Suppose $0<\alpha<\beta$, and $\Omega$ is a bounded subset of $\mathbb{R}^n$. Then the Holder space $C^{\beta}(\Omega)$ is compactly embedded into $C^{\alpha}(\Omega)$. But if $\Omega=\mathbb{R}^n$, then the compact embedding is not true.

However, if we consider the weaker weighted Holder space $C^{\alpha, -\delta}(\mathbb{R}^n)$ (for any $\delta>0$) instead of $C^{\alpha}(\mathbb{R}^n)$. Then is $C^{\beta}(\mathbb{R}^n)$ compactly embedded to $C^{\alpha, -\delta}(\mathbb{R}^n)$?

Here $$ \|f\|_{C^{\alpha, -\delta}}=\|(1+|\cdot|^2)^{-\frac{\delta}{2}}f\|_{C^{\alpha}}. $$

I could not find a precise reference from some books on functional analysis. Thanks in advance.


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