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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
DOI: https://doi.org/10.1090/S0002-9939-1984-0735561-7
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.


References [Enhancements On Off] (What's this?)

  • [1] A. M. Bruckner, R. J. O'Malley and B. S. Thomson, Path derivatives: a unified view of certain generalized derivatives, Trans. Amer. Math. Soc. (to appear). MR 807973 (87b:26008)
  • [2] I. P. Natanson, Theory of functions of a real variable, Vol. I, Ungar, New York, 1961. MR 0148805 (26:6309)
  • [3] S. Saks, Theory of the integral, Monograf. Mat., PWN, Warszawa-Lwów, 1937.
  • [4] H. Whitney, On totally differentiable and smooth functions, Pacific J. Math. 1 (1951), 143-159. MR 0043878 (13:333d)

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Additional Information

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

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