Convex domains of finite type. (English) Zbl 0777.31007

McNeal, Jeffery D.

J. Funct. Anal. 108, No. 2, 361-373 (1992).

Let \(\Omega\subset\subset\mathbb{C}^ n\) be a smoothly bounded domain and \(p\in\partial\Omega\). Let \(p\) have a neighbourhood \(U\) in which \(\Omega\) is convex. Suppose that the line type of \(p\) is \(L<\infty\). The author proves that for each \(z\in\Omega\cap U\), there exists a uniformly bounded \(C^ \infty\)-plurisubharmonic function on \(\Omega\) with maximally large Hessian on a polydisc \(P_ \delta(z)\). As a consequence, it is deduced that the variety type of \(p\) is also finite and equals \(L\). This corollary is also known to Fornaess-Sibony and Boas-Straube by different methods.

Reviewer: V.Anandam (Riyadh)

MSC:

31C10	Pluriharmonic and plurisubharmonic functions
32A10	Holomorphic functions of several complex variables

Keywords:

type of a boundary point; variety type

References:

[1]	\scH. Boas and E. Straube, On the equality of line type and variety type of real hyper-surfaces in \(C\)^n, preprint. · Zbl 0749.32009
[2]	Catlin, D.W, Necessary conditions for subellipticity of the \( \̄\)t6-Neumann problem, Ann. of math. (2), 117, 147-171, (1983) · Zbl 0552.32017
[3]	Chen, J.-H, Estimates of the invariant metrics on convex domains, ()
[4]	D’Angelo, J.P, Real hypersurfaces, orders of contact, and applications, Ann. of math. (2), 115, 615-637, (1982) · Zbl 0488.32008
[5]	\scJ. E. Fornæss, personal communication.
[6]	Fefferman, C; Kohn, J.J, Hölder estimates on domains of complex dimension two and on three dimensional CR manifolds, Adv. math., 69, 233-303, (1988) · Zbl 0649.35068
[7]	Fornæss, J.E; Sibony, N, Construction of P.S.H. functions on weakly pseudoconvex domains, Duke math. J., 58, 633-655, (1989) · Zbl 0679.32017