A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

Abstract

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces for and . More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from to . Here denotes the set of all functions of bounded variation on .

1. Introduction

The purpose of this paper is to present new results related to the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator, in both the continuous and discrete setting. We start with a recall of some recent developments on the general regularity theory of maximal operators. In 1997, Kinnunen [1] first studied the Sobolev regularity of the usual centered Hardy-Littlewood maximal function and showed that is bounded on the first-order Sobolev spaces for all (the same conclusion also holds for the uncentered version by a simple modification of Kinnunen’s arguments or [2, Theorem 1]). Later on, Kinnunen’s result was extended to a local version in [3], to a fractional version in [4] and to a multisublinear version in [5, 6] and to a one-sided version in [7]. Due to the lack of the sublinearity for the derivative of the maximal function, the continuity of for is certainly a nontrivial issue. This problem was addressed by Luiro [8] in the affirmative and was later extended to the local version in [9] and the multisublinear version in [5, 10].

Since the maximal operator is not of type , the -regularity for maximal operator is more delicate. A crucial question was posed by Hajłasz and Onninen in [2]: Is the operator bounded from to ? A complete solution was achieved only in dimension in [11–14] and partial progress on the general dimension was given by Hajłasz and Malý [15] and Luiro [16]. Tanaka [14] first observed that if , then is weakly differentiable and Inequality (1) with the sharp constant was later proved by Liu et al. in [13]. An important improvement of Tanaka’s result was given by Aldaz and Pérez Lázaro [11] who proved that if is of bounded variation on , then is absolutely continuous and Here denotes the total variation of . Notice that the constant in inequality (2) is sharp. Recently, inequality (2) was extended to a fractional setting in [17, Theorem 1] and to a multisublinear fractional setting in [18, Theorems 1.3-1.4]. In the centered setting, Kurka [12] showed that if is of bounded variation on , then inequality (2) holds for (with constant ). It was also shown in [12] that if , then is weakly differentiable and (1) holds for with constant . Very recently, Carneiro et al. [19] proved that the operator is continuous from to . It is currently unknown whether inequality (1) with also holds for and the map is also continuous from to . For other interesting works related to this theory, we refer the reader to consult [20, 21], among others.

In order to address the -regularity of the one-dimensional Hardy-Littlewood maximal function, Tanaka [14] first studied the -regularity of the one-sided Hardy-Littlewood maximal function and proved that if , then the distributional derivative of is integrable, and It is observed that is also absolutely continuous on by a combination of arguments in [13, 14]. We remark that maps into boundedly by similar arguments to those in [1, Remark (iii)]. Recently, Liu and Mao [7] proved the following result.

Theorem A (see [7]). Let . Then is bounded and continuous on .

In this paper we focus on the regularity of two-dimensional one-sided Hardy-Littlewood maximal operator. In 2011, Forzani et al. [22] first introduced and studied the weighted weak estimates for the two-dimensional one-sided Hardy-Littlewood maximal operator , which is defined by where . It is not difficult to see that This together with the bounds for implies that is of type for and of weak type . One can easily check that is a sublinear operator and commutes with translations. These facts together with Theorem 1 in [2] imply that maps boundedly with . On the other hand, by the arguments similar to those used in deriving [8, Theorem 4.1] we can obtain that maps continuously with . It was shown in [23, Lemma 2.2] that for all . Here . Combining (7) with (6) yields for all . On the other hand, one can easily check that and for arbitrary functions defined on . Here . By using (8)-(10) and the arguments similar to those used to derive [21, Theorems 1 and 3] we can obtain that is bounded and continuous on the inhomogeneous Triebel-Lizorkin spaces and the inhomogeneous Besov spaces for and . For other interesting works on this topic we refer the readers to consult [24–28]. We denote by the fractional Sobolev spaces defined by the Bessel potentials. Since for all and and , the above analyses lead to the following result.

Theorem 1. is bounded and continuous from to for all and .

Our main motivation of this paper is to investigate the regularity of the discrete version of , which is defined by where and . Let us recall some pertinent definitions, notations, and background. For a discrete function and , we define its -norm by and -norm by . We also define the total variation of on by where and . We denote by the set of functions of bounded variation defined on . We also denote by the set of functions of bounded variation defined on satisfying .

