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

Zhang, Daiqing

doi:https://doi.org/10.1155/2020/1369454

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Research Article | Open Access

Volume 2020 | Article ID 1369454 | https://doi.org/10.1155/2020/1369454

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

Daiqing Zhang¹

Academic Editor: Stanislav Hencl

Received07 Apr 2020

Accepted10 Jun 2020

Published13 Jul 2020

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 mapbounded fromto? 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 On the other hand, one can easily check that which together with (70) and (25) yields that

Proof of Theorem 13. (i) It is clear that One can easily check that for all . In light of (73) and (74) we would have On the other hand, by Theorem 9 and (73), (69), and (25), we have Combining (76) with (75) yields that the map is bounded. (iii)We now prove the continuity result. Let , , and in as . We want to show thatBy the sublinearity of and (25), (73), and (74), it holds that for all . From (78), we see that uniformly for . Therefore, to prove (77), it suffices to show that By (73) and (25), we get which yields that in as . This together with Theorem 9 implies that On the other hand, by Theorem 9 again, we get By (69), (74), and (25) and the sublinearity for , we have This together with (82) yields that Combining (84) with (81) implies that which proves (79) and finishes the Proof of Theorem 13.

Proof of Theorem 14. We only prove Theorem 14 for since another one is analogous. The proof will be divided into two steps:

Step 1. Proof of the boundedness part. Let . Without loss of generality we may assume since . For convenience, we define the function by for any . It is clear that is decreasing on and . Fix and , it holds that This yields that for any fixed , there exists such that Let Then, we have We can write By Hölder’s inequality with exponents and and the fact that for all , it holds that In light of (90) and (92), we would have By (91) and (93), we have It follows from (89), (94), and (95) that Fix , one can easily check that This together with (96) yields that This proves (41).

Step 2. Proof of the continuity part. Let , , and in when . Without loss of generality, we may assume that all and since . We want to show that By the sublinearity of , we can get which yields that uniformly in . Fix , there exists depending only on such that It follows from (100) that for all . By the boundedness part, we have that . This together with the classical Brezis-Lieb lemma in [3] implies that (99) reduces to the following: By (103) and Fatou’s lemma, one can get Therefore, to prove (104), it suffices to show that We now prove (106). Since , then there exists a sufficiently large positive integer depending only on such that There exists an integer such that if . Since , then there exists a large positive integer depending only on such that Let . By (103), there exists depending only on such that It follows from (109) that for all . Fix , we set Since , then for , there exists such that . Then, we can write Similar arguments to those used in deriving (94) and (95) may yield that It follows from (102), (108), and (112)–(114) that By (101) and (107), we can get It follows from (115) and (116) that Inequality (117) together with (110) implies that for all . Here, depends only on . This leads to (106) and completes the Proof of Theorem 14.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that she have no conflicts of interest.

Acknowledgments

The author want to express her sincerely thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable. This work was funded by the Natural Science Foundation of University Union of Science and Technology Department of Fujian Province (No. 2019J01784) and the Natural Science Foundation of Fujian University of Technology (Nos. GY-Z15124 and GY-Z160129).

