Removable singularities ofLp CR-functions on hypersurfaces

Let H be a C2 hypersurface in ℂn, n ≥ 3, and let M be a generic submanifold of H of real codimension one. We describe classes of compact removable singularities K for Lp-solutions of the tangential Cauchy-Riemann equations on H under the conditions K ⊂ M, 1 ≤ p ≤ ∞. Removability is understood here in the classical sense, but new effects occur based on compulsory analytic extension and envelopes of holomorphy. The classical theory gives results only in the case p > 1. But even for p > 1, removable singularities for Lp-solutions of the tangential Cauchy-Riemann equations may be metrically much more massive than the classical theory predicts. The results for p = 1 are close to corresponding results on removability in the sense of analytic extension justifying the name “removability” for the latter subject. No Levi-form condition on H is posed and the description is given intrinsically in terms of the CR-structure of M. This may be interesting in connection with generalizations, for example, to more general CR-manifolds instead of H.