An Approximation Theorem for CR distributions

Abstract: In our work we present a modification of the known Baouendi-Treves
Approximation Theorem. Instead of working with a general $N$ -dimensional smooth
manifold we will use a quadric manifold $M$.
While the original theorem deals with any locally integrable structure $\mathcal{L}$ of $\mathbb{C}T M$
we will focus on the CR-structure of $M$ and its solutions: the CR distributions.
With this restrictions we achieve convergence in any compact set instead of local
approximation. More precisely: our main theorem states that if $u\in$CR$(M)$ then one can find
smooth CR-polynomials $P_{n}(w, t)$ such that for every compact subset $K$ we can
approximate $u$ by $P_{n}$ in the distribution sense. In addition to that
(1) if $u\in C^{k}(M )$, $k = 0, 1,\ldots$, then the convergence occurs in the topology of $C^{k}(K)$;
(2) if $u\in h^{p}(M )$, $0 < p < \infty$, then the approximation occurs in the topology of $h^{p}(K)$.
If there is enough time, we will present an application.