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Can anybody give a reference to the following two facts?

The embeddings $$H_0^{1,2}(\mathbb R^n)\to L^2(\partial B_1(0))$$ and $$H^{1,2}(\partial B_1(0))\to H^{1/2,2}(\partial B_1(0))$$ are compact?


  • $B_1(0)$ denotes the unit ball centered at $0$ in $\mathbb R^n$,

  • for a domain $\Omega\subset \mathbb R^n$, $H_0^{1,2}(\mathbb \Omega)$ denotes the closure of $C^\infty(\Omega)$ functions with respect to the norm $\int_\Omega |u|^2\mathrm d x+\int_\Omega|\nabla u|^2\mathrm d x$

  • $H^{1/2,2}$ is the fractional Sobolev space.

I've never met anything concerning this, so I am looking for good places to begin with the study. Thank you very much in advance


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