##
**Microlocal hypo-analyticity and extension of CR functions.**
*(English)*
Zbl 0575.32019

Let M be a submanifold of \({\mathbb{C}}^ n\) and f a smooth function on M. It is natural to ask if f is the restriction of some function holomorphic in a neighborhood of M. Simple examples show that when M is a real hypersurface there may be functions holomorphic on only one side of M which have smooth restrictions to M. So it is even more natural to ask if f is the restriction of the boundary values of some function holomorphic on some set which contains M in its boundary. A necessary condition is that f satisfy any induced Cauchy-Riemann equation. The basic (local) result is that of H. Lewy [Ann. Math. II. Ser. 64, 514-522 (1956; Zbl 0074.062)], who showed that if M is a strictly pseudoconvex hypersurface in \({\mathbb{C}}^ 2\) then f extends to one side of M. Until now, the ”analytic discs” method of Lewy has been used in all results on this problem and it appears that this method has been pushed almost as far as possible. Fortunately, the paper under review provides a new approach. This approach is based on recent advances in partial differential equations and uses those elements that could naively be expected. Namely, since one-sided extensions occur even in the simplest cases, it is to be expected that microlocal analysis should be employed. In particular, one should expect microlocal analogues of the analytic wave front set and the Bros-Iagolnitzer-Sjöstrand integral to be important. Also the general framework should be that of Trèves’s RC structures and so M can be a more general subset than a submanifold. An the final results should still involve a Levi form.

This paper develops the general theory of microlocal hypo-analyticity and shows that it is natural to seek the extension of f by representing f as the sum of the boundary values of a finite number of distributions defined in various cones containing M in their boundaries. These results will probably be the starting point for a large part of future progress on the extension problem. Already Baouendi and Trèves have obtained results which show that there are real hypersurfaces in \({\mathbb{C}}^ 2\) which are not pseudoconvex at a given point p but still all CR functions extend to one side and some CR function does not extend to a full neighborhood of p.

This paper develops the general theory of microlocal hypo-analyticity and shows that it is natural to seek the extension of f by representing f as the sum of the boundary values of a finite number of distributions defined in various cones containing M in their boundaries. These results will probably be the starting point for a large part of future progress on the extension problem. Already Baouendi and Trèves have obtained results which show that there are real hypersurfaces in \({\mathbb{C}}^ 2\) which are not pseudoconvex at a given point p but still all CR functions extend to one side and some CR function does not extend to a full neighborhood of p.

### MSC:

32V40 | Real submanifolds in complex manifolds |

58J15 | Relations of PDEs on manifolds with hyperfunctions |

32A45 | Hyperfunctions |

46F15 | Hyperfunctions, analytic functionals |

32E35 | Global boundary behavior of holomorphic functions of several complex variables |

32D15 | Continuation of analytic objects in several complex variables |

42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |