We obtain local estimates (also called propogation of smallness, or Remez-type inequalities) for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudo-convex domains and give a sufficient condition on more general domains.