We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-y|^\alpha}$. Let $*$ be the usual convolution operator. Let $C^{2, \alpha}_b (\mathbb R^d)$ be the Hölder space containing $f \in C^2_b (\mathbb R^d)$ such that $[\nabla^2 f]_\alpha < \infty$. Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (x) \, \mathrm d x=1$ and $\|f\|_\infty < \infty$.
Are there $\rho \in C^{2, \alpha}_b (\mathbb R^d) \cap \mathcal D_1$ and a sequence $(f_n) \subset \mathcal D_1$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?
Thank you so much for your elaboration!
