Geodesic
Sharpness of Sobolev Inequalities for a~Class of Irregular Domains
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 218-222

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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. 
@article{TM_2001_232_a17,
     author = {D. A. Labutin},
     title = {Sharpness of {Sobolev} {Inequalities} for {a~Class} of {Irregular} {Domains}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {218--222},
     publisher = {mathdoc},
     volume = {232},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2001_232_a17/}
}
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D. A. Labutin. Sharpness of Sobolev Inequalities for a~Class of Irregular Domains. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 218-222. http://geodesic.mathdoc.fr/item/TM_2001_232_a17/