Campos localmente resolúveis, espaços de Hardy e extensão de funções CR

Abstract

Suppose that M is a smooth submanifold of CN and that L = ∪z∈CnLz is the Cauchy- Riemann structure associated to the N-dimensional complex space. For each p ∈ M we can consider the vector space Ap = CTpM ∩ Lp. If the reunion of those fibers originates a locally integrable structure, then we are going to say that M is a CR submanifold with CR structure A = ∪p∈MAp. It immediately follows that if h is a holomorphic application defined in a certain neighborhood U ⊃ M, then A(h|M) = 0. If we consider a distribution u ∈ D′(M) such that Au = 0, then one can asks: is there a holomorphic application h defined in certain open set U such that h|M = u? The question, as it is, can be paraphrased as: is there any analytic extension of the CR distribution u? The answer is negative and there are several examples one can create. Consider a quadric application q : Cm × Cm −→ Cd and the manifold given by M = {(w, t) ∈ Cm × Cd,ℑt = q(w,w)}. Boggess has proved in [Bog01] that all Lp CR distributions in M (with p ≥ 1) admit a holomorphic extension to the interior of the convex hull of M. In this work, we are going to address the same question, but we are going to deal with CR distributions that are in hp with p > 0. Since hp(M) = Lp(M) if p > 1, then our result can be understood as an extension of the original Boggess theorem. The main ingredient of your work is a version of the Baouendi-Treves Approximation Theorem.