Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform

Consider the multivariate smoothing transform fixed-point equation: $\eta =$ law of ${\sum }_{i=1}^N{A}_i{Z}_i$, where $N\ge 0$ is a random integer, $(A_i{)}_{i\ge 1}$ are $d\times d$ random nonnegative matrices, $(Z_i{)}_{i\ge 1}$ is a sequence of ${\mathbb{R}}_{+}^d$-valued random variables independent of $(N,A_1,A_2,\cdots )$, and all $Z_i$ have the same law $\eta$. For each fixed point $\eta$, under suitable conditions, we describe its support, establish its absolute continuity, and prove the existence of its harmonic moments.