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)