Question: What does an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ look like locally?
As formulated, the question might be a bit difficult to answer since the Fourier transform of a function f ∈ L∞(ℝ) is a distribution, and it is not easy to "write down" a distribution. So let me first illustrate the situation at hand with an easy example:
Example: The Fourier transform of the Heaviside function H(x) (i.e. the characteristic function of the positive reals) is given by a linear combination of the function 1/x and of the Dirac delta function (see this Wikipaedia entry for the exact formula, as well as for the meaning of the distribution "1/x").
The formalism of distributions is bit overkill for talking about measures, and things that look like 1/x. For example, the primitive of an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ is always a function (well defined outside of a set of measure zero). Using the above observation, we get the following
Reformulation of the question:
Let f(x) ∈ L∞(ℝ) be a function, and let g(x) be a primitive of its Fourier transform.
• What can g(x) look like locally?
• What local conditions must g satisfy in order to have a chance of coming from some f ∈ L∞(ℝ)?
• On what kind of sets can g fail to be continuous?