ Ak, z ∈ Aj and j ≤ k − 2, then|x− z| ∼ |x|. By the virtue of the generalized Holder’s inequality, we have

µ∗,ρΩ,λ(fj)(x) ≤ C2−kn∥fj∥Lp1(·)(wp1(·))∥χj∥(Lp1(·)(wp1(·)))′ .






Applying Lemma 13 and Lemma 17, we take ∥.∥Lp1(·)(wp1(·)) for each side, we have

∥µ∗,ρΩ,λ(fj)(x)χk∥Lp1(·)(wp1(·))

≤ C2−kn∥fj∥Lp1(·)(wp1(·))∥χj∥(Lp1(·)(wp1(·)))′∥χk∥Lp1(·)(wp1(·))

≤ C2−kn∥fj∥Lp1(·)(wp1(·))∥χBj∥(Lp1(·)(wp1(·)))′∥χBk∥Lp1(·)(wp1(·))

≤ C∥fj∥Lp1(·)(wp1(·))∥χBj∥(Lp1(·)(wp1(·)))′∥χBk∥−1(Lp1(·)(wp1(·)))′

≤ C∥fj∥Lp1(·)(wp1(·))

∥χBj∥(Lp1(·)(wp1(·)))′

∥χBk∥(Lp1(·)(wp1(·)))′

≤ C2(j−k)nδ2∥fj∥Lp1(·)(wp1(·)).

On the other side, we have the following fact:

∥fj∥Lp1(·)(wp1(·)) = 2−jα(2jαq1∥fj∥q1Lp1(·)(wp1(·))

)1/q1≤ C2−jα

(j∑

n=−∞2nαq1∥fn∥q1Lp1(·)(wp1(·))

)1/q1

= C2j(λ−α)

⎛⎝2−λj

(j∑

n=−∞2nαq1∥fn∥q1Lp1(·)(wp1(·))

)1/q1⎞⎠

≤ C2j(λ−α)∥f∥MKα,q1p1(·)

(wp1(·)).

Thus, by using condition α < λ+ nδ2, it follows that

F1 = supk0∈z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞


⎞⎠q1

≤ C supk0∈z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞

2(j−k)nδ2∥fj∥Lp1(·)(wp1(·))

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞

2(j−k)nδ22j(λ−α)

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

k0∑k=−∞

2λq1k

⎛⎝ k−2∑j=−∞

2(k−j)(α−λ−nδ2)

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

(k0∑

k=−∞2λq1k

).

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·)).






Finally, we estimate F3. Noting that for x ∈ Ak, y ∈ Aj and j ≥ k + 2, then|y − x| ∼ |y|. By (3) and the virtue of the generalized Holder’s inequality, we showthat

µ∗,ρΩ,λ(fj)(x) ≤ C2−jn∥fj∥Lp1(·)(wp1(·))∥χj∥(Lp1(·)(wp1(·)))′ .

Applying Lemma 13 and Lemma 17, we can take ∥.∥Lp1(·)(wp1(·)) for each side, wehave

∥µ∗,ρΩ,λ(fj)(x)χk∥Lp1(·)(wp1(·))

≤ C2−jn∥fj∥Lp1(·)(wp1(·))∥χj∥(Lp1(·)(wp1(·)))′∥χk∥Lp1(·)(wp1(·))

≤ C2−jn∥fj∥Lp1(·)(wp1(·))∥χBj∥(Lp1(·)(wp1(·)))′∥χBk∥Lp1(·)(wp1(·))

≤ C∥fj∥Lp1(·)(wp1(·))∥χBk∥(Lp1(·)(wp1(·)))∥χBj∥

−1(Lp1(·)(wp1(·)))

≤ C∥fj∥Lp1(·)(wp1(·))

∥χBk∥(Lp1(·)(wp1(·)))

∥χBj∥(Lp1(·)(wp1(·)))

≤ C2(k−j)nδ1∥fj∥Lp1(·)(wp1(·)).

