**Preliminaries:** Let $u$ be the solution of the Cauchy problem for the heat equation with initial datum $u_0 \in L^1 \cap L^p$. Then I know that the following estimates hold:
$$\Vert u(t,\cdot)\Vert_{L^p} \le Ct^{-\frac{N}{2}\left(1-\frac{1}{p}\right)}\Vert u_0\Vert_{L^1}$$
and $$\Vert u(t,\cdot)\Vert_{L^p} \le \Vert u_0 \Vert_{L^p}$$
where $N$ is the space dimension, $p \in [0,\infty]$ and $C$ a constant.

**Question 1:** Now let $q \in [1,2]$. and $u_0 \in L^q$
I'm not able to prove the following estimate (which should be classical)
$$\Vert \partial^k_t D^\alpha u(t,\cdot)\Vert_{L^p} \le Ct^{-\frac{N}{2}\left(\frac{1}{p}-\frac{1}{q}\right)-\frac{|\alpha|}{2} - k}\Vert u_0\Vert_{L^q},$$
where $p \in [q,\infty]$, $C$ is a constant, and $|\alpha|$ is the lenght of the multi-index $\alpha$.
How does one do it?

I've tried to compute $$\partial_{x_k} K(t,x) = \frac{-x_k}{t}K(t,x)$$ $$\partial_{x_kx_k} K(t,x) = \left(\frac{x^2_k}{t^2}- \frac{1}{t}\right)K(t,x)$$ $$\partial_{t} K(t,x) = \frac{1}{2}\left(\frac{|x|^2}{t^2}- \frac{N}{t}\right)K(t,x)$$

(where $K$ is the heat kernel), but I don't see where this leads.

**Question 2:** To obtain the first estimates that I stated, I used the explicit form of the heat kernel. I guess one should use it to derive the last one too. Can we also obtain these estimates without making use of the form of the heat kernel K? That is, can we also obtain these estimates using the fact that for the heat kernel we have $$\mathcal{F} K_t(\xi) = (\pi/t)^{\frac{N}{2}}K_{(4t)^{-1}}(\xi),$$ where $\mathcal{F}$ is the Fourier transform.