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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
  • Mathematics

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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