Let Ω be aC∞bounded strongly pseudoconvex domain inCn. Forbe the space of functions on Ωthat are square integrable with respect to the measure |ρ|vdm. Letbe the subspace ofL2(|ρ|vdm) consisting of the holomorphic functions. We consider the weighted Bergman projection. We prove thatPvadmits both local and global regularity estimates in Sobolev norms, and that the weighted kernel functionKv(z,w) is smooth up to the boundary off the boundary diagonal. As a consequence we prove estimates for the solution to the-equation which is minimal in theL2(|ρ|vdm)-norm.