I'm studying partial derivatives equations, developing the case of solving Klaine-Gordon equation - after expansion of solution into Fourier integral there appears the algebraic equation $g(x)f(x) = 0, x$ in terms of generalized function. It is said that the solution is $f(x) = \delta(g(x))h(x), x\in\Bbb R^4$, where $h(x)$ - arbitrary function.
Now I'm looking forward any proof of this result and will appreciate any help
That equation has some ambiguities in interpretation...
One precise thing that is true, is that for smooth $g$ and distribution $u$, if $g\cdot u=0$ (under the natural action of smooth functions on distributions), then the support of $u$ is contained in the zero-set of $g$.
Another precise thing that is true, is that (e.g., for ease-of-precision) if smooth $g$'s zero-set $Z$ is a compact smooth submanifold of the ambient Euclidean space, then every distribution supported on that zero-set is a finite linear combination of compositions of derivatives transverse to $Z$ with distributions on $Z$. That is, first differentiate transverse to $Z$, then apply a distribution on $Z$, as a smooth manifold in its own right. This is an old, standard theorem.
Although these examples certainly do not touch the most general case, they do suggest to me that the principle quoted in the original question cannot be quite right...