Two-Weight Norm Inequality for the One-Sided Hardy-Littlewood Maximal Operators in Variable Lebesgue Spaces

Abstract

The authors establish the two-weight norm inequalities for the one-sided Hardy-Littlewood maximal operators in variable Lebesgue spaces. As application, they obtain the two-weight norm inequalities of variable Riemann-Liouville operator and variable Weyl operator in variable Lebesgue spaces on bounded intervals.

1. Introduction and Main Results

The one-sided Hardy-Littlewood maximal operators and are defined bySawyer [1] showed that is bounded from to if the pairs of nonnegative functions satisfy Sawyer-type two-weight condition for the one-sided maximal operator. In [2] Martín-Reyes et al. generalized this result to , wherewith is positive locally integrable function. The similar results are also true for and (see [1–3]).

Let be a measurable set in . Given a measurable function , we denoteIf , we will simply note that and

Definition 1. Given , is a locally integrable function such that . We say that if there exists a constant such that for every , with ,

Definition 2 (see [4]). It is given that We say that , if there exist constants and such that for all

Let The variable Lebesgue space is the set of measurable functions on such thatThis is a Banach space (see [4–7]) with the normIf , we will write as . The variable Lebesgue space is the special case of the Musielak-Orlicz space (see [7, 8]). For the detail of we refer to [4–7] and so on.

Let be a nonnegative locally integrable function on . The weighted variable Lebesgue space is the set of measurable functions on such that and . When is a constant, coincides with the classical weighted Lebesgue space .

Edmunds et al. [9] investigated the boundedness of and in the variable Lebesgue spaces . The two-weight weak type modular inequalities of and in were discussed in [10]. In [11], Kokilashvili et al. acquired the sufficient condition such that and are bounded from to , where is constant on some interval and is bounded in with

Throughout this paper, and are nonnegative locally integrable functions and is a positive constant whose value may change from one occurrence to the next. For exponent function with , its conjugate exponent will be denoted by with . For a Lebesgue measurable set , will be its characteristic function.

Definition 3. It is given that such that . We say that if there exists a constant such that for every interval , with ,where and .

Definition 4. It is given that such that . We say that if there exist constants and such that for every interval , with ,where and .

and can be considered to be the generalization of Sawyer-type two-weight condition (see [1]) for the one-sided maximal operator in variable exponents case. If we will simply write as and as . We can define , , and in similar ways.

Our main results are the following theorems.

Theorem 5. Let , , and such that . If and with , then

Theorem 6. Let , , and such that . If and with , then

Theorem 7. Let be a bounded interval and let such that . If and with , then

Corollary 8. It is given that such that . If there exists a constant such that and with , then

Corollary 9. It is given that such that , , and with . If(a)there exists a constant such that and for ,(b)there exist constants and such that for every interval , with ,Then

Remark 10. In Theorems 5 and 6, the set is not empty. In fact, if and , when , then .

Remark 11. If we change the conditions , , to , , and , respectively, in above theorems, we will obtain similar results of .

Remark 12. If we take and for whenever is an open interval, then (see Definition 18 below) and the results of this paper coincide with those of [9].

Remark 13. The Sawyer-type condition was used earlier in [12] to characterize the two-weight boundedness of the classical Hardy-Littlewood maximal operator . Corresponding results for variable Lebesgue spaces can be found in [13–15].

2. Proof of the Main Results

In order to establish our main results, we will need following lemmas.

Lemma 14 (see [4]). It is given that and such that . (a)If , then .(b)If , then .In particular, if , then

Lemma 15 (see [4]). If , the set of bounded functions with compact support is dense in .

Lemma 16 (see [1]). Suppose is integrable function with compact support on . If is a component interval of the open set with , then

Lemma 17 (see [16]). It is given that a set and two exponents and such that Then for every there exists a constant such that for all functions with ,where is a given nonnegative measure.

