We were given to show that: no estimate of the type $|u|_{p} \leq C_{pqn}|u|_{q}$ with $p>q\geq 1$ can hold for all $u \in C_c(\mathbb{R}^n)$.
As far as I know the idea that I am trying to use is to write u as a convolution product and consider scalling the spatial variable. But I can`t seem to figure this part out.
After this proof, we need to use it to get $f^a$ and $f^b$ such that
$ |f|_{a+b}^{a+b} \leq |f|_{ap}^a|f|_{bq}^b$. Any ideas on using the previous result on this part would also be appreciated.
Suppose such an inequality holds. Let $f \in L^{q}$. There exist $(f_k) \subset C_c(\mathbb R^{n})$ such that $\|f_k-f\|_q \to 0$. By Fatou's Lemma it follows that $\|f\|_p <\infty$. But it is easy to see that $L^{q}$ is not contained in $L^{p}$