Bound for $fg\in L^{1}$ in terms of $f,g \in L^{1} $

Question

Let's assume that $fg\in L^{1}$ with $f\in L^{1}$ and $g\in L^{1}$. Is there a common way to bound $\Arrowvert fg \Arrowvert_{1}$ in terms of the norms $\Arrowvert f \Arrowvert_{1}$ and $\Arrowvert g \Arrowvert_{1}$? Are there any examples where such bound doesn't exist?

Thank you.

Answer 1

Consider the unit interval $[0,1]$ and $$f_\epsilon(x) = \frac 1 {\sqrt x} \chi_E = g_{\epsilon}(x)$$

where $E = [\epsilon, 1]$ and $\chi_E$ is a characteristic function. Then $f_{\epsilon}g_{\epsilon} \in L^1$, and we have

$$\|f_{\epsilon}\|_1 = \int_{\epsilon}^1 \frac{1}{\sqrt x} dm(x) \le \int_0^1 \frac{1}{\sqrt x} dm(x) = 1$$ and likewise for $g_{\epsilon}$. However,

$$\|f_{\epsilon}g_{\epsilon}\|_1 = \int_{\epsilon}^1 \frac 1 x dm(x) = -\ln \epsilon \to \infty$$ as $\epsilon \to 0$. So no uniform bound can exist.