Let $f:\mathbb{R}^{N}\to\mathbb{R}$ be the initial data of the following linear heat equation : \begin{align*}\tag{1.1} \begin{cases} \partial_{t}u(t,x) = \Delta u(t,x)\quad x\in\mathbb{R}^{N},t>0\\ u(0,x)= f(x)\quad x\in\mathbb{R}^{N} \end{cases} \end{align*} Then, we let $G_{t}(x):=\frac{1}{(4\pi t)^{N/2}}\exp\big(\frac{-|x|^{2}}{4t}\big)$ and we can write the the solution of (1.1) as follows \begin{align*}\tag{1.2} u(t,x):= \int_{\mathbb{R}^{N}}G_{t}(x-y)f(y)dy \end{align*}
I want to prove the following theorem using Young's Inequality for convolution.
Theorem ($L^{p}-L^{q}$ Estimates)
Let $u$ be the solution of (1.1) with initial data $f$. For $1\leq q \leq p\leq\infty$, there exists a constant $C=C(p,q,N)$ such that
\begin{align*}\tag{1.3}
\|\partial_{x_{j}}u(t)\|_{p}\leq Ct^{-\frac{N}{2}\big(\frac{1}{q}-\frac{1}{p}\big)+\frac{1}{2}}||f||_{q}\quad j=1,2,...,N,\,t>0
\end{align*}
\begin{align*}\tag{1.4}
\|\partial_{t}u(t)\|_{p}\leq Ct^{-\frac{N}{2}\big(\frac{1}{q}-\frac{1}{p}\big)+1}||f||_{q}\quad j=1,2,...,N,\,t>0
\end{align*}
Moreover, for higher derivatives, there exists a constant $C=C(p,q,N,k,\alpha)$ such that
\begin{align*}\tag{1.5}
||\partial_{t}^{k}\partial_{x}^{\alpha}u(t)||_{p}\leq Ct^{-\frac{N}{2}\big(\frac{1}{q}-\frac{1}{p}\big)+k+\frac{|\alpha|}{2}}\|f\|_{q}
\end{align*}
where $k\in\mathbb{N}$ and $\alpha$ is a multi-index $(\alpha_{1},\alpha_{2},...,\alpha_{N})$ with $|\alpha|=\alpha_{1}+\alpha_{2}+...+\alpha_{N}$.
For convenience, I will also include the following Theorem.
Theorem (Young Inequality)
Let $1\leq p,q,r\leq\infty$ such that $\frac{1}{p}=\frac{1}{r}+\frac{1}{q}-1$. Then, for any $h\in L^{r}(\mathbb{R}^{N})$ and $f\in L^{q}(\mathbb{R}^{N})$, we have $h*f\in L^{p}(\mathbb{R}^{N})$ and
\begin{align*}\tag{Y-in}
\|h*f\|_{p}\leq\|h\|_{r}\|f\|_{q}
\end{align*}
This is my attempt so far:
First, we will prove (1.3) and (1.4). Observe that we have the following calculations
\begin{align*}
\partial_{x_{j}}u&= \partial_{x_{j}}G_{t}*f\\
\partial_{t}u&= \partial_{t}G_{t}*f
\end{align*}
This means that
\begin{align*}\tag{1.6a}
|\partial_{x_{j}}G_{t}(x)| &=\bigg|(4\pi t)^{-N/2}\big(\frac{-2x_{j}}{4t}\big)\exp\big(\frac{-|x|^{2}}{4t}\big)\bigg|\\
&\leq (4\pi t)^{-N/2}\frac{|x|}{2t}\exp\big(\frac{-|x|^{2}}{4t}\big)
\end{align*}
and
\begin{align*}\tag{1.6b}
|\partial_{t}G_{t}(x)| &=(4\pi t)^{-N/2}\bigg|-\frac{N}{2}(4\pi t)^{-1}4\pi + \frac{|x|^{2}}{4t^{2}} \bigg|\exp\big(\frac{-|x|^{2}}{4t}\big)\\
