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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

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

Author(s): J. M. Aldaz; J. Pérez Lázaro
Journal: Trans. Amer. Math. Soc. 359 (2007), 2443-2461.
MSC (2000): Primary 42B25, 26A84
Posted: December 19, 2006
MathSciNet review: 2276629
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Abstract | References | Similar articles | Additional information

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 $ \Vert DMf\Vert _{L^1(I)}\le \vert Df\vert(I)$. This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.

References:

[AlPe]
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.

[AFP]
Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, (2000). MR 1857292 (2003a:49002)

[Bu]
Buckley, Stephen M., Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math. 24 (1999), 519-528. MR 1724375 (2001e:42025)

[Ha]
Haj\lasz, Piotr, A new characterization of the Sobolev space. Studia Math. 159 (2003), no. 2, 263-275. MR 2052222 (2005d:46075)

[HaOn]
Haj\lasz, Piotr; Onninen, Jani, On boundedness of maximal functions in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 167-176. MR 2041705 (2005a:42010)

[Ka]
Ka\lamajska, Agnieszka, Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces. Studia Math. 108 (1994), no. 3, 275-290. MR 1259280 (94k:46059)

[Ki]
Kinnunen, Juha, The Hardy-Littlewood maximal function of a Sobolev function. Israel J. Math. 100 (1997), 117-124. MR 1469106 (99a:30029)

[KiLi]
Kinnunen, Juha; Lindqvist, Peter, The derivative of the maximal function. J. Reine Angew. Math. 503 (1998), 161-167. MR 1650343 (99j:42027)

[KiSa]
Kinnunen, Juha; Saksman, Eero, Regularity of the fractional maximal function. Bull. London Math. Soc. 34 (2003),no. 4, 529-535. MR 1979008 (2004e:42035)

[Ko1]
Korry, Soulaymane, A class of bounded operators on Sobolev spaces. Arch. Math. (Basel) 82 (2004), no. 1, 40-50. MR 2034469 (2004k:42033)

[Ko2]
Korry, Soulaymane, Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces. Rev. Mat. Complut. 15 (2002), no. 2, 401-416. MR 1951818 (2004a:42020)

[Lu]
Luiro, Hannes, Continuity of the maximal operator in Sobolev spaces. Proc. Amer. Math. Soc. 135 (2007), 243-251.

[MaSh1]
Maz $ \textprime$ya, Vladimir; Shaposhnikova, Tatyana, On pointwise interpolation inequalities for derivatives. Math. Bohem. 124 (1999), no. 2-3, 131-148. MR 1780687 (2001h:26026)

[MaSh2]
Maz $ \textprime$ya, V. G.; Shaposhnikova, T. O., Pointwise interpolation inequalities for derivatives with best constants. (Russian) Funktsional. Anal. i Prilozhen. 36 (2002), no. 1, 36-58, 96; translation in Funct. Anal. Appl. 36 (2002), no. 1, 30-48 MR 1898982 (2003c:42020)

[Ta]
Tanaka, Hitoshi, A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function. Bull. Austral. Math. Soc. 65, no. 2, (2002), 253-258. MR 1898539 (2002m:42017)

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Additional Information:

J. M. Aldaz
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, La Rioja, Spain
Email: aldaz@dmc.unirioja.es

J. Pérez Lázaro
Affiliation: Departamento de Matemáticas e Informática, Universidad de La Rioja, 26004 Logroño, La Rioja, Spain
Email: javier.perezl@unirioja.es

DOI: 10.1090/S0002-9947-06-04347-9
PII: S 0002-9947(06)04347-9
Keywords: Maximal function, functions of bounded variation.
Received by editor(s): December 30, 2005
Posted: December 19, 2006
Additional Notes: The authors were partially supported by Grant BFM2003-06335-C03-03 of the D.G.I. of Spain
The second author thanks the University of La Rioja for its hospitality.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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