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Transactions of the American Mathematical Society

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Pointwise differentiability and absolute continuity


Authors: Thomas Bagby and William P. Ziemer
Journal: Trans. Amer. Math. Soc. 191 (1974), 129-148
MSC: Primary 26A54; Secondary 46E35
DOI: https://doi.org/10.1090/S0002-9947-1974-0344390-6
MathSciNet review: 0344390
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Abstract: This paper is concerned with the relationships between $ {L_p}$ differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in $ {L_p}$, and $ 0 \leq l \leq k$, then f has an $ {L_p}$ derivative of order l everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an $ {L_p}$ derivative everywhere except for a set small in capacity and if these derivatives are in $ {L_p}$, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0344390-6
Keywords: $ {L_p}$ derivatives, Sobolev functions, capacity, normal currents, sets of finite perimeter, Gauss-Green theorem
Article copyright: © Copyright 1974 American Mathematical Society

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