Let $\Omega\subset\mathbb{R}^d$ be a bounded domain with Lipschitz smooth boundary and $\delta>0$ sufficiently small so that $ \Omega_\delta = ${ $x\in\Omega : dist(x,\partial\Omega)>\delta $ }$ $ is also a domain with Lipschitz smooth boundary.

For sufficiently small $\delta<0$ and $\tau>d/2$ is there a linear extension operator $ E: H^\tau(\Omega_\delta) \rightarrow H^\tau(\Omega) $ such that

(i) $Ef(x) = f(x)$ for $x\in\Omega_\delta$

(ii) $||E f||\_\{H^\tau(\Omega_\delta)\}\leq C||f||\_\{H^\tau(\Omega)\}$

where the constant $C$ is independent of $\delta$?

In response to Tapio Rajala: $\Omega$ is a domain meaning it is an open connected subset of $\mathbb{R}^d$. From the definition of $\Omega_\delta$ and $\delta$ being sufficiently small this makes $\Omega_\delta$ a connected subset. I also want $\Omega_\delta$ to have a Lipschitz smooth boundary, that is locally a graph of a Lipschitz smooth function.

  • 1
    $\begingroup$ Are the norms in (ii) taken with respect to the wrong spaces? And should the constant C be inserted on the right? $\endgroup$ Commented Jan 28, 2011 at 18:57
  • 3
    $\begingroup$ Have you looked at Chapter VI of Stein's book, "Singular Integrals and Differentiability Properties of Functions"? $\endgroup$
    – Deane Yang
    Commented Jan 28, 2011 at 19:03
  • $\begingroup$ What do you mean by $\Omega_\delta$ has a Lipschitz smooth boundary? In particular, can $\Omega_\delta$ have infinitely many connected components? I think the answer to the question depends on this. From the classical results of Calderón and Stein we know that Lipschitz domains are Sobolev extension domains for basically all the possible parameters. On the other hand, if we are not talking about domains, then you can construct a counterexample by taking smaller and smaller balls (and then considering their capacities). $\endgroup$ Commented Jan 28, 2011 at 19:19
  • $\begingroup$ @Tapio Rajala: Yes Lipschitz domains are Sobolev extension domains but does the constant of extension depend on the domain? $\endgroup$
    – alext87
    Commented Jan 29, 2011 at 12:36
  • $\begingroup$ @alext87: Yes, I know what a domain is. But on the choice of $\delta$ you only assumed that $\Omega_\delta$ has Lipschitz boundary - not that it is a domain. And yes, the constant of the extension depends on the domain. $\endgroup$ Commented Jan 29, 2011 at 13:23


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