Reference for compact embedding between (weighted) Holder space on $\mathbb{R}^n$

Suppose $$0<\alpha<\beta<1$$, 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. Any comment is welcome.

1 Answer

Since the norm is quite specific, I am not sure if you can find it in any book. However, you can prove compactness of the embedding directly. Given a sequence $$f_k\in C^{\beta}(\mathbb{R}^n)$$, the compactness for bounded domains and a standard diagonal argument shows that you can find a subsequence $$f_{k_\ell}$$ that converges to some $$f$$ in $$C^\alpha$$ on any ball $$\mathbb{B}^n(0,R)$$. Then it is easy to prove that this subsequence converges to $$f$$ in $$C^{\alpha,-\delta}$$ because roughly speaking: on a large ball the $$C^\alpha$$ norm of $$f-f_{k_\ell}$$ is small and on the complement on a large ball the $$C^{\alpha,-\delta}$$ norm is small.