Embedding of Sobolev Spaces on Hölder Domains

It is well known that the embedding $W^1_p(\Omega)\hookrightarrow L_q(\Omega)$, $1\leq p<q\leq\infty$, is equivalent to certain isoperimetric or capacity inequalities for the subsets of $\Omega$. P. Hajłasz with P. Koskela and T. Kilpeläinen with J. Malý have proved in their recent works the inequalities of this type for a wide class of $s$+John domains. In the present paper, we prove the exact isoperimetric inequality and the embedding $W^1_p(\Omega)\hookrightarrow L_q(\Omega)$ with the best index $q$ for a narrower class of Hölder domains. A Hölder domain locally coincides with the epigraph of a function satisfying the Hölder condition. The improvement of the index $q$ as compared with the case considered in the aforementioned works is achieved due to the application of special coverings of the subsets of $\Omega$.