Are squares of square integrable functions square integrable?

Question

Given a complex-valued square integrable function $f$. Is its absolute square $|f|^2=f^\ast{}f$ square integrable as well?

I could not find an example, that isn't, but could not find a way to prove the statement in general.

Answer 1

If I understand your question correctly, you want $f$ such that $\int |f|^2$ is finite, but $\int|f^2|^2$ is infinite. If so, then I pick a positive function to get rid of the bars, and I pick a real-valued function to make life easier.

Under these conditions, such a function which is in $L^2(X)$ but not $L^4(X)$ is $ f(x) = 1/\sqrt[3]{x}$ on the interval $(0,1)$. This is square integrable because $\int_0^1|f(x)|^2 = \int_0^1 x^{-2/3} dx = x^{1/3}|^1_0 = 1$ which is finite, but $\int_0^1|f(x)|^4 dx = \int_0^1x^{-4/3} dx = 3x^{-1/3}|^1_0$ which diverges.