Boundedness of some sublinear operators on Herz-Morrey spaces

arXiv:1404.1627v1 [math.FA] 6 Apr 2014

Boundedness of some sublinear operators on Herz-Morrey spaces with variable exponent Jianglong WU Department of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, China Georgian Math. J. 2014; 21 (1):101-111

Abstract: In this paper, the boundedness of some sublinear operators is proved on homogeneous Herz-Morrey spaces with variable exponent. Keywords: sublinear operator; Lebesgue space with variable exponent; Herz-Morrey space with variable exponent AMS(2010) Subject Classification: 42B20; 42B35

1

Introduction

Function spaces with variable exponent are being watched with keen interest not in real analysis but also in partial differential equations and in applied mathematics because they are applicable to the modeling for electrorheological fluids and image restoration. The theory of function spaces with variable exponent has rapidly made progress in the past twenty years since some elementary properties were established by Kov´aˇcik-R´akosn´ık [18]. One of the main problems on the theory is the boundedness of the Hardy-Littlewood maximal operator on variable Lebesgue spaces. By virtue of the fine works [4–6, 8–10, 15, 19, 23, 24], some important conditions on variable exponent, for example, the log-H¨ older conditions and the Muckenhoupt type condition, have been obtained. The class of the Herz spaces is arising from the study on characterization of multipliers on the classical Hardy spaces. The well-known Morrey spaces is used to show that certain systems of partial differential equations (PDEs) had H¨ older continuous solutions. And the homogeneous Herz-Morrey spaces α,λ n ˙ M Kp,q (R ) coordinate with the homogeneous Herz space K˙ qα,p (Rn ) when λ = 0. One of the important problems on Herz spaces and Herz-Morrey spaces is the boundedness of sublinear operators. Hern´andez, Li, Lu and Yang [11,20,22] have proved that if a sublinear operator T is bounded on Lp (Rn ) and satisfies the size condition Z |T f (x)| ≤ C |x − y|−n |f (y)|dy, a. e. x ∈ / supp f (1.1) Rn

for all f ∈ L1 (Rn ) with compact support, then T is bounded on the homogeneous Herz space K˙ qα,p (Rn ). In 2005, Lu and Xu [21] established the boundedness for some sublinear operators on homogeneous Herz-Morrey spaces. In 2010, Izuki [14] proves the boundedness of some sublinear operators on Herz spaces with variable exponent; and recently Izuki [12, 13] also considers the boundedness of some operators on Herz-Morrey spaces with variable exponent. Motivated by the study on the Herz spaces and Lebesgue spaces with variable exponent, when (1.1) is replaced by some more general size conditions, the main purpose of this paper is to establish some boundedness results of sublinear operators on Herz-Morrey spaces with variable exponent. This size condition is satisfied by many important operators in harmonic analysis. 1

Let us explain the outline of this article. In Section 2 we state some important properties of Lq(·) (Rn ) based on [8, 12, 18, 23], and give some lemmas which will be needed for proving our main theorem. In Section 3 we prove the boundedness of sublinear operators on Herz-Morrey spaces with variable exponent α,λ M K˙ p,q(·) (Rn ), and obtain the corresponding corollaries. Throughout this paper, we will denote by |S| the Lebesgue measure and by χS the characteristic function for a measurable set S ⊂ Rn . C denotes a constant that is independent of the main parameters involved but whose value may differ from line to line. For any index 1 < q(x) < ∞, we denote by q ′ (x) q(x) its conjugate index, namely, q ′ (x) = q(x)−1 . For A ∼ D, we mean that there is a constant C > 0 such −1 thatC D ≤ A ≤ CD.

2

Preliminaries and Lemmas

In this section, we give the definition of Lebesgue and Herz-Morrey spaces with variable exponent, and state their properties. Let E be a measurable set in Rn with |E| > 0. We first define Lebesgue spaces with variable exponent. Definition 2.1 Let q(·) : E → [1, ∞) be a measurable function. 1) The Lebesgue spaces with variable exponent Lq(·) (E) is defined by Z  |f (x)| q(x) q(·) dx < ∞ for some constant η > 0}. L (E) = {f is measurable function : η E q(·)

2) The space Lloc (E) is defined by q(·)

Lloc (E) = {f is measurable function : f ∈ Lq(·) (K) for all compact subsets K ⊂ E}. Lq(·) (E) is a Banach space with the norm defined by Z  n o |f (x)| q(x) kf kLq(·) (E) = inf η > 0 : dx ≤ 1 . η E Now, we define two classes of exponent functions. Given a function f ∈ L1loc (E), the Hardy-Littlewood maximal operator M is defined by Z −n M f (x) = sup r |f (y)|dy (x ∈ E), r>0

