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: 


1. Under what condition on $F$ is $V$ an absolutely continuous measure? For example, is absolute continuity of $F$ enough?
2. Can we say anything about $V$ based on the tail behavior of $F$?
3. Can we say anything about $V$ based on the tail behavior of $\phi(t)$?
4. 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.



Questions about Levy measure in the canonical representation of infinitely divisible distributions
I would discuss your questions partially under the following condition:
$$ \int\_{-\infty}^\infty x^2\,dF(x)<\infty, \tag{1}$$
or equivalently $\varphi^{\prime\prime}(0)$ exists and finite or $\int\_{-\infty}^\infty y^2\,dV(y)<\infty$. Let $K(x)\triangleq\int\_{-\infty}^x y^2\,dV(y)<\infty$, it is easy to obtain the following expressions:
\begin{gather}
-(\log\varphi(t))^{\prime\prime}=\sigma^2+\int\_{-\infty}^{\infty}\cos(tx)dK(x),\\
-\lim\_{T\to\infty}\frac1{2T}\int\_{-T}^T (\log\varphi(t))^{\prime\prime} dt =\sigma^2,\\
\psi(t)\triangleq -(\log\varphi(t))^{\prime\prime}-\sigma^2=\int\_{-\infty}^\infty \cos(tx)\,dK(x).\tag{2}
\end{gather}
Now from the inversion formula of characteristic functions we have following conclusion: Under (1),
\begin{gather} \int\_{-\infty}^\infty|\psi(t)|\,dt<\infty \quad\Rightarrow\quad
dK(x)\ll d\lambda\quad \Rightarrow \quad dV(x)\ll d\lambda\;(\text{$V$ has density})\\
\begin{aligned}\int x^{2n}dF(x)<\infty \quad &\Rightarrow \quad \varphi^{(2n)}(0)\; \text{exists and finite}\\
&\Rightarrow \quad \psi^{(2n)}(0)\; \text{exists and finite}\\
&\Rightarrow \quad \int x^{2n}\,dV(x)<\infty, \qquad n\in\mathbb{N}
\end{aligned} 
\end{gather}


