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I'm looking for a simple example where an inequality of the form $||u||_{L^q} \leq C||u||_{W^{1,p}}$ fails for some $1 \leq q \leq p^*$ (ie. within the acceptable range for which the bound should hold) on a bounded domain $\Omega$ which must necessarily have a cusp of some sort. I am interested in an example for $1 \leq p < n$ of course. I know how to find examples for the $p > n$ case.

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Hi, I don't know such an example, but the weaker condition i know is the following

The Weak Cone Condition: Given $x\in \Omega$, let $R(x)$ consist of all points $y\in \Omega$ such that the line segment from $x$ to $y$ lies in $\Omega$. Let $$\Gamma(x)=\{y \in R ( x )\hbox { s.t. } \vert y - x \vert < 1\}.$$ We say that $\Omega$ satisfies the weak cone condition if there exists $\delta > 0$ such that $$ \vert (\Gamma(x))\vert \geq \delta \hbox{ for all } x\in \Omega .$$

You will find a proof in: Adams, R. A.; Fournier, John Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61 (1977), no. 3, 713--734.

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