Let $ u_0\in L^\infty (\mathbb{R^d})\cap C^\alpha(\mathbb{R}^d), \alpha \in (0,1), \Phi $ the fundamental solution of the heat equation and $u(t,\cdot ):= \Phi (t,\cdot )*u_0(\cdot)$ for $t>0$. Let $ T,R >0$. Then for every $t\in (0,T), x\in\mathbb{R}^d$ with $|x|<R$, and $i,j\in \{ 1,...,d\} $
$$ |\partial_{ij} u(t,x)|\leq \frac{c}{t^{1-\frac{\alpha}{2}}},$$
with $ c\geq 1 $ constant.
Can someone give me hints?