On Lelong-Bremermann Lemma

Abstract

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma: Let u be a continuous plurisubharmonic function on a Stein manifold. of dimension n. Then there exists an integer m C-m, s = 1, 2,..., such that the sequence of functions u(s) (z) = 1/p(s) max (ln vertical bar g(j)((s)) (z)vertical bar : j = 1,..., m converges to u uniformly on each compact subset of Omega. In the case when Omega is a domain in the complex plane, it is shown that one can take m = 2 in the theorem above (Section 3); on the other hand, for n-circular plurisubharmonic functions in C-n the statement of this theorem is true with m = n + 1 (Section 4). The last section contains some remarks and open questions.