# Estimates for the derivatives of the solution of heat equation

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?

• a good place to start would be to write down $u$ – qbert Jan 5 '18 at 12:02
• $$u(t,x)= \frac{1}{(4\pi t)^{(d/2)}}\int_{\mathbb{R}^d} exp(-\frac{|x-y|^2}{4t})u_0(y) dy$$ and i know that $\partial_{ij}u=\partial_{ij}\Phi*u_0$, i can calculate this. But then i dont know how to use that $u_0\in C^\alpha$. – user494831 Jan 5 '18 at 13:28
• Try doing the computation in the Fourier domain. The condition $u_0 \in C^\alpha$ tells you something about the decay of the Fourier transform of $u_0$. – Jeff Jan 5 '18 at 22:20
• Sorry, but what is the Fourier domain? We dont speak about the Fourier transormation in the lectures, only about convolution. – user494831 Jan 6 '18 at 9:14