# Unique continuation for solutions to the induced Cauchy-Riemann equations

- Authors
- Journal
- Journal of Differential Equations 0022-0396
- Publisher
- Elsevier
- Publication Date
- Volume
- 23
- Issue
- 3
- Identifiers
- DOI: 10.1016/0022-0396(77)90121-8
- Disciplines

## Abstract

Abstract Let M be a real infinitely differentiable closed hypersurface in X, a complex manifold of dimension n ⩾ 2, and let \ ̄ t6 M denote the induced Cauchy-Riemann operator on M. The problem considered in this paper is unique continuation for distribution solutions to the equation \ ̄ t6 Mu = 0 (these solutions are called CR distributions). In a local version of the problem it is shown that a CR distribution u in an open set U ⊂ M which vanishes on one side of a C 1 hypersurface S ⊂ U which is noncharacteristic at a point p ϵ S necessarily vanishes in a neighborhood of p. If the CR distribution u is a continuous function on U, then it is only necessary to assume that u vanishes on S in order to prove that u vanishes in a neighborhood of p in M. It is also proved that if u is a CR distribution on M, then the boundary of the support of u is foliated by complex hypersurfaces. Thus a global unique continuation theorem is obtained by assuming that such a set is not contained in M.

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