B(x,r)∩E

n

where B(x, r) = {y ∈ R : |x − y| < r}. Definition 2.2 1) The set P(E) consists of all measurable functions q(·) satisfying 1 < ess inf q(x) = q− , q+ = ess sup q(x) < ∞. x∈E

x∈E

2) The set B(E) consists of all measurable functions q(·) ∈ P(E) such that the Hardy-Littlewood maximal operator M is bounded on Lq(·) (E). Next we define the Herz-Morrey spaces with variable exponent. Let Bk = B(0, 2k ) = {x ∈ Rn : |x| ≤ 2k }, Ak = Bk \ Bk−1 and χk = χAk for k ∈ Z. Definition 2.3 Let α ∈ R, 0 ≤ λ < ∞, 0 < p < ∞, and q(·) ∈ P(Rn ). The Herz-Morrey space α,λ with variable exponent M K˙ p,q(·) (Rn ) is defined by q(·) α,λ M K˙ p,q(·) (Rn ) = {f ∈ Lloc (Rn \{0}) : kf kM K˙ α,λ

p,q(·)

where kf kM K˙ α,λ

p,q(·)

(Rn )

= supk0 ∈Z 2−k0 λ

P

k0 k=−∞

2kαp kf χk kp q(·) L

2

(Rn )

 p1

.

(Rn )

< ∞},

α,λ Compare the Herz-Morrey space with variable exponent M K˙ p,q(·) (Rn ) with the Herz space with α,p variable exponent K˙ q(·) (Rn ) [13], where ∞ n o X q(·) α,p K˙ q(·) (Rn ) = f ∈ Lloc (Rn \{0}) : 2kαp kf χk kpLq(·) (Rn ) < ∞ . k=−∞

α,0 α,p Obviously, M K˙ p,q(·) (Rn ) = K˙ q(·) (Rn ). In 2012, Almeida and Drihem [1] discussed the boundedness of a wide class of sublinear operators, α(·),p including maximal, potential and Calder´ on-Zygmund operators, on variable Herz spaces Kq(·) (Rn ) and α(·),p (Rn ). Meanwhile, they also established Hardy-Littlewood-Sobolev theorems for fractional integrals K˙ q(·)

on variable Herz spaces. In other papers [2,16,17] the boundedness of operators of harmonic analysis are established in variable exponent Morrey spaces. In this paper, the author considers Herz-Morrey space α(·),λ M K˙ p,q(·) (Rn ) with variable exponent q(·) for fixed α ∈ R and p ∈ (0, ∞). However, for the case of the exponent α(·) is variable as well, we can refer to the furthermore work of the author of this paper. Next we state some properties of variable exponent. Cruz-Uribe et al [6] and Nekvinda [23] proved the following sufficient conditions for the boundedness of M in variable exponent space independently. We note that Nekvinda [23] gave a more general condition in place of (2.2). Proposition 2.1 [23] Suppose that E is an open set. If q(·) ∈ P(E) satisfies the inequality −C ln(|x − y|) C |q(x) − q(y)| ≤ ln(e + |x|)

|q(x) − q(y)| ≤

if |x − y| ≤ 1/2,

(2.1)

if |y| ≥ |x|,

(2.2)

where C > 0 is a constant independent of x and y, then we have q(·) ∈ B(E). The next proposition is due to Diening [8](see Theorem 8.1). We remark that Diening has also proved general results on Musielak-Orlicz spaces. We only describe partial results we need in this paper. Proposition 2.2 [8] Suppose that q(·) ∈ P(Rn ), then q(·) ∈ B(Rn ) iff q ′ (·) ∈ B(Rn ). In order to prove our main theorems, we also need the following result which is the Hardy-LittlewoodSobolev theorem on Lebesgue spaces with varible expoonent due to Capone, Cruz-Uribe and Fiorenza [3](see Theorem 1.8). We remark that this result was proved by Diening [7] provided that q1 (·) is constant outside of a large ball. Proposition 2.3 [3] Suppose that q1 (·) ∈ P(Rn ) satisfies conditions (2.1) and (2.2) in Proposition 2.1. 0 < β < n/(q1 )+ and define q2 (·) by 1 β 1 − = . q1 (x) q2 (x) n Then we have kIβ f kLq2 (·) (Rn ) ≤ Ckf kLq1 (·) (Rn ) , R f (y) where fractional integral Iβ is defined by Iβ (f )(x) = Rn |x−y| n−β dy. In addition, by Proposition 2.3, we can obtain

Proposition 2.4 Let q1 (·), q2 (·) and β be the same as in Proposition 2.3. Then we have kχBk kLq2 (·) (Rn ) ≤ C2−kβ kχBk kLq1 (·) (Rn ) for all balls Bk = {x ∈ Rn : |x| ≤ 2k } with k ∈ Z. 3

Proof It is easy to see that Z

Iβ (χBk )(x) ≥ Iβ (χBk )(x) · χBk (x) =

Bk

dy · χBk (x) ≥ C2kβ χBk (x). |x − y|n−β

Thus, applying Proposition 2.3, we get kχB k k

L

q (·) 2

(Rn )

≤ C2

−kβ

kIβ (χB )k k

L

q (·) 2

(Rn )

≤ C2

−kβ

kχB k k

L

q (·) 1

(Rn )

.

