Some counterexamples for the theory of sobolev spaces on bad domains

Abstract

It is shown that some well-known properties of the Sobolev spaceL ^l _p (Ω) do not admit extension to the spaceL ^l _p (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

• (i) L ^l _p (Ω)∩L _∞(Ω) is not dense inL ^l _p (Ω),

• (ii) L ^l _p (Ω)∩L _∞(Ω) is not a Banach algebra.

• (iii) the strong capacitary inequality for the norm inL ^l _p (Ω) fails.

In the Appendix necessary and sufficient conditions are given for the imbeddingsL ^l _p (Ω)⊂L _q (Ω, μ) andH ^l _p (R ⁿ)⊂L _q (R ⁿ, μ), wherep≥1,p>q>0, μ is a measure andH ^l _p (Ω) is the Bessel potential space, 1<p<∞,l>0.