Sharpness of Sobolev inequalities for a class of irregular domains (Q2783077)

Recently, \textit{O.\,V.\thinspace Besov} [Sb.\ Math.\ 192, No.\,3, 323--346 (2001); translation from Mat.\ Sb.\ 192, No.\,3, 3--26 (2001; Zbl 1020.46007)] proved the embedding for the Sobolev spaces of higher orders \(m=2,3,\dots\) over a domain \(\Omega\subset\mathbb R^n\) satisfying the \(s\)-John condition. We show that the number \(q\) obtained by Besov in this embedding is maximal over the class of \(s\)-John domains. An unimprovable embedding of the Sobolev spaces \(W^1_p(\Omega )\) was found earlier in works of \textit{P.\,Hajłasz} and \textit{P.\,Koskela} [J.~Lond.\ Math.\ Soc., II.\ Ser.\ 58, No.\,2, 425--450 (1998; Zbl 0922.46034)] and of \textit{T.K\,ilpeläinen} and \textit{J.\,Malý} [Z.\ Anal.\ Anwend.\ 19, No.\,2, 369--380 (2000; Zbl 0959.46020)].NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].