This completes the proof of Proposition 2.4. The next lemma describes the generalized H¨ older’s inequality and the duality of Lq(·) (E). The proof can be found in [18]. Lemma 2.1 [18] Suppose that q(·) ∈ P(E). Then the following statements hold. ′ 1) (generalized H¨ older’s inequality) For all f ∈ Lq(·) (E) and all g ∈ Lq (·) (E), we have Z |f (x)g(x)|dx ≤ rq kf kLq(·) (E) kgkLq′ (·) (E) ,

(2.3)

E

where rq = 1 + 1/q− − 1/q+ . 2) For all f ∈ Lq(·) (E), we have kf kLq(·) (E) ≤ sup

nZ

o |f (x)g(x)|dx : kgkLq′ (·) (E) ≤ 1 .

E

Lemma 2.2 [12] If q(·) ∈ B(Rn ), then there exist positive constants δ ∈ (0, 1) and C > 0 such that  |S| δ kχS kLq(·) (Rn ) ≤C kχB kLq(·) (Rn ) |B|

(2.4)

holds for all balls B in Rn and all measurable subsets S ⊂ B. Lemma 2.3 [12] If q(·) ∈ B(Rn ), then there exists a positive constant C > 0 such that C −1 ≤

1 kχB kLq(·) (E) kχB kLq′ (·) (E) ≤ C. |B|

(2.5)

holds for all balls B in Rn .

3

Main theorems and their proofs

Let q(·) ∈ P(Rn ) satisfy conditions (2.1) and (2.2) in Proposition 2.1. Then so does q ′ (·). In particular, we can see that q(·), q ′ (·) ∈ B(Rn ) from Proposition 2.1. Therefore applying Lemma 2.2, we can take constants δ1 ∈ (0, 1/(q ′ )+ ), δ2 ∈ (0, 1/(q)+ ) such that kχS kLq′ (·) (Rn ) kχB kLq′ (·) (Rn )



|S| ≤C |B|

δ1

 δ kχS kLq(·) (Rn ) |S| 2 ≤C kχB kLq(·) (Rn ) |B|

,

(3.1)

holds for all balls B in Rn and all measurable subsets S ⊂ B. And when q1 (·), q2 (·) ∈ P(Rn ), applying Lemma 2.2, we can take constants δ3 ∈ (0, 1/(q2′ )+ ), δ4 ∈ (0, 1/(q1 )+ ) such that kχS kLq1′ (·) (Rn ) kχB kLq1′ (·) (Rn )

≤C



|S| |B|

δ3

kχS kLq2 (·) (Rn )

,

kχB kLq2 (·) (Rn ) 4

≤C



|S| |B|

δ4

(3.2)

holds for all balls B in Rn and all measurable subsets S ⊂ B. In this section, we will give some size condition which are more general than (1.1), and prove the boundedness of some sublinear operators, satisfying this size condition on Herz-Morrey spaces with variable exponent. This size condition is satisfied by many important operators in harmonic analysis. Our main result can be stated as follows. Theorem 3.1 Let q(·) ∈ P(Rn ) satisfies conditions (2.1) and (2.2) in Proposition 2.1, and 0 < p < ∞, λ > 0, λ−nδ2 < α < λ+nδ1 , where δ1 ∈ (0, 1/(q ′ )+ ) and δ2 ∈ (0, 1/(q)+ ) are the constants satisfying (3.1). Suppose that a sublinear operator T satisfies (i) T is bounded on Lq(·) (Rn ); (ii) for suitable function f with supp f ⊂ Ak and |x| ≥ 2k+1 with k ∈ Z, |T f (x)| ≤ C|x|−n kf kL1 (Rn ) ;

(3.3)

(iii) for suitable function f with supp f ⊂ Ak and |x| ≤ 2k−2 with k ∈ Z, |T f (x)| ≤ C2−kn kf kL1(Rn ) .

