Suppose that $f$ is a given (smooth) function defined on $B\subset \mathbb R^n$. (For simplicity, take $B$ to be the unit ball but more generally we can take $B$ to be some other measurable set). How can we lower bound the following quantity? $$ C_f:=\inf_{g:\mathbb R^n\to \mathbb R, g|_B = f} \int |\omega||\hat g(\omega)|\,d\omega $$ where $g$ is restricted to be in some nice class of functions, e.g. Schwartz functions. I'm having trouble even coming up with a nonzero bound.

Functions with small $C_f$ can be approximated well on $B$ by neural networks with 1 hidden layer; see A. Barron, 1993, Universal approximation bounds for superpositions of a sigmoidal function (http://pages.cs.wisc.edu/~brecht/cs838docs/93.Barron.Universal.pdf). All he writes on the topic of finding this minimum is that it is an interesting open question (p. 932).

More generally, I would appreciate any references that deal with problems of this sort: minimize some function $L(\hat g)$ of the Fourier transform, over all possible extensions $g$ of $f:B\to \mathbb R$.