&=4\pi(4\pi t)^{-N/2-1}\bigg|-\frac{N}{2} + \frac{|x|^{2}}{4t} \bigg|\exp\big(\frac{-|x|^{2}}{4t}\big)\\
&\leq 4\pi(4\pi t)^{-N/2-1}\bigg(\frac{N}{2} + \frac{|x|^{2}}{4t} \bigg)\exp\big(\frac{-|x|^{2}}{4t}\big)
\end{align*}
Next, we begin by considering the case $p=\infty$ and $q=1$. From (1.6a),(1.6b), and $z=\frac{x}{2\sqrt{t}}$, we see that
\begin{align*}
|\partial_{x_{j}}G_{t}(x)|&\leq \frac{(4\pi t)^{-N/2}}{\sqrt{t}}|z|\exp(-|z|^{2})\\
\|\partial_{x_{j}}G_{t}\|_{\infty}&\leq (4\pi t)^{-N/2-\frac{1}{2}}\frac{1}{\sqrt{2e}}
\end{align*}
and
\begin{align*}
|\partial_{t}G_{t}(x)|&\leq 4\pi(4\pi t)^{-N/2 -1}\bigg(\frac{N}{2}+|z|^{2}\bigg)\exp(-|z|^{2})\\
\|\partial_{t}G_{t}\|_{\infty}&\leq 4\pi(4\pi t)^{-N/2 -1}(\frac{N}{2}+\frac{1}{e})
\end{align*}
Chosing $C(p,q,N)=\max\bigg\{\frac{(4\pi)^{-N/2-\frac{1}{2}}}{\sqrt{2e}},4\pi(4\pi)^{-N/2 -1}(\frac{N}{2}+\frac{1}{e})\bigg\}$, we obtain (1.3) and (1.4) for the case $p=\infty$ and $q=1$ since (Y-in) implies
\begin{align*}
\|\partial_{x_{j}}u(t)\|_{\infty}&\leq \|\partial_{x_{j}}G_{t}\|_{\infty}\|f\|_{1}\\
\|\partial_{t}u(t)\|_{\infty}&\leq \|\partial_{t}G_{t}\|_{\infty}\|f\|_{1}
\end{align*}
Then, we can consider other case. From (1.6a) and (1.6b), again we can obtain
\begin{align*}
|\partial_{x_{j}}G_{t}(x)|^{r}&\leq \frac{2^{r}}{(4\pi t)^{Nr/2}}\big|\frac{|x|}{4t}\big|^{r}\exp\big(\frac{-|x|^{2}r}{4t}\big)\\
\|\partial_{x_{j}}G_{t}\|_{r}^{r}&\leq \frac{2^{r}}{(4\pi t)^{Nr/2}}\int_{\mathbb{R}^{N}}t^{-r/2}(2\sqrt{t})^{N}|z|^{r}\exp(-|z|^{2}r)dz\\
\|\partial_{x_{j}}G_{t}\|_{r}^{r}&\leq C_{1}t^{-\frac{N}{2}(r-1)-\frac{r}{2}}\int_{\mathbb{R}^{N}}|z|^{r}\exp(-|z|^{2}r)dz\\
\|\partial_{x_{j}}G_{t}\|_{r}&\leq C_{A}t^{-\frac{N}{2}(1-\frac{1}{r})-\frac{1}{2}}\\
\|\partial_{x_{j}}G_{t}\|_{r}&\leq C_{A}t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})-\frac{1}{2}}
\end{align*}
and similarly
\begin{align*}
\|G_{t}(x)\|_{r}&\leq (4\pi t)^{-N/2 -1}4\pi\bigg[\frac{N}{2}\|exp(-\frac{-|\,\cdot\,|^{2}}{4t})\|_{r} + \|\frac{-|\,\cdot\,|^{2}}{4t}exp(-\frac{-|\,\cdot\,|^{2}}{4t})\|_{r}\bigg]\\
&\leq C_{B}t^{-\frac{N}{2}-1+\frac{N}{2r}}\\
&= C_{B}t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})-1}
\end{align*}
By using (Y-in), again, we obtain the (1.3) and (1.4) for the other case of $p$ and $q$.
Now, my question is how to proceed to obtain the higher derivative case in (1.5)? I have tried to use induction and strong induction but it does not work quite well with manual calculation. Any help or hint to enable me to proceed to (1.5) is much appreciated!
Thank you!