The CR Hartogs separate analyticity theorem for convex domains

摘要

In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in with real analytic boundary. Given such a domain D, and given an function f on which has holomorphic extensions on the slices of D by complex lines parallel to the coordinate axes, f must be CR-i.e. f has a holomorphic extension to D which is in the Hardy space . This is the first general result of CR Hartogs type which is not for the ball, or some other domain with symmetries, and holds for functions. As a corollary, the SzegA kernel is shown to be the strong operator limit of , where is the projection onto holomorphically extendible functions (in , with a slightly more complicated formula in ).