How is this inequality called? (And how to improve this process)

Question

I am reading a book and it mentions the following:

Let $u \in H^1_0(G)$; then $$\lVert u\rVert ^2_{L^\infty(G)} \le C \lVert u \rVert_{L^2(G)}\lVert u'\rVert_{L^2(G)}$$

Note: Here $G = (a,b) \subset \mathbb R$ is a bounded interval, $H^1_0(G)$ is the sobolev space of functions null at the boundary of $G$ and $C$ is a constant that depends on $b-a$

How is this inequality called / how to show that it holds? I'm guessing Poincarè inequality plays a role but I am not sure how to deal with the $\lVert \cdot \rVert_{L^\infty(G)}$ norm

P.S. This really seems like something I should be able to find out just by browsing the internet, instead of having to ask here. Is there a way to search the web such that this kind of things show up?

Answer 1

Following Daniel Fischer comments I'm trying to post an answer:

Let $G = (a,b)$

We have that $$\lVert u\rVert_{L^\infty(G)}^2 = \lVert u^2\rVert_{L^\infty(G)} \le \int_a^b 2 |u(t)u'(t)|dt \le 2 \lVert u\rVert_{L^2(G)}\lVert u'\rVert_{L^2(G)}$$

where the first inequality is justified by the fact that $u^2$ is absolutely continuos and $u(a) = 0$ so that we can write $u^2(x) = \int_a^x 2u(t)u'(t)dt$ and the second inequality is just Holder.