Literature: Regularity of maximal functions

Here is a collection of the literature on the regularity of maximal functions. This list is likely to be incomplete and will need periodical updating. Please email me with any omissions. When possible, I link to the published version of the article.

B. Bojarski and P. Hajlasz, Pointwise inequalities for Sobolev functions and some applications, Studia Math 106 (1993), 77-92.
J. Kinnunen, The Hardy--Littlewood maximal operator of a Sobolev function, Israel Journal of Math. 199 (1997), 117-12. MR 1469106 (99a:30029)
J. Kinnunen and P. Lindqvist, The derivative of the maximal function, J. Reine Angew. Math. 503 (1998), 161-167. MR1650343 (99j:42027)
S. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 2, 519–528.
J. Kinnunen and M. Olli, Maximal operator and superharmonicity. Function spaces, differential operators and nonlinear analysis (Pudasjärvi, 1999), 157–169, Acad. Sci. Czech Repub., Prague, 2000.
H. Tanaka, A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function, Bull. Austral. Math. Soc. 65 (2002), no. 2, 253–258.
J. Kinnunen and V. Latvala, Lebesgue points for Sobolev functions on metric spaces, Rev. Mat. Iberoamericana 18 (2002), no. 3, 685–700.
S. Korry, Boundedness of Hardy–Littlewood maximal operator in the framework of Lizorkin–Triebel spaces, Rev. Mat. Complut. 15 (2) (2002) 401–416.
J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529–535. MR1979008 (2004e:42035)
P. Hajlasz and J. Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 167-176. MR2041705 (2005a:42010)
S. Korry, A class of bounded operators on Sobolev spaces, Arch. Math. (Basel) 82 (1) (2004) 40–50.
H. Luiro, Continuity of the maximal operator in Sobolev spaces, Proc. Amer. Math. Soc. 135 (2007), no. 1, 243-251. MR2280193 (2007i:42021)
J.M. Aldaz and J. Perez Lazaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2443-2461. MR2276629
J. Kinnunen and H. Tuominen, Pointwise behaviour of M1,1 Sobolev functions, Math. Z. 257 (2007), no. 3, 613–630.
J.M. Aldaz and J. Perez Lazaro, Boundedness and unboundedness results for some maximal operators on functions of bounded variation,
E. Carneiro and D. Moreira, On the regularity of maximal operators, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4395–4404.
D. Aalto and J. Kinunnen, Maximal functions in Sobolev spaces. Sobolev spaces in mathematics. I, 25–67, Int. Math. Ser. (N. Y.), 8, Springer, New York, 2009.
J.M. Aldaz and J. Perez Lazaro, Regularity of the Hardy-Littlewood maximal operator on block decreasing functions, Studia Math. 194 (2009), no. 3, 253–277.
D. Aalto and J. Kinunnen, The discrete maximal operator in metric spaces. J. Anal. Math. 111 (2010), 369–390.
P. Haj lasz and J. Maly, On approximate differentiability of the maximal function, Proc. Amer. Math. Soc. 138 (2010), 165–174.
H. Luiro, On the regularity of the Hardy-Littlewood maximal operator on subdomains of Rd, Proc. Edinburgh Math. Soc. 53 (2010), no 1, 211–237.
J.M. Aldaz, L. Colzani and J. Perez Lazaro, Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function, J. Geom. Anal. 22 (2012), no. 1, 132–167.
J. Bober, E. Carneiro, K. Hughes and L. B. Pierce, On a discrete version of Tanaka’s theorem for maximal functions, Proc. Amer. Math. Soc. 140 (2012), 1669–1680.
E. Carneiro and K. Hughes, On the endpoint regularity of discrete maximal operators, Math. Res. Lett. 19, no. 6 (2012), 1245–1262.
E. Carneiro and B. F. Svaiter, On the variation of maximal operators of convolution type, J. Funct. Anal. 265 (2013), 837–865.
O. Kurka, On the variation of the Hardy-Littlewood maximal function, Ann. Acad. Sci. Fenn. Math. 40 (2015), 109–133.
T. Heikkinen, J. Kinnunen, J. Korvenpaa and H. Tuominen, Regularity of the local fractional maximal function. Ark. Mat. 53 (2015), no. 1, 127–154. 42B25 (35L05)
E. Carneiro and J. Madrid, Derivative bounds for fractional maximal functions, Trans. Amer. Math. Soc., to appear.
F. Temur, On regularity of the discrete Hardy-Littlewood maximal function, preprint at http://arxiv.org/abs/1303.3993.
O. Saari, Poincaré inequalities for the maximal function, https://arxiv.org/abs/1605.05176
Carneiro, Finder, Sousa - On the variation of maximal operators of convolution type II
Madrid - Sharp inequalities for the variation of the discrete maximal function (Jose obtains the sharp bounds and extremizers from BCHP)