Littlewood-Paley Operators on Morrey Spaces with Variable Exponent

By applying the vector-valued inequalities for the Littlewood-Paley operators and their commutators on Lebesgue spaces with variable exponent, the boundedness of the Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g μ*-functions, and their commutators generated by BMO functions, is obtained on the Morrey spaces with variable exponent.

As = 1, we denote , ( ) as ( ). Now let us turn to the introduction of the other two Littlewood-Paley operators. It is well known that the Littlewood-Paley operators include also the Littlewood-Paley -functions and the Littlewood-Paley * -functions besides the Lusin area integrals. The Littlewood-Paley -functions, which can be viewed as a "zero-aperture" version of , and * -functions, which can be viewed as an "infinite-aperture" version of , are, respectively, defined by If we take to be the Poisson kernel, then the functions defined above are the classical Littlewood-Paley operators. Letting ∈ 1 loc (R ), ≥ 1, the corresponding -order commutators of Littlewood-Paley operators above generated by a function are defined by 2 The Scientific World Journal where > 0. The Littlewood-Paley operators are a class of important integral operators. Due to the fact that they play very important roles in harmonic analysis, PDE, and the other fields (see [1][2][3]), people pay much more attention to this class of operators. In 1995, Lu and Yang investigated the behavior of Littlewood-Paley operators in the space CBMO p (R ) in [4]. In 2005, Zhang and Liu proved the commutator [ , ] is bounded on ( ) in [5]. In 2009, Xue and Ding gave the weighted estimate for Littlewood-Paley operators and their commutators (see [6]). There are some other results about Littlewood-Paley operators in [7][8][9] and so forth.
On the other hand, Lebesgue spaces with variable exponent (⋅) (R ) become one class of important research subject in analysis filed due to the fundamental paper [10] by Kováčik and Rákosník. In the past twenty years, the theory of these spaces has made progress rapidly, and the study of which has many applications in fluid dynamics, elasticity, calculus of variations, and differential equations with nonstandard growth conditions (see [11][12][13][14][15]). In [16], Cruz-Uribe et al. stated that the extrapolation theorem leads the boundedness of some classical operators including the commutator on (⋅) (R ). Karlovich and Lerner also independently obtained the boundedness of the singular integrals commutator on Lebesgue spaces with variable exponent in [17]. In 2009 and 2010, Izuki considered the boundedness of vector-valued sublinear operators and fractional integrals on Herz-Morrey spaces with variable exponent in [18,19], respectively. In 2013, Ho in [20] introduced a class of Morrey spaces with variable exponent M (⋅), and studied the boundedness of the fractional integral operators on M (⋅), .
Inspired by the results mentioned previously, in this paper we will consider the vector-valued inequalities of the Littlewood-Paley operators and their -order commutators on Morrey spaces with variable exponent. Before stating our main results, we need to recall some relevant definitions and notations.
Let be a Lebesgue measurable set in R with measure | | > 0.
The Lebesgue space with variable exponent (⋅) ( ) is defined by The space for all compact subsets ⊂ } .
The Lebesgue space (⋅) ( ) is a Banach space with the norm defined by (1) Note that if the function ( ) = 0 is a constant function, then (⋅) (R ) equals 0 (R ). This implies that the Lebesgue spaces with variable exponent generalize the usual Lebesgue spaces. And they have many properties in common with the usual Lebesgue spaces.
(3) The Hardy-Littlewood maximal operator is defined by Denote B( ) to be the set of all functions (⋅) ∈ P( ) satisfying the condition that is bounded on (⋅) ( ).
The Scientific World Journal 3 Definition 4 (see [20]). Let ( ) ∈ B(R ), ( , ) ∈ W (⋅) . Then the Morrey spaces with variable exponent M (⋅), (R ) are defined by where (2) Notice that if ( ) ≡ , 1 < < ∞, is a constant function, then formula (12) can be rewritten as an integral in form. To be precise, formula (12) can be rewritten in the following form (see [20]): Let 0 < < . By the the conditions of Morrey weight functions mentioned in [21] ∫ ∞ ( , ) and Hölder's inequality, via simple calculation, we have From this, it follows that if ( ) ≡ , 1 < < ∞, is a constant function, then condition (12) is weaker than condition (16). Thus, the class of the Morrey spaces introduced in Definition 4 is more wide than that satisfying condition (1.8) in [21]. More studies of common Morrey spaces can be seen in [22,23] and so forth.

