On the problem of the stability of the decomposition of the normal law into components

The following theorem is proved: Let $\Phi$ be the distribution function of $N(0,1)$, $L$ the Levy metric and $F=F_1*F_2$ a distribution function such that
$$ L(F,\Phi)\le\varepsilon<1. $$
Then, there can be found a normal distribution $\Phi_1$ such that
$$ C_1\biggl(\log\frac1\varepsilon\biggr)^{-1/2}<L(F_1,\Phi_1)<C_2\biggl(\log\frac1\varepsilon\biggr)^{-1/11}, $$
where $C_1$ and $C_2$ are positive constants.