Regularity of the -Neumann problem at point of infinite type

We introduce general estimates for “gain of regularity” of solutions of the -Neumann problem and relate it to the existence of weights with large Levi form at the boundary. This enables us to discuss in a unified framework the classical results on fractional ellipticity (= subellipticity), superlogarithmic ellipticity and compactness. For each case, we exhibit a corresponding class of domains.