Theorem 6. Suppose that function
Theorem 7. Suppose that is defined by (3). Then under the same condition as the one in Theorem 6, there exists a constant > 0 independent of such that, for any function Theorem 8. Suppose that * is defined by (4) and > 3 + 2( + )/ , 0 < < . Then under the same condition as the one in Theorem 6, there exists a constant > 0 independent of such that, for any function sequences For commutators [ , , ], [ , ], and [ , * ], we have the following results.
Theorem 9. Suppose that function ∈ 1 (R ) satisfies (i)-(iii) and [ , , ] is defined by (5). Let ∈ BMO(R ), (⋅) ∈ B(R ), ≥ 1, 1 < < ∞. If, for any ∈ R and 0 > 0, function satisfies then there exists a constant > 0 independent of such that, for any function sequences Theorem 10. Suppose that [ , ] is defined by (6). Then under the same condition as the one in Theorem 9, there exists 4 The Scientific World Journal a constant > 0 independent of such that, for any function Theorem 11. Suppose that [ , * ] is defined by (7), > 3 + 2( + )/ , and 0 < < . Then under the same condition as the one in Theorem 9, there exists a constant > 0 independent of such that, for any function sequences Remark 12. (1) It is easy to see that condition (22) in Theorem 9 is stronger than condition (12) in Definition 3. Therefore, if a function satisfies condition (22), then ∈ W (⋅) .

Preliminary Lemmas
In this section, we introduce some conclusions which will be used in the proofs of our main results.

Proofs of Main Results
Next, let us show the proofs of Theorems 6-11, respectively.
Proof of Theorem 6. Let ‖{ ℎ } ℎ ‖ ∈ M (⋅), (R ); for any 0 ∈ R , 0 > 0, denote Noting that, in order to prove Theorem 6, it is enough to show that the following inequality holds: Thus, For the term 1 , notice that supp 0 ℎ ⊂ ( 0 , 2 0 ); using Lemma 18 and (32), it is easy to see that We now turn to estimate 2 . To do this, we need to consider , first. Without loss of generality, we may assume that ≥ 1. Let 6 The Scientific World Journal Then, by Minkowski's inequality, we have Therefore, it follows from condition (ii) that On the other hand, we denote Note that if > 2 +1 0 , then > | − |/ ≥ (| | − | |)/ > (2 +1 0 − 0 )/ ≥ 2 0 / . Thus, by condition (iii), we get And by condition (ii), similar to the estimate of , we obtain Hence, from the estimates above, it follows that Thus, The Scientific World Journal 7 Therefore, applying the generalized Hölder's inequality, Lemma 16,and (12), we have Adding up the estimates of 1 , 2 , we obtain This completes the proof of Theorem 6.

Now let us prove Theorems 7 and 8 in brief.
Proof of Theorem 7. For , similar to the estimate of , , via a simple calculation, we get that (see [26]) if ∈ ( 0 , 0 ), Hence, similar to the proof of Theorem 6, it follows from inequality (2) in Lemma 18 that This accomplishes the proof of Theorem 7.
Proof of Theorem 8. For * , by the definitions of , and * , we have * ( ) ( ) According to the estimate of , in the proof of Theorem 6, we know that if Thus, as > 3 + 2( + )/ , we obtain * ( ℎ ) ( ) ≤ 3 /2+ + (1 + Hence, also similar to the proof of Theorem 6, and from inequality (3) in Lemma 18, it follows that This finishes the proof of Theorem 8.
For any 0 ∈ R , 0 > 0, denote The Scientific World Journal To finish the proof of Theorem 9, we only need to prove Thus, For the term 1 , notice that supp 0 ℎ ⊂ ( 0 , 2 0 ); by Lemma 19 and inequality (32), we have Now we turn to estimate 2 . According to the estimate of , in the proof of Theorem 6, we see that if ∈ ( 0 , 0 ), ∈ ( 0 , 2 +1 0 ) \ ( 0 , 2 0 ), ≥ 1, then Therefore, Thus, Using Hölder's inequality and Lemma 17, we get And then, it follows from Lemma 16 and (22) that Hence, Adding up the results of 1 , 2 , we have The proof of Theorem 9 is accomplished.
Hence, similar to the proof of Theorem 9, and from inequality (2) in Lemma 19, it follows that .
The proof of Theorem 10 is completed.
Hence, also similar to the proof of Theorem 9, it follows from inequality (3) in Lemma 19 that .
The proof of Theorem 11 is accomplished.

Conflict of Interests
The authors declare that they have no conflict of interests.