(3.4)

α,λ (Rn ). Then T is also bounded on M K˙ p,q(·)

Note that if T satisfies the size condition (1.1), then T satisfies (3.3) and (3.4). Thus, by Theorem 3.1, we have Corollary 3.1 Let q(·), p, λ, α, δ1 and δ2 be the same as in Theorem 3.1. If a sublinear operator T α,λ satisfies the size condition (1.1) and is bounded on Lq(·) (Rn ), then T is also bounded on M K˙ p,q(·) (Rn ). Remark 1 For any q(·) ∈ P(Rn ) satisfies conditions (2.1) and (2.2) in Proposition 2.1, the HardyLittlewood maximal operator M , defined by Z 1 |f (y)|dy, M (f )(x) = sup B∋x |B| B∩Ω also satisfies the assumptions of Theorem 3.1, where the supremum is taken over all balls B containing x, and Ω ⊂ Rn is an open set. α,λ Proof (Proof of Theorem 3.1) For all f ∈ M K˙ p,q(·) (Rn ). If we denote fj := f · χj = f · χAj for each j ∈ Z, then we can write ∞ ∞ X X fj (x). f (x) · χj (x) = f (x) = j=−∞

j=−∞

We have kT f kp

˙ α,λ (Rn ) MK p,q(·)

≤ C sup 2−k0 λp k0 ∈Z

k0 ∈Z

 X k0

+C sup 2−k0 λp k0 ∈Z

= sup 2−k0 λp

k=−∞

2kαp kT (f ) · χk kpLq(·) (Rn )

k=−∞

2kαp

k=−∞

 X k0

 X k0



k−2 X

kT (fj ) · χk kLq(·) (Rn )

j=−∞

k+1

p

 X 

2kαp T fj · χk q(·) j=k−1

5

L

p 

(Rn )





+C sup 2−k0 λp k0 ∈Z

 X k0

2kαp

k=−∞

∞  X

kT (fj ) · χk kLq(·) (Rn )

j=k+2

p 

= C(E1 + E2 + E3 ). First we estimate E2 . Applying the Lq(·) (Rn )-boundedness of T , we have E2

=

sup 2

−k0 λp

k0 ∈Z

≤

 X k0

C sup 2

−k0 λp

k0 ∈Z

≤

C sup 2−k0 λp

Lq(·) (Rn )

j=k−1

 X k0

2

 X k0

2kαp kf · χk kpLq(·) (Rn )

k=−∞

k0 ∈Z

=

2

k=−∞

p

 k+1  X

f j · χk

T

kαp

 k+1

p  X

fj · χk q(·)

kαp

L

j=k−1

k=−∞ p Ckf kM K˙ α,λ (Rn ) . p,q(·)



(Rn )





For E1 , we notice the facts that j ≤ k − 2 and a. e. x ∈ Ak with k ∈ Z, then using the size condition (3.3) and the generalized H¨ older’s inequality(see (2.3) in Lemma 2.1), we have |T (fj )(x)| ≤ C|x|−n kfj kL1 (Rn ) ≤ C2−kn kfj kL1 (Rn ) ≤ C2−kn kfj kLq(·) (Rn ) kχj kLq′ (·) (Rn ) .

(3.5)

Using Proposition 2.1, Proposition 2.2, Lemma 2.2, Lemma 2.3 and (3.1), we obtain 2−kn kχk kLq(·) (Rn ) kχj kLq′ (·) (Rn ) ≤ 2−kn kχBk kLq(·) (Rn ) kχBj kLq′ (·) (Rn ) −1 ′ ≤ CkχBk kL q′ (·) (Rn ) kχBj kLq (·) (Rn )

=C

kχBj kLq′ (·) (Rn ) kχBk kLq′ (·) (Rn )

(3.6)

≤ C2(j−k)nδ1 .

On the other hand, note the following fact  kfj kLq(·) (Rn ) = 2−jα 2jαp kfj kp q(·) L

≤2

−jα

 X j

2

iαp

(Rn )

1/p

kfi kpLq (·) (Rn )

i=−∞



= 2j(λ−α) 2−jλ

j  X

2iαp kfi kp q(·)

i=−∞

≤ C2j(λ−α) kf kM K˙ α,λ

p,q(·)

6

1/p

(Rn ) .

L

(3.7) (Rn )

1/p 

Therefore, combining (3.5), (3.6) and (3.7), and using α < λ + nδ1 , it follows that E1 ≤ C sup 2−k0 λp k0 ∈Z

≤ C sup 2

−k0 λp

≤

 X k0

−k0 λp

2

 X k0

 k−2 X

2−kn kfj kLq(·) (Rn ) kχj kLq′ (·) (Rn ) kχk kLq(·) (Rn )

j=−∞

kαp

 k−2 X

2

(j−k)nδ1

2

(j−k)nδ1 j(λ−α)

kfj kLq(·) (Rn )

j=−∞

k=−∞

k0 ∈Z

≤

2kαp

k=−∞

k0 ∈Z

≤ C sup 2

 X k0

2

kαp

 k−2 X

2

p 

kf kM K˙ α,λ

p,q(·)

j=−∞

k=−∞

Ckf kp ˙ α,λ n sup 2−k0 λp M Kp,q(·) (R ) k ∈Z 0

 X k0

Ckf kp ˙ α,λ n sup 2−k0 λp M Kp,q(·) (R ) k ∈Z 0

 X k0

2

kλp

 k−2 X

2

(Rn )

(j−k)(nδ1 +λ−α)

j=−∞

k=−∞

2

kλp

k=−∞



p 

≤ Ckf kp

˙ α,λ (Rn ) MK p,q(·)

p 

p 

.

