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Proper maps which are Lipschitz α up to the boundary
Authors:Berit Stensones
Institution:1. Department of Mathematics, University of Michigan, USA
Abstract:In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cn into the unit ball in CN which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see 2], 3], and 5]). On the other hand, if F is Ck on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see 1] and 4]). We are interested in maps F that are less than CN ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bn → BN that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C -boundary in Cn, then we can find a map F: D → BN, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f1, …, FN): Bn → BN that is Lipschitz α and fN(z) = c(1 - Z1)1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bn.
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