Estimate of L2 norm of a product of functions

Question

Assume that $f,g, f\cdot g\in L^2(\mathbb{R}^n)$. Is the estimate $\|fg\|_{L^2(\mathbb{R}^n)}\leq \|f\|_{L^2(\mathbb{R}^n)}\|g\|_{L^2(\mathbb{R}^n)} $ correct?

Thank you!

Answer 1

By holders inequality it would be:

$\|fg\|_1\le\|f\|_2\|g\|_2$, rather than $\|fg\|_2\le\|f\|_2\|g\|_2$, i.e. you need the $1$ norm.

Answer 2

No, this isn't true, and there's not even a universal $C$ for which $\|fg\|_{L^2} \le C \|f\|_{L^2} \|g\|_{L^2}$. If you consider

$$f_{\epsilon}(x) = \frac{1}{\sqrt x} \chi_{(\epsilon, 1]}(x) = g_{\epsilon}(x)$$

for very small $\epsilon$, we have that $\|f_{\epsilon}\|_{L^2} \le 1$. However, $\|f_{\epsilon} g_{\epsilon}\|_{L^2} \to \infty$ as $\epsilon \to 0$, so a universal constant can't exist.

Note that this could be easily modified to force $f_{\epsilon}$ to be continuous, or even smooth.