LAUSR

In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in $$\mathbf{C}^{\mathbf{n}}$$Cn with real analytic boundary. Given such a domain D, and given an $$L^p$$Lp function f on $$\partial D$$∂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 $$H^p(D)$$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 $$L^1$$L1 functions. As a corollary, the Szegő kernel is shown to be the strong operator limit of $$(\pi _1 \pi _2)^n$$(π1π2)n, where $$\pi _i$$πi is the projection onto $$z_i$$zi holomorphically extendible $$L^2(\partial D)$$L2(∂D) functions (in $$\mathbf{C}^{\mathbf{2}}$$C2, with a slightly more complicated formula in $$\mathbf{C}^{\mathbf{n}}$$Cn).