Regularity of Commutators of the One-Sided Hardy-Littlewood Maximal Functions

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.


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 f . The question was first studied by Kinnunen [1], who showed that the usual centered Hardy-Littlewood maximal function M is bounded on the first order Sobolev spaces W 1,p ðℝ d Þ for all 1 < p ≤ ∞. Recall that the Sobolev spaces W 1,p ðℝ d Þ, 1 ≤ p ≤ ∞, are defined by where ∇f is the weak gradient of f . It was noted that the W 1,p -bound for the uncentered maximal operatorM 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 M at the derivative level, the continuity of M : W 1,p ðℝ d Þ ⟶ W 1,p ðℝ d Þ for 1 < p < ∞ 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 M : L 1 ðℝ d Þ ⟶ L 1 ðℝ d Þ is not bounded, the W 1,1 -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 f ↦ j∇Mf j bounded from W 1,1 ðℝ d Þ to L 1 ðℝ d Þ ? A complete solution was obtained only in dimension d = 1 (see [13][14][15][16] for an example), and partial progress on the general dimension d ≥ 2 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][20][21][22], among others. Very recently, Liu et al. [23] investigated the regularity of commutators of the Hardy-Littlewood maximal function. Precisely, let b be a locally integrable function defined on ℝ n , we define the commutator of the Hardy-Littlewood maximal function ½b, M by The maximal commutator of M with b is defined by where Bðx, rÞ is the open ball in ℝ d centered at x with radius r and volume jBðx, rÞj.
We now list the main result of [23] as follows: Theorem 1 (see [23] for almost every 1 : ð5Þ 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 0 ≤ β < 1, the one-sided fractional maximal operators M + β and M − β are defined by When β = 0, the operators M + β (resp., M − β ) reduce to the one-sided Hardy-Littlewood maximal functions M + (resp., M − ). 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 M + β is of type ðp, qÞ for 1 < p < ∞, 0 ≤ β < 1/p and q = p/ð1 − pβÞ. For p = 1 we have M + β : L 1 ðℝÞ ⟶ L 1/ð1−βÞ,∞ ðℝÞ bounded. The same conclusions hold for M − β . In order to establish the W 1,1 -regularity for the onedimensional uncentered Hardy-Littlewood maximal function, Tanaka [16] first studied the regularity of M + and M − . Precisely, Tanaka proved that if f ∈ W 1,1 ðℝÞ, then the distributional derivatives of M + f and M − f are integrable functions, and A combination of arguments in [15,16] yields that both M + f and M − f are absolutely continuous on ℝ. Later on, Liu and Mao [6] proved that both M + and M − map W 1,p ðℝÞ ⟶ W 1,p ðℝÞ boundedly and continuously for 1 < p < ∞. Similar arguments to those in Remark (iii) in [1] can be used to conclude that both M + and M − map W 1,∞ ðℝÞ ⟶ W 1,∞ ðℝÞ 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] for almost every x ∈ ℝ. The same conclusions hold for the operator M − β .
Now we introduce two classes of commutators of the one-sided fractional maximal functions.
It should be pointed out that the following facts are useful in proving our main results.
Remark 5. (i) The operator ½b, M + β is neither positive nor sublinear. By Hölder's inequality and the L p -bounds and continuity for M + β , we have that the map ½b, M + β : L p 1 ðℝÞ ⟶ L q ðℝÞ is bounded and continuous, provided that 1 < p 1 , p 2 ,

