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.