Philippe Charpentier, Yves Dupain
Extremal Basis, Geometrically Separated Domains and Applications
Preprint series:
ESI preprints
- MSC:
- 32H15 Invariant metrics and pseudodistances
Abstract: In this paper we introduce the notion of extremal basis of tangent
vector fields at a boundary point of finite type of a pseudo-convex
domain in $\mathbb{C}^{n}$, $n\geq3$. Using this notion we define
the class of geometrically separated domains at a boundary point and
we give a description of their complex geometry. Examples of such
domains are given, for instance, by locally convex domains, domains
with locally diagonalizable Levi form at a point or domains for which
the Levi form have comparable eigenvalues near a point and moreover
we show that geometrically separated domains can be localized. Next
we define what we call {}``adapted pluri-subharmonic function and
give sufficient conditions, related to extremal basis, for their existence.
Then, for these domains, when such functions exist, we prove global
and local sharp estimate for the Bergman and Szeg\"o projections. As
an application, we strengthen a result by C. Fefferman, J. J. Kohn
and M. Machedon (\cite{F-K-M-d-bar-Diag-Levi-Form}) for the local
H\"older estimate of the Szeg\"o projection removing the arbitrary small
loss in the H\"older index and giving a stronger non-isotropic estimate.
Keywords: finite type, extremal basis, complex geometry, adapted pluri-subharmonic function, Bergman and Szego projections