Necessary geometric and analytic conditions for general estimates in the $bar{partial}$-Neumann problem

We show the geometric and analytic consequences of a general estimate in the (bar{partial})-Neumann problem: a “gain” in the estimate yields a bound in the “type” of the boundary, that is, in its order of contact with an analytic curve as well as in the rate of the Bergman metric. We also discuss the potential-theoretical consequence: a gain implies a lower bound for the Levi form of a bounded weight.