This paper deals with the extension of $\text{CR}$ functions from a manifold $M\,\subset \,{{\mathbb{C}}^{n}}$ into directions produced by higher order commutators of holomorphic and antiholomorphic vector fields. It uses the theory of complex “sectors” attached to real submanifolds introduced in recent joint work of the authors with D. Zaitsev. In addition, it develops a new technique of approximation of sectors by smooth discs.

In this note we consider $\bar{\partial }$-problem in line bundles over complex projective space $\mathbb{C}{{\mathbb{P}}^{1}}$ and prove that the equation can be solved for (0, 1) forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{C}{{\mathbb{P}}^{2}}$ since by removing a point from it we get a line bundle over $\mathbb{C}{{\mathbb{P}}^{1}}$.

Let M be a circular CR manifold and let N be a rigid CR manifold in some complex vector spaces. The problem of the existence of local CR mappings from M into N is considered. Conditions are given which ensure that the space of such CR mappings depends on a finite number of parameters. The idea of the proof of the main result relies on a Bishop type equation for CR mappings. Roughly speaking, we look for CR mappings from M into N in the form F = (ƒ,g), we assume that g is given, then we find ƒ in terms of g and some parameters, and finally we look for conditions on g. It works independently of assumptions on the Levi forms of M and N, and there is also some freedom on the codimension of the manifolds.