For all real value function $f$ in $C^\infty_0(D)$ where $D$ is bounded domain of $\mathbb{R}^n$
If norm $\|f\|^2_0 = \int_Df^2dx$ and $\|f\|^2_1 = \int_D\sum f_i^2 dx$ (where $f_i$ is $\partial_if$)
Now prove the following ineuqality:
$$\|f\|_0 \le d\|f\|_1$$
where $d$ is the width of domain $D$
I have shown for each direction $i$ ,we have the following inequality:
$$f^2(x)\le d\int_{\mathbb{R}}|f_i|^2dx_i$$
How to combine them togethter?(this looks quite like Sobolev inequality)
My attempt:since $\int_D f^2(x) dx \le d^2\int_D |\partial_1f|^2 dx \le d^2\int_D\sum_i|\partial_i f|^2 dx$ which is exactly the result.
Since all the function are non-negative we can use Fubini here
Hint
First, try to prove it when $D\subset \mathbb R$, and use the fact that since $f\in \mathcal C_0(D)$, then there is $a\in D$ s.t. $f(x)=0$, i.e. $$f(x)=\int_a^x f'(u)\,\mathrm d u.$$
Jensen inequality allow you to conclude. I let you generalize when $D\subset \mathbb R^n$.