I am struggling with a problem I am providing below. Honestly, I don't know if it could be solved at all. Here it is.
Let $f\in L^2$ is a function of two variables (say $x$ and $y$) and $\hat f$ is its Fourier transformation. One could easily check that $|\widehat{f_{xy}}|\leq|\widehat{f_{xx}}|+|\widehat{f_{yy}}|$. Here $|\widehat{f_{xy}}|$, etc. are the mixed derivatives with respect to $x$ and $y$.
And now my question is whether a similar inequality exists if $f$ is a smooth function (i.e. whether $\widehat{f_{xy}}$ could be evaluated by $\widehat{f_{xx}}$ and $\widehat{f_{yy}}$)?
If anyone has ever crashed into something at least similar, I would be thankful if s/he shares experience. :)
Thank you all!