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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 fL(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 fL(ℝ)?
• On what kind of sets can g fail to be continuous?

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up vote 5 down vote accepted

It is pretty much the same as to describe the class $G$ of functions $g$ on the circle whose Fourier coefficients decay as $O(|k|^{-1})$. There is no nice "space side" property $P$ that would characterize them but for every nice "space side" property $P$ one can figure out in finite time if it holds for all such functions or not.

As to your particular questions, the answers are

1) On the circle being in this class it is a local property (this needs compactness of the circle) because if $g\in G$, then the product of $g$ and any sufficiently smooth function is in $G$ and you can do partitions of unity.

2) There are obvious inclusions $BV\subset G\subset BMO$ ("bounded variation" and "bounded mean oscillation"). If you need something tighter than that, tell the family of comparison spaces you want to use.

3) Since $\sum_{k\ge 1} \frac 1kz^k$ is unbounded at $1$ and continuous everywhere else, we can move such spikes around to create a function that is locally unbounded on any closed set we want and discontinuous on any $F_\sigma$ set we want including the entire circle.

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