Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 0344390
Full-text PDF Free Access

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.

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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A54, 46E35

Retrieve articles in all journals with MSC: 26A54, 46E35

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