#### Abstract

In this paper, the regularity properties of two classes of commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants are investigated. Some new bounds for the derivatives of the above commutators and the boundedness and continuity for the above commutators on the Sobolev spaces will be presented. The corresponding results for the discrete analogues are also considered.

#### 1. Introduction

The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis. One of the driving questions in this theory is whether a given maximal operator improves, preserves, or destroys a priori regularity of an initial datum . The question was first studied by Kinnunen [1], who showed that the usual centered Hardy-Littlewood maximal function is bounded on the first order Sobolev spaces for all . Recall that the Sobolev spaces , , are defined by
where is the weak gradient of . It was noted that the -bound for the uncentered maximal operator also holds by a simple modification of Kinnunen’s arguments or Theorem 1 of [12]. Later on, Kinnunen’s result was extended to a local version in [2], to a fractional version in [3], to a multisublinear version in [4, 5], and to a one-sided version in [6]. Due to the lack of sublinearity for at the derivative level, the continuity of for is certainly a nontrivial issue. This problem was addressed by Luiro [7] in the affirmative and was later extended to the local version in [8] and the multisublinear version in [4, 9]. Other works on the regularity of maximal operators can be consulted in [10, 11]. Since the map is not bounded, the -regularity for the maximal operator seems to be a deeper issue. A crucial question was posed by Hajłasz and Onninen in [12]: *Is the map**bounded from**to**?* A complete solution was obtained only in dimension (see [13–16] for an example), and partial progress on the general dimension was given by Hajłasz and Malý [17] and Luiro [18]. For other interesting works related to this theory, we suggest the readers to consult [19–22], among others.

Very recently, Liu et al. [23] investigated the regularity of commutators of the Hardy-Littlewood maximal function. Precisely, let be a locally integrable function defined on , we define the commutator of the Hardy-Littlewood maximal function by

The maximal commutator of with is defined by where is the open ball in centered at with radius and volume .

We now list the main result of [23] as follows:

Theorem 1 (see [23]). *Let and . If , then
*(i)*The map is bounded and continuous. In particular, if , then
for almost every . Moreover,
*(ii)*The map is bounded. Moreover, if , then
*

The main motivations of this work not only extend Theorem 1 to a one-sided setting but also investigate the regularity properties of the discrete analogue for commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants. Let us recall some definitions and backgrounds. For , the one-sided fractional maximal operators and are defined by

When , the operators (resp., ) reduce to the one-sided Hardy-Littlewood maximal functions (resp., ). The study of the one-sided maximal operators originated ergodic maximal operator (see [24]). The one-sided fractional maximal operators have a close connection with the well-known Riemann-Liouville fractional integral operator and the Weyl fractional integral operator (see [25]). It was known that is of type for , and . For we have bounded. The same conclusions hold for .

In order to establish the -regularity for the one-dimensional uncentered Hardy-Littlewood maximal function, Tanaka [16] first studied the regularity of and . Precisely, Tanaka proved that if , then the distributional derivatives of and are integrable functions, and

A combination of arguments in [15, 16] yields that both and are absolutely continuous on . Later on, Liu and Mao [6] proved that both and map boundedly and continuously for . Similar arguments to those in Remark (iii) in [1] can be used to conclude that both and map boundedly. Recently, the main result of [6] was extended to the fractional version in [26] and to the multilinear case in [27]. We now introduce the partial result of [26] as follows:

Theorem 2 (see [26]). *Let , , and . Then, the map is bounded and continuous. Moreover, if , then
for almost every . The same conclusions hold for the operator .*

Now we introduce two classes of commutators of the one-sided fractional maximal functions.

*Definition 3. *Let be a locally integrable function and . The commutators of the one-sided fractional maximal function and are defined by

*Definition 4. *Let be a locally integrable function and . The maximal commutators of and with are defined, respectively, by
It should be pointed out that the following facts are useful in proving our main results.

*Remark 5. *(i) The operator is neither positive nor sublinear. By Hölder’s inequality and the -bounds and continuity for , we have that the map is bounded and continuous, provided that , , , , and . Moreover,
The same conclusions also hold for .

(ii) The operator is positive and sublinear. Clearly
Inequality (13) together with Hölder’s inequality, the bounds, and sublinearity of yields that the map is bounded if , , , , and . Moreover,
The same conclusions also hold for .

Based on the above, it is a natural question to ask whether the commutators , , , and have somewhat regularity properties. This is one main motivation of this paper, which can be addressed by the following results.

Theorem 6. *Let , , , and . If , then the map is bounded and continuous. In particular, if , it holds that
for almost every . Moreover,
**The same conclusions also hold for the operator .*

Theorem 7. *Let , , , and . If and , then
for almost every . Moreover,
**The same conclusions also hold for the operator .*

On the other hand, the investigation on the regularity of discrete maximal operators has also attracted the attention of many authors (see [6, 19, 28–33]). Let and be a discrete function, we define the -norm and the -norm of by

Formally, we define the discrete analogue of the Sobolev spaces by where is the first derivative of . It is clear that

Estimate (21) implies that the discrete Sobolev space is just the classical with an equivalent norm. Hence, the () regularity for discrete maximal operators is trivial. However, the situation is highly nontrivial. We define the total variation of by

We also write for the variation of on the interval , where and are integers (or possibly , or ). It is clear that . Denote by the set of functions of bounded variation defined on , which is a Banach space with the norm where . Clearly,

