The moment problem with bounded density

Let $\mu$ be a given Borel measure on $\K\subseteq\R^n$ and let $y=(y_\alpha)$, $\alpha\in\N^n$, be a given sequence. We provide several conditions linking $y$ and the moment sequence $z=(z_\alpha)$ of $\mu$, for $y$ to be the moment sequence of a Borel measure $\nu$ on $\K$ which is absolutely continuous with respect to $\mu$ and such that its density is in $L_\infty(\K,\mu)$. The conditions are necessary and sufficient if $\K$ is a compact basic semi-algebraic set, and sufficient if $\K\equiv\R^n$. Moreover, arbitrary finitely many of these conditions can be checked by solving either a semidefinite program or a linear program with a single variable