主要研究截断Hardy-Littlewood极大函数的连续性。首先证明截断Hardy-Littlewood极大函数是下半连续的，随后通过研究截断参数γ的变化对截断Hardy-Littlewood极大函数的影响，得到截断Hardy-Littlewood极大函数连续性的一个等价条件。
This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function. We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous. Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameter γ changes, we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.
[1] Hardy G H, Littlewood J E. A maximal theorem with function-theoretic applications[J]. Acta Mathematica, 1930, 54(1): 81-116. DOI: 10.1007/BF02547518.
[2] Wiener N. The ergodic theorem[J]. Duke Mathematical Journal, 1939, 5(1): 1-18. DOI: 10.1215/s0012-7094-39-00501-6.
[3] Grafakos L. Classical Fourier analysis[M]. Graduate texts in mathematics: vol 249. 3rd ed. New York, NY: Springer New York, 2014.
[4] Rudin W. Real and complex analysis[M]. 3rd ed. New York, NY: McGraw-Hill, 1987.
[5] Wei M Q, Nie X D, Wu D, et al. A note on Hardy-Littlewood maximal operators[J]. Journal of Inequalities and Applications, 2016, 2016(1):21. DOI: 10.1186/s13660-016-0963-x.
[6] Kinnunen J. The Hardy-Littlewood maximal function of a Sobolev function[J]. Israel Journal of Mathematics, 1997, 100(1): 117-124. DOI: 10.1007/BF02773636.
[7] Aldaz J M, Pérez Lázaro J. Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities[J]. Transactions of the American Mathematical Society, 2007, 359(5): 2443-2461. DOI: 10.1090/s0002-9947-06-04347-9.
[8] Wu D, Yan D Y. Continuity of Hardy-Littlewood maximal function[J]. Acta Mathematicae Applicatae Sinica, English Series, 2020, 36(4): 982-990. DOI: 10.1007/s10255-020-0983-z.