Then, by using condition α > λ− nδ1, it follows that

F3 = supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ ∞∑j=k+2


⎞⎠q1

≤ C supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ ∞∑j=k+2

2(k−j)nδ1∥fj∥Lp1(·)(wp1(·))

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ ∞∑j=k+2

2(k−j)nδ12j(λ−α)

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

k0∑k=−∞

2λq1k

⎛⎝ ∞∑j=k+2

2(k−j)(α−λ+nδ1)

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

(k0∑

k=−∞2λq1k

)≤ C∥f∥q1

MKα,q1p1(·)

(wp1(·)).

This completes the proof of Theorem 22.






4 Fractional integral operator with variable kernel onMKα,q

p(·)(wp(·)) spaces

Let 0 < µ < n and Ω ∈ L∞(Rn)×Lr(Sn−1) satisfies (i)-(ii), the fractional integraloperator with variable kernel TΩ,µ is defined by

TΩ,µf(y) =

∫Rn

Ω(y, y − z)

|y − z|n−µf(z)dz.

In this section, we will prove some main results of this paper. We begin twopreliminary Lemmas.

Lemma 24 ([29],[30]). Let Ω ∈ L∞(Rn)×Lr(Sn−1)(r ≥ 1) and wr′(x) ∈ A(p/r′, q/r′).If 0 < µ < n, 1 ≤ r′ < p < n

µ ,1p − 1

q = nµ , then we have

(∫Rn

[TΩ,µf(x)w(x)]qdx

) 1q

≤(∫

Rn

|f(x)w(x)|pdx) 1

p

.

Lemma 25 ([19]). Assume that 1 < p0 ≤ q0 < ∞ and every w0 ∈ Ap0,q0,⎛⎝∫Rn

f(x)q0w0(x)q0dx

⎞⎠ 1q0

≤ C

⎛⎝∫Rn

g(x)p0w0(x)p0dx

⎞⎠ 1p0

, (f, g) ∈ F .

Given p(·), q(·) ∈ P(Rn) and

1

p(x)− 1

q(x)=

1

p0− 1

q0.

Define σ ≥ 1 by 1σ = 1

p0− 1

q0. If w ∈ Ap(·),q(·) and (q(·)/σ,wσ) is an M -pair then

∥f∥Lq(·)(w) ≤ C∥g∥Lp(·)(w), (f, g) ∈ F ,

Lemma 25 holds for p0 = 1 and the maximal operator is bounded on L(q(·)/q0)′(w−q0).Izuki and Noi [20] proved the Hardy−Littlewood maximal operator M is boundedon Lq(·)/σ(wσ.q(·)/σ) and L(q(·)/σ)′(w−σ(q(·)/σ)′), then (q(·)/σ,wσ) is an M -pair whenw ∈ Ap(·),q(·) and p(·), q(·) ∈ LH(Rn) ∩ P(Rn).

From Lemma 24 and Lemma25, we have the following conclusion.

Corollary 26. If p1(·) ∈ LH(Rn)∩P(Rn), Ω ∈ L∞(Rn)×Lr(Sn−1)(r ≥ 1), 0 < µ <n/p+1 and 1/p1(·) − 1/p2(·) = µ/n. Then for w ∈ Ap1(·),p2(·), the fractional integral

operator with variable kernel TΩ,µ is bounded from Lp1(·)(wp1(·)) to Lp2(·)(wp2(·)).






Theorem 27. Let 0 ≤ λ < ∞, 0 < q1 ≤ q2 < ∞, p2(·) ∈ LH(Rn) ∩ P(Rn) andΩ ∈ L∞(Rn) × Lr(Sn−1)(r ≥ 1). If wp2(·) ∈ A1 and λ − nδ1 < α < nδ2 − n

r + λ,where δ1, δ2 are the constants in Lemma 17. Define p2(·) by 1/p1(·)−1/p2(·) = µ/n,

then the operator TΩ,µ is bounded from MKα,λq2,p2(·)(w

p2(·)) to MKα,λq1,p1(·)(w

p1(·)).

