This is Problem 9.24 from Teschl's "Partial Differential Equations". Show that for $f \in H^{1}_0((a, b))$ we have
$$\lVert f \rVert^{2}_{\infty} ≤ 2\lVert f \rVert_2 \lVert f'\rVert_2.$$
Show that the inequality continues to hold if $f \in H^{1}(\mathbb{R})$ or $f \in H^{1}((0, \infty))$.
(Hint: Start by differentiating $|f(x)|^{2}$.)
Can anybody help me with that? I know that $\lVert f \rVert^{2}_{\infty} = \text{sup} \lvert f(x) \rvert^{2}$ and differentiating $\lvert f(x) \rvert^{2}$ gives me $2 \lvert f(x) \rvert \lvert f'(x) \rvert$. Furthermore I know that $\lvert f(x) \rvert \leq \lVert f(x) \rVert_{2}$ as well as $\lvert f'(x) \rvert \leq \lVert f'(x) \rVert_2$, so at least we have $2 \lvert f(x) \rvert \lvert f'(x) \rvert \leq 2 \lVert f(x) \rVert_2 \lVert f'(x) \rVert_2$.
But how does that give me my required inequality? And could anybody explain me how to show that his holds especially for the given spaces?
Differentiating yields $$ \frac{d}{dx} \vert f(x) \vert^2 = 2 f(x)f'(x)$$ for all $x\in (a,b)$. Now, integrating from $a$ to $x$ yields, using that $f\in H^1_0(a,b)$, $$\vert f(x) \vert^2 = 2\int_a^x f(x) f'(x)\,dx \leq 2 \int_a^b \vert f(x)f'(x)\vert\,dx \leq 2\Vert f\Vert_{L^2(a,b)}\Vert f'\Vert_{L^2(a,b)}.$$ Taking the supremum over $x\in (a,b)$ finishes the proof.
To show that the same inequality holds for $f\in H^1(\mathbb{R})$, you can argue by density. So we can find $(f_n)\subset C_c^1(\mathbb{R})$ such that $f_n \rightarrow f$ in $H^1(\mathbb{R})$. Then as above, we get $$ \Vert f_n \Vert_{L^\infty(\mathbb{R})}^2 \leq 2 \Vert f_n \Vert_{L^2(\mathbb{R})}\Vert f_n'\Vert_{L^2(\mathbb{R})}.$$ Clearly, the right-hand side converges to $2 \Vert f \Vert_{L^2(\mathbb{R})}\Vert f'\Vert_{L^2(\mathbb{R})}$ as $n\rightarrow\infty$. For the left-hand side, note that $$ \Vert v \Vert_{L^\infty(\mathbb{R})}\leq C \Vert v \Vert_{H^1(\mathbb{R})} \quad\forall v\in C_c^1(\mathbb{R}).$$ Thus we also have $f_n \rightarrow f$ in $L^\infty(\mathbb{R})$ as $n\rightarrow\infty$.