I have a function $a(u,v) := \int_\Omega c(x) u(x)v(x)dx$ where $\Omega \subseteq \mathbb{R}^n$ is open and bounded, $c \in C(\bar{\Omega})$, $c\geq 0$, $u,v \in L^2(\Omega)$. Now I want to show that there exists some constant $C>0$ s.t. $|a(u,v)| \leq C ||u||_{L^2}||v||_{L^2}$. We can apply Cauchy-Schwarz to get: $$|a(u,v)|\leq \int_\Omega |c(x) u(x)| \cdot |v(x)|dx \leq ||cu||_{L^2} \cdot ||v||_{L^2}$$
Now it would be lovely if $||cu||_{L^2} \leq ||cu||_{L^1}$, because then we could apply Hölder's inequality and would be done. However I don't think that $|| \cdot ||_{L^2} \leq || \cdot ||_{L^1}$ holds, so I don't really know what to do here. In the lecture we only mentioned that Hölder's inequality has to be used here, but I don't see how. Could someone give me a hint?
Edit: Just as I was reading the wikipedia article of Hölder's inequality, I saw the generalized inequality, since $c \in L^\infty$, could we just say $||cu||_{L^2} \leq ||c||_{L^\infty} \cdot ||u||_{L^2}$?
Best Answer
Since $c \in C(\overline{\Omega})$, $||c||_{\infty} < \infty$. No need for generalized inequalities. Im just using the fact that continuous functions on compact domains in $\mathbb{R}^n$ attain a max and hence are bounded.
Thus: $$|a(u,v)| \leq \int_{\Omega} |c(x)u(x)v(x)| \: dx \leq \int_{\Omega} ||c||_{\infty} |u(x)| \: |v(x)| \: dx$$ $$ = ||c||_{\infty} \langle |u|, |v| \rangle \leq ||c||_{\infty} ||u||_{L^2(\Omega)} \: ||v||_{L^2(\Omega)}.$$
So the constant $C > 0$ you want is the max of $c$.