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Let $\Omega$ be a bounded Lipschitz domain. It is well known that $H^1(\Omega)$ can be compactly embedded into $L^2(\Omega)$. I also found references for the compact embedding $H^\delta(\Omega)\hookrightarrow L^2(\Omega)$, where $0<\delta<1$.

The following is my question. IS the embedding $H^1(\Omega)\hookrightarrow H^\delta(\Omega)$ compact? If so, is there any reference?

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    $\begingroup$ See this related question. (Note that the $W^{s,2}$ and the $H^s$ scale coincide, on $R^n$ in general and on $\Omega$ because of your supposed Lipschitz regularity.) Essentially, you want to obtain a multiplicative inequality $\|u\|_{H^\delta(\Omega)} \leq C \|u\|_{L^2(\Omega)}^{1-\delta} \|u\|_{H^1(\Omega)}^\delta$, from which your desired compactness follows via Rellich-Kondrachov. $\endgroup$
    – Hannes
    Commented Jul 18, 2019 at 7:42
  • $\begingroup$ Great. The multiplicative/interpolation inequality is quite instructive. In fact, I found related theorems from page 26, "Elliptic problems in nonsmooth domains" by Pierre Grisvard. $\endgroup$
    – mathyul
    Commented Jul 18, 2019 at 18:17

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