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Transactions of the American Mathematical Society

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Uniqueness properties of CR-functions

Author: L. R. Hunt
Journal: Trans. Amer. Math. Soc. 231 (1977), 329-338
MSC: Primary 32D15; Secondary 32C05
MathSciNet review: 0450610
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Abstract: Let M be a real infinitely differentiable closed hypersurface in X, a complex manifold of complex dimension $ n \geqslant 2$. The uniqueness properties of solutions to the system $ {\bar \partial _M}u = f$, where $ {\bar \partial _M}$ is the induced Cauchy-Riemann operator on M, are of interest in the fields of several complex variables and partial differential equations. Since dM is linear, the study of the solution to the equation $ {\bar \partial _M}u = 0$ is sufficient for uniqueness. A $ {C^\infty }$ solution to this homogeneous equation is called a CR-function on M. The main result of this article is that a CR-function is uniquely determined, at least locally, by its values on a real k-dimensional $ {C^\infty }$ generic submanifold $ {S^k}$ of M with $ k \geqslant n$. The facts that $ {S^k}$ is generic and $ k \geqslant n$ together form the lower dimensional analogue of the concept of noncharacteristic.

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Keywords: Uniqueness, partial differential equations, CR-functions, generic manifold
Article copyright: © Copyright 1977 American Mathematical Society