Lipschitz function in $L^2(\Omega)$.

Question

I read a statement in a paper and I do not think it is correct.

Let $X = L^2(\Omega) \times L^2(\Omega) \times L^2(\Omega)$ where $\Omega \subset \mathbb{R}^2$ is a smooth bounded domain. For $u = (f,g,h) \in X$ define $G(u) = (fg, fh, gh).$

Is $G: X \to X$ Lipschitz?

In my opinion, it is not simply because $fg$ might not even belong to $L^2.$ And even if it does, the Lipschitz estimate does not work.

Am I correct?