We consider the Gaussian heat kernel $(g_t)_{t>0}$ on $\mathbb R^d$, i.e., $$ g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \quad t>0, x \in \mathbb R^d, $$

Let $f : \mathbb R^d \to \mathbb R$ belong to the Hölder space $C^{0, \alpha}_b (\mathbb R^d)$ for some $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f$, i.e., $$ [f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-y|}. $$

We define $F : \mathbb R^d \to \mathbb R$ by $$ F(x) := \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y, $$ where $\partial_1$ is the partial derivative w.r.t. the first coordinate.

Can we upper bound $[F]_\alpha$ in terms of $[f]_\alpha$?

Thank you so much for your elaboration!