Proof of Theorem 5. By the homogeneity, Lemma 15, the Fatou lemma for variable Lebesgue spaces [4] and Lemma 14, we only need to provefor the nonnegative bounded function with compact support and . Let be a positive integer and for , define Obviously each is a bounded open set. Let be the component intervals of , where is an integer. Applying Lemma 16 to a fixed , we haveLet , then the sets are pairwise disjoint and for every Therefore whereWe show that for every and the inequalityis valid, where , ( is the same as that in (10)), and is independent of and .
Let , then . By and Lemma 14 we get Let , wherethenSince , by (27) we can getfor . ThenIf , by (4), we haveIf , thenThereforeThus, by the Hölder inequality and we can get Next, we will estimate . If , by , we have On the other hand, if , noticing that andthusApplying (38), and the Hölder inequality,By and Lemma 17, we getHence, combining (36) and (40)This completes the proof of (26) by (29), (35), and (41). Applying (26) to (24), we obtain Let be the following linear operator:and . Since , if we show that the operator is bounded from to , we obtainIt is easy to see that is bounded on . Therefore, we only need to show that is bounded from to to complete our proof by the Marcinkiewicz interpolation theorem. For arbitrary and each pair , defineObviously, are pairwise disjoint. Let and . It is clear that any two intervals of are disjoint or one is contained in the other. By the definition of , we also haveLet be the maximal elements of the family . These maximal elements exist since the intervals have uniformly bounded length. The intervals also satisfy (46). Then, by (9) and (46), we obtain This has proved the weak inequality for . Hence the estimate for is completed.
Since linear operator is bounded from to , by the second inequality of (10),Similarly, by (10), we have Therefore, by (42), (44), (48), and (49), we havewhere is independent of . Let tend to infinity and the proof of Theorem 5 is finished.

Theorem 6 can be proved similarly.

Proof of Theorem 7. We can assume to be a bounded open interval and to be nonnegative with . It is sufficient to prove thatFor , defineLet be the component intervals of and , where is an integer. Using the same procedure as (24), we obtainwhereLet and . The estimate for is the same as (35). Since , we haveCombining (53) and (55), we obtainSimilarly as we can estimate to get . Applying (9) to , we have

The Corollary 8 can be obtained by the results of Theorems 5, 6, and 7 directly.

Proof of Corollary 9. Without loss of generality, we can assume that is nonnegative and bounded with a compact support and . Let . Due to Theorem 7 we only need to proveLet be a positive integer and . For , defineLet be the component intervals of and , where is an integer. Using the same procedure as (24), we obtain where and have the similar definitions as those in the proof of Theorem 5. Let and . The estimate for is also the same as (35). Let , then and for . Since for every and , by the Hölder inequality and Lemma 17,By (35) and (61), we get Therefore we have By the similar estimates of and , we get andLet tend to infinity and the proof is complete.

3. Applications

In this section, we assume that with Letwhere . If is a constant function, is the classical Riemann-Liouville operator and is the classical Weyl operator. In [17], Andersen and Sawyer obtained the two weighted norm inequalities of and from to . Other results about and on can be seen in [18–21] and so forth.

Edmunds et al. [9] studied the boundedness of , , , and in variable Lebesgue spaces. Kokilashvili et al. [11] discussed the two weighted norm inequalities of and from to . In this section, we will discuss the two-weight inequalities of and in .

Definition 18 (see [9]). Given , we say that if there exists a constant such that for all , with , We also say that if there exists a constant such that for all with ,

Our results in this section are the following theorems:

Theorem 19. It is given that such that and . If and , where and , then

Theorem 20. Given such that and . If and , where and , then

Remark 21. The sets and are not empty. Let (resp., ). If is lower Ahlfors -regular (), which means there exists a constant such that for every bounded interval , then (resp., ).

In order to prove Theorem 19, we need the following lemma.

Lemma 22 (see [9]). It is given that , , and . If , then there exists a positive constant depending only on and such that

Proof of Theorem 19. By the homogeneity of norm, we can assume that and . It is sufficient to proveApplying (70) and Theorem 7 we have

Using the similar proving method as that of Lemma   in [9], we can prove the following lemma.

Lemma 23. It is given that , , and . If , then there exists a positive constant depending only on and such that

By this lemma and the two weighted results of , we can get the result of Theorem 20 directly.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research is supported by NNSF-China (Grants nos. 11171345 and 51234005).