References

J. Kinnunen, “The Hardy-Littlewood maximal function of a Sobolev function,” Israel Journal of Mathematics, vol. 100, no. 1, pp. 117–124, 1997.
View at: Publisher Site | Google Scholar
P. Lindqvist and J. Kinnunen, “The derivative of the maximal function,” Journal fur die Reine und Angewandte Mathematik, vol. 1998, no. 503, pp. 161–167, 1998.
View at: Publisher Site | Google Scholar
J. Kinnunen and E. Saksman, “Regularity of the fractional maximal function,” The Bulletin of the London Mathematical Society, vol. 35, no. 4, pp. 529–535, 2003.
View at: Publisher Site | Google Scholar
E. Carneiro and D. Moreira, “On the regularity of maximal operators,” Proceedings of the American Mathematical Society, vol. 136, no. 12, pp. 4395–4404, 2008.
View at: Publisher Site | Google Scholar
F. Liu and H. Wu, “On the regularity of the multisublinear maximal functions,” Canadian Mathematical Bulletin, vol. 58, no. 4, pp. 808–817, 2015.
View at: Publisher Site | Google Scholar
F. Liu and S. Mao, “On the regularity of the one-sided Hardy-Littlewood maximal functions,” Czechoslovak Mathematical Journal, vol. 67, no. 1, pp. 219–234, 2017.
View at: Publisher Site | Google Scholar
H. Luiro, “Continuity of the maximal operator in Sobolev spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 1, pp. 243–251, 2007.
View at: Publisher Site | Google Scholar
H. Luiro, “On the regularity of the Hardy-Littlewood maximal operator on subdomains of ,” Proceedings of the Edinburgh Mathematical Society, vol. 53, no. 1, pp. 211–237, 2010.
View at: Publisher Site | Google Scholar
F. Liu, “Continuity and approximate differentiability of multisublinear fractional maximal functions,” Mathematical Inequalities & Applications, vol. 21, no. 1, pp. 25–40, 2018.
View at: Publisher Site | Google Scholar
J. Hart, F. Liu, and Q. Xue, “Regularity and continuity of local multilinear maximal type operators,” The Journal of Geometric Analysis, 2020.
View at: Publisher Site | Google Scholar
S. Korry, “Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces,” Revista Matemática Complutense, vol. 15, no. 2, pp. 401–416, 2002.
View at: Publisher Site | Google Scholar
P. Hajłasz and J. Onninen, “On boundedness of maximal functions in Sobolev spaces,” Annales Academiae Scientiarum Fennicae Mathematica, vol. 29, no. 1, pp. 167–176, 2004.
View at: Google Scholar
J. M. Aldaz and J. Pérez Lázaro, “Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities,” Transactions of the American Mathematical Society, vol. 359, no. 5, pp. 2443–2462, 2007.
View at: Publisher Site | Google Scholar
O. Kurka, “On the variation of the Hardy-Littlewood maximal function,” Annales Academiae Scientiarum Fennicae Mathematica, vol. 40, pp. 109–133, 2015.
View at: Publisher Site | Google Scholar
F. Liu, T. Chen, and H. Wu, “A note on the endpoint regularity of the Hardy-Littlewood maximal functions,” Bulletin of the Australian Mathematical Society, vol. 94, no. 1, pp. 121–130, 2016.
View at: Publisher Site | Google Scholar
H. Tanaka, “A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function,” Bulletin of the Australian Mathematical Society, vol. 65, no. 2, pp. 253–258, 2002.
View at: Publisher Site | Google Scholar
P. Hajłasz and J. Malý, “On approximate differentiability of the maximal function,” Proceedings of the American Mathematical Society, vol. 138, no. 1, pp. 165–174, 2010.
View at: Publisher Site | Google Scholar
H. Luiro, “The variation of the maximal function of a radial function,” Arkiv för Matematik, vol. 56, no. 1, pp. 147–161, 2018.
View at: Publisher Site | Google Scholar
E. Carneiro and J. Madrid, “Derivative bounds for fractional maximal functions,” Transactions of the American Mathematical Society, vol. 369, no. 6, pp. 4063–4092, 2017.
View at: Publisher Site | Google Scholar
E. Carneiro, J. Madrid, and L. B. Pierce, “Endpoint Sobolev and BV continuity for maximal operators,” Journal of Functional Analysis, vol. 273, no. 10, pp. 3262–3294, 2017.
View at: Publisher Site | Google Scholar
E. Carneiro and B. F. Svaiter, “On the variation of maximal operators of convolution type,” Journal of Functional Analysis, vol. 265, no. 5, pp. 837–865, 2013.
View at: Publisher Site | Google Scholar
H. Luiro and J. Madrid, “The variation of the fractional maximal function of a radial function,” International Mathematics Research Notices, vol. 17, pp. 5284–5298, 2019.
View at: Publisher Site | Google Scholar
F. Liu, Q. Xue, and P. Zhang, “Regularity and continuity of commutators of the Hardy-Littlewood maximal function,” Mathematische Nachrichten, vol. 293, no. 3, pp. 491–509, 2020.
View at: Publisher Site | Google Scholar
G. H. Hardy and J. E. Littlewood, “A maximal theorem with function-theoretic applications,” Acta Mathematica, vol. 54, pp. 81–116, 1930.
View at: Publisher Site | Google Scholar
K. F. Andersen and E. T. Sawyer, “Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators,” Transactions of the American Mathematical Society, vol. 308, no. 2, pp. 547–558, 1988.
View at: Publisher Site | Google Scholar
F. Liu, “On the regularity of one-sided fractional maximal functions,” Mathematica Slovaca, vol. 68, no. 5, pp. 1097–1112, 2018.
View at: Publisher Site | Google Scholar
F. Liu and L. Xu, “Regularity of one-sided multilinear fractional maximal functions,” Open Mathematics, vol. 16, no. 1, pp. 1556–1572, 2018.
View at: Publisher Site | Google Scholar
J. Bober, E. Carneiro, K. Hughes, and L. B. Pierce, “On a discrete version of Tanaka's theorem for maximal functions,” Proceedings of the American Mathematical Society, vol. 140, no. 5, pp. 1669–1680, 2012.
View at: Publisher Site | Google Scholar
E. Carneiro and K. Hughes, “On the endpoint regularity of discrete maximal operators,” Mathematical Research Letters, vol. 19, no. 6, pp. 1245–1262, 2012.
View at: Publisher Site | Google Scholar
F. Liu, “A remark on the regularity of the discrete maximal operator,” Bulletin of the Australian Mathematical Society, vol. 95, no. 1, pp. 108–120, 2017.
View at: Publisher Site | Google Scholar
F. Liu and H. Wu, “A note on the endpoint regularity of the discrete maximal operator,” Proceedings of the American Mathematical Society, vol. 147, no. 2, pp. 583–596, 2019.
View at: Publisher Site | Google Scholar
J. Madrid, “Sharp inequalities for the variation of the discrete maximal function,” Bulletin of the Australian Mathematical Society, vol. 95, no. 1, pp. 94–107, 2017.
View at: Publisher Site | Google Scholar
F. Temur, “On regularity of the discrete Hardy-Littlewood maximal function,” http://arxiv.org/abs/1303.3993v1.
View at: Google Scholar
J. Madrid, “Endpoint Sobolev and BV continuity for maximal operators, II,” Revista Matemática Iberoamericana, vol. 35, no. 7, pp. 2151–2168, 2019.
View at: Publisher Site | Google Scholar
H. Brezis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983.
View at: Publisher Site | Google Scholar
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2nd edition, 1983.

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Copyright © 2020 Daiqing Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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