On infinitely divisible distributions

Let $F$ be a $\sigma$-finite measure with the property (3), § 2, in a separable Banach space $\mathscr B$. $F$ belongs to $\mathfrak G$ iff the infinitely divisible distributions in $\mathscr B$ with the ch.f.
$$ \exp\biggl\{2\int_{|x|\ge\varepsilon}\cos(\langle t,x\rangle-1)F(dx)\biggr\} $$
have a weak limit $e(\widetilde F)$ as $\varepsilon\to0$.
If $F$ of class $\mathfrak G$ is concentrated in a bounded set,
$$ \int\exp(\gamma|x|)e(F)(dx) $$
is finite for some $\gamma>0$; $\int\langle t,x\rangle^2F(dx)\le C|t|^2$.
For $\mathscr B=l_p$, $p\ge2$, this leads to a characterization of $\mathfrak G$ (Theorem 3).
In the general case, condition
$$ \int_{|x|\le1}|x|F(dx) $$
is shown to imply $F\in\mathfrak G$.