Holder inequality for 3 functions

Question

Suppose $f \in L^{\infty}(\Omega)$, $g \in L^2(\Omega)$, and $\Omega \subset \mathbb{R}^n$ is bounded domain. Is the following a correct application for Holder inequality?

$$ \int_{\Omega} f^2 \cdot g dx = \int_{\Omega} f \cdot f \cdot g dx \leq \|f\|_{L^\infty(\Omega)} \|f\|_{L^\infty(\Omega)} \|g\|_{L^1(\Omega)}$$

Using the fact that $L^2 \subset L^1$ in bounded domains.

Answer 1

Indeed, you can recursively apply the Hölder inequality using associativity to get $$\|f^2g\|_{L^1(\Omega)}=\|f\cdot (fg)\|_{L^1(\Omega)}\leq \|f\|_{L^\infty(\Omega)}\cdot \|fg\|_{L^1(\Omega)}\leq \|f\|_{L^\infty(\Omega)}\cdot \|f\|_{L^\infty(\Omega)}\cdot \|g\|_{L^1(\Omega)}$$