Holomorphic extension from weakly pseudoconcave CR manifolds

The goal of the paper is to improve known sufficient conditions on a generic CR manifold $M\subset\mathbb C^n$ for local holomorphic extensions of CR functions to a full neighbourhood of a point $p\in M$. Conditions of this kind date back to the important extension result of {\it J. M. Trépreau} [Invent. Math. 83, 583--592 (1986; Zbl 0586.32016)] on monolateral extension of CR functions on a class $C^2$ hypersurface $M$. A known sufficient condition is strict pseudoconvexity of $M$, i.e., the Levi form of $M$ has at least one negative eigenvalue in each conormal direction. The general problem of characterizing weakly pseudoconcave manifolds which admit full local extensions is still far from being completely solved. Some sufficient conditions (sector and ray property) are known for weakly pseudoconcave hypersurfaces of finite type. In this paper, the higher codimension case is taken under consideration showing that the situation is much richer than in codimension one. A CR manifold $M$ is said to be trace pseudoconcave at a point $p\in M$ if, for every $\xi\in H^0_pM$ (the holomorphic tangent), the Levi form $\mathcal L_\xi$ is either zero or has eigenvalues of both signs. Thus trace pseudoconcavity is a condition stronger than weak pseudoconcavity and weaker than strong pseudoconcavity [see {\it C. D. Hill} and {\it M. Nacinovich}, Invent. Math. 142, No. 2, 251--283 (2000; Zbl 0973.32018)]. Let $\mathcal G_1$ be the sheaf of sections of the holomorphic tangent $HM$ (i.e., smooth germs of CR vector fields), and let $\mathcal G_k$, $k\geq2$, be defined as the sheaf generated by $\mathcal G_{k-1}$ and $[\mathcal G_1,\mathcal G_{k-1}]$. $M$ is said to be of kind $k_p$ at $p$ if $k_p$ is the smallest $j\in\mathbb N$ for which the evaluation of $\mathcal G_j$ at $p$ is the tangent space $T_pM$. $M$ is said to satisfy the constant rank condition if $k_p$ is independent of $p\in M$. The main result of the paper is the following: Let $M\subset\mathbb C^n$ be a smooth generic CR manifold and $p_0\in M$. Assume that in a neighbourhood of $p_0$ $M$ is trace pseudoconvex, satisfies the constant rank condition and is of kind $k\leq 3$. Then for every open neighbourhood $U\ni p_0$ in $M$, there is an open neighbourhood $V\ni p_0$ in $\mathbb C^n$ such that every CR distribution on $U$ is smooth on $M\cap V$ and has a unique holomorphic extension to $V$. The authors conjecture that the main theorem extends to arbitrary finite kind. The paper is devoted to the proof of the main theorem and to various applications, extensions of the theorem and examples.