The hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type

Boggess, A.; Dwilewicz, R.; Nagel, A.

doi:10.1090/S0002-9947-1991-1079050-X

The hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type
HTML articles powered by AMS MathViewer

by A. Boggess, R. Dwilewicz and A. Nagel PDF

Trans. Amer. Math. Soc. 323 (1991), 209-232 Request permission

Abstract:

We show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in ${C^n}$ contains an open set in ${C^n}$ which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball. In addition, we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions.

References

M. S. Baouendi and F. Trèves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), no. 2, 387–421. MR 607899, DOI 10.2307/2006990
Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21. MR 200476
Thomas Bloom and Ian Graham, A geometric characterization of points of type $m$ on real submanifolds of $\textbf {C}^{n}$, J. Differential Geometry 12 (1977), no. 2, 171–182. MR 492369
A. Boggess and J. Pitts, CR extension near a point of higher type, Duke Math. J. 52 (1985), no. 1, 67–102. MR 791293, DOI 10.1215/S0012-7094-85-05206-8
A. Boggess, J. T. Pitts, and J. C. Polking, The local hull of holomorphy of semirigid submanifolds of codimension two, Michigan Math. J. 34 (1987), no. 1, 105–118. MR 873025, DOI 10.1307/mmj/1029003488
Josip Globevnik and Walter Rudin, A characterization of harmonic functions, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 4, 419–426. MR 976525
C. Denson Hill and Geraldine Taiani, Families of analytic discs in $\textbf {C}^{n}$ with boundaries on a prescribed CR submanifold, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 327–380. MR 501906
J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. MR 322365
Hans Lewy, 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 81952, DOI 10.2307/1969599
Alexander Nagel, Vector fields and nonisotropic metrics, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 241–306. MR 864374
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 11, 6596–6599. MR 634936, DOI 10.1073/pnas.78.11.6596
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539