When is the fourier transform of a function locally L1 integrable?

Question

What condition can I put on a function f to guarantee its fourier transform is locally L1 integrable? General intuition is also appreciated.

Answer 1

If $f$ is integrable then its Fourier transform $\hat {f}$ is continuous, hence locally integrable. If $f \in L^{2}$ then $\hat {f} \in L^{2}$ so $\hat {f}$ is again locally integrable.

Answer 2

We have that $||\hat{f}||_{\infty} \leq ||f||_1$ if $f \in L^1$

So $\hat{f}$ is integrable on every compact set.