Surjective isometries of weighted Bergman spaces

Let $\Omega$ be a bounded, simply connected domain in ${{\mathbf{C}}^n} = {R^{2n}}$, let $F \in {L^1}({m_\Omega })$ be positive and continuo on $\Omega$, and let $B_F^P(\Omega ) = {L^p}(Fdm) \cap H(\Omega )(0 < p < \infty )$ denote the weighted Bergman space over $\Omega$. We characterize those automorphisms $\Phi$ of $\Omega$ such that the map $f \to g \cdot (f \circ \Phi )$ is a surjective isometry of $B_F^P(\Omega )$, including an explicit description of $\vert g\vert$.