Microlocal hypo-analyticity and extension of CR functions
type: Item
instance of: scholarly article
title: Microlocal hypo-analyticity and extension of CR functions
zbMATH Open document ID: 0575.32019
author: Mohamed Salah Baouendi
author: Chin-Huei Chang
author: François Treves
published in: Journal of Differential Geometry
publication date: 1983-01-01T00:00:00Z
review text: 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 \textit{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 …
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 58J15
Mathematics Subject Classification ID: 32A45
Mathematics Subject Classification ID: 46F15
Mathematics Subject Classification ID: 32E35
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 3918804
zbMATH Keywords: extension of CR functions
zbMATH Keywords: Levi form
zbMATH Keywords: microlocal hypo-analyticity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214437782
OpenAlex ID: W1486307825
Wikidata QID: Q115187937
DOI: 10.4310/JDG/1214437782
description: scientific article; zbMATH DE number 3918804