  Consider the random matrix model $A^{1/2} UBU^* A^{1/2},$ where $A$ and $B$
are two $N \times N$ deterministic matrices and $U$ is either an $N \times N$
Haar unitary or orthogonal random matrix. It is well-known that on the
macroscopic scale, the limiting empirical spectral distribution (ESD) of the
above model is given by the free multiplicative convolution of the limiting
ESDs of $A$ and $B,$ denoted as $\mu_\alpha \boxtimes \mu_\beta,$ where
$\mu_\alpha$ and $\mu_\beta$ are the limiting ESDs of $A$ and $B,$
respectively. In this paper, we study the asymptotic microscopic behavior of
the edge eigenvalues and eigenvectors statistics. We prove that both the
density of $\mu_A \boxtimes \mu_B,$ where $\mu_A$ and $\mu_B$ are the ESDs of
$A$ and $B,$ respectively and the associated subordination functions have a
regular behavior near the edges. Moreover, we establish the local laws near the
edges on the optimal scale. In particular, we prove that the entries of the
resolvent are close to some functionals depending only on the eigenvalues of
$A, B$ and the subordination functions with optimal convergence rates. Our
proofs and calculations are based on the techniques developed for the additive
model $A+UBU^*$ in [3,5,6,8], and our results can be regarded as the
counterparts of [8] for the multiplicative model.