Proof. Let f ∈ MKα,λq1,p1(·)(w

p1(·)). By the Jensen’s inequality, we have

∥TΩ,µf∥q1MKα,λ

q2,p2(·)(wp2(·))

≤ supk0∈Z

2−k0λq1

(k0∑

k=−∞2αq2k∥(TΩ,µf)χk∥q2Lp2(·)(wp2(·))

)q1/q2

≤ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k∥(TΩ,µf)χk∥q1Lp2(·)(wp2(·)).

Denote fj := fχj for each j ∈ z, then f :=∑∞

j=−∞ fj , so we have

∥TΩ,µf∥q1MKα,λ

q2,p2(·)(wp2(·))

≤ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k∥(TΩ,µf)χk∥q1Lp2(·)(wp2(·))

≤ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞

∥(TΩ,µfj)χk∥Lp2(·)(wp2(·))

⎞⎠q1

≤ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k+2∑j=k−2


⎞⎠q1

≤ supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ ∞∑j=k+2


⎞⎠q1

=: H1 +H2 +H3.

First, we consider H2. Using Corollary 26 and −2 ≤ k − j ≤ 2, it is easy to get

H2 = supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k+2∑j=k−2


⎞⎠q1

≤ C supk0∈Z

2−k0λq1

k0∑k=−∞

⎛⎝ k+2∑j=k−2

2α(k−j)2αj∥fj∥Lp1(·)(wp1(·))

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·)).






Now we turn to estimate of H1. Noting that j ≤ k − 2 for every j, k ∈ Z, thenwe have the following inequality by applying the generalized Holder’s inequality

|TΩ,µfj | ≤ C2−k(n−µ)∥Ω(y, y − z)∥Lr(Rn)∥fj(z)∥Lr′ (Rn).

By virtue of the generalized Holder’s inequality, we have

∥fj(z)∥Lr′ ≤ |Bj |−1r ∥fjw∥Lp1(·)∥w−1χj∥Lp′1(·)

. (4)

for any Ω ∈ L∞(Rn)× Lr(Sn−1), we get by (4)

|TΩ,µfj | ≤ C2−k(n−µ)2(k−j)nr ∥fjw∥Lp1(·)∥w−1χj∥Lp′1(·)

. (5)

By 5 and taking ∥.∥Lp2(·)(wp2(·)) for each side, we have


≤ C2−k(n−µ)2(k−j)nr ∥fjw∥Lp1(·)∥w−1χj∥Lp′1(·)

∥χk∥Lp2(·)(wp2(·))

≤ C2−k(n−µ)2(k−j)nr ∥fjw∥Lp1(·)∥χBj∥(Lp1(·)(wp1(·)))′∥χBk

∥Lp2(·)(wp2(·)).

Noting that χBk(x) ≤ C2−kµTΩ,µ(χBk

)(x), by the Corollary 26, we obtain

∥χBk∥Lp2(·)(wp2(·)) ≤ C2−kµ∥TΩ,µ(χBk

)∥Lp2(·)(wp2(·))

≤ C2−kµ∥χBk∥Lp1(·)(wp1(·)).

From this and using Lemma 13 and Lemma 17, we have


≤ C2−k(n−µ)2(k−j)nr ∥fjw∥Lp1(·)∥χBj∥(Lp1(·)(wp1(·)))′2

−kµ∥χBk∥Lp1(·)(wp1(·))

≤ C2(k−j)nr ∥fjw∥Lp1(·)∥χBj∥(Lp1(·)(wp1(·)))′2

−kn∥χBk∥Lp1(·)(wp1(·))

≤ C2(k−j)nr ∥fjw∥Lp1(·)∥χBj∥(Lp1(·)(wp1(·)))′∥χBk

∥−1(Lp1(·)(wp1(·)))′

≤ C2(k−j)nr ∥fjw∥Lp1(·)

∥χBj∥(Lp1(·)(wp1(·)))′

∥χBk∥(Lp1(·)(wp1(·)))′

≤ C2(k−j)(nr−nδ2)∥fjw∥Lp1(·) .






