I want to prove that if $f \in C^{1}(\mathbb{R})$ is compactly supported then its Fourier transform is integrable. I was able to prove the result for $f \in C^{2}(\mathbb{R})$ and compactly supported. I used the fact that if $f \in C^{2}(\mathbb{R})$, then $\hat{f}$ is bounded by $\frac{c}{1+{x}^{2}}$. So it is integrable. I failed to prove it if $f \in C^{1}(\mathbb{R})$

$\begingroup$ This is probably false: there must exist compactly supported $C^1$functions whose Fourier transform is not integrable. I'll look for an example. $\endgroup$– Denis SerreApr 21 '11 at 6:13

1$\begingroup$ Does page 4 of the following link help? math.unc.edu/Faculty/met/s14.pdf $\endgroup$– SuvritApr 21 '11 at 7:35

$\begingroup$ No, but page 6 does :), thanks a lot,... I'm trying to see now if this result can be upgraded to $\mathbb{R}^{n}$ $\endgroup$– jessicaApr 21 '11 at 9:07

$\begingroup$ @Suvrit: The link is broke. Could you please write out the author and title of the paper? $\endgroup$– HansFeb 19 '19 at 0:59
Here is a link to the paper about necessary conditions for the integrability of the Fourier transform:
http://www.heldermannverlag.de/gmj/gmj16/gmj16043.pdf
It is stated in that paper that sufficient conditions for the integrability of the Fourier transform are given in the book
R. M. Trigub and E. S. Bellinsky, Fourier analysis and approximation of functions. Kluwer Academic Publishers, Dordrecht, 2004.

1$\begingroup$ The link is broke. Could you please write out the author and title of the paper? $\endgroup$– HansFeb 19 '19 at 0:43

$\begingroup$ I think the paper is: Elijah Liflyand, Necessary Conditions for Integrability of the Fourier Transform, Georgian Mathematical Journal, Volume 16 (2009), Number 3, 553–559 $\endgroup$ Nov 20 '19 at 20:18