The study of regularity properties of discrete maximal operators began with Bober et al. [28] who studied the endpoint regularity of one dimensional discrete centered and uncentered Hardy-Littlewood maximal operators and , which are defined by where . It was shown in [28] that

It was noted that inequality (27) is sharp and inequality (27) for was proven by Temur in [33] (with constant ). Inequality (28) was improved by Madrid [32] who obtained the sharp constant . Recently, Carneiro and Madrid [19] extended (28) to the fractional setting and showed that if , , and is a discrete function such that and , then where is the discrete uncentered fractional maximal operator defined by

It is currently unknown whether inequality (29) holds for the discrete centered fractional maximal operator

It was pointed out in [30] that both the maps and (for ) are bounded and continuous from to . Moreover, if , then and the constants are the best possible. Liu [30] pointed out that the operator is not bounded from to for all . The continuity of and was proven by Madrid [34] and Carneiro et al. [20], respectively. Recently, Liu and Mao [6] studied the regularity of the discrete one-sided Hardy-Littlewood maximal operators and proved them.

Theorem 8 (see [6]). *Let be a discrete function such that , then
**Moreover, the map is continuous from to . The same results also hold for . Here
**Very recently, Liu [26] extended Theorem 8 to the fractional setting.*

Theorem 9 (see [26]). *Let . Then, the map is bounded and continuous from to. Moreover, if , then
and the constant is the best possible. The same results hold for . Here
*

*The second aim of this paper is to study the regularity of the discrete analogues of and . Let us introduce some definitions.*

*Definition 10. *Let be a discrete function and . The commutators of the discrete one-sided fractional maximal function and are defined by

*Definition 11. *Let be a discrete function. For , the maximal commutators of and with are defined, respectively, by
We now formulate the rest of main results as follows:

Theorem 12. *Let . Then, the map is bounded. Moreover,
**The same conclusions also hold for the operator .*

Theorem 13. *Let and . Then, the map is bounded and continuous. Moreover, for any , it holds that
**The same conclusions also hold for the operator .*

Theorem 14. *Let be a discrete function such that and . Then, the map is bounded and continuous from to . Moreover,
**The same conclusions also hold for the operator .*

This paper will be organized as follows. Section 2 is devoted to proving Theorems 6 and 7. The proofs of Theorems 12–14 will be given in Section 3. We remark that the proofs of Theorems 6 and 7 are motivated by [21, 28]. The main ideas in the proofs of Theorems 12–14 are motivated by [6, 26], but some techniques are needed. In particular, in the proof of the continuity part of Theorem 14, we give a useful application of the Brezis-Lieb lemma in [35].

Throughout this paper, the letter will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables. In particular, the letter denotes the positive constants that depend on the parameters .

#### 2. Proofs of Theorems 6 and 7

In this section, we shall prove Theorems 6 and 7. Before giving our proofs, let us give some notations and lemmas. Let with . For all with , we define

For convenience, we set

According to Section 7.11 in [36], one has that for ,

Moreover, for functions for , we have (see [36], Section 7.11) that in when .

In order to prove Theorems 6 and 7, we need the following lemma, which follows from [23].

Lemma 15. (see [23]). *Let and . If and , then . Moreover,
for almost every . In particular, it holds that
**Now we are in a position to prove Theorems 6 and 7.*

*Proof of Theorem 6. *We only prove Theorem 6 for since another case can be obtained similarly. Let , , , , , and . Let be such that and . It is clear that , , and .
(i)We first prove the bounds for . By Theorem 2, we have . By Lemma 15, we have that and . By Theorem 2 again, we see that . Hence, (ii)We now prove (16). Applying Theorem 2, one has
which together with Lemma 15 yields that
By Lemma 15 again, it holds that
which together with Theorem 2 implies that
Combining (50) with (48) leads to (16).
(iii)We now prove the continuity part. Let in . It suffices to show thatBy Lemma 15 and applying the continuity result in Theorem 2, one can get
Combining (53) with the continuity result in Theorem 2 implies that
Then, (51) follows from (52) and (54).
(iv)It remains to prove (15). By Lemma 15, we havefor almost every . By Theorem 2, one can get
for almost every . It follows from (55)–(58) that
for almost every . This proves (15) and completes the Proof of Theorem 6.

*Proof of Theorem 7. *We only prove Theorem 7 for since another case can be obtained similarly. Let , , , , , and . Let be such that and . It is clear that , and .
(i)We first prove that . Fix , it is easy to see thatwhich gives that
By (61), Hölder’s inequality, and the bounds for and , one can get
Because of and , then by (44), we can get
It follows from (62) and (63) that
Combining (64) with (44) and the bounds for yields .
(ii)We now prove (17). Since , and , then in as , and in as , in as , in as , in as , in as , in when . Therefore, there exists a sequence of real numbers satisfying and a measurable set satisfying such that , , , , and as for all . From (61) and (13) we have that for all It follows that
for any , which gives (17).
(iii)By (17), the bounds for , and Hölder’s inequality, one can getwhich together with (14) yields (18).

#### 3. Proofs of Theorems 12–14

This section is devoted to presenting the proofs of Theorems 12–14.

*Proof of Theorem 12. *It is clear that
for all . By (68) one has
By (69) and Theorem 8, it holds that