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.