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

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 inequa...

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Bibliographic Details
Published in:	Transactions of the American Mathematical Society Vol. 359; no. 5; pp. 2443 - 2461
Main Authors:	Aldaz, J. M., Lázaro, J. Pérez
Format:	Journal Article
Language:	English
Published:	Providence, RI American Mathematical Society 01-05-2007
Subjects:	Continuous functions Eigenfunctions Exact sciences and technology Fourier analysis Interior points Mathematical analysis Mathematical functions Mathematical inequalities Mathematical intervals Mathematical theorems Mathematics Open intervals Real functions Sciences and techniques of general use Sobolev spaces functions of bounded variation Maximal function Regularity Continuous function
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Summary:	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.
ISSN:	0002-9947 1088-6850
DOI:	10.1090/S0002-9947-06-04347-9