The famous Parseval equality states that \[ \int|f|^{2} = \int |F|^2, \] where $f$ and $F$ are related to one another by \[ \int f e^{-2 \pi i r y } dr = F(y). \]
My question is, whether there is any result for \[ \int |f| \] given its Fourier transform.
There is a general result called the Babenko–Beckner inequality:
$$\left(\sqrt{q}\int |F|^q\right)^{\frac{1}{q}} \leq \left(\sqrt{p}\int |f|^p\right)^{\frac{1}{p}}$$ where $\frac{1}{p}+\frac{1}{q}=1$.
The equality is only reached in the case $p=2=q$.
In the limit where $p=1$, you might get something like: $$\max F \leq \int |f|$$ or you most probably remains with the disappointing $$0 \leq \int |f|$$ (I should check whether we can commute the order of the limit $q\longrightarrow \infty$ and the limit of the Riemann sum.)