Extremal bases, geometrically separated domains and applications

The notion of an extremal basis of tangent vector fields is introduced for a boundary point of finite type of a pseudo-convex domain in $\mathbb C^n$, $n\geq3$. By using this notion, the class of geometrically separated domains at a boundary point is defined and a description of their complex geometry is presented. Examples of such domains are given, for instance, by locally lineally convex domains, domains with locally diagonalizable Levi form at a point, or by domains for which the Levi form has comparable eigenvalues near a point. Moreover, it is shown that geometrically separated domains can be localized. An example of a not geometrically separated domain is presented. Next, the so-called “adapted plurisubharmonic functions” are defined and sufficient conditions, related to extremal bases, for their existence are given. Then, for these domains, when such functions exist, global and local sharp estimates are proved for the Bergman and Szegö projections. As an application, a result by C. Fefferman, J. J. Kohn, and M. Machedon for the local Hölder estimate of the Szegö projection is refined, by removing the arbitrarily small loss in the Hölder index and giving a stronger nonisotropic estimate.