I am reading a paper which deals with the densities of $(X_t)$ where $X_t$ is typically a Levy process.
The paper assumes that the density of, say, $X_1$, is square-integrable, i.e. in $L^2$. Other times, it will assume that the characteristic function of $X_1$ is integrable, i.e. in $L^1$.
It provides no reference to these facts.
Is this known to be true? Is this in general true for densities that one typically encounters? Or are these results particularly true for Levy processes? If so, does it hold for all of them, or must some conditions be satisfied?
Not all densities are square-integrable, and not all characteristic functions are integrable. Counterexamples are not hard to construct. The fact that the paper is stating an assumption rather than a fact should tell you something.