Pointwise differentiability and absolute continuity

Abstract

This paper is concerned with the relationships between L p {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 {L_p} , and 0 ≤ l ≤ k 0 \leq l \leq k , then f has an L p {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 {L_p} derivative everywhere except for a set small in capacity and if these derivatives are in L p {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.