Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities

Aldaz, J.; Pérez Lázaro, J.

doi:10.1090/S0002-9947-06-04347-9

Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities
HTML articles powered by AMS MathViewer

by J. M. Aldaz and J. Pérez Lázaro PDF

Trans. Amer. Math. Soc. 359 (2007), 2443-2461 Request permission

Abstract:

We prove that if $f:I\subset \mathbb {R}\to \mathbb {R}$ is of bounded variation, then the uncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_{L^1(I)}\le |Df|(I)$. This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.

References

Aldaz, J.M.; Pérez Lázaro, J., Boundedness and unboundedness results for some maximal operators on functions of bounded variation. Submitted. Available at the Mathematics ArXiv: arXiv:math.CA/0605272.
Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
Stephen M. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 2, 519–528. MR 1724375
Piotr Hajłasz, A new characterization of the Sobolev space, Studia Math. 159 (2003), no. 2, 263–275. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). MR 2052222, DOI 10.4064/sm159-2-7
Piotr Hajłasz and Jani Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 167–176. MR 2041705
Agnieszka Kałamajska, Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces, Studia Math. 108 (1994), no. 3, 275–290. MR 1259280, DOI 10.4064/sm-108-3-275-290
Juha Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math. 100 (1997), 117–124. MR 1469106, DOI 10.1007/BF02773636
Juha Kinnunen and Peter Lindqvist, The derivative of the maximal function, J. Reine Angew. Math. 503 (1998), 161–167. MR 1650343
Juha Kinnunen and Eero Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529–535. MR 1979008, DOI 10.1112/S0024609303002017
Soulaymane Korry, A class of bounded operators on Sobolev spaces, Arch. Math. (Basel) 82 (2004), no. 1, 40–50. MR 2034469, DOI 10.1007/s00013-003-0416-x
Soulaymane Korry, Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces, Rev. Mat. Complut. 15 (2002), no. 2, 401–416. MR 1951818, DOI 10.5209/rev_{R}EMA.2002.v15.n2.16899
Luiro, Hannes, Continuity of the maximal operator in Sobolev spaces. Proc. Amer. Math. Soc. 135 (2007), 243–251.
Vladimir Maz′ya and Tatyana Shaposhnikova, On pointwise interpolation inequalities for derivatives, Math. Bohem. 124 (1999), no. 2-3, 131–148. MR 1780687
V. G. Maz′ya and T. O. Shaposhnikova, Pointwise interpolation inequalities for derivatives with best constants, Funktsional. Anal. i Prilozhen. 36 (2002), no. 1, 36–58, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 36 (2002), no. 1, 30–48. MR 1898982, DOI 10.1023/A:1014478100799
Hitoshi Tanaka, A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function, Bull. Austral. Math. Soc. 65 (2002), no. 2, 253–258. MR 1898539, DOI 10.1017/S0004972700020293