For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such that $$ \|f\|_{C_b^n}:=\sum_{i=0}^n\left\|\nabla^i f\right\|_{\infty}<\infty . $$
For any $n \in \mathbb{Z}^{+}$and $\alpha \in(0,1)$, let $C_b^{n+\alpha}\left(\mathbb{R}^d\right)$ be the space of functions $f \in C_b^n\left(\mathbb{R}^d\right)$ such that $$ \|f\|_{C_b^{n+\alpha}}:=\|f\|_{C_b^n}+\sup _{x \neq y} \frac{\left|\nabla^n f(x)-\nabla^n f(y)\right|}{|x-y|^\alpha}<\infty . $$
At the end of page 582 of this paper, the author said that
The next lemma contains two classical estimates on the operator $1-\Delta$ and the heat semigroup $P_t=\mathrm{e}^{t \Delta}$.
Lemma 5.2.
(1) For any $\beta>0$, there exists a constant $c>0$ such that $$ \left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta} . \tag{$*$} \label{*} $$
(2) For any $\alpha, \beta, k \geq 0$, there exists a constant $c>0$ such that $$ \left\|(1-\Delta)^{-k} P_t f\right\|_{C_b^{\alpha+\beta}} \leq c t^{-\left(\frac{\alpha}{2}-k\right)^{+}}\|f\|_{C_b^\beta}, \quad t>0 . $$
I would like to use (\ref{*}) of Lemma 5.2(1) in my paper. Could you elaborate on the references that contain this result so that I can cite it?
Thank you so much for your help!
