Let $j \in \mathbb N$ and $\alpha \in (0, 1)$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\mathbb R}^d)$ the usual Hölder space. For convenience, we denote $H^{\alpha} := H^{j + \alpha}$ for the case $j = 0$. I have encounter a "classical" result from this paper.
Lemma 5.2. Let $P_t = e^{t \Delta}$.
- For $\alpha \in (0, 1)$, there is a constant $c >0$ such that $$ \| (1-\Delta)^{\frac{\alpha}{2}} f\|_\infty \le c \| f \|_{H^{\alpha}}, \quad f \in H^{\alpha}. $$
- For $k \ge 0 , j \in \mathbb N$ and $\alpha \in (0, 1)$, there is a constant $c >0$ such that $$ \| (1-\Delta)^{-k} P_t f\|_{H^{j + \alpha}} \le c t^{-(\frac{j}{2} - k)^+} \| f \|_{H^{\alpha}}, \quad t > 0. $$
Could you elaborate on the references for this result? Thank you so much for your help!