Now, let us estimate E3 . For every j ≥ k + 2 and a. e. x ∈ Ak with k ∈ Z, applying the size condition (3.4) and the generalized H¨ older’s inequality(see (2.3) in Lemma 2.1), we have |T (fj )(x)| ≤ C2−jn kfj kL1 (Rn ) ≤ C2−jn kfj kLq(·) (Rn ) kχj kLq′ (·) (Rn ) .

(3.8)

Using Proposition 2.1, Lemma 2.2, Lemma 2.3 and (3.1), we obtain 2−jn kχk kLq(·) (Rn ) kχj kLq′ (·) (Rn ) ≤ 2−jn kχBk kLq(·) (Rn ) kχBj kLq′ (·) (Rn ) ≤ CkχBk kLq(·) (Rn ) kχBj k−1 Lq(·) (Rn ) =C

kχBk kLq(·) (Rn )

(3.9)

≤ C2(k−j)nδ2 .

kχBj kLq(·) (Rn )

Thus, combining (3.7), (3.8) and (3.9), and using α > λ − nδ2 , it follows that E3 = sup 2

−k0 λp

k0 ∈Z

 X k0

2

kαp

k=−∞

≤ C sup 2

−k0 λp

k0 ∈Z

∞  X

 X k0

2

 X k0

2kαp

 X k0

2kαp

kαp

k=−∞

≤ C sup 2−k0 λp k0 ∈Z

k0 ∈Z

k=−∞

≤ Ckf kp

∞  X

2

∞  X

2(k−j)nδ2 kfj kLq(·) (Rn )

∞  X

2(k−j)nδ2 2j(λ−α) kf kM K˙ α,λ

p,q(·)

p 

p,q(·)

j=k+2

sup 2−k0 λp

 X k0

sup 2−k0 λp

 X k0

(Rn ) k ∈Z 0

kfj kLq(·) (Rn ) kχj kLq′ (·) (Rn ) kχk kLq(·) (Rn )

j=k+2

˙ α,λ (Rn ) MK p,q(·) k0 ∈Z

≤ Ckf kpM K˙ α,λ

−jn

j=k+2

k=−∞

≤ C sup 2−k0 λp

kT (fj ) · χk kLq(·) (Rn )

j=k+2

p 

2kλp

∞  X

2kλp



k=−∞

2(j−k)(λ−α−nδ2 )

j=k+2

k=−∞

≤ Ckf kpM K˙ α,λ

p,q(·)

Combining the estimates for E1 , E2 and E3 yields kT f kM K˙ α,λ

p,q(·)

(Rn )

(Rn )

≤ Ckf kM K˙ α,λ

p,q(·)

and then completes the proof of Theorem 3.1. 7

(Rn )

(Rn )

.

p 

p 

p 

Now, let us turn to consider the fractional singular integrals. We first have the following theorem similar to 3.1. Theorem 3.2 Let q1 (·) ∈ P(Rn ) satisfies conditions (2.1) and (2.2) in Proposition 2.1. Define the variable exponent q2 (·) by 1 β 1 − = . q1 (x) q2 (x) n And let 0 < p1 ≤ p2 < ∞, λ > 0, 0 < β < n/(q1 )+ , λ − nδ4 < α < λ + nδ3 , where δ3 ∈ (0, 1/(q1′ )+ ) and δ4 ∈ (0, 1/(q2 )+ ) are the constants appearing in (3.2). Suppose that a sublinear operator Tβ satisfies (i) Tβ maps from L

q1 (·)

(Rn ) to L

q2 (·)

(Rn );

(ii) for any function f with supp f ⊂ Ak and any |x| ≥ 2k+1 with k ∈ Z, |Tβ (f )(x)| ≤ C|x|β−n kf kL1(Rn ) ;

(3.10)

(iii) for any function f with supp f ⊂ Ak and any |x| ≤ 2k−2 with k ∈ Z, |Tβ (f )(x)| ≤ C2k(β−n) kf kL1(Rn ) . Then Tβ (f ) is bounded from M K˙ pα,λ,q 1

1 (·)

(Rn ) to M K˙ pα,λ,q 2

2 (·)

(3.11)

(Rn ).

