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 | References | Similar Articles | Additional Information
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
Keywords:
<IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${L_p}$"> derivatives,
Sobolev functions,
capacity,
normal currents,
sets of finite perimeter,
Gauss-Green theorem
Article copyright:
© Copyright 1974
American Mathematical Society