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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Path derivatives and growth control


Authors: A. M. Bruckner and K. G. Johnson
Journal: Proc. Amer. Math. Soc. 91 (1984), 46-48
MSC: Primary 26A24; Secondary 28A12
MathSciNet review: 735561
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Abstract: We show that a standard theorem relating the growth of a function on a measurable set to constraints on its Dini derivates extends to a class of generalized derivatives. More precisely, we show that if $ \left\vert {{{\bar F'}_E}} \right\vert \leqslant M$ and $ \left\vert {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{F} '}_E}} \right\vert \leqslant M$ on a measurable set $ A$, then $ \lambda (F(A)) \leqslant M\lambda (A)$. Here $ \lambda $ denotes Lebesgue measure and $ {\bar F'_E}$ and $ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{F} '_E}$ are the extreme derivates of $ F$ relative to a system of paths which satisfy the Intersection Condition [1]. In particular, the result holds in the setting of unilateral differentiation, approximate differentiation, preponderant differentiation and qualitative differentiation.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0735561-7
Keywords: Generalized derivatives, path derivatives, Intersection Condition, generalized absolute continuity
Article copyright: © Copyright 1984 American Mathematical Society