Let $X$ be a random variable with infinitely divisible and symmetric distribution $F$ distributed on $\mathbb{R}$.

It is well known that the characteristic function of $X$ has a canonical representation of the form: \begin{align} \phi(t)=e^{ -\frac{\sigma^2}{2}t^2-\int_{-\infty}^\infty (1-cos(tx) ) dV(x)} \end{align} where $V$ is a non-negative measure such that $V(\{0\})=0$ and $\int_{-\infty}^\infty \min(1,x^2) dV(x)<\infty$.

The measure $V$ is called Levy measure and here we are interested in its properties.

My questions are:

- Under what condition on $F$ is $V$ an absolutely continuous measure? For example, is absolute continuity of $F$ enough?
- Can we say anything about $V$ based on the tail behavior of $F$?
- Can we say anything about $V$ based on the tail behavior of $\phi(t)$?
I know it is generally difficult to determine $V$ but what can generally be said about $V$ from basic properties of $F$ and $\phi(t)$?

Also, any good reference would be appreciated.