Solution
= Solution
\b[No.] Define the standard smooth bump
$$
\eta(t)=
\begin{cases}
e^{-1/(1-t)},&t<1,\\
0,&t\geq1,
\end{cases}
$$
and set, for $x\in\mathbb R^2$,
$$
f(x)=\eta(|x|^2),
\qquad
g(x)=\eta(|x-(3,0)|^2).
$$
These are nonzero smooth functions supported in disjoint unit discs, so $fg=0$ identically.