A kernel approach to the local solvability of the tangential Cauchy Riemann equations

Authors:
A. Boggess and M.-C. Shaw

Journal:
Trans. Amer. Math. Soc. **289** (1985), 643-658

MSC:
Primary 32F20; Secondary 35C15, 35N15

MathSciNet review:
784007

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in . In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when is not satisfied.

**[**A. Andreotti and C. D. Hill,**AH**]*E. E. Levi convexity and the Hans Lewy problem*. I, II, Ann. Scuola Norm. Sup. Pisa Cl. Sci.**26**(1972), 325-363; ibid.**26**(1972), 747-806.**[**Al Boggess,**B1**]*Kernels for the tangential Cauchy-Riemann equations*, Trans. Amer. Math. Soc.**262**(1980), no. 1, 1–49. MR**583846**, 10.1090/S0002-9947-1980-0583846-0**[**Al Boggess,**B2**]*Kernels for the local solvability of the tangential Cauchy-Riemann equations*, Duke Math. J.**47**(1980), no. 4, 903–921. MR**596119****[**Reese Harvey and John Polking,**HP1**]*Fundamental solutions in complex analysis. I. The Cauchy-Riemann operator*, Duke Math. J.**46**(1979), no. 2, 253–300. MR**534054****[**Reese Harvey and John Polking,**HP2**]*Fundamental solutions in complex analysis. II. The induced Cauchy-Riemann operator*, Duke Math. J.**46**(1979), no. 2, 301–340. MR**534055****[**C. Denson Hill,**Hi**]*A hierarchy of nonsolvability examples*, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 301–305. MR**0425376****[**G. M. Henkin,**He**]*H. Lewy’s equation and analysis on pseudoconvex manifolds*, Uspehi Mat. Nauk**32**(1977), no. 3(195), 57–118, 247 (Russian). MR**0454067****[**G. B. Folland and J. J. Kohn,**FK**]*The Neumann problem for the Cauchy-Riemann complex*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 75. MR**0461588****[**Hans Lewy,**L**]*On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables*, Ann. of Math. (2)**64**(1956), 514–522. MR**0081952****[**François Trèves,**T1**]*A remark on the Poincaré lemma in analytic complexes with nondegenerate Levi form*, Comm. Partial Differential Equations**7**(1982), no. 12, 1467–1482. MR**679951**, 10.1080/03605308208820259**[**François Trèves,**T2**]*On the local solvability and the local integrability of systems of vector fields*, Acta Math.**151**(1983), no. 1-2, 1–38. MR**716369**, 10.1007/BF02393203

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
32F20,
35C15,
35N15

Retrieve articles in all journals with MSC: 32F20, 35C15, 35N15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0784007-7

Article copyright:
© Copyright 1985
American Mathematical Society