2
Journal of Function Spaces The same conclusions also hold for ½b, M − β . (ii) The operator M + b,β is positive and sublinear. Clearly Inequality (13) together with Hölder's inequality, the bounds, and sublinearity of M + β yields that the map M + b,β : The same conclusions also hold for M − b,β . Based on the above, it is a natural question to ask whether the commutators ½b, properties. This is one main motivation of this paper, which can be addressed by the following results.
for almost every x ∈ ℝ. Moreover, The same conclusions also hold for the operator ½b, M − β .
for almost every x ∈ ℝ. Moreover, The same conclusions also hold for the operator M − b,β .
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][29][30][31][32][33]). Let 1 ≤ p < ∞ and f : ℤ ⟶ ℝ be a discrete function, we define the ℓ p -norm and the ℓ ∞ -norm of f by Formally, we define the discrete analogue of the Sobolev spaces by where Estimate (21) implies that the discrete Sobolev space W 1,p ðℤÞ is just the classical ℓ p ðℤÞ with an equivalent norm. Hence, the W 1,p ðℤÞ (1 < p≤∞) regularity for discrete maximal operators is trivial. However, the situation p = 1 is highly nontrivial. We define the total variation of f by We also write for 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 where ℕ = f0, 1, 2, 3, ⋯g. It was shown in [28] that Var Mf It was noted that inequality (27) is sharp and inequality (27) for M was proven by Temur in [33] (with constant C = 294,912,004). Inequality (28) was improved by Madrid [32] who obtained the sharp constant C = 2. Recently, Carneiro and Madrid [19] extended (28) to the fractional setting and showed that if 0 ≤ β < 1, q = 1/ð1 − βÞ, and f : whereM β is the discrete uncentered fractional maximal operator defined bỹ 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 f ↦ ðM β f Þ ′ and f ↦ ðM β f Þ ′ (for 0 ≤ β < 1) are bounded and continuous from ℓ 1 ðℤÞ to ℓ 1 ðℤÞ. Moreover, if f ∈ ℓ 1 ðℤÞ, then and the constants C = 2 are the best possible. Liu [30] pointed out that the operator f ′ ↦ ðM β f Þ′ is not bounded from ℓ 1 ðℤÞ to ℓ r ðℤÞ for all 1 ≤ r < 1/ð1 − βÞ. The continuity of M : BVðℤÞ ⟶ BVðℤÞ andM : BVðℤÞ ⟶ BVðℤÞ 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 9 (see [26]). Let 0 ≤ β < 1. Then, the map f ↦ ðM + β f Þ′ is bounded and continuous from ℓ 1 ðℤÞ to ℓ 1 ðℤÞ. Moreover, if f ∈ ℓ 1 ðℤÞ, then and the constant C = 2 is the best possible. The same results hold for M − β . Here The second aim of this paper is to study the regularity of the discrete analogues of ½b, M + β and M + b,β . Let us introduce some definitions.  Journal of Function Spaces We now formulate the rest of main results as follows: Theorem 12. Let b ∈ BVðℤÞ. Then, the map ½b, M + : BVðℤÞ ⟶ BVðℤÞ is bounded. Moreover, The same conclusions also hold for the operator ½b, M − .
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 C will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables. In particular, the letter C α,β denotes the positive constants that depend on the parameters α, β.

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 f ∈ L p ðℝÞ with p ≥ 1. For all h ∈ ℝ with h ≠ 0, we define For convenience, we set According to Section 7.11 in [36], one has that for 1 < p < ∞, Moreover, for functions f ∈ W 1,p ðℝÞ for 1 < p≤∞, we have (see [36], Section 7.11) that f h ⟶ f ′ in L p ðℝÞ when j hj ⟶ 0.
In order to prove Theorems 6 and 7, we need the following lemma, which follows from [23].
(iii) We now prove the continuity part. Let f j ⟶ f in W 1,p 1 ðℝÞ. It suffices to show that By Lemma 15 and applying the continuity result in Theorem 2, one can get bf j − bf 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 have for almost every x ∈ ℝ. By Theorem 2, one can get for almost every x ∈ ℝ. It follows from (55)-(58) that for almost every x ∈ ℝ. This proves (15) and completes the Proof of Theorem 6.
(i) We first prove that M + b,β f ∈ W 1,q ðℝÞ. Fix x, h ∈ ℝ, it is easy to see that which gives that Because of f ∈ W 1,p 1 ðℝÞ and b ∈ W 1,p 2 ðℝÞ, then by (44), we can get It follows from (62) and (63) that Combining (64) with (44) and the bounds for M + b,β yields M + b,β f ∈ W 1,q ðℝÞ.
(ii) We now prove (17). Since M + b,β f ∈ W 1,q ðℝÞ, f ∈ W 1,p 1 ðℝÞ and b ∈ W 1,p 2 ðℝÞ, then when h ⟶ 0. Therefore, there exists a sequence of real numbers fh k g satisfying lim k→∞ h k = 0 and a measurable set E satisfying jℝ \ Ej = 0 such (61) and (13) we have that for all h ∈ ℝ It follows that for any x ∈ E, which gives (17).

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 n ∈ ℤ. By (68) one has bf ð Þ ′ On the other hand, one can easily check that which together with (70) and (25) yields