For $n \in \mathbb N$ and $\alpha \in (0, 1)$, let $C^{n, \alpha}_b (\mathbb R^d)$ be the space of $n$-times continuously differentiable real-valued functions $f$ on $\mathbb R^d$ that admit the finite norm $$ \| f \|_{C^{n , \alpha}_b} := \sup_{\substack{x, y \in {\mathbb R}^d \\ 0 <|x-y| \le 1}} \frac{| \nabla^n f (x) - \nabla^n f (y)|}{|x-y|^\alpha} + \sum_{k=0}^n \|\nabla^k f\|_{\infty}, $$ where $\nabla^k f$ is the $k$-th order Fréchet derivative of $f$. For simplicity, we denote $C^{\alpha}_b (\mathbb R^d) := C^{0 , \alpha}_b (\mathbb R^d)$.
For $\kappa >0$, the Gaussian heat kernel $p^\kappa_t$ on $\mathbb R^d$ and its induced operator $P^\kappa_t$ are defined by $$ \begin{align*} p^\kappa_t (x) &:= (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}}, \quad t>0, x \in \mathbb R^d, \\ P^\kappa_t f (x) &:= \int_{\mathbb R^d} p^\kappa_t ( x- y) f(y) \, \mathrm d y. \end{align*} $$
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 $n \in \mathbb N$ and $\alpha, \beta \in (0, 1)$, there is a constant $c >0$ such that $$ \| (1-\Delta)^{- \alpha} P^4_t f\|_{C^{n, \beta}_b} \le c t^{-(\frac{n}{2} - \alpha)^+} \| f \|_{C^{\beta}_b}, \quad t > 0, f \in C^{\alpha}_b (\mathbb R^d). \tag{$*$} \label{*} $$
Above, $s^+ := \max \{s, 0\}$ for $s \in \mathbb R$. Thank you so much for your elaboration!