Notice that if Tβ (f ) satisfies the size condition Z |f (y)| |Tβ (f )(x)| ≤ C dy, |x − y|n−β n R

x∈ / supp f

(3.12)

for any integrable function f with compact support, then Tβ (f ) obviously satisfies the assumptions of Theorem 3.2. Therefore, by Theorem 3.2, we have Corollary 3.2 Let q1 (·), q2 (·), p1 , p2 , λ, β, α, δ3 and δ4 be the same as in Theorem 3.2. If a q1 q2 sublinear operator Tβ satisfies the size condition (3.12) and is bounded from L (Rn ) to L (Rn ), then Tβ is also bounded from M K˙ pα,λ,q (·) (Rn ) to M K˙ pα,λ,q (·) (Rn ). 1

1

2

2

Remark 2 We can see that when β = 0, Theorem 3.2 is just Theorem 3.1. In particular, if Tβ (f ) is a (standard) fractional integral, then Tβ (f ) obviously satisfies (3.12). The fractional maximal function Mβ (f ), defined by Z 1 |f (y)|dy, Mβ (f )(x) = sup 1−β/n B∋x |B| B∩Ω also satisfies the conditions of Theorem 3.2, where the supremum is again taken over all balls B which contain x, and Ω ⊂ Rn is an open set. Proof (Proof of Theorem 3.2) For all f ∈ M K˙ pα,λ,q (·) (Rn ). If we denote fj := f · χj = f · χAj for 1 1 each j ∈ Z, then we can write f (x) =

∞ X

f (x) · χj (x) =

∞ X

fj (x).

j=−∞

j=−∞

Because of 0 < p1 /p2 ≤ 1, we apply inequality  X ∞

i=−∞

|ai |

p1 /p2 8

≤

∞ X

i=−∞

|ai |p1 /p2 ,

(3.13)

and obtain p1

kTβ (f )kM K˙ α,λ

p ,q (·) 2 2

≤ C sup 2

−k0 λp1

k0 ∈Z

(Rn )

= sup 2

−k0 λp1

k0 ∈Z

 X k0

 X k0

≤ C sup 2 k0 ∈Z

 X k0

2

kαp1

2

kαp1

kTβ (f ) · χk k

+C sup 2 k0 ∈Z

+C sup 2

 X k0

2

k=−∞

−k0 λp1

k0 ∈Z

 k−2 X

p1 Lq2 (·) (Rn )

kTβ (fj ) · χk kLq2 (·) (Rn )

 X k0

2

p1

 k+1  X

f j · χk q

Tβ

kαp1 kαp1

k=−∞

L

j=k−1

∞  X

p2 Lq2 (·) (Rn )

p1 /p2



j=−∞

k=−∞

−k0 λp1

kTβ (f ) · χk k

k=−∞

k=−∞

−k0 λp1

2

kαp2

2

(·)

p1 

(Rn )

kTβ (fj ) · χk kLq2 (·) (Rn )

j=k+2

 p1 

= C(E1 + E2 + E3 ). For E2 , using the boundedness of Tβ from L E2 = sup 2

−k0 λp1

k0 ∈Z

≤ C sup 2

 X k0

2

k=−∞

−k0 λp1

k0 ∈Z

≤ C sup 2

q1 (·)

 X k0

2

k=−∞

−k0 λp1

k0 ∈Z

 X k0

2

(Rn ) to L

p1 ,q1 (·)

(Rn )

(Rn ), we have

p1

 k+1  X

f j · χk q

Tβ

kαp1

L

j=k−1

p1

 k+1  X

f j · χk q

kαp1 kαp1

L

j=k−1

kf · χk k

k=−∞

= Ckf kpM1 K˙ α,λ

q2 (·)

p1 Lq1 (·) (Rn )

1

(·)

2

(·)

(Rn )

(Rn )







.

For E1 . Note that j ≤ k − 2 and a. e. x ∈ Ak with k ∈ Z, then using the size condition (3.10) and the generalized H¨ older’s inequality(see (2.3) in Lemma 2.1), we have |Tβ (fj )(x)| ≤ C|x|β−n kfj kL1 (Rn ) ≤ C2k(β−n) kfj kL1 (Rn ) ≤ C2k(β−n) kfj kLq1 (·) (Rn ) kχj kLq1′ (·) (Rn ) .

(3.14)

Using Proposition 2.1, Proposition 2.2, Proposition 2.4, Lemma 2.2, Lemma 2.3 and (3.2), we obtain 2k(β−n) kχk kLq2 (·) (Rn ) kχj kLq1′ (·) (Rn ) ≤ C2−kn 2kβ kχBk kLq2 (·) (Rn ) kχBj kLq1′ (·) (Rn ) ≤ C2−kn kχBk kLq1 (·) (Rn ) kχBj kLq1′ (·) (Rn ) ≤ CkχBk k−1q′ (·) L

=C

9

1

(Rn )

kχBj kLq1′ (·) (Rn )

kχBj kLq1′ (·) (Rn ) kχBk kLq1′ (·) (Rn )

≤ C2(j−k)nδ3 .

