Bounded holomorphic function with some boundary behavior
by
Toshio Matsushima
Ishikawa National College of Technology, Ishikawa, Japan

DEFINITION 1.     Let z be a point of ¶W.

(1) z is linearly accessible in W if there exists a nonzero vector v such that

{z Î Cn : z = z+ sv,  0 < s £ 1 } Ì W.

We set V(z) as the set which consists of such vectors.

(2) Let f(z) be a function defined in W, and let z be linearly accessible. Define the linear cluster set of f at z as

\romanCL(f:z, v\roman) = Ç T < 1  { f((1-t)v + z) : T < t < 1 }   ,

where v Î V(z).

DEFINITION 2.     Let z be a boundary point of W. z is C-convex point if there exists a complex hyperplane of Cⁿ that contains z and does not intersect W. W is C-convex if every point of ¶W is C-convex point. When n = 1, we consider { z} as the hyperplane.

DEFINITION 3.     Let D be a domain of C. D is L-connected if D satisfies the following three conditions:

(1) D is simply connected,

(2) 0 \not Î D, and

(3) if z Î D, then argz £ Q for some constant Q.

DEFINITION 4.     Let z be a C-convex point in ¶W. Set

H = { z=(z1, ¼zn) Î Cn : c1z1 + ¼+ cnzn + c0 = 0}

as the hyperplane which goes through z and does not intersect W, and let P_z(z) = c₁z₁ + ¼+ c_nz_n + c₀ for z=(z₁, ¼z_n) Î Cⁿ. z satisfies CL-condition by H if P_z(W) = { w Î C : w = P_z(z), z Î W} is L-connected. Note that 0 = P_z(z). We remark that CL-condition by H does not depend on the choice of the polynomial P_z(z).

Our main result in terms of the function is the following theorem:

THEOREM.     Let { z_k }_k=1^m be a discrete subset of the boundary of W, where z_k is linearly accessible for all k and 1 £ m £ +¥. Suppose that there exists a complex hyperplane H_k for all k that goes through z_k and does not intersect W, and every z_k satisfies CL-condition by H_k. Assume that H_k satisfies either

(1.0)     H_k ÇH_l \not ' z_k, z_l or

(1.1)     H_k = H_l

if k ¹ l. Then there exists a bounded holomorphic function f(z) in W whose arbi trary linear cluster set at z_k contains a closed disk of positive radius, or a closed annulus of positive thickness.

Date received: August 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-68.