On the other side, we have the following fact:

∥fj∥Lp1(·)(wp1(·)) = 2−jα(2jαq1∥fj∥q1Lp1(·)(wp1(·))

)1/q1≤ C2−jα

(j∑

n=−∞2nαq1∥fn∥q1Lp1(·)(wp1(·))

)1/q1

= C2j(λ−α)

⎛⎝2−λj

(j∑

n=−∞2nαq1∥fn∥q1Lp1(·)(wp1(·))

)1/q1⎞⎠

≤ C2j(λ−α)∥f∥MKα,q1p1(·)

(wp1(·)).

Thus, by using condition α < λ+ nδ2 − nr , it follows that

H1 = supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞


⎞⎠q1

≤ C supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞

2(k−j)(nr−nδ2)∥fj∥Lp1(·)(wp1(·))

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

k0∑k=−∞

2αq1k

⎛⎝ k−2∑j=−∞

2(k−j)(nr−nδ2)2j(λ−α)

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

k0∑k=−∞

2λq1k

⎛⎝ k−2∑j=−∞

2(k−j)(α−λ−nδ2+nr)

⎞⎠q1

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·))supk0∈Z

2−k0λq1

(k0∑

k=−∞2λq1k

).

≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·)).

Finally, we estimate U3. For every j, k ∈ Z with j ≥ k + 2 and applying thegeneralized Holder’s inequality assures that

|TΩ,µfj | ≤ C2−j(n−µ)∥Ω(y, y − z)∥Lr(Rn)∥fj(z)∥Lr′ (Rn).

For any Ω ∈ L∞(Rn)× Lr(Sn−1), we get by (4)

|TΩ,µfj | ≤ C2−j(n−µ)∥fjw∥Lp1(·)∥w−1χj∥Lp′1(·).






from this and applying Lemma 17 and taking ∥.∥Lp2(·)(wp2(·)) for each side, we have

∥TΩ,µfj(x)χk∥Lp2(·)(wp2(·))

≤ C2−j(n−µ)∥fjw∥Lp1(·)∥w−1χj∥Lp′1(·)∥χk∥Lp2(·)(wp2(·))

≤ C2−j(n−µ)∥fjw∥Lp1(·)∥χBj∥(Lp1(·)(wp1(·)))′∥χBk∥Lp2(·)(wp2(·))

≤ C2−j(n−µ)∥fjw∥Lp1(·)∥χBj∥(Lp1(·)(wp1(·)))′∥χBj∥Lp2(·)(wp2(·))

∥χBk∥Lp2(·)(wp2(·))

∥χBj∥Lp2(·)(wp2(·))

≤ C2−j(n−µ)2nδ1(k−j)∥fjw∥Lp1(·)∥χBj∥(Lp1(·)(wp1(·)))′∥χBj∥Lp2(·)(wp2(·))

From the definition of Ap1(·),p2(·) (Definition 6), we get

∥χBj∥(Lp1(·)(wp1(·)))′∥χBj∥Lp2(·)(wp2(·))

≤ ∥w−1χBj∥Lp′1(·)∥wχBj∥Lp2(·)

≤ |Bj |n−µn

= 2j(n−µ).

Then, we obtain that

∥TΩ,µfj(x)χk∥Lp2(·)(wp2(·)) ≤ C2(k−j)nδ1∥fjw∥Lp1(·) .

The rest of the proof is the same as the proof of F3 in Theorem 22, it is not difficultto obtain

H3 ≤ C∥f∥q1MK

α,q1p1(·)

(wp1(·)).

Thus, we omit the details there. Then the proof of Theorem 27 is finished.

Acknowledgement. The author wishes to express profound gratitude to the re-viewers for their useful comments on the manuscript.

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Afif Abdalmonem

Faculty of Science, University of Dalanj, Dalanj, Sudan.

e-mail:[email protected]

Omer Abdalrhman

College of Education, Shendi University, Shendi, Sudan.

e-mail: [email protected]

Shuangping Tao

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China.

e-mail: [email protected]

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