Endpoint Sobolev Regularity for Composition of Maximal Operators and Commutators

Abstract

In this paper, we study the endpoint Sobolev regularity for composition of maximal operators and commutators. Some new endpoint Sobolev bounds and continuity of the above composition operators are established.

Keywords

Composition operator / maximal function / maximal commutator / W^1,1(ℝ) / boundedness and continuity / 42B25 / 46E35

References

[1] Aldaz JM, Pérez-Lázaro J. Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Amer. Math. Soc., 2007, 359(5): 2443-2461

[2] Beltran D, González-Riquelme C, Madrid J, Weigt J. Continuity of the gradient of the fractional maximal operator on W^1,1(ℝ^d). Math. Res. Lett., 2023, 30(3): 689-707

[3] Beltran D, Madrid J. Regularity of the centered fractional maximal function on radial functions. J. Funct. Anal., 2020, 279(8): 108686 28 pp.

[4] Calderón AP. Algebras of singular integral operators. Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), 1967, Providence, RI, American Mathematical Society18-55

[5] Calderón AP. Algebras of singular integral operators. Proc. Internat. Congr. Math. (Moscow, 1966), 1968, Moscow, Izdat. “Mir”393-395

[6] Calderón AP, Zygmund A. Algebras of certain singular operators. Amer. J. Math., 1956, 78: 310-320

[7] Carneiro E, González-Riquelme C, Madrid J. Sunrise strategy for the continuity of maximal operators. J. Anal. Math., 2022, 148(1): 37-84

[8] Carneiro E, Madrid J. Derivative bounds for fractional maximal functions. Trans. Amer. Math. Soc., 2017, 369(6): 4063-4092

[9] Carneiro E, Madrid J, Pierce LB. Endpoint Sobolev and BV continuity for maximal operators. J. Funct. Anal., 2017, 273(10): 3262-3294

[10] Carneiro E, Moreira D. On the regularity of maximal operators. Proc. Amer. Math. Soc., 2008, 136(12): 4395-4404

[11] Carozza M, Passarelli di Napoli A. Composition of maximal operators. Publ. Mat., 1996, 40(2): 397-409

[12] Chen T, Liu F. Derivative bounds and continuity of maximal commutators. Studia Math., 2022, 266(1): 93-119

[13] Chen T., Liu F., Endpoint Sobolev regularity of the fractional maximal commutators. J. Fourier Anal. Appl., 2022, 28(5): Paper No. 74, 39 pp.

[14] Evans LC, Gariepy RF. Measure Theory and Fine Properties of Functions, 1992, Boca Raton, FL, CRC Press

[15] González-Riquelme C., Continuity for the one-dimensional centered Hardy–Littlewood maximal operator at the derivative level. J. Funct. Anal., 2023, 285(9): Paper No. 110097, 14 pp.

[16] Hajłasz P, Malý J. On approximate differentiability of the maximal function. Proc. Amer. Math. Soc., 2010, 138(1): 165-174

[17] Hajłasz P, Onninen J. On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math., 2004, 29(1): 167-176

[18] Hu G. Quantitative weighted bounds for the composition of Calderón–Zygmund operators. Banach J. Math. Anal., 2019, 13(1): 133-150

[19] Hu G. Weighted weak type endpoint estimates for the compositions of Calderón–Zygmund operators. J. Aust. Math. Soc., 2020, 109(3): 320-339

[20] Hu G, Lai X, Xue Q. On the composition of rough singular integral operators. J. Geom. Anal., 2021, 31(3): 2742-2765

[21] Jiang X, Liu F. Continuity of the maximal commutators in Sobolev spaces. J. Korean Math. Soc., 2024, 61(3): 461-494

[22] Kinnunen J. The Hardy–Littlewood maximal function of a Sobolev function. Israel J. Math., 1997, 100: 117-124

[23] Kinnunen J, Lindqvist P. The derivative of the maximal function. J. Reine Angew. Math., 1998, 503: 161-167

[24] Kinnunen J, Saksman E. Regularity of the fractional maximal function. Bull. London Math. Soc., 2003, 35(4): 529-535

[25] Kurka O. On the variation of the Hardy–Littlewood maximal function. Ann. Acad. Sci. Fenn. Math., 2015, 40(1): 109-133

[26] Liu D, Zhao J. Weighted endpoint estimates for the composition of Calderón–Zygmund operators on spaces of homogeneous type. Collect. Math., 2022, 73(3): 359-382

[27] Liu F. Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequal. Appl., 2018, 21(1): 25-40

[28] Liu F., Ma Y., Endpoint Sobolev regularity of higher order maximal commutators. Banach J. Math. Anal., 2023, 17(4): Paper No. 63, 25 pp.

[29] Liu F, Wang G. Regularity of commutators of maximal operators with Lipschitz symbols. Taiwanese J. Math., 2021, 25(5): 1007-1039

[30] Liu F, Wen Y, Zhang X. Endpoint Sobolev regularity of higher order multilinear maximal commutators. Anal. Math., 2025, 51(3): 903-939

[31] Liu F., Wu H., Xue Q., Yabuta K., Endpoint Sobolev boundedness and continuity of multilinear fractional maximal functions. Bull. Sci. Math., 2022, 179: Paper No. 103155, 39 pp.

[32] Liu F., Xi S., Sobolev regularity for commutators of the fractional maximal functions. Banach J. Math. Anal., 2021, 15(1): Paper No. 5, 36 pp.

[33] Liu F, Xue Q, Zhang P. Regularity and continuity of commutators of the Hardy–Littlewood maximal function. Math. Nachr., 2020, 293(3): 491-509

[34] Liu F, Zhang X, Zhang H. Endpoint Sobolev regularity of multilinear maximal operators. Taiwanese J. Math., 2023, 27(5): 889-912

[35] Luiro H. The variation of the maximal function of a radial function. Ark. Mat., 2018, 56(1): 147-161

[36] Luiro H, Madrid J. The variation of the fractional maximal function of a radial function. Int. Math. Res. Not. IMRN, 2019, 2019(17): 5284-5298

[37] Madrid J. Endpoint Sobolev and BV continuity for maximal operators, II. Rev. Mat. Iberoam., 2019, 35(7): 2151-2168

[38] Oberlin R. Estimates for compositions of maximal operators with singular integrals. Canad. Math. Bull., 2013, 56(4): 801-813

[39] Ramos JPG. Sharp total variation results for maximal functions. Ann. Acad. Sci. Fenn. Math., 2019, 44(1): 41-64

[40] Tan J., Xue Q., Composition of rough singular integral operators on rearrangement invariant Banach type spaces. Nonlinear Anal., 2025, 254: Paper No. 113742, 17 pp.

[41] Weigt J. Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators. Math. Z., 2022, 301(3): 2317-2337

RIGHTS & PERMISSIONS

Peking University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.