A particular product of distributions Suppose you have two continuous functions $f,g: \mathbb{R}\to\mathbb{R}$; is the product $f'g$ as a distribution, at least locally? I am interested in a local result, actually, so you can as well assume that $f$ and $g$ are compactly supported.
Obivously, $f'$ is, as $f\in L^1_{\mathrm{loc}}$.
Moreover, I am aware of the following particular cases:


*

*if $f\in W^{1,1}(I)$, then $f'g$ is well defined on $I$

*if $f'\in L^p(I)$ and $g\in L^q(I)$ for $p^{-1}+q^{-1}=1$, then $f'g$ is defined on $I$;

*if $g\in W^{1,1}(I)$, then again $f'g$ is well defined on $I$


I am interested in a general result or in a counterexample. Thank you!
 A: Edited after questioner's clarifications: The analogous question for functions on a circle is somewhat easier to answer decisively, since all distributions are tempered (and have Fourier series expansions), and since the index for Fourier series is discrete. Further, the local question on the line or on $\mathbb R^n$ reduces to the analogous question on products $\mathbb T^n$ of circles, since we can smoothly truncate.
The $L^2$ Levi-Sobolev spaces $H^s$ on the circle [or products of circles, similarly] can be characterized/defined as distributions $u$ with Fourier coefficients $\hat{u}(n)$ satisfying $\sum_n (1+n^2)^s\cdot |\hat{u}(n)|^2<\infty$. A distribution (provably) lies in some $H^s$, so has Fourier coefficients of polynomial growth. [This is true on $\mathbb T^n$.]
The product of two Fourier series, one in $H^s$ the other in $H^t$, certainly converges for $s+t\ge 0$, and converse, I think, by Banach-Steinhaus/uniform-boundedness. [Similarly on $\mathbb T^n$.]
While (Levi-Sobolev imbedding) $H^{1/2+\epsilon}\subset C^o$, and while Fourier series of functions in $H^{1/2+\epsilon}$ converge uniformly pointwise to the functions, Baire category (and Riemann-Weierstrass counterexamples) show that the converse fails significantly. [Edit:] The traditional Baire category argument to prove that typical continuous functions' Fourier series fail to converge to them pointwise (hence, note, are in no $H^{1/2+\epsilon}$ for any $\epsilon>0$), as in http://www.math.umn.edu/~garrett/m/fun/notes 2012-13/05b_banach_fourier.pdf does not produce an explicit function, but is perhaps more compelling/persuasive. I hesitate somewhat to try to recap the Riemann-Weierstrass explicit versions, unless "explicit example" is essential to your/the-questioner's context.
Thus, in terms of the Levi-Sobolev spaces, for $f$ continuous but not in $H^{1/2}$ [or not in $H^{{\rm dim}/2}$, for $\mathbb T^n$] there should exist many $g\in C^o$ such that the apparent Fourier series product for $f'\cdot g$ does not make sense because the expression for the $0th$ coefficient diverges (beyond repair).
(If this kind of answer is interesting/useful, I can amplify it and worry about the details.) [Edit:] a bit added, per request... 
