Об абсолютной непрерывности безгранично делимых распределений при сдвигах

Random variables $\xi$ with values in a separable Hilbert space $H$ with infinitely divisible distributions are considered. Some sufficient conditions for the absolute continuity of the measure corresponding to $\xi+a$ ($a\in H$) with respect to the measure corresponding to $\xi$ are obtained.
Let now $H$ denote the real line and let the characteristic function of $\xi$ be
$$ \exp\{\int(e^{ixt}-1-\frac{ixt}{1+x^2})\Pi(dx)\}. $$
It is proved that in this case $\xi$ has a density when the condition $\int_{-1}^1|x|\Pi(dx)=\infty$ is satisfied.