Q:

Relationship between max of a function and Fourier transform of that function

I'm trying to see if any relationship can be inferred between the maximum of a function, and the Fourier transform of that function. Say I know the FT expression, can I infer the maximum in real domain? 
I've been attempting several calculus but I'm wondering now if there's in fact nothing much to be said and if I'm wasting my time :/
I will post some calculation examples when i figure how to convert my mathtype into mathjax (gosh it is tedious). Apologies for now.
Thank you for any tip on this topic!

A:

Here's a pretty easy relationship to derive.  Suppose that $\hat{f}(k)$ is integrable.  Then, $f(x)$ is continuous and bounded, and 
$$
\max_{x\in\Bbb{R}}\vert f(x)\vert \leq C \int_{-\infty}^\infty\vert\hat{f}(k)\vert dk
$$
To see why, just write the inverse Fourier transform (for a particular choice of definition):
$$
f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \hat{f}(k)\exp(ikx)dk
$$
Then, by the triangle inequality,
$$
\vert f(x)\vert \leq \frac{1}{2\pi}\int_{-\infty}^\infty\vert \hat{f}(k)\vert dk
$$ 

