Let X be a bounded subset of $\mathbb{R}$ and let $f, g,$ and $h$ be real valued functions in $L^2(X)$. Consider $$\| fgh\|_{L^1(X)}.$$ The hope is to get an upper bound in terms of $\|\cdot\|_{L^2(X)}$ of the individual functions; however, I don't actually think this is possible. Any idea on how close one can get? Here are some upper bounds, does anyone else have suggestions?
Cauchy-Schwarz\Hölder gives $$\| fgh\|_{L^1(X)} \le \|f\|_{L^2(X)}\|gh\|_{L^2(X)} =\|f\|_{L^2(X)}\|g^2h^2\|^{1/2}_{L^1(X)} \le\|f\|_{L^2(X)}\|g\|_{L^4(X)}\|h\|_{L^4(X)}$$
Or if one of the functions is in $L^{\infty}(X)$ we have $$\| fgh\|_{L^1(X)} \le \|f\|_{L^2(X)}\|gh\|_{L^2(X)} \le\|f\|_{L^2(X)}\|g\|_{L^2(X)}\|h\|_{L^\infty(X)}$$ Also, I suppose we could use some similar arguments for arbitrary $p$ and $q$ satisfying $1/p + 1/q = 1$.
Thanks in advance.
Let $$ f=g=h=\frac1{t^{1/3}}\,1_{(0,1]}^\vphantom{2}(t). $$ Then $$ \|f\|_2=\|g\|_2=\|h\|_2=\left(\int_0^1\frac1{t^{2/3}}\,dt\right)^{1/2}=\sqrt3, $$ and $$ \int_{\mathbb R}fgh=\int_0^1\frac1{t}\,dt=+\infty. $$