Recently, the investigation of the regularity properties of the discrete maximal operators has also attracted the attention of many authors. A good start was due to Bober et al. [29] in 2012 when they proved that if , and if . Here the operators and are the one-dimensional discrete centered and uncentered Hardy-Littlewood maximal operators, respectively, which are defined by We notice that inequality (13) is sharp. Subsequently, Temur [30] proved (13) for (with constant ) following Kurka’s breakthrough [12]. Inequality (14) is not optimal, and it was asked in [29] whether the sharp constant for inequality (14) is in fact ; this question was resolved in the affirmative by Madrid in [31]. Later on, the above results were extended to a fractional case in [17, 32, 33], to a one-sided case in [7], and to a high dimensional case in [34]. For other interesting works we can consult [19, 20, 35–37]. In particular, Liu and Mao [7] investigated the regularity of the one-dimensional discrete one-sided Hardy-Littlewood maximal operator and proved the following result.

Theorem B (see [7]). The operator is bounded and continuous from to . Moreover, if , then and the constant is the best possible.

Based on the above, it is natural to ask whether is bounded and continuous, which can be addressed by the following.

Theorem 2. The operator is bounded and continuous from to . In particular, if , then and the constant is the best possible one.

Remarks. (i) It should be pointed out that To see this, let us consider the characteristic function . It is clear that and . One can easily check that . This together with Theorem 2 yields the claim.

(ii) The main results of this paper can be extended to the general dimension by using similar arguments.

The rest of this paper is organized as follows. In Section 2 we shall prove Theorem 2. We remark that the proof of the boundedness part in Theorem 2 is motivated by the method in [31], but our proof is simpler and more direct than that of [31]. The proof of the continuity part in Theorem 2 relies on the previous boundedness result and an useful application of the Brezis-Lieb lemma in [38]. Throughout this paper, the letter , sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables.

2. Proof of Theorem 2

In this section we shall prove Theorem 2. For convenience, for any and a discrete function , we define the average function by

We shall divide the proof of Theorem 2 into two steps.

Step 1 (the boundedness part). Let . Without loss of generality we may assume that since and . To prove (18), it suffices to show that and We only prove (21) since (22) is analogous. Fix , and let Then (21) reduces to the following inequalities: We now prove (24). Since , then for any and , there exists such that . It follows that Note that, for any and , the following holds:This together with (26) yields which gives (24).
It remains to prove (25). We want to show that This can be seen by the following: since, for fixed , the following holds: Combining (29) with (24) yields (25).

Step 2 (the continuity part). Let in as . Without loss of generality we may assume that , for all , and that since . It suffices to show that since can be proved analogously.
We now prove (32). By the sublinearity of , we have which implies that pointwise as and for all . Applying (34) and the classical Brezis-Lieb lemma in [38], to get (32), it suffices to show that By (34) and Fatou’s lemma we get Thus, (35) reduces to the following: By Step 1 we have . Therefore, given , there exists an integer such that For the above , since in as , then there exists an integer such that for all . By (34) again, there exists a positive integer such that And (40) and (38) yield the following: for all . Therefore, (37) reduces to the following: Below we shall prove (42). Fix . We can write We only estimate since is analogous. Fix with . We denote Then we can write Since , then for any with and , there exists such that . Then we can write Observe that which together with (46) and (38)-(39) yields the following: We now estimate . Since , then, for any with and , there exists such that . Then we have By similar arguments to those in deriving (48) we can obtain On the other hand, we get from (38) and (39) that It follows from (49)-(51) that Combining (52) with (45) and (48) implies that Similarly, we can get Also (43) together with (53)-(54) yields (42) and completes the proof of Theorem 2.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

The authors read and approved the final manuscript.

Acknowledgments

This work was partially supported by the NNSF of China (Grant no. 11701333) and Support Program for Outstanding Young Scientific and Technological Top-Notch Talents of College of Mathematics and Systems Science (Grant no. Sxy2016k01).