Some counterexamples for the theory of sobolev spaces on bad domains

It is shown that some well-known properties of the Sobolev spaceL (Ω) do not admit extension to the spaceL (Ω) of the functions withl-th order derivatives inL p (Ω),l>1, without requirements to the domain Ω. Namely, we give examples of Ω such that In the Appendix necessary and sufficient conditions are given for the imbeddingsL (Ω)⊂L q (Ω, μ) andH (R n )⊂L q (R n , μ), wherep≥1,p>q>0, μ is a measure andH (Ω) is the Bessel potential space, 1 0.