Abstract
In the paper, we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Littlewood maximal function Mf and the centered Hardy-Littlewood maximal function Mcf on ℝn. As two applications, we can easily deduce that Mcf and Mf are continuous if f is continuous, and Mf is continuous if f is of local bounded variation on ℝ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubert, G., Kornprobst and P. Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, second ed. Springer, New York, 2006
Evans, L.C., Gariepy, R.F. Measure Theory and Fine Properties of Functions. CRC Press, Inc., 1992
Hardy, G.H., Littlewood, J.E. A maximal theorem with function-theoretic applications. Acta Math., 54: 81–116 (1930)
Lu, S.Z., Ding, Y., Yan, D.Y. Singular Integral and Related Topics. World Scientific Publishing Co. Pte. Ltd., 2007
Wiener, N. The ergodic theorem. Duke Math. J., 5: 1–18 (1939)
Wheeden, R.L., Zygmund, A. Measure and integral. Marcel Dekker, Inc., New York, 1977
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is supported by NSF of Zhejiang Province of China (Grant No. LQ18A010002 and No. LQ17A010002) and in part by National Natural Foundation of China (Grant Nos. 11871452 and 12001488).
Rights and permissions
About this article
Cite this article
Wu, D., Yan, Dy. Continuity of Hardy-Littlewood Maximal Function. Acta Math. Appl. Sin. Engl. Ser. 36, 982–990 (2020). https://doi.org/10.1007/s10255-020-0983-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-020-0983-z