In previous work, we have
considered the problem of showing that a continuous function on a real hypersurface
Γ in ℂN satisfies the tangential Cauchy-Riemann equations provided that its slices
satisfy conditions of Morera type. For instance, these results imply that if Ω ⊂ ℂN is
a bounded convex domain with smooth boundary, strictly convex at z0∈ 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 coefficients, 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.
