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WHICH POWERS OF HOLOMORPHIC FUNCTIONS ARE INTEGRABLE?

By János Kollár

Abstract

The aim of this lecture is to investigate the following, rather elementary, problem: Question 1. Let f(z1,..., zn) be a holomorphic function on an open set U ⊂ C n. For which t ∈ R is |f | t locally integrable? The positive values of t pose no problems, for these |f | t is even continuous. If f is nowhere zero on U then again |f | t is continuous for any t ∈ R. Thus the question is only interesting near the zeros of f and for negative values of t. More generally, if h is an invertible function then |f | t locally integrable iff |fh | t is locally integrable. Thus the answer to the question depends only on the hypersurface (f = 0) but not on the actual equation. (A hypersurface (f = 0) is not just the set where f vanishes. One must also remember the vanishing multiplicity for each irreducible component.) It is traditional to change the question a little and work with s = −t/2 instead. Thus we fix a point p ∈ U and study the values s such that |f | −s is L 2 in a neighborhood of p. It is not hard to see that there is a largest value s0 (depending on f and p) such that |f | −s is L 2 in a neighborhood of p for s < s0 but not L 2 fo

Year: 2008
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