Abstract

In previous work, we have considered the problem of showing that a continuous function on a real hypersurface Γ in C^N satisfies the tangential Cauchy-Riemann equations provided that its slices satisfy conditions of Morera type. For instance, these results imply that if Ω ⊂ C^N is a bounded convex domain with smooth boundary, strictly convex at z₀ in bD, if L₀ is a complex line tangent to bΩ at z₀ and if f is a continuous function on bΩ such that ∫ _L∩bΩfω = 0 for all complex lines L close to L₀ which meet Ω and for all (1,0) forms with constant coeficients, then f is a CR function in a neighbourhood of z₀. This fails to hold if L₀ is a complex line that meets Ω even under much stronger assumption of holomorphic extendibility along complex lines. Indeed, let B be the open unit ball in C², and define a function f on bB ∖{z = 0} by f(z,w) = 1 ∕ z. It is easy to verify that for each complex line L close to the z-axis,  f|L ∩ bB has a continuous extension to L ∩B which is holomorphic on L ∩ B, yet there is no open set in bB on which f is a CR function. So to conclude that f is a CR function one has to assume the holomorphic extension property for a larger family of analytic discs.

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