Abstract
Consider a regular domain 12 & SUB; RN and let d(x) = dist(x, partial differential 12). Denote L1,& INFIN;a(12) the space of functions from L1,& INFIN;(12) having absolutely continuous quasinorms.
This set is essentially smaller than L1,& INFIN;(12) but, at the same time, essentially larger than the union of all L1,q(12), q & ISIN; [1,& INFIN;).A classical result of late 1980's states that for p & ISIN; (1, & INFIN;) and m & ISIN; N, u belongs to the Sobolev space Wm,p 0 (12) if and only if u/dm & ISIN; Lp(12) and | backward difference mu| & ISIN; Lp(12). During the consequent decades, several authors have spent considerable effort in order to relax the characterizing condition.
Recently, it was proved that u & ISIN;W0m,p(12) if and only if u/dm & ISIN; L1(12) and | backward difference mu| & ISIN; Lp(12). In this paper we show that for N & GE; 1 and p & ISIN;(1,& INFIN;) we have u & ISIN;Wo,p(12) if and only if u/d & ISIN;aL1,& INFIN;(12) and | backward difference u| & ISIN;Lp(12).
Moreover, we present a counterexample which demonstrates that after relaxing the condition u/d & ISIN; L1,& INFIN;a(12) to u/d & ISIN; L1,& INFIN;(12) the equivalence no longer holds.& COPY; 2023 Elsevier Inc. All rights reserved.