Take a probability measure $\nu$ on $\mathbb{R}^2$ and assume
$$\int e^{\lambda_1x+\lambda_2y} \, d\nu(x,y) < +\infty \quad \forall (\lambda_1,\lambda_2)\in\mathbb{R}^2.$$
Assume that $\nu$ has a density w.r.t. the Lebesgue measure. Is there any way to know if the characteristic function of $\nu$ is in some $L^p$ space?