Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Aldo Andreotti and C. Denson Hill, E. E. Levi convexity and the Hans Lewy problem. I. Reduction to vanishing theorems, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 325–363. MR 0460725
  • [2] E. Bedford, Totally real submanifolds and the edge of the wedge theorem (to appear).
  • [3] Reese Harvey and John Polking, Removable singularities of solutions of linear partial differential equations, Acta Math. 125 (1970), 39–56. MR 0279461
  • [4] L. R. Hunt, J. C. Polking, and M. J. Strauss, Unique continuation for solutions to the induced Cauchy-Riemann equations, J. Differential Equations 23 (1977), no. 3, 436–447. MR 0590067
  • [5] L. R. Hunt and Monty J. Strauss, Uniqueness of analytic continuation: necessary and sufficient conditions, J. Differential Equations 21 (1976), no. 2, 279–296. MR 0409891
  • [6] L. R. Hunt and R. O. Wells Jr., Holomorphic extension for nongeneric 𝐶𝑅-submanifolds, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 81–88. MR 0385167
  • [7] L. R. Hunt and R. O. Wells Jr., Extensions of CR-functions, Amer. J. Math. 98 (1976), no. 3, 805–820. MR 0432913
  • [8] J. C. Polking and R. O. Wells, Jr., Boundary values of Dolbeault cohomology classes, Abh. Math. Sem. Univ. Hamburg (to appear).
  • [9] Monty Strauss and François Trèves, First-order linear PDEs and uniqueness in the Cauchy problem, J. Differential Equations 15 (1974), 195–209. MR 0330739

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32D15, 32C05

Retrieve articles in all journals with MSC: 32D15, 32C05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0450610-2
Keywords: Uniqueness, partial differential equations, CR-functions, generic manifold
Article copyright: © Copyright 1977 American Mathematical Society