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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: J. M. Aldaz and J. Pérez Lázaro
Journal: Trans. Amer. Math. Soc. 359 (2007), 2443-2461
MSC (2000): Primary 42B25, 26A84
Published electronically: December 19, 2006
MathSciNet review: 2276629
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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.


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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: http://dx.doi.org/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
Published electronically: 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.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.