Sharpness of Sobolev Inequalities for a Class of Irregular Domains

Recently, O. V. Besov proved the embedding $W^{m}_p(\Omega)\subset L_q(\Omega)$ for the Sobolev spaces of higher orders $m=2,3,\ldots $ over a domain $\Omega\subset\mathbb R^n$ satisfying $s$-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 Hajłasz and Koskela and of Kilpeläinen and Malý.