For $n \in \mathbb{N}$ and $\alpha \in(0,1)$, let $C^{n, \alpha} (\mathbb{R}^d)$ be the Hölder space of real-valued functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\{\nabla^i f\}_{0 \leq i \leq n}$ that admit the finite norm $$ \|f\|_{C^{n, \alpha}} := \sup _{x \neq y} \frac{|\nabla^n f(x)-\nabla^n f(y)|}{|x-y|^\alpha} + \sum_{i=0}^n \|\nabla^i f\|_{\infty}. $$
I am investigating the Hölder regularity of the marginal density of a stochastic differential equation. I would like to ask whether the following estimate is true or not, i.e.,
For $\alpha \in (0, 1)$, there is a constant $c>0$ such that $$ \|(1-\Delta)^{\frac{\alpha}{2}} f\|_{\infty} \leq c\|f\|_{C^{0, \alpha}}, \quad \forall f \in C^{0, \alpha} (\mathbb{R}^d). \tag{$*$} \label{*} $$
Thank you so much for your elaboration!