(3.15)

On the other hand, note the following fact  kfj kLq1 (·) (Rn ) = 2−jα 2jαp1 kfj kp1q1 (·) L

≤2

−jα

 X j

2

iαp1

(Rn )

1/p1

kfi kpL1q1 (·) (Rn )

i=−∞

1/p1

 j  X −jλ j(λ−α) 2iαp1 kfi kp1q1 (·) 2 =2 L

i=−∞

≤ C2j(λ−α) kf kM K˙ α,λ

p1 ,q1 (·)

(3.16)

(Rn )

1/p1 

(Rn ) .

Hence, combining (3.14), (3.15) and (3.16), and using α < λ + nδ3 , it follows that E1 = sup 2

 X k0

−k0 λp1

k0 ∈Z

2

kαp1

k0 ∈Z

 X k0

≤ C sup 2−k0 λp1 k0 ∈Z

 k−2 X

2k(β−n) kfj kLq1 (·) (Rn ) kχj kLq1′ (·) (Rn ) kχk kLq2 (·) (Rn )

2kαp1

 k−2 X

2(j−k)nδ3 kfj kLq1 (·) (Rn )

j=−∞

k=−∞

≤ Ckf kp1

˙ α,λ MK p ,q 1

1 (·)

≤ Ckf kp1

˙ α,λ MK p ,q 1

1 (·)

≤ Ckf kp1

˙ α,λ MK p ,q 1

1 (·)

j=−∞

sup 2−k0 λp1

 X k0

2kαp1

sup 2−k0 λp1

 X k0

2kλp1

k0 X

2kλp1

(Rn ) k ∈Z 0

(Rn ) k ∈Z 0

(Rn )

p1 

2kαp1

k=−∞

 X k0

kTβ (fj ) · χk kLq2 (·) (Rn )

j=−∞

k=−∞

≤ C sup 2−k0 λp1

 k−2 X

sup 2−k0 λp1 k0 ∈Z

 k−2 X

2(j−k)nδ3 2j(λ−α)

 k−2 X

2(j−k)(λ−α+nδ3 )

j=−∞

k=−∞



p1 

j=−∞

k=−∞

k=−∞



p1 

≤ Ckf kp1

˙ α,λ MK p ,q 1

1 (·)

(Rn )

p1 

p1 

.

Now, let us estimate E3 . For every j ≥ k + 2 and a. e. x ∈ Ak with k ∈ Z, applying the size condition (3.11) and the generalized H¨ older’s inequality(see (2.3) in Lemma 2.1), we have |Tβ (fj )(x)| ≤ C2j(β−n) kfj kL1 (Rn ) ≤ C2j(β−n) kfj kLq1 (·) (Rn ) kχj kLq1′ (·) (Rn ) .

(3.17)

Using Proposition 2.1, Proposition 2.2, Proposition 2.4, Lemma 2.2, Lemma 2.3 and (3.2), we obtain 2j(β−n) kχk kLq2 (·) (Rn ) kχj kLq1′ (·) (Rn ) ≤ 2j(β−n) kχBk kLq2 (·) (Rn ) kχBj kLq1′ (·) (Rn ) ≤ CkχBk kLq2 (·) (Rn ) · 2jβ 2−jn kχBj kLq1′ (·) (Rn ) ≤ CkχBk kLq2 (·) (Rn ) · 2jβ kχBj k−1 Lq1 (·) (Rn ) ≤ CkχBk kLq2 (·) (Rn ) · =C

kχBk kLq2 (·) (Rn ) kχBj kLq2 (·) (Rn )

10

kχBj k−1 Lq2 (·) (Rn ) ≤ C2(k−j)nδ4 .

(3.18)

Therefore, combining (3.16), (3.17) and (3.18), and using α > λ − nδ4 , it follows that  X k0

E3 = sup 2−k0 λp1 k0 ∈Z

k=−∞

≤ C sup 2−k0 λp1 k0 ∈Z

≤ C sup 2

−k0 λp1

2kαp1

 X k0

kαp1

Ckf kpM1 K˙ α,λ p ,q

1 (·)

Ckf kpM1 K˙ α,λ p ,q 1

≤ Ckf kpM1 K˙ α,λ

1 (·)

p1 ,q1 (·)

kTβ (fj ) · χk kLq2 (·) (Rn )

j=k+2

 X k0

∞  X

p1 

2j(β−n) kfj kLq1 (·) (Rn ) kχj kLq1′ (·) (Rn ) kχk kLq2 (·) (Rn )

j=k+2

2

k=−∞

1

≤

∞  X

k=−∞

k0 ∈Z

≤

2kαp1

sup 2

(Rn ) k ∈Z 0

sup 2

(Rn ) k ∈Z 0

∞  X

kfj kLq1 (·) (Rn )

j=k+2

−k0 λp1

 X k0

2

kαp1

k=−∞

−k0 λp1

sup 2−k0 λp1

(Rn ) k ∈Z 0

2

(k−j)nδ4

∞  X

2

p1 

(k−j)nδ4 j(λ−α)

2

j=k+2

 X k0

2

k0  X

2kλp1

kλp1

k=−∞

∞  X

2

(k−j)(α−λ+nδ4 )

j=k+2

k=−∞



p1 

≤ Ckf kpM1 K˙ α,λ

p1 ,q1 (·)

(Rn )

p1 

p1 

.

