In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in Cn with real analytic boundary. Given such a domain D, and given an Lp function f on ∂D 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 Hp(D). 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 L1 functions. As a corollary, the Szegő kernel is shown to be the strong operator limit of (π1π2)n, where πi is the projection onto zi holomorphically extendible L2(∂D) functions (in C2, with a slightly more complicated formula in Cn).
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