Let $\Omega$ be a sufficiently smooth open domain of $\mathbb R^d$. Is any embedding of the Sobolev spaces $W^{s,p}(\Omega)$, $s>0$, into the target space $C^0(\overline{\Omega})$ (the space of bounded, uniformly continuous functions on $\Omega$) but NOT into any smaller space of bounded, Hölder continuous functions known? Even the case of $p=2$ would be interesting.
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1$\begingroup$ Such a space exists and is not really a Sobolev space but rather a Besov space, namely for any Lipschitz domains $\Omega\subset \mathbb{R}^n$, $\mathrm{B}^{n/p}_{p,1}(\Omega) \hookrightarrow \mathrm{C}^0_b(\overline{\Omega})$. Notice that in this case one recovers usual embbedings since we can prove $\mathrm{W}^{s,p}(\Omega) \hookrightarrow \mathrm{B}^{n/p}_{p,1}(\Omega)$ for any $s>n/p$. $\endgroup$– ToGleCommented Jul 23, 2023 at 11:35
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