Combining the estimates for E1 , E2 and E3 , we conclude that kTβ (f )kM K˙ α,λ

p ,q (·) 2 2

(Rn )

≤ Ckf kM K˙ α,λ

p ,q (·) 1 1

(Rn )

and then completes the proof of Theorem 3.2. Remark 3 It is easy to see that when λ = 0, the above results are also true on the Herz space with variable exponent, and containing some main results for [14]. Acknowledgments The author cordially thank the referees for their valuable suggestions and useful comments which have lead to the improvement of this paper. This work was supported by the Project (No. SY201313) of Mudanjiang Normal University and the Research Project (No.12531720) for Department of Education of Heilongjiang Province.

References [1] A. Almeida and D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl. 394 (2012), no. 2, 781–795. [2] A. Almeida, J. Hasanov and S. Samko, Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J. 15 (2008), no. 2, 195–208. [3] C. Capone, D. Cruz-Uribe and A. Fiorenza, The fractional maximal operator and fractional integrals on variable Lp spaces, Rev. Mat. Iberoam. 23 (2007), no. 3, 743–770. [4] D. Cruz-Uribe, L. Diening and A. Fiorenza,A new proof of the boundedness of maximal operators on variable Lebesgue spaces, Boll. Unione Mat. Ital. (9) 2 (2009), no. 1, 151–173. [5] D. Cruz-Uribe, A. Fiorenza, J. M. Martell and C. P´erez,The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239–264.

11

[6] D. Cruz-Uribe, A. Fiorenza and C. J. Neugebauer,The maximal function on variable Lp spaces,Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223–238. [7] L. Diening,Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and W k,p(·) , Math. Nachr. 268 (2004), 31–43. [8] L. Diening,Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces,Bull. Sci. Math. 129 (2005), no. 8, 657–700. [9] L. Diening,Maximal function on generalized Lebesgue spaces Lp(·) ,Math. Inequal. Appl. 7 (2004), no. 2, 245–253. [10] L. Diening, P. Harjulehto, P. H¨ ast¨ o, Y. Mizuta and T. Shimomura,Maximal functions in variable exponent spaces: limiting cases of the exponent,Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 503–522. [11] E. Hern´andez and D. Yang,Interpolation of Herz spaces and applications,Math. Nachr. 205 (1999), 69–87. [12] M. Izuki,Fractional integrals on Herz–Morrey spaces with variable exponent,Hiroshima Math. J. 40 (2010), no. 3, 343–355. [13] M. Izuki,Boundedness of vector-valued sublinear operators on Herz–Morrey spaces with variable exponent,Math. Sci. Res. J. 13 (2009), no. 10, 243–253. [14] M. Izuki,Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization,Anal. Math. 36 (2010), no. 1, 33–50. [15] T. Kopaliani,Infimal convolution and Muckenhoupt Ap(·) condition in variable Lp spaces,Arch. Math. (Basel) 89 (2007), no. 2, 185–192. [16] V. Kokilashvili and A. Meskhi,Boundedness of maximal and singular operators in Morrey spaces with variable exponent,Armen. J. Math. 1 (2008), no. 1, 18–28. [17] V. Kokilashvili and A. Meskhi,Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure,Complex Var. Elliptic Equ. 55 (2010), no. 8-10, 923–936. [18] O. Kov´aˇcik and J. R´ akosn´ık,On spaces Lp(x) and W k,p(x) ,Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. [19] A. K. Lerner,On some questions related to the maximal operator on variable Lp spaces,Trans. Amer. Math. Soc. 362 (2010), no. 8, 4229–4242. [20] X. Li and D. Yang,Boundedness of some sublinear operators on Herz spaces,Illinois J. Math. 40 (1996), no. 3, 484–501. [21] S. Lu and L. Xu,Boundedness of rough singular integral operators on the homogeneous Morrey-Herz spaces,Hokkaido Math. J. 34 (2005), no. 2, 299–314. [22] S. Lu and D. Yang,The decomposition of weighted Herz space on Rn and its applications,Sci. China Ser. A 38 (1995), no. 2, 147–158. [23] A. Nekvinda,Hardy-Littlewood maximal operator on Lp(x) (R), Math. Inequal. Appl. 7 (2004), no. 2, 255–265. 12

[24] L. Pick and M. Ru˘zi˘cka, An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded, Expo. Math. 19 (2001), no. 4, 369–371.

13