Pointwise differentiability and absolute continuity

Bagby, Thomas; Ziemer, William P.

doi:10.1090/S0002-9947-1974-0344390-6

Pointwise differentiability and absolute continuity
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by Thomas Bagby and William P. Ziemer PDF

Trans. Amer. Math. Soc. 191 (1974), 129-148 Request permission

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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