Let $u \in L^2(\mathbb{R}^2)$ be a function of two variables $x$ and $y$. I want to know if there is a relation between the Fourier tranform (with respect to $x$) of the $L^2$ norm (with respect to $y$) of $u$ and the $L^2$-norm (with respect to $y$) of the Fourier transform of $u$ (with respect to $x$). In particular, I am interested in knowing if something like the following holds: $$\bigg|\int e^{-i\xi .x}\left(\int|u(x,y)|^2dy\right)^{1/2}dx\bigg| \leq \bigg(\int\bigg|\int e^{-i\xi .x}u(x,y)dx\bigg|^2dy\bigg)^{1/2}$$
Thanks a lot for your help!
Take $f(x,y):=g(x)h(y)$, where $g,h\in L^2\cap L^1(\Bbb R)$ and $\int_{\Bbb R}g(x)dx=0$. With $\xi=0$, the LHS is $$\lVert h\rVert_{L^2}\cdot \int_{\Bbb R}|g(x)|dx,$$ while the RHS is $0$.
However, replacing $\leqslant$ by $\geqslant $ make the inequality true by Minkowski inequality.