If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and
$$
\|\widehat{f}\|\_{L^{p'}}\lesssim \|f\|\_{L^p}.
$$
We note that $f\in L^p$ is equivalent to $|f|\in L^p$, so we find $\widehat{|f|}\in L^{p'}$ and
$$
\|\widehat{|f|}\|\_{L^{p'}}\lesssim \|f\|\_{L^p}.
$$
So my question arises, does there exist connection between $\|\widehat{f}\|\_{L^{p'}}$ and $\|\widehat{|f|}\|\_{L^{p'}}$ ? Precisely, does the following
$$
\|\widehat{f}\|\_{L^{p'}}\lesssim \|\widehat{|f|}\|\_{L^{p'}}
$$
holds? Note that $p=2$ it is a trivial by Plancherel theorem. 



Connection between the Fourier transform of f and |f|
No. The inequality $||\hat{f}||\_{L^p} \lesssim || \hat{|f|} ||\_{L^{p}}$ does not hold for $p \neq 2$. This is, perhaps, easier to see in the case of Fourier series on the circle. A sketch of a contstruction is given in my answer to another mathoverflow quesion [here](https://mathoverflow.net/questions/143342/lp-norm-of-fourier-series-in-terms-of-coefficients-p-neq-2/143393#143393). This example can be modified to work in Euclidean space as well.


