It seems not true in general. However, we assume $a$, $b$ are two functions (or two operators) which are integrable over $\Omega$, then $\|a b\| \le \|a\| \|b\|$ seems a correct inequality. For a basic example, let $a=x$ and $b=x^2$, then $ab=x^3$, it is obviously true that $\|x^3\|_{L^2(\Omega)} \le \|x\|_{L^2(\Omega)}\|x^2\|_{L^2(\Omega)}$ for $\Omega = [0,1]$. But I am unable to conclude the specific conditions to make the inequality true.
It looks like what the Cauchy–Schwarz inequality claims, but not the same. The Cauchy–Schwarz inequality says an inner product of two elements is smaller or equal to the product of the norms of the two elements. So the key point is if we can define $\|ab\|$ as an inner product $